Tempo Scales in Polytempo Music

by John Greschak

Introduction

Analogous to a pitch scale, a tempo scale is a collection of tempi arranged in order from lowest to highest or from highest to lowest.

This article is a survey of the different forms of tempo scales that have been used in polytempo music (i.e. music in which two or more tempi occur simultaneously).

Tempo-Term Scales

In a tempo-term scale, tempi are indicated by a word or phrase. Often, Italian words such as allegro or presto are used for this purpose. Generally, the meaning of a given tempo term is ambiguous in the sense that it does not imply a specific rate of speed such as MM quarter note = 72, or range of speeds such as MM quarter note = 72-84. In current practice, usually terms are ordered by speed from slowest to fastest, and each term is taken to mean some range of tempi that are faster than those associated with the preceding term and slower than those of the succeeding term.

For example, some scales of this type, from slowest to fastest, would be:

Tempo-Term Scales
Example 1 Source: The New Harvard Dictionary of Music. (1986).	Example 2 Source: Seth Thomas, Metronome de Maelzel #10, model no. E873-006. (ca. 1974)	Example 3 Source: The American Heritage Dictionary of the English Language, Fourth Edition. (2000).
adagio andante allegretto allegro presto	largo larghetto adagio andante allegro presto	largo larghetto adagio andante allegretto allegro presto prestissimo

The first example was taken from the tempo entry of The New Harvard Dictionary of Music where it is given as an example of a sequence of tempo terms for which "there was general agreement about the relative position" by the 18th century. The second example was taken from the scale written on a particular metronome manufactured by Seth Thomas. The third example was obtained from the The American Heritage Dictionary of the English Language, Fourth Edition (2000) by following the chain of definitions, forward and backward, beginning with that of the word adagio which is given as: "In a slow tempo, usually considered to be slower than andante but faster than larghetto".

Tempo terms have rarely been used to specify the simultaneous different tempi of a polytempo piece. Examples can be found in the works of Charles Ives in instances where parts of different tempi are to be coordinated loosely. For example, in Ives' Central Park in the Dark, the tempo of the strings is Molto Adagio throughout, while the rest of the orchestra accelerates gradually from Molto Adagio to Allegro molto during measures 64 through 118. And, in The Unanswered Question, the trumpet and strings are written at MM quarter note = 50 throughout, while each of six successive flute phrases is faster, and the last accelerates according to the instructions: Adagio, Andante, Allegretto, Allegro, Allegro molto, and Allegro - Accel. to Presto. Finally, in Section 8 of the second movement of Ives' Symphony No. 4, some instruments continue at Adagio while others begin playing Allegro and then gradually accelerate through the entire section. Note that the tempo-term scale used by Ives for the accelerando in The Unanswered Question is the same as that which was given in Example 1, with the exception that Ives inserts the term Allegro molto between Allegro and Presto.

The Metronome Scale

In 1815, Johann Nepomuk Maelzel patented a mechanical device that he called the "metronome" that could be used to indicate various musical tempi. Tempi could be specified in terms of some number of beats per minute. For this device, Maelzel established a scale of tempi that consisted of the following rates (beats per minute): 50, 52, 54, 56, 58, 60, 63, 66, 69, 72, 76, 80, 84, 88, 92, 96, 100, 104, 108, 112, 116, 120, 126, 132, 138, 144, 152 and 160. Later, in the 1820's, Maelzel extended the lower and upper limits of this scale to include the following rates: 40, 42, 44, 46, 48, 168, 176, 184, 192, 200 and 208. Since then, this has been the most common numerical scale by which musical tempi are specified.

Metronome Scale (beats per minute)

40 42 44 46 48 50 52 54 56 58 60 63 66 69 72 76 80 84 88 92 96 100 104 108 112 116 120 126 132 138 144 152 160 168 176 184 192 200 208

The rationale for Maelzel's metronome scale is given in the following excerpt from the second chapter of the second section of Notice sur Le Métronome (an English translation follows the original French version):

French Version

Manière de s'en servir pour les Compositeurs

Il a fallu pour déterminer la mesure du tems musical, ainsi calculée par minute, établir combien de degrés de vîtesse il fallait pour indiquer avec exactitude le tems de tous les mouvemens de musique possible, depuis le prestissimo le plus accéléré, jusqu'au lento le plus lent.

Après avoir examine et parcouru les oeures de tous les compositeurs classiques, M. J. Maëlzel s'est convaincu que dans le lento le plus lent, 50 croches absorbent une minute, et que dans le prestissimo le plus accéléré, moins de 160 croches remplissaient également une minute. Les nos. 50 et 160 furent en conséquence adoptés comme les deux extrêmes du mouvement. Les nos. intermédiaires seront trouvés suffisans pour indiquer tous les autres degrés de vîtesse de chaque tempo.

L'auteur a trouvé que le no. 80 peut être considéré comme le terme moyen, de l'adagio avec [eighth note] =, de l'andante avec [quarter note] =, de l'allegro avec [half note] =, et du presto avec [whole note] =.

On remarquera que du no. 50 le Métronome va de suite au no. 52, et de 60 à 63, et ainsi de suite. L'auteur, instruit par sa propre expérience, se connaissances en mathématiques, et surtout par les lois certaines et invariables du compas, a démontré que les nos. intermédiaires donneraient une complication inutile et sans but. En effet, la différence d'un no. au no. le plus prochain, comme par exemple de 50 à 51, de 60 à 61, et même à 62, n'est pas sensible.

Quelle différence encore présente une pièce indiquée no. 50 en [eighth note] = d'avec une au no. 52 aussi [eighth note] =, seulement une [quarter note] = dans l'espace d'une minute: différence imperceptible à l'oreille la plus exercée, et qui certainement ne peut ajouter ni ôter du mérite de la phrase exécutée, pendant la durée d'une minute.

C'est par cette raison que les nos. depuis 50 jusqu'à 60, vont de 2 en 2, ceux de 60 à 72, de 3 en 3, ceux de 72 à 120, de 4 en 4, ceux de 120 à 144, de 6 en 6, et ceux depuis 144 jusqu'à 160, de 8 en 8. Au fait, les nos. 100 jusqu'à 160 ne sont autre chose que le double de ceux 50 jusqu'à 80, et ils indiquent un mouvement exactement du double plus vîte que ces derniers: de manière que si des notes noires sont indiquées au mouvement avec 50, 58, 66, etc., on peut marquer les choches au même mouvement avec les nos. 100, 116, 132, etc.

English Translation (by John Greschak)

How it serves the interests of composers

In order to determine a scale of musical beats per minute, it was necessary to establish how many degrees of speed are needed to accurately indicate the beats of all possible musical tempi, from the fastest prestissimo to the slowest lento.

After having examined the works of all the classical composers, Maëlzel is convinced that in the slowest lento, there are more than 50 eighth notes per minute, and in the fastest prestissimo, there are less than 160 eighth notes per minute. Consequently, the numbers 50 and 160 were adopted as the two extremes of tempo. The intermediate numbers will be found to be sufficient to indicate all the other degrees of speed of each tempo.

The author has found that the number 80 can be considered as the medium term; adagio with MM eighth note = 80, andante with MM quarter note = 80, allegro with MM half note = 80, and presto with MM whole note = 80.

One will notice that, on the metronome, the number 50 is followed by 52, and 60 is followed by 63, and so on. The author, from experience, knowledge in mathematics, and especially by the certain and invariable laws of measurement, has demonstrated that it would be pointless and an unnecessary complication to include the intermediate numbers. The difference in effect from one number to the next, for example 50 to 51, 60 to 61, and even to 62, is not noticeable.

What difference would it make to perform a piece marked as MM eighth note = 50, at MM eighth note = 52 instead; only a quarter note in the time of a minute; an imperceptible difference even to the most trained ear, and certainly, a duration for which one would not be able to add or remove from the merit of the musical material performed during a minute.

It is for this reason that the numbers from 50 to 60 proceed in steps of 2, those from 60 to 72, in steps of 3, those of 72 to 120, in steps of 4, those of 120 to 144, in steps of 6, and those from 144 to 160, in steps of 8. By the way, the numbers 100 to 160 are just double those from 50 to 80, and they indicate a tempo exactly twice as fast as the latter. So, if the quarter notes indicate a given tempo with 50, 58, 66, etc., one could mark the eighth notes of the same tempo with the numbers 100, 116, 132, etc.

Maelzel's mention that the tempo of 80 beats per minute may be used to indicate various tempi in the way in which he describes might have been derived from Johann Joachim Quantz's Versuch einer Anweisung die Flöte traversiere zu spielen (1752) wherein Quantz discusses a tempo scale that is based upon a human pulse rate of 80 beats per minute.

In his review of David Epstein's Shaping Time: Music, the Brain, and Performance, Anthony Pople describes the metronome scale as an integer sequence with a non-decreasing difference sequence that approximates a 16-step, equal tempered tempo scale from 40 to 80. The difference sequence (i.e. the sequence of differences between consecutive integers) for the metronome scale is: 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8. The equal tempered scale to which Pople refers is as follows (rounded to two decimal places): 40.00, 41.77, 43.62, 45.55, 47.57, 49.67, 51.87, 54.17, 56.57, 59.07, 61.69, 64.42, 67.27, 70.25, 73.36, 76.61 and 80.00. The ratio between any two successive tempi of this scale is equal to the sixteenth root of two. Its structure is similar to that of the scale which is discussed in the section Equal Tempered Scales.

The metronome tempo scale has been used often to specify the simultaneous different tempi of a polytempo piece. For example, in the first movement of György Ligeti's Chamber Concerto for 13 Instrumentalists, the following tempi are played simultaneously: MM quarter note = 54, 60, 66, 76, 80, 84, 92 and 100. Later, in the third movement of the same work, the following tempi are performed at once: MM quarter note = 56, 60, 66 and 72. Other examples of the use of the metronome scale in polytempo music can be found throughout the works of Henry Brant.

Harmonic Scales

In his book New Musical Resources, Henry Cowell noted that when a harmonic interval of two different pitches is sounded, two different rates of speed occur at once, and the ratio between the rates of speed equals the ratio between the frequencies of the pitches. Based on this observation, Cowell suggested that polytempo textures be formed by using a tempo scale that is constructed from ratios that are ordinarily associated with pitch scales. His rationale for using the ratios of the overtone series instead of those of an equal tempered scale is given in the following excerpt from pages 98 and 99 of New Musical Resources:

Since our appreciation has been limited, for the most part, to the simplest rhythms, and since it is difficult to play accurately more complex ones, it is necessary to form rhythmic scales of the simplest possible ratios. Therefore, instead of using the ratios of customary systems of temperament upon which to base rhythmic scales, we employ the simplest overtone ratios which can be found to approximate each interval. The series of ratios upon which these scales are formed, then, is as follows:

C:C C:C# C:D C:Eb C:E C:F C:Gb C:G C:Ab C:A C:Bb C:B C:C

= = = = = = = = = = = = =

1:1 14:15 (C:Db = 15:16) 8:9 5:6 4:5 3:4 5:7 2:3 5:8 3:5 4:7 8:15 1:2

With these ratios, Cowell builds two tempo scales that consist of the following rates (beats per minute):

Harmonic Tempo Scales
Scale No. 1	Scale No. 2
60 64 2/7 67 1/2 72 75 80 84 90 96 100 105 112 1/2 120	48 51 3/7 54 57 2/5 60 64 67 1/5 72 76 4/5 80 84 90 96

The number 60 is used as a base in the first scale because, in tuning systems for which middle C is 256 Hz, there are 60 cycles per minute in the low C that is eight octaves below middle C. Note: In the standard equal tempered tuning system that is used today, for which the frequency of the A above middle C is 440 Hz, the frequency of middle C is approximately 261.63 Hz.

Regarding the fractions in "Scale No. 1", Cowell states that they "could be accurately determined and marked on an ordinary metronome", or they could be eliminated in one of three ways. First, one might use a similar scale that is based on the number 48 (see: "Scale No. 2" in the table). Cowell notes that, unlike the fractions in the first tempo scale, the fractions in this scale do not occur on diatonic scale degrees, and this might be advantageous because diatonic scale degrees are used more often. Second, fractional tempi might be rounded to the nearest whole number. Third, alternative ratios might be used to eliminate fractions. Specifically, for a scale that is based on 48, the ratios 24:25, 6:7, 8:11, and 16:25 may be used in place of 14:15, 5:6, 5:7, and 5:8, respectively, to generate the following scale: 48, 50, 54, 56, 60, 64, 66, 72, 75, 80, 84, 90 and 96.

Throughout the works of Conlon Nancarrow, there are simultaneous different tempi that are in the ratios suggested by Cowell. For example, the following tempi occur in Nancarrow's Study No. 37 for Player Piano: MM quarter note = 150, 160-5/7, 168-3/4, 180, 187-1/2, 200, 210, 225, 240, 250, 262-1/2 and 281-1/4. At several points in the piece, all twelve of these tempi occur simultaneously. The ratios between the tempo numbers and the lowest number 150 are as follows: 1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4 and 15/8. This series of ratios is identical to that which was proposed by Cowell. The tempo scale used by Nancarrow in this work differs from that of Cowell in that it is based on 150 rather than 48 or 60.

Equal Tempered Scales

In the article "How Time Passes", Karlheinz Stockhausen puts forward an "equal tempered" tempo scale. Here, he is motivated by a desire to have a scale of durations that has the same equal tempered structure as that of a scale of pitches, for use in the composition of serial music. He uses the following metronome markings in combination with standard note symbols (i.e. sixteenth note, eighth note, quarter note, half note, whole note, double whole-note, quadruple whole-note, and octal whole-note) to represent the various durations in an eight-octave equal tempered scale of durations:

Equal Tempered Scale (MM whole note =)

60 63.6 67.4 71.4 75.6 80.1 84.9 89.9 95.2 100.9 106.9 113.3 120

In this tempo scale, there are twelve tempi per octave and the ratio between any two successive tempi is equal to the twelfth root of two.

The shortest duration in the duration scale is that of a sixteenth note at MM whole note = 120, or equivalently, one thirty-second of a second, while the longest duration is that of a octal whole-note at MM whole note = 60, or equivalently, eight seconds. At a tempo of MM whole note = 120, the frequency at which notes occur in a sequence of sixteenth notes will be 32 notes per second, or 32 Hz. Thus, as Stockhausen notes, the duration scale may be thought of a low-frequency extension of an equal tempered pitch scale for which the lowest frequency is 32 Hz. Note: In tuning systems for which middle C is 256 Hz, the frequency of the C that is three octaves below middle C will be 32 Hz. In the standard equal tempered tuning system that is used today, for which the frequency of the A above middle C is 440 Hz, the frequency of this low C and the B below it is approximately 32.70 Hz and 30.87 Hz, respectively.

In Stockhausen's polytempo work Gruppen (Groups) for 3 Orchestras, a tempo scale is used that consists of the following tempi: MM quarter note = 60, 63.5, 67, 71, 75.5, 80, 85, 90, 95, 101, 107, 113.5 and 120. Rounded to two decimal places, the tempi in a 12-step equal-tempered scale from 60 to 120 would be: 60.00, 63.57, 67.35, 71.35, 75.60, 80.09, 84.85, 89.90, 95.24, 100.91, 106.91, 113.26 and 120.00. With some exceptions, the tempo scale used in Gruppen may be derived from this equal tempered tempo scale by rounding each tempo to the nearest half. Here, the exceptions are as follows: 67.35 is rounded to 67 instead of 67.5, and 71.35 is rounded to 71 instead of 71.5. In this piece, almost all tempi are derived from this scale or a multiple of 2 or 1/2 thereof.

Augmentation/Diminution Scales

By using augmentation or diminution (i.e. uniform lengthening or shortening of note durations), it is possible to notate polytempo music as monotempo music whereby all parts are written with a common reference tempo and meter, and changes in tempo or meter occur in all parts simultaneously. With this notational technique, standard note symbols (e.g. sixteenth note, eighth note, quarter note, half note, and whole note) and tuplets (e.g. triplets, quintuplets, septuplets, etc.) are used in combination with augmentation dots and ties, to represent a given tempo that stands in a given ratio to the reference tempo.

To this end, in Figure 14 of the article "Multiple Tempi: A Survey and Method", Timothy Sullivan gives an extensive table that indicates the note duration that would be required to notate one beat, or various fractions of a beat (i.e. 1/5, 1/4, 1/3, 1/2, 2/3 and 3/4), of a tempo that stands in a given ratio to a reference tempo, given that the duration of a beat in the reference tempo equals that of a quarter note. Sullivan's table contains entries for 55 tempo ratios of which 25 are faster than the reference tempo and 29 slower. These are listed in the following table:

Augmentation/Diminution Tempo Ratios
Tempo Ratio	Beat Duration
2:9 1:4 3:11 2:7 1:3 4:11 3:8 2:5 3:7 4:9 5:11 1:2 5:9 4:7 3:5 5:8 2:3 5:7 8:11 3:4 4:5 5:6 6:7 7:8 8:9 9:10 10:11 11:12 12:13 1:1 9:8 8:7 7:6 6:5 5:4 9:7 4:3 7:5 3:2 8:5 5:3 7:4 9:5 2:1 9:4 7:3 5:2 8:3 3:1 7:2 4:1 9:2 5:1 7:1 9:1	w + e w h. + 3[q] h. + e h. h + e. h + 3[q] h + e h + 3[e] h + s h + 5[s] h q + 5[q] q + e. q + 3[q] q + 5[e.] q. q + 5[e] q + s. q + 3[e] q + s 5[q.] q + 6[s] q + 7[s] q + t q + 9[s] q + 10[t] q + 11[t] q + 12[t] q 9[e. + e. + e] e.. 7[q.] 3[e + e + s] 5[q] 9[e. + e. + s] e. 7[q + s] 3[q] e + t 5[e.] 7[q] 9[e. + e] e 9[e. + s] 7[e.] 5[e] s. 3[e] 7[e] s 9[e] 5[s] 7[s] 9[s]

Note: The table given here differs slightly from that given by Sullivan. Here: beat durations have been represented symbolically using alphanumeric characters rather than with common music notation symbols; the columns of Sullivan's table that give the durations for various fractions of a beat have been omitted; and some minor corrections have been made to the ratios (Specifically, 6:11, 7:9 and 4:2 of Sullivan's table have been replaced by 5:9, 9:7 and 2:1, respectively.).

The symbols used in the "Beat Duration" column of this table have the following meaning:

Beat Duration Symbol Key
Symbol	Meaning
w h q e s t . + 3[x] 5[x] 6[x] 7[x] 9[x] 10[x] 11[x] 12[x]	whole note half note quarter note eighth note sixteenth note thirty-second note augmentation dot tie tuplet: 3 x's in the time of 2 x's tuplet: 5 x's in the time of 4 x's tuplet: 6 x's in the time of 4 x's tuplet: 7 x's in the time of 4 x's tuplet: 9 x's in the time of 4 x's tuplet: 10 x's in the time of 8 x's tuplet: 11 x's in the time of 8 x's tuplet: 12 x's in the time of 8 x's

By using these ratios, one could construct a tempo scale about any given reference tempo.

Instances where such techniques have been used are quite common in the literature. For example, see Bach's fourteenth canon Canon a 4 per Augmentationem et Diminutionem of the Goldberg Canons (BWV 1087), Elliott Carter's String Quartet No. 5, and the first movement of Henry Cowell's Quartet Romantic. Also, Timothy Sullivan has used these techniques extensively in his works Three Etudes in Multiple Tempi and Terrains I for Orchestra.

References

Cowell, Henry 1930. New Musical Resources. New York: Alfred A. Knopf.

Maelzel, Johann 1816. Notice sur Le Métronome. Paris: C. Ballard. Available: The New York Public Library for the Performing Arts (call number: Drexel 3612).

Pople, Anthony 1997. "Review of David Epstein, Shaping Time: Music, the Brain, and Performance (New York: Schirmer Books, 1995)." Music Theory Online 3(3). Available: http://www.societymusictheory.org/mto/issues/mto.97.3.3/mto.97.3.3.pople.html.

Stockhausen, Karlheinz 1959. "How Time Passes." die Reihe 3 (Musical Craftsmanship): 10-40. Bryn Mawr: T. Presser. English translation of the original German version by Cornelius Cardew.

Sullivan, Timothy 1997. "Multiple Tempi: A Survey and Method." Conference of the Canadian University Music Society.

Update History

September 20, 2001: Completed the first version of this page.

Technology-Assisted Conducting of Polytempo Music

by John Greschak

Introduction

This article proposes some suggestions as to how technology might be used to coordinate the performance of polytempo music (i.e. music in which two or more tempi occur simultaneously). It expounds on some of the possibilities that were presented in the brief article Viewing a Finale Document on a Wearable Computer for Performance Purposes in which a particular technology-based conducting system was put forward for consideration.

For an overview of the ways in which technology has been used to coordinate polytempo music, see the section Technology-Assisted Conducting in the article Facilitating the Performance of Polytempo Music: An Overview.

Semi-Automatic Conducting

Various automated conducting systems have been used to coordinate the performance of polytempo music. In these systems, the cue-signal mechanism operates autonomously. For example, for systems in which cue signals are sounded from audio tape, the tape player operates independently of the performers or conductor (between the times at which playback is started and stopped). Similarly, for MIDI-based systems where cue signals are sounded by sequencer-controlled synthesizers, the sequencer operates in a self-directed manner, outside of the control of anyone and unaffected by the performance.

For fully automated conducting systems such as these, the cue-signal mechanism is in effect, the main conductor, and all performers and conductors must follow its lead. Consequently, the pace of cue signals, and thus the overall tempo of the performance, is not controlled by anyone during the performance. Instead, the tempo is set prior to a performance at the time at which the cue signals are encoded on an audio tape or in a MIDI file.

However, this need not be the case. Technology exists that could enable a human conductor or lead performer to serve as the main conductor. For example, a human conductor could be equipped with a wired baton (e.g. one of the available Baton Interfaces such as the Mathews/Boie Radio Baton or the Buchla Lightning) or a lead performer could be equipped with a suitable controller such as a wired foot pedal. Another device could receive the beat messages sent from the human leader's controller, and transmit appropriate beat-cue signals to each performer. The beat-cue information which is supplied to each performer would be derived from, but not necessarily equal to, the beat established by the human leader. In a semi-automatic conducting system such as this, the cue-signal mechanism would serve as an automated assistant conductor.

Alternatively, the cue-signal mechanism might automatically determine the appropriate pacing of beat-cue information by applying a beat detection algorithm to a collection of audio signals from some selected performers, or from the composite audio output of the entire ensemble.

For such systems, before the performance, one would need to specify the times at which conductor beats are to occur. For example, with the radio baton, this could be done by inserting appropriate note-on MIDI messages for MIDI key numbers in the range 0 through 11. This feature of the radio baton is described in the section "Updates on the Radio-Baton Program" by Max V. Mathews on the page "Controllers for Computers and Musical Instruments" that is available at http://www-ccrma.stanford.edu/overview/controllers.html.

When doing this, one might specify conductor beat times that coincide with recognizable beats of the overall resultant tempo that is heard when the simultaneous different tempi of the work are performed. Or, one might choose to have the times be irregularly spaced. For example, this might be done to avoid boredom on the part of the main conductor, or to exaggerate or diminish the impact of certain anticipated tempo changes that might be caused by the conductor. For example, if one expects that a conductor will play a given passage faster than desired, this tendency might be counterbalanced by increasing the duration between consecutive conductor beats during the passage. To this end, one might experiment with various changes in the duration between consecutive conductor beats or the time of such beats relative to beats in the overall resultant tempo, and observe the effect that such variations have on the conductor and the tempo of the performance.

Content-Rich Audio Cues

In an automated conducting system where audio cues are employed, cue signals are provided to performers (or conductors) through headphones. For most systems of this type, audio cue signals consist of a sequence of "clicks", each of which is sounded at the time at which a downbeat is to occur. For most systems, the click sound is constant throughout. That is, all clicks have the same pitch, duration, intensity, timbre and spatial position. For example, each click might be an audio sample of a cowbell that is centered in the stereo field.

