# 503edo

 ← 502edo 503edo 504edo →
Prime factorization 503 (prime)
Step size 2.38569¢
Fifth 294\503 (701.392¢)
Semitones (A1:m2) 46:39 (109.7¢ : 93.04¢)
Consistency limit 7
Distinct consistency limit 7

503 equal divisions of the octave (abbreviated 503edo or 503ed2), also called 503-tone equal temperament (503tet) or 503 equal temperament (503et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 503 equal parts of about 2.39 ¢ each. Each step represents a frequency ratio of 21/503, or the 503rd root of 2.

## Theory

503et tempers out 2401/2400 in the 7-limit; 56723625/56689952, 100663296/100656875, 10333575/10307264, 46656/46585, 166698/166375, 759375/758912, 496125/495616, 226492416/226474325, 2359296/2358125, 172032/171875, 369140625/369098752, 4302592/4296875, 24057/24010, 8019/8000, 16808715/16777216, 766656/765625, 391314/390625, 3294225/3294172, 43923/43904 and 102487/102400 in the 11-limit.

### Prime harmonics

Approximation of prime harmonics in 503edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.56 +0.17 -0.24 -0.22 -0.77 +0.01 +0.70 -0.84 +1.04 +0.09
Relative (%) +0.0 -23.6 +7.0 -10.0 -9.4 -32.1 +0.6 +29.2 -35.2 +43.6 +3.9
Steps
(reduced)
503
(0)
797
(294)
1168
(162)
1412
(406)
1740
(231)
1861
(352)
2056
(44)
2137
(125)
2275
(263)
2444
(432)
2492
(480)

### Subsets and supersets

503edo is the 96th prime edo.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-797 503 503 797] 0.1777 0.1777 7.45
2.3.5 [38 -2 -15, [-5 31 -19 503 797 1168] 0.0944 0.1869 7.83
2.3.5.7 2401/2400, 1959552/1953125, 26873856/26796875 503 797 1168 1412] 0.0920 0.1619 6.79
2.3.5.7.11 2401/2400, 8019/8000, 24057/24010, 43923/43904 503 797 1168 1412 1740] 0.0866 0.1452 6.09
2.3.5.7.11.13 729/728, 1001/1000, 1716/1715, 4225/4224, 6656/6655 503 797 1168 1412 1740 1861] 0.1066 0.1400 5.87
2.3.5.7.11.13.17 729/728, 936/935, 1001/1000, 1716/1715, 2401/2400, 2601/2600 503 797 1168 1412 1740 1861 2056] 0.0909 0.1352 5.67

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 11\503 26.24 1594323/1562500 Sfourth (5-limit)
1 50\503 119.28 15/14 Septidiasemi
1 81\503 193.24 262144/234375 Luna
1 127\503 302.98 25/21 Quinmite
1 146\503 348.31 57344/46875 Subneutral

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

Francium