Introduction
Tempo-Term Scales
The Metronome Scale
Harmonic Scales
Equal Tempered Scales
Augmentation/Diminution Scales
References
Update History

Introduction

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

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

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

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

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:
 

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

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 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

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:
 

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:
 

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

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.

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