810edo

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← 809edo 810edo 811edo →
Prime factorization 2 × 34 × 5
Step size 1.48148¢ 
Fifth 474\810 (702.222¢) (→79\135)
Semitones (A1:m2) 78:60 (115.6¢ : 88.89¢)
Consistency limit 9
Distinct consistency limit 9

810 equal divisions of the octave (abbreviated 810edo or 810ed2), also called 810-tone equal temperament (810tet) or 810 equal temperament (810et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 810 equal parts of about 1.48 ¢ each. Each step represents a frequency ratio of 21/810, or the 810th root of 2.

Theory

810 = 270 × 3, and 810edo has three copies of 270edo in the 13-limit (and the 2.3.5.7.11.13.19 subgroup). It makes for a reasonable 17-, 19- and 23-limit system, and perhaps beyond. It is, however, only consistent to the 9-odd-limit. 11/9, 13/12, 13/9, 13/10, and their octave complements are all mapped inconsistently in this edo.

As an equal temperament, it tempers out 4914/4913 in the 17-limit; and 2024/2023, 2737/2736, and 3520/3519 in the 23-limit. Although it does quite well in these limits, it is way less efficient than 270edo's or 540edo's mappings, as it has greater relative errors (→ #Regular temperament properties). It is therefore a question of whether one thinks these tuning improvements and differently supplied essentially tempered chords are worth the load of all the extra notes.

Prime harmonics

Approximation of prime harmonics in 810edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47
Error Absolute (¢) +0.000 +0.267 +0.353 +0.063 -0.207 -0.528 +0.230 +0.265 -0.126 +0.052 +0.150 +0.508 +0.567 -0.407 -0.321
Relative (%) +0.0 +18.0 +23.8 +4.3 -14.0 -35.6 +15.5 +17.9 -8.5 +3.5 +10.1 +34.3 +38.3 -27.4 -21.7
Steps
(reduced)
810
(0)
1284
(474)
1881
(261)
2274
(654)
2802
(372)
2997
(567)
3311
(71)
3441
(201)
3664
(424)
3935
(695)
4013
(773)
4220
(170)
4340
(290)
4395
(345)
4499
(449)

Subsets and supersets

Since 810 factors into 2 × 34 × 5, 810edo has subset edos 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3.5.7.11.13.17 676/675, 1001/1000, 1716/1715, 3025/3024, 4096/4095, 4914/4913 [810 1284 1881 2274 2802 2997 3311]] -0.0281 0.1025 6.92
2.3.5.7.11.13.17.19 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728, 4914/4913 [810 1284 1881 2274 2802 2997 3311 3441]] -0.0324 0.0966 6.52
2.3.5.7.11.13.17.19.23 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728, 2024/2023, 2737/2736 [810 1284 1881 2274 2802 2997 3311 3441 3664]] -0.0257 0.0930 6.28