List of just intonation intervals

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Revision as of 03:55, 14 December 2025 by Vector (talk | contribs) (Created page with " This is a list of just ratios, similar to the list of EDOs and the list of regular temperaments. It exists to compile information on a number of just ratios. '''Commas should not redirect here!''' They should instead redirect to their corresponding entry on the temperament page, or if present, a page for the temperament itself. {{Adv|The formula for "good edos" is the edos that satisfy, for interval x in cents and edo n: <br> abs(round(xn/1200)-xn/1200)*...")
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This is a list of just ratios, similar to the list of EDOs and the list of regular temperaments. It exists to compile information on a number of just ratios.

Commas should not redirect here! They should instead redirect to their corresponding entry on the temperament page, or if present, a page for the temperament itself.

The formula for "good edos" is the edos that satisfy, for interval x in cents and edo n:
abs(round(xn/1200)-xn/1200)*sqrt(12/n)<(1/16) for n < or equal to 31
abs(round(xn/1200)-xn/1200)*((n^2)/1024)*sqrt(12/31)<(1/16) for n > 31

EDOs below 5 are excluded.

3-limit

(MOS) diatonic intervals
Fifthspan Ratio Cents Good edos Name Notes
-7 4096/2187 1086.3 10, 11, 21, 22, 31, 32, 42, 53, 74, 95, 190 Diatonic diminished octave
-6 1024/729 588.3 47, 49, 51, 53, 102 Diatonic diminished fifth Found in a diminished triad. A stack of two diatonic minor thirds
-5 256/243 90.2 13, 14, 26, 27, 40, 53, 80, 93, 133, 306 Diatonic semitone Small step of the diatonic scale.
-4 128/81 792.2 6, 44, 47, 50, 53, 56, 103 Diatonic minor sixth
-3 32/27 294.1 8, 12, 33, 37, 41, 45, 49, 53, 57, 102 Diatonic minor third Middle interval in a Pythagorean minor chord
-2 16/9 996.1 6, 12, 18, 24, 29, 30, 35, 41, 47, 53, 59, 100, 153 Diatonic minor seventh
-1 4/3 498 12, 17, 24, 29, 36, 41, 53, 94, 200 Perfect fourth
0 1/1 0 (All) Perfect unison Represents a multiplication by 1, i.e. no change in pitch
+1 3/2 702 12, 17, 24, 29, 36, 41, 53, 94, 200 Perfect fifth Generator of Pythagorean tuning, most consonant interval within the octave after 2/1 itself
+2 9/8 203.9 6, 12, 18, 24, 29, 30, 35, 41, 47, 53, 59, 100, 153 Diatonic major second Large step of the diatonic scale.
+3 27/16 905.9 8, 12, 33, 37, 41, 45, 49, 53, 57, 102 Diatonic major sixth
+4 81/64 407.8 6, 44, 47, 50, 53, 56, 103 Diatonic major third Middle interval in a Pythagorean major chord
+5 243/128 1109.8 13, 14, 26, 27, 40, 53, 80, 93, 133, 306 Diatonic major seventh
+6 729/512 611.7 47, 49, 51, 53, 102 Diatonic augmented fourth Stack of 3 tones (tritone)
+7 2187/2048 113.7 10, 11, 21, 22, 31, 32, 42, 53, 74, 95, 190 Chromatic semitone Chroma of the diatonic scale.

5-limit

Ratio Cents Good edos Name Notes
128/125 41.1 27, 28, 29, 30, 31, 32, 58, 59, 117, 146, 292 augmented diesis
25/24 70.7 16, 17, 18, 33, 34, 35, 51, 68, 85, 102
135/128 92.2 13, 25, 26, 27, 39, 52, 65, 78, 91, 104 major chroma
16/15 111.7 11, 21, 22, 32, 33, 43, 54, 86 classical diatonic semitone
27/25 133.2 9, 18, 27, 36, 45, 54, 63, 72, 81, 90
10/9 182.4 13, 20, 26, 33, 46, 79, 125, 250 grave whole tone
75/64 274.6 13, 22, 26, 31, 35, 48, 70, 83, 118 classical subminor third Also an augmented second.
6/5 315.6 15, 19, 23, 34, 38, 42, 57, 76 classical minor third
5/4 386.3 22, 25, 28, 31, 34, 56, 59, 87, 146, 292 classical major third
32/25 427.4 14, 17, 28, 31, 42, 45, 59, 73, 87, 146, 219, 292 classical supermajor third Also a diminished fourth.
27/20 519.6 7, 14, 16, 23, 30, 37, 44, 60, 67, 97, 194 acute fourth
45/32 590.2 59, 61, 63, 122 classical narrow tritone
64/45 609.8 59, 61, 63, 122 classical wide tritone
40/27 680.4 7, 14, 16, 23, 30, 37, 44, 60, 67, 97, 194 grave fifth
25/16 772.6 14, 17, 28, 31, 42, 45, 59, 73, 87, 146, 219, 292 classical subminor sixth
8/5 813.7 22, 25, 28, 31, 34, 56, 59, 87, 146, 292 classical minor sixth
5/3 884.4 15, 19, 23, 34, 38, 42, 57, 76 classical major sixth
128/75 925.4 13, 22, 26, 31, 35, 48, 70, 83, 118 classical supermajor sixth
9/5 1017.6 13, 20, 26, 33, 46, 79, 125, 250

