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Some applications of the two-dimensional tuning matrix

John S. Allen

Gear trains | Just intonation | Tuning by fourths and fifths

In the previous article, equal temperament was defined as a two-dimensional matrix with one dimension representing octaves and the other, semitones. The two-dimensional matrix is widely used in electronic instruments, and can make it possible to define nonstandard tunings.

Gear trains in electronic organs and tuning aids

The two-dimensional matrix approach, as described in the previous article, is common in electronic organs and synthesizers: pitches are established for one octave, and frequency multipliers or dividers transpose them to other octaves.

The mathematical series underlying the two-dimensional matrix approach,

2nFm = 2(m/k) + n FR (mL = 0, mH = k - 1)

generates a series of pitches in each octave, whose frequencies are in the order as the values of m.

A very accurate approximation of the equal-tempered scale my be generated by alternation of two gear ratios, shown below.


Gear ratio Decimal equivalent Ratio relative to 21/12
107/101 1.059406 0.999946
21/12 1.059463 1.000000
89/84 1.059524 1.000057

As shown in the table above, 107/101 is smaller than 21/12 by 54 parts in 106, and 89/84 is greater than 21/12 by 57 parts in 106. Alternating these two ratios, as in the table below, minimizes the accumulation of error. The greatest relative error between any two degrees of the resulting scale is approximately 70 parts in 106, or 0.12 cent (.0012 semitone), compared with mathematically exact equal temperament. This approximation is more accurate than can be achieved in tuning by ear, or even by many digital frequency dividers, -- the gear train has the advantage that it can use multiplication as well as division.

The table below compares a 12-tone gear train temperament with exact equal temperament. The gear train runs from higher frequencies (at the bottom of the table) to lower frequencies, so the stress on the gears is as equal as possible in spite of friction.


Note name Ratio relative to next higher semitone Exact equal tempera-
ment, 2n/12
Gear train frequency Equal tempera-
ment frequency
Gear train/ equal tempera-
ment
Error in cents

A#/Bb

101/107

0.529732

233.09

233.08

1.000037

0.064

B

84/89

0.561231

246.94

246.94

0.999983

-0.029

C

101/107

0.594604

261.64

261.63

1.000041

0.070

C#/Db

84/89

0.629961

277.18

277.18

0.999987

-0.023

D

101/107

0.667420

293.68

293.66

1.000044

0.076

D#/Eb

84/89

0.707107

311.12

311.13

0.999990

-0.017

E

101/107

0.749154

329.64

329.63

1.000047

0.082

F

84/89

0.793701

349.23

349.23

0.999993

-0.012

F#/Gb

101/107

0.840896

370.01

369.99

1.000051

0.088

G

84/89

0.890899

391.99

392.00

0.999997

-0.006

G#/Ab

101/107

0.943874

415.33

415.30

1.000054

0.093

A

1

1.000000

440.00

440.00

1.000000

0.000


Driving the gears using a synchronous electric clock motor furthers assures tuning accuracy. A gear ratio of 11/10 sets the frequency of A to 440 Hz. when the power line frequency is 50 Hz. A gear ratio of 22/15 is used when the power line frequency is 60 Hz. (This same 22/15 ratio has another common, though surprising application -- it is the ratio between feet per second and miles per hour in the English system of measurement.)

This arrangement uses a long series of gear ratios, with potential problems of high loads on the earlier gears in the series, and additive frequency jitter due to gear backlash and loose bearings. However, early versions of the Conn Strobotuner, which used gears to drive its strobe wheels, may have used a gear system somewhat like this.

The Hammond organ's electromechanical tone generator uses different gearing, shown in the table below for the same octave. (A detailed theory of operation of the Hammond is available on another Web site). The gearing is chosen for reasonably accurate ratios, avoiding the need for compound gearing. The A of the Hammond is exactly 440 Hz, but the other pitches differ from those of equal temperament by as much as -0.7 cent and +0.2 cent. The average pitch of the Hammond's 12 gear ratios is 0.34 cents below A440 pitch, corresponding to A439.92.

The deviation of the Hammond's tuning from exact equal temperament is sufficient that irregularities in the progression of beats of fourths and fifths would be noticeable, if these beats were not avoided through a some distinctive features of the Hammond. Its tone wheels (except those for bass pedal notes on some Hammonds, whose beats are very slow) and electrical filtering generate essentially pure sine waves. All partials which are nominally on the same note,, including the 3rd, 5th and 6th, which normally would differer from the tempered pitches, are generated by the same tone wheels, and and so are at the same frequency and phase, eliminating beating and assuriing that all partials at the same pitch will add rather than cancel. The inharmonic, tempered intonation of the upper partials and the absence of beating are characteristic of the Hammond's distinctive sound.