In a few cases, clicks have been used to communicate information other than beat times. For example, in works by Emmanuel Ghent, audio cue signals are used for the following purposes: to indicate downbeats and upbeats (with signals for downbeats being longer than those for upbeats), to establish a new tempo before notes are to be played (as a few preparatory beats), to signal the entire rhythmic pattern of a complicated phrase before it is to be played (e.g. for klangfarbenmelodie that has an aperiodic rhythm), and to provide a reassuring checkpoint cue or an alerting cue just before the end of a long rest (in the form of a rapid succession of three cue signals).

In Larry Austin's Life Pulse Prelude, the click pitch and the duration between clicks is varied to distinguish the first and last beats of a measure from intermediate beats. For example, the sequence of click pitches used for the 31 beats of a 31/8 measure is |C---c---c---c---c---c---c---ceg| where 'C' is the pitch that is an octave above the pitch 'c', the pitches 'e' and 'g' are in the octave between 'c' and 'C', and the symbol '-' represents a beat for which no click is sounded. Similarly, the sequence used for 13/4 measures is |Ccccccccccceg|. In each case, a high 'C' is used to distinguish clicks that coincide with the first beat of a measure, and the pitches 'e' and 'g' are used to signal the last two beats of each measure.

Beyond this, audio cue signals could be used to transmit other types of messages as well. For example, audio cues could provide textual information in the form of words that are spoken automatically, such as a lead-in measure consisting of the words "mark 95...one, two, three, four", or instructions such as the words "con sordino" to remind a given performer to play a particular passage with the mute on. Similarly, expression markings or performance notes could be spoken as cues. (Note: For spoken cues, it might be possible to use existing software that is capable of automatically converting text to speech at specified times.) Also, representational sounds could be used to communicate various messages. For example, the sound of a rapid succession of two taps of a baton on the edge of a table might be used as an alerting cue. And, audio cues could indicate the pitch, duration, intensity and timbre of notes that are to be played. Pitch cues might be especially useful for microtonal works in which unfamiliar pitches are employed, or for tuning purposes. Some audio cues could be spoken into a microphone by a conductor and then transmitted directly to the headphones of a given performer.

Additionally, audio cues could consist of music (with no clicks) to which a performer would add his part. To guide a performer, a composer might specify the intended relationship between the cue music and the added part. For example, the added part might be thought of as a solo that is to be added to an accompaniment, as an accompaniment to be added to a solo, or as a part to be blended into the cue music. The cue music could be different for each performer and would not be heard by the audience. In addition to providing a pulse, it could serve to put the performer in a particular frame of mind for a given phrase that is to be performed. This approach would have the advantage of providing audio cues that are similar to those that a performer would hear ordinarily when performing in a traditional group.

Potentially, audio cue signals could be disadvantageous. Since they are transmitted via the same medium (i.e. sound) through which the creative output of the performer is carried, they might be distracting. For example, it is possible that some performers would find it difficult to create a non-metronomic style of artistic expression while listening to (and following) a sequence of evenly spaced click sounds. Also, headphones or earphones that are used to transmit cue signals can act as an unwanted attenuator or filter, affecting a performer's perception of other sounds such as those emitted from the performer's own instrument as well as sounds from other performers in an ensemble, all of which might be used by a performer to control pitch, intensity and timbre.

There is another difficulty associated with the use of audio cues. For most works, consideration must be given to the relative loudness of cues and musical sounds. Soft audio cues can be difficult to hear over musical sounds. And, loud cues can drown out musical sounds or bleed from headphones into a neighboring microphone used for amplification or recording purposes. For works that vary over a wide dynamic range, it might be necessary to vary the intensity of cue signals throughout the course of a performance.

Visual Cues from Artificial Conductors

Visual cues are often used in performance, especially to communicate a pulse. For example, performers follow a tempo established by the motions of a conductor or the gestures of a lead performer such as the first violinist in a string quartet. Given this, one would expect that automatically generated visual cues might be used for a similar purpose in a conducting system for coordinating the performance of polytempo music. But, outside of a few examples where blinking-light metronomes have been employed, visual cues have rarely been used in this way.

With existing technology, it would be possible to devise an artificial conductor that would imitate gestures typical of human conductors, to indicate the time at which downbeats are to occur. For example, an artificial conductor could be realized as a hologram to be displayed before a performer or subgroup of performers. Or, it could be a more abstract representation such as a colored disc that moves up and down on a standard computer monitor, like the tip of a human conductor's baton. Or, a video recording of a human conductor could be used. (Note: A video recording of a human conductor has been used to coordinate the performance of electro-acoustic music. For example, see the project Animated Visual Score for Electro-Acoustic Music by Chad Kirby that was done during the Fall of 1998 at the research center CARTAH of the University of Washington.) Another possibility would be to use robot technology to simulate a human conductor. (Note: A robot conductor was used by artists Rob Gonsalves and William Tremblay in their work Y2K Pops (1999) of which an image is available at http://members.aol.com/y2kpops/show_front_wide.jpg.)

A visual mechanism such as this might have the following advantages over audio systems that employ a sequence of click sounds for the same purpose:

• With visual cues, beat times would be indicated by gestures similar to those that are used to coordinate a traditional ensemble. Because of the familiarity of this type of cueing mechanism, performers might willingly accept it and adapt to it easily.


• With visual cues that imitate a human conductor, the precise time at which a downbeat is to occur is relatively indistinct, and as a result, players have some freedom to choose the times at which downbeats are to occur in their own part. With visual cues such as this, the moment at which one beat ends and another beat begins cannot be sharply delineated because the speed of a conductor's downward motion must gradually decelerate and accelerate about the extreme lowest position in order to change direction. Under such conditions, a performer may play a beat at any time while the conductor is in the neighborhood of the bottom of a down stroke, and still feel that he is in conformance with the conductor's indication. One consequence of this would be that it might be easy for a player to deviate from a metronomic style of performance when following visual cues that imitate a human conductor.


• It is possible to predict the onset of a pending downbeat, when downbeats are indicated by the motion of some object such as a conductor's baton that moves up and down in such a way that the lowest point of each stroke occurs at the same vertical position. For a given downbeat, one may estimate the time at which the baton will arrive at its low point by observing the vertical position from which the down stroke began and the speed at which the baton descends to the low point. This can be beneficial, especially when the tempo is to be changed as in an accelerando or decelerando.


• Another possible advantage of using visual beat cues relates to the ease with which a performer might synchronize to such cues. This stems from observations made by Edward T. Hall in a chapter titled "Rhythm and Body Movement" (p.61-73) of his book Beyond Culture (1976). There, Hall notes that the members of a group of people automatically synchronize their body motions among themselves when they are interacting. For example, he describes a study in which the analysis of a video recording of children playing in a playground revealed that the children's motions were synchronized as if they were all dancing to the same song, although no music was playing and the children were not singing together. And throughout, one of the children moved about the group affecting the motions of neighboring children, much like a conductor. To play a note on an instrument, one must move in some way. Usually, the time at which a note is to occur is related in a simple way to the times at which downbeat and upbeat gestures are made by the conductor and performers. Thus, in performing a work, each member of an ensemble synchronizes their motions to those of a conductor, lead performer, or in some cases, another performer playing a part that serves as a cue. From Hall's observations, one might hypothesize that people are naturally suited to the task of synchronizing their motions to those of others, and consequently, performers might be coordinated with ease by using visual cues that imitate the gestures of a conductor.


• Visual cues do not have the disadvantages of audio cues that were cited in the previous section. Specifically, since visual cues are silent and are perceived by one's eyes rather than ears: they may be perceived regardless of the loudness of the music; they do not interfere with a performer's perception of the sound of their own instrument or the sounds of others; and they do not generate unwanted sonic noise in a recording or performance situation. (Here, it is assumed that the mechanism used to generate visual cues is silent.)

For systems in which an artificial conductor is some type of image, musical scores could be included in the image, as well. To cue the entry of notes and beats, or to indicate the current position in a score, a "follow-the-bouncing-ball" approach could be used whereby some moving object such as a scrolling vertical line or bouncing disc could be displayed. As noted in the article Viewing a Finale Document on a Wearable Computer for Performance Purposes, additional benefits would come from integrating the display of musical scores into an automated conducting system. For example, it might facilitate the use of multi-colored scores, and musicians could continue playing through page turns, which could be accomplished almost immediately upon request (e.g. by using a foot mouse) or automatically. The designers of such a system could draw from the experience of recent explorations on the use of computer monitors to display scores for the performance of monotempo music. These would include the following efforts:

• The MOODS Project.

This project is directed by Paolo Nesi at the University of Florence and is described in the article "Managing Music in Orchestras". The musicians and conductor are equipped with specially designed electronic lecterns on which common music notation is displayed and automatically scrolled during a performance.

• The Muse

This is a digital music stand that was designed as part of the 1995 Apple Design Project by Christopher Graefe, Derek Wahila, Justin Maguire and Orya Dasna of Carnegie Mellon University. It is described in the article "Designing the muse: A Digital Music Stand for the Symphony Musician". Images of the muse are available at http://www.idsa.org/whatis/seewhat/idea/winners/muse.htm.

• A Study on the Automation of Page Turns

For an Honours Project at the University of Canterbury at Christchurch New Zealand, John R. McPherson did a detailed study of issues pertaining to the automation of page turns in a system in which music is displayed on a computer monitor. Results are given in the report "Page Turning - Score Automation for Musicians" (1999).

• The Chamber Orchestra of the Bamberg Symphony Orchestra.

This orchestra has given performances in which each musician and conductor is equipped with a laptop resting on a music stand. Performers control page turns with a foot mouse. See the articles "Notebook statt Notenblatt - Bamberger Symphoniker lesen im Konzertsaal vom Computer ab / Bildschirm mit dezenter Hintergrundbeleuchtung" (Notebook instead of sheet music: Bamberg Symphony read from the computer in concert hall / Screen with a subdued back-lighting) and "Noten aus dem Notebook", and the image http://info.bamberg.de/magazin/kultur/symphoniker/SiemensFoto.jpg.

• Harry Connick Jr..

Connick's jazz band has used Macintosh computers on stage to display scores. Musicians trigger page turns by pressing a key. This setup is described in the articles "A Chart A Day" and "Concert Technology Debuts in Ames".

Also, along with beat-cues and scores, visual cues could include electronic messages consisting of text, signs or symbols that would be sent by a conductor to be displayed before a given performer. For this, it might be possible to use a speech recognition system to translate commands spoken by a conductor into electronic messages.

Wearable computer displays might be used in place of standard computer monitors to display beat-cue information, scores and messages. As side benefits, music stands and monitors could be eliminated from the stage which would result in a simpler setup with an aesthetically pleasing clean appearance, and musicians equipped with wearable computers would be mobile (although a given performer's mobility would still be limited by the mobility of their instrument). For more information on wearable computers, see: Wearables Links.

Tactile Cues

Tactile cues have rarely been used for the purpose of coordinating the performance of polytempo music, if at all.

It is possible to perceive beats with the tactile sense. For example, one is able to measure a person's pulse rate by placing the fingertips on various locations on the body (e.g. at the wrist, neck, top of the foot, or temple near the ear). And, during a performance, especially if it is loud, one can sense the vibrations and literally feel the beat. Deaf musicians such as percussionist Evelyn Glennie use the tactile sense extensively in live performances. This suggests that it might be possible to transmit beat cues as periodic movements or pulsations of some device that could be felt during a performance.

For example, an inflatable rubber cuff similar to that which is used for measuring blood pressure could be placed around a leg above the ankle. With a solenoid-operated pump, a small amount of air could be pumped into the cuff on downbeats and released on upbeats. The resulting pressure changes in the cuff would be felt by the performer. Alternatively, a band with a winding mechanism (like a tourniquet) could be used to produce pressure changes. With systems of this type, safeguards would be needed to ensure that the pressure placed on a performer would not stop blood flow, cause injury or be painful in any way.

Alternatively, a solenoid-based device could be attached to the front of a performer's thigh and positioned so that its line of motion is perpendicular to the surface of the leg. The solenoid could be thrust slightly toward the leg on downbeats, and withdrawn on upbeats. Safeguards would be needed to ensure that the trusting would not cause pain.

The devices suggested here are somewhat related to those that have been proposed to provide haptic feedback to a musician that is using an electronic controller. For example, see the section "Incorporating Haptic Feedback into Music Controllers" by Sile O'Modhrain on the page "Controllers for Computers and Musical Instruments" that is available at http://www-ccrma.stanford.edu/overview/controllers.html.

The tactile sense might be of relatively limited use for receiving musical cues, because, as compared to the visual and auditory senses, it is capable of perceiving much less complex information. However, tactile cues might be very helpful in a multi-sensory cueing system where they could reinforce simultaneous visual and audio cues.

Multi-Sensory Cueing

In a traditional performance by a Western music ensemble, coordination is accomplished with a complex system of multimodal cues that are transmitted simultaneously. Each ensemble member sees their own part score as well as gestures of the conductor, lead performer, other performers, and the audience. Each hears the sound that they generate together with that of other performers. And, they feel the vibrations of that sound.

Most automated conducting systems use audio or visual cues, but usually not both. Automated conducting systems could be developed that use audio and visual cues in combination, along with tactile cues. Such a system could simulate much of what a given musician perceives in a traditional performance, but on an individual basis with independent cues for each musician. Along with cues supplied by an automated conducting system, all of the traditional audio, visual and tactile cues (mentioned in the previous paragraph) could be used as well.

Any work in this area would be related to and could benefit from that which is currently being done in the areas of multimodal interface design, multimodal presentation or output, and multimodal virtual reality systems.

Wireless Connections

All connections could be wireless. This was suggested by Emmanuel Ghent in the article "Programmed Signals to Performers: A New Compositional Resource" (1967) for an automated conducting system involving audio cue signals derived from tape to be distributed to musicians through miniature earphones. Ghent's suggestion could be extended to include the connection between a controller that a conductor might use to transmit beat messages (e.g. a baton), and the device to which such messages would be sent. Also, regardless of the mode of cue signals (e.g. audio, visual or tactile), such signals could be transmitted to musicians by wireless means.

Coda

Most likely, any automated conducting system that is adopted for use by traditional Western music ensembles, if any system is adopted at all, will satisfy the following criteria: it will be the simplest system that accomplishes the task, it will be robust, it will gather and present information in a way that seems natural and is least disturbing, it will be suitable for ensembles ranging in size from a duo to a full orchestra, it will not detract from the ambiance of the performance environment, and it will be relatively inexpensive. In short, probably, it will be a system that represents an evolutionary step in performance technology that is simple, reliable, scalable, aesthetically pleasing and affordable.

References

Austin, Larry 1985. “Charles Ives's Life Pulse Prelude for Percussion Orchestra: A Realization for Modern Performance from Sketches for his Universe Symphony.” Percussive Notes: Research Edition 23(6): 58-84.

Ghent, Emmanuel 1967. “Programmed Signals to Performers: A New Compositional Resource.” Perspectives of New Music 6(1): 96-106. Reprinted in Boretz, Benjamin and Edward T. Cone (Eds.) 1976. Perspectives on Notation and Performance. New York: Norton.

Greschak, John 2000. “Viewing a Finale Document on a Wearable Computer for Performance Purposes.” Finale: IWBNI Suggestion No. 644. Available: http://www.greschak.com/notation/finale/iwbni/fs644.htm.

Greschak, John 2001. “Facilitating the Performance of Polytempo Music: An Overview.” Polytempo Music Articles. Available: http://www.greschak.com/polytempo/ptperf.htm.

Hall, Edward T. 1976. Beyond Culture. Garden City: Doubleday.

Ligeti, Lukas 2000. “Beta Foly: Experiments with Tradition and Technology in West Africa.” Leonardo Music Journal 10: 41-47. Available: http://muse.jhu.edu/demo/lmj/10.1ligeti.html.

Update History

August 30, 2001: Completed the first version of this page.

Facilitating the Performance of Polytempo Music

by John Greschak

Introduction

In monotempo music, part scores are written against or synchronized with a common sequence of pulses (or points in time), and some individual, be it a conductor or lead player, sets the pace of the underlying pulses. A traditional Western art music ensemble such as a (conductorless) string quartet or an orchestra lead by a single conductor, is well suited to the task of producing a coordinated performance of monotempo music.

In polytempo music, two or more tempi occur simultaneously. In its most general form, polytempo music consists of a set of parts where each part is written against a different sequence of pulses, and the pulse sequences bear no simple relationship to one another. It is difficult (and in some cases, impossible) for a traditional ensemble to produce a coordinated performance of such music.

This article gives an overview of the various approaches that have been taken by composers to facilitate the performance of polytempo music. For detailed descriptions of the techniques and works discussed here, see the article Polytempo Music: An Annotated Bibliography.

Loose Coordination

A composer may facilitate the performance of a given piece of polytempo music by specifying that the work be performed in a less than fully coordinated manner. This might be accomplished through an explicit instruction to perform in an uncoordinated manner, or it might be achieved indirectly as a consequence of some other instruction or method of notation.

In an uncoordinated polytempo texture, some tempo is performed independently, without any regard for the other tempi that occur simultaneously. An instruction to perform in an uncoordinated manner might apply to all performers throughout an entire work, or to just one performer during one section of a work. In the case when an ensemble is partitioned into smaller groups, the groups might be uncoordinated with each other, or the players of a given group might be uncoordinated among themselves.

Sometimes, a conductor is expected to freely select (and cue) the start and stop times of uncoordinated phrases. More often, start and stop times are specified relative to (and thus are to be coordinated with) a musical event that occurs in another part. Here, the sound of some event might serve as a start or stop cue for a conductor or performer. Or, a conductor or performer might be asked to determine start and stop times based on the relative horizontal position of phrases as they appear in the full score. The latter approach is used for start times more often than stop times. In some cases, an uncoordinated phrase is to be halted abruptly, on cue. In other cases, a player is instructed to continue playing up to the next breath mark, stop mark or fermata in their part, after having received a stop cue. Sometimes, an uncoordinated phrase is extended, as needed, by repeating the entire phrase or the last few notes. In some cases, the conductor is permitted to vary the tempo of a conducted part to ensure that it ends at a specified time relative to an uncoordinated phrase.

In many cases, the layout of phrases in the full score of an uncoordinated work does not reflect the coincidence of musical events. Often, only the start time is indicated relative to other parts.

For some uncoordinated works, multiple conductors are used where each conductor beats an independent tempo. Sometimes, an ensemble member temporarily serves as an assistant conductor. Usually, a conductor is assigned to a fixed group of performers. However, in some cases, the conductor to which a given performer is assigned changes frequently throughout a piece. In some cases, a line of sight must be maintained between two given conductors as one is to receive visual start or stop cues from the other. After being cued to start, a conductor might begin by beating a silent measure to establish a new tempo and meter, and prepare his group for entry.

There are many examples from the literature where uncoordinated parts are specified using the techniques described thus far. Charles Ives used uncoordinated parts in The Unanswered Question, Central Park in the Dark and Symphony No. 4. Numerous examples may be found throughout the spatial works of Henry Brant (e.g. Divinity: Dialogues in the Form of Secret Portraits for Harpsichord and Brass Quintet and Prevailing Winds for Invisible Woodwind Quintet). Note: In some works, Brant specifies that parts be uncoordinated even in the case when all performers have the same tempo marking (e.g. American Weather). Examples by György Ligeti are Chamber Concerto for 13 Instrumentalists and Magyar Etüdök (Hungarian Studies) for Mixed A Cappella Choir (16 part). Note: Ligeti suggests that the latter work may be performed in a coordinated manner with the assistance of conducting technology which is discussed later in this article. Other examples would be Karlheinz Stockhausen's Gruppen (Groups) for 3 Orchestras, Luciano Berio's Folk Songs for Viola, Voice, Harp and Cello, Emmanuel Ghent's Helices for Violin, Piano and Tape, Brian Ferneyhough's Missa Brevis and Firecycle Beta, Pierre Boulez's Rituel: In Memoriam Maderna and Michael Colgrass' Letter from Mozart for Orchestra.

Sometimes, explicit instructions are given to perform simultaneous different tempi in an uncoordinated manner. However, in many cases, uncoordination (or at least, loose coordination) is a byproduct of some other instruction. Here are some examples of ways by which composers have indirectly specified that ensemble playing be loosely coordinated:

• Rubato tempo markings (or equivalent directions to freely deviate from the prevailing tempo) might be used.


• Some form of proportional notation might be used to indicate approximately the time at which notes are to occur as in Krzysztof Penderecki's Dimensionen der Zeit und der Stille, Luciano Berio's Tempi Concertati and Emmanuel Ghent's Helices for Violin, Piano and Tape.


• The duration of notes might be specified relative to the extreme limits of the capabilities of individual performers. For example, a wind instrumentalist or vocalist might be asked to hold a note as long as possible. Or, a given passage might be marked "Play as fast as possible" as in György Ligeti's Chamber Concerto for 13 Instrumentalists.


• The duration of notes and rests might be left to the discretion of individual performers. For example, an instruction might be given to play notes or rests of irregular length as in Henry Brant's Prevailing Winds for Invisible Woodwind Quintet.


• A section of a work might be marked "Senza tempo" as in György Ligeti's Chamber Concerto for 13 Instrumentalists.


• An instruction might be given to gradually increase (or decrease) the tempo of a given part until the moment at which the part becomes aligned in time with another simultaneous part as in Steve Reich's Violin Phase for Violin and Pre-recorded Tape or Four Violins.

Notational Solutions

Various methods have been used to notate polytempo music in such a way that it may be performed in a coordinated manner by a traditional ensemble, with or without a conductor. In general, each of these methods involves notating the music in such a way that some temporal characteristics (e.g. tempo and meter; barline placement; or the duration of a given type of note) are the same in all parts while other temporal characteristics are allowed to be independent.

An unusual approach is taken by Charles Ives in the second of the Three Harvest Home Chorales and in the piccolo part in Section 8 of the second movement of Symphony No. 4. In these sections, parts share a common meter, tempo, and barline placement, but the durational meaning of a given type of note symbol is not the same in all parts. A form of rhythmic cue line is provided that is aligned with the beats of one part. Arrows are drawn from notes on the cue line to notes of another part to indicate the time at which such notes are to be played relative to the part upon which the cue line is based.

Many composers have chosen to notate polytempo music as monotempo music. With this method, at all points in time, all parts are notated against a common tempo and meter, and changes in tempo or meter occur in all parts simultaneously. Tempi that differ from the common underlying tempo are represented by various means. For some cases, augmentation or diminution (i.e. uniform lengthening or shortening of note durations) may be used to notate the desired multi-rate effects. Similarly, tuplets may be used for this purpose. For example, if the common tempo is MM quarter note = 84, the tempi MM quarter note = 147 may be represented by using tuplets with the ratio 7:4. This approach is discussed in Timothy Sullivan's article "Multiple Tempi: A Survey and Method" (1997) which contains a table of note durations that one could use to notate a beat, or various fractions of a beat, of a tempo that stands in a given ratio to the common tempo, assuming that the duration of a beat in the common tempo equals that of a quarter note. Other tempi for which the ratio might be too complicated to be represented in this way, may be approximated by writing the part in the common tempo, for example, to the nearest sixteenth note. Instances where such techniques have been used are quite common in the literature. For example, see Bach's fourteenth canon Canon a 4 per Augmentationem et Diminutionem of the Goldberg Canons (BWV 1087) (augmentation and diminution), Elliott Carter's String Quartet No. 5 (tuplets), the first movement of Henry Cowell's Quartet Romantic (tuplets), and Elliott Carter's String Quartet No. 2 (approximation). These methods may be extended to represent accelerandos and decelerandos approximately as in Elliott Carter's String Quartet No. 2 (e.g. see the first measure of the cello part).

