Insatinismic: Difference between revisions
m Little things |
29 and 31 sting like sons of a bitch, declare the pain despite the low badness and extremely small error |
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Insatinismic is a [[rank-5 temperament]] that is a [[weak extension]] of [[5767