433edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|433}}
{{EDO intro|433}}
== Theory ==
== Theory ==
433et tempers out 4096000/4084101, 95703125/95551488 and [[1640558367/1638400000]] in the 7-limit; 161280/161051, 25165824/25109315, 1019215872/1019046875, [[4000/3993]], 2359296/2358125, [[6250/6237]], 180224/180075, 17537553/17500000, [[3025/3024]] and 1362944/1361367 in the 11-limit.
443edo is only [[consistent]] to the [[5-odd-limit]] since [[harmonic]] [[7/1|7]] is about halfway between its steps. To start with, the [[patent val]] {{val| 433 686 1005 '''1216''' }} as well as the 433d [[val]] {{val| 433 686 1005 '''1215''' }} are worth considering.
===Subsets and supersets===
 
Using the patent val, the equal temperament [[tempering out|tempers out]] [[19683/19600]] and 4096000/4084101 in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], 161280/161051, and 180224/180075 in the 11-limit.
 
=== Odd harmonics ===
{{Harmonics in equal|433}}
 
=== Subsets and supersets ===
433edo is the 84th [[prime edo]].
433edo is the 84th [[prime edo]].
===Odd harmonics===
 
{{Harmonics in equal|433}}
== 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|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|-686 433}}
| {{monzo| -686 433 }}
|{{val|433 686}}
| {{mapping| 433 686 }}
| 0.2525
| 0.2525
| 0.2525
| 0.2525
| 9.11
| 9.11
|-
|-
|2.3.5
| 2.3.5
|{{monzo|-21 3 7}}, {{monzo|-29 52 -23}}
| 2109375/2097152, {{monzo| -29 52 -23 }}
|{{val|433 686 1005}}
| {{mapping| 433 686 1005 }}
| 0.3254
| 0.3254
| 0.2306
| 0.2306
| 8.32
| 8.32
|-
|-
|2.3.5.7
| 2.3.5.7
|19683/19600, 4096000/4084101, 2109375/2097152
| 19683/19600, 4096000/4084101, 2109375/2097152
|{{val|433 686 1005 1216}}
| {{mapping| 433 686 1005 1216 }}
| 0.1414
| 0.1414
| 0.3759
| 0.3759
| 13.56
| 13.56
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|3025/3024, 6250/6237, 30375/30184, 180224/180075
| 3025/3024, 6250/6237, 30375/30184, 180224/180075
|{{val|433 686 1005 1216 1498}}
| {{mapping| 433 686 1005 1216 1498 }}
| 0.1026
| 0.1026
| 0.3451
| 0.3451
| 12.45
| 12.45
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|2080/2079, 625/624, 3025/3024, 18954/18865, 41472/41405
| 2080/2079, 625/624, 3025/3024, 18954/18865, 41472/41405
|{{val|433 686 1005 1216 1498 1602}}
| {{mapping| 433 686 1005 1216 1498 1602 }}
| 0.1217
| 0.1217
| 0.3179
| 0.3179
| 11.47
| 11.47
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|2080/2079, 375/374, 715/714, 936/935, 1377/1372, 76032/75803
| 2080/2079, 375/374, 715/714, 936/935, 1377/1372, 76032/75803
|{{val|433 686 1005 1216 1498 1602 1770}}
| {{mapping| 433 686 1005 1216 1498 1602 1770 }}
| 0.0919
| 0.0919
| 0.3033
| 0.3033
Line 65: Line 71:
|+Table of rank-2 temperaments by generator
|+Table of rank-2 temperaments by generator
! Periods<br>per 8ve
! Periods<br>per 8ve
! Generator<br>(reduced)
! Generator*
! Cents<br>(reduced)
! Cents*
! Associated<br>ratio
! Associated<br>Ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|98\433
| 98\433
|271.594
| 271.594
|75/64
| 75/64
|[[Orson]]
| [[Orson]]
|}
|}
<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==
== Music ==
* [https://www.youtube.com/watch?v=M4CFR4yYF_U sleeping as we don't know] by Francium
; [[User:Francium|Francium]]
* [https://www.youtube.com/watch?v=M4CFR4yYF_U ''sleeping as we don't know''] (2023)

Revision as of 08:18, 3 November 2023

← 432edo 433edo 434edo →
Prime factorization 433 (prime)
Step size 2.77136 ¢ 
Fifth 253\433 (701.155 ¢)
Semitones (A1:m2) 39:34 (108.1 ¢ : 94.23 ¢)
Consistency limit 5
Distinct consistency limit 5

Template:EDO intro

Theory

443edo is only consistent to the 5-odd-limit since harmonic 7 is about halfway between its steps. To start with, the patent val ⟨433 686 1005 1216] as well as the 433d val ⟨433 686 1005 1215] are worth considering.

Using the patent val, the equal temperament tempers out 19683/19600 and 4096000/4084101 in the 7-limit; 3025/3024, 4000/3993, 6250/6237, 161280/161051, and 180224/180075 in the 11-limit.

Odd harmonics

Approximation of odd harmonics in 433edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -0.80 -1.09 +1.15 +1.17 +0.18 -0.80 +0.88 +0.36 -0.98 +0.35 +0.82
Relative (%) -28.9 -39.5 +41.5 +42.2 +6.6 -29.0 +31.6 +12.9 -35.3 +12.7 +29.8
Steps
(reduced) 		686
(253) 		1005
(139) 		1216
(350) 		1373
(74) 		1498
(199) 		1602
(303) 		1692
(393) 		1770
(38) 		1839
(107) 		1902
(170) 		1959
(227)

Subsets and supersets

433edo is the 84th prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-686 433⟩ [⟨433 686]] 0.2525 0.2525 9.11
2.3.5 2109375/2097152, [-29 52 -23⟩ [⟨433 686 1005]] 0.3254 0.2306 8.32
2.3.5.7 19683/19600, 4096000/4084101, 2109375/2097152 [⟨433 686 1005 1216]] 0.1414 0.3759 13.56
2.3.5.7.11 3025/3024, 6250/6237, 30375/30184, 180224/180075 [⟨433 686 1005 1216 1498]] 0.1026 0.3451 12.45
2.3.5.7.11.13 2080/2079, 625/624, 3025/3024, 18954/18865, 41472/41405 [⟨433 686 1005 1216 1498 1602]] 0.1217 0.3179 11.47
2.3.5.7.11.13.17 2080/2079, 375/374, 715/714, 936/935, 1377/1372, 76032/75803 [⟨433 686 1005 1216 1498 1602 1770]] 0.0919 0.3033 10.94

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 98\433 271.594 75/64 Orson

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

Music

Francium
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