Another common approach is one in which polytempo music is written as polymetric music whereby each part has an independent tempo marking and meter, but common barlines are used whenever possible. Here, coordination is facilitated by having the first beat of most measures occur at the same time in all parts. For example, with this approach one might find two simultaneous measures (of equal width) for which the barlines are aligned horizontally, and the meters and tempi are 4/4 at MM quarter note = 84, and 7/4 at MM quarter note = 147. This approach is used less frequently than that in which polytempo music is written as monotempo music. Still, it is quite common. For example, see the sketches for Charles Ives' unfinished Universe Symphony, Elliott Carter's String Quartet No. 1, Henry Brant's American Debate: An Antiphonal Overture and Verticals Ascending: After the Rodia Towers: For Two Separated Instrumental Groups, Section I of György Ligeti's Magyar Etüdök (Hungarian Studies) for Mixed A Cappella Choir (16 part), and Mark Applebaum's Tlön (where only conductors are coordinated as this piece calls for no players).

Another way by which common barlines may be used to simplify coordination is found in Section III of György Ligeti's Magyar Etüdök (Hungarian Studies) for Mixed A Cappella Choir (16 part). Here, the tempi are independent but the meter (4/4) is common to all parts. The tempi used in this work (MM quarter note = 90, 110, 140, 160 and 190) are such that the duration (or width) of some whole number of bars in one part equals the duration (or width) of some other whole number of bars in another part. As a result, some barlines of different parts coincide periodically. More generally, this will occur in cases for which the tempo numbers have a common divisor (other than one). A further simplification occurs in Ligeti's case, because one of the tempo numbers (i.e. 160) is evenly divisible by the product (40) of the greater common divisor (10) and the number of beats in a measure (4). Thus, in this case, four measures of the part which is written in the tempo MM quarter = 160 serves as a unit of organization to which the tempi of the other parts may be coordinated.

In other cases, polytempo music has been notated as polymetric music with independent meters, barlines and tempi for each part, but where the tempi are such that the duration of a given type of note (e.g. a quarter note) is the same in all parts, at all points in time. In this case, since each part has its own meter, the duration of beats, and consequently the tempo, is not the same in all parts. For example, one might find three measures of meter 4/4 at MM quarter note = 84, occurring simultaneously with (and having the same total width as) four measures of meter 6/8 at MM dotted-eighth note = 56. This approach is used in the final section of George Crumb's Black Angels which is titled Threnody III: Night of the Electric Insects.

For some polymetric works such those discussed here, it has been suggested that one conductor be used to beat two different patterns simultaneously as in Charles Ives' Three Places in New England (Orchestral Set No. 1) (Note: According to the version of the score edited by by James B. Sinclair, this appears to have been more the idea of the conductor Nicholas Slonimsky than of Ives, and for this work, Ives preferred to have all the material notated against a common meter and tempo). This approach has limited applicability as it is not well suited to cases involving more than two simultaneous different tempi, and it is difficult for a conductor to beat two sequences of pulses when the pulse sequences do not bear a simple relationship to one another. For other polymetric works, it has been suggested that two or more conductors might be used to facilitate coordination. For example, this is done in Elliott Carter's Double Concerto for Harpsichord and Piano with Two Chamber Orchestras.

Finally, for some works, ossia lines have been used to notate a given part with an alternative tempo and meter. In most cases, one of the two representations of the part is written in the meter and tempo of the other parts, and sometimes this is only an approximation of how the part is to be played. For example, ossia lines of this type are given in Charles Ives' Three Places in New England (Orchestral Set No. 1) and Elliott Carter's String Quartet No. 2.

Technology-Assisted Conducting

Various technology-based conducting systems have been used to help ensembles perform polytempo music in a coordinated way. With these devices, some number of automated conductors are employed to lead the performance.

For most of the devices that have been used thus far, cue signals are transmitted to players by sound or light. For sound-based systems, players are equipped with headphones through which they may hear various cue signals. If light is used, players are provided with a light that blinks. Sound has been used more often than light for transmitting cues.

In most cases, the cue signals for a given work are defined before a performance. Signals might be recorded on audio tape or as MIDI data in a sequencer.

Most often, cue signals have been used to indicate the time at which downbeats are to occur. However, in some cases, they have been used for other purposes as well such as: to indicate the time of sub-beats (or upbeats); to establish a new tempo before notes are to be played; to signal the entire rhythmic pattern of a complicated phrase before it is to be played; and to provide a reassuring checkpoint cue or an alerting cue just before the end of a long rest.

For a given performance, the number of automated conductors used and the way in which they are configured depends upon the particular demands of the work being performed. In some cases, each player of an ensemble might be provided with a dedicated automated conductor. In arrangements that involve multiple groups, each member of a given group of players might be lead by a single automated conductor, identical signals of which might be distributed to each player of the group (e.g. through headphones). Or, if each group has its own human conductor, each human conductor might follow the lead of a different automated conductor.

For some electro-acoustic works, the mechanism that is used to generate cue signals has also been used to automatically play additional electronic parts (which might have their own tempo). Depending upon the type of mechanism that is used, such parts might be sounded from tape or played by a synthesizer that is controlled by a sequencer. In some cases, the parts that are played automatically might be of a type that could not possibly be played by human performers in real time.

The following composers have written compositions and articles related to the use of automatic conducting technology (Note: A more detailed discussion of the works mentioned here is given in the article Polytempo Music: An Annotated Bibliography.):

• Emmanuel Ghent.

In the 1960's, Ghent began to use automatic conducting technology for his works. This is documented in his article "Programmed Signals to Performers: A New Compositional Resource" (1967) and to a lesser extent in his article "The Coordinome in Relation to Electronic Music" (1967). Automatic conductors are used in the following works by Ghent: Triality: for Violin, Trumpet and Bassoon: in Conjunction with a Polynome: 2 Studies (1964), Dithyrambos for Brass Quintet (1965) and Hex: An Ellipsis for Trumpet, Instruments and Tape (1966). Also, Ghent prepared a click-track tape that was used in the 1978 recording of Henry Cowell's Quartet Romantic (New World Records: Catalog #80285, see: "Producer's Note" in Liner Notes).

• Lejaren Hiller.

In Emmanuel Ghent's article "Programmed Signals to Performers: A New Compositional Resource" (1967), it is mentioned that "Lejaren A. Hiller at the University of Illinois is currently developing a system similar to that described in this paper. One difference will be the use of visual (opaque lights) in place of auditory stimuli" (see p.143, Footnote 10).

• Iannis Xenakis.

In Xenakis' polytempo work Persephassa pour Six Percussionistes (1969), a performance note specifies that on page 14, measure 191, "the 6 metronomes must be exactly synchronized (or else electronically controlled)" (see the English "Directions").

• Henry Brant.

In the performance notes for Brant's polytempo work Verticals Ascending: After the Rodia Towers: For Two Separated Instrumental Groups (1969), it is suggested that a click-track or flashing light might be used to facilitate coordination.

• Godfried-Willem Raes.

Automatic conductors were used in Raes' works Logos 3/5 for Violin, Cello, Oboe, and Piano with Electronic Conductor (1969) and Fall '95 for Violin, Flute, Clarinet, Horn, Euphonium, Tuba, Player Piano, Computer and Polymetronome (1995).

• Larry Austin.

The following works by Austin call for the use of automatic conductors: Canadian Coastlines: Canonic Fractals for Musicians and Computer Band (1981), Sonata Concertante (1983-4), Life Pulse Prelude (1984), Life Pulse Prelude for Live and Recorded Percussionists (1984/96), and Charles Ives' Universe Symphony as realized and completed by Larry Austin (1974-93). The use of automatic conductors in his work Life Pulse Prelude is documented in the article "Charles Ives's Life Pulse Prelude for Percussion Orchestra: A Realization for Modern Performance from Sketches for his Universe Symphony" (1985). The article also mentions an early version of the work that was recorded in 1975 with the use of automatic conductors.

• Kyle Gann.

In Gann's biography that is available at http://home.earthlink.net/~kgann/longbio.html, it is mentioned that "silent (blinking-light) metronomes" have been used for the performance of some of his polytempo works (ca. 1981).

• György Ligeti.

In the performance notes for Section III of Ligeti's Magyar Etüdök (Hungarian Studies) for Mixed A Cappella Choir (16 part) (1983), it is suggested that either a light-based or sound-based automatic conducting system may be used to synchronize the performance of the simultaneous different tempi.

• Lukas Ligeti.

In his article "Beta Foly: Experiments with Tradition and Technology in West Africa" (2000), Lukas Ligeti describes a MIDI-based automatic conducting system. Automatic conductors have been used in Lukas Ligeti's Groove Magic for 11 Instruments and Computer Controlled Click Tracks (1993).

• Stefan Dragostinov.

A device which Dragostinov called a photopolymetronome was once mentioned (during 2001) on the page http://www.angelfire.com/bc/dragostinov/ (but the information has been deleted).

Electronic Realizations

By using recording technology in a studio, some composers have realized their polytempo works as an audio recording that may be performed at a later time. With this process, it is possible to record each tempo, one at a time. Depending upon the structure of the piece, this might involve recording each instrument, one at a time. Or, it might mean recording each phrase one at a time, where any given phrase might be performed by one instrument or a group of instruments. During the recording process, sequencer-controlled electronic instruments may be used in combination with traditional instruments, or either may be used exclusively. After each component part has been recorded, the total work may be assembled from the parts by digital means. Using a digital signal processing technique know as time-scale modification (TSM), individual elements may be scaled in time to increase or decrease their tempo as needed (without pitch change). These techniques may be used to realize works for traditional acoustic instruments with human players that might be difficult or impossible to perform in real time.

In some cases, a special score called a "recording score" might be devised that is tailored more to the needs of performance in the recording studio than real-time performance. For example, in the case when a piece consists of a sequence of overlapping phrases where each phrase may have a different instrumentation and tempo, the recording score might consist of a sequence of pages on which there is one phrase per page. Additional instructions would be provided to indicate how the individual phrases are to be assembled to form the composite work. For example, phrases might be numbered and coincidence points (in time) could be labeled in each phrase.

For a composition that is intended to be realized by using recording technology, one might impose limitations during the process of composition that would ensure that the work will be (at least theoretically) possible to perform in real time by human players. For example, one might use the constraint that an instrument that usually plays only one note at a time, never sounds more than one note at the same time after independent phrases have been overlapped to form the composite recording.

As an alternative to using recording technology, polytempo music may be performed in real-time before an audience by using automated instruments only, such as an ensemble of synthesizers that are centrally controlled by a sequencer, or a player piano. In this case, musical information is encoded prior to a performance, for example, as a MIDI file or a paper piano roll.

There are many composers who have realized polytempo music as electronic music. Some examples would be: Conlon Nancarrow's recordings and live performances of his Studies for Player Piano; Larry Austin's recordings of acoustic instruments for an early version of his Life Pulse Prelude (1975) and later synthetic realizations of the same work (1981); Frank Zappa's use of the technique that he called xenochrony (strange synchronizations), for example, in his recordings Friendly Little Finger from Zoot Allures (1976), Rubber Shirt from Sheik Yerbouti (1979), and Packard Goose and Keep it Greasy from Joe's Garage (1979), whereby various recorded tracks of unrelated material were assembled into new composite recordings; Emmanuel Ghent's synthetic realization of his Dithyrambos for Brass Quintet (1977); and Paul Dolden's recording L'Ivresse de la Vitesse (Intoxication by Speed) (1992-3) in which various separate human performances are processed digitally (e.g. speeded up) and then combined.

Coda

Any of the approaches discussed in this article may be used to facilitate the performance of polytempo music. In some cases, one might use more than one approach for a given work. For example, one might notate a given piece in more than one way: as a recording score that would be well suited to the task of recording the work in a studio, and as a set of part scores that would be tailored to the task of real-time performance before an audience. Or, one might notate a given polytempo piece as a polymetric piece with common barlines throughout, and coordinate the performance with technology-assisted conductors.

As an alternative, one might choose not to facilitate performance in any way. At this time, there is no standard way to realize polytempo music in a coordinated manner, and one cannot possibly envision the requirements of systems that might be used in the future to perform such works. Under these conditions, one might choose to notate a polytempo piece in an abstract way that most naturally represents the musical material with no concern for the degree to which it serves the needs of a particular mode of performance. Or, perhaps any abstract representation of the music would be sufficient.

Update History

August 6, 2001: Completed the first version of this page.

August 23, 2001: Added mention of Emmanuel Ghent's click track for Henry Cowell's Quartet Romantic.

September 1, 2001: Added mention of Timothy Sullivan's article "Multiple Tempi: A Survey and Method".

Polytempo Music

An Annotated Bibliography by John Greschak

Introduction

This is a list of polytempo music compositions, and books or articles that discuss polytempo music. The list is limited to those pieces in which the composer has notated the music by using two or more different and simultaneous tempo markings, and consequently, the duration of a given type of note (e.g. a quarter note) is not the same in all parts at some point in time. Here, "tempo marking" means a metronome marking or a tempo term (e.g. allegro or accelerando).

Pieces for which more than one tempo might be perceived but for which all parts are written against a common tempo, no tempo markings are specified, or performers are instructed to deviate freely from specified tempi (e.g. as in rubato or senza tempo), have been excluded. More specifically:

• Pieces where multi-rate effects are notated with tuplets against a global tempo as in Elliott Carter's String Quartet No. 5 (tuplets 3, 5 and 7), Brian Ferneyhough's Epicycle for Twenty Solo Strings, or the first movement of Henry Cowell's Quartet Romantic (tuplets 6 2/3, 5 1/3, 2 2/15, 5 3/5, etc.), have been excluded.


• Pieces where multi-rate effects are notated with polymeters against a global tempo, have been excluded. For example, in the final section of George Crumb's Black Angels titled Threnody III: Night of the Electric Insects, there is a duration of 36 quarter notes at a tempo of MM quarter note = 60 that is partitioned with six different meters simultaneously and where at least three different rates of speed may be perceived (i.e. half-note rate, dotted-quarter rate, and dotted-half rate). And, later in this section, a metronome marking of MM dotted-quarter note = 36 in one part is written against other parts that have a metronome marking of MM quarter note = 54. Since the duration of a quarter note is the same in all parts, at all points in time, Crumb's piece has been excluded. Similarly, in measures 93 through 103 of Elliott Carter's Double Concerto for Harpsichord and Piano with Two Chamber Orchestras, 6/8 measures at a tempo of MM dotted-quarter note = 70 occur simultaneously with 3/4 measures at MM quarter note = 105. A similar polymetric structure occurs in measures 619 through 689. Carter's piece has been excluded because the duration of a quarter note is the same in all parts, at all points in time.


• Pieces have been excluded for which multi-rate effects are written with proportional notation (also called: spatial notation, time-space notation, proportionate notation or analog notation) and no tempo markings are given. For example, Conlon Nancarrow's Study No. 8 for Player Piano has been excluded. Throughout this piece, accelerandos occur simultaneously with decelerandos. But, the entire piece is written using a form of proportional notation whereby staccato tones are indicated with flagged eighth notes, the duration of sustained tones is indicated with horizontal lines that are appended after noteheads, and the time at which a given note is to begin is indicated by its horizontal position within the staff. Similarly, Nancarrow's Study No. 21 for Player Piano (Canon X) in which one of two voices gradually accelerates throughout, while the other voice gradually decelerates, has been excluded. And, Nancarrow's Study No. 41 for Player Piano has been excluded. This work consists of three sections, the first two of which are for two voices that have different tempi, and the third section consists of the first two sections performed simultaneously on two pianos. Similarly, a 22-second section on page 19 of Krzysztof Penderecki's Dimensionen der Zeit und der Stille during which six simultaneous unsynchronized decelerandos occur, has been excluded because the start times for notes in this passage have been indicated graphically using proportional notation and no tempo markings are given. Likewise, Luciano Berio's Tempi Concertati has been excluded.


• Mensuration canons of the Renaissance period in which multiple mensuration signs are provided on a given staff to indicate that a line is to be performed at different speeds simultaneously, have been excluded. For example, Johannes Ockeghem's Missa Prolationum has been excluded.


• Pieces that contain the multi-rate effects of augmentation or diminution where such effects have been notated by uniformly lengthening or shortening the note durations of a given theme, have been excluded. For example, Bach's fourteenth canon Canon a 4 per Augmentationem et Diminutionem of the Goldberg Canons (BWV 1087) has been excluded.


• Instances of rubato tempo markings or equivalent directions to freely deviate from the prevailing tempo have been excluded. For example, cases where a solo part is marked rubato against an accompaniment part that is to remain in strict tempo have been excluded. Also, "tempo glissandos" (i.e. written, controlled or measured accelerandos or decelerandos that are embraced by a broken slur that ends with an arrowhead) such as those found in Elliott Carter's String Quartet No. 2 (e.g. see the first measure of the cello part) have been excluded. In the case of this piece, rubato is implied by the textual directions which specify that such notation indicates an accelerando where the first and last note of the passage must be played with a duration as written and "the intervening notes are to be played as a continuous accelerando (or in other cases, a ritardando, depending on the notation), the notation indicating approximately whether the accelerando or ritardando is regular, or more active at the beginning or end of the passage." (Note: Carter's String Quartet No. 2 has been included in the list given here for other reasons.)


• Instances of the marking "senza tempo" have been excluded. For example, the long "senza tempo" cadenzas beginning in measures I:19 and IV:8 of the first and fourth movements of György Ligeti's Chamber Concerto for 13 Instrumentalists have been excluded. (Note: Ligeti's Chamber Concerto has been included in the list given here for other reasons.) In this work, the following instructions are given for these passages in a footnote:


...After attacking, the cadenzas are played as fast as possible, independent of the metre and bar lines. The rests in brackets are not to be taken in consideration when playing; they indicate the imaginary remaining duration of the bars, the cadenzas being notated as though they had no duration. When a cadenza is finished, the player waits for the beginning of the next bar; from that point on, the previously suspended meter is resumed.

NB. When cadenzas begin simultaneously in two or more instruments, only the attack is simultaneous; after that, the cadenzas are individual, each instrument playing its notes independently as fast as possible, and finishing independently. In this way, little time-shifts can occur; the notation in the score is only an indication, and the individual cadenzas may end a bit earlier or later.


• Pieces have been excluded for which the score appears to indicate different simultaneous tempi but it is generally believed that the composer did not intend this to be the case. For example, this might occur if any of the following conditions is believed to be true: there was an error on the part of the composer or copyist; the composer considered two different tempo terms to be equivalent; the composer used different tempo terms to suggest various moods (or some other characteristics) at a given tempo; the composer used tempo terms but not to specify tempo. For example, in his book The Mass in B Minor: The Great Catholic Mass (p. 295 n.59), Stauffer mentions that he has found forty-nine instances in the original performance parts for Bach's Mass in B Minor where different tempo indications (e.g. adagio and lente) have been used simultaneously. Since it is generally believed that Bach did not intend to have these parts played at different tempi, this piece has been excluded.


• Any polytempo piece for which there is no known score has been excluded. For example, electronic pieces for which there is no score have been excluded.

Works such as Henry Brant's American Weather and Pierre Boulez's Rituel: In Memoriam Maderna in which the same tempo marking is specified for all parts, but the parts are to be performed in an uncoordinated manner (for example, by using independent conductors), have been included. Also, works such as Charles Ives' Three Harvest Home Chorales and Symphony No. 4 (specifically, the fourth movement) where at all points in time, one tempo marking is specified that applies to all parts, but the duration of a given type of note is not the same in all parts at some point in time, have been included.

The limitations or restrictions placed on the list given here are not intended to be taken as definitive properties of the class of music that is polytempo. Instead, the imposed limitations merely reflect the interests of the author and also serve to bound the size of the list.

For additional examples of pieces where more than one tempo might be perceived but that fall outside of the scope of this bibliography, see the books Modern Rhythmic Notation by Gardner Read and Music Notation in the Twentieth Century: A Practical Guidebook by Kurt Stone.

Detailed information regarding instances of simultaneous different tempi are described in the annotations. Also, the methods by which (and the degrees to which) performers are to be coordinated or synchronized are discussed.

Compositions

Applebaum, Mark. 1995. Tlön for 3 Conductors and No Players.

This piece calls for 3 conductors that face the audience and lead imaginary ensembles.
A graphic file of a portion of the score (p.14: marks R, S and T) is available at http://www.markapplebaum.com/tlon01.html. In this excerpt, there are three staves, each of which indicates the meter and tempo (as well as dynamic markings and expression text) for one of the conductors. For example, the following measures occur between marks R and S: the first conductor has two measures of meter 9/8 at a tempo of MM quarter note = 135 (MM dotted-quarter note = 90); the second conductor has four measures with meters 1/4, 2/8, 1/4 and 2/8 all at a tempo of MM quarter note = 60; and the third conductor has two measures of 7/8 at MM quarter note = 105 (MM dotted-quarter note = 70). For these measures, all barlines are common and drawn through all parts, except the right barlines of the two 1/4 measures in the part of the second conductor. A description of the piece is available at http://www.markapplebaum.com/acoustic.html.

Arrigo, Girolamo 1963. Fluxus per Nove Strumenti. Firenze: Aldo Bruzzichelli.

On page 25, the nine instruments are partitioned into three groups of three. The first group (trombone, bass clarinet, bassoon) plays a 3-measure phrase with meters 7/8, 3/4 and 5/8 at MM eighth note = about 114. At the fifth eighth note of this phrase, the second group (viola, cello, double bass) begins a 4-measure phrase with meters 4/4, 7/8, 4/4 and 7/8 at MM eighth note = 152. At the fourth eighth note of the third measure of the first phrase, the third group (flute, clarinet, harp) begins a 3-measure phrase with meters 7/8, 4/4 and 7/8 at MM eighth note = about 133. None of the barlines of these simultaneous phrases coincide. Each phrase ends with a long fermata. This page is quoted in the book Modern Rhythmic Notation by Gardner Read (p. 170).

Arrigo-Nelson, John 2000. Temporal Quartets.

The eight instruments are organized as a string quartet and a mixed quartet (oboe, alto flute, vibraphone and trombone). The tempo markings for the quartets are independent. Over the course of the piece, the tempo of the string quartet decreases in a step-wise manner through the following sequence of tempi: MM quarter note = 112, 84, 70, 58 and 48. Concurrently, the tempo of the mixed quartet increases step-wise through the following sequence: MM quarter note = 48, 58, 70, 84 and 112. (Note: All tempi are in the metronome scale, except 70.) The times at which the quartets change tempo do not coincide.

The performance notes call for two conductors. Usually, the quartets play in an uncoordinated manner, relative to each other. There are two times at which the beats of the quartets are to be coincident. During 5 measures in the middle of the piece, both quartets play at the tempo MM quarter note = 70, with common barlines. The first beat of this section is coordinated by having the members of the string quartet play unmeasured, trilled glissandi while the mixed quartet plays two (preparatory) measures at MM quarter note = 70. At the end of the piece, the quartets play the final chord simultaneously. One of the conductors provides a cue for this event. Here, the duration of rests that immediately precede the final chord in each quartet, is adjusted, as needed.

There are beats or sub-beats that would coincide if the tempo of each quartet were followed precisely. Dashed vertical lines are used to indicate these. (Note: Coincident barlines are not drawn through from one quartet to another.) In some cases, a rhythm line is provided between the quartet parts to indicate sub-beats associated with tuplets. Such beats and sub-beats are not required to coincide in performance. In the performance notes, Arrigo-Nelson writes:

I have placed occasional dotted vertical lines in the score where certain beats or events line up. These vertical lines, however, are simply a guide as to where these events occur. My overall concern is the effect of the two tempi existing simultaneously.

