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Equal temperaments as mathematical series John S. Allen Standard 12tone equal temperament  Nonstandard integer equal temperaments Musical tunings, including the conventional 12tone equal temperament, may be derived mathematically in a number of different ways. Different instruments may generate tunings differently, making these instruments more or less adaptable to other, unconventional tunings. An understanding of the relationships between tunings, instrument architectures and keyboard designs makes it clear which are most compatible with one another. Chapter 5 of Scott Wilkinson's book Tuning In describes nonstandard tunings on electronic instruments. The present series of articles covers similar ground, but focuses on the theory which underlies tunings. The presentation here is mathematical, so that it may be generalized to tunings, instruments and keyboards beyond those covered here. No understanding of mathematics beyond algebra is required, however. We will start by examining equal temperaments as onedimensional mathematical series; in later articles, we will expand the discussion to other mathematical definitions and other tunings. A caution is in order: it is useful to describe musical structures through mathematics, but musical flexibility and creativity surpasses what can be described by simple mathematics. For example, the scale of a wind instrument established by the fingering may be represented through mathematics: but in the hands of a skillful player, the predefined scale serves only as the basis for a more flexible intonation controlled through the embouchure and breath control. Standard 12tone equal temperament A musical equal temperament is most simply described as a mathematical series, or onedimensional matrix, whose terms are as F_{m} = 2^{m/k} F_{R}, where F is the fundamental frequency of a musical tone, k is a positive integer constant; m is an integer variable, which may be positive, zero or negative; and F_{R} is a standard reference frequency such as A, 440 Hz. In standard 12tone equal temperament, k is 12, and so F_{m} = 2^{m/12} F_{R}. We may limit the frequency range of musical pitches described by this series either directly, F_{m} = 2^{m/12} F_{R}, F_{L} < F < F_{H} or by setting limits to the values of m, describing, for example, the number of keys below and above the reference pitch on a musical instrument keyboard: F_{m} = 2^{m/12} F_{R}, m_{L} < m < m_{H}. The following table describes these relationships for a twooctave range centered around the reference pitch of A, 440 Hz (highlighted in yellow). Numbers used in deriving the musical pitches mathematically are shown in green. The cells of the table which represent musical pitches are colored black and white like the corresponding keys of a musical keyboard. Think of the mathematical formula as inside the instrument, behind the keyboard, manifesting itself in the pitches which the keys play. The values in musical cents in this and following tables are with relation to the next lower A, 220, 440 or 880 Hz. 
m= 
2^{m/12} 
2^{m/12}= 
F_{m} 
cents 
note name 

12 
2^{2} 
2.0000 
880.00 
0.0 
A 

11 
2^{11/12} 
1.8877 
830.61 
1100.0 
G# 

10 
2^{5/6} 
1.7818 
783.99 
1000.0 
G 

9 
2^{3/4} 
1.6818 
739.99 
900.0 
F# 

8 
2^{2/3} 
1.5874 
698.46 
800.0 
F 

7 
2^{7/12} 
1.4983 
659.26 
700.0 
E 

6 
2^{1/2} 
1.4142 
622.25 
600.0 
Eb 

^ 
5 
2^{5/12} 
1.3348 
587.33 
500.0 
D 
 
4 
2^{1/3} 
1.2599 
554.37 
400.0 
C# 
Higher 
3 
2^{1/4} 
1.1892 
523.25 
300.0 
C 
2 
2^{1/6} 
1.1225 
493.88 
200.0 
B 

1 
2^{1/12} 
1.0595 
466.16 
100.0 
Bb 

F_{R} 
0 
2^{0} 
1.0000 
440.00 
0.0 
A 
1 
2^{1/12} 
0.9439 
415.30 
1100.0 
G# 

2 
2^{1/6} 
0.8909 
392.00 
1000.0 
G 

Lower 
3 
2^{1/4} 
0.8409 
369.99 
900.0 
F# 
 
4 
2^{1/3} 
0.7937 
349.23 
800.0 
F 
v 
5 
2^{5/12} 
0.7492 
329.63 
700.0 
E 
6 
2^{1/2} 
0.7071 
311.13 
600.0 
Eb 

