Frequencies in Pythagorean tuning used for Gagaku instruments


My piece Twelfth Aitake I: Beyond Eleven for the Gagaku Sho was originally written for the modern Sho, which is tuned to 442Hz. When I had to rewrite the piece for the traditional Sho, which is tuned to 430Hz, I tried to find a list of the frequencies in the Pythagorean system on the Web because the differences between an equal temperament and Pythagorean tuning are not negligible:

The opening of Twelfth Aitake I at A4=442Hz in equal temperament:




The opening of the same piece at A4=430Hz in Pythagorean tuning:




To my surprise, I couldn't find such a list, however. So, I had to calculate all the frequencies and made the list myself.

I make the list available here because it may be useful for those composers who would like to write a piece for the traditional Sho.

The following is part of the score file for RTcmix that generated the 430Hz version shown above:

=================================================================

// VOCODE2 takes channel 0 as carrier and channel 1 as modulator.
bus_config("MIX", "in 0", "aux 0 out")
bus_config("WAVETABLE", "aux 1 out")
bus_config("VOCODE2", "aux 0-1 in", "out 0-1")

outSkip = 1.0
inSkip = 0.0
dur = outSkip + 112 + 30
amp = 50.0
ampenv = maketable("linebrk", 1024, 0,12,0.7,200,1,800,1,12,0)

// Using a table of center frequencies:
numfilt = 0 // flag indicating that we're using cftab instead of interval stack
lowcf = 0
spacemult = 1

// A stack of 4ths in Pythagorean tuning with lowcf=143.33, or D3
cftab = maketable("literal", "nonorm", 0,
143.33,
191.11,
254.81,
339.75,
453.00,
603.99,
805.34,
1073.76,
1451.25,
1935.00,
2580.00,
3440.00,
4586.64,
6115.52,
8153.92,
10871.89,
14495.86
)

cartransp = 0.00
bw = 0.0001
resp = 0.02
hipass = 0.00
hpcf = 3000
noise = 0.001
noisubsamp = 8

VOCODE2(outSkip, inSkip, dur, amp*ampenv, numfilt, lowcf, spacemult, cartransp, bw, resp, hipass, hpcf, noise, noisubsamp, pan=0.0, cftab)
VOCODE2(outSkip, inSkip, dur, amp*ampenv, numfilt, lowcf, spacemult, cartransp, bw, resp, hipass, hpcf, noise, noisubsamp, pan=1.0, cftab)

=================================================================

The following is part of the score file for RTcmix that generated the 442Hz version shown above. In this score file, the center frequencies of the filter bank of a channel vocoder instrument are specified by the interval of perfect 4th, or 0.05 in the octave/pitch-class notation, in the equal temperament, and a base pitch, namely, 7.02, or D3.

=================================================================

// VOCODE2 takes channel 0 as carrier and channel 1 as modulator.
bus_config("MIX", "in 0", "aux 0 out")
bus_config("WAVETABLE", "aux 1 out")
bus_config("VOCODE2", "aux 0-1 in", "out 0-1")

outSkip = 1.0
inSkip = 0.0
dur = outSkip + 112 + 30
amp = 50.0
ampenv = maketable("linebrk", 1024, 0,12,0.7,200,1,800,1,12,0)

numfilt = 16
lowcf = 7.02 // D3
interval = 0.05 // perfect 4th in the oct.pc notation
spacemult = cpspch(interval) / cpspch(0)
cartransp = 0.00
bw = 0.0001
resp = 0.02
hipass = 0.00
hpcf = 3000
noise = 0.001
noisubsamp = 8

VOCODE2(outSkip, inSkip, dur, amp*ampenv, numfilt, lowcf, spacemult, cartransp, bw, resp, hipass, hpcf, noise, noisubsamp, pan=0.0)
VOCODE2(outSkip, inSkip, dur, amp*ampenv, numfilt, lowcf, spacemult, cartransp, bw, resp, hipass, hpcf, noise, noisubsamp, pan=1.0)

=================================================================


I'm so grateful to John Gibson for his brilliant design of the VOCODE2 instrument, which accepts not only a base frequency and an interval but also a table of frequencies to specify the center frequencies of its filter bank.

Akira Takaoka
New York
May, 2015


=================================================================

The followings are lists of frequencies in the Pythagorean system:

/* A stack of perfect 5ths based on A4=430Hz Bb8 = 7346.95
Eb8 = 4897.97
Ab7 = 3265.31
Db7 = 2176.88
Gb6 = 1451.25
B5 = 967.50
E5 = 645.00
A4 = 430.00
D4 = 286.67
G3 = 191.11
C3 = 127.41
F2 = 84.94
*/

/* Pythagorean tuning based on A4=430Hz
C2 = 63.70
C3 = 127.41
C4 = 254.81
C5 = 509.62
C6 = 1019.24
C7 = 2038.48
C8 = 4076.96
C9 = 8153.92
C10 = 16307.84

Db2 = 67.11
Db3 = 134.22
Db4 = 268.44
Db5 = 536.88
Db6 = 1073.76
Db7 = 2147.52
Db8 = 4295.04
Db9 = 8590.08
Db10= 17180.16