System breaks are placed in the score at barlines of the quartet that is playing at the faster tempo. The measures of the slower quartet are broken across systems. In broken measures, the following notational devices are used: broken beams; arrows pointing to the right that are written on the middle staff line at the end of one system and at the beginning of the subsequent system; and broken tuplet brackets with an arrow pointing right after the tuplet number at the end of a system, and an arrow pointing left before the tuplet number at the beginning of the subsequent system.

Arrigo-Nelson, John 2002. Solo/Duo for Violin and Piano.

Often in this work, the tempi of the violin and piano are different. For example, in the opening, there is a ritardando in the piano from MM quarter note = 112 to MM quarter note = 69, while the violin continues at MM quarter note = 112. Later, the tempo of the violin is MM quarter note = 45 while that of the piano is MM quarter note = 120. During another period, the tempi of the violin and piano are MM quarter note = c. 80 and MM quarter note = c. 36, respectively.

Throughout, the parts are notated so that the horizontal position of events indicates the relative times at which events are to occur. Generally, the instruments are to be loosely coordinated. In the performance notes, Arrigo-Nelson writes:

In sections of the score where the instruments are playing in independent tempi, the music of each part is written in correct proportion to the other. It is not expected that every event in these areas will occur exactly as written on the page. The events should occur, however, in the general area in which they are written.

Several events, which are listed in the performance notes, are to be coordinated between the two parts. This is accomplished by adjusting the duration (or number of repetitions) of material that precedes the coordinated events, as needed.

Austin, Larry 1981. Canadian Coastlines: Canonic Fractals for Musicians and Computer Band.

This 11-minute work is an eight-voice canon in which eight musicians perform four voices along with four voices of tape music that was generated with a Synclavier Digital Music System. Each voice enters as an exact melodic and rhythmic imitation but at a tempo that is different from all other voices. Voices gradually (and independently) accelerate and decelerate over long periods of time. Four audio beat-cue tracks are played from tape and distributed to the eight musicians to enable the players to remain synchronized with each other and with the tape music. At five points in time, all eight voices momentarily play in unison. Program notes for this piece may be found at http://www.music.unt.edu/cemi/LApnotes.htm#canadian. The score is available at the American Music Center (call number: AMC M1613.3 A936 C2).

Berio, Luciano 1964. Folk Songs for Viola, Voice, Harp and Cello. London: Universal Edition.

The viola opens the first song Black is the Colour... with some introductory material at MM quarter note = about 72 after which the voice and harp enter at MM quarter note = 54 while the viola continues at MM quarter note = 72 "indipendente dal canto" (independent from the singing). Then the viola plays alone at MM quarter note = about 90. The voice and harp join in at MM quarter note = 54 and the viola accompanies them at MM quarter note = 94. Again the viola plays alone at MM quarter note = 72 but then switches to MM quarter note = 92 as the voice and harp enter. Gradually, over the course of this passage the viola slows down to MM quarter note = 54, and for a brief period, all play at the same tempo. The song concludes with seven 3/4 measures for the violin and cello at MM quarter note = 92. Most of the viola passages in this song are unmeasured.

In the song Motettu de Tristura (for viola, cello, 2 percussionists, voice, harp and piccolo), in the section that follows mark number 2, the voice is at MM eighth note = 60 while the piccolo is at MM eighth note = about 72 "liberamente, senza badare al canto" (freely, without paying attention to the singing). The following footnote is given: "It is not necessary to synchronize the parts. At the entrance of ottavino [piccolo] the conductor should stop giving cues. Viola and Cello should stop playing last.".

Boulez, Pierre 1974-5. Rituel: In Memoriam Maderna. London: Universal Edition.

The orchestra is divided into eight groups, each with a percussionist. The piece consists of a sequence of 15 segments or sections. In the odd-numbered segments, Group 8 (which has two percussionists) plays at MM half note = about 50-54 (très lent) while the conductor improvises the cue time for each of a sequence of chords to be played by the rest of the orchestra. The even-numbered segments are played asynchronously. In these segments, each of the groups 1 through 7 is given a phrase to be played at a tempo of MM eighth note = about 80-92 (moderato). The conductor cues the entry of each phrase at a time and order as he chooses, with the constraints that the first phrase that is played must be the longest and that all phrases must begin and end within the duration of this longest phrase. The percussionist in each group maintains the beat for his group. All beats are of equal duration but the times at which beats occur is not necessarily the same in all groups.

Brant, Henry 1953, revised 1968. Antiphony One for Symphony Orchestra Divided into 5 Separated Groups. New York: Carl Fischer.

The orchestra is divided into five widely spatially-separated groups: strings (or clarinet choir), woodwinds, horns, muted brass, percussion. Each group is positioned in a different part of the hall and is notated with a distinct tempo, meter and barline scheme. The strings are led by a principal conductor and there are four section-conductors for the other groups. An optional voice group may be added with its own conductor. The principal conductor cues the entrances of the section-conductors. In the "Explanatory Remarks", Brant writes:

Each section conductor must take his own tempo (specifically indicated) from the beginning of each entrance, and then proceed, clearly and steadily, entirely independently of all the other conductors.

Also, he writes:

It is not expected that the relationships between the different barlines in the score will be carried out precisely as indicated. What is intended is an approximate relationship, but the approximation should be as close to the score as possible, although it will be apparent that a certain small amount of leeway is feasible, in the timing of entrance cues, and in permitting certain groups to finish their phrases slightly before or after the points indicated.

This was the first spatial (or antiphonal) work by Brant. He has composed dozens of pieces of this type. In all cases, the various ensembles are to be widely separated, and in most cases, a different metronome marking is specified for each simultaneous musical phrase or fragment. A biography of Henry Brant may be found at http://www.carlfischer.com/brantbio.html and a list of his works which contains a subsection titled "Spatial Works" may be found at http://www.carlfischer.com/brantworks.html. The scores for most of the works that are discussed in the bibliography given here are available at The New York Public Library for the Performing Arts. A complete set of Brant's manuscripts is contained in the collection of the Paul Sacher Foundation in Basel, Switzerland.

Brant, Henry 1954. Millenium 2. New York: Carl Fischer.

This work contains early instances of many devices that Brant has used again in later works. For example, there are looped phrases that are to begin on cue and then be played in an uncoordinated manner. Upon hearing the sound of some musical event, such phrases are to be stopped after a stop mark has been encountered in the score. Often, several phrases of this type overlap. In some cases, they begin sequentially, one by one, while in other cases they begin at the same time. Also, there are phrases that are to be played once (with no repeat), in an uncoordinated manner. Simultaneously with uncoordinated phrases such as these, there are coordinated phrases that are conducted. Each phrase is allowed to have a tempo that is independent of that of any other phrase that is being performed simultaneously.

In the notes, Brant writes the following:

The horns, baritone horns and tuba, seated on the stage, follow the conductor precisely, in the normal way, from beginning to end. The timpani, gongs and thundersheet likewise; and from (M) to the end, the bell instruments also.

However, trumpets 1 to 9, and trombones 1 to 9, from the beginning to (V), all work on a different co-ordination plan. Each of these players enters on a direct, individual cue from the conductor, after which he ignores the conductor and all other players, and proceeds for himself, making no effort to maintain ensemble. The individual player, under these conditions, should nevertheless make every effort to perform in as steady and accurate a manner as possible. In most cases these entrances are divided up into phrases which are indicated by the termination sign [right bracket symbol]. After several phrases a repeat sign appears. The player is instructed to play through each such sequence, repeating if necessary in order to keep playing until a particular designated point, an obvious place such as a loud percussion entrance. When he hears this "cue", he continues playing until the end of his current phrase [right bracket symbol] and there makes a clean and definite exit. No "fade-out" or sudden "cut-off on cue."

At (S) and elsewhere a simpler device is used, there are no indicated phrase divisions or repeats; the player merely enters on cue, plays everything in the sequence once and then stops.

At (G) the 9 trumpets and 9 trombones enter precisely together, with the conductor, and then immediately go their separate ways. The trumpets alone at (Q), and the trombones at (S) have this same briefly simultaneous, and then promptly scattered, procedure. The bell instruments have this same kind of thing at letter (B).

In other sections, especially in the long (M) sequence, the conductor has the task of maintaining precise ensemble for the horns, tubas and percussion, and at the same time, giving direct individual cues to particular trumpet-trombone pairs or sections. This is workable because these "free" entrances always occur in relation to specific down-beats of the particular tempo which the conductor happens to be maintaining at these points.

From (V) to the end, the tempo system becomes normal, the entire group moving together in the same metre and tempo.

Here, the right bracket symbol that Brant uses as a stop mark is similar to the mark that has been used by others to indicate the end of a passage that is to be prominent as in Schoenberg's Hauptstimme and Nebenstimme.

Brant, Henry 1956. On the Nature of Things (after Lucretius): Spatial Tone Poem. New York: Carl Fischer.

This work is for strings, woodwind trio (flute, clarinet, bassoon), solo flute, solo horn, solo oboe, glockenspiel and one conductor. There is also a version for wind ensemble. The conductor leads a unison part for strings (violins, violas and cellos) that consists of a sequence of five sections that have the following tempi: MM quarter note = about 104, about 60, about 104, about 60, and about 104. Sometimes, the woodwind trio is synchronized with the strings, and when used, the basses are synchronized with the strings, though not in unison. Other parts which have a different tempo are cued by the conductor and then continue unconducted. The following tempi are used for such parts: MM quarter note = about 84 (oboe), MM half note = 72 or slower (horn), MM quarter note = about 80 (flute), MM quarter note = 72 or slower (woodwind trio, when not synchronized with strings).

Phrases are repeated in order to control the total duration of a given passage and to ensure that a passage ends before or after the start or end of some other passage. To achieve this form of control in the string part, which has no repeats, the conductor may continuously adjust the tempo as needed. In repeated phrases, stop marks in the form of breath marks or bold, nearly-horizontal, upward-sloping slash symbols are placed over rests. In such phrases, when the conductor issues a stop cue, the player continues to the next stop mark and stops, or jumps to a repeat ending, depending on the case.

Brant, Henry 1969. Verticals Ascending: After the Rodia Towers: For Two Separated Instrumental Groups. New York: MCA Music.

This work calls for two spatially separated groups of instruments. Throughout, Group I plays in the meter 4/4 at MM quarter note = 94-100, and Group II plays in the meter 3/4 at MM quarter note = 72-80. All parts share a common barline throughout. That is, the duration of each 4/4 measure for Group I equals that of a simultaneous 3/4 measure for Group II. In a section titled "Rhythmic Co-ordination" of the performance notes, Brant writes:

Each group has its own conductor, and the seatings are so arranged that each musician sees only his own conductor. Group I plays in 4/4 meter, Group II in 3/4, throughout. One measure of 4/4 is equal to one measure of 3/4; the first beat of each 4/4 measure must coincide with the first beat of the corresponding 3/4 measure, and the resulting effect will be that of "3 against 4" throughout, whenever both groups are heard simultaneously. In order to achieve this 3:4 ratio as easily and securely as possible, it is essential that the two conductors face each other directly.

The entire piece is in a single overall tempo which does not change, although a minimal flexibility is expected; it should be kept even and steady. This tempo is MM quarter note = 94-100 for the 4/4 group, which corresponds to MM quarter note = 72-80 for the 3/4 group.

Depending on the acoustic properties of the hall and the distance between the two groups, the conductors may wish to devise further aids to co-ordination, such as a "click-track" or a flashing light at the beginning of each bar (24 times per minute); or instead, a cuing system so that either conductor may stop beating during extended rests, and be cued in by his colleague one bar before his next entrance. These are suggestions only. Any accurate solution of the 3:4 ratio is satisfactory.

The Rodia Towers (also known as the Watts Towers) were constructed in the Watts neighborhood of Los Angeles by Simon Rodia (also known as Simon Rodilla, Sabato "Sam" Rodia, and Don Simon) over a period of thirty-three years (1921-1954). For more information on these towers, see: http://www.trywatts.com/towers.htm and http://www.wattstowers.net/towers/index.htm.

Brant, Henry 1973. Divinity: Dialogues in the Form of Secret Portraits for Harpsichord and Brass Quintet, Spatially Separated. New York: Carl Fischer.

No conductor is required. Each player reads from the full score. In the score, entry times are indicated relative to the music of instruments that are already playing. In the "Directions for Performance", Brant discusses the terms co-ordinated and un-co-ordinated:

The term CO-ORDINATED means that all the players have the same metre, tempo and barlines; and that they maintain exact rhythmic ensemble together in the usual way. UN-CO-ORDINATED, or NON-CO-ORDINATED, as used here, means that each instrument, or group of instruments, has an individual tempo and metre. Each such tempo should be maintained steadily and accurately, but the players should make no attempt to "keep together". Each tempo ignores the others.

Uncoordinated playing is specified when there are simultaneous different tempi. And, it is used in some cases when all members of an ensemble have the same tempo marking (which is individually specified for each instrument in the ensemble).

Each page of the score is divided vertically into sections by bold horizontal lines, where each section contains a set of independent systems that are to occur simultaneously.

As in all of Brant's works (that have been seen by this author), traditional metronome numbers are used for all tempi. For example, the following numbers are used here: 63, 66, 69, 76, 80, 84, 92, 116, 132, 144 and 176.

Brant, Henry 1974. Prevailing Winds for Invisible Woodwind Quintet, Spatially Separated. New York: Carl Fischer.

In the introductory notes, Brant writes the following regarding the mode of performance and the structure of the score:

This piece should, if at all possible, be performed in a dark hall, with all instruments invisible to the audience. It may thus happen that the players will be unable to see each other, but this will not create a problem, since no exact rhythmic co-ordination between the players is intended, and no visual contact between them necessary. All can be accomplished via sound cues, but this obviously makes it essential for the players to be able to hear each other.

There can be no "score" of this work, in the usual meaning of the term, because the five parts are completely independent of each other in tempo and rhythmic co-ordination, so that no attempt to represent vertically what takes place when the instruments all play, can be accurate. The present form of the score gives the entire musical material of the piece fully written out, the musical "events" indicated in the sequence in which they occur.

The piece consists of a sequence of 7 sections. A structure that consists of a sequence of progressively faster sections is used thematically throughout the work. The tempo markings for the first three subsections of Section 1 are: "slow", "moderate" and "faster still". In each of Sections 2 through 6, one of the winds plays a "solo" part. The tempo of each solo is faster than (or the same as) that of the previous solo, and is as follows:

Section 2: Oboe Solo, MM quarter note = around 60 (free)
Section 3: Clarinet Solo, MM quarter note = around 66 (free)
Section 4: Flute Solo, MM quarter note = around 69
Section 5: Bassoon Solo, MM quarter note = around 69
Section 6: Horn Solo, MM quarter note = around 72

In Section 7 (which contains 6 subsections), the tempo markings for the even-numbered subsections are: "moderate", "moderately fast" and "fast".

In Sections 2 through 6, the soloist is accompanied by the other four instruments which enter sequentially and play looped phrases that are in a tempo that is different from that of the soloist. The accompaniment passages begin at one of following types of points: a 1-measure rest within the solo part, after a "jazz-lick" in the previous solo part, shortly after another accompaniment begins, or (more rarely) during a sustained solo note. The accompanists complete their current figure upon hearing the "jazz-lick" that ends each solo. In each section, all accompaniment passages are based upon a particular type of figure (e.g. a trill, sustained note, grace note or repeated notes). The figures are separated by pauses of irregular or varying length.

Brant, Henry 1975/8. Curriculum: Spatial Tone Poem. New York: Carl Fischer.

Repeated phrases with breath marks or slash symbols are used as in Brant's On the Nature of Things. Here, fermatas are also used as stop marks. Start and stop cues are issued by the conductor (who does not beat time at all during the piece) or by easily recognizable phrases such as a loud oboe chord or a piccolo motif. Time intervals during which the conductor is to cue the entry of an instrument are indicated with text above or below the staff of an instrument that is playing at the time of entry (e.g. "CUE HARP").

A linear polytempo texture consisting of the sequential accumulation of different repeated phrases, occurs throughout the piece. For example, in the second segment, a 7-measure phrase is played repeatedly on the bassoon at MM quarter note = 69. During the fourth measure of this phrase, the conductor cues the harp to begin playing a 4-measure phrase repeatedly at MM quarter note = 66. During the second or third measure of the harp, the conductor cues a trombone to begin playing an 11-measure phrase repeatedly at MM half note = 76. Then, the conductor cues a short piccolo phrase which is used throughout the piece as a cueing device. Upon hearing the piccolo, the bassoon, harp and trombone complete their phrases through to their repeat barlines.

Vertical (or harmonic) textures occur in which all instruments play at the same tempo but in an uncoordinated manner. For example, upon hearing the piccolo (at the end of the previous example given here), the viola, cello, double bass and piano begin playing different 2-measure phrases repeatedly at the tempo MM quarter note = 66. Here, the players are instructed to "begin as soon as possible but without attempting to start at the same exact moment".

A canon-like texture occurs in the violins. Here, a 20-measure phrase is played repeatedly and simultaneously by 10 violins at MM half note = 63, where each violin begins playing at a different odd-numbered measure within the phrase. The violins are instructed to begin "'at once' but not attempting an exact simultaneous attack".

At various points throughout the piece, these textures (linear, vertical and canon) are performed simultaneously.

Brant, Henry 1976. American Weather: Antiphonal Motet for Two Widely Separated Choral Ensembles. New York: Carl Fischer.

In this piece there are two groups. Group I consists of a full choir with a trombone and chimes. Group II consists of eight solo voices, a trumpet and a glockenspiel. Group I is lead by the conductor and performs throughout as a coordinated ensemble at the tempo MM half note = 96. At four points in the piece (marks 5, 12a, 20a and 28a), the conductor cues the entry of the trumpet. After the trumpet plays a few measures, the entrance of the eight solo voices is cued by the glockenspiel. The eight solo voices attack their opening note together and are instructed to "then proceed independently, making no effort to keep together". The conductor does not beat time for the trumpet, glockenspiel or solo voices. The tempo marking for all the parts in Group II is the same as that of Group I (i.e. MM half note = 96), but Group II is not coordinated with Group I and in general, members of Group II are not coordinated among themselves.

Brant, Henry 1976. Spatial Concerto: "Questions from Genesis" for Pianoforte Solo with Orchestra Divided into Widely Separated Groups, and Ensemble of Women's Voices (8 Sopranos, 8 Altos). New York: Carl Fischer.

Uncoordinated repeated phrases occur at a tempo that is different from that of music with which they are simultaneous. Also, there are phrases that are uncoordinated after the first note where each instrument of the group has the same tempo marking.

Brant, Henry 1977. Cerberus: after "A Report to an Academy" (Kafka) for Piccolo and Double Bass. New York: Carl Fischer.

The bass plays throughout and serves to cue the entry of the piccolo. During the course of the piece, the tempo of the bass is changed twice. It begins at MM quarter note = 63, changes to MM quarter note = about 152, and then returns to MM quarter note = 63. Throughout, the piccolo plays various phrases each of which ends with a final barline. Each piccolo phrase begins at a rest in the bass part, usually at a down beat in the bass. When the tempo of the bass is MM quarter note = 63, the piccolo plays at MM quarter note = 160. When the bass changes to MM quarter note = about 152, the piccolo changes to MM quarter note = 92. Whenever the piccolo enters, the bass rests a given number of beats in its own tempo, and then continues playing. Generally, the bass begins playing again before the piccolo has completed its phrase, and therefore at these times, the instruments play simultaneously at a different tempo.

Brant, Henry 1977. American Debate: An Antiphonal Overture. New York: Carl Fischer.

One conductor leads two separated groups (or an assistant conductor may be used to lead one group, in which case the assistant duplicates the principal conductor's beats exactly). One group is located on stage while the other is at the back of the hall. The tempo of Group I is MM dotted-quarter note = 84 and the meter is 6/8. The tempo of Group II is MM quarter note = 84 and the meter is 2/4. There are no meter or tempo changes. The barlines of the groups are horizontally aligned and the ensembles are to be coordinated in the normal way. The piece consists of a sequence of 34 subsections that are divided into 7 sections.

Brant, Henry 1978. Antiphonal Responses for 3 Bassoons, Orchestra, 8 Widely Separated Instruments and Piano Obbligato. New York: Carl Fischer.

The eight widely separated instruments are: piccolo, clarinet, English horn, trumpet, trombone, {timpani, xylophone or chimes}, {glockenspiel or vibraphone} and harp. Throughout various sections, these instruments play repeated phrases at a tempo that is different from that of other instruments. The conductor cues their entry after which each continues to play without direction, uncoordinated with other music. For example, at mark 4 the conductor cues the entry of the harp, timpani and vibraphone in relation to the bassoons. Here the bassoons have a tempo of MM half note = 56. The three entering instruments (whose entry is staggered) have a tempo marking of "Slowly, in irregular, free rhythm". At another point, the five wind and brass solo instruments are cued by the conductor, one at a time, in relation to the bassoons. Here the bassoons play at MM half note = 76 and the five entering instruments have a tempo marking of "Moderately fast" or "Fast 16ths".

Brant, Henry 1978. Trinity of Spheres: 3 Orchestral Groups and 3 Conductors. New York: Carl Fischer.

This work calls for an orchestra that is to be divided into three mixed, spatially separated groups (A, B and C). Each group has its own conductor and plays in the same tempo and meter throughout as follows: Group A: meter 4/4 at MM quarter note = 84, Group B: alternating between meters 2/4 and 3/4 at MM quarter note = 100, and Group C: meter 3/4 at MM quarter note = 60. An exception to this occurs in Sections 4, 7, 14 and 16 where Group A may be slowed to MM quarter note = 76 if its normal tempo (MM quarter note = 84) is uncomfortable.

The work consists of a sequence of phrases each of which is played by a given group. The entry of each (child) phrase is cued by a conductor in response to a percussive signal that is heard in a (parent) phrase that is being played by some other group. For any sound cues that are found to be inaudible, the conductor of the parent phrase may give a visual cue to the conductor of a child phrase that is about to enter.

In the "Notes on Performance", Brant writes:

Each conductor should make every effort to retain his tempo and to keep it steady, but it is not intended that an exact correspondence of the barlines of the 3 groups should be maintained. The result will be that the groups, although playing in accurate ensemble separately, will not be in precise rhythmic synchronization all together.

Accordingly, each conductor, upon hearing the percussion signal and seeing that his musicians are ready to play, may begin immediately without trying to make his first attack coincide with the downbeat of another group as indicated in the score. Once begun, each conductor should lead each passage through steadily to its end without being influenced by what the other groups may be playing.

There are, however, a few places where the conductors should make every effort to make entrances coincide as far as possible with the downbeat of another group as indicated in the score. These are at 1a and 1b, and at 17a and 17b. See also the note in Group C at 16.

Brant, Henry 1981. Meteor Farm: A Spatial Concert of Ceremonies for Instrumental and Vocal Groups. New York: Carl Fischer.

The following forces are used: an orchestra, a "wall brass" (i.e. 3 trumpets and 3 trombones in single file against a side wall), two percussion ensembles, a jazz orchestra, two soprano soloists, a chorus with saxophones, a chorus with piccolos, a Javanese gamelan orchestra, a West African drumming ensemble, and a South Indian trio. The principal conductor leads the orchestra, wall brass, the two percussion ensembles and sometimes the saxophones and piccolos that are associated with the choral groups. Each choral group has its own conductor. The jazz orchestra is lead by its drummer.

The principal conductor cues the entry of the solo sopranos, the South Indian trio, the gamelan orchestra, the African drummers, and the jazz drummer. In Appendix II (p.53) titled "Notes on Cues and Signals", 14 cue gestures are suggested, each of which may be used to cue a particular ensemble or individual. These gestures have grown out of Brant's experience with spatial music and background music for live radio. It is suggested that a cue gesture be issued with the left hand to gain the attention of the entering player(s) and that this be followed by an up and down-beat with the (right hand) baton alone. For example, for "cueing a large group out in the hall, at the conductor's right", the following gesture is suggested: "left arm curved over the head, with index finger pointing to the right".