Note name Gear ratio Oct- ave multi- plier Fre-
quency
A440 E.T. Fre-
quency
Gear train/ E.T. Error, cents Error, cents re: aver- age Error re: fourth below Error re: min. 3rd below Error re: maj. 3rd below
A#/Bb 67/46 16 233.04 233.08 0.999835 -0.285 -0.057 -0.396 -0.305 -0.311
B 54/35 16 246.86 246.94 0.999658 -0.593 -0.250 -0.619 0.114 -0.613
C 85/52 16 261.54 261.63 0.999667 -0.576 -0.234 -0.597 -0.576 0.130
C#/Db 71/82 32 277.07 277.18 0.999605 -0.684 -0.342 0.023 -0.399 -0.684
D 67/73 32 293.70 293.66 1.000115 0.200 0.542 0.200 0.792 0.485
D#/Eb 35/36 32 311.11 311.13 0.999949 -0.088 0.254 0.197 0.488 0.504
E 69/67 32 329.55 329.63 0.999772 -0.396 -0.053 0.197 0.288 0.181
F 12/11 32 349.09 349.23 0.999607 -0.681 -0.339 -0.104 -0.880 0.003
F#/Gb 37/32 32 370.00 369.99 1.000015 0.026 0.368 0.710 0.114 -0.174
G 49/40 32 392.00 392.00 1.000012 0.020 0.362 -0.179 0.416 0.108
G#/Ab 48/37 32 415.14 415.30 0.999592 -0.707 -0.365 -0.619 -0.026 -0.311
A 11/8 32 440.00 440.00 1.000000 0.000 0.342 0.396 -0.026 0.681

Before the digital era, most electronic organs other than the Hammond used free-running oscillators in the top octave, with frequency dividers to generate the pitches of the lower octaves. The free-running oscillators were subject to frequency drift, and required occasional tuning. Today's digital synthesizers use digital frequency synthesis to avoid the need for tuning. With clock frequencies below 10 MHz, the accuracy of the temperament is often not as good with the Hammond organ, since only division of the clock frequency, not multiplication, is possible in a digital computer. Some synthesizers have overcome this problem with "frequency dithering," using unequal numbers of clock cycles that average out to the desired musical frequency. This approach improves frequency accuracy, but the digital output has significant non-harmonic frequency content.unless it is processed through an analog smoothing circuit, typically a phase locked loop.

The matrix and just intonation

The two-dimensional matrix may describe various systems of just intonation, in which all pitches are related to each other through integer ratios. To accommodate just intonation on a conventional keyboard, we may modify the formula from the previous article,

2nFm = 2(m/k) + n FR

which is

2nFm = 2n 2(m/k)FR

to

2nFx = 2n x(k)FR

where x(k) is an arbitrary multiplier, different for each value of the integer variable k. Commonly, k has 12 values, one for each of the 12 keys per octave of the conventional keyboard. The 12 independent values of x may, for example, correspond to 12 degrees of a scale in just intonation. Many electronic organs and synthesizers make this possible through adjustment of the frequency of the master oscillator of each octave divider chain. The table below shows a scale in just intonation in A major, unstretched as would be typical of an electronic organ or other instrument with octave frequency dividers and harmonic partials.


