433edo: Difference between revisions

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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|433}}
{{ED intro}}
 
== 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===
 
433edo is the 84th [[prime edo]].
Using the patent val, it [[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===
 
=== Odd harmonics ===
{{Harmonics in equal|433}}
{{Harmonics in equal|433}}
==Regular temperament properties==
 
=== Subsets and supersets ===
433edo is the 84th [[prime edo]]. It might be interesting due to being the smallest subset edo of the nanotemperament [[2901533edo]], an extremely high-precision/complexity [[microtemperament]].
 
== 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" |[[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! rowspan="2" | [[Subgroup]]
![[TE simple badness|Relative]] (%)
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br />8ve stretch (¢)
! colspan="2" | Tuning error
|-
|-
|2.3
! [[TE error|Absolute]] (¢)
|{{monzo|-686 433}}
! [[TE simple badness|Relative]] (%)
|{{val|433 686}}
|-
| 0.2525
| 2.3
| {{monzo| -686 433 }}
| {{mapping| 433 686 }}
| +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
| 10.94
| 10.94
Line 63: Line 70:
=== 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
|-
! Generator<br>(reduced)
! Periods<br />per 8ve
! Cents<br>(reduced)
! Generator*
! Associated<br>ratio
! Cents*
! Associated<br />ratio*
! Temperaments
! Temperaments
|-
|-
|1
| 1
|98\433
| 98\433
|271.594
| 271.594
|75/64
| 75/64
|[[Orson]]
| [[Orson]]
|}
|}
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
== Music ==
; [[Francium]]
* [https://www.youtube.com/watch?v=M4CFR4yYF_U ''sleeping as we don't know''] (2023)


==Music==
[[Category:Listen]]
* [https://www.youtube.com/watch?v=M4CFR4yYF_U sleeping as we don't know] by Francium

Latest revision as of 06:01, 21 February 2025

← 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

433 equal divisions of the octave (abbreviated 433edo or 433ed2), also called 433-tone equal temperament (433tet) or 433 equal temperament (433et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 433 equal parts of about 2.77 ¢ each. Each step represents a frequency ratio of 21/433, or the 433rd root of 2.

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, it 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. It might be interesting due to being the smallest subset edo of the nanotemperament 2901533edo, an extremely high-precision/complexity microtemperament.

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 distinct

Music

Francium
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