The piece consists of a sequence of 17 sections. Different forms and combinations of ensembles are used for each section. Various polytempo structures are used, some of which occur in earlier works by Brant. For example, in Section 5, loops of repeated phrases with the indication "fast 32nds, irregular 1/4 rests" are played by the percussion groups while the orchestra plays at MM quarter note = 72. Later, in Section 11, the strings begin looped phrases at MM quarter note = about 104. Then, on cue, the flutes and clarinets begin (together) two different staggered canons (like that of the violins in Brant's Curriculum) at MM quarter note = about 116. A chord which is repeated three times by the oboes, bassoons and horns is then used to cue, in sequence: a looped phrase in the solo sopranos (with no tempo marking), a looped phrase for Chorus II at MM quarter note = 80, and a looped phrase in Chorus I at MM quarter note = 56. All parts (except the cue chords) continue to play through to the end of the section and then stop after the West African drumming ensemble enters to start Section 12.

Throughout the full score, musical material is abbreviated. Some phrases are abbreviated symbolically as "X", "Y" or "Z" (e.g. see: p.33) or with numbers (e.g. see numbered songs: p.88). The complete material for such phrases is contained in Appendix I.

In all metronome markings, Brant uses traditional numerical values consistently. For example, in Section 15 (p.82) the jazz orchestra is marked with MM half note = 104 [or] MM quarter note = 200 (rather than MM half note = 104 [or] MM quarter note = 208, or MM half note = 100 [or] MM quarter note = 200).

In Section 16, the percussionists are instructed to look up upon encountering a fermata in their part to see if the conductor is issuing a (sustained) stop cue. Here, the text "look up" is written above each fermata (see: Appendix I: pp.39-42).

Brant, Henry 1982. Inside Track: Concerto for Solo Piano with Three Separated Instrumental Groups and Projected Images. New York: Carl Fischer.

There are three separate scores: a piano part score; a score for a woodwind group, string group and lights (for projected slides of photographs of paintings by Leslie Parke); and a score for band (trumpet, trombone, horn, alto/baritone saxophone, drummer and sopranino voice). The woodwind and string groups are lead by a principal conductor and the band is lead by an assistant conductor who may be the vocalist.

Like Brant's Meteor Farm, the piece consists of a sequence of 17 sections. The band plays alone in the odd-numbered sections through section 15. The piano, woodwinds and strings play the even-numbered sections as well as the last section (number 17).

The band and piano scores contains no cue information from other parts. A one or two-measure ossia of the beginning of the piano part is given at the beginning of each even section in the woodwind/string score, and the entire piano part for section 17 is notated in this score with a meter and tempo that matches that of the winds and strings. There are no meter or tempo changes within a section in any part, except in the piano part where changes of meter (but not tempo) occur often within a section.

In the even sections, the winds and strings play synchronously through material that is repeated a number of times (though either the winds or strings are omitted from some repeats). Against this background, the piano plays in an independent meter and tempo. The tempo combinations for the 17 sections are as follows (p = piano, w/s = winds/strings):

1. March I: MM quarter note = 132 (band)
2. MM half note = 60 (p), MM quarter note = 60 (w/s)
3. Boggie: MM quarter note = 116 (band)
4. MM quarter note = 184 (p), MM quarter note = 96 (w/s)
5. Waltz: MM quarter note = 126 (band)
6. MM eighth note = 112 (p), MM quarter note = 66 (w/s)
7. Rumba: MM quarter note = 144 (band)
8. MM quarter note = 56 (p), MM quarter note = 88 (w/s)
9. March II: MM quarter note = 132 (band)
10. MM quarter note = 76 (p), MM quarter note = 69 (w/s)
11. Galop: MM quarter note = 120 (band)
12. MM quarter note = 192 (p), MM half note = 76 (w/s)
13. Polka: MM quarter note = 116 (band)
14. MM quarter note = 176 (p), MM quarter note = 84 (w/s)
15. Jump: MM quarter note = 192 (band)
16. MM quarter note = 132 (p), MM half note = 76 (w/s)
17. A single marking of MM quarter note = 126 is given and this applies to (p) and (w/s).

Brant, Henry 1983. Desert Forests: Spatial Panoramas for Separated Orchestral Groups. New York: Carl Fischer.

The string section is lead by the principal conductor who beats time for this group. A group of high woodwinds (3 piccolos, 3 oboes, and 3 clarinets) which are to be positioned in the topmost balcony at the back of the hall, are coordinated by an assistant conductor. Generally, the other players do not follow a conductor's beat. Instead, their entry is cued by the principal conductor or by the sound of some musical event (without a visual cue), and thereafter they play in an uncoordinated manner. The time at which conductor cues are to be given is not indicated in the score by vertical lines that relate one part to another, though for some parts, textual notes instruct the conductor to be guided by the relative horizontal position of an entering part and other parts that are playing, in which case, such entries are to be made "approximately at the point indicated". In many cases, the members of a group that has been cued are to enter in such a way that their first notes are not necessarily coordinated.

Simultaneous different tempi occur throughout the piece. For example, the first section contains the following sequence of events:


• On a cue from the principal conductor, the timpani begins a 10-measure phrase at MM quarter note = 76-80 (meter 3/4) and then continues independently, without being lead by the conductor. The phrase contains one fermata.
• At some time during the first few measures of the timpani part, the low woodwind group (bass clarinet, 2 bassoons, and contrabassoon) and tuba are cued by the principal conductor to begin playing looped phrases at MM quarter note = 92 (meter 4/4). This group plays their first notes together, and then they play independently of each other thereafter. Each instrument has a phrase of a different length (bass clarinet and contrabassoon: 17 measures; bassoons: 18 measures; tuba: 15 measures) and each phrase has two to four fermatas which serve as stop marks.
• Near the end of the timpani phrase, the conductor cues the tom-tom and bass drum to begin playing a 10-measure phrase at MM quarter note = 76-80 (meter 3/4). The two instruments are to be coordinated among themselves but are not lead by the conductor. The phrase contains one fermata.
• During the first few measures of the tom-tom and bass drum phrase, the conductor cues the five horns to begin playing material that is structurally similar to that which the low woodwinds and tuba have been playing, and like this other group, the horns play at MM quarter note = 92 (meter 4/4).
• Before the end of the tom-tom/bass drum phrase, the conductor cues the entry of the strings which play a 22-measure phrase at MM half note = 60 ("or a little slower - but not faster") (meter 2/2). The conductor beats time for the strings. The string phrase has no fermatas.
• At the 7th measure of the string phrase, the conductor cues the timpani to begin a 4-measure phrase at MM half note = 76-80 (meter 3/4).
• At the 11th measure of the string phrase, the conductor cues the tom-tom and bass drum to begin a 4-measure phrase at MM half note = 76-80 (meter 3/4).
• At the 17th measure of the string phrase, the conductor cues the timpani to begin a 4-measure phrase at MM half note = 76-80 (meter 3/4).
• The first section of the piece ends with the strings holding a fermata at their 22nd measure, and the low winds, tuba, and horns all continuing their looped phrases. During this time, the conductor cues the entry of the "Bell Trio" (i.e. glockenspiel, vibraphone and chimes) to begin playing the second section of the piece at MM quarter note = 76. Upon hearing the bell trio, each of the instruments that is playing a looped phrase stops after playing through to the next fermata in their own part.


The prime numbers 7 and 17 appear often in the works of Brant. For example, many of his works consist of a sequence of 7 or 17 sections. This piece has 7 sections, and in the first section, cued phrases are to begin at the 7th and 17th measures of the string phrase. Also, sometimes Brant uses the double of a prime number. For example, some works have 34=2*17 sections. The string phrase in the first section of this piece is 22=2*11 measures long.

In the section titled "Co-ordination and Ensemble" of the "Performance Notes", Brant writes the following regarding the term "unco-ordinated":

The term "unco-ordinated" is not intended to suggest rhythmically casual or imprecise playing. Accurate rhythm is desired with careful attention to all indications of tempo, articulation, and dynamics. However, in playing an "unco-ordinated" passage, the performer is asked to maintain an individual correctness in execution, without attempting to "match" or "fit in", and without following a beat.

Brant, Henry 1984. Orbits: Spatial Symphonic Ritual for 80 Trombones, Organ and Sopranino Voice. New York: Carl Fischer.

The 80 trombones are partitioned into eight spatially separated groups (A through H) of ten each. Some uncoordinated textures occur in this work. For example, in Section 1c, which is a 1-measure section for Groups A through D, the instruction "Unco-ordinated (after first note)" is given. Here, ten phrases are played simultaneously with four trombones assigned to each phrase. The phrases contain a mix of straight eighth notes as well as eighth-note tuplets with the ratios 5:4, 3:2 and 7:4. This is followed by a similar uncoordinated 1-measure Section 1d for Groups E through H. Later, in the 1-measure Section 7a, 20 different repeated phrases are played simultaneously by all 80 trombones, with four trombones per phrase, and all play in an uncoordinated manner (after the first note) at MM quarter note = 80. A similar texture is used in Section 9a at a tempo of MM quarter note = 176. Outside of these uncoordinated measures, all parts share the same tempo marking and play in a coordinated manner throughout.

Carter, Elliott 1951. String Quartet No. 1. New York: Associated Music Publishers.

There are many measures in this piece for which the duration of a given type of note is not the same in all parts. Some instances where this occurs in the first movement are as follows: In measure 30, the tempo of the viola is MM dotted half note = 60 while the other instruments play at MM quarter note = 120; In measure 37, the cello plays at MM half note = 90 while the other instruments play at MM 5 eighth notes = 90; In measure 45, violin I plays at MM quarter note = 150 while the others play at MM 5 eighth notes = 60; In measure 46 violin I plays at MM 5 eighth notes = 120 while the others play at MM half note = 60. Similar instances also occur in the second movement. For example, in measures 85 through 125 of the second movement, the violins play at MM eighth note = 108 while the viola and cello play at MM eighth note = 72. In all the measures cited here, all parts share a common right barline but not a common meter. For example, in measure 37 of the first movement, the meter of the cello is 3/2 while the meter of the other instruments is 15/8. In all these cases, a tempo marking is given to explicitly indicate the difference in tempo that results from such a polymetric configuration.

Carter, Elliott 1959. String Quartet No. 2. New York: Associated Music Publishers.

In the second movement, the part of the second violin has been approximated and written in the meter and tempo of all other parts (i.e. 5/4 at the tempo MM quarter note = 175). A rhythmic cue line and explanatory footnote are given to indicate that this part is to be played in the meter 4/4 at the tempo MM quarter note = 140.

Carter, Elliott 1971. String Quartet No. 3. New York: Associated Music Publishers.

The instruments are partitioned into two groups of two: Duo I (violin I and cello) and Duo II (violin II and viola). Generally, the piece is notated in such a way that the duration of a given type of note is the same in both parts at all times. However, an exception occurs in measures 26 through 28 where Duo I plays a 4/4 measure and 2/4 measure at MM half note = 63 followed by a 3/4 measure at MM quarter note = 108, and Duo II plays a 2/4 measure at MM quarter note = 189 followed by a 3/4 measure and 4/4 measure at MM quarter note = 108. The left barline of measure 26 and the right barline of measure 28 coincide. This is not an approximation (because 6/126 = 2/189 + 4/108).

Colgrass, Michael 1976. Letter from Mozart for Orchestra. Carl Fischer.

The piece calls for two conductors and for most of the time these conductors establish two simultaneous independent tempi. The orchestra is not divided into two fixed groups. Instead, any given instrument might be conducted by a different conductor at different points in the piece. In the "Conductors" section of the "Program Notes", Colgrass writes:

This work requires 2 conductors because of multi-tempo musics. The orchestral musicians have either a I or a II shown in their parts next to each entrance so that they may simply look to the appropriate conductor for each cue. Although the music as written should be played as accurately as possible, I do not expect the 2 conductors to be necessarily in exact coincidence, bar for bar, at all times. Music-making is more important to me here than computer precision. There will usually be a fermata of other obvious "catch-up" point for the conductors to become re-oriented.

Constant, Marius 1968. Winds. Paris: Editions Salabert.

On pages 18 and 19 of the 29-page score, the following tempi occur simultaneously: MM quarter note = 69 in 2/4 (horns), MM quarter note = 126 in 3/4 (trumpets), MM dotted-quarter note = 69 in 6/8 (trombones), and MM quarter note = 63 in 4/4 (tuba). A rhythmic schematic of this is given in the book Modern Rhythmic Notation by Gardner Read (p. 172).

Crumb, George 1977. Star-Child: A Parable for Soprano, Antiphonal Children's Voices, Male Speaking Choir and Bell Ringers, and Large Orchestra. New York: C. F. Peters.

This work calls for four conductors: two primary and two secondary. The performers are spatially separated. Some groups are positioned in the audience. In the "Program Note", Crumb writes:

Star-Child is continuous, despite sectional divisions. The germinal idea, "Music of the Spheres" (strings, pianissimo), moves throughout the work in a circular music over which the human drama is enacted. This idea consists of a continuum of chords built upon the interval of a perfect fifth. Over these slow-moving strains of "suspended" music I have superimposed (in the manner of Charles Ives!) a sequence of boldly contrasting musics. The necessity for four conductors arose from the fact that each music has its own tempo and metrics (metrics tend to be odd-numbered: the opening string music is in 11/4 time, the entire Apocalyptica in 5/16, and there are other sections based on sevens and threes). The four conductors do not synchronize and therefore all sense of vertical alignment is erased. I had even imagined that the "visual counterpoint" of the four-fold conducting would produce a choreography of its own!

The "circular" music to which Crumb refers is music that is notated on two curved staves that together form a circle. Against this background of looped phrases (that are structurally similar to the looped phrases found in the works of Henry Brant) music of a different tempo is played. The looped phrases are conducted by one of the conductors II, III or IV, or in one case, performed by one instrument (i.e. the organ) without a conductor. Such phrases are repeated until a stop cue is issued. At a stop cue, players are instructed to complete the circle and then, depending upon the particular phrase, either stop playing, or proceed to a repeat ending.

The visual counterpoint of the conductors that Crumb mentions in this note is presented in a pure form in Mark Applebaum's Tlön for 3 Conductors and No Players.

Ferneyhough, Brian 1969. Missa Brevis. London: Edition Peters.

This work is for three spatially separated choirs of four soloists each (SATB). It consists of five sections titled "Kyrie", "Gloria", "Sanctus", "Benedictus" and "Agnus Dei". In the first four sections, all parts share the same meter and tempo throughout. The 6-page "Agnus Dei" consists of three subsections. In the middle subsection, all parts share a common meter and tempo. There are simultaneous different tempi in the first and last subsections of the "Agnus Dei".

In the first subsection of the "Agnus Dei", Choir I repeats a one-measure 10/8 phrase, four times. The first pass is taken at MM eighth note = 56. During the course of the second and third passes, the tempo is increased gradually from MM eighth note = 56 to MM eighth note = 72. The fourth pass is taken at MM eighth note = 72. At the fifth eighth note of the Choir I part, Choir III begins a one-measure 12/8 phrase that is repeated four times. The first pass begins at MM eighth note = 70 and gradually decelerates until reaching the tempo MM eighth note = 50 at the beginning of the fourth pass. The fourth pass is taken at MM eighth note = 50. The following instruction is given: "After conductor starts groups I + III in their respective tempi, their future internal co-ordination should be assigned to a chosen member of each group". Twelve seconds after Choir III begins (or roughly between the 5th and 6th eighth note of Choir III), Choir II sings a 3-measure phrase at MM eighth note = 60, and then, after a fermata of six to eight seconds, it sings a 3-measure phrase that begins at MM sixteenth note = 100, accelerates to MM sixteenth note = 112, and then ends with a rallentando to an unspecified tempo. The subsection ends with all singers of Groups I and II holding their last note. Group I is instructed to hold its last note through that of Group III.

The third and final subsection of the "Agnus Dei" is sung by the sopranos and altos. This subsection begins with the three altos holding their last note from the previous subsection. Together, each soprano begins a one-measure 10/8 phrase that is repeated 3 times. The first pass is at MM eighth note = 66. The second and third passes are marked "rall.ad lib." for Groups I and III and "ad lib." for Group II. After cueing this entry, the conductor cues the simultaneous entry of the three altos. Each has a different phrase. Alto 1 sings his phrase "3 times: Molto rall. 2+3rd times", Alto 2 sings his phrase "3 times: Poco rall. 2+3rd times", and Alto 3 sings his phrase "Twice rall. 2nd time". When all sopranos have completed their repetitions, they give a signal and then sing a final phrase. The start times of the notes of the first repetition of the alto phrases and the final soprano phrase occur in a specified order across all parts using a form of proportional notation.

In the "Performing Notes", Ferneyhough writes:

In the Agnus Dei the three vocal groups work largely independently of each other. In the passages involving multiple superimposed and conflicting tempi, each group should assign the task of conductor to one of its own members. No extra conductor is required for group two.

In the last [sub]section each individual singer pursues his or her own course to the end, after repeating their material for the given number of times. Any extra instructions are provided in context.

A graphic image of the first page of the score for the "Agnus Dei" section is available at http://www.edition-peters.de/gifs/ferney_back.gif.

Ferneyhough, Brian 1969-71. Firecycle Beta: Symphonic Torso for Two Pianos and Orchestra with Five Conductors. Ricordi.

This piece calls for a main conductor and four sub-conductors. The players are partitioned into two groups (Orchestra 1 and 2) and each group is partitioned into two sub-groups: a string ensemble and a mixed ensemble of soloists. The main conductor (MC) is flanked by sub-conductors 1 (SC1) and 2 (SC2) who are positioned in front of orchestra 1 and 2, respectively. Sub-conductors 3 (SC3) and 4 (SC4) are positioned within orchestra 1 and 2, respectively. Multiple conductors are used in the following sections: Processional(MC, SC1, SC2), Sequence a (SC1, SC2, SC3, SC4), Sequence b (MC, SC3), Alleluia IIIb (all 5 conductors) and Amen a (MC, SC3, SC4). An introductory section titled "Special Instructions for the Conductors" provides details regarding cueing and coordination. For example, the subsection that pertains to sub-conductors 3 and 4 begins as follows:

These [SC3 and SC4] remain, in large part, dependant upon [SC1] and [SC2] for their points of entry. The line of sight between each pair should, therefore, be very clear [SC3 coordinates with SC1, SC4 with SC2].

Sequence a, b. Although fully conventionally notated, the materials (lower strings) are nevertheless dependant upon the speed of movement of Sub-conductors 1 and 2 for their frequency of entry. Either these latter give signals for each new entry, or else Sub-conductors 3 and 4 take their cues from the score itself at appropriate moments. As soon as entry has been made, each conductor continues in his own, independently prescribed tempo.

The section concludes with the following note:

In sections where several conductors are acting at one and the same time, the vertical layout of the material does not necessarily reflect the coincidence of sound-actions. Within the scope of the general guidelines given above, the conductors move at their own individual speeds through the material assigned to them.

Ghent, Emmanuel 1964. Triality: for Violin, Trumpet and Bassoon: in Conjunction with a Polynome: 2 Studies.

The performance notes for this piece include the following instructions: "It is intended that this music be played with the assistance of a polynome ... an instrument which can simultaneously beat up to 7 different tempi". The score is available at the University of Chicago Library (call number: fM362.G411T81) and the SUNY at Buffalo Music Library (call numbers: Triality: M 5546 G341-3 t, Triality II: M 5546 G341-3 t(2)).

Ghent, Emmanuel 1965. Dithyrambos for Brass Quintet. New York: Oxford University Press.

This piece uses the audio cueing system and notation conventions described in the article Programmed Signals to Performers: A New Compositional Resource by the same author. It is also available in its original form under the title Quintet for Brass Instruments at the University of Chicago Library (call number: fM557.4.G41Q72) and Columbia University Library (call number: 51 G341 Qu). A version titled Double Brass Quintet which is an arrangement of Dithyrambos for two brass quintets is available at the The New York Public Library for the Performing Arts (call number: JNJ 80-1). The GROOVE System of Bell Telephone Laboratories was used to generate a four-channel realization of the piece and a two-channel mixdown of this is available under the title Five Brass Voices on the Society of Composers, Inc. CD Series album titled Potpourri (Capstone Records CPS-8609 (SCI CD # 2)), and as the RealAudio file http://composerver.sss.arts.ohio-state.edu/Ghent.ram at Composerver. A biography of Emmanuel Ghent may be found at http://www.weststreetrecords.com/eg_music_bio_short.html.

Ghent, Emmanuel 1966. Hex: An Ellipsis for Trumpet, Instruments and Tape.

Like Ghent's earlier work Dithyrambos, this piece employs pre-recorded audio beat cues which are distributed to each player during a performance. The ensemble is partitioned into six pairs of instruments. The instruments in a pair share a common tempo throughout and the tempo of each pair is allowed to change independently of that of any other pair. The audio cue signals enable the pairs to remain synchronized with each other and with three channels of pre-recorded music. The six initial metronome markings are in the ratio 5:3 (i.e. MM thirty-second note = 555.6, MM sixteenth note = 333.3, MM sixteenth note = 200, MM eighth note = 120, MM quarter note = 72, and MM half note = 43.2). The score is available at The New York Public Library for the Performing Arts (call number: JNG 79-227).

Ghent, Emmanuel 1969. Helices for Violin, Piano and Tape.

In some sections of this piece, two different and simultaneous metronome markings are indicated, and at some points, accelerandos and decelerandos are specified for one instrument while the other instrument proceeds at a constant rate. Most of the notation conventions described in Ghent's article Programmed Signals to Performers: A New Compositional Resource are not used here. Instead, synchronization is achieved by other means. For example, where the piano is to play at a tempo of MM quarter note = about 64 while the violin plays at MM quarter note = 116, the following instruction is given: "Piano: Play even sixteenths at MM quarter note = about 64 attacca into the next measure. If too soon, cut a few sixteenths; if late, repeat as required.". Later, when the violin is to play at a tempo of MM quarter note = 96 while the piano plays septuplets at MM quarter note = 109.7, the violin is given the instruction: "The tempo and note placement are intended as guides and need not be slavishly followed.". Also, at one point the page turner is instructed to use a stop-watch during a 16-second duration of tape sound to cue the entry of the piano. And at one point, a 25-second tape solo is notated against a timeline that runs from 0 through 25. The score was published in Volume 8 of The SCI Journal of Music Scores and is available at The New York Public Library for the Performing Arts (call number: JMM 76-29 v. 8).

Greschak, John 2003. Platonic Dice: Dodecahedron: for 12 Woodwinds. Available: http://www.greschak.com/pdicedod.pdf.

The performance notes for this work include the following instructions: "This piece is an example of polytempo music (i.e. music in which two or more tempi occur simultaneously). As such, the techniques discussed in the articles Facilitating the Performance of Polytempo Music: An Overview and Technology Assisted Conducting of Polytempo Music: Some Possibilities might be of use for creating a realization of this piece. For more information, see http://www.greschak.com/polytempo.".

The piece consists of a sequence of 659 overlapping phrases, each with an independent tempo selected from the metronome scale. The following tempi are used: MM 4-note = 52, 58, 66, 76, 88, 100, 112, 126 and 144. Note: In this score, a given note that starts at a given time-scale point i is said to be an "n-note" (e.g. 1-note, 2-note, 3-note, etc.) if the note ends at the time-scale point i+n.

No particular tempo predominates or serves as a reference. Each phrase is notated on a separate page using a timetable. A zero-based time scale of numbered points is given along the left and right edges of each timetable. The time at which a given phrase is to occur, relative to other phrases, is specified by an equation that indicates coincident time-scale points. For example, for phrase 16, "1 = ¶15.3" indicates that time-scale point 1 of phrase 16 occurs at the same time as time-scale point 3 of phrase 15. (Note: In this score, the symbol ¶ means "phrase".)

