345edo

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345edo

346edo →

Prime factorization	3 × 5 × 23
Step size	3.47826 ¢
Fifth	202\345 (702.609 ¢)
Semitones (A1:m2)	34:25 (118.3 ¢ : 86.96 ¢)
Consistency limit	5
Distinct consistency limit	5

Theory

345et is only consistent to the 5-limit. Using the patent val, it tempers out 2460375/2458624, 134217728/133984375, 5120/5103, 95703125/95551488 and 359661568/358722675 in the 7-limit; 56723625/56689952, 20155392/20131375, 14348907/14348180, 1835008/1830125, 46656/46585, 2734375/2725888, 2657205/2656192, 16384/16335, 514714375/514434888, 540/539, 9765625/9732096, 369140625/369098752, 200704/200475, 1375/1372, 537109375/536870912, 19712/19683, 968000/964467, 3025/3024, 532400/531441, 202397184/201768035 and 20614528/20588575 in the 11-limit. It provides the optimal patent val for quartonic and kwai.

Odd harmonics

Approximation of odd harmonics in 345edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.65	-0.23	+1.61	+1.31	+1.73	+1.21	+0.43	-0.61	+1.62	-1.22	+1.29
	Relative (%)	+18.8	-6.5	+46.3	+37.6	+49.6	+34.8	+12.3	-17.5	+46.5	-35.0	+37.1
Steps (reduced)		547 (202)	801 (111)	969 (279)	1094 (59)	1194 (159)	1277 (242)	1348 (313)	1410 (30)	1466 (86)	1515 (135)	1561 (181)

Subsets and supersets

345 factors into 3 × 5 × 23, with subset edos 3, 5, 15, 23, 69, and 115.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[547 -345⟩	[⟨345 547]]	-0.2062	0.2062	5.93
2.3.5	[3 -18 11⟩, [47 -15 -10⟩	[⟨345 547 801]]	-0.1050	0.2210	6.35
2.3.5.7	5120/5103, 16875/16807, 95703125/95551488	[⟨345 547 801 969]]	-0.2220	0.2788	8.02
2.3.5.7.11	540/539, 3025/3024, 5120/5103, 26796875/26763264	[⟨345 547 801 969 1194]]	-0.2773	0.2728	7.84
2.3.5.7.11.13	540/539, 625/624, 3025/3024, 5120/5103, 4225/4224	[⟨345 547 801 969 1194 1277]]	-0.2857	0.2497	7.18

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator (reduced)*	Cents (reduced)*	Associated Ratio*	Temperaments
1	38\345	132.17	[-38 5 13⟩	Astro
1	143\345	497.39	4/3	Kwai
5	106\345 (32\345)	368.70 (111.30)	1024/891 (16/15)	Qintosec

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