[contact John S. Allen by e-mail]

Bikexprt.com Web Site

Equal temperaments as mathematical series

John S. Allen

Standard 12-tone equal temperament | Nonstandard integer equal temperaments
Non-integer equal-interval tunings | Graph of equal-interval tunings

Musical tunings, including the conventional 12-tone 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 one-dimensional 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 12-tone equal temperament

A musical equal temperament is most simply described as a mathematical series, or one-dimensional 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 12-tone 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 two-octave 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

2²

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

2⁰

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 = v₀ + 1/log 2(log F_m - log F_R)

or

v = v₀ + 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 = v₀ + 1/log 2 (m/k log 2), or

v = v₀ + 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 equally-tempered tunings of different cultures. The slendro scale used in the Javanese gamelan has -- in essence -- 5 equal-tempered 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 equal-tempered scales with any chosen number of pitches per octave. There are 7 equally-tempered 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.

5-tone equal temperament maps well to the black keys of the conventional keyboard, as shown in the table below. 7-tone 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 equal-tempered slendro scale. The named pitches do not correspond exactly to the key names. Rather, each step is somewhat larger than a whole step in 12-tone 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 12-tone 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 5-tone scale structure any more than similar anomalies in tuning of a piano affect the structure of the 12-tone equal temperament.

5

2.0200

888.80

1217.2

G#/Ab

4

1.7550

772.21

973.8

F#/Gb

3

1.5248

670.91

730.3

D#/Eb

^

2

1.3248

582.90

486.9

C#/Db

|

Higher

1

1.1510

506.43

243.4

A#/Bb

F_R

0

1.0000

440.00

0.0

G#/Ab

-1

0.8688

382.28

-243.4

F#/Gb

Lower

|

v

-2

0.7548

332.13

-486.9

D#/Eb

-3

0.6558

288.56

-730.3

C#/Db

-4

0.5698

250.71

-973.8

A#/Bb

-5

0.4950

217.82

-1217.2

G#/Ab

Non-integer equal-interval scales

The keys of a resistor ladder keyboard or other equal-ratio keyboard can map to musical pitches in which there are no exact octaves, according to the formula

v = v₀ + 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 equal-interval scales in which no interval need correspond to the octave.

Composer Wendy Carlos has undertaken a mathematical analysis which showed that certain non-integer 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.

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

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.

Non-integer equal-interval scales are one useful implementation of non-standard 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 Equal-Interval 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 moderate-sized file) which depicts all integer and non-integer equal temperaments with up to 64 pitches per octave.

The 19-tone scale on the graph is fairly close to the Carlos alpha scale, and the 34-tone scale, to the Carlos gamma scale. Though I did not contemplate non-integer scales when I prepared this graph in 1975, you may nonetheless examine non-integer 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 two-dimensional mathematical matrices.

[contact John S. Allen by e-mail]

Contents © 1997 John S. Allen

Last revised 6 May 2003