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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