144edo: Difference between revisions

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-'''144edo''' is the [[EDO|equal division of the octave]] into 144 parts of 8.3333 cents each, making it the square of [[12edo]] and the 12th number in the fibonacci sequence. It is closely related to [[72edo]], but the patent vals differ on the mapping for 13 and 17. It is contorted in the 11-limit, tempering out 225/224, 243/242, 385/384, 441/440, and 4000/3993. Using the patent val, it tempers out 847/845, 1188/1183, 1701/1690, 1875/1859, and 4225/4224 in the 13-limit; 273/272, 715/714, 833/832, 875/867, 891/884, and 1275/1274 in the 17-limit; 210/209, 325/323, 343/342, 363/361, 400/399, 513/512, and 665/663 in the 19-limit. It can produce extremely precise approximations of both the linear and logarithmic versions of the golden ratio, at 89/144 and 100/144 steps respectively.
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Edo]]
+edo's step size is called a '''farab''' when used as an [[interval size unit]].<ref>http://tonalsoft.com/enc/f/farab.aspx</ref>
+== Theory ==
+edo is closely related to [[72edo]], but the [[patent val]]s differ on the mapping for [[13/1|13]] and [[17/1|17]]. It is [[enfactoring|enfactored]] in the 11-limit, [[tempering out]] [[225/224]], [[243/242]], [[385/384]], [[441/440]], and [[4000/3993]]. Using the [[patent val]], it tempers out [[847/845]], [[1188/1183]], 1701/1690, 1875/1859, and [[4225/4224]] in the 13-limit. It [[support]]s [[hemisecordite]], the {{nowrap|41 &amp; 103}} temperament, though [[103edo]] is better suited for this purpose.
+Although the patent val comes out on top accuracy in the 13-limit, in the 17-limit 144 falls behind to 144g. The 144g val tempers out [[170/169]], [[289/288]], [[375/374]], [[561/560]], [[595/594]]. It supports [[semihemisecordite]], the {{nowrap|62 &amp; 82f}} temperament. The patent val tempers out [[273/272]], [[715/714]], [[833/832]], 875/867, 891/884, and [[1275/1274]], supporting 17-limit hemisecordite. In the 19-limit the patent val tempers out 210/209, 325/323, [[343/342]], [[363/361]], [[400/399]], [[513/512]], and 665/663.
+Besides all these, the 144eff val supports hemimiracle, the {{nowrap|41 &amp; 103e}} temperament. 144ee supports oracle, the {{nowrap|31 &amp; 113e}} temperament. 144cf supports necromanteion, the {{nowrap|31 &amp; 113cf}} temperament.
+=== Prime harmonics ===
+{{Harmonics in equal|144}}
+=== Subsets and supersets ===
+edo is the square of world-dominant [[12edo]].
+Since 144 factors into 2<sup>4</sup> × 3<sup>2</sup>, 144edo has subset edos {{EDOs| 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, and 72 }}.
+=== Approximation to φ ===
+edo is the 12th Fibonacci edo. As a consequence of being a Fibonacci edo, it can produce extremely precise approximation of the [[Logarithmic phi|logarithmic golden ratio]] at 89 steps. Coincidentally, it ''also'' excellently represents the [[Acoustic phi|acoustic golden ratio]] by 100 steps.
+== 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]] (%)
+|-
+| 2.3.5.7.11.13
+| 225/224, 243/242, 385/384, 847/845, 1875/1859
+| {{mapping| 144 228 334 404 498 533 }}
+| +0.560
+| 0.595
+| 7.13
+|- style="border-top: double;"
+| 2.3.5.7.11.13.17
+| 225/224, 243/242, 273/272, 325/324, 847/845, 875/867
+| {{mapping| 144 228 334 404 498 533 589 }} (144)
+| +0.362
+| 0.734
+| 8.80
+|- style="border-top: double;"
+| 2.3.5.7.11.13.17
+| 170/169, 225/224, 243/242, 289/288, 375/374, 385/384
+| {{mapping| 144 228 334 404 498 533 588 }} (144g)
+| +0.653
+| 0.596
+| 7.15
+|}
+=== 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
+| 7\144
+| 58.33
+| 27/26
+| [[Hemisecordite]] (144)
+|-
+| 2
+| 7\144
+| 58.33
+| 27/26
+| [[Semihemisecordite]] (144g)
+|-
+| 12
+| 1\144
+| 8.33
+| 129/128
+| [[Substitute harmonic#Dotcom|Dotcom]]
+|}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+== Music ==
+; [[Hideya]]
+* [https://www.youtube.com/watch?v=p71oF-A9gF8 ''Like an endless uphill''] (2022)
+== References ==
+<references />
+== External links ==
+* [http://tonalsoft.com/enc/number/144edo.aspx 144-tone equal-temperament / 144-edo] on [[Tonalsoft Encyclopedia]]
+[[Category:Listen]]