935edo

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935edo

936edo →

Prime factorization	5 × 11 × 17
Step size	1.28342 ¢
Fifth	547\935 (702.032 ¢)
Semitones (A1:m2)	89:70 (114.2 ¢ : 89.84 ¢)
Consistency limit	27
Distinct consistency limit	27

935 equal divisions of the octave (abbreviated 935edo or 935ed2), also called 935-tone equal temperament (935tet) or 935 equal temperament (935et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 935 equal parts of about 1.28 ¢ each. Each step represents a frequency ratio of 2^1/935, or the 935th root of 2.

Theory

935edo is a very strong 23-limit system, and is consistent through to the 27-odd-limit. It does reasonably well in the higher limits, though the sharply tuned 11 and 23 and the flatly tuned 29 and 31 create inconsistencies together, those being 29/22, 29/23, 31/22, 31/23 and their octave complements; it is otherwise consistent to the 39-odd-limit. It is a zeta peak edo.

As an equal temperament, it tempers out the [39 -29 3⟩ (alphatricot comma), [-52 -17 34⟩ (septendecima), and [91 -12 -31⟩ (astro comma) in the 5-limit; 4375/4374 and 52734375/52706752 in the 7-limit; 117649/117612, 151263/151250, 161280/161051 in the 11-limit; 2080/2079, 4096/4095, 4225/4224 in the 13-limit; 2058/2057, 2500/2499, 4914/4913 in the 17-limit; 2432/2431, 3136/3135, 3250/3249, 4200/4199 in the 19-limit; and 2025/2024, 2300/2299, 2646/2645 among others in the 23-limit.

Prime harmonics

Approximation of prime harmonics in 935edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	+0.077	-0.004	+0.158	+0.554	+0.114	+0.285	+0.241	+0.603	-0.272	-0.223
	Relative (%)	+0.0	+6.0	-0.3	+12.3	+43.1	+8.9	+22.2	+18.8	+47.0	-21.2	-17.4
Steps (reduced)		935 (0)	1482 (547)	2171 (301)	2625 (755)	3235 (430)	3460 (655)	3822 (82)	3972 (232)	4230 (490)	4542 (802)	4632 (892)

Approximation of prime harmonics in 935edo (continued)

Harmonic		37	41	43	47	53	59	61	67	71	73	79
Error	Absolute (¢)	+0.207	-0.399	+0.568	+0.590	+0.506	-0.348	-0.307	+0.265	-0.017	-0.624	-0.045
	Relative (%)	+16.1	-31.1	+44.2	+45.9	+39.4	-27.1	-23.9	+20.7	-1.4	-48.6	-3.5
Steps (reduced)		4871 (196)	5009 (334)	5074 (399)	5194 (519)	5356 (681)	5500 (825)	5545 (870)	5672 (62)	5750 (140)	5787 (177)	5894 (284)

Subsets and supersets

Since 935 factors into primes as 5 × 11 × 17, 935edo has subset edos 5, 11, 17, 55, 85, and 187.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[1482 -935⟩	[⟨935 1482]]	−0.0243	0.0243	1.89
2.3.5	[39 -29 3⟩, [-52 -17 34⟩	[⟨935 1482 2171]]	−0.0157	0.0233	1.82
2.3.5.7	4375/4374, 52734375/52706752, [36 -5 0 -10⟩	[⟨935 1482 2171 2625]]	−0.0259	0.0268	2.08
2.3.5.7.11	4375/4374, 117649/117612, 131072/130977, 161280/161051	[⟨935 1482 2171 2625 3235]]	−0.0527	0.0588	4.58
2.3.5.7.11.13	2080/2079, 4096/4095, 4375/4374, 78125/78078, 117649/117612	[⟨935 1482 2171 2625 3235 3460]]	−0.0490	0.0543	4.23
2.3.5.7.11.13.17	2058/2057, 2080/2079, 2500/2499, 4096/4095, 4375/4374, 4914/4913	[⟨935 1482 2171 2625 3235 3460 3822]]	−0.0520	0.0508	3.96
2.3.5.7.11.13.17.19	2058/2057, 2080/2079, 2432/2431, 2500/2499, 3136/3135, 4375/4374, 4914/4913	[⟨935 1482 2171 2625 3235 3460 3822 3972]]	−0.0526	0.0475	3.70
2.3.5.7.11.13.17.19.23	2025/2024, 2058/2057, 2080/2079, 2300/2299, 2432/2431, 2500/2499, 2646/2645, 4375/4374	[⟨935 1482 2171 2625 3235 3460 3822 3972 4230]]	−0.0616	0.0515	4.01

• 935et has lower absolute errors than any previous equal temperaments in the 13-, 17-, 19- and 23-limit. It is the first to beat 764 in the 13-limit, 814 in the 17- and 23-limit, and 742 in the 19-limit, only to be bettered by 954h in all of those subgroups.

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	103\935	132.19	[-38 5 13⟩	Astro
1	339\935	435.08	9/7	Supermajor
1	442\935	567.27	104/75	Alphatrillium
17	194\935 (26\935)	248.98 (33.37)	[-23 5 9 -2⟩ (100352/98415)	Chlorine

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