For most phrases the tempo is constant. For some phrases, the tempo changes continuously as an accelerando or decelerando. Such transitions are either geometric or sinusoidal. For example, the tempo of phrase 33 decreases geometrically from MM 4-note = 144 to MM 4-note = 76.

Immediately after phrase 1 ends, phrases 2 and 3 begin simultaneously at a tempo of MM 4-note = 66 and MM 4-note = 52, respectively. While these phrases are being played, phrase 5 begins at a tempo of MM 4-note = 112. Immediately after phrase 8 ends, phrases 9 and 10 begin simultaneously at a tempo of MM 4-note = 58 and MM 4-note = 112, respectively. The tempo of phrase 9 increases geometrically to MM 4-note = 126, while phrase 10 continues at a constant tempo. Simultaneous different tempo markings of this type occur throughout the piece.

Henze, Hans Werner 1971. Compases para Preguntas Ensimismadas for Solo Viola and Orchestra. Mainz; New York: Schott.

On pages 71 through 80 of this work (marks 46 through 50), the solo viola and other instruments are written in a meter of 3/4 at a tempo of MM quarter note = 66. Simultaneously, the harp and harpsichord are written in 4/4 at MM quarter note = 80. In this section of the score, the harp and harpsichord parts have been reduced in size (both vertically and horizontally) by an appropriate amount in order to scale the width of measures. Page breaks have been placed at viola barlines and at various beat fractions in the measures for the harp and harpsichord.

Hiller, Lejaren 1962. String Quartet No. 5 (in Quarter-tones). Bryn Mawr: T. Presser.

In Variation Twelve (p.47), the following tempi occur simultaneously: MM quarter note = 120 (Allegro), MM eighth note = 110 (Adagio), MM 5 eighth notes = 50 (Scherzando), and MM quarter note = 90 (Risoluto). A rhythmic schematic of this is given in the book Modern Rhythmic Notation by Gardner Read (p. 169).

Ives, Charles E. 1906. Central Park in the Dark. Hillsdale: Mobart.

The strings which represent night sounds and darkness play Molto Adagio throughout while the rest of the orchestra (which represents a variety of other sounds that one might hear in the late 1800's in Central Park) accelerates gradually from Molto Adagio to Allegro molto during measures 64 through 118. In a note in the score, Ives writes:

From measure 64 on, until the rest of the orchestra has played measure 118, the relation of the string orchestra's measures to those of the other instruments need not and cannot be written down exactly, as the gradual accelerando of all but the strings cannot be played in precisely the same tempi each time.

Ives, Charles E. 1908. The Unanswered Question for Chamber Orchestra or Chamber Ensemble. New York: Southern Music Publishing.

This is a programmatic piece in which a theme that represents "The Perennial Question of Existence" is posed repeatedly by the trumpet and the four flutes respond with material that represents the human hunt for "The Invisible Answer". Throughout, a muted string ensemble represents "The Silences of the Druids - Who Know, See and Hear Nothing". The trumpet and strings are written in 4/4 at MM quarter note = 50 throughout. The six flute answers, each of which is two to six bars long, are also written in 4/4. Each successive answer is faster and the last accelerates, according to the instructions: Adagio, Andante, Allegretto, Allegro, Allegro molto, and Allegro - Accel. to Presto. All three barlines of the two-measure first answer coincide with (and are connected through) the barlines of the trumpet. In the same way, the leading barlines of the first two measures of the four-measure second answer coincide with those of the trumpet. No barlines of the fourth, fifth or sixth answers coincide with the trumpet. The following temporal instructions are provided by Ives in the Foreward:

This part [the flute answers] need not be played in the exact time position indicated. It is played in somewhat of an impromptu way; if there is no conductor, one of the flute players may direct their playing.

The flutes will end their part approximately near the position indicated in the string score; but in any case, "The Last Question" should not be played by the trumpet until "The Silences" of the strings in the distance have been heard for a measure or two. The strings will continue their last chord for two measures or so after the trumpet stops. If the strings shall have reached their last chord before the trumpet plays "The Last Question", they will hold it through and continue after, as suggested above.

"The Answers" may be played somewhat sooner after each "Question" than indicated in the score, but "The Question" should be played no sooner for that reason.

Ives, Charles E. 1898-1912. Three Harvest Home Chorales.

This work consists of three chorales. On pages 3 through 12 of the second chorale, "Lord of the Harvest", the bass (and at times some other instruments and voices) play at a speed that is 50% faster than that of the other instruments and voices. All parts are written in 4/4 but those which are faster are written as three consecutive half-note triplets that span two 4/4 measures. Only one tempo term is specified at any given time, but the duration of a given type of note is not the same in all parts at all points in time. Ives uses arrows in the score to indicate the times at which the half-note triplets are to begin in the faster part, relative to half-note triplets in the slower part.

Ives, Charles E. 1909-16. Symphony No. 4.

The first page of the second movement contains a number of simultaneous different tempi. (Note: The description given here refers to the version of the score that was published in 1929 in New Music Edition 2(2). This version differs from that published in 1965 by Associated Music Publishers. In the later version, meters and barlines have been changed and two conductors are required.) On the first page, one tempo term Allegretto is used along with one metronome marking MM dotted-quarter note (of 6/8) = about 50. Other tempi are indicated by the use of polymeters. For example, in the time of five 6/8 measures, the primo orchestra piano has five 2/4 measures, the bassoons have two 7/4 measures, and the strings have three 4/4 measures. At the same time, an accelerando in the solo piano is indicated by a sequence of seven 6/8 measures of decreasing width. In the "Conductor's Note", Ives writes:

The recitative of the basses controls this page...The solo piano from the third measure to end of page may not be played in the exact time relation indicated by the measure divisions, but there is rather a brief accelerando and crescendo and an easing down towards end of bass recitative...

Later, in Section 8 of the second movement, some instruments continue at Adagio while others begin playing Allegro and gradually accelerate through the entire section. For this section, Ives writes:

The instruments are divided here into two separate orchestras; the lower continuing the proceeding adagio, while the upper, including woodwind, brass, tympani and both pianos, breaks suddenly in [at allegro]...Both groups may keep in the time relation indicated on this page, but at about the beginning of the next page the upper orchestra begins to play gradually faster and faster until the "collapse" indicated on page 29, but which will occur sooner - perhaps towards the end of page 28...During this passage it may be advisable to have one of the players in the upper orchestra act as a separate conductor.

Also, in the third and fourth measures of Section 8, the piccolo part has a sequence of three quarter-note triplets that span two 4/4 measures. As in Three Harvest Home Chorales, the times at which these triplets begin is indicated by arrows, which in this case, are directed from ordinary eighth-note triplets that begin on a beat in 4/4 and are positioned above the piccolo staff.

For Section 20, Ives writes:

All the percussion with the saxophones or bassoons play in Sec. 20 as a single and independent group. There may be a slight ritardando as well as a decrescendo in this passage which may extend into Sec. 22. If so, one of their number acting as leader for these few measures will simplify the playing.

Also, in the "Conductor's Note", Ives suggests that the various rhythms "would better stand out in their perspective" if the various players were spatially separated.

In the fourth movement, there are polymetric textures for which the parts do not share a common barline. The movement begins with a group of percussionists playing "very slowly" in 4/4. Against this, the orchestra plays in 6/2. Each 6/2 measure of the orchestra spans two 4/4 measures of the percussion ensemble. Here, a whole note in the percussion ensemble has the same duration as a dotted whole note in the orchestra, and thus, these two ensembles play at a different tempo (though independent tempo markings are not explicitly given in Ives' original manuscript).

Ives, Charles E. 1903-14/1929. Three Places in New England (Orchestral Set No. 1). Bryn Mawr: T. Presser.

The "Preface" of the full orchestral version edited by James B. Sinclair (1976) contains the following notes on page vii which pertain to the second section titled Putnam's Camp, Redding, Connecticut (Here, Sinclair has added information (in brackets []) regarding the conductor Nicholas Slonimsky who premiered the work in 1931.):

[This "Conductor's Note" was written in the early 1930's by Ives with some collaboration by Slonimsky...]

[The following note is Slonimsky's except perhaps the last paragraph. Slonimsky was particularly intrigued by the juxtaposition of tempi here. He argued in numerous letters to Ives that the passage should be engraved as an actual bi-tempo event, requiring the conductor to beat conflicting patterns. Ives' refusals were politely firm until he allowed Slonimsky his wish on the condition that an "ossia" line appear in the score and parts showing how the passage can be performed in relation to a single beat pattern. This edition restores Ives' original notation.]

At the measure after H, Page 35 [ii., m.68], the orchestra is divided into two parts going on in two different rhythms until the last two measures, Page 38 [ii., m.82-83]: the alla breve of the piano, drums, bassoon, trumpet and second viola (or horn) is going 25% faster than the rest of the orchestra, the result being that four beats of the former equals three beats of the latter.

The conductor may beat the four-four time with his right hand, and the alla breve with his left hand. The down-beat of the first bar of the alla breve (left hand) falls in with the third beat of the 4/4 (right hand); the down-beat of the second bar of the alla breve (left hand) falls in with the second beat of the right hand; the down-beat of the third bar of the alla breve (left hand) falls in with the first beat of the right hand, - in other words, here both down-beats come together. In the following bar, the downbeat of the left hand falls with the fourth beat of the right hand, then again with the third beat, etc. The piano, drums, bassoon, trumpet and horn follow the conductor's left hand, the rest - his right hand.

The up-beat of the left hand falls between the fourth beat of the corresponding 4/4 bar and the first beat of the following bar, and so forth, half of the alla breve bar being equal to 3/4 divided by two, that is, 3/8. It is not difficult if the conductor will at first place a mental accent on the third beat of 4/4, then on the second beat, and so forth, as if following a waltz time while going through the motions of conducting a quickstep.

Should the conductor prefer, he may beat only the main line without bothering about the second rhythm part, a provision having been made in the orchestral parts in case this is done.

Ives, Charles E. 1911-51. Universe Symphony.

Sketches of this unfinished work which are in the Collection of Ives Manuscripts at the Irving S. Gilmore Music Library of Yale University Library, indicate that a large number of simultaneous different meters were to be set against a common barline. It is not known whether Ives would have notated a finished version of this with various tuplets against a common meter and common tempo (as in Elliott Carter's String Quartet No. 5), or with simultaneous different meters each with different tempo markings (as in Elliott Carter's String Quartet No. 1). Realizations of this work have been created by Larry Austin and Johnny Reinhard. (Note: The Reinhard realization which is titled Charles Ives Universe Symphony Realized Exclusively on the Extant Sketches by Johnny Reinhard (1994) has been excluded from this bibliography because it is notated using a common tempo throughout.) An index to Ives manuscripts that was compiled by John Kirkpatrick is available at Yale University Library (call number: Ref. ML134 I95 K51+).

Ligeti, György 1962. Poème Symphonique for 100 Metronomes. Mainz: Schott.

The score for this work consists of four pages of German text by Ligeti along with translations in English (by Sid McLauchlan) and French (by Louise Duchesneau). There are "Instructions for Performance" and "Notes". The piece calls for 100 mechanical metronomes and two or more "interpreters". The metronomes are divided evenly among the interpreters. Without the audience present, the interpreters initialize the metronomes for performance. They set the metronomes to different speeds between MM 50 and MM 144. They wind each with four half-turns (from an unwound state) and place them on stage on top of chairs, tables, pianos, or other "suitable resonators". Then, they set all metronomes in motion "as simultaneously as possible", after which the interpreters leave the stage and the audience enters the hall. After 15 to 20 minutes, the piece ends when all metronomes have stopped ticking.

Since all metronomes are wound by roughly the same amount, the faster metronomes stop before slower ones. In his "Notes", Ligeti observes that the piece may be thought of as consisting of the following three phases: uniformity resulting from the "collective blur of the individual periods" which is heard as continuous sound, gradual structuring when complex rhythms become perceptible after some metronomes have stopped, and uniformity when the last metronome plays alone.

Ligeti, György 1969-70. Chamber Concerto for 13 Instrumentalists. Mainz: Schott.

In measures 14 through 31 of the first movement of this four-movement work, different tempo markings are indicated simultaneously. Here, a new tempo is introduced with each successive measure. The conductor cues each new tempo while some instrument is instructed to remain at the previous tempo. The process continues until measure 21 at which time the following tempi are in play (the tempi are listed by the order in which they are introduced): MM quarter note = 92 (clarinet), 84 (bass clarinet), 76 (flute), 100 (piano), 60 (horn), 80 (organ), 66 (oboe d'amore) and 54 (trombone). A footnote (p.34) states that each player should do their best to estimate their proper tempo and "should not be synchronized by the conductor". Also, here it is indicated that "The vertical relationships of the score are of no relevance for these instruments (Fl., Clar., Clar. bass, Org., Pf.)...". A similar pattern occurs in measures 35 through 39 of the third movement where the tempi MM quarter note = 56, 60, 66, and 72 are used simultaneously. Later in the same movement, simultaneous tempi accumulate in the same way in measures 46 through 59. A footnote (p.80) for this passage states the following:

The players left by the conductor on their own keep as accurately as possible to their individual tempo (MM quarter note = 60 or MM quarter note = 80): however, since they cannot measure the tempo, but only estimate it, tempo fluctuations may occur. Synchronization of the (non-conducted) parts is not to be striven for; in fact, slight shifts in the metre are welcome.

Ligeti, György 1983. Magyar Etüdök (Hungarian Studies) for Mixed A Cappella Choir (16 part). Mainz: Schott.

This work is in three sections. Throughout, the choir is divided into two sub-choirs with the following composition: Section I: Choir I = Choir II = {S1,S2;A;T1,T2;B}; Section II: Choir I = Choir II = {S1,S2;A1,A2;T1,T2;B1,B2}; Section III: Choir I = {S1,S2;A1,A2;T;B} and Choir II = {S+A,T+B}.

Throughout Section I, the tempo of Choir I is MM dotted-half note = 64 while that of Choir II is MM half note = 64. The meter for Choir I is 6/4 while that of Choir II is 2/2, and all parts share a common barline. All measures are partitioned into 2 equal parts with a dashed barline. At the beginning, the following tempo marking is given for each choir: [dotted-half note] = [half note] = 64. Moderato meccanico.

In Section II, the two choirs share the same tempo marking throughout.

In Section III, five simultaneous tempi are used. The Bass of Choir I begins at MM quarter note = 90 (Tempo giusto). On the third beat of measure 5 of the Bass, the Alto of Choir I begins at MM quarter note = 160 (Allegro Vivace). At measure 11 of the Alto, the Soprano of Choir I begins at MM quarter note = 140 (Allegro con anima). At measure 17 of the Alto, the Tenor of Choir I begins at MM quarter note = 110 (Allegro moderato). At measure 61 of the Alto, the Soprano and Alto of Choir II begin a unison part at MM quarter note = 190 (Vivacissimo). The Tenor and Bass of Choir II sing the same line as the Soprano and Alto through to the end, but lag the Soprano and Alto by one quarter note. Arrows are used to indicate entry times. All parts are in 4/4 and each part has an independent sequence of measure numbers and every measure is numbered. Measures are frequently fractured across system breaks. In each system, a left barline is drawn through the beginning of all staves immediately after the opening clefs. It is solid through groups of staves that begin on a barline and otherwise, it is dashed. A similar matching barline is drawn at the end of the previous system. Since the five tempi have a common divisor of 10, a duration consisting of four measures of the Alto of Choir I (i.e. 16 beats) serves as a unit of organization to which the tempi of the other parts may be coordinated.

In the (German) performance notes for Section III, Ligeti indicates that each of the five groups (S (#1), A (#2), T (#3) and B (#4) of Choir I and Choir II (#5)) is to have an assistant conductor which can be one of the singers in the group if desired. The main conductor cues the entry of each of the five groups, after which each assistant conductor maintains the tempo for their group with the assistance of a metronome (a light metronome is suggested). The main conductor may serve as the conductor of Group 2 (Alto of Choir I) while cueing the other groups along the way. Alternatively, it is suggested that a 4-track tape player may be used to play pre-recorded pulses for the assistant conductors of Groups 1 through 4 while the main conductor leads Group 5 separately, or an 8-track tape player (or a digital recorder) may be used to mechanically coordinate all five tempi. In either case, the conductors would follow clicks heard in headphones or blinking lights triggered by the pre-recorded pulses. Here, Ligeti indicates a preference for lights.

Messiaen, Olivier 1987-91. Éclairs sur l'Au-Delà... (Illuminations of the Beyond) for Large Orchestra. Paris: Alphonse Leduc.

This work consists of eleven sections. A common meter and tempo is used throughout, except in the following sections (Note: The page numbers given here are for the 1998 edition of the score published by Alphonse Leduc, which is in two parts):

Section II. La constellation du Sagittaire. Part I: pp. 28-30 and 42-45.
Section VIII. Les étoiles et la Gloire: Part II. pp. 23-29, 37, 45-59, 62, 65.
Section IX. Plusieurs oiseaux des arbres de Vie. Part II: pp. 103-126.

Section II. On pages 28 through 30, the following tempo terms are used simultaneously for six flutes: "Presque vif" (Fl. 1), "Un peu vif" (Fl. 2), "Vif" (Fl. 3), "Modéré" (Fl. 4), "Un peu lent" (Fl. 5), "Bien modéré" (Fl. 6). Each flute is associated with a different bird: Orphean warbler (Fl. 1), spotted morning warbler (Fl. 2), chorister robin (Fl. 3), Natal robin (Fl. 4), Rupell's robin-chat (Fl. 5), and musician wren (Fl. 6). Throughout the passage, the tempo of each flute is constant, except the fifth flute, which plays through the following sequence of tempi: "Un peu lent", "Vif", "Un peu vif", "Un peu lent", "Un peu vif", "Un peu lent", "Un peu vif", "Vif", "Un peu vif", "Très modéré". The flutes enter in numerical order, staggered. Their entries are cued by the conductor. In each part, a bold arrow is directed up or down to the note or rest that is to be cued. A similar passage occurs on pages 42 through 45. Here again, the six flutes play at different tempi, and the tempo of the fifth flute changes throughout while that of the other flutes remains constant. In these sections, no parts are metered. Common music notation is spaced to indicate the time at which notes are to occur.

Section VIII. On pages 23 through 29, flute 1 plays an unmeasured solo passage at "(Vif) hors tempo - au signe du chef" while the rest of the orchestra plays measured material at "Bien modéré (MM eighth note = 100)". A similar structure occurs on pages 45 through 59. However, here flute 2 begins a short time after flute 1 at "(Presque vif) hors tempo - au signe du chef". In these passages, the conductor cues the entries of the flutes. Here, the flutes are associated with garden warblers. At other times in this section, there are passages in which one instrument plays in a different tempo while other instruments hold a note throughout (pp. 37, 62 and 65).

Section IX. This section (pp. 103-126) is polytempo throughout. It is scored for 18 woodwinds (3 piccolos, 6 flutes, a solo flute, 2 piccolo clarinets in Eb, 6 clarinets in Bb), a triangle, and a small cymbal. Each woodwind instrument is associated with one or more of 25 different species of bird. Each species is represented by some characteristic thematic material of a given tempo, which is indicated by a tempo term. The entry of each bird-song phrase is cued by the conductor. Different bird songs overlap in time. A wide variety and large number of different tempi occur simultaneously. For example, at the end of page 10, the following tempi are used simultaneously: "Vif" (Picc. 2, Fl. 3, Fl. 4); "Modéré, un peu vif" (Fl. 1, Fl. 2); "Modéré" (Fl. 5, Picc. Cl. 2, Cl. 2, Cl. 4); "Un peu vif" (Fl. 6, Cl. 5); "Un peu lent" (Solo Flute); "Presque vif" (Cl. 3). Although some simultaneous phrases have the same tempo, those which do are not synchronized. That is, their beats do not occur at the same time. Throughout this section, parts are unmeasured, and the time at which notes are to occur is indicated spatially.

Musgrave, Thea 1966. Chamber Concerto No. 2. London: Chester.

In various unmeasured sections of this piece, simultaneous different tempi are specified. For example, on page 51, the clarinet has a tempo marking of "sempre MM half note = 84 (faster than violin)" where the prevailing tempo marking for the violin is "Moderato MM half note = 60". On page 54, the clarinet and violin play at MM half note = 100 while the cello plays at "MM half note = about 76 (slower than Cl./Vl.)" and the piano is at "MM quarter note = 132 Tempo di presto misterioso". In this section, the piano part is measured with dashed barlines while the other parts are unmeasured.

Nancarrow, Conlon. Study No. 2 for Player Piano. Santa Fe: Soundings.

The following tempo markings are used (Note: Here, the tempi are listed in the order in which they first appear and meters are given in parentheses): MM quarter note = 69 (3/4), MM eighth note = 115 (5/8), MM quarter note = 86-1/4 (2/4), MM dotted-eighth note = 115 (6/16), MM five sixteenth notes = 69 (10/16), and MM dotted-quarter note = 57-1/2 (6/8). The tempo numbers are in the ratio 10:12:15:20. During one interval of time, five different tempo markings are in effect simultaneously. These accumulate in the following order: MM quarter note = 69, MM eighth note = 115, MM quarter note = 86-1/4, MM five sixteenth notes = 69, and MM dotted-quarter note = 57-1/2. Each tempo occurs in a constant meter throughout with the exception of a 5/16 measure where the tempo of one voice changes from MM dotted-eighth note = 115 to MM five sixteenth notes = 69. Common barlines occur frequently.

Nancarrow, Conlon. Study No. 6 for Player Piano. Santa Fe: Soundings.

A single tempo marking of "measure = 60" is given which indicates that there are to be 60 measures per minute. Each part is measured with a different meter that remains constant throughout, and all parts share a common barline throughout. Thus, the duration of a given type of note is different in each part. The simultaneous meters are 4/8, 5/8 and 3/4. The implied simultaneous tempi are: MM eighth note = 240 (= 4*60), MM eighth note = 300 (= 5*60), and MM quarter note = 180 (= 3*60).

Nancarrow, Conlon. Study No. 9 for Player Piano. Santa Fe: Soundings.

There are up to three simultaneous independent parts. The parts have independent metrical schemes and various combinations of two or three tempi are used simultaneously, in the following order:

{MM quarter note = 144, 192; MM dotted-quarter note = 160} 9:10:12
{MM dotted-quarter note = 96, 128} 3:4
{MM dotted-quarter note = 96; MM quarter note = 240} 2:5
{MM dotted-quarter note = 80, 96} 5:6
{MM dotted-quarter note = 80; MM quarter note = 192} 5:12
{MM dotted-quarter note = 80, 128} 5:8
{MM dotted-quarter note = 80; MM quarter note = 192} 5:12
{MM quarter note = 144, 192} 9:10
{MM quarter note = 120, 144} 5:6
{MM half note = 96; MM quarter note = 120} 4:5
{MM eighth note = 288, 384, 480} 3:4:5
{MM dotted-quarter note = 96, 128, 160} 3:4:5
{MM quarter note = 144, 192, 240} 3:4:5
{MM dotted-quarter note = 96, 128, 160} 3:4:5

To indicate that an unmeasured pause is to occur after a given phrase, an upward-sloping line is drawn through the middle three staff lines after the last barline of the phrase. This notational device is used in many of Nancarrow's scores.

Nancarrow, Conlon. Study No. 14 for Player Piano (Canon 4/5). Santa Fe: Soundings.

There are two simultaneous parts with independent tempi and meter. The tempo of one part is MM quarter note = 88 while that of the other part is MM quarter note = 110. The ratio between the two tempi is 4:5. The faster part begins after, and ends before, the slower part. Each part has two voices or staves. The staves of the faster part are positioned above those of the slower part. Very few barlines are common to both parts. System breaks are placed at a barline of the slower part more often than at a barline of the faster part.

Nancarrow, Conlon. Study No. 15 for Player Piano (Canon 3/4). Santa Fe: Soundings.

