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The previous article described equal temperaments as mathematical series. This article will describe them as two-dimensional mathematical matrices.

Exact musical octaves, 2/1 ratios, are among the results of the formula

F_m = 2^m/k F_R

in which m and k are integers. The octaves are as

2ⁿF_m = F_m+nk/F_m = 2 ^{(m + nk)/k} F_R

and we may also revise the formula to

2ⁿF_m = 2^{(m/k) + n} F_R

Using this revised formula, we may generate the octaves through a separate variable, n. To avoid generating redundant solutions, we also require that m_H - m_L < k.

If m_H = k - 1 and m_L = 0, the k values of m define the equal-tempered pitches within one octave. By using different integer values of n, we may transpose these pitches to other octaves and define an equal-tempered scale of as many octaves as we wish. Mathematically speaking, we are now defining the equal temperament as a two-dimensional matrix, even though it can be defined more simply as a one-dimensional series according to our earlier formula. The more complicated formula is useful because it is often desirable in music to categorize musical pitches in groups of octaves.

The table below gives the nominal fundamental frequencies of the notes sounded by the 88 keys of a piano. The horizontal dimension represents octave intervals, and the vertical dimension represents semitones within each octave. The mathematical derivation of the octaves is shown in red, and that for the semitones within each octave, in green. The reference frequency, A, 440 Hz, is highlighted in a yellow table cell. 

m =

2^m/12

F_m,n

cents

note name

11

2^11/12

51.91

103.83

207.65

415.30

830.61

1661.22

3322.44

1100

G#/Ab

10

2^5/6

49.00

98.00

196.00

392.00

783.99

1567.98

3135.96

1000

G

9

2^3/4

46.25

92.50

185.00

369.99

739.99

1479.98

2959.96

900

F#/Gb

8

2^2/3

43.65

87.31

174.61

349.23

698.46

1396.91

2793.83

800

F

7

2^7/12

41.20

82.41

164.81

329.63

659.26

1318.51

2637.02

700

E

6

2^1/2

38.89

77.78

155.56

311.13

622.25

1244.51

2489.02

600

D#/Eb

5

2^5/12

36.71

73.42

146.83

293.66

587.33

1174.66

2349.32

500

D

4

2^1/3

34.65

69.30

138.59

277.18

554.37

1108.73

2217.46

400

C#/Db

3

2^1/4

32.70

65.41

130.81

261.63

523.25

1046.50

2093.00

4186.01

300

C

2

2^1/6

30.87

61.74

123.47

246.94

493.88

987.77

1975.53

3951.07

200

B

1

2^1/12

29.14

58.27

116.54

233.08

466.16

932.33

1864.66

3729.31

100

A#/Bb

0

2⁰

27.50

55.00

110.00

220.00

440.00

880.00

1760.00

3520.00

0

A

2ⁿ=

1/16

1/8

1/4

1/2

1

2

4

8

2ⁿ

2^-4

2^-3

2^-2

2^-1

2⁰

2¹

2²

2³

n=

-4

-3

-2

-1

0

1

2

3

The scale of the piano is stretched more at its extremes, where the thickness of the strings is greater in relation to their length, increasing the bending stiffness. The formula applied in the table below modifies the nominal, equal-tempered fundamental frequencies as:

F_s = (F_(m,n) /440)^1.002 · 0.1[log₂ (F_(m,n) /440)³]

where F_s is the stretched frequency and F_(m,n) is the nominal frequency. With a nominal equal-tempered scale where F_R = 440, the above formula is equivalent to:

F_s = (F_(m,n) /440)^1.002 · 0.1[(m + n/12)/440]³]

This formula is intended only as an example, since the actual stretching depends on the characteristics of the strings of each individual piano. (Another article on this site describes the vibrations of stretched wires in more detail. The topic of that article is the tensioning of bicycle spokes, which, as vibrating systems, are essentially identical with piano strings.)

m =

2^m/12

F_s(m,n)

cents (approx.)

note name

11

2^11/12

51.35

103.31

207.28

415.26

831.81

1668.30

3354.63

1100

G#/Ab

10

2^5/6

48.43

97.48

195.61

391.90

785.00

1574.17

3164.52

1000

G

9

2^3/4

45.68

91.97

184.59

369.86

740.83

1485.38

2985.25

900

F#/Gb

8

2^2/3

43.08

86.77

174.20

349.06

699.15

1401.61

2816.21

800

F

7

2^7/12

40.64

81.87

164.38

329.43

659.82

1322.60

2656.81

700

E

6

2^1/2

38.32

77.24

155.12

310.90

622.70

1248.05

2506.50

600

D#/Eb

5

2^5/12

36.14

72.87

146.38

293.41

587.68

1177.73

2364.74

500

D

4

2^1/3

34.08

68.74

138.13

276.91

554.63

1111.38

2231.05

400

C#/Db

3

2^1/4

32.14

64.85

130.34

261.33

523.43

1048.78

2104.96

4237.57

300

C

2

2^1/6

30.31

61.17

122.98

246.62

494.00

989.72

1986.04

3996.96

200

B

1

2^1/12

28.58

57.71

116.05

232.75

466.22

934.00

1873.88

3770.13

100

A#/Bb

0

2⁰

26.95

54.43

109.50

219.65

440.00

881.42

1768.09

3556.27

0

A

2ⁿ=

1/16

1/8

1/4

1/2

1

2

4

8

2ⁿ

2^-4

2^-3

2^-2

2^-1

2⁰

2¹

2²

2³

n=

-4

-3

-2

-1

0

1

2

3

The next article in this series describes some uses of the two-dimensional matrix approach in electronic instruments.

[contact John S. Allen by e-mail]

Last revised 8 September 1997