# Hammond Organ tuning

[Joe Monzo]

A notorious example of an RI (rational intonation) which is not synonymous with just intonation, is the tuning of the Hammond organ. All 12 pitches bear rational relationships, with terms of one or two digits (thus, this tuning is a 100-integer-limit tuning), but which sounds for all intents and purposes like 12-edo.

```Hammond Organ tuning
--------------------

absolute 1/1 = 320 Hz (internal gear only; does not get heard)

"absolute ratio" is the list of ratios of the notes to the
unheard 320 Hz

"relative ratio" fixes each one of the 12 notes as 1/1 in turn
and measures all others from that.

"c" marks ratios which appear in the table of
"Some rational convergents to 12-edo degrees".

the maximum error from true 12-edo is just over __ cent.

Hammond Organ tuning

unheard 1/1 = 320 Hz

absolute ratios from unheard 1/1 :

-------------------------------- monzo ------------------------------
2  3,  5  7 11, 13 17 19, 23 29 31, 37 41 43, 47 53 59, 61 67 71, 73
x  x   x  x  x   x  x      x         x  x                   x  x   x
85/52    [-2  0,  1  0  0, -1  1 >
54/35    [ 1  3, -1 -1  0, >
67/46    [-1  0,  0  0  0,  0  0  0, -1  0  0,  0  0  0,  0  0  0,  0  1 >
11/8     [-3  0,  0  0  1, >
48/37    [ 4  1,  0  0  0,  0  0  0,  0  0  0, -1 >
49/40    [-3  0, -1  2 >
37/32    [-5  0,  0  0  0,  0  0  0,  0  0  0,  1 >
12/11    [ 2  1,  0  0 -1, >
69/67    [ 0  1,  0  0  0,  0  0  0,  1  0  0,  0  0  0,  0  0  0,  0 -1 >
35/36    [-2 -2,  1  1 >
67/73    [ 0  0   0  0  0,  0  0  0,  0  0  0,  0  0  0,  0  0  0,  0  1  0, -1 >
71/82    [-1  0,  0  0  0,  0  0  0,  0  0  0,  0 -1  0,  0  0  0,  0  0  1, >

relative ratios from each note as tonic:

absolute    relative                   ~cents error
note    Hz         ratio      ratio         ~cents       from 12-edo

C
C   523.0769231   85 / 52      1 / 1          0            0
B   493.7142857   54 / 35   2808 / 2975    1099.983921     0
Bb  466.0869565   67 / 46   1742 / 1955    1000.29122    + 2/7
A   440           11 / 8     143 / 170     900.5764808   + 4/7
G#  415.1351351   48 / 37   2496 / 3145    799.8695005   - 1/8
G   392           49 / 40    637 / 850     700.5966375   + 3/5
F#  370           37 / 32    481 / 680     600.6025772   + 3/5
F   349.0909091   12 / 11    624 / 935     499.8955969   - 1/9
E   329.5522388   69 / 67   3588 / 5695    400.180858    + 1/6
Eb  311.1111111   35 / 36     91 / 153     300.488157    + 1/2
D   293.6986301   67 / 73   3484 / 6205    200.7760963   + 7/9
C#  277.0731707   71 / 82   1846 / 3485    99.89267627   - 1/9

B
C   523.0769231   85 / 52   2975 / 2808
B   493.7142857   54 / 35      1 / 1
Bb  466.0869565   67 / 46   2345 / 2484
A   440           11 / 8     385 / 432
G#  415.1351351   48 / 37    280 / 333
G   392           49 / 40    343 / 432
F#  370           37 / 32   1295 / 1728
F   349.0909091   12 / 11     70 / 99
E   329.5522388   69 / 67    805 / 1206
Eb  311.1111111   35 / 36   1225 / 1944
D   293.6986301   67 / 73   2345 / 3942
C#  277.0731707   71 / 82   2485 / 4428

Bb
C   523.0769231   85 / 52   1955 / 1742
B   493.7142857   54 / 35   2484 / 2345
Bb  466.0869565   67 / 46      1 / 1
A   440           11 / 8     253 / 268
G#  415.1351351   48 / 37   2208 / 2479
G   392           49 / 40   1127 / 1340
F#  370           37 / 32    851 / 1072
F   349.0909091   12 / 11    552 / 737
E   329.5522388   69 / 67   3174 / 4489
Eb  311.1111111   35 / 36    805 / 1206
D   293.6986301   67 / 73     46 / 73
C#  277.0731707   71 / 82   1633 / 2747

A                                               2  3,  5  7 11, 13 17 19, 23 29 31, 37 41 43, 47 53 59, 61 67 71, 73-monzo
x  x   x  x  x   x  x      x         x  x                   x  x   x
C   523.0769231   85 / 52    170 / 143    [ 1  0,  1  0 -1, -1  1 >
B   493.7142857   54 / 35    432 / 385    [ 4  3, -1 -1 -1, >
Bb  466.0869565   67 / 46    268 / 253    [ 2  0,  0  0 -1,  0  0  0, -1  0  0,  0  0  0,  0  0  0,  0  1 >
A   440           11 / 8       1 / 1      [ 0 >
G#  415.1351351   48 / 37    384 / 407    [ 7  1,  0  0 -1,  0  0  0,  0  0  0, -1 >
G   392           49 / 40     49 / 55     [ 0  0, -1  2 -1, >
F#  370           37 / 32     37 / 44     [-2  0,  0  0 -1,  0  0  0,  0  0  0,  1 >
