684edo: Difference between revisions

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-{{Infobox ET
+{{Infobox ET}}
-| Prime factorization = 2<sup>2</sup> × 3<sup>2</sup> × 19
+{{ED intro}}
-| Step size = 1.75439¢
-| Fifth = 400\684 (701.75¢) (→ [[171edo|100\171]])
-| Semitones = 64:52 (112.28¢ : 91.23¢)
-| Consistency = 17
-}}
-{{EDO intro|684}}
 == Theory ==
-edo divides the steps of [[171edo]] into four. It is [[consistent]] to the 17-odd-limit, tempering out [[2401/2400]], [[3025/3024]], [[4225/4224]], [[4375/4374]], and [[32805/32768]] in the 13-limit; [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], 2058/2057, 2500/2499, 8624/8619, and 14875/14872 in the 17-limit.
+edo divides the steps of [[171edo]] into four. It is [[consistent]] to the [[17-odd-limit]], [[tempering out]] [[2401/2400]], [[3025/3024]], [[4225/4224]], [[4375/4374]], and [[32805/32768]] in the 13-limit; [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], [[2058/2057]], [[2500/2499]], 8624/8619, and 14875/14872 in the 17-limit.
 === Prime harmonics ===
 {{Harmonics in equal|684|columns=11}}
+=== Subsets and supersets ===
+Since 684 factors into {{factorization|684}}, 684edo has subset edos {{EDOs| 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, and 342 }}.
+== Approximation to JI ==
+=== Zeta peak index ===
+{{ZPI
+| zpi = 5818
+| steps = 683.938934890938
+| step size = 1.75454260429165
+| tempered height = 14.267321
+| pure height = 7.268914
+| integral = 1.773752
+| gap = 20.109967
+| octave = 1200.10714133549
+| consistent = 18
+| distinct = 18
+}}
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
 ! rowspan="2" | Optimal<br>8ve stretch (¢)
@@ Line 27: / Line 40: @@
 | 2.3.5.7.11.13
 | 2401/2400, 3025/3024, 4225/4224, 4375/4374, 32805/32768
-| [{{val| 684 1084 1588 1920 2366 2531 }}]
+| {{mapping| 684 1084 1588 1920 2366 2531 }}
 | +0.0994
 | 0.0558
@@ Line 33: / Line 46: @@
 |-
 | 2.3.5.7.11.13.17
-| 1089/1088, 1701/1700, 2025/2023, 2058/2057, 4225/4224, 13013/13005
+| 1089/1088, 1225/1224, 1701/1700, 2025/2023, 4225/4224, 13013/13005
-| [{{val| 684 1084 1588 1920 2366 2531 2796 }}]
+| {{mapping| 684 1084 1588 1920 2366 2531 2796 }}
 | +0.0744
 | 0.0800
 | 4.56
 |}
-* 684et is the first equal temperament with a lower 13-limit absolute error after 494. The next equal temperament that is better tuned is 764.
+* 684et is the first equal temperament past [[494edo|494]] with a lower 13-limit absolute error. The next equal temperament that is better tuned is [[764edo|764]].
 === Rank-2 temperaments ===
+Note: 11-limit temperaments supported by [[342edo|342et]] are not shown.
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per Octave
+|-
-! Generator<br>(Reduced)
+! Periods<br>per 8ve
-! Cents<br>(Reduced)
+! Generator*
-! Associated<br>Ratio
+! Cents*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 55: / Line 71: @@
 | 1053/800<br>(1287/1280)
 | [[Semihemiennealimmal]]
+|-
+| 38
+| 151\684<br>(7\684)
+| 264.91<br>(12.28)
+| 500/429<br>(144/143)
+| [[Semihemienneadecal]]
 |}
+<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 ==
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+* [[15-odd-limit|Diamond15]]: 64 4 5 6 7 8 10 12 16 9 11 13 15 18 7 15 18 10 11 25 22 8 7 11 20 11 7 8 22 25 11 10 18 15 7 18 15 13 11 9 16 12 10 8 7 6 5 4 64