773edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|773}} == Theory == 773edo is consistent to the 7-odd-limit. Using the patent val, it tempers out 2401/2400, 328..."
 
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|773}}
{{ED intro}}


== Theory ==
== Theory ==
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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" |[[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]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo|-1225 773}}
| {{monzo|-1225 773}}
| {{mapping|773 1225}}
| {{mapping|773 1225}}
| 0.0862
| +0.0862
| 0.0862
| 0.0862
| 5.55
| 5.55
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| 32805/32768, {{monzo|65 85 -86}}
| 32805/32768, {{monzo|65 85 -86}}
| {{mapping|773 1225 1795}}
| {{mapping|773 1225 1795}}
| 0.0241
| +0.0241
| 0.1125
| 0.1125
| 7.25
| 7.25
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| 2401/2400, 32805/32768, {{monzo|20 22 -20 -3}}
| 2401/2400, 32805/32768, {{monzo|20 22 -20 -3}}
| {{mapping|773 1225 1795 2170}}
| {{mapping|773 1225 1795 2170}}
| 0.0299
| +0.0299
| 0.0979
| 0.0979
| 6.31
| 6.31
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| 2401/2400, 32805/32768, 137781/137500, 766656/765625
| 2401/2400, 32805/32768, 137781/137500, 766656/765625
| {{mapping|773 1225 1795 2170 2674}}
| {{mapping|773 1225 1795 2170 2674}}
| 0.0365
| +0.0365
| 0.0886
| 0.0886
| 5.71
| 5.71
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=== 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
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
! Periods<br>per 8ve
|-
! Periods<br />per 8ve
! Generator*
! Generator*
! Cents*
! Cents*
! Associated<br>Ratio*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
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| 498.318
| 498.318
| 4/3
| 4/3
| [[Helmholtz]]
| [[Helmholtz (temperament)|Helmholtz]]
|}
|}
<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]]
* "Elvonna Chelf" from ''Don't Give Your Kids These Names'' (2025) − [https://open.spotify.com/track/4xjDobEhkET26uWYP4lgsi Spotify] | [https://francium223.bandcamp.com/track/elvonna-chelf Bandcamp] | [https://www.youtube.com/watch?v=YACJumzzMb0 YouTube]

Latest revision as of 13:31, 13 March 2026

← 772edo 773edo 774edo →
Prime factorization 773 (prime)
Step size 1.55239 ¢ 
Fifth 452\773 (701.682 ¢)
Semitones (A1:m2) 72:59 (111.8 ¢ : 91.59 ¢)
Consistency limit 7
Distinct consistency limit 7

773 equal divisions of the octave (abbreviated 773edo or 773ed2), also called 773-tone equal temperament (773tet) or 773 equal temperament (773et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 773 equal parts of about 1.55 ¢ each. Each step represents a frequency ratio of 21/773, or the 773rd root of 2.

Theory

773edo is consistent to the 7-odd-limit. Using the patent val, it tempers out 2401/2400, 32805/32768, 137781/137500 and 766656/765625 in the 11-limit. It supports counterultrakleismic.

Prime harmonics

Approximation of prime harmonics in 773edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 -0.273 +0.232 -0.133 -0.218 -0.683 +0.607 +0.546 +0.445 -0.340 +0.631
Relative (%) +0.0 -17.6 +15.0 -8.5 -14.1 -44.0 +39.1 +35.2 +28.7 -21.9 +40.6
Steps
(reduced) 		773
(0) 		1225
(452) 		1795
(249) 		2170
(624) 		2674
(355) 		2860
(541) 		3160
(68) 		3284
(192) 		3497
(405) 		3755
(663) 		3830
(738)

Subsets and supersets

773edo is the 137th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [-1225 773⟩ [⟨773 1225]] +0.0862 0.0862 5.55
2.3.5 32805/32768, [65 85 -86⟩ [⟨773 1225 1795]] +0.0241 0.1125 7.25
2.3.5.7 2401/2400, 32805/32768, [20 22 -20 -3⟩ [⟨773 1225 1795 2170]] +0.0299 0.0979 6.31
2.3.5.7.11 2401/2400, 32805/32768, 137781/137500, 766656/765625 [⟨773 1225 1795 2170 2674]] +0.0365 0.0886 5.71

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 113\773 175.420 448/405 Sesquiquartififths
1 321\773 498.318 4/3 Helmholtz

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

Music

Francium
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