31-odd-limit

Odd limits:

---

• 1-odd-limit
• 3-odd-limit
• 5-odd-limit
• 7-odd-limit
• 9-odd-limit
• 11-odd-limit
• 13-odd-limit
• 15-odd-limit
• 17-odd-limit
• 19-odd-limit
• 21-odd-limit
• 23-odd-limit
• 25-odd-limit
• 27-odd-limit
• 29-odd-limit
• 31-odd-limit

The 31-odd-limit is the set of all rational intervals for which neither the numerator nor the denominator of the frequency ratio exceeds 31, once all powers of 2 are removed. To the 29-odd-limit, it adds 15 interval pairs involving 31.

Below is a list of all octave-reduced intervals in the 31-odd-limit.

• 1/1
• 32/31, 31/16
• 31/30, 60/31
• 30/29, 29/15
• 29/28, 56/29
• 28/27, 27/14
• 27/26, 52/27
• 26/25, 25/13
• 25/24, 48/25
• 24/23, 23/12
• 23/22, 44/23
• 22/21, 21/11
• 21/20, 40/21
• 20/19, 19/10
• 19/18, 36/19
• 18/17, 17/9
• 17/16, 32/17
• 16/15, 15/8
• 31/29, 58/31
• 15/14, 28/15
• 29/27, 54/29
• 14/13, 13/7
• 27/25, 50/27
• 13/12, 24/13
• 25/23, 46/25
• 12/11, 11/6
• 23/21, 42/23
• 34/31, 31/17
• 11/10, 20/11
• 32/29, 29/16
• 21/19, 38/21
• 31/28, 56/31
• 10/9, 9/5
• 29/26, 52/29
• 19/17, 34/19
• 28/25, 25/14
• 9/8, 16/9
• 26/23, 23/13
• 17/15, 30/17
• 25/22, 44/25
• 8/7, 7/4
• 31/27, 54/31
• 23/20, 40/23
• 15/13, 26/15
• 22/19, 19/11
• 29/25, 50/29
• 36/31, 31/18
• 7/6, 12/7
• 34/29, 29/17
• 27/23, 46/27
• 20/17, 17/10
• 13/11, 22/13
• 32/27, 27/16
• 19/16, 32/19
• 25/21, 42/25
• 31/26, 52/31
• 6/5, 5/3
• 29/24, 48/29
• 23/19, 38/23
• 17/14, 28/17
• 28/23, 23/14
• 11/9, 18/11
• 38/31, 31/17
• 27/22, 44/27
• 16/13, 13/8
• 21/17, 34/21
• 26/21, 21/13
• 31/25, 50/31
• 36/29, 29/18
• 5/4, 8/5
• 34/27, 27/17
• 29/23, 46/29
• 24/19, 19/12
• 19/15, 30/19
• 14/11, 11/7
• 23/18, 36/23
• 32/25, 25/16
• 9/7, 14/9
• 40/31, 31/20
• 31/24, 48/31
• 22/17, 17/11
• 13/10, 20/13
• 30/23, 23/15
• 17/13, 26/17
• 38/29, 29/19
• 21/16, 32/21
• 25/19, 38/25
• 29/22, 44/29
• 4/3, 3/2
• 31/23, 46/31
• 27/20, 40/27
• 23/17, 34/23
• 42/31, 31/21
• 19/14, 28/19
• 34/25, 25/17
• 15/11, 22/15
• 26/19, 19/13
• 11/8, 16/11
• 40/29, 29/20
• 29/21, 42/29
• 18/13, 13/9
• 25/18, 36/25
• 32/23, 23/16
• 7/5, 10/7
• 38/27, 27/19
• 31/22, 44/31
• 24/17, 17/12

Ratio	Size (¢)	Color name		Name
32/31	54.964	31u2	thiwu 2nd	lesser tricesimoprimal quartertone
31/30	56.767	31og1	thiwogu unison	greater tricesimoprimal quartertone
31/29	115.458	31o29u1	thiwotwenu unison	tricesimoprimal augmented unison
34/31	159.920	31u17o3	thiwuso 3rd	tricesimoprimal diminished third
31/28	176.210	31or1	thiworu unison	tricesimoprimal superaugmented unison
31/27	239.171	31o2	thiwo 2nd	tricesimoprimal supermajor second
36/31	258.874	31u3	thiwu 3rd	tricesimoprimal inframinor third
31/26	304.508	31o3u2	thiwothu 2nd	tricesimoprimal augmented second
38/31	352.477	31u19o4	thiwuno 4th	tricesimoprimal diminished fourth
31/25	372.408	31ogg3	thiwogugu 3rd	tricesimoprimal submajor third
40/31	441.278	31uy4	thiwuyo 4th	tricesimoprimal subminor fourth
31/24	443.081	31o3	thiwo 3rd	tricesimoprimal supermajor third
31/23	516.761	31o23u3	thiwotwethu 3rd	tricesimoprimal augmented third
42/31	525.745	31uz5	thiwuzo 5th	tricesimoprimal diminished fifth
31/22	593.718	31o1u4	thiwolu 4th	lesser tricesimoprimal tritone
44/31	606.282	31u1o5	thiwulo 5th	greater tricesimoprimal tritone
31/21	674.255	31or4	thiworu 4th	tricesimoprimal augmented fourth
46/31	683.239	31u23o6	thiwotwetho 6th	tricesimoprimal subminor sixth
48/31	756.919	31u6	thiwu 6th	tricesimoprimal supermajor sixth
31/20	758.722	31og5	thiwogu 5th	tricesimoprimal supramajor fifth
50/31	827.592	31uyy6	thiwuyoyo 6th	tricesimoprimal supraminor sixth
31/19	847.523	31o19u5	thiwonu 5th	tricesimoprimal augmented fifth
52/31	895.492	31u3o7	thiwutho 7th	tricesimoprimal diminished seventh
31/18	941.126	31o6	thiwo 6th	tricesimoprimal ultramajor sixth
54/31	960.829	31u7	thiwu 7th	tricesimoprimal subminor seventh
56/31	1023.790	31uz8	thiwuzo octave	tricesimoprimal supradiminished octave
31/17	1040.080	31o17u6	thiwosu 6th	tricesimoprimal augmented sixth
58/31	1084.542	31u29o8	thiwutweno octave	tricesimoprimal diminished octave
60/31	1143.233	31uy8	thiwuyo octave	lesser tricesimoprimal infraoctave
31/16	1145.036	31o7	thiwo 7th	greater tricesimoprimal infraoctave

The smallest equal division of the octave which is consistent to the 31-odd-limit is 311edo (by virtue of it being consistent through the 41-odd-limit); that which is distinctly consistent to the same is 1600edo (by virtue of it being distinctly consistent through the 37-odd-limit).