1277edo

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← 1276edo1277edo1278edo →
Prime factorization 1277 (prime)
Step size 0.939702¢ 
Fifth 747\1277 (701.958¢)
Semitones (A1:m2) 121:96 (113.7¢ : 90.21¢)
Consistency limit 11
Distinct consistency limit 11

1277 equal divisions of the octave (abbreviated 1277edo or 1277ed2), also called 1277-tone equal temperament (1277tet) or 1277 equal temperament (1277et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 1277 equal parts of about 0.94 ¢ each. Each step represents a frequency ratio of 21/1277, or the 1277th root of 2.

Theory

1277edo is consistent to the 11-odd-limit. The equal temperament tempers out 4375/4374, 52734375/52706752, 645700815/645657712 (starscape comma) and [51 -13 -1 -10 (technologisma) in the 7-limit; 151263/151250, 759375/758912, and 2097152/2096325 in the 11-limit. It supports monzismic, supermajor, revopent, as well as the rank-3 temperament bragi.

Prime harmonics

Approximation of prime harmonics in 1277edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.003 -0.096 +0.007 +0.287 -0.434 +0.291 +0.373 +0.387 +0.337 +0.462
Relative (%) +0.0 +0.3 -10.2 +0.8 +30.6 -46.2 +31.0 +39.7 +41.1 +35.8 +49.1
Steps
(reduced)
1277
(0)
2024
(747)
2965
(411)
3585
(1031)
4418
(587)
4725
(894)
5220
(112)
5425
(317)
5777
(669)
6204
(1096)
6327
(1219)

Subsets and supersets

1277edo is the 206th prime edo. 2554edo, which divides the edostep in two, is the smallest edo distinctly consistent through the 41-odd-limit, and provides correction for harmonics 11 through 41.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [2024 -1277 [1277 2024]] -0.0009 0.0009 0.10
2.3.5 [54 -37 2, [-67 -9 35 [1277 2024 2965]] +0.0132 0.0199 2.12
2.3.5.7 4375/4374, 52734375/52706752, [51 -13 -1 -10 [1277 2024 2965 3585]] +0.0093 0.0186 1.98
2.3.5.7.11 4375/4374, 151263/151250, 759375/758912, 2097152/2096325 [1277 2024 2965 3585 4418]] -0.0092 0.0405 4.31

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 265\1277 249.021 [-27 11 3 1 Monzismic
1 380\1277 357.087 768/625 Dodifo
1 463\1277 435.082 9/7 Supermajor

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

Music

Francium
  • "mututhery" from albumwithoutspaces (2024) – Spotify | Bandcamp | YouTube – monzismic[19] in 1277edo tuning