111edo

← 110edo

111edo

112edo →

Prime factorization	3 × 37
Step size	10.8108 ¢
Fifth	65\111 (702.703 ¢)
Semitones (A1:m2)	11:8 (118.9 ¢ : 86.49 ¢)
Consistency limit	21
Distinct consistency limit	15

Theory

111edo is consistent through to the 21-odd-limit, and is the smallest edo uniquely consistent through the 15-odd-limit, marking it as an important higher limit tuning. With harmonics 3 through 19 all tuned sharp, 111edo is somewhat related to 37edo, with which it shares the mappings for 5, 7, 11, and 13.

It is also significant for lower limits, especially in terms of what it tempers out in its patent val; for example, it tempers out 176/175 and gives an excellent optimal patent val for the corresponding 11-limit rank-4 temperament.

In fact in the 7-limit it tempers out 1728/1715, 3136/3125 and 5120/5103, and in the 11-limit, 176/175, 540/539, 1331/1323, 1375/1372, and notably the quartisma.

It is a particularly good tuning for the 11- or 13-limit versions of semisept, the 31 & 80 temperament, and buzzard, the 53 & 58 temperament. Gene Ward Smith's trio in #Music section is in guanyin temperament, the planar temperament tempering out 176/175 and 540/539, for which 111 also provides the optimal patent val.

Prime harmonics

Approximation of prime harmonics in 111edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.75	+2.88	+4.15	+0.03	+2.72	+3.15	+5.19	-1.25	-2.55	+0.91
	Relative (%)	+0.0	+6.9	+26.6	+38.4	+0.3	+25.1	+29.2	+48.0	-11.5	-23.6	+8.4
Steps (reduced)		111 (0)	176 (65)	258 (36)	312 (90)	384 (51)	411 (78)	454 (10)	472 (28)	502 (58)	539 (95)	550 (106)

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[176 -111⟩	[⟨111 176]]	-0.236	0.236	2.18
2.3.5	78732/78125, 67108864/66430125	[⟨111 176 258]]	-0.570	0.510	4.72
2.3.5.7	1728/1715, 3136/3125, 5120/5103	[⟨111 176 258 312]]	-0.797	0.591	5.47
2.3.5.7.11	176/175, 540/539, 1331/1323, 5120/5103	[⟨111 176 258 312 384]]	-0.639	0.615	5.69
2.3.5.7.11.13	176/175, 351/350, 540/539, 676/675, 1331/1323	[⟨111 176 258 312 384 411]]	-0.655	0.562	5.21
2.3.5.7.11.13.17	176/175, 256/255, 351/350, 442/441, 540/539, 715/714	[⟨111 176 258 312 384 411 454]]	-0.672	0.523	4.84
2.3.5.7.11.13.17.19	176/175, 256/255, 286/285, 324/323, 351/350, 400/399, 476/475	[⟨111 176 258 312 384 411 454 472]]	-0.740	0.521	4.83

Rank-2 temperaments

Note: 2.5.7.11.13 subgroup temperaments supported by 37edo are not listed.

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperament
1	11\111	118.92	15/14	Subsedia
1	13\111	140.54	13/12	Quanic
1	14\111	151.35	12/11	Browser
1	16\111	172.97	400/363	Undetrita
1	20\111	216.22	17/15	Tremka
1	23\111	248.65	15/13	Hemikwai
1	31\111	335.14	17/14	Cohemimabila
1	35\111	378.38	56/45	Subpental
1	41\111	443.24	162/125	Sensipent / warrior
1	43\111	464.86	17/13	Semisept
1	44\111	475.68	21/16	Vulture / buzzard
1	46\111	497.30	4/3	Kwai
1	49\111	529.73	19/14	Tuskaloosa
1	55\111	594.59	55/39	Gaster
3	7\111	75.68	24/23	Terture
3	12\111	129.73	14/13	Trimabila
3	13\111	140.54	243/224	Septichrome
3	17\111	183.55	10/9	Mirkat
3	23\111 (14\111)	248.65 (151.35)	231/200 (12/11)	Hemimist
3	46\111 (9\111)	497.30 (97.30)	4/3 (18/17~19/18)	Misty

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Music

Gene Ward Smith

• Trio for SoftSaturn, NebulaSing and TromBonehead (archived 2010) – SoundCloud | details | play – guanyin[22] in 111edo tuning