7 
2^{7/12} 
0.6674 
293.66 
500.0 
D 

8 
2^{2/3} 
0.6300 
277.18 
400.0 
C# 

9 
2^{3/4} 
0.5946 
261.63 
300.0 
C 

10 
2^{5/6} 
0.5612 
246.94 
200.0 
B 

11 
2^{11/12} 
0.5297 
233.08 
100.0 
Bb 

12 
2^{1} 
0.5000 
220.00 
0.0 
A 
The derivation of the equal temperament by sequential steps corresponds directly to the control of pitch in analog synthesizers, whose control voltage varies logarithmically with frequency, usually at the rate of 1 volt per octave, v = v_{0} + 1/log 2(log F_{m}  log F_{R}) or v = v_{0} + 1/log 2 [log (2^{ m/12} F_{R} /F_{R})] Nonstandard integer equal temperaments The keyboard of an analog synthesizer typically uses a resistor ladder to generate an equal voltage increment for each key. Varying the voltages at the ends of the resistor ladder changes the upper and lower limits of the keyboard's pitch range, and changes all of the steps equally. The most common nonstandard tunings substitute values of k other than 12 in the formula v = v_{0} + 1/log 2 (m/k log 2), or v = v_{0} + m/k, which corresponds to the first formula in this article, F_{m} = 2^{m/k} F_{R}, in which the number of keys per octave is k. The article on this site on the Fibonacci series has described different integer values of n used in equallytempered tunings of different cultures. The slendro scale used in the Javanese gamelan has  in essence  5 equaltempered pitches, and so the formula for that scale is F_{m} = 2^{m/5} F_{R}. By varying the voltage at the ends of the resistor ladder, or by an analogous procedure, a keyboard can be mapped to equaltempered scales with any chosen number of pitches per octave. There are 7 equallytempered pitches per octave in Siamese music; 12 in European equal temperament; 19 in Joseph Yasser's proposed system, and 31 in Dr. Adriaan Fokker's realized system. The integers 2, 5, 12, 19 and 31 are terms of the same Fibonacci series, as described in the previous article. Other values of m also have musical significance, as we shall see later. 5tone equal temperament maps well to the black keys of the conventional keyboard, as shown in the table below. 7tone equal temperament maps equally well to the white keys, and 12 pitches map to all of the keys. As described in an article on this site about keyboards, scales with more than 12 pitches per octave do not map well to the conventional keyboard. Special keyboards designed for more than 12 pitches per octave, are preferable for use with such scales. The general keyboard described in another article on this site is one such keyboard. The following table represents an equaltempered slendro scale. The named pitches do not correspond exactly to the key names. Rather, each step is somewhat larger than a whole step in 12tone equal temperament. This is true of the intervals with two white keys between them and to those with only one white key between them. Reflecting the inharmonic partials of the instruments of the gamelan, the octaves are slightly stretched, to a ratio of 2.02/1, or 17 cents (100^{ths} of a semitone in 12tone equal temperament) per octave . Stretched octaves generally reflect the inharmonicity of overtones of gongs, bells and strings in free vibration; the octave sounds more harmonious when tuned so frequencies of vibration coincide, rather than when tuned exactly. The slightly stretched octave of the table below is derived from measurements of gamelan instruments. It does not affect the essential 5tone scale structure any more than similar anomalies in tuning of a piano affect the structure of the 12tone equal temperament. 
m=  2.02^{m/5}  2.02^{m/5}=  F_{m}  cents  note name  
5 
2.02^{1}  2.0200 
888.80 
1217.2 
G#/Ab 

4 
2.02^{4/5}  1.7550 
772.21 
973.8 
F#/Gb 

3 
2.02^{3/5}  1.5248 
670.91 
730.3 
D#/Eb 

^ 
2 
2.02^{2/5}  1.3248 
582.90 
486.9 
C#/Db 
 

Higher 

1 
2.02^{1/5}  1.1510 
506.43 
243.4 
A#/Bb 

F_{R} 
0 
2.02^{0}  1.0000 
440.00 
0.0 
G#/Ab 
1 
2.02^{1/5}  0.8688 
382.28 
243.4 
F#/Gb 

Lower 

 

v 
2 
2.02^{2/5}  0.7548 
332.13 
486.9 
D#/Eb 
3 
2.02^{3/5}  0.6558 
288.56 
730.3 
C#/Db 

4 
22.02^{4/5}  0.5698 
250.71 
973.8 
A#/Bb 

5 
2.02^{1}  0.4950 
217.82 
1217.2 
G#/Ab 
Noninteger equalinterval scales The keys of a resistor ladder keyboard or other equalratio keyboard can map to musical pitches in which there are no exact octaves, according to the formula v = v_{0} + m/k. In this formula, k, representing the number of pitches per octave, need not be even nearly an integer as in the slendro scale. Since the pitch ratios between adjacent keys remain equal, this formula describes equal temperaments or equalinterval scales in which no interval need correspond to the octave. Composer Wendy Carlos has undertaken a mathematical analysis which showed that certain noninteger values of k result in better approximations some common musical intervals than in conventional equal temperament. Carlos selected and named scales based on three intervals, as follows:
The following table shows intervals of the Alpha scale either side of the reference pitch A, 220 Hz. While the ratios corresponding to thirds and fifths are quite accurate (for example, the ratio of the perfect fifth in the Alpha tuning is 1.5000 or 3/2, accurate to the fourth decimal place), there is no close approximation to certain other intervals of diatonic scales, including the perfect fourth and the octave. 
m =  2^{m/15.385}=  F_{m}  cents  note name  
16 
2.0562 
452.36 
1248.0 

