742edo

← 741edo

742edo

743edo →

Prime factorization	2 × 7 × 53
Step size	1.61725 ¢
Fifth	434\742 (701.887 ¢) (→ 31\53)
Semitones (A1:m2)	70:56 (113.2 ¢ : 90.57 ¢)
Consistency limit	21
Distinct consistency limit	21

Theory

742edo is a very strong 19-limit system and a zeta peak edo, and is distinctly consistent in the 21-odd-limit. As an equal temperament, it tempers out the vishnuzma and the fortune comma in the 5-limit, supporting vishnu and fortune; 2401/2400 in the 7-limit, 9801/9800 in the 11-limit, 4096/4095, 6656/6655, 10648/10647 in the 13-limit, 1701/1700, 2058/2057, 2601/2600, 4914/4913, 5832/5831 in the 17-limit, 2376/2375, 2432/2431, 2926/2925, 3136/3135, 4200/4199, 5776/5775, 5929/5928, 5985/5984, 6860/6859 in the 19-limit.

Prime harmonics

Approximation of prime harmonics in 742edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.000	-0.068	+0.209	-0.093	+0.165	+0.443	+0.166	+0.061	-0.781	+0.611	-0.022
	Relative (%)	+0.0	-4.2	+12.9	-5.7	+10.2	+27.4	+10.3	+3.8	-48.3	+37.8	-1.4
Steps (reduced)		742 (0)	1176 (434)	1723 (239)	2083 (599)	2567 (341)	2746 (520)	3033 (65)	3152 (184)	3356 (388)	3605 (637)	3676 (708)

Subsets and supersets

Since 742 factors into 2 × 7 × 53, 742edo has subset edos 2, 7, 14, 53, 106, and 371, of which 7edo, 14edo and 53edo are very notable. It supports silicon (224 & 518) with 14 periods per octave in the 13-limit, and iodine (159& 583f) with 53 periods per octave in the 17-limit.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3.5	[23 6 -14⟩, [-84 53⟩	[⟨742 1176 1723]]	−0.0157	0.0555	3.43
2.3.5.7	2401/2400, 14348907/14336000, [23 6 -14⟩	[⟨742 1176 1723 2083]]	−0.0035	0.0525	3.24
2.3.5.7.11	2401/2400, 9801/9800, 172032/171875, 1240029/1239040	[⟨742 1176 1723 2083 2567]]	−0.0123	0.0501	3.10
2.3.5.7.11.13	2401/2400, 4096/4095, 6656/6655, 9801/9800, 39366/39325	[⟨742 1176 1723 2083 2567 2746]]	−0.0302	0.0608	3.76
2.3.5.7.11.13.17	1701/1700, 2058/2057, 2401/2400, 2601/2600, 4096/4095, 6656/6655	[⟨742 1176 1723 2083 2567 2746 3033]]	−0.0317	0.0564	3.49
2.3.5.7.11.13.17.19	1701/1700, 2058/2057, 2376/2375, 2401/2400, 2432/2431, 2601/2600, 3213/3211	[⟨742 1176 1723 2083 2567 2746 3033 3152]]	−0.0295	0.0531	3.28
2.3.5.7.11.13.17.19.23	1197/1196, 1496/1495, 1701/1700, 2025/2024, 2058/2057, 2401/2400, 2601/2600, 3213/3211	[⟨742 1176 1723 2083 2567 2746 3033 3152 3357]] (742i)	−0.0468	0.0699	4.32

• 742et has a lower 19-limit relative error than any previous equal temperaments. It is only bettered by 935 in terms of absolute error, and by 1178 in terms of relative error.
• 742et (742i val) is also notable in the 17- and 23-limit, where it has lower absolute errors than any previous equal temperaments. In the 17-limit it beats 581 and is bettered by 764; in the 23-limit it beats 718 and is bettered by 814.

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	137\742	221.563	8388608/7381125	Fortune
1	303\742	490.026	896/675	Surmarvelpyth
2	44\742	71.159	25/24	Vishnu
14	434\742 (10\742)	701.886 (16.173)	3/2 (105/104)	Silicon
53	239\742 (1\742)	386.523 (1.617)	5/4 (32805/32768)	Mercator
53	565\742 (5\742)	913.746 (8.086)	441/260 (196/195)	Iodine

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