277edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|277}}
{{EDO intro|277}}
==Theory==
277et tempers out 32805/32768 (schisma) and |-11 -37 30> in the 5-limit.


Using the patent val, it tempers out 4375/4374, 65625/65536, and 829440/823543 in the 7-limit; 540/539, 6250/6237, 15488/15435, and 35937/35840 in the 11-limit; 625/624, 729/728, 1573/1568, 2080/2079, and 2200/2197 in the 13-limit.  
== Theory ==
277edo is a good 5-limit tuning; however, it is in[[consistent]] in the [[7-odd-limit]]. The equal temperament [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and {{monzo| -11 -37 30 }} in the 5-limit.  


Using the 277d val, it tempers out 1029/1024, 10976/10935, and 48828125/48771072 in the 7-limit; 385/384, 441/440, 19712/19683, and 234375/234256 in the 11-limit; 625/624, 847/845, 1001/1000, and 1575/1573 in the 13-limit.  
The [[patent val]] {{val| 277 439 643 778 }} tempers out [[4375/4374]], [[65625/65536]], and 829440/823543 in the 7-limit; [[540/539]], [[6250/6237]], 15488/15435, and 35937/35840 in the 11-limit; [[625/624]], [[729/728]], [[1573/1568]], [[2080/2079]], and [[2200/2197]] in the 13-limit. It [[support]]s [[pontiac]].  


The patent val [[support]]s the [[pontiac]], and the 277d val supports the [[guiron]] and the [[widefourth]].
The 277d val {{val| 277 439 643 '''777''' }} tempers out [[1029/1024]], [[10976/10935]], and 48828125/48771072 in the 7-limit; [[385/384]], [[441/440]], [[19712/19683]], and 234375/234256 in the 11-limit; 625/624, [[847/845]], [[1001/1000]], and [[1575/1573]] in the 13-limit. It supports [[guiron]] and [[widefourth]].
===Prime harmonics===
 
=== Prime harmonics ===
{{Harmonics in equal|277}}
{{Harmonics in equal|277}}
===Subsets and supersets===
 
277edo is the 59th [[prime EDO]].
=== Subsets and supersets ===
==Regular temperament properties==
277edo is the 59th [[prime edo]].
 
== 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|-439 277}}
| {{monzo| -439 277 }}
|{{val|277 439}}
| {{mapping| 277 439 }}
| 0.0473
| 0.0473
| 0.0473
| 0.0473
| 1.09
| 1.09
|-
|-
|2.3.5
| 2.3.5
|32805/32768, {{monzo|-11 -37 30}}
| 32805/32768, {{monzo| -11 -37 30 }}
|{{val|277 439 643}}
| {{mapping| 277 439 643 }}
| 0.1398
| 0.1398
| 0.1364
| 0.1364
| 3.15
| 3.15
|}
|}
=== Rank-2 temperaments ===
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|115\277
| 115\277
|498.19
| 498.19
|4/3
| 4/3
|[[Helmholtz]]
| [[Helmholtz]]
|}
|}
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
[[Category:Prime EDO]]

Revision as of 06:28, 8 March 2024

← 276edo 277edo 278edo →
Prime factorization 277 (prime)
Step size 4.33213 ¢ 
Fifth 162\277 (701.805 ¢)
Semitones (A1:m2) 26:21 (112.6 ¢ : 90.97 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

Theory

277edo is a good 5-limit tuning; however, it is inconsistent in the 7-odd-limit. The equal temperament tempers out 32805/32768 (schisma) and [-11 -37 30⟩ in the 5-limit.

The patent val ⟨277 439 643 778] tempers out 4375/4374, 65625/65536, and 829440/823543 in the 7-limit; 540/539, 6250/6237, 15488/15435, and 35937/35840 in the 11-limit; 625/624, 729/728, 1573/1568, 2080/2079, and 2200/2197 in the 13-limit. It supports pontiac.

The 277d val ⟨277 439 643 777] tempers out 1029/1024, 10976/10935, and 48828125/48771072 in the 7-limit; 385/384, 441/440, 19712/19683, and 234375/234256 in the 11-limit; 625/624, 847/845, 1001/1000, and 1575/1573 in the 13-limit. It supports guiron and widefourth.

Prime harmonics

Approximation of prime harmonics in 277edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 -0.15 -0.75 +1.57 -1.14 -0.09 -0.98 +1.40 -0.12 +1.47 -1.35
Relative (%) +0.0 -3.5 -17.4 +36.3 -26.3 -2.2 -22.7 +32.4 -2.7 +33.9 -31.2
Steps
(reduced) 		277
(0) 		439
(162) 		643
(89) 		778
(224) 		958
(127) 		1025
(194) 		1132
(24) 		1177
(69) 		1253
(145) 		1346
(238) 		1372
(264)

Subsets and supersets

277edo is the 59th prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-439 277⟩ [⟨277 439]] 0.0473 0.0473 1.09
2.3.5 32805/32768, [-11 -37 30⟩ [⟨277 439 643]] 0.1398 0.1364 3.15

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 115\277 498.19 4/3 Helmholtz

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

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