202edo: Difference between revisions

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 {{Infobox ET}}
-The '''202 equal temperament''' divides the octave into 202 equal parts of 5.941 cents each.
+{{ED intro}}
-et tempers out [[2401/2400]], 19683/19600 and 65625/65536 in the 7-limit, and [[243/242]], [[441/440]], [[4000/3993]] in the 11-limit. It also notably tempers out the [[quartisma]]. It is the [[optimal patent val]] for the 11-limit rank-2 temperaments [[harry]] and [[tertiaseptal]], the rank-3 temperament [[jove]] tempering out 243/242 and 441/440, and the rank-4 rastmic temperament tempering out 243/242.
+== Theory ==
-{{harmonics in equal|202|start=2|prec=3}}
+edo is [[consistent]] to the [[9-odd-limit]] with a flat tendency in harmonics [[3/1|3]], [[5/1|5]], and [[7/1|7]]. It also has a decent harmonic [[11/1|11]], though it is sharp unlike the previous harmonics, with [[11/9]] barely exceeding 50% [[relative interval error|relative error]]. Despite this, it is most notable in the [[11-limit]], providing the [[optimal patent val]] for many temperaments tempering out [[243/242]].
+Using the patent val, 202et [[tempering out|tempers out]] [[2401/2400]], [[19683/19600]] and [[65625/65536]] in the [[7-limit]], and [[243/242]], [[441/440]], [[4000/3993]] in the 11-limit. It also notably tempers out the [[quartisma]], equating a stack of five [[33/32]] quartertones with [[7/6]]. It is the [[optimal patent val]] for the 11-limit rank-2 temperaments [[harry]] and [[tertiaseptal]], the rank-3 temperament [[jove]] tempering out 243/242 and 441/440, which also tempers out [[540/539]], and the rank-4 [[rastmic]] temperament, which tempers out 243/242.
+It extends less well to the [[13-limit]], with harmonic [[13/1|13]] being about halfway between its steps. Nonetheless, the patent val tempers out [[351/350]], [[364/363]], [[676/675]], [[729/728]], and [[2080/2079]], supporting [[breed family #Jovial|jovial]] and [[breed family #Jovis|jovis]], as well as 13-limit harry. Primes [[17/1|17]] and [[23/1|23]] are quite sharp, but prime [[19/1|19]] is accurate. 202edo can thus be considered a 2.3.5.7.11.13.19-subgroup temperament with a mostly flat tendency, with the exception of prime 11. The intervals [[11/9]], [[13/11]], and their octave complements are the only inconsistencies in the no-17 [[21-odd-limit]], and the no-11 no-17 21-odd limit is completely consistent, though one may also want to exclude prime 13 given its inaccuracy, giving us the 2.3.5.7.19 subgroup.
+=== Prime harmonics ===
+{{Harmonics in equal|202}}
+=== Subsets and supersets ===
+Since 202 factors into {{nowrap| 2 × 101 }}, 202edo contains [[2edo]] and [[101edo]] as subset edos.
+== 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
+| {{Monzo| -13 17 -6 }}, {{monzo| 23 6 -14 }}
+| {{Mapping| 202 320 469 }}
+| +0.2280
+| 0.2710
+| 4.56
+|-
+| 2.3.5.7
+| 2401/2400, 19683/19600, 65625/65536
+| {{Mapping| 202 320 469 567 }}
+| +0.2164
+| 0.2356
+| 3.97
+|-
+| 2.3.5.7.11
+| 243/242, 441/440, 4000/3993, 65625/65536
+| {{Mapping| 202 320 469 567 699 }}
+| +0.1061
+| 0.3049
+| 5.13
+|}
+=== 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
+| 13\202
+| 77.23
+| 256/245
+| [[Tertiaseptal]]
+|-
+| 1
+| 51\202
+| 302.97
+| 25/21
+| [[Quinmite]]
+|-
+| 1
+| 85\202
+| 504.95
+| 104976/78125
+| [[Countermeantone]]
+|-
+| 1
+| 87\202
+| 516.83
+| 27/20
+| [[Larry]]
+|-
+| 2
+| 12\202
+| 71.29
+| 25/24
+| [[Narayana]]
+|-
+| 2
+| 87\202<br>(14\202)
+| 516.83<br>(83.17)
+| 27/20<br>(21/20)
+| [[Harry]]
+|}
+<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 ==
-* [[Jove1]]
+* [[Jove1]], [[jove2]], [[jove3]], [[jove4]], [[jove5]], [[jove6]]
-* [[Jove2]]
+* [[Elfjove7]], [[elfjove8d]], [[elfjove10]], [[elfjove11c]], [[elfjove12]]
-* [[Jove3]]
+* [[Oktone]]
-* [[Jove4]]
-* [[Jove5]]
+== Music ==
-* [[Jove6]]
+; [[Mundoworld]]
+* [https://www.youtube.com/watch?v=_bNbb2o5K80 ''Home Planet Nostalgia''] – in Oktone scale
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Harry]]
 [[Category:Tertiaseptal]]
 [[Category:Jove]]
 [[Category:Rastmic]]
-[[Category:Quartismic]]
+[[Category:Listen]]