167edo: Difference between revisions

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-'''167edo''' is the [[EDO|equal division of the octave]] into 167 parts of 7.18562874251 [[cent]]s each. It [[tempering_out|tempers out]] the [[Würschmidt family|würschmidt comma]], 393216/390625 and 10737418240/10460353203 in the [[5-limit]]; 2401/2400, 3136/3125, and 179200/177147 in the [[7-limit]]; 896/891, 2200/2187, and 3388/3375 in the [[11-limit]]; 325/324, 352/351, 364/363, 1001/1000, and 1716/1715 in the [[13-limit]]; 256/255, 442/441, 595/594, 715/714, and 936/935 in the [[17-limit]]. It provides the [[optimal patent val]] for 11-limit [[Breedsmic temperaments|unthirds temperament]] and 13- and 17-limit [[Porwell temperaments|polypyth temperament]].
+{{Infobox ET}}
+{{ED intro}}
-edo is the 39th [[prime EDO]].
+== Theory ==
+et [[tempering out|tempers out]] the [[würschmidt comma]], 393216/390625, and the leapday comma, {{monzo| 31 -21 1 }}, in the [[5-limit]]; [[2401/2400]], [[3136/3125]], [[6144/6125]], and 179200/177147 in the [[7-limit]]; [[896/891]], [[2200/2187]], [[3025/3024]], [[3388/3375]], and [[4000/3993]] in the [[11-limit]]; [[325/324]], [[352/351]], [[364/363]], [[1001/1000]], and [[1716/1715]] in the [[13-limit]], providing the [[optimal patent val]] for 11- and 13-limit [[polypyth]] temperament; [[256/255]], [[442/441]], [[595/594]], [[715/714]], and [[936/935]] in the [[17-limit]]. It also [[support]]s the 11-limit [[unthirds]] temperament.
-[[Category:Edo]]
+edo also has a very close approximation to the [[golden magic]] scale.
-[[Category:Prime EDO]]
-[[Category:Theory]]
+=== Prime harmonics ===
+{{Harmonics in equal|167|intervals=prime|columns=12}}
+{{Harmonics in equal|167|intervals=prime|columns=12|start=13|title=Approximation of prime harmonics in 167edo (continued)|collapsed=1}}
+=== Subsets and supersets ===
+edo is the 39th [[prime edo]].
+== Intervals ==
+{{Main|Table of 167edo intervals}}
+== Regular temperament properties ==
+{| 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| 265 -167 }}
+| {{mapping| 167 265 }}
+| −0.7056
+| 0.7052
+| 9.81
+|-
+| 2.3.5
+| 393216/390625, {{monzo| 31 -21 1 }}
+| {{mapping| 167 265 388 }}
+| −0.7158
+| 0.5759
+| 8.01
+|-
+| 2.3.5.7
+| 2401/2400, 3136/3125, 179200/177147
+| {{mapping| 167 265 388 469 }}
+| −0.6467
+| 0.5129
+| 7.14
+|-
+| 2.3.5.7.11
+| 896/891, 2200/2187, 2401/2400, 3136/3125
+| {{mapping| 167 265 388 469 578 }}
+| −0.6315
+| 0.4598
+| 6.40
+|-
+| 2.3.5.7.11.13
+| 325/324, 352/351, 364/363, 1001/1000, 1716/1715
+| {{mapping| 167 265 388 469 578 618 }}
+| −0.5349
+| 0.4721
+| 6.57
+|-
+| 2.3.5.7.11.13.17
+| 256/255, 325/324, 352/351, 364/363, 442/441, 1001/1000
+| {{mapping| 167 265 388 469 578 618 683 }}
+| −0.5573
+| 0.4405
+| 6.13
+|}
+=== 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
+| 27\167
+| 194.01
+| 28/25
+| [[Hemiwürschmidt]]
+|-
+| 1
+| 44\167
+| 316.17
+| 6/5
+| [[Counterhanson]]
+|-
+| 1
+| 54\167
+| 388.02
+| 5/4
+| [[Würschmidt]]
+|-
+| 1
+| 58\167
+| 416.77
+| 14/11
+| [[Unthirds]] (11-limit)
+|-
+| 1
+| 63\167
+| 452.69
+| 125/96
+| [[Majo]]
+|-
+| 1
+| 69\167
+| 495.81
+| 4/3
+| [[Polypyth]]
+|-
+| 1
+| 70\167
+| 502.99
+| 147/110
+| [[Quadrawürschmidt]]
+|-
+| 1
+| 78\167
+| 560.48
+| 242/175
+| [[Whoops]]
+|}
+<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
+== Scales ==
+* [[Helayo19]]
+== Music ==
+; [[Francium]]
+* "way too random partying" from ''Helayo EP'' (2023) – [https://open.spotify.com/track/4yf5R4eVOxK2fgZEZRfCqU Spotify] | [https://francium223.bandcamp.com/track/way-too-random-partying Bandcamp] | [https://youtu.be/33T11NI7EQQ?si=mZ57p2EN4uvPCVo7 YouTube] – in Helayo, 167edo tuning
+* "moving on" from ''hope in dark times'' (2024) – [https://open.spotify.com/track/5h0JcJ4YTQV20CB9N8S8Af Spotify] | [https://francium223.bandcamp.com/track/moving-on Bandcamp] | [https://www.youtube.com/watch?v=FSjU0-w6XVE YouTube]
+* "ordering the universal theme on wish" from ''End of Sartorius Membranes'' (2024) – [https://open.spotify.com/track/00S85fGWQBI19kRwC9GrJ2 Spotify] | [https://francium223.bandcamp.com/track/ordering-the-universal-theme-on-wish Bandcamp] | [https://www.youtube.com/watch?v=g70V2NIPq1I YouTube]
+* "Funky Man's Love" from ''Abbreviations Gone Wrong'' (2024) – [https://open.spotify.com/track/0ILOgCY4pzx7S3B51wA9ee Spotify] | [https://francium223.bandcamp.com/track/funky-mans-love Bandcamp] | [https://www.youtube.com/watch?v=4Evj3vX8ZDY YouTube]
+* "Don't Bother" from ''Don't'' (2025) – [https://open.spotify.com/track/5B9LMtfG3wTNgQX0PKBFO3 Spotify] | [https://francium223.bandcamp.com/track/dont-bother Bandcamp] | [https://www.youtube.com/watch?v=kzlP4bWfQf8 YouTube]
+[[Category:Listen]]