465edo

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← 464edo465edo466edo →
Prime factorization 3 × 5 × 31
Step size 2.58065¢
Fifth 272\465 (701.935¢)
Semitones (A1:m2) 44:35 (113.5¢ : 90.32¢)
Consistency limit 5
Distinct consistency limit 5

465 equal divisions of the octave (abbreviated 465edo or 465ed2), also called 465-tone equal temperament (465tet) or 465 equal temperament (465et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 465 equal parts of about 2.581 ¢ each. Each step represents a frequency ratio of 21/465, or the 465th root of 2.

Theory

465edo is only consistent to the 5-odd-limit, and the errors of harmonics beyond 3 tend to be quite large. It can be considered for the 2.3.5.11.13.17 subgroup, tempering out 936/935, 1377/1375, 71874/71825, 131648/131625 and 225000/224939. It supports counterschismic in the 5-limit, and birds and belobog in the 7-limit using the patent val.

Prime harmonics

Approximation of prime harmonics in 465edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error absolute (¢) +0.00 -0.02 +0.78 -1.08 +0.94 +0.76 +0.85 -0.74 -1.18 +0.10 +0.77
relative (%) +0 -1 +30 -42 +36 +30 +33 -29 -46 +4 +30
Steps
(reduced)
465
(0)
737
(272)
1080
(150)
1305
(375)
1609
(214)
1721
(326)
1901
(41)
1975
(115)
2103
(243)
2259
(399)
2304
(444)

Subsets and supersets

Since 465 factors into 3 × 5 × 31, 465edo has subset edos 3, 5, 15, 31, 93, and 155. 930edo, which doubles it, gives a good correction to the harmonic 7.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-737 465 [465 737]] +0.0062 0.0062 0.24
2.3.5 [25 15 -21, [-22 30 -11 [465 737 1080]] -0.1083 0.1619 6.27

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 193\465 498.06 4/3 Counterschismic
5 322\465
(43\465)
830.97
(110.97)
80/49
(15/14)
Qintosec (465)

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