This is a two-voice canon. The voices begin and end together, and both sound throughout the piece. The material for each voice is played twice, once at one tempo and then again at a second tempo. For one voice, the first pass is at MM half note = 165 and the second pass is at MM half note = 220. For the other voice, the first pass is at MM half note = 220 and the second pass is at MM half note = 165. The ratio between the tempo numbers is 3:4. Changes in meter that occur in one voice do not coincide with those of the other voice. Common barlines are drawn wherever the barlines of the voices coincide exactly.

Nancarrow, Conlon. Study No. 16 for Player Piano. Santa Fe: Soundings.

This piece consists of a sequence of three sections. The tempo of the first section is MM half note = 84. For the second section, the tempo is MM half note = 140. In the third section, the first two sections are played simultaneously. The ratio between the two tempi is 3:5. In each section, there are four voices. Each voice has a constant meter that is different from that of the other voices and equal to one of the following: 4/2, 5/2, 6/2 and 7/2.

Nancarrow, Conlon. Study No. 17 for Player Piano (Canon 12/15/20). Santa Fe: Soundings.

In this work, there are three independent voices. All voices sound throughout the piece. The following tempi occur simultaneously: MM half note = 138, 172-1/2 and 230. The ratio between these tempi is 12:15:20. Sometimes, two of the voices share the same tempo, but at no time during the piece are all three voices at the same tempo. The three voices begin and end together (i.e. the first and last notes occur at the same time in all voices). The piece begins with the tempi assigned as follows: MM half note = 172-1/2 (voice 1), 138 (voice 2) and 230 (voice 3). During the course of the piece, tempo changes are made in the following order: the tempo of the second voice is changed to MM half note = 230; the tempo of the third voice is changed to MM half note = 138; the tempo of the second voice is changed to MM half note = 172-1/2; and finally, the tempo of the first voice is changed to MM half note = 230. Thus, the tempo of the second voice is swapped with that of the third voice, and then swapped with that of the first voice. Changes in meter that occur in one voice do not coincide with those of the other voices. Common barlines are drawn wherever the barlines of the voices coincide exactly, which is rarely the case.

Nancarrow, Conlon. Study No. 18 for Player Piano (Canon 3/4). Santa Fe: Soundings.

This is a two-voice canon. The tempo of one voice is MM half note = 168 while that of the other voice is MM half note = 224. The ratio between these tempi is 3:4. The slower voice begins alone and then is joined by the faster voice. The two voices end together (i.e. the last note of the slower voice occurs at the same time as that of the faster voice). Changes in meter that occur in one voice do not coincide with those of the other voice. Common barlines are drawn wherever the barlines of the voices coincide exactly, which is frequently the case.

Nancarrow, Conlon. Study No. 19 for Player Piano (Canon 12/15/20). Berkeley: Soundings.

This a canon for three voices each with its own tempo. The ratio between the three tempi is 12:15:20 and the tempi are as follows: MM quarter note = 144 (voice number 1), MM quarter note = 180 (voice number 2), and MM quarter note = 240 (voice number 3). The voices enter, one at a time, from slowest to fastest. The entry times of the faster voices, each of which is a transposed version of the first voice, are delayed so that all voices end the piece at almost the same time. The first voice is unmeasured until the second voice enters, after which all parts are measured, though time signatures are not indicated explicitly. Common barlines occur across all parts before and after blocks (or hypermeasures) consisting of the simultaneous combination of twelve eighth notes at MM quarter note = 144 (voice 1), fifteen eighth notes at MM quarter note = 180 (voice 2), and twenty eighth notes at MM quarter note = 240 (voice 3). Also, common barlines are drawn through voices 1 and 2 at every four eighth notes of voice 1 and five eighth notes of voice 2. And there are common barlines through voices 2 and 3 at every three eighth notes of voice 2 and four eighth notes of voice 3.

Nancarrow, Conlon. Study No. 24 for Player Piano (Canon 14/15/16). Santa Fe: Soundings.

In some sections, the following tempi occur simultaneously: MM dotted-quarter note = 149-1/3, 160 and 170-2/3. In other sections, the following tempi are used together: MM dotted-quarter note = 224, 240 and 256. In either case, the ratios between the tempi are 14:15:16. All parts are metered, and at many points in time, all parts have the same meter. Common barlines occur infrequently.

Nancarrow, Conlon. Study No. 27 for Player Piano (Canon 5% / 6% / 8% / 11%). Berkeley: Soundings.

One voice has a constant tempo while each of the other voices accelerates or decelerates at one of four possible rates that are indicated by tempo markings such as "accel. - 6%" or "rit. - 11%". Different rates of acceleration and deceleration occur simultaneously. Tempo markings are positioned above the staff with which they are associated. The accelerations and decelerations are geometric. That is, in a given passage for which a gradual change in tempo is to occur, the ratio between successive durations is a constant that is determined by the indicated percentage. Proportional notion is used throughout for all parts. Barlines are used in some sections, and where they are used, they are positioned independently in each part and no time signature is indicated.

Nancarrow, Conlon. Study No. 31 for Player Piano (Canon 21/24/25). Berkeley: Soundings.

This is a three-voice canon where each voice has a different tempo. The ratio between the three tempi is 21:24:25 and the tempi are as follows: MM quarter note = 105 (voice number 1), MM quarter note = 120 (voice number 2), and MM quarter note = 125 (voice number 3). The voices enter, one at a time, from slowest to fastest, and the entry times of the faster voices are such that all voices end the piece together. As is generally the case in Nancarrow's studies for player piano, the staves are ordered in each system by tempo, with the topmost staff being the fastest. All voices are measured with a meter of 2/4 throughout. Common barlines occur across all parts before and after blocks (or hypermeasures) consisting of the simultaneous combination of 21 measures at MM quarter note = 105 (voice 1), 24 measures at MM quarter note = 120 (voice 2), and 25 measures at MM quarter note = 125 (voice 3). Also, common barlines are drawn through voices 1 and 2 at every seven measures of voice 1 and eight measures of voice 2.

Nancarrow, Conlon. Study No. 33 for Player Piano (Canon sqrt(2)/2). Santa Fe: Soundings.

This piece has two simultaneous parts with one or two voices per part. The tempo of one part is MM quarter note = 140*sqrt(2) and that of the other part is MM quarter note = 280. Here, sqrt(2) means the square root of 2. The ratio between the tempi is sqrt(2):2. Both parts are measured, and meters are changed frequently and independently in each part. Since the tempo ratio is irrational, there are few barlines that are common to both parts. Common barlines occur at the left barline of the first measure, and before a measure that is near the middle of the piece during a phrase in one part that, after an unmeasured pause, begins at a time that is not a beat or (rational) sub-beat of the other part.

Nancarrow, Conlon. Study No. 35 for Player Piano. Berkeley: Soundings.

At almost all times during this work, there are simultaneous different tempi. At times, as many as five different tempi occur at once. For example, at the beginning of the piece, five tempi accumulate, one at a time, in the following order: MM quarter note = 119, 170, 204, 283-1/3 and 340. During the course of the piece, the following tempi are used: MM quarter note = 79-1/3, 85, 102, 113-1/3, 119, 141-2/3, 170, 204, 238, 283-1/3 and 340. The ratios between the tempi may be seen more clearly by dividing the tempo numbers by 5-2/3 to yield the quotients: 14, 15, 18, 20, 21, 25, 30, 36, 42, 50 and 60. Parts are metered independently and common barlines are drawn whenever the barlines of parts coincide exactly. One section of the piece is written with proportional notation.

Nancarrow, Conlon. Study No. 36 for Player Piano (Canon 17/18/19/20). Berkeley: Soundings.

This is a four-voice canon where each voice has a different tempo. The ratio between the three tempi is 17:18:19:20 and the tempi are as follows: MM whole note = 85 (voice number 1), MM whole note = 90 (voice number 2), MM whole note = 95 (voice number 3), and MM whole note = 100 (voice number 4). The voices enter, one at a time, from slowest to fastest, and the entry times of the faster voices are such that all voices end the piece together. All system breaks occur at a barline in the slowest part which is the lowest staff in each system. All voices are measured throughout with various meters (e.g. 4/4, 2/4 and 6/4). Common barlines occur across all parts before and after blocks (or hypermeasures) consisting of the simultaneous combination of 17 measures at MM whole note = 85 (voice 1), 18 measures at MM whole note = 90 (voice 2), 19 measures at MM whole note = 95 (voice 3), and 20 measures at MM whole note = 100 (voice 4).

Nancarrow, Conlon. Study No. 37 for Player Piano (Canon 150 / 160-5/7 / 168-3/4 / 180 / 187-1/2 / 200 / 210 / 225 / 240 / 250 / 262-1/2 / 281-1/4). Santa Fe: Soundings.

The following tempi are used: MM quarter note = 150, 160-5/7, 168-3/4, 180, 187-1/2, 200, 210, 225, 240, 250, 262-1/2 and 281-1/4. At several points in the piece, all twelve tempi occur simultaneously. For example, the piece begins with twelve voices sounding, one for each tempo. The ratios between the tempo numbers and lowest number 150 are as follows: 1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 7/4 and 15/8. This series of ratios is identical to that which is associated with a tempo scale proposed by Henry Cowell in his book New Musical Resources. Cowell's scale is based upon ratios that occur in an overtone series.

Throughout the piece, phrases are metered independently in each voice and common barlines are rare. The piece consists of a sequence of 12 sections, though they are not labeled as such. In some sections, the entry times of successive phrases are staggered. In such cases, the first beat of the first measure, or at least the first beat of some measure, of a (child) phrase that is to enter coincides with a beat or sub-beat of a (parent) phrase in another voice. For example, on page 14, the first beat of the second measure of the child phrase at tempo MM quarter note = 180 coincides with the first beat of a measure of the parent phrase at tempo MM quarter note = 168-3/4. Such times of coincidence are indicated with a common barline or a dotted vertical line that runs from the parent phrase to the child phrase. In some sections, all phrases are aligned at the first barline of the section (e.g. Section 1: p.1-4, or Section 2: p.4-8) or at the last barline of the section (e.g. Section 3: p.8-11).

Nancarrow, Conlon. Study No. 40 for 2 Player Pianos (Canon e/pi). Berkeley: Soundings.

This work calls for two non-synchronized pianos. The piece consists of two sections labeled A and B. There are no tempo markings. Instead, tempo is specified indirectly in terms of the duration of each section.

Section A is a two-voice canon with a duration of 4 minutes. The ratio between the tempi of the voices is e:pi (or e/pi which may be pronounced "e over pi") where "e" is the base of the natural system of logarithms (approx. 2.71828) and "pi" is the ratio of the circumference to the diameter of a circle (approx. 3.14159). In his book The Music of Conlon Nancarrow, Kyle Gann notes that this tempo ratio may be approximated by 13:15 (Note: This approximation is slightly better than 19:22 which would be obtained by dividing a common rational approximation for e (19/7) by a common rational approximation for pi (22/7)). The entry of the faster voice is delayed so that the two voices end simultaneously. Both voices are metered throughout and meters are changed frequently (at almost every bar). Except for the left barline of the final bar, there are no common barlines.

In Section B, Section A is performed again while simultaneously, on a second player piano, Section A is performed more slowly. The slow version of Section A is to have a duration that is 20 seconds longer than the original. Thus, the ratio between the tempi in the slow version and the original is 12:13. In Section B, the entry of the original version of Section A occurs after that of the slow version.

Raxach, Enrique 1969, 1989. Inside Outside. Donemus.

On page 2, the following tempi occur simultaneously: MM quarter note = 76 in 2/4 (cello 1), MM quarter note = 60 in 5/16 (cello 2), MM quarter note = 60 in 3/4 (cello 3), MM dotted-quarter note = 60 in 6/8 (cello 4), MM quarter note = 63 in 3/4 (double bass 3), MM dotted-quarter note = 60 in 3/8 (double bass 4), MM quarter note = 48 in 2/4 (double bass 5), and MM eighth note = 104 in 2/8 (double bass 6). These instruments begin together and play in this way for about 17 seconds while the violins and some violas and cellos play sustained notes (some with quarter-tone vibrato) at the prevailing tempo. Also, during this passage double basses 1 and 2 play different unmetered phrases that contain accelerandos and decelerandos (indicated with feathered beams) which are at times simultaneous and opposing. This page is quoted in the book Modern Rhythmic Notation by Gardner Read (p. 170).

Reich, Steve 1967. Piano Phase for Two Pianos or Two Marimbas. London: Universal Edition.

One metronome marking of MM dotted-quarter note = about 72 is given.

The piece begins with the first piano repeatedly playing a sequence of twelve sixteenth notes on the pitches: E / F# / B / C# / D \ F# \ E / C# \ B \ F# / D \ C#. Note: Here "X / Y" ("X \ Y") indicates that Y is a higher (lower) pitch than X by an octave or less. The odd-numbered notes of the sequence are played with the left hand while the even-numbered notes are played with the right. Gradually, the second piano fades in, in unison with piano 1. Then, the second piano is instructed to "accel very slightly" until the following two notes are occurring at the same time: the first note of the phrase played by piano 1 (E), and the second note of the phrase played by piano 2 (F#). When the players become aligned in this way, the second piano is instructed to "hold tempo 1". During the times at which it is accelerating, the part of the second piano is notated with dotted staff lines only. Next, piano 2 accelerates again (henceforth, indicated with the abbreviation "a.v.s.") until its phrase is two sixteenths ahead of piano 1. Again, at this time piano 2 is instructed to hold tempo 1. This process is repeated twelve times until the two pianos are in unison again. Then the second piano fades out and rests.

After some number of repetitions of the twelve-note phrase, piano 1 begins repeating an eight-note phrase that is a sub-phrase of the twelve-note phrase, consisting of the 1st through 6th notes followed by the 9th and 12th notes (E / F# / B / C# / D \ F# / B / C#). After some repetitions, piano 2 fades in and plays the following eight-note phrase against piano 1: E / E \ A / B / D / E \ A / B. Then, piano 2 performs eight acceleration transformations as before until the two pianos are aligned again as they were before the transformations. Then piano 1 fades out while piano 2 continues to repeat its eight-note phrase.

Piano 2 then plays the first six notes of its eight-note phrase, once (E / E \ A / B / D / E), after which it plays the last four notes of this, repeatedly (A / B / D / E). Piano 1 fades in, in unison with Piano 2. Piano 2 performs four acceleration transformations as before until the two pianos are in unison again. Then after some repetitions, both players end together.

Throughout the piece, the number of repetitions for each bar is specified approximately as a range. For example, "x12-18" is used to indicate that a given bar is to be repeated about 12 to 18 times. Players are instructed to proceed to the next bar after having done the following: "listen to the two voice relationship and as you hear it clearly and have absorbed it, move on to the next bar".

Reich, Steve 1967. Violin Phase for Violin and Pre-recorded Tape or Four Violins. London: Universal Edition.

A single metronome marking of MM quarter note = about 144 is given. During several intervals of time, a given instrument is instructed to accelerate very slightly while the other instruments remain in tempo. Instruments that accelerate do so until the one-measure looped phrase that is being played is advanced in time by one eighth note. An instruction such as "(x6-16)" is given to indicate that an acceleration is to occur over the duration of 6 to 16 repetitions of the looped phrase. In sections where a given instrument is to accelerate, the 5-line staff for the part is replaced by a 5-line, 2-dimensional pattern of dots.

Reich, Steve 1970. Phase Patterns for Four Electric Organs. London: Universal Edition.

This piece contains one metronome marking of MM quarter note = about 176. When an organ is not at rest, it is playing one of a small set of one-measure (4/4) patterns repeatedly. This piece begins with the first and second organs playing the same pattern in unison. At various points in the piece, the second organ is instructed to "accel. very slightly". In the "Directions for Performance", it is specified that in these sections the tempo of the second organ is to be increased gradually "as slowly as possible" until it is playing one eighth note earlier than before, while all other instruments continue at the original tempo. After the eighth-note out-of-phase relationship is established between the first and second organ, the second organ returns to the original tempo. After eight transformations of this type, the second organ is once again playing in unison with the first organ and the piece ends.

Reich, Steve 1971. Drumming Part Two for Three Marimbas and Female Voices.

In this work, there are nine marimba players positioned around three marimbas. At various points during the piece, one marimba player is instructed to slightly increase their tempo so as to gradually move ahead of the other players by a full quarter note. At other times, players that have become out of phase in this way, are instructed to slightly increase or decrease their tempo in order to slowly move back into unison with other players.

Stockhausen, Karlheinz 1955-6. Zeitmasse (Time-measures) for Five Woodwinds. London: Universal Edition.

In various sections of this work, simultaneous different tempo markings are specified. For example, this occurs on the following pages in the 77-page full score (Universal Edition 12697LW): pp. 10-12, 22-24, 37-42 and 44-53.

The tempo of some passages is indicated with a metronome marking while that of others is indicated with a German textual phrase such as: "so schnell wie möglich" (as fast as possible), "so langsam wie möglich" (as slow as possible), "schnell - verlangsamen" (fast - slowing down), "langsam - beschleunigen" (slow - quickening). Regarding these various tempo markings, Stockhausen writes the following in the "Performing Directions" (Note: Here, the term "group" means passage or phrase):

1. Tempi in the form of metronome-markings are to be observed exactly.

2. The tempo "as fast as possible" applies to the shortest note-value of a group. Within this group the chosen mensural unit remains constant, while the effective speed varies with the note-values employed.

3. The tempo "as slow as possible" is determined by the breathing-capacity of the woodwind-player, who has to perform a particular series of groups in one breath. The greatest total duration for the series determines the constant mensural value.

4. When the tempo is "fast - slowing down", the mensural units (mainly [sixteenth note]) at the beginning and end of the group are to stand roughly in the proportion 4:1 (e.g. [sixteenth note] ca. 240 --> [sixteenth note] ca. 60). Thus the length of the group determines the rate of slowing-down. The smallest note-value in the group decides how quickly the beginning can be played. While the mensural units become steadily longer, the effective speed can, because of the varying note-values, alter irregularly.

5. When the tempo is "slow - quickening", the shortest possible mensural unit at the end of the group is first to be settled: from this is determined the mensural unit, four times as long, for the beginning of the group (e.g. [sixteenth note] ca. 70 <-- [sixteenth note] ca. 280).

As an approximate control measure the groups with relative tempi (2.-5.) are drawn in the score to proportionate scales.

At some points, simultaneous different metronome markings are specified. For example, this occurs on page 44 where the following markings are used: MM eighth note = 64 (oboe), MM eighth note = 80 (English horn), MM eighth note = 112 (clarinet). The same markings are used simultaneously in one measure on page 46 for the English horn, bassoon and clarinet, respectively. Here, the three parts share common barlines with meters 4/16, 5/16, and 7/16, and they occur simultaneously with an unmeasured part for oboe that has the marking "schnell - verlangsamen" (fast - slowing down). Later, on pages 52-3 the following metronome markings are used simultaneously: MM eighth note = 64 (oboe and flute), 80 (English horn), 96 (clarinet), and 112 (bassoon).

Dashed vertical lines are used to indicate common barlines or beats. Such lines are drawn through a staff at the point at which a common barline or beat occurs.

Stockhausen, Karlheinz 1955-7. Gruppen (Groups) for 3 Orchestras. London: Universal Edition.

This piece requires three orchestras, each with its own conductor. In general, some note or sound within the part of one of the sounding orchestras serves as a cue for a conductor to begin beating a silent measure to establish a new tempo and meter, and prepare his orchestra for entry.

A tempo scale is used that consists of the tempi MM quarter note = 60, 63.5, 67, 71, 75.5, 80, 85, 90, 95, 101, 107, 113.5 and 120. Analogous to pitch scales, this is approximately an "equal-tempered" tempo scale from 60 through 120 with 12 steps where the ratio between any two successive tempi in the scale is roughly equal to the 12th root of 2. (More precisely, the tempi in an equal-tempered scale of this type would be: 60.00, 63.57, 67.35, 71.35, 75.60, 80.09, 84.85, 89.90, 95.24, 100.91, 106.91, 113.26 and 120.00.) In this piece, almost all tempi are derived from this scale or a multiple of 2 or 1/2 thereof. One exception occurs at mark 54 on page 48 in orchestra 1. Here the tempo MM eighth note = 157.5 is used in order to align the barlines of a 7/8 measure of orchestra 1 with that of a 4/4 measure at MM quarter note = 90 in orchestra 2.

Approximations in timing have been made throughout the score to simplify the notation. For example, on page 33, the right barline of a measure of 4 beats at MM quarter note = 90 in orchestra 3 is made to coincide with that of a measure with 5 beats at MM quarter note = 113.5 in orchestra 2. Actually, these two durations are not equal, but treating them as such greatly simplifies the placement of system breaks as measures are not fractured across systems. Another example occurs on page 47 where the right barline of a 7/8 measure at MM quarter note = 80 in orchestra 1 is made to coincide with that of a 4/4 measure at MM quarter note = 90 in orchestra 2. In this case, the approximation is clarified to be in the tempo of orchestra 1 as this is written as MM quarter note = 80 (78.75).

Xenakis, Iannis 1969. Persephassa pour Six Percussionistes. Paris: Editions Salabert.

The six percussionists are positioned at the corners of a hexagon around the audience. In the English "Directions" section, Xenakis writes: "p. 14 m. 191: The 6 metronomes must be exactly synchronized (or else electronically controlled).". During various sections of the piece, the tempi of the players are independent. For example, in measures 193 to 215, tempo changes are gradually and independently introduced into each part. The tempi that occur simultaneously are MM quarter note = 38, 40, 42, 58, 74 and 78. From measures 220 to 227, these tempi are used again, but here they begin simultaneously at measure 220 after a 6-second silence which is presumably used to allow each player to receive a preparatory cue to establish their individual tempo. In measures 250 and 296-297, the following tempi are juxtaposed: MM quarter note = 44, 52, 56, 60, 64 and 68.

Articles

Austin, Larry 1985. “Charles Ives's Life Pulse Prelude for Percussion Orchestra: A Realization for Modern Performance from Sketches for his Universe Symphony.” Percussive Notes: Research Edition 23(6): 58-84.

Austin describes the process by which his Life Pulse Prelude was created based upon sketches by Ives. Copies of relevant sketch pages are included and their significance discussed. Also, the method by which synchronization was accomplished in live performance is described (p.78) as well as other methods used in earlier recorded versions that were created for tape performance (p.73). In this piece, different simultaneous tempi occur because all parts share a common right barline but not a common meter. The number of beats per measure which occur simultaneously are: 1, 2, 4, 8, 3, 6, 9, 12, 5, 10, 7, 14, 11, 22, 13, 17, 19, 23, 29 and 31. In performance, 12 pre-recorded tape tracks of independent beat-cue patterns were used to generate cues for the following tempi: MM quarter note = 60, 75, 82.5, 90, 105 and 97.5; MM half note = 67.5; MM eighth note = 127.5, 142.5, 172.5, 217.5 and 232.5. The click pitch and the duration between clicks was varied to distinguish the first and last beats of a measure from intermediate beats. For example, the sequence of click pitches used for the 31 beats of a 31/8 measure was |C---c---c---c---c---c---c---ceg| where 'C' is the pitch that is an octave above the pitch 'c', 'e' and 'g' are in the octave between 'c' and 'C', and '-' represents a beat for which no click was sounded. Similarly, the sequence used for 13/4 measures was |Ccccccccccceg|. In performance, it was found that open air headphones which were used to deliver cue signals to performers, were unacceptable during extremely loud passages.

Cowell, Henry 1927. “The Impasse of Modern Music.” The Century Magazine 114(6): 671-7.

