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{{Infobox ET}}
{{Infobox ET}}
{{EDO intro|233}}
{{EDO intro|233}}
== Theory ==
== Theory ==
233et has a generally flat tendency, in the sense that if the octave is pure, 3, 5, 7, 11, 13, and 17 are all flat. 233edo is accurate for the 5th harmonic (only 0.0476 cents flat), but less for the third harmonic (1.5258 cents flat).  
233et has a generally flat tendency, in the sense that if the [[octave]] is pure, [[prime harmonic]]s 3 through 17 are all flat. 233edo is accurate for the [[5/1|5th harmonic]] (only 0.0476 cents flat), but less for the [[3/1|third harmonic]] (1.5258 cents flat).
 
The equal temperament [[tempering out|tempers out]] [[78732/78125]] and {{monzo| -53 32 1 }} in the 5-limit; [[2401/2400]], [[65625/65536]], and 177147/175616 in the 7-limit (supporting [[tertiaseptal]] and [[catafourth]]). Using the [[patent val]], it tempers out [[243/242]], [[441/440]], 35937/35840, and 78408/78125 in the 11-limit; [[351/350]], [[1001/1000]], [[1575/1573]], [[4225/4224]], and [[6656/6655]] in the 13-limit.


It tempers out 78732/78125 and |-53 32 1> in the 5-limit; 2401/2400, 65625/65536, and 177147/175616 in the 7-limit (supporting [[Breedsmic temperaments|tertiaseptal]] and [[Breedsmic temperaments|catafourth]]). Using the patent val, it tempers out 243/242, 441/440, 35937/35840, and 78408/78125 in the 11-limit; 351/350, 1001/1000, 1575/1573, 4225/4224, and 6656/6655 in the 13-limit.
=== Odd harmonics ===
=== Odd harmonics ===
{{Harmonics in equal|233}}
{{Harmonics in equal|233}}
===Subsets and supersets ===
 
=== Subsets and supersets ===
233edo is the 51st [[prime edo]].
233edo is the 51st [[prime edo]].
==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|-369 233}}
| {{monzo| -369 233 }}
|{{val|233 369}}
| {{mapping| 233 369 }}
| 0.4813
| 0.4813
| 0.4815
| 0.4815
| 9.35
| 9.35
|-
|-
|2.3.5
| 2.3.5
|78732/78125, {{monzo|-53 32 1}}
| 78732/78125, {{monzo| -53 32 1 }}
|{{val|233 369 541}}
| {{mapping| 233 369 541 }}
| 0.3277
| 0.3277
| 0.4492
| 0.4492
| 8.72
| 8.72
|-
|-
|2.3.5.7
| 2.3.5.7
|2401/2400, 65625/65536, 78732/78125
| 2401/2400, 65625/65536, 78732/78125
|{{val|233 369 541 654}}
| {{mapping| 233 369 541 654 }}
| 0.2979
| 0.2979
| 0.3924
| 0.3924
| 7.62
| 7.62
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|243/242, 441/440, 540/539, 2401/2400
| 243/242, 441/440, 540/539, 2401/2400
|{{val|233 369 541 654 806}}
| {{mapping| 233 369 541 654 806 }}
| 0.2525
| 0.2525
| 0.3625
| 0.3625
| 7.04
| 7.04
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|243/242, 351/350, 441/440, 540/539, 1001/1000
| 243/242, 351/350, 441/440, 540/539, 1001/1000
|{{val|233 369 541 654 806 862}}
| {{mapping| 233 369 541 654 806 862 }}
| 0.2574
| 0.2574
| 0.3311
| 0.3311
| 6.43
| 6.43
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|351/350, 441/440, 540/539, 561/560, 936/935, 1156/1155
| 351/350, 441/440, 540/539, 561/560, 936/935, 1156/1155
|{{val|233 369 541 654 806 862 952}}
| {{mapping| 233 369 541 654 806 862 952 }}
| 0.2888
| 0.2888
| 0.3161
| 0.3161
| 6.14
| 6.14
|}
|}
=== 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
|+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
|15\233
| 15\233
|77.25
| 77.25
|256/245
| 256/245
|[[Tertiaseptal]]
| [[Tertiaseptal]]
|-
|-
|1
| 1
|22\233
| 22\233
|113.30
| 113.30
|16/15
| 16/15
|[[Misneb]]
| [[Misneb]]
|-
|-
|1
| 1
|55\233
| 55\233
|283.26
| 283.26
|189/160
| 189/160
|[[Neominor]]
| [[Neominor]]
|-
|-
|1
| 1
|77\233
| 77\233
|396.57
| 396.57
|98304/78125
| 98304/78125
|[[Squarschmidt]]
| [[Squarschmidt]]
|-
|-
|1
| 1
|86\233
| 86\233
|442.92
| 442.92
|9/7
| 9/7
|[[Sensi]]
| [[Sensi]]
|-
|-
|1
| 1
|95\233
| 95\233
|489.27
| 489.27
|250/189
| 250/189
|[[Catafourth]]
| [[Catafourth]]
|}
|}
 