x= x as ratio Fn,x               cents note name

1.8750

15/8

51.56

103.13

206.25

412.50

825.00

1650.00

3300.00

6600.00

1088.27

G#Ab

1.7778

16/9

48.89

97.78

195.56

391.11

782.22

1564.44

3128.89

6257.78

996.09

G

1.6667

5/3

45.83

91.67

183.33

366.67

733.33

1466.67

2933.33

5866.67

884.36

F#/Gb

1.6000

8/5

44.00

88.00

176.00

352.00

704.00

1408.00

2816.00

5632.00

813.69

F

1.5000

3/2

41.25

82.50

165.00

330.00

660.00

1320.00

2640.00

5280.00

701.96

E

1.4063

45/32

38.67

77.34

154.69

309.38

618.75

1237.50

2475.00

4950.00

590.22

D#/Eb

1.3333

4/3

36.67

73.33

146.67

293.33

586.67

1173.33

2346.67

4693.33

498.04

D

1.2500

5/4

34.38

68.75

137.50

275.00

550.00

1100.00

2200.00

4400.00

386.31

C#/Db

1.2000

6/5

33.00

66.00

132.00

264.00

528.00

1056.00

2112.00

4224.00

315.64

C

1.1250

9/8

30.94

61.88

123.75

247.50

495.00

990.00

1980.00

3960.00

203.91

B

1.0667

16/15

29.33

58.67

117.33

234.67

469.33

938.67

1877.33

3754.67

111.73

A#/Bb

1.0000

1

27.50

55.00

110.00

220.00

440.00

880.00

1760.00

3520.00

0.00

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    

Just intonation is typically applied to diatonic scales of 7 tones. Even a 12-tone keyboard, however, does not have enough keys to allow a wide range of harmonic modulation. There are a few perfectly consonant intervals, but many of the others are seriously out of tune. This is the primary justification for equal temperament as well as for the extended systems of just intonation such as the 43-tone system of Harry Partch. These extended systems allow modulation and transposition by providing all of the pitches needed for consonance in to various key signatures. Strictly speaking, however, such tunings are multidimensional rather than only two-dimensional, as we shall see in connection with a forthcoming article.

The matrix by fourths and fifths

Another application of the two-dimensional matrix is to generate the pitches of a tempered octave in a non-monotonic sequence. We may change our formula to

2nFm = 2 (pm/k- q) + n FR (mL = 0, mH = k - 1, 0 < p < k)

where, in addition to the definitions already stated,

p is an integer constant less than k,

p and k do not have a common factor, and

q is an integer variable.

Values of p which have a common factor with k must be excluded, as they do not generate all of the pitches of the temperament. In conventional 12-tone equal temperament, p may only be 1, 5, 7 or 11, since all other integers between 1 and 11 have a common factor with 12. For example, if p is 3, only the four pitches of the diminished chord will be generated.

Taking p as 7 and k as 12 defines the 12 musical fifths in equal temperament as equal to 7 octaves. Values of p = 5 or p = 7 correspond to the usual approach in tuning a 12-tone equal-tempered keyboard instrument by ear, since:

27/12 is 1.4983+, a close approximation to a 3/2 ratio, the musical interval of the perfect fifth, and

25/12 is 1.3348+, a close approximation to a 4/3 ratio, the musical interval of the perfect fourth.

The coincidence that values of p which approximate these ratios do not have a common factor with 12 is central to the harmonic structure of Western music, with key signatures progressing through fourths and fifths. This coincidence also allows equal temperament to be established by ear by adjusting the rate of beats between the overtones of pitches a fourth or fifth apart, as described in standard texts on piano and organ tuning. (With different beat rates, the same tuning approach may also be applied to other scales tuned in fourths and fifths, for example, those in the Fibonacci series described in an earlier article on this site.)

In order to keep all of the pitches within the range of a single octave in a sequence of fourths and fifths, we may require that whenever pm/k >= 1 or pm/k < 0, q is the integer power of two which restores (pm/k) - q to a value between 0 and 1.

The following table describes an unstretched twelve-tone equal temperament as a sequence of fourths and fifths within each octave. The progression through fourths and fifths places the sharps and flats all together in the middle of the table.


m = 7m 2(7m/12)-q Fm,n               cents note name

11

77

25/12

36.71

73.42

146.83

293.66

587.33

1174.66

2349.32

4698.64

500

D

10

70

25/6

49.00

98.00

196.00

392.00

783.99

1567.98

3135.96

6271.93

1000

G

9

63

21/4

32.70

65.41

130.81

261.63

523.25

1046.50

2093.00

4186.01

300

C

8

56

22/3

43.65

87.31

174.61

349.23

698.46

1396.91

2793.83

5587.65

800

F

7

49

21/12

29.14

58.27

116.54

233.08

466.16

932.33

1864.66

3729.31

100

A#/Bb

6

42

21/2

38.89

77.78

155.56

311.13

622.25

1244.51

2489.02

4978.03

600

D#/Eb

5

35

211/12

51.91

103.83

207.65

415.30

830.61

1661.22

3322.44

6644.88

1100

G#/Ab

4

28

21/3

34.65

69.30

138.59

277.18

554.37

1108.73

2217.46

4434.92

400

C#/Db

3

21

23/4

46.25

92.50

185.00

369.99

739.99

1479.98

2959.96

5919.91

900

F#/Gb

2

14

21/6

30.87

61.74

123.47

246.94

493.88

987.77

1975.53

3951.07

200

B

1

7

27/12

41.20

82.41

164.81

329.63

659.26

1318.51

2637.02

5274.04

700

E

0

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    

This article has shown several applications of the two-dimensional matrix. The next article will extend the concept of the two-dimensional matrix with pitches out of sequence.

[Top: John S. Allen's Home Page]
[Up: Mathematical representations of tunings]

[Previous: Defining octaves separately]
[Next: The permutation lattice]

[contact John S. Allen by e-mail]

Contents 1997 John S. Allen

Last revised 28 September 2003