D2 = 71.67
D3 = 143.33
D4 = 286.67
D5 = 573.33
D6 = 1146.66
D7 = 2293.32
D8 = 4586.64
D9 = 9173.28
D10 = 18346.56

Eb2 = 75.50
Eb3 = 151.00
Eb4 = 302.00
Eb5 = 603.99
Eb6 = 1207.99
Eb7 = 2415.98
Eb8 = 4831.95
Eb9 = 9663.90
Eb10= 19327.81

E2 = 80.63
E3 = 161.25
E4 = 322.50
E5 = 645.00
E6 = 1290.00
E7 = 2580.00
E8 = 5160.00
E9 = 10320.00

F2 = 84.94
F3 = 169.87
F4 = 339.75
F5 = 679.49
F6 = 1358.99
F7 = 2717.97
F8 = 5435.95
F9 = 10871.89

Gb2 = 90.70
Gb3 = 181.41
Gb4 = 362.81
Gb5 = 725.63
Gb6 = 1451.25
Gb7 = 2902.50
Gb8 = 5805.00
Gb9 = 11610.00

G2 = 95.56
G3 = 191.11
G4 = 382.22
G5 = 764.44
G6 = 1528.88
G7 = 3057.76
G8 = 6115.52
G9 = 12231.04

Ab1 = 50.33
Ab2 = 100.67
Ab3 = 201.34
Ab4 = 402.67
Ab5 = 805.34
Ab6 = 1610.68
Ab7 = 3221.36
Ab8 = 6442.72
Ab9 = 12885.44

A1 = 53.75
A2 = 107.50
A3 = 215.00
A4 = 430.00
A5 = 860.00
A6 = 1720.00
A7 = 3440.00
A8 = 6880.00
A9 = 13760.00

Bb1 = 56.62
Bb2 = 113.25
Bb3 = 227.50
Bb4 = 453.00
Bb5 = 905.99
Bb6 = 1811.98
Bb7 = 3623.96
Bb8 = 7247.93
Bb9 = 14495.86

B1 = 60.47
B2 = 120.94
B3 = 241.88
B4 = 483.75
B5 = 967.50
B6 = 1935.00
B7 = 3870.00
B8 = 7740.00
B9 = 15480.00
*/



/* Chromatic scale in Pythagorean tuning based on A4=430Hz
B1 = 60.47
Bb1 = 56.62
A1 = 53.75
Ab1 = 50.33
C2 = 63.70
Db2 = 67.11
D2 = 71.67
Eb2 = 75.50
E2 = 80.63
F2 = 84.94
Gb2 = 90.70
G2 = 95.56
Ab2 = 100.67
A2 = 107.50
Bb2 = 113.25
B2 = 120.94
C3 = 127.41
Db3 = 134.22
D3 = 143.33
Eb3 = 151.00
E3 = 161.25
F3 = 169.87
Gb3 = 181.41
G3 = 191.11
Ab3 = 201.34
A3 = 215.00
Bb3 = 226.50
B3 = 241.88
C4 = 254.81
Db4 = 268.44
D4 = 286.67
Eb4 = 302.00
E4 = 322.50
F4 = 339.75
Gb4 = 362.81
G4 = 382.22
Ab4 = 402.67
A4 = 430.00
Bb4 = 453.00
B4 = 483.75
C5 = 509.62
Db5 = 536.88
D5 = 573.33
Eb5 = 603.99
E5 = 645.00
F5 = 679.49
Gb5 = 725.63
G5 = 764.44
Ab5 = 805.34
A5 = 860.00
Bb5 = 905.99
B5 = 967.50
C6 = 1019.24
Db6 = 1073.76
D6 = 1146.66
Eb6 = 1207.99
E6 = 1290.00
F6 = 1358.99
Gb6 = 1451.25
G6 = 1528.88
Ab6 = 1610.68
A6 = 1720.00
Bb6 = 1811.98
B6 = 1935.00
C7 = 2038.48
Db7 = 2147.52
D7 = 2293.32
Eb7 = 2415.98
E7 = 2580.00
F7 = 2717.97
Gb7 = 2902.50
G7 = 3057.76
Ab7 = 3221.36
A7 = 3440.00
Bb7 = 3623.96
B7 = 3870.00
C8 = 4076.96
Db8 = 4295.04
D8 = 4586.64
Eb8 = 4831.95
E8 = 5160.00
F8 = 5435.95
Gb8 = 5805.00
G8 = 6115.52
Ab8 = 6442.72
A8 = 6880.00
Bb8 = 7247.93
B8 = 7740.00
C9 = 8153.92
Db9 = 8590.08
D9 = 9173.28
Eb9 = 9663.90
E9 = 10320.00
F9 = 10871.89
Gb9 = 11610.00
G9 = 12231.04
Ab9 = 12885.44
A9 = 13760.00
Bb9 = 14495.86
B9 = 15480.00
C10 = 16307.84
Db10= 17180.16
D10 = 18346.56
Eb10= 19327.81
*/




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