1277edo: Difference between revisions

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Review. The listing of temps was particularly lame.
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== Theory ==
== Theory ==
1277edo is [[consistent]] to the [[11-odd-limit]]. The equal temperament [[tempering out|tempers out]] [[4375/4374]], 52734375/52706752 and {{monzo|51 -13 -1 -10}} in the 7-limit; 4375/4374, 759375/758912, 151263/151250 and 2097152/2096325 in the 11-limit. It [[support]]s [[nanismic]], [[nanic]], [[ragismic]], [[bragi]], [[revopentic]], [[revopent]], [[sasaquinbizo-atriyo]], [[starscape]], [[nommismic]], [[technologismic]], [[supermajor]] and [[monzismic]].
1277edo is [[consistent]] to the [[11-odd-limit]]. The equal temperament [[tempering out|tempers out]] [[4375/4374]], 52734375/52706752, 645700815/645657712 ([[starscape comma]]) and {{monzo| 51 -13 -1 -10 }} ([[technologisma]]) in the 7-limit; 151263/151250, 759375/758912, and 2097152/2096325 in the 11-limit. It [[support]]s [[monzismic]], [[supermajor]], [[revopent]], as well as the rank-3 temperament [[bragi]].


=== Prime harmonics ===
=== Prime harmonics ===
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=== Subsets and supersets ===
=== Subsets and supersets ===
1277edo is the 206th [[prime EDO]].
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.
 
[[2554edo]], which divides the edostep in two, is the smallest EDO [[distinctly consistent]] through the [[41-odd-limit]], hence provides correction for harmonics 11 through 41.


== Regular temperament properties ==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|2024 -1277}}
| {{monzo| 2024 -1277 }}
|{{mapping|1277 2024}}
| {{mapping| 1277 2024 }}
| -0.0009
| -0.0009
| 0.0009
| 0.0009
| 0.10
| 0.10
|-
|-
|2.3.5
| 2.3.5
|{{monzo|54 -37 2}}, {{monzo|-67 -9 35}}
| {{monzo| 54 -37 2 }}, {{monzo| -67 -9 35 }}
|{{mapping|1277 2024 2965}}
| {{mapping| 1277 2024 2965 }}
| +0.0132
| +0.0132
| 0.0199
| 0.0199
| 2.12
| 2.12
|-
|-
|2.3.5.7
| 2.3.5.7
|4375/4374, 52734375/52706752, {{monzo|51 -13 -1 -10}}
| 4375/4374, 52734375/52706752, {{monzo| 51 -13 -1 -10 }}
|{{mapping|1277 2024 2965 3585}}
| {{mapping| 1277 2024 2965 3585 }}
| +0.0093
| +0.0093
| 0.0186
| 0.0186
| 1.98
| 1.98
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|4375/4374, 759375/758912, 151263/151250, 2097152/2096325
| 4375/4374, 151263/151250, 759375/758912, 2097152/2096325
|{{mapping|1277 2024 2965 3585 4418}}
| {{mapping| 1277 2024 2965 3585 4418 }}
| -0.0092
| -0.0092
| 0.0405
| 0.0405
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! Temperaments
! Temperaments
|-
|-
|1
| 1
|265\1277
| 265\1277
|249.021
| 249.021
|{{monzo|-27 11 3 1}}
| {{monzo| -27 11 3 1 }}
|[[Monzismic]]
| [[Monzismic]]
|-
|-
|1
| 1
|380\1277
| 380\1277
|357.087
| 357.087
|768/625
| 768/625
|[[Dodifo]]
| [[Dodifo]]
|-
|-
|1
| 1
|463\1277
| 463\1277
|435.082
| 435.082
|9/7
| 9/7
|[[Supermajor]]
| [[Supermajor]]
|}
|}
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

Revision as of 10:56, 17 May 2024

← 1276edo 1277edo 1278edo →
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

Template:EDO intro

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

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