# 403edo

Jump to navigation Jump to search
 ← 402edo 403edo 404edo →
Prime factorization 13 × 31
Step size 2.97767¢
Fifth 236\403 (702.73¢)
Semitones (A1:m2) 40:29 (119.1¢ : 86.35¢)
Consistency limit 5
Distinct consistency limit 5

403 equal divisions of the octave (abbreviated 403edo or 403ed2), also called 403-tone equal temperament (403tet) or 403 equal temperament (403et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 403 equal parts of about 2.98 ¢ each. Each step represents a frequency ratio of 21/403, or the 403rd root of 2.

## Theory

403edo is only consistent to the 5-odd-limit, since the error of harmonic 7 is quite large. To start with, the 403def val 403 639 936 1132 1395 1492], the 403df val 403 639 936 1132 1394 1492], and the patent val 403 639 936 1131 1394 1491] are worth considering.

The equal temperament tempers out 1600000/1594323 (amity comma) and [70 0 -31 (31-5, or birds comma) in the 5-limit.

Using the 403d val, it tempers out 4375/4374, 5120/5103, and 6144/6125 in the 7-limit, so that it supports septimal amity, the 152 & 251 temperament. Extending it by the 403def val, it tempers out 540/539, 5632/5625, 6250/6237, and 19712/19683 in the 11-limit, supporting 11-limit amity; and 1575/1573, 1716/1715, 2200/2197, 3584/3575 in the 13-limit. Extending it by the alternative 403df val, 1375/1372, 14641/14580 in the 11-limit; 352/351, 847/845, and 2080/2079 in the 13-limit.

Using the patent val, it tempers out 3136/3125, 2100875/2097152, and 78125000/78121827 in the 7-limit; 3025/3024, 3388/3375, 12005/11979, 14641/14580, 42875/42768, and 131072/130977 in the 11-limit; 2080/2079, 4096/4095, 4225/4224, 6656/6655, and 10648/10647 in the 13-limit.

### Odd harmonics

Approximation of odd harmonics in 403edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) +0.77 +0.78 -1.08 -1.43 -0.45 -0.83 -1.42 -0.74 +0.25 -0.31 +0.01
Relative (%) +26.0 +26.3 -36.4 -48.0 -15.1 -27.7 -47.7 -24.8 +8.5 -10.4 +0.5
Steps
(reduced)
639
(236)
936
(130)
1131
(325)
1277
(68)
1394
(185)
1491
(282)
1574
(365)
1647
(35)
1712
(100)
1770
(158)
1823
(211)

### Subsets and supersets

Since 403 factors into 13 × 31, 403edo contains 13edo and 31edo as subsets.

## Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [639 -403 [403 639]] -0.2443 0.2443 8.20
2.3.5 1600000/1594323, [81 -13 -26 [403 639 936]] -0.2753 0.2042 6.86
2.3.5.7 3136/3125, 1600000/1594323, 2100875/2097152 [403 639 936 1131]] (403) -0.3751 0.2473 8.31
2.3.5.7.11 3025/3024, 3136/3125, 12005/11979, 131072/130977 [403 639 936 1131 1394]] (403) -0.0621 0.3160 10.61
2.3.5.7.11.13 2080/2079, 3025/3024, 3136/3125, 4096/4095, 12005/11979 [403 639 936 1131 1394 1491]] (403) -0.0146 0.3074 10.32
2.3.5.7.11.13.17 595/594, 833/832, 1225/1224, 3025/3024, 3136/3125, 4096/4095 [403 639 936 1131 1394 1491 1647]] (403) +0.0133 0.2927 9.83

### Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 53\403 157.82 36756909/33554432 Hemiegads
1 106\403 315.63 6/5 Egads
1 114\403 339.45 243/200 Amity (403defff)

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