877edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|877}} == Theory == 877edo is consistent to the 15-odd-limit. It tempers out 3025/3024, 496125/495616, 420175/419904 and 4096000..."
 
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|877}}
{{ED intro}}


== Theory ==
== Theory ==
877edo is [[consistent]] to the [[15-odd-limit]]. It [[tempers out]] [[3025/3024]], 496125/495616, [[420175/419904]] and 40960000/40920957 in the 11-limit; [[2080/2079]], 3025/3024, [[123201/123200]], 91125/91091 and [[65625/65536]] in the 13-limit. Using the 2.3.7.11.23.43 [[subgroup]], it tempers out [[3312/3311]]. The equal temperament [[support]]s [[quartic]], [[quarterframe]] and pulsar temperaments.
877edo is [[consistent]] to the [[15-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] [[3025/3024]], 496125/495616, [[420175/419904]], 40960000/40920957, as well as the [[quartisma]] in the 11-limit; [[2080/2079]], [[123201/123200]], 91125/91091 and [[65625/65536]] in the 13-limit. Using the 2.3.7.11.23.43 [[subgroup]]<!-- explain why this subgroup is good to consider -->, it tempers out [[3312/3311]]. It [[support]]s the [[quarterframe]] temperament.  


=== Prime harmonics ===
=== Prime harmonics ===
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== 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]]
! 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|-1390 877}}
| {{monzo| -1390 877 }}
| {{mapping|877 1390}}
| {{mapping| 877 1390 }}
| 0.0052
| 0.0052
| 0.0052
| 0.0052
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|-
|-
| 2.3.5
| 2.3.5
| {{monzo|-20 -24 25}}, {{monzo|54 -37 2}}
| {{monzo| -20 -24 25 }}, {{monzo| 54 -37 2 }}
| {{mapping|877 1390 2036}}
| {{mapping| 877 1390 2036 }}
| 0.0685
| 0.0685
| 0.0896
| 0.0896
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|-
|-
| 2.3.5.7
| 2.3.5.7
| 65625/65536, 420175/419904, {{monzo|18 -18 13 -7}}
| 65625/65536, 420175/419904, {{monzo| 18 -18 13 -7 }}
| {{mapping|877 1390 2036 2462}}
| {{mapping| 877 1390 2036 2462 }}
| 0.0575
| 0.0575
| 0.0799
| 0.0799
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| 2.3.5.7.11
| 2.3.5.7.11
| 3025/3024, 496125/495616, 420175/419904, 40960000/40920957
| 3025/3024, 496125/495616, 420175/419904, 40960000/40920957
| {{mapping|877 1390 2036 2462 3034}}
| {{mapping| 877 1390 2036 2462 3034 }}
| 0.0398
| 0.0398
| 0.0797
| 0.0797
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| 2.3.5.7.11.13
| 2.3.5.7.11.13
| 2080/2079, 3025/3024, 123201/123200, 91125/91091, 65625/65536
| 2080/2079, 3025/3024, 123201/123200, 91125/91091, 65625/65536
| {{mapping|877 1390 2036 2462 3034 3245}}
| {{mapping| 877 1390 2036 2462 3034 3245 }}
| 0.0508
| 0.0508
| 0.0768
| 0.0768
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! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br>ratio*
! Temperaments
! Temperaments
|-
|-
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| 182\877
| 182\877
| 249.031
| 249.031
| {{monzo|-26 18 -1}}
| {{monzo| -26 18 -1 }}
| [[Monzismic]]
| [[Monzismic]]
|-
|-
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| [[Sesesix]]
| [[Sesesix]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct


<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
== Music ==
; [[Francium]]
* "Stay In the Eyes of the House." from ''Random Sentences'' (2025) – [https://open.spotify.com/track/1fi1zIHjccUpBin6WacaoA Spotify] | [https://francium223.bandcamp.com/track/stay-in-the-eyes-of-the-house Bandcamp] | [https://www.youtube.com/watch?v=P0nCXyQdYCg YouTube]
* "My Throat Is Not An Egg" from ''Eggs'' (2025) – [https://open.spotify.com/track/4LEaU8QXI2BUv1gMcNIl5S Spotify] | [https://francium223.bandcamp.com/track/my-throat-is-not-an-egg Bandcamp] | [https://www.youtube.com/watch?v=G-HZx5Rv7QE YouTube]
* "Ghost In the Attic" from ''Void'' (2025) – [https://open.spotify.com/track/1hecNwT1QIrlUZqzTsx2i7 Spotify] | [https://francium223.bandcamp.com/track/ghost-in-the-attic Bandcamp] | [https://www.youtube.com/watch?v=VJO_Ysm50AY YouTube]

Latest revision as of 13:32, 13 March 2026

← 876edo 877edo 878edo →
Prime factorization 877 (prime)
Step size 1.3683 ¢ 
Fifth 513\877 (701.938 ¢)
Semitones (A1:m2) 83:66 (113.6 ¢ : 90.31 ¢)
Consistency limit 15
Distinct consistency limit 15

877 equal divisions of the octave (abbreviated 877edo or 877ed2), also called 877-tone equal temperament (877tet) or 877 equal temperament (877et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 877 equal parts of about 1.37 ¢ each. Each step represents a frequency ratio of 21/877, or the 877th root of 2.

Theory

877edo is consistent to the 15-odd-limit. As an equal temperament, it tempers out 3025/3024, 496125/495616, 420175/419904, 40960000/40920957, as well as the quartisma in the 11-limit; 2080/2079, 123201/123200, 91125/91091 and 65625/65536 in the 13-limit. Using the 2.3.7.11.23.43 subgroup, it tempers out 3312/3311. It supports the quarterframe temperament.

Prime harmonics

Approximation of prime harmonics in 877edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.017 -0.453 -0.069 +0.107 -0.391 +0.404 -0.592 -0.224 -0.615 +0.232
Relative (%) +0.0 -1.2 -33.1 -5.0 +7.8 -28.6 +29.5 -43.2 -16.4 -44.9 +17.0
Steps
(reduced) 		877
(0) 		1390
(513) 		2036
(282) 		2462
(708) 		3034
(403) 		3245
(614) 		3585
(77) 		3725
(217) 		3967
(459) 		4260
(752) 		4345
(837)

Subsets and supersets

877edo is the 151st prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-1390 877⟩ [⟨877 1390]] 0.0052 0.0052 0.38
2.3.5 [-20 -24 25⟩, [54 -37 2⟩ [⟨877 1390 2036]] 0.0685 0.0896 6.55
2.3.5.7 65625/65536, 420175/419904, [18 -18 13 -7⟩ [⟨877 1390 2036 2462]] 0.0575 0.0799 5.84
2.3.5.7.11 3025/3024, 496125/495616, 420175/419904, 40960000/40920957 [⟨877 1390 2036 2462 3034]] 0.0398 0.0797 5.82
2.3.5.7.11.13 2080/2079, 3025/3024, 123201/123200, 91125/91091, 65625/65536 [⟨877 1390 2036 2462 3034 3245]] 0.0508 0.0768 5.61

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 182\877 249.031 [-26 18 -1⟩ Monzismic
1 231\877 316.078 6/5 Counterhanson
1 359\877 491.220 8388608/6328125 Sesesix

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

Music

Francium
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