197edo: Difference between revisions

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-'''197edo''' is the [[EDO|equal division of the octave]] into 197 parts of 6.0914 cents each. It gives excellent results for tuning both marvel, the planar temperament tempering out 225/224, and [[Kleismic_family|catakleismic]], the temperament [[tempering_out|tempering out]] both 225/224 and 4375/4374, which has wedgie &lt;&lt;6 5 22 -6 18 37||. Among [[Patent_val|patent vals]], in fact, it gives the best results for both. In fact, the [[11-limit]] patent val &lt;197 312 457 553 682| has a comma basis [225/224, 4375/4374, 441/440, 65536/65219], so taking 225/224 and 441/440 together ([[Marvel family|prodigy temperament]]) also works well with 197edo, and taking 225/224, 441/440, and 4375/4374 (an alternative 11-limit catakleismic) is once again excellently tuned by 197edo.
+{{Infobox ET}}
+{{ED intro}}
-If we use 197e, the &lt;197 312 457 553 681| val, we can also use 197edo as an excellent tuning for the 11-limit version of marvel temperament, tempering out 385/384 as well as 225/224. If we add 4375/4374 to the comma list for 11-limit [[Marvel family|marvel]], we get 11-limit catakleismic, and 197edo with the above val is also an excellent tuning for that.
+== Theory ==
+edo gives excellent results for tuning both [[marvel]], the planar temperament [[tempering out]] [[225/224]], and [[catakleismic]], the temperament tempering out both 225/224 and [[4375/4374]]. Among [[patent val]]s, it gives the best results for both. In fact, the [[11-limit]] patent val {{val| 197 312 457 553 682 }} has a [[comma basis]] {225/224, 441/440, 4375/4374, 65536/65219}, so taking 225/224 and [[441/440]] together ([[prodigy]] temperament) also works well with 197edo, and taking 225/224, 441/440, and 4375/4374 (an alternative 11-limit catakleismic) is once again excellently tuned by 197edo.
-edo is the 45th [[prime_numbers|prime]] edo.
+If we use 197e, the {{val| 197 312 457 553 681 }} val, we can also use 197edo as an excellent tuning for the 11-limit version of marvel temperament, tempering out [[385/384]] as well as 225/224. If we add 4375/4374 to the comma list for 11-limit marvel, we get 11-limit catakleismic, and 197edo with the above val is also an excellent tuning for that. The 197ef val, {{val| 197 312 457 553 681 728}}, is an excellent tuning for the 13-limit version of catakleismic.
-==Scales==
+=== Odd harmonics ===
-<ul><li>[[Prismarv]]</li><li>[[Marveldene]]</li><li>[[Pump12_1|Pump12_1]]</li></ul>[http://micro.soonlabel.com/pump_tunings/pump1/daily20111029-a-pump1-woodwinds-no-verb.mp3 Pump1 by Chris Vaisvil]
+{{Harmonics in equal|197}}
-<ul><li>[[Pump12_2|Pump12_2]]</li><li>[[Pump13]]</li><li>[[Pump14]]</li><li>[[Pump15]]</li><li>[[Pump16]]</li><li>[[Pump17]]</li><li>[[Pump18]]</li></ul>
+=== Subsets and supersets ===
+edo is the 45th [[prime 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| -312 197 }}
+| {{mapping| 197 312 }}
+| +0.4566
+| 0.4568
+| 7.50
+|-
+| 2.3.5
+| 15625/15552, {{monzo| -53 32 1 }}
+| {{mapping| 197 312 457 }}
+| +0.6717
+| 0.4813
+| 7.90
+|-
+| 2.3.5.7
+| 225/224, 4375/4374, {{monzo| -25 6 -3 8 }}
+| {{mapping| 197 312 457 553 }}
+| +0.5302
+| 0.4834
+| 7.94
+|}
+=== 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
+| 52\197
+| 316.75
+| 6/5
+| [[Catakleismic]]
+|-
+| 1
+| 53\197
+| 322.84
+| 3087/2560
+| [[Seniority]]
+|}
+<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 ==
+* [[Prismarv]]
+* [[Marveldene]]
+* [[Pump12 1]]
+* [[Pump12 2]]
+* [[Pump13]]
+* [[Pump14]]
+* [[Pump15]]
+* [[Pump16]]
+* [[Pump17]]
+* [[Pump18]]
+== Music ==
+; [[Francium]]
+* "Have You Ever Eaten A Square Cutlet?" from ''Questions'' (2024) – [https://open.spotify.com/track/7cA0T79hEkzEoMGvufGZ5R Spotify] | [https://francium223.bandcamp.com/track/have-you-ever-eaten-a-square-cutlet Bandcamp] | [https://www.youtube.com/watch?v=_LkriwwqbIY YouTube]
+; [[Chris Vaisvil]]
+* [http://micro.soonlabel.com/pump_tunings/pump1/daily20111029-a-pump1-woodwinds-no-verb.mp3 ''Pump1''] – in pump12 1
-[[Category:197edo| ]] <!-- main article -->
 [[Category:Catakleismic]]
-[[Category:Edo]]
 [[Category:Marvel]]
-[[Category:Prime EDO]]
-[[Category:todo:add_definition]]