15 
1.9656 
432.43 
1170.0 

14 
1.8790 
413.38 
1092.0 
G# 

13 
1.7962 
395.17 
1014.0 
G 

12 
1.7171 
377.76 
936.0 

11 
1.6415 
361.12 
858.0 

10 
1.5692 
345.21 
780.0 

9 
1.5000 
330.01 
702.0 
E 

8 
1.4339 
315.47 
624.0 

7 
1.3708 
301.57 
546.0 

6 
1.3104 
288.28 
468.0 

^ 
5 
1.2527 
275.58 
390.0 
C# 
 
4 
1.1975 
263.44 
312.0 
C 
Higher 
3 
1.1447 
251.84 
234.0 

2 
1.0943 
240.74 
156.0 

1 
1.0461 
230.14 
78.0 

F_{R} 
0 
1.0000 
220.00 
0.0 
A 
1 
0.9559 
210.31 
78.0 

2 
0.9138 
201.04 
156.0 

Lower 
3 
0.8736 
192.19 
234.0 

 
4 
0.8351 
183.72 
312.0 
F# 
V 
5 
0.7983 
175.63 
390.0 
F 
6 
0.7631 
167.89 
468.0 

7 
0.7295 
160.49 
546.0 

8 
0.6974 
153.42 
624.0 

9 
0.6667 
146.66 
702.0 
D 

10 
0.6373 
140.20 
780.0 

11 
0.6092 
134.03 
858.0 

12 
0.5824 
128.12 
936.0 

13 
0.5567 
122.48 
1014.0 
B 

14 
0.5322 
117.08 
1092.0 
Bb 

15 
0.5087 
111.92 
1170.0 

16 
0.4863 
106.99 
1248.0 
In a noninteger tuning, an interval may be consonant,
while its inversion is not. For example, the major third of the Alpha scale is
nicely in tune while the minor sixth is not. The pattern of consonant pitches above and
below a note is symmetrical. Octave transpositions and doublings are not possible in
noninteger scales. The composition Beauty in the Beast, published in the recorded album of the same name, uses the Alpha and Beta scales. As Carlos states in the album notes, "While both scales have nearly perfect triads (two remarkable coincidences!) neither can build a standard diatonic scale, and so the melodic motion is strange and exotic." In other compositions, Carlos has used "stretched" tunings in the same way as in the slendro scale, to increase consonance of tones with inharmonic overtones, such as are produced by gongs and bells. Descriptions of the Carlos tunings have been published on pages 50 ff. and 81 ff. of Scott Wilkinson's book Tuning In and in the Spring 1987 issue of Computer Music Journal. Noninteger equalinterval scales are one useful implementation of nonstandard tunings, though it is more common to use octaves which are only very slightly "stretched" to accommodate the slight inharmonicity of plucked or hammered strings. The EqualInterval Tuning Graph The image below is an "elephant's thumbnail" version of a large graph (126 KB GIF, 1786 x 1260 pixels very large image but moderatesized file) which depicts all integer and noninteger equal temperaments with up to 64 pitches per octave. 
The vertical axis of the graph corresponds to the number of scale degrees per octave,
and the horizontal dimension corresponds to a oneoctave range of musical pitch. The
boldface numbers indicate the scales which approximate just intervals more closely; the
small numbers inside the chart grid give the deviation of intervals from just intonation. The 19tone scale on the graph is fairly close to the Carlos alpha scale, and the 34tone scale, to the Carlos gamma scale. Though I did not contemplate noninteger scales when I prepared this graph in 1975, you may nonetheless examine noninteger scales by laying a straightedge horizontally across the graph between the horizontal lines. You will have to print the graph out on several pages and paste them together to do this, unless your computer has a real monster of a monitor. *** This article has described equal temperaments as mathematical series. The next article will describe equal temperaments as twodimensional mathematical matrices. 
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Allen's Home Page] [Up: Tunings, introduction] [Previous: the Fibonacci Series] [Next: Defining octaves separately] 
Contents © 1997 John S. Allen Last revised 6 May 2003 