2777edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|2777}}
{{ED intro}}


== Theory ==
== Theory ==
2777edo is [[consistent]] to the [[7-odd-limit]] and its [[harmonic]] [[3/1|3]] is about halfway its steps. Using the 2.9.5.7.11.13.17.23.29.31 [[subgroup]] it [[tempering out|tempers out]] [[12376/12375]], [[14400/14399]], [[25025/25024]], [[123201/123200]], 20736/20735, [[194481/194480]], 16445/16443, 27625/27621 and 23716/23715. It [[support]]s [[joshuavoic]].
2777edo is [[consistent]] to the [[7-odd-limit]] and its [[harmonic]] [[3/1|3]] is about halfway its steps. Using the 2.9.5.7.11.13.17.23.29.31 [[subgroup]], it [[tempering out|tempers out]] [[12376/12375]], [[14400/14399]], [[25025/25024]], [[123201/123200]], 20736/20735, [[194481/194480]], 16445/16443, 27625/27621 and 23716/23715. Using the 2.5.7.11.13.17.23 subgroup, it tempers out [[25025/25024]].


=== Odd harmonics ===
=== Odd harmonics ===
Line 13: Line 13:
== 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" |[[Comma list|Comma List]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Comma list]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | [[Mapping]]
! colspan="2" |Tuning Error
! rowspan="2" | Optimal<br />8ve stretch (¢)
|-
! colspan="2" | Tuning error
![[TE error|Absolute]] (¢)
|-
![[TE simple badness|Relative]] (%)
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.9
| 2.9
| {{monzo|8803 -2777}}
| {{monzo|8803 -2777}}
| {{mapping|2777 8803}}
| {{mapping|2777 8803}}
| -0.0081
| −0.0081
| 0.0081
| 0.0081
| 1.87
| 1.87
Line 32: Line 33:
| {{monzo|146 -38 -11}}, {{monzo|-89 -21 67}}
| {{monzo|146 -38 -11}}, {{monzo|-89 -21 67}}
| {{mapping|2777 8803 6448}}
| {{mapping|2777 8803 6448}}
| -0.0057
| −0.0057
| 0.0074
| 0.0074
| 1.71
| 1.71
Line 39: Line 40:
| {{monzo|2 -10 14 -1}}, {{monzo|-1 -9 -3 13}}, {{monzo|-48 0 11 8}}
| {{monzo|2 -10 14 -1}}, {{monzo|-1 -9 -3 13}}, {{monzo|-48 0 11 8}}
| {{mapping|2777 8803 6448 7796}}
| {{mapping|2777 8803 6448 7796}}
| -0.0033
| −0.0033
| 0.0076
| 0.0076
| 1.76
| 1.76
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| 151263/151250, 184549376/184528125, 35156250/35153041, 3487704605/3486784401
| 151263/151250, 184549376/184528125, 35156250/35153041, 3487704605/3486784401
| {{mapping|2777 8803 6448 7796 9607}}
| {{mapping|2777 8803 6448 7796 9607}}
| -0.0066
| −0.0066
| 0.0095
| 0.0095
| 2.20
| 2.20
Line 53: Line 54:
| 123201/123200, 6656/6655, 151263/151250, 8859375/8859136, 43061200/43046721
| 123201/123200, 6656/6655, 151263/151250, 8859375/8859136, 43061200/43046721
| {{mapping|2777 8803 6448 7796 9607 10276}}
| {{mapping|2777 8803 6448 7796 9607 10276}}
| -0.0032
| −0.0032
| 0.0116
| 0.0116
| 2.68
| 2.68
Line 60: Line 61:
| 12376/12375, 14400/14399, 123201/123200, 194481/194480, 4685824/4685625, 81331250/81310473
| 12376/12375, 14400/14399, 123201/123200, 194481/194480, 4685824/4685625, 81331250/81310473
| {{mapping|2777 8803 6448 7796 9607 10276 11351}}
| {{mapping|2777 8803 6448 7796 9607 10276 11351}}
| -0.0045
| −0.0045
| 0.0112
| 0.0112
| 2.59
| 2.59
|}
|}
== Music ==
; [[Francium]]
* "Joekalille" from ''Naughty Girl Era'' (2024) − [https://open.spotify.com/track/3JkOxSgwBe9dUtxNwI86qa Spotify] | [https://francium223.bandcamp.com/track/joekalille Bandcamp] | [https://www.youtube.com/watch?v=bIwZCmfz7HM YouTube] – joshuavoic in 2777edo

Latest revision as of 12:46, 21 February 2025

← 2776edo 2777edo 2778edo →
Prime factorization 2777 (prime)
Step size 0.432121 ¢ 
Fifth 1624\2777 (701.764 ¢)
Semitones (A1:m2) 260:211 (112.4 ¢ : 91.18 ¢)
Dual sharp fifth 1625\2777 (702.197 ¢)
Dual flat fifth 1624\2777 (701.764 ¢)
Dual major 2nd 472\2777 (203.961 ¢)
Consistency limit 7
Distinct consistency limit 7

2777 equal divisions of the octave (abbreviated 2777edo or 2777ed2), also called 2777-tone equal temperament (2777tet) or 2777 equal temperament (2777et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 2777 equal parts of about 0.432 ¢ each. Each step represents a frequency ratio of 21/2777, or the 2777th root of 2.

Theory

2777edo is consistent to the 7-odd-limit and its harmonic 3 is about halfway its steps. Using the 2.9.5.7.11.13.17.23.29.31 subgroup, it tempers out 12376/12375, 14400/14399, 25025/25024, 123201/123200, 20736/20735, 194481/194480, 16445/16443, 27625/27621 and 23716/23715. Using the 2.5.7.11.13.17.23 subgroup, it tempers out 25025/25024.

Odd harmonics

Approximation of odd harmonics in 2777edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.191 +0.002 -0.011 +0.051 +0.068 -0.052 -0.188 +0.050 -0.214 -0.201 +0.030
Relative (%) -44.1 +0.6 -2.5 +11.8 +15.8 -12.1 -43.5 +11.6 -49.5 -46.5 +6.8
Steps
(reduced) 		4401
(1624) 		6448
(894) 		7796
(2242) 		8803
(472) 		9607
(1276) 		10276
(1945) 		10849
(2518) 		11351
(243) 		11796
(688) 		12197
(1089) 		12562
(1454)

Subsets and supersets

2777edo is the 404th prime EDO. 5554edo, which doubles it, gives a good correction to the harmonic 3.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.9 [8803 -2777⟩ [⟨2777 8803]] −0.0081 0.0081 1.87
2.9.5 [146 -38 -11⟩, [-89 -21 67⟩ [⟨2777 8803 6448]] −0.0057 0.0074 1.71
2.9.5.7 [2 -10 14 -1⟩, [-1 -9 -3 13⟩, [-48 0 11 8⟩ [⟨2777 8803 6448 7796]] −0.0033 0.0076 1.76
2.9.5.7.11 151263/151250, 184549376/184528125, 35156250/35153041, 3487704605/3486784401 [⟨2777 8803 6448 7796 9607]] −0.0066 0.0095 2.20
2.9.5.7.11.13 123201/123200, 6656/6655, 151263/151250, 8859375/8859136, 43061200/43046721 [⟨2777 8803 6448 7796 9607 10276]] −0.0032 0.0116 2.68
2.9.5.7.11.13.17 12376/12375, 14400/14399, 123201/123200, 194481/194480, 4685824/4685625, 81331250/81310473 [⟨2777 8803 6448 7796 9607 10276 11351]] −0.0045 0.0112 2.59

Music

Francium
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