757edo: Difference between revisions

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Created page with "{{Infobox ET}} {{EDO intro|757}} == Theory == 757edo is consistent to the 7-odd-limit, tempering out 4375/4374, 67108864/66976875 and 282475249/281250000. Usi..."
 
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|757}}
{{ED intro}}


== Theory ==
== Theory ==
757edo is [[consistent]] to the [[7-odd-limit]], [[tempering out]] [[4375/4374]], 67108864/66976875 and 282475249/281250000. Using the [[patent val]], it tempers out [[5632/5625]], 4375/4374, [[160083/160000]] and 5764801/5749920 in the [[11-limit]].
757edo is [[consistent]] to the [[7-odd-limit]]. As an equal temperament, it [[tempering out|tempers out]] [[4375/4374]], 67108864/66976875 and 282475249/281250000. Using the [[patent val]], it tempers out [[5632/5625]], [[160083/160000]] and 5764801/5749920 in the [[11-limit]].


=== 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" |[[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|1200 -757}}
| {{monzo| 1200 -757 }}
| {{mapping|757 1200}}
| {{mapping| 757 1200 }}
| -0.0917
| −0.0917
| 0.0917
| 0.0917
| 5.78
| 5.78
|-
|-
| 2.3.5
| 2.3.5
| {{monzo|24 -21 4}}, {{monzo|78 5 -37}}
| {{monzo| 24 -21 4 }}, {{monzo| 78 5 -37 }}
| {{mapping|757 1200 1758}}
| {{mapping| 757 1200 1758 }}
| -0.1295
| −0.1295
| 0.0920
| 0.0920
| 5.80
| 5.80
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| 2.3.5.7
| 2.3.5.7
| 4375/4374, 67108864/66976875, 282475249/281250000
| 4375/4374, 67108864/66976875, 282475249/281250000
| {{mapping|757 1200 1758 2125}}
| {{mapping| 757 1200 1758 2125 }}
| -0.0735
| −0.0735
| 0.1256
| 0.1256
| 7.92
| 7.92
|-
|-
| 2.3.5.7.11
| 2.3.5.7.11
| 5632/5625, 4375/4374, 160083/160000, 5764801/5749920
| 4375/4374, 5632/5625, 160083/160000, 5764801/5749920
| {{mapping|757 1200 1758 2125 2619}}
| {{mapping| 757 1200 1758 2125 2619 }}
| -0.0780
| −0.0780
| 0.1127
| 0.1127
| 7.11
| 7.11
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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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| [[Vulture]]
| [[Vulture]]
|}
|}
<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>*</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]]
* "developeed" from ''wiloliquy'' (2025) – [https://open.spotify.com/track/1RaOpdvBzCxAED154qb5UT Spotify] | [https://francium223.bandcamp.com/track/developeed Bandcamp] | [https://www.youtube.com/watch?v=rskZavKFwv4 YouTube] – in Vulture, 757edo tuning

Latest revision as of 13:31, 13 March 2026

← 756edo 757edo 758edo →
Prime factorization 757 (prime)
Step size 1.5852 ¢ 
Fifth 443\757 (702.246 ¢)
Semitones (A1:m2) 73:56 (115.7 ¢ : 88.77 ¢)
Consistency limit 7
Distinct consistency limit 7

757 equal divisions of the octave (abbreviated 757edo or 757ed2), also called 757-tone equal temperament (757tet) or 757 equal temperament (757et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 757 equal parts of about 1.59 ¢ each. Each step represents a frequency ratio of 21/757, or the 757th root of 2.

Theory

757edo is consistent to the 7-odd-limit. As an equal temperament, it tempers out 4375/4374, 67108864/66976875 and 282475249/281250000. Using the patent val, it tempers out 5632/5625, 160083/160000 and 5764801/5749920 in the 11-limit.

Prime harmonics

Approximation of prime harmonics in 757edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.291 +0.476 -0.266 +0.333 -0.369 -0.332 +0.505 -0.533 -0.779 -0.518
Relative (%) +0.0 +18.3 +30.0 -16.8 +21.0 -23.3 -20.9 +31.9 -33.6 -49.2 -32.7
Steps
(reduced) 		757
(0) 		1200
(443) 		1758
(244) 		2125
(611) 		2619
(348) 		2801
(530) 		3094
(66) 		3216
(188) 		3424
(396) 		3677
(649) 		3750
(722)

Subsets and supersets

757edo is the 134th prime edo.

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [1200 -757⟩ [⟨757 1200]] −0.0917 0.0917 5.78
2.3.5 [24 -21 4⟩, [78 5 -37⟩ [⟨757 1200 1758]] −0.1295 0.0920 5.80
2.3.5.7 4375/4374, 67108864/66976875, 282475249/281250000 [⟨757 1200 1758 2125]] −0.0735 0.1256 7.92
2.3.5.7.11 4375/4374, 5632/5625, 160083/160000, 5764801/5749920 [⟨757 1200 1758 2125 2619]] −0.0780 0.1127 7.11

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 300\757 475.561 320/243 Vulture

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

Music

Francium
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