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]

Revision as of 09:35, 26 March 2024

← 232edo 233edo 234edo →
Prime factorization 233 (prime)
Step size 5.15021 ¢ 
Fifth 136\233 (700.429 ¢)
Semitones (A1:m2) 20:19 (103 ¢ : 97.85 ¢)
Consistency limit 7
Distinct consistency limit 7

Template:EDO intro

Theory

233et has a generally flat tendency, in the sense that if the octave is pure, prime harmonics 3 through 17 are all flat. 233edo is accurate for the 5th harmonic (only 0.0476 cents flat), but less for the third harmonic (1.5258 cents flat).

The equal temperament tempers out 78732/78125 and [-53 32 1⟩ in the 5-limit; 2401/2400, 65625/65536, and 177147/175616 in the 7-limit (supporting tertiaseptal and catafourth). Using the patent val, it tempers out 243/242, 441/440, 35937/35840, and 78408/78125 in the 11-limit; 351/350, 1001/1000, 1575/1573, 4225/4224, and 6656/6655 in the 13-limit.

Odd harmonics

Approximation of odd harmonics in 233edo
Harmonic 3 5 7 9 11 13 15 17 19 21 23
Error Absolute (¢) -1.53 -0.05 -0.59 +2.10 -0.24 -1.04 -1.57 -1.95 +1.20 -2.11 +0.05
Relative (%) -29.6 -0.9 -11.4 +40.7 -4.8 -20.2 -30.6 -37.9 +23.3 -41.0 +1.0
Steps
(reduced) 		369
(136) 		541
(75) 		654
(188) 		739
(40) 		806
(107) 		862
(163) 		910
(211) 		952
(20) 		990
(58) 		1023
(91) 		1054
(122)

Subsets and supersets

233edo is the 51st prime edo.

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [-369 233⟩ [⟨233 369]] 0.4813 0.4815 9.35
2.3.5 78732/78125, [-53 32 1⟩ [⟨233 369 541]] 0.3277 0.4492 8.72
2.3.5.7 2401/2400, 65625/65536, 78732/78125 [⟨233 369 541 654]] 0.2979 0.3924 7.62
2.3.5.7.11 243/242, 441/440, 540/539, 2401/2400 [⟨233 369 541 654 806]] 0.2525 0.3625 7.04
2.3.5.7.11.13 243/242, 351/350, 441/440, 540/539, 1001/1000 [⟨233 369 541 654 806 862]] 0.2574 0.3311 6.43
2.3.5.7.11.13.17 351/350, 441/440, 540/539, 561/560, 936/935, 1156/1155 [⟨233 369 541 654 806 862 952]] 0.2888 0.3161 6.14

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
Ratio* 		Temperaments
1 15\233 77.25 256/245 Tertiaseptal
1 22\233 113.30 16/15 Misneb
1 55\233 283.26 189/160 Neominor
1 77\233 396.57 98304/78125 Squarschmidt
1 86\233 442.92 9/7 Sensi
1 95\233 489.27 250/189 Catafourth

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

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