523edo

← 522edo

523edo

524edo →

Prime factorization	523 (prime)
Step size	2.29446¢
Fifth	306\523 (702.103¢)
Semitones (A1:m2)	50:39 (114.7¢ : 89.48¢)
Consistency limit	15
Distinct consistency limit	15

523 equal divisions of the octave (abbreviated 523edo or 523ed2), also called 523-tone equal temperament (523tet) or 523 equal temperament (523et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 523 equal parts of about 2.29 ¢ each. Each step represents a frequency ratio of 2^1/523, or the 523rd root of 2.

Theory

523et tempers out 2401/2400 in the 7-limit; 56723625/56689952, 26214400/26198073, 78675968/78594219, 759375/758912, 496125/495616, 6250/6237, 42875/42768, 19712/19683, 1479016/1476225, 16808715/16777216, 3294225/3294172, 43923/43904, 102487/102400 and 322102/321489 in the 11-limit.

Prime harmonics

Approximation of prime harmonics in 523edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	absolute (¢)	+0.00	+0.15	-0.85	-0.57	-0.65	-0.76	+0.59	+0.77	+0.41	+0.63	-0.10
	relative (%)	+0	+6	-37	-25	-28	-33	+26	+33	+18	+28	-4
Steps (reduced)		523 (0)	829 (306)	1214 (168)	1468 (422)	1809 (240)	1935 (366)	2138 (46)	2222 (130)	2366 (274)	2541 (449)	2591 (499)

Subsets and supersets

523edo is the 99th prime edo.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[829 -523⟩	[⟨523 829]]	-0.0468	0.0468	2.04
2.3.5	1600000/1594323, [-101 11 36⟩	[⟨523 829 1214]]	+0.0901	0.1974	8.60
2.3.5.7	2401/2400, 1600000/1594323, 341796875/339738624	[⟨523 829 1214 1468]]	+0.1180	0.1776	7.74
2.3.5.7.11	6250/6237, 2401/2400, 19712/19683, 166375/165888	[⟨523 829 1214 1468 1809]]	+0.1319	0.1613	7.03
2.3.5.7.11.13	2080/2079, 625/624, 4459/4455, 19712/19683, 50193/50176	[⟨523 829 1214 1468 1809 1935]]	+0.1440	0.1497	6.52

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	37\523	84.89	21/20	Amicable
1	52\523	119.31	15/14	Septidiasemi
1	148\523	339.58	243/200	Amity

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

Music

Francium

• "(not) stuck" from hope in dark times (2024) – Spotify | Bandcamp | YouTube