401edo

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401edo

402edo →

Prime factorization	401 (prime)
Step size	2.99252 ¢
Fifth	235\401 (703.242 ¢)
Semitones (A1:m2)	41:28 (122.7 ¢ : 83.79 ¢)
Dual sharp fifth	235\401 (703.242 ¢)
Dual flat fifth	234\401 (700.249 ¢)
Dual major 2nd	68\401 (203.491 ¢)
Consistency limit	3
Distinct consistency limit	3

Theory

401edo is inconsistent to the 5-odd-limit and harmonic 3 is about halfway between its steps. Otherwise, it has a reasonable approximation to harmonics 5, 7, 9, 11, and 13, making it suitable for a 2.9.5.7.11.13 subgroup interpretation.

Using the patent val nonetheless, the equal temperament tempers out 283115520/282475249 and 703125/702464 in the 7-limit; 35156250/35153041, 2097152/2096325, 117649/117612, 226492416/226474325, 9765625/9732096, 42875/42768, 1375/1372, 5632/5625, 15488/15435, 202397184/201768035, 102487/102400 and 805255/802816 in the 11-limit. It provides the optimal patent val for diatessic, the 140 & 261 temperament.

Odd harmonics

Approximation of odd harmonics in 401edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+1.29	-0.28	+0.75	-0.42	-0.69	+0.37	+1.01	-0.22	-1.25	-0.96	+0.15
	Relative (%)	+43.0	-9.3	+25.1	-14.0	-23.2	+12.4	+33.7	-7.3	-41.9	-31.9	+5.2
Steps (reduced)		636 (235)	931 (129)	1126 (324)	1271 (68)	1387 (184)	1484 (281)	1567 (364)	1639 (35)	1703 (99)	1761 (157)	1814 (210)

Subsets and supersets

401edo is the 79th prime edo

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[636 -401⟩	[⟨401 636]]	-0.4060	0.4058	13.56
2.3.5	15625/15552, [107 -66 -1⟩	[⟨401 636 931]]	-0.2307	0.4139	13.83
2.3.5.7	10976/10935, 15625/15552, 67108864/66706983	[⟨401 636 931 1126]]	-0.2400	0.3588	11.99
2.3.5.7.11	2200/2187, 1375/1372, 5632/5625, 102487/102400	[⟨401​ 636 ​931 ​1126 ​1387 ​]]	-0.1518	0.3661	12.23
2.3.5.7.11.13	325/324, 352/351, 625/624, 1375/1372, 3276800/3270267	[⟨401 ​636 ​931 ​1126 ​1387 ​1484]]	-0.1431	0.3348	11.19

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	169\401	505.74	75/56	Diatessic

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

Scales

• Starlingtet11
• Diatessic27

Music

Francium

• Diana Tessa (2023) – diatessic in 403edo tuning