Cowell discusses various forms of innovation that characterize the Western art music of the early 20th century. Regarding the use of simultaneous different tempi, he writes:

In orchestral music we may very practically evolve combinations in musical tempo which cannot well be written for any single instrument. As a matter of musical feeling a slow note in allegro is very different from a fast note in adagio, and music which is to convey more than a simple emotion, can be written far more effectively by using different tempos in its different parts. The effort of conveying different emotions in the same tempo is a tremendous handicap to a composer, and is largely responsible for the stilted and artificial quality of most opera. Take, for example, the quartette from Rigoletto, where Verdi attempts to portray the emotions of four different characters. It would be far more realistic if each part were written in the tempo called for by the emotion of the character. The contralto part should be sung in allegro, the tenor in allegretto, the baritone in adagio and the soprano in andante. Instead, all are sung in the same tempo, the conflict of emotions is reduced to an average and the standardized tempo reduces, rather than advances, the dramatic action.

Ghent, Emmanuel 1967. “Programmed Signals to Performers: A New Compositional Resource.” Perspectives of New Music 6(1): 96-106. Reprinted in Boretz, Benjamin and Edward T. Cone (Eds.) 1976. Perspectives on Notation and Performance. New York: Norton.

This article describes a system for transmitting audio cue signals to musicians to coordinate the performance of rhythmically complex music. An associated notation system is proposed whereby conventional music notation is set against and scaled to a timeline that runs throughout the score above each system. A rhythmic cue line is provided above the staff for each part to indicate beat groupings and the times at which various cues are to be transmitted. Cue signals are used for the following purposes: to indicate downbeats and upbeats (with signals for downbeats being longer than those for upbeats), to establish a new tempo before notes are to be played (as a few preparatory beats), to signal the entire rhythmic pattern of a complicated phrase before it is to be played (e.g. for klangfarbenmelodie that has an aperiodic rhythm), and to provide a reassuring checkpoint cue or an alerting cue just before the end of a long rest (in the form of a rapid succession of three cue signals). Initially, cue signals were generated during performance by a device called the coordinome. In later performances, signals were played back and decoded from a single track of magnetic tape on which cue signals for all instruments had been recorded, with the signal for each instrument being at a different frequency.

Ghent, Emmanuel 1967. “The Coordinome in Relation to Electronic Music.” Electronic Music Review 1(1): 33-43.

This article describes the coordinome which is a device that was designed and developed at New York University by Ghent and Stein G. Raustein. The device was originally intended to be used to coordinate performers of rhythmically complex music, or to synchronize performers with tape music. This article discusses these uses along with additional applications in the areas of rhythmic training in education, the control of theatrical elements (e.g. lighting), and the real-time control of sound generating and processing equipment.

Greschak, John 2000. “Viewing a Finale Document on a Wearable Computer for Performance Purposes.” Finale: IWBNI Suggestion No. 644. Available: http://www.greschak.com/notation/finale/iwbni/fs644.htm.

This brief article proposes a system consisting of centrally-coordinated wearable computer displays on which musical scores and beat indicators could be shown to facilitate the performance of rhythmically complex pieces such as polytempo music. In this system, beat-cue information which is supplied to each performer would be derived from, but not necessarily equal to, the beat established by a human conductor equipped with a wired baton (e.g. one of the available Baton Interfaces such as the Mathews/Boie Radio Baton or the Buchla Lightning) or a lead performer equipped with a suitable controller such as a wired foot pedal. The article lists the musical implications and potential advantages of such a system. Note: This article is contained in the web site Finale: IWBNI (It Would Be Nice If) which is a set of 724 suggestions for improving the music notation program Finale by Coda Music, a division of Net4Music. Thus, in this context the system is presented as a potential feature for the music notation software program Finale.

Greschak, John 2001. “The Word ‘Polytempo’.” Polytempo Music Articles. Available: http://www.greschak.com/polytempo/ptword.htm.

The derivation, spelling, usage and meaning of the word polytempo is given. Related words such as multi-tempo, polytempic, polychronic and polytemporal, which have been used in place of polytempo by various authors are also discussed. Non-English spellings of the word polytempo are also given.

Greschak, John 2001. “Facilitating the Performance of Polytempo Music: An Overview.” Polytempo Music Articles. Available: http://www.greschak.com/polytempo/ptperf.htm.

This is an overview of the various approaches that have been taken by composers to facilitate the performance of polytempo music. The following subjects are discussed: loose coordination, notational solutions, technology-assisted conducting, and electronic realizations.

Greschak, John 2001. “Technology-Assisted Conducting of Polytempo Music: Some Possibilities.” Polytempo Music Articles. Available: http://www.greschak.com/polytempo/pttac.htm.

This article proposes some suggestions as to how technology might be used to coordinate the performance of polytempo music. The following subjects are discussed: semi-automatic conducting, content-rich audio cues, visual cues from artificial conductors, tactile cues, multi-sensory cues, and wireless connections.

Greschak, John 2001. “Tempo Scales in Polytempo Music: A Survey.” Polytempo Music Articles. Available: http://www.greschak.com/polytempo/ptts.htm.

This article is a survey of the different forms of tempo scales that have been used in polytempo music. A tempo scale is defined as a collection of tempi arranged in order from lowest to highest or from highest to lowest. The following types of scales are discussed: tempo-term scales, the metronome scale, harmonic scales, equal tempered scales, and augmentation/diminution scales.

Ligeti, Lukas 2000. “Beta Foly: Experiments with Tradition and Technology in West Africa.” Leonardo Music Journal 10: 41-47. Available: http://muse.jhu.edu/demo/lmj/10.1ligeti.html.

In the subsection "Sound of No Restraint", a MIDI-based click-track system is described that enables an ensemble to remain synchronized while each player performs at a different tempo. First, a MIDI file is created that contains one track for each player. A unique MIDI channel is assigned to each track. An appropriate sequence of click notes is inserted into each track. During performance, a sequencer program plays the MIDI file. The MIDI output of the sequencer is routed to the MIDI input of a sampler that has multiple monophonic audio outputs. The sampler is configured so that a unique MIDI channel is assigned to each audio output. The audio outputs are routed through small headphone amplifiers to open air headphones. A cowbell sample is used for click sounds. In the concluding section "Some Present and Future Experiments", Ligeti proposes a system by which a set of drum pads could be used to conduct an ensemble. In this case, the audio output derived from each pad would be routed to a different player.

Stockhausen, Karlheinz 1959. "How Time Passes." die Reihe 3 (Musical Craftsmanship): 10-40. Bryn Mawr: T. Presser. English translation of the original German version by Cornelius Cardew.

In this article, Stockhausen puts forward an equal-tempered, eight-octave duration scale that is a low-frequency extension of an equal-tempered pitch scale for which the frequency of the lowest pitch is 32 Hz. To represent this duration scale, he uses the following metronome markings MM whole note = 60, 63.6, 67.4, 71.4, 75.6, 80.1, 84.9, 89.9, 95.2, 100.9, 106.9, 113.3 and 120. These tempi are used in combination with the following note symbols: sixteenth note, eighth note, quarter note, half note, whole note, double whole-note, quadruple whole-note, and octal whole-note. In this tempo scale, there are twelve tempi per octave and the ratio between any two successive tempi is equal to the twelfth root of two. A tempo scale that is closely related to this is used in Stockhausen's Gruppen for 3 Orchestras.

In discussing realizations of works based upon this scale of durations, Stockhausen writes (p.25):

...the realization of our example requires three chordal instruments, or, better, three orchestras, that would at times play independently of each other, at differing tempi, orientating themselves to the others only at the points where they entered. Before each entry, each group - or each conductor - could prepare for the next tempo without difficulty (with a metronome). Spatial separation would result naturally from the need to make various time-strata appreciable.

On page 32, he considers the notion of four simultaneous passages with different tempo structures: a passage with a constant tempo, a passage that is to be played as fast as possible, a passage during which there is an accelerando, and a passage during which there is a decelerando.

Sullivan, Timothy 1997. “Multiple Tempi: A Survey and Method.” Conference of the Canadian University Music Society.

This article discusses a method by which simultaneous different tempi may be notated so that all parts are written against at common reference tempo, and in each measure, all parts have the same meter. The method consists of a thorough application of the techniques of augmentation and diminution, and uses notational devices similar to those found in scores by Elliott Carter. In Figure 14, an extensive table is given to indicate the note duration that would be required to notate one beat, or various fractions of a beat (i.e. 1/5, 1/4, 1/3, 1/2, 2/3 and 3/4), of a tempo that stands in a given ratio to the reference tempo, given that the duration of a beat in the reference tempo equals that of a quarter note. A wide range of possible tempo ratios are listed in this table. Excerpts are quoted from the literature in which different tempo markings are indicated simultaneously, including: Karlheinz Stockhausen's Gruppen (Figure 7: five measures around mark 28), Charles Ives' Putnam's Camp from Three Places in New England (Figure 11: mark H to I), György Ligeti's Chamber Concerto for 13 Instrumentalists (Figure 19: measures 13-23), and Harry Somers' Those Silent, Awe Filled Spaces (Figure 21: pages 16-21), and the proposed method of notation is used to renotate these excerpts (except Gruppen). With this method, the duration of a given type of note is the same in all parts, at each point in time. Sullivan has used these techniques in his works Three Etudes in Multiple Tempi and Terrains I for Orchestra (1999).

Willey, Robert 2003. “An Analysis and Realization of Conlon Nancarrow's Study No. 37 for Player Piano.” Available: http://www.willshare.com/willeyrk/creative/papers/study37.

Willey uses the melody of Frère Jaques to illustrate various tempo ratios between two voices. The concepts of consonant tempo ratios (e.g. 2:1), dissonant tempo ratios (e.g. 32:31), convergence (i.e. voices end synchronized), and divergence (i.e. voices begin synchronized) are described. The relative degree of dissonance of a given tempo ratio is expressed (loosely) in terms of the number of beats that occur between consecutive coincident beats.

A detailed description of the structure of Nancarrow's Study No. 37 is given in a table that includes the following entries for each section of the work: page number, duration, [entry] order of tempi, type of canon (i.e. convergent, divergent, or "internal synch"), order of ranges [or entry order of voices], interval of entrance, ambitus of voice material, dynamic level, duration of entry notes, melodic character, texture, perceptibility of canon, and contrast with previous section. Three excerpts are given from this study (specifically, page 1 and two portions of Section V), all of which depict instances of simultaneous different tempo markings.

The use of contrasting timbres and spatial positions for different voices is explored in order to make the underlying canonic forms of Study No. 37 more apparent. Consideration is given to the relative degree to which tempo and various note properties (i.e. start time, pitch, duration, dynamic level, timbre and spatial position) are a determining factor for one to perceive a given passage as melodic (i.e. horizontal) vs. harmonic (i.e. vertical). For this, concepts of Max Wertheimer's Gestalt psychology are applied (e.g. proximity, similarity, direction [continuity] and common destiny [common fate]).

Books, Theses and Dissertations

Cole, Hugo 1974. Sounds and Signs: Aspects of Musical Notation. London: Oxford University Press.

The section "Accelerando and rallentando" of the chapter titled "Time" (p. 73) contains a discussion of various methods for notating a gradual change of tempo in one part while other parts continue in strict tempo. A "tempo glissando" in Elliott Carter's String Quartet No. 2 is quoted.

Cowell, Henry 1930. New Musical Resources. New York: Alfred A. Knopf.

In Section 6 "Tempo" and Section 7 "Scales of Rhythm" of Part II "Rhythm", Cowell discusses the notion of combining different tempi simultaneously. Also, some matters pertaining to situations in which the tempo of one part remains constant while that of another simultaneous part is varied continuously, are considered. The musical excerpts given in the figures for Examples 24 (p.93) and 26 (p.97) contain simultaneous different metronome markings. It is suggested that the tempi of such polytempo textures be drawn from a tempo scale that is constructed using ratios that are ordinarily associated with the overtone series.

Epstein, David 1995. Shaping Time: Music, the Brain, and Performance. New York: Schirmir.

This is an extensive scientific and artistic examination of various topics related to time and tempo in music. It includes numerous case studies of works from the Classical and Romantic Periods. The following excerpt from "Chapter Eleven: Rubato" (p. 373) discusses simultaneous different tempi in romantic rubato (Here, "romantic rubato" refers to a rubato style whereby all elements (melody, harmony, rhythm) of a passage are affected simultaneously):

Further implicit in both views of romantic rubato is the fascinating phenomenon of two separate and different time controls operating simultaneously as the phrase or other musical segment progresses. There is the strict metric control of the beat - the hypothetically invariant time unit that serves as the ground element for timing. Moving in parallel with this is the pulse of various segments of the phrase, a pulse distorted by rubato as a means of enhancing musical expression.

These two time controls, really systems of time control, rapidly become dissynchronous and thus in conflict, thereby adding excitement to the performance, for we feel ourselves pulled by each system, internally stressed or contorted by the tensions that each exerts. A large part of the gratification in good rubato playing lies in the eventual reconciliation of these two systems, their return to phase synchrony.

Gann, Kyle 1995. The Music of Conlon Nancarrow. Cambridge UK: Cambridge University Press.

This book contains biographical information on Conlon Nancarrow and analyses of sixty-five of his works. The author maintains a web page on Nancarrow at http://home.earthlink.net/~kgann/index2.html.

Read, Gardner 1978. Modern Rhythmic Notation. Bloomington: Indiana University Press (1978); London: Victor Gollancz (1980).

In the chapter titled "Polymetric Notation" (pp. 168-173), the following examples of pieces that contain simultaneous different tempi are discussed: Lejaren Hiller's String Quartet No. 5, Hans Werner Henze's Compases para preguntas ensimismadas for solo viola and orchestra, Girolamo Arrigo's Fluxus, Marius Constant's Winds and Enrique Raxach's Inside Outside. Single-page excerpts of the Arrigo and Raxach works are quoted. Rhythmic schematics are given for small portions of the Hiller and Constant pieces.

Saunders, James E. 1996. “The Final Frontier: The Development of Polytempo in the Music of Conlon Nancarrow.” Masters dissertation, Royal Northern College of Music.

Abstract excerpt:
The development of Conlon Nancarrow's use of polytempo is traced from the simple micro-pulse relationships in his early music through to the more elaborate irrational number relationships of the later pieces. The effects of his various approaches on the resulting music are examined.

Stauffer, George B. 1997. The Mass in B Minor: The Great Catholic Mass. New York: Schirmer; London: Prentice Hall.

Instances in which the part scores for Bach's Mass in B Minor contain simultaneous different tempi are discussed (p. 234 and p. 295 n.59). A related article by Bernard D. Sherman titled Bach's Notation of Tempo and Early-Music Performance: some reconsiderations is available at http://www.kdsi.net/~sherman/bachtempo.htm.

Stone, Kurt 1980. Music Notation in the Twentieth Century: A Practical Guidebook. New York; London: Norton.

The notation of music with different simultaneous speeds is discussed in Chapter III: "Duration and Rhythm: Preliminary Survey" and Chapter IV: "Duration and Rhythm: Individual Items". In the section "Spatial Notation versus Traditional (Symbolic) Notation" (p. 137), a figure is given to show how spatial notation may be used to notate a simultaneous ritardando and accelerando. In the section "Different Rates of Speed" (pp. 119-120), the approximations used in Elliott Carter's String Quartet No. 2 are discussed (see also: p. 115). The multi-rate nature of this work is described in greater detail in the section "Polymetric Combinations: Notation by Reference Meter" (pp. 90-3) where two excerpts of the score are quoted.

Thomas, Margaret Elida 1996. “Conlon Nancarrow's 'Temporal dissonance': Rhythmic and Textural Stratification in the Studies for Player Piano.” Ph.D. dissertation, Yale University, UMI No. DA9632450.

This 317-page dissertation explores the rhythmic and textural techniques of Conlon Nancarrow's Studies for Player Piano in general, and his temporal dissonance in particular. It includes the following sections: I. Introduction; II. Temporal Stratification in the Works of Other Composers; III. Multidimensionality and Textural Strategies; IV. Temporal Dissonance; V. Study No. 41; VI. Concluding Remarks.

Works Under Consideration

This is a list of works that are known or suspected to be suitable for inclusion in the bibliography of polytempo music given here. They are being researched.

• Austin, Larry 1983-4. Sonata Concertante.
• Austin, Larry. Charles Ives' Universe Symphony as realized and completed by Larry Austin (1974-93). Also, related works: Life Pulse Prelude (1984) and Life Pulse Prelude for Live and Recorded Percussionists (1984/96).
• Brant, Henry. Any work by Brant is a candidate for the bibliography given here, especially spatial pieces and works completed after the initial version of Antiphony One (1953), with the exception of the following pieces which have been reviewed and judged to be unqualified for inclusion in this bibliography: The Marx Brothers: 3 Faithful Portraits, Capriccio by J. S. Bach, All Souls Carnival, Concerto for Alto Saxophone Solo or Trumpet Solo with Nine Instruments, Stresses, Whoopee in D Major, The $1,000,000 Confessions, American Commencement, American Requiem, Galaxy 2, Requiem in Summer, Millennium 1, Signs and Alarms, Partita for Flute and Piano, From Bach's Menagerie, Four Short Nature Pieces, and Atlantis.
• Dolden, Paul 1992-3. L'Ivresse de la Vitesse (Intoxication by Speed).
• Dragostinov, Stefan 1980. Politempy n.4 per Pianoforte e Orchestra (Concerto for Piano and Orchestra Polytempi IV).
• Dragostinov, Stefan. A device called a photopolymetronome.
• Drelas, Nikos 1986-7. Strophen.
• Dubrovay, László. Symphonia.
• Gann, Kyle. Long Night for Three Pianos.
• Gann, Kyle. The Convent at Tepoztlan for Piano and Tape, or Two Pianos.
• Gisby, Steven. “Timephase: Multiple Tempo in Music.” Department of Performing Arts Seminar. Brunel University, West London. October 18, 2000.
• Grella-Mozejko, Piotr 1983. The First February the Last January for One or Two Pianos and One to Four Optional Melodic Instruments.
• Harrison, Bryn 1999. Fields of Red for Two Groups of Instruments, Piano Solo and Doubling Keyboard.
• Harvey, Jonathan Dean. Timepiece for Orchestra with Two Conductors.
• Jönsson, Kenneth 1993-5. Fugue No. 7.
• Jönsson, Kenneth 1993-5. Fugue No. 8.
• Jönsson, Kenneth 1993-5. Good Mourning.
• Jönsson, Kenneth 1993-5. Preludium No. 2.
• Käser, Mischa 1994/5. Napoleonskurve for Chamber Orchestra.
• Kostitsyn, Evgeni. Symphony No. 1 for Two Conductors and Large Orchestra.
• Kostitsyn, Evgeni. Symphony No. 2 for Quintet and Orchestra.
• Kostitsyn, Evgeni. Symphony No. 3 for 10 Conductors, Choir and Orchestra.
• Kulesha, Gary. Symphony for Two Conductors and Orchestra.
• Levinas, Michaël. 3 Pensées pour Coma.
• Ligeti, György. Entrelacs (Twelfth Piano Etude).
• Ligeti, Lukas 1993. Groove Magic for 11 Instruments and Computer Controlled Click Tracks.
• Maupoint, Andrés 1996. Miró...matiz for Flute and Piano.
• McTee, Cindy 1996. Einstein's Dreams.
• MIDI Manufacturers Association 1996. The Complete MIDI 1.0 Detailed Specification: Standard MIDI Files 1.1: Format 2.
• Mizelle, Dary John 1979. Polytempus II for Marimba and Computer.
• Morris, Robert 1975. In Different Voices for Five Wind Ensembles.
• Myers, Jeff. Five Parametric Etudes for Disklavier.
• Raes, Godfried-Willem 1969. Logos 3/5 for Violin, Cello, Oboe, and Piano with Electronic Conductor.
• Raes, Godfried-Willem 1994. A device called a polymetronome.
• Raes, Godfried-Willem 1995. Fall '95 for Violin, Flute, Clarinet, Horn, Euphonium, Tuba, Player Piano, Computer and Polymetronome.
• Reich, Steve 1967. Reed Phase.
• Reich, Steve 1979. Music for a Large Ensemble.
• Riley, Terry 1959. Spectra for Sextet.
• Somers, Harry 1975. Zen, Yeats and Emily Dickinson for Soprano, Flute, Piano, Actress and Actor.
• Somers, Harry 1977. Death of Enkidu.
• Somers, Harry 1978. Those Silent, Awe Filled Spaces.
• Viñao, Alejandro 1995. Phrase and Fiction for String Quartet and Computer.
• Wilson, Charles M. 1997. Psycho Red.
• Wolpe, Stefan 1950-3. Enactments for Three Pianos.
• Zappa, Frank. Friendly Little Finger from Zoot Allures (1976), Rubber Shirt from Sheik Yerbouti (1979). Packard Goose and Keep it Greasy from Joe's Garage (1979), and other examples of xenochrony.

Update History

June 3, 2001: Completed the first version of this page. Frequent updates shall be posted as works under consideration are processed and as new candidates are discovered or suggested. A backlog of such items already exists thanks to the help of Bob Kosovsky (a librarian in the Music Division of The New York Public Library) and many others on the Society for Music Theory List who have responded to a query that he posted there.

June 8, 2001: Added several works under consideration.

June 10, 2001: Added the article “Beta Foly: Experiments with Tradition and Technology in West Africa” by Lukas Ligeti and more works under consideration. Excluded mensuration canons, augmentation and diminution.

June 22, 2001: Added over a dozen entries including works by Austin (article), Berio, Boulez, Brant, Carter, Colgrass, Epstein (book), Ferneyhough, Ives, Ligeti, Musgrave, Stockhausen and Xenakis. Excluded rubato and senza tempo passages.

July 1, 2001: Added an Acknowledgments section and some works under consideration.

July 11, 2001: Added ten works by Henry Brant.

July 15, 2001: Added the article “The Word ‘Polytempo’” by John Greschak.

August 2, 2001: Added two works by Charles Ives (Three Harvest Home Chorales and Symphony No. 4), four works by Henry Brant (Verticals Ascending, Spatial Concerto, Trinity of Spheres and Orbits), Brian Ferneyhough's Missa Brevis, and Steve Reich's Violin Phase. Added Henry Cowell's article “The Impasse of Modern Music” and book New Musical Resources.

August 5, 2001: Added the article “Facilitating the Performance of Polytempo Music: An Overview” by John Greschak.

August 22, 2001: Added seventeen works by Conlon Nancarrow.

August 30, 2001: Added the article “Technology-Assisted Conducting of Polytempo Music: Some Possibilities” by John Greschak.

September 1, 2001: Added the article “Multiple Tempi: A Survey and Method” by Timothy Sullivan.

September 8, 2001: Added two works by Henry Brant (Desert Forests and Millenium 2), George Crumb's Star-Child, and Steve Reich's Drumming.

September 13, 2001: Added the article “How Time Passes” by Karlheinz Stockhausen.

September 20, 2001: Added the article “Tempo Scales in Polytempo Music: A Survey” by John Greschak.

December 17, 2001: Added György Ligeti's Poème Symphonique, Steve Reich's Piano Phase, Karlheinz Stockhausen's Zeitmasse, and Olivier Messiaen's Éclairs sur l'Au-Delà.

April 20, 2003: Added John Greschak's Platonic Dice: Dodecahedron.

April 23, 2003: Added the article “An Analysis and Realization of Conlon Nancarrow's Study No. 37 for Player Piano” by Robert Willey.

May 23, 2003: Added John Arrigo-Nelson's Temporal Quartets and Solo/Duo.

Acknowledgments

I thank the following composers for responding to my inquiries regarding their polytempo works: Emmanuel Ghent, Henry Brant, Timothy Sullivan and Evgeni Kostitsyn. Also, I thank the following individuals who have suggested works to be included in this bibliography: Bob Kosovsky, John Snyder, Stephen Taylor, Mike Baker, Zac Cairns, Robert Coburn, Mark Williams, Michael D. Golden, Eliot Handelman, Timothy Sullivan, Joseph Harris, Joshua Cohen, Emmanuel Ghent, Jonathan Dawe and John Arrigo-Nelson.