653edo: Difference between revisions

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-'''653edo''' is the [[EDO|equal division of the octave]] into 653 parts of 1.837672 [[cent]]s each. It is consistent to the [[21-odd-limit|21-limit]], tempering out 68719476736000/68630377364883 ([[tricot comma]]) and |-20 -24 25&gt; ([[counterhanson comma]]) in the 5-limit; 2401/2400, 65625/65536, and 7656250000000/7625597484987 in the 7-limit; 3025/3024, 41503/41472, 496125/495616, and 1953125/1948617 in the 11-limit; 2080/2079, 4459/4455, 6656/6655, 10985/10976, and 170625/170368 in the 13-limit; 1225/1224, 2058/2057, 2431/2430, 2500/2499, 4914/4913, and 11271/11264 in the 17-limit; 1445/1444, 1521/1520, 1540/1539, 1729/1728, 3136/3135, 4200/4199, and 4394/4389 in the 19-limit.
+{{Infobox ET}}
+{{ED intro}}
-edo is the 119th [[prime EDO]].
+== Theory ==
+edo is [[consistency|distinctly consistent]] to the [[21-odd-limit]], and the [[23-odd-limit]] if not for the [[23/13]] and its [[octave complement]] barely missing the mark. Although the [[25/1|25]] is flat enough to create more inconsistencies, the [[29/1|29]] and [[31/1|31]] blend well with the lower primes, together making it a fairly strong [[31-limit]] system.
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+As an equal temperament, it [[tempering out|tempers out]] {{monzo| 39 -29 3 }} ([[alphatricot comma]]) and {{monzo| -20 -24 25 }} ([[counterhanson comma]]) in the [[5-limit]]; [[2401/2400]], [[65625/65536]], and {{monzo| 7 -27 13 2 }} in the [[7-limit]]; [[3025/3024]], [[41503/41472]], 496125/495616, and 1953125/1948617 in the [[11-limit]]; [[2080/2079]], [[4459/4455]], [[6656/6655]], [[10985/10976]], and 170625/170368 in the [[13-limit]]; [[1225/1224]], [[2058/2057]], [[2431/2430]], [[2500/2499]], [[4914/4913]], and 11271/11264 in the [[17-limit]]; [[1445/1444]], [[1521/1520]], [[1540/1539]], [[1729/1728]], [[3136/3135]], [[4200/4199]], and 4394/4389 in the [[19-limit]]; [[875/874]], [[1105/1104]] among others in the [[23-limit]].
-[[Category:Prime EDO]]
+=== Prime harmonics ===
+{{Harmonics in equal|653|columns=11}}
+{{Harmonics in equal|653|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 653edo (continued)}}
+=== Subsets and supersets ===
+edo is the 119th [[prime edo]]. As such, it does not contain any nontrivial subset edo.
+== 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| 1035 -653 }}
+| {{Mapping| 653 1035 }}
+| −0.0113
+| 0.0113
+| 0.61
+|-
+| 2.3.5
+| {{Monzo| 39 -29 3 }}, {{monzo| -20 -24 25 }}
+| {{Mapping| 653 1035 1516 }}
+| +0.0503
+| 0.0875
+| 4.76
+|-
+| 2.3.5.7
+| 2401/2400, 65625/65536, {{monzo| 7 -27 13 2 }}
+| {{Mapping| 653 1035 1516 1833 }}
+| +0.0709
+| 0.0838
+| 4.56
+|-
+| 2.3.5.7.11
+| 2401/2400, 3025/3024, 65625/65536, 1953125/1948617
+| {{Mapping| 653 1035 1516 1833 2259 }}
+| +0.0576
+| 0.0795
+| 4.33
+|-
+| 2.3.5.7.11.13
+| 2080/2079, 2401/2400, 3025/3024, 10985/10976, 65625/65536
+| {{Mapping| 653 1035 1516 1833 2259 2416 }}
+| +0.0801
+| 0.0882
+| 4.80
+|-
+| 2.3.5.7.11.13.17
+| 1225/1224, 2058/2057, 2080/2079, 2401/2400, 4914/4913, 10985/10976
+| {{Mapping| 653 1035 1516 1833 2259 2416 2669 }}
+| +0.0759
+| 0.0823
+| 4.48
+|-
+| 2.3.5.7.11.13.17.19
+| 1225/1224, 1445/1444, 1521/1520, 1540/1539, 2058/2057, 2080/2079, 2401/2400
+| {{Mapping| 653 1035 1516 1833 2259 2416 2669 2774 }}
+| +0.0608
+| 0.0867
+| 4.72
+|-
+| 2.3.5.7.11.13.17.19.23
+| 875/874, 1105/1104, 1225/1224, 1445/1444, 1521/1520, 1540/1539, 2058/2057, 2080/2079
+| {{Mapping| 653 1035 1516 1833 2259 2416 2669 2774 2954 }}
+| +0.0489
+| 0.0884
+| 4.81
+|}
+=== 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
+| 21\653
+| 38.59
+| 45/44
+| [[Hemitert]]
+|-
+| 1
+| 42\653
+| 77.18
+| 256/245
+| [[Tertiaseptal]]
+|-
+| 1
+| 172/653
+| 316.08
+| 6/5
+| [[Counterhanson]]
+|-
+| 1
+| 308/653
+| 566.00
+| 81920/59049
+| [[Alphatricot]]
+|}
+<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]]
+* "Seamless Toggle-Style" from ''Take Advantage'' (2024) – [https://open.spotify.com/track/6nGhaXwzE85erYFtxGB9Dt Spotify] | [https://francium223.bandcamp.com/track/seamless-toggle-style Bandcamp] | [https://www.youtube.com/watch?v=TQP9W0vIvqw YouTube]