7-limit

2.3.7

Ratio Cents Good edos Name Notes
49/48 35.7 31, 32, 33, 34, 35, 36, 67, 101, 168 interseptimal diesis
28/27 63 18, 19, 20, 37, 38, 39, 57, 76
8/7 231.2 5, 21, 26, 31, 36, 52, 57, 83, 109, 218 septimal supermajor second
7/6 266.9 9, 18, 27, 36, 45, 54, 63, 72 septimal subminor third
9/7 435.1 11, 22, 25, 33, 36, 44, 47, 58, 69, 80, 91, 171 septimal supermajor third
21/16 470.8 5, 18, 23, 28, 33, 46, 51, 56, 79, 130, 209 septimal subfourth
32/21 729.2 5, 18, 23, 28, 33, 46, 51, 56, 79, 130, 209 septimal superfifth
14/9 764.9 11, 22, 25, 33, 36, 44, 47, 58, 69, 80, 91, 171 septimal subminor sixth
12/7 933.1 9, 18, 27, 36, 45, 54, 63, 72 septimal supermajor sixth
7/4 968.8 5, 21, 26, 31, 36, 52, 57, 83, 109, 218 harmonic seventh
27/14 1137 18, 19, 20, 37, 38, 39, 57, 76

2.3.5.7

Ratio Cents Good edos Name Notes
50/49 35 31, 32, 33, 34, 35, 36, 68, 69, 103, 137, 240 jubilisma
36/35 48.8 23, 24, 25, 26, 49, 50, 74, 123
21/20 84.5 14, 15, 28, 29, 42, 43, 57, 71, 142, 213
15/14 119.4 10, 20, 30, 31, 40, 50, 201 septimal major semitone
35/32 155.1 8, 15, 16, 23, 31, 39, 54, 62, 85, 116, 147, 294 septimal neutral second
28/25 196.2 6, 12, 18, 24, 25, 30, 31, 37, 43, 49, 55, 61, 104, 159, 263
60/49 350.6 7, 17, 24, 31, 34, 41, 48, 65, 89 septimal artoneutral third
49/40 351.3 17, 24, 31, 34, 41, 58, 82 septimal tendoneutral third
63/50 400.1 6, 9, 12, 15, 18, ... 99, 102, 105
35/27 449.3 8, 16, 24, 32, 40, 48, 56, 211 septimal ultramajor third
64/49 462.3 13, 18, 26, 31, 39, 44, 52, 109, 122, 244
48/35 546.8 11, 22, 24, 33, 35, 44, 46, 57, 68, 79, 90 septimal neutral fourth
7/5 582.5 29, 31, 33, 35, 37, 68, 70, 103 septimal narrow tritone
10/7 617.5 29, 31, 33, 35, 37, 68, 70, 103 septimal wide tritone
35/24 653.2 11, 22, 24, 33, 35, 44, 46, 57, 68, 79, 90 septimal neutral fifth
49/32 737.7 13, 18, 26, 31, 39, 44, 52, 109, 122, 244
54/35 750.7 8, 16, 24, 32, 40, 48, 56, 211 septimal inframinor sixth
100/63 799.9 6, 9, 12, 15, 18, ... 99, 102, 105
25/14 1003.8 6, 12, 18, 24, 25, 30, 31, 37, 43, 49, 55, 61, 104, 159, 263
64/35 1044.9 8, 15, 16, 23, 31, 39, 54, 62, 85, 116, 147, 294 septimal neutral seventh
35/18 1151.2 23, 24, 25, 26, 49, 50, 74, 123

11-limit

2.3.5.11

Ratio Cents Good edos Name Notes
11/8 551.3 11, 13, 24, 26, 35, 37, 50, 61, 74, 111
11/9 347.4 7, 14, 21, 24, 31, 38, 45, 76, 114, 152
11/10 165 7, 15, 22, 29, 36, 44, 51, 58, 80, 160, 240
11/6 1049.4 8, 16, 24, 32, 40, 48, 56, 247
12/11 150.6 8, 16, 24, 32, 40, 48, 56, 247
16/11 648.7 11, 13, 24, 26, 35, 37, 50, 61, 74, 111
18/11 852.6 7, 14, 21, 24, 31, 38, 45, 76, 114, 152
20/11 1035 7, 15, 22, 29, 36, 44, 51, 58, 80, 160, 240
27/22 354.5 10, 17, 27, 34, 37, 44, 61, 88, 132
22/15 663 9, 18, 20, 27, 29, 38, 47, 67, 76, 181
15/11 537 9, 18, 20, 27, 29, 38, 47, 67, 76, 181
33/32 53.3 21, 22, 23, 24, 44, 45, 46, 68, 90, 135
33/25 480.6 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55
33/20 867 11, 18, 25, 29, 36, 47, 54, 191
44/27 845.5 10, 17, 27, 34, 37, 44, 61, 88, 132
25/22 221.3 11, 16, 22, 27, 32, 33, 38, 49, 65, 76, 103, 141
44/25 978.7 11, 16, 22, 27, 32, 33, 38, 49, 65, 76, 103, 141
55/54 31.8 35, 36, 37, 38, 39, 40, 75, 76, 113, 151

2.3.5.7.11

Ratio Cents Good edos Name Notes
44/35 396.2 6, 9, 12, 15, 18, 21, 24, 27, 30, 103, 106
66/35 1098.1 12, 23, 24, 35, 36, 47, 59, 106
35/33 101.9 12, 23, 24, 35, 36, 47, 59, 106
33/28 284.4 17, 21, 25, 34, 38, 42, 59, 76, 135, 173
21/11 1119.5 14, 15, 29, 30, 31, 44, 45, 60, 149, 164
22/21 80.5 14, 15, 29, 30, 31, 44, 45, 60, 149, 164
14/11 417.5 20, 23, 26, 29, 43, 46, 69, 92
11/7 782.5 20, 23, 26, 29, 43, 46, 69, 92

13-limit

2.3.5.7.13

Ratio Cents Good edos Name Notes

2.3.5.7.11.13

Ratio Cents Good edos Name Notes

Higher limits

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