# 509edo

 ← 508edo 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

509 equal divisions of the octave (abbreviated 509edo or 509ed2), also called 509-tone equal temperament (509tet) or 509 equal temperament (509et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 509 equal parts of about 2.36 ¢ each. Each step represents a frequency ratio of 21/509, or the 509th root of 2.

## 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.

The equal temperament 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 it is distinct

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