509edo

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509edo

510edo →

Prime factorization	509 (prime)
Step size	2.35756 ¢
Fifth	298\509 (702.554 ¢)
Semitones (A1:m2)	50:37 (117.9 ¢ : 87.23 ¢)
Consistency limit	7
Distinct consistency limit	7

Theory

509edo has a sharp tendency in lower harmonics. The 13-limit optimal tuning of this temperament is consistent to the 15-integer-limit, so one might want to keep the octave compression tight.

It tempers out 1600000/1594323 (amity comma) in the 5-limit; 2401/2400 and 29360128/29296875 in the 7-limit; and 3025/3024, 5632/5625, 41503/41472, 42592/42525, 151263/151250, 172032/171875, 180224/180075, 322102/321489, 422576/421875, 456533/455625, and 1953125/1948617 in the 11-limit. It provides the optimal patent val for petrtri, the 2.11/5.13/5 subgroup temperament tempering out 2200/2197.

Odd harmonics

Approximation of odd harmonics in 509edo

Harmonic		3	5	7	9	11	13	15	17	19	21	23
Error	Absolute (¢)	+0.60	+0.33	+0.13	-1.16	+0.35	+1.12	+0.93	+1.13	-0.46	+0.73	-1.16
	Relative (%)	+25.4	+13.9	+5.6	-49.2	+14.9	+47.6	+39.3	+48.1	-19.5	+31.0	-49.3
Steps (reduced)		807 (298)	1182 (164)	1429 (411)	1613 (86)	1761 (234)	1884 (357)	1989 (462)	2081 (45)	2162 (126)	2236 (200)	2302 (266)

Subsets and supersets

509edo is the 97th prime edo.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[807 -509⟩	[⟨509 807]]	−0.1890	0.1889	8.01
2.3.5	[9 -13 5⟩, [93 -3 -38⟩	[⟨509 807 1182]]	−0.1729	0.1559	6.61
2.3.5.7	2401/2400, 1600000/1594323, 29360128/29296875	[⟨509 807 1182 1429]]	−0.1415	0.1456	6.18
2.3.5.7.11	2401/2400, 3025/3024, 5632/5625, 1600000/1594323	[⟨509 807 1182 1429 1761]]	−0.1335	0.1312	5.57
2.3.5.7.11.13	2080/2079, 2200/2197, 2401/2400, 3025/3024, 5632/5625	[⟨509 807 1182 1429 1761 1884]]	−0.1618	0.1354	5.74
2.3.5.7.11.13.17	1225/1224, 2080/2079, 2200/2197, 2401/2400, 2431/2430, 4914/4913	[⟨509 807 1182 1429 1761 1884 2081]]	−0.1784	0.1318	5.59

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	18\509	42.44	40/39	Humorous
1	36\509	84.87	21/20	Amicable
1	115\509	271.12	1024/875	Quasiorwell
1	144\509	339.49	243/200	Amity

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

Music

Francium

• "Modern Legends" from Mysteries (2023) Spotify | Bandcamp | YouTube