F   349.0909091   12 / 11     96 / 121    [ 5  1,  0  0 -2, >
E   329.5522388   69 / 67    552 / 737    [ 3  1,  0  0 -1,  0  0  0,  1  0  0,  0  0  0,  0  0  0,  0 -1 >
Eb  311.1111111   35 / 36     70 / 99     [ 1 -2,  1  1 -1, >
D   293.6986301   67 / 73    536 / 803    [ 3  0,  0  0 -1,  0  0  0,  0  0  0,  0  0  0,  0  0  0,  0  1  0, -1 >
C#  277.0731707   71 / 82    284 / 451    [ 2  0,  0  0 -1,  0  0  0,  0  0  0,  0 -1  0,  0  0  0,  0  0  1, >

G#
C   523.0769231   85 / 52   3145 / 2496
B   493.7142857   54 / 35    333 / 280
Bb  466.0869565   67 / 46   2479 / 2208
A   440           11 / 8     407 / 384
G#  415.1351351   48 / 37      1 / 1
G   392           49 / 40   1813 / 1920
F#  370           37 / 32   1369 / 1536
F   349.0909091   12 / 11     37 / 44
E   329.5522388   69 / 67    851 / 1072
Eb  311.1111111   35 / 36   1295 / 1728
D   293.6986301   67 / 73   2479 / 3504
C#  277.0731707   71 / 82   2627 / 3936

G
C   523.0769231   85 / 52    850 / 637
B   493.7142857   54 / 35    432 / 343
Bb  466.0869565   67 / 46   1340 / 1127
A   440           11 / 8      55 / 49
G#  415.1351351   48 / 37   1920 / 1813
G   392           49 / 40      1 / 1
F#  370           37 / 32    185 / 196
F   349.0909091   12 / 11    480 / 539
E   329.5522388   69 / 67   2760 / 3283
Eb  311.1111111   35 / 36     50 / 63
D   293.6986301   67 / 73   2680 / 3577
C#  277.0731707   71 / 82   1420 / 2009

F#
C   523.0769231   85 / 52    680 / 481
B   493.7142857   54 / 35   1728 / 1295
Bb  466.0869565   67 / 46   1072 / 851
A   440           11 / 8      44 / 37
G#  415.1351351   48 / 37   1536 / 1369
G   392           49 / 40    196 / 185
F#  370           37 / 32      1 / 1
F   349.0909091   12 / 11    384 / 407
E   329.5522388   69 / 67   2208 / 2479
Eb  311.1111111   35 / 36    280 / 333
D   293.6986301   67 / 73   2144 / 2701
C#  277.0731707   71 / 82   1136 / 1517

F
C   523.0769231   85 / 52    935 / 624
B   493.7142857   54 / 35     99 / 70
Bb  466.0869565   67 / 46    737 / 552
A   440           11 / 8     121 / 96
G#  415.1351351   48 / 37     44 / 37
G   392           49 / 40    539 / 480
F#  370           37 / 32    407 / 384
F   349.0909091   12 / 11      1 / 1
E   329.5522388   69 / 67    253 / 268
Eb  311.1111111   35 / 36    385 / 432
D   293.6986301   67 / 73    737 / 876
C#  277.0731707   71 / 82    781 / 984

E
C   523.0769231   85 / 52   5695 / 3588
B   493.7142857   54 / 35   1206 / 805
Bb  466.0869565   67 / 46   4489 / 3174
A   440           11 / 8     737 / 552
G#  415.1351351   48 / 37   1072 / 851
G   392           49 / 40   3283 / 2760
F#  370           37 / 32   2479 / 2208
F   349.0909091   12 / 11    268 / 253
E   329.5522388   69 / 67      1 / 1
Eb  311.1111111   35 / 36   2345 / 2484
D   293.6986301   67 / 73   4489 / 5037
C#  277.0731707   71 / 82   4757 / 5658

Eb
C   523.0769231   85 / 52    153 / 91
B   493.7142857   54 / 35   1944 / 1225
Bb  466.0869565   67 / 46   1206 / 805
A   440           11 / 8      99 / 70
G#  415.1351351   48 / 37   1728 / 1295
G   392           49 / 40     63 / 50
F#  370           37 / 32    333 / 280
F   349.0909091   12 / 11    432 / 385
E   329.5522388   69 / 67   2484 / 2345
Eb  311.1111111   35 / 36      1 / 1
D   293.6986301   67 / 73   2412 / 2555
C#  277.0731707   71 / 82   1278 / 1435

D
C   523.0769231   85 / 52   6205 / 3484
B   493.7142857   54 / 35   3942 / 2345
Bb  466.0869565   67 / 46     73 / 46
A   440           11 / 8     803 / 536
G#  415.1351351   48 / 37   3504 / 2479
G   392           49 / 40   3577 / 2680
F#  370           37 / 32   2701 / 2144
F   349.0909091   12 / 11    876 / 737
E   329.5522388   69 / 67   5037 / 4489
Eb  311.1111111   35 / 36   2555 / 2412
D   293.6986301   67 / 73      1 / 1
C#  277.0731707   71 / 82   5183 / 5494

C#
C   523.0769231   85 / 52   3485 / 1846
B   493.7142857   54 / 35   4428 / 2485
Bb  466.0869565   67 / 46   2747 / 1633
A   440           11 / 8     451 / 284
G#  415.1351351   48 / 37   3936 / 2627
G   392           49 / 40   2009 / 1420
F#  370           37 / 32   1517 / 1136
F   349.0909091   12 / 11    984 / 781
E   329.5522388   69 / 67   5658 / 4757
Eb  311.1111111   35 / 36   1435 / 1278
D   293.6986301   67 / 73   5494 / 5183
C#  277.0731707   71 / 82      1 / 1
```
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