233edo: Difference between revisions

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 {{Infobox ET}}
-'''233edo''' is the [[EDO|equal division of the octave]] into 233 parts of 5.1502 [[cent]]s each. It 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). It tempers out 78732/78125 and |-53 32 1&gt; 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.
+{{ED intro}}
+== Theory ==
+et 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{{c}} flat), but less for the [[3/1|third harmonic]] (1.5258{{c}} 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.
+=== Odd harmonics ===
+{{Harmonics in equal|233}}
+=== Subsets and supersets ===
 edo is the 51st [[prime edo]].
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+== Regular temperament properties ==
-[[Category:Prime EDO]]
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3
+| {{Monzo| -369 233 }}
+| {{Mapping| 233 369 }}
+| +0.4813
+| 0.4815
+| 9.35
+|-
+| 2.3.5
+| 78732/78125, {{monzo| -53 32 1 }}
+| {{Mapping| 233 369 541 }}
+| +0.3277
+| 0.4492
+| 8.72
+|-
+| 2.3.5.7
+| 2401/2400, 65625/65536, 78732/78125
+| {{Mapping| 233 369 541 654 }}
+| +0.2979
+| 0.3924
+| 7.62
+|-
+| 2.3.5.7.11
+| 243/242, 441/440, 540/539, 2401/2400
+| {{Mapping| 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
+| {{Mapping| 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
+| {{Mapping| 233 369 541 654 806 862 952 }}
+| +0.2888
+| 0.3161
+| 6.14
+|}
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br>per 8ve
+! Generator*
+! Cents*
+! Associated<br>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
+| 162/125
+| [[Sensipent]]
+|-
+| 1
+| 95\233
+| 489.27
+| 250/189
+| [[Catafourth]]
+|}
+<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
+== Music ==
+; [[Francium]]
+* "Cuckoo Cider Mouth" from ''Cursed Cuckoo Creations'' (2024) – [https://open.spotify.com/track/7BizYOfoxjMncUJaGEfL0u Spotify] | [https://francium223.bandcamp.com/track/cuckoo-cider-mouth Bandcamp] | [https://www.youtube.com/watch?v=t1hczh8fRQ4 YouTube]
+* "goodblucky" from ''albumwithoutspaces'' (2024) – [https://open.spotify.com/track/6uq6TWCajp0WueDjfr77ox Spotify] | [https://francium223.bandcamp.com/track/goodblucky Bandcamp] | [https://www.youtube.com/watch?v=4YATiqIwlqE YouTube]
+* "Desks For Ladies" from ''Check Us Out'' (2026) – [https://open.spotify.com/track/0tLYtnNlBluBNapUOS322P Spotify] | [https://francium223.bandcamp.com/track/desks-for-ladies Bandcamp] | [https://www.youtube.com/watch?v=MLd7ij1C3kc YouTube]
+[[Category:Listen]]