5767168/5767125: Difference between revisions

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-'''5767168/5767125''', the '''insanisma''', is a [[13-limit]] interval, one of the smallest 13-limit [[comma]]s. It is an [[ultraparticular]], the difference between [[4096/4095]] and [[4225/4224]], or S64/S65 [[S-expression|S-expressed]].
+'''5767168/5767125''', the '''insanisma''', is a [[13-limit]] interval. It is one of the smallest 13-limit [[comma]]s and the smallest 13-limit [[ultraparticular]], the difference between [[4096/4095]] and [[4225/4224]], or S64/S65 [[S-expression|S-expressed]].
 As an ultraparticular, it is the difference between [[22/21]] and ([[65/64]])<sup>3</sup>. It also splits the [[olympia]] in three and the [[ibnsinma]] in two, each for 4096/4095~4225/4224.
+In terms of commas, it is also ([[1990656/1990625]])/([[123201/123200]]).
 == Temperaments ==
-This comma is trivially tempered out in [[olympic]], as 4096/4095 and 4225/4224 are also tempered out there. Thus, any temperament of olympic also tempers it out. See [[Olympic clan|here]] for a compendium of temperaments and edos that temper out this comma due to being olympian.
+This comma is trivially tempered out in [[olympic|(tridecimal) olympic]], as 4096/4095 and 4225/4224 are also tempered out there. Thus, any temperament of olympic also tempers it out. See [[Olympic clan|here]] for a compendium of temperaments and edos that temper out this comma due to being olympic.
 Other non-olympian edos that temper out the insanisma by patent val, but not either of S64 and S65 include {{EDOs| 35, 40, 49 }}, and some hyperfine edos like [[3395edo|3395]] and [[8539edo|8539]].
+=== Insanismic ===
+[[Subgroup]]: 2.3.5.7.11.13
+[[Comma list]]: 5767168/5767125
+[[Mapping]]:
+{| class="right-all"
+|[⟨
+|1
+|0
+|0
+|0
+|2
+|7
+|],
+|-
+|⟨
+|0
+|1
+|0
+|0
+|0
+|0
+|],
+|-
+|⟨
+|0
+|0
+|1
+|0
+|0
+| -1
+|],
+|-
+|⟨
+|0
+|0
+|0
+|1
+|1
+|0
+|],
+|-
+|⟨
+|0
+|0
+|0
+|0
+|3
+|1
+|]]
+|}
+Mapping generators: ~2, ~3, ~5, ~7, ~65/128
+[[Optimal tuning|Optimal tunings]]:
+: [[WE]]: ~2 = 1199.999558¢, ~3/2 = 701.955501¢, ~5/4 = 386.314974¢, ~7/4 = 968.826974¢, ~65/128 = -1173.1542¢
+: error map: -0.000442 0.000058 0.000376 0.000183 -0.000278 0.000955
+: [[CWE]]: ~2 = 1200.¢, ~3/2 = 701.955606¢, ~5/4 = 386.315031¢, ~7/4 = 968.827097¢, ~65/128 = -1173.154680¢
+: error map: 0.000000 0.000605 0.001318 0.001190 0.000720 0.002627
+[[Optimal ET sequence]]: [[7edo|7]], [[10edo|10e]], [[12edo|12e]], [[30edo|30b]], [[31edo|31e]], [[34edo|34]], [[36edo|36]], [[41edo|41]], [[46edo|46]], [[53edo|53]], [[84edo|84]], [[87edo|87]], [[130edo|130]], [[183edo|183]], [[217edo|217]], [[224edo|224]], [[270edo|270]], [[494edo|494]], [[764edo|764]], [[935edo|935]], [[1075edo|1075]], [[1205edo|1205]], [[1609edo|1609]], [[1696edo|1696]], [[1920edo|1920]], [[2190edo|2190]], [[2684edo|2684]], [[3395edo|3395]], [[5144edo|5144]], [[5585edo|5585]], [[6079edo|6079]], [[8269edo|8269]], [[8539edo|8539]], [[11664edo|11664]], [[14124edo|14124]], [[14348edo|14348]], [[14618edo|14618]], [[20203edo|20203]]
+Badness (Sintel): 0.241
+For the [[19-limit]] weak extension, see [[insatinismic]].
+==== 2.3.5.7.11.13.29 subgroup ====
+[[Comma list]]: 5767168/5767125, [[176176/176175]]
+{| class="right-all"
+|[⟨
+|1
+|0
+|0
+|0
+|2
+|7
+|15
+|],
+|-
+|⟨
+|0
+|1
+|0
+|0
+|0
+|0
+| -3
+|],
+|-
+|⟨
+|0
+|0
+|1
+|0
+|0
+| -1
+| -3
+|],
+|-
+|⟨
+|0
+|0
+|0
+|1
+|1
+|0
+|3
+|],
+|-
+|⟨
+|0
+|0
+|0
+|0
+|3
+|1
+|7
+|]]
+|}[[Optimal tuning|Optimal tunings]]:
+: [[WE]]: ~2 = 1199.999527¢, ~3/2 = 701.95558¢, ~5/4 = 386.314433¢, ~7/4 = 968.825693¢, ~65/128 = -1173.154534¢
+: error map: -0.000473  0.000579  0.000719 -0.000214 -0.001218  0.000062  0.001014
+Badness (Sintel): 0.211
 == Etymology ==
-This comma was named by [[Eufalesio]] in 2026, on the basis that tempering out this comma alone purportedly entails an ''insane'' level of precision.
+This comma was named by [[Eufalesio]] in 2026, on the basis that tempering out this comma alone purportedly entails a ''insane'' level of precision.
 [[Category:Insanismic]]
 [[Category:Commas named for other reasons]]