277edo: Difference between revisions

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'''277EDO''' is the [[EDO|equal division of the octave]] into 277 parts of 4.3321 [[cent]]s each. It 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. 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 supports the [[pontiac]], and the 277d val supports the [[guiron]] and the [[widefourth]].
{{Infobox ET}}
{{ED intro}}


277EDO is the 59th [[prime EDO]].
== Theory ==
277edo is a good [[5-limit]] tuning; however, it is in[[consistent]] in the [[7-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and {{monzo| -11 -37 30 }} in the 5-limit.  


[[Category:Edo]]
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]].
[[Category:Prime EDO]]
 
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 ===
{{Harmonics in equal|277}}
 
=== Subsets and supersets ===
277edo is the 59th [[prime edo]].
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| -439 277 }}
| {{mapping| 277 439 }}
| +0.0473
| 0.0473
| 1.09
|-
| 2.3.5
| 32805/32768, {{monzo| -11 -37 30 }}
| {{mapping| 277 439 643 }}
| +0.1398
| 0.1364
| 3.15
|}
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br />per 8ve
! Generator*
! Cents*
! Associated<br />ratio*
! Temperaments
|-
| 1
| 115\277
| 498.19
| 4/3
| [[Helmholtz (temperament)|Helmholtz]]
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 
== Music ==
; [[Francium]]
* "Rizzardry" from ''The Scallop Disco Accident'' (2025) – [https://open.spotify.com/track/1NqjyKORN0c2AIOhTMMUK7 Spotify] | [https://francium223.bandcamp.com/track/rizzardry Bandcamp] | [https://www.youtube.com/watch?v=HZ67knYwALA YouTube] – in Zangaric, 277edo tuning
* "It's a Smell." from ''Random Sentences'' (2025) – [https://open.spotify.com/track/35sVZfy1mDsPHZwjgh8Sql Spotify] | [https://francium223.bandcamp.com/track/its-a-smell Bandcamp] | [https://www.youtube.com/watch?v=L9Tn7n-THCM YouTube] – in Yerkesic, 277edo tuning

Latest revision as of 02:30, 17 April 2025

← 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

277 equal divisions of the octave (abbreviated 277edo or 277ed2), also called 277-tone equal temperament (277tet) or 277 equal temperament (277et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 277 equal parts of about 4.33 ¢ each. Each step represents a frequency ratio of 21/277, or the 277th root of 2.

Theory

277edo is a good 5-limit tuning; however, it is inconsistent in the 7-odd-limit. As an equal temperament, it 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 distinct

Music

Francium
  • "Rizzardry" from The Scallop Disco Accident (2025) – Spotify | Bandcamp | YouTube – in Zangaric, 277edo tuning
  • "It's a Smell." from Random Sentences (2025) – Spotify | Bandcamp | YouTube – in Yerkesic, 277edo tuning
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