32/31

Interval information

Ratio	32/31
Subgroup monzo	2.31 [5 -1⟩
Size in cents	54.96443 ¢
Names	octave-reduced 31st subharmonic, small tricesimoprimal quartertone
Color name	31u2, thiwu 2nd
FJS name	[math]\displaystyle{ \text{m2}_{31} }[/math]
Special properties	superparticular, reduced, reduced subharmonic
Tenney norm (log2 nd)	9.9542
Weil norm (log2 max(n, d))	10
Wilson norm (sopfr(nd))	41
[sound info]
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32/31 is the small tricesimoprimal quartertone measuring about 55 ¢. It differs from 33/32, the undecimal quartertone, by 1024/1023 (approx. 1.69 ¢). It differs from 31/30, another tricesimoprimal quartertone, by 961/960 (approx. 1.80 ¢); they together make 16/15.

Approximation

This interval is approximated well (about 0.4 ¢ off) by 1\22 and extremely closely (about 0.0025 ¢ off) by 6\131.

Edo approximations for 32/31 (54.96 ¢)
≤ 80edo, relative error ≤ 10%

Edo	Step size	Cents (¢)	Absolute error (¢)	Relative error (%)
2	0\2	0.00	-54.96	-9.16
20	1\20	60.00	+5.04	+8.39
21	1\21	57.14	+2.18	+3.81
22	1\22	54.55	-0.42	-0.77
23	1\23	52.17	-2.79	-5.35
24	1\24	50.00	-4.96	-9.93
42	2\42	57.14	+2.18	+7.62
43	2\43	55.81	+0.85	+3.04
44	2\44	54.55	-0.42	-1.54
45	2\45	53.33	-1.63	-6.12
64	3\64	56.25	+1.29	+6.86
65	3\65	55.38	+0.42	+2.28
66	3\66	54.55	-0.42	-2.30
67	3\67	53.73	-1.23	-6.88

Notation

This interval is significant in Helmholtz–Ellis notation as the formal comma to translate a Pythagorean interval to a nearby tricesimoprimal (31-limit) interval. The symbols are adapted from Persian quartertones accidentals invented by Ali-Naqi Vaziri.

Temperaments

This interval can serve as a generator for the escapade temperament (which both 22 and 131edo, among others, support).

See also

• 31/16 – its octave complement
• 31/24 – its fourth complement
• Gallery of just intervals