733edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|733}}
{{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|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 error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
| 2.3
| 2.3
| {{monzo|1162 -733}}
| {{monzo|1162 -733}}
| {{mapping|733 1162}}
| {{mapping|733 1162}}
| -0.1149
| −0.1149
| 0.1149
| 0.1149
| 7.02
| 7.02
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| {{monzo|38 -2 -15}}, {{monzo|42 -47 14}}
| {{monzo|38 -2 -15}}, {{monzo|42 -47 14}}
| {{mapping|733 1162 1702}}
| {{mapping|733 1162 1702}}
| -0.0829
| −0.0829
| 0.1042
| 0.1042
| 6.36
| 6.36
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| 420175/419904, 67108864/66976875, 48828125/48771072
| 420175/419904, 67108864/66976875, 48828125/48771072
| {{mapping|733 1162 1702 2058}}
| {{mapping|733 1162 1702 2058}}
| -0.0926
| −0.0926
| 0.0918
| 0.0918
| 5.61
| 5.61
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| 3025/3024, 19712/19683, 180224/180075, 1953125/1951488
| 3025/3024, 19712/19683, 180224/180075, 1953125/1951488
| {{mapping|733 1162 1702 2058 2536}}
| {{mapping|733 1162 1702 2058 2536}}
| -0.0965
| −0.0965
| 0.0824
| 0.0824
| 5.03
| 5.03
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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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| [[Luna]]
| [[Luna]]
|}
|}
<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]]
* "Slow Down Or Die" from ''I Want To'' (2025) – [https://open.spotify.com/track/2vzrMXihetN2DhUWBLUcZs Spotify] | [https://francium223.bandcamp.com/track/slow-down-or-die Bandcamp] | [https://www.youtube.com/watch?v=7lPwEesZbsI YouTube]


[[Category:Luna]]
[[Category:Luna]]

Latest revision as of 13:31, 13 March 2026

← 732edo 733edo 734edo →
Prime factorization 733 (prime)
Step size 1.63711 ¢ 
Fifth 429\733 (702.319 ¢)
Semitones (A1:m2) 71:54 (116.2 ¢ : 88.4 ¢)
Consistency limit 11
Distinct consistency limit 11

733 equal divisions of the octave (abbreviated 733edo or 733ed2), also called 733-tone equal temperament (733tet) or 733 equal temperament (733et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 733 equal parts of about 1.64 ¢ each. Each step represents a frequency ratio of 21/733, or the 733rd root of 2.

Theory

733edo is consistent to the 11-odd-limit, tempering out 3025/3024, 19712/19683, 180224/180075 and 1953125/1951488. It supports hemiluna and French decimal.

Prime harmonics

Approximation of prime harmonics in 733edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.000 +0.364 +0.044 +0.342 +0.387 -0.691 -0.181 +0.441 +0.375 +0.164 -0.697
Relative (%) +0.0 +22.2 +2.7 +20.9 +23.7 -42.2 -11.0 +26.9 +22.9 +10.0 -42.6
Steps
(reduced) 		733
(0) 		1162
(429) 		1702
(236) 		2058
(592) 		2536
(337) 		2712
(513) 		2996
(64) 		3114
(182) 		3316
(384) 		3561
(629) 		3631
(699)

Subsets and supersets

733edo is the 130th prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [1162 -733⟩ [⟨733 1162]] −0.1149 0.1149 7.02
2.3.5 [38 -2 -15⟩, [42 -47 14⟩ [⟨733 1162 1702]] −0.0829 0.1042 6.36
2.3.5.7 420175/419904, 67108864/66976875, 48828125/48771072 [⟨733 1162 1702 2058]] −0.0926 0.0918 5.61
2.3.5.7.11 3025/3024, 19712/19683, 180224/180075, 1953125/1951488 [⟨733 1162 1702 2058 2536]] −0.0965 0.0824 5.03

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperaments
1 118\733 193.179 262144/234375 Luna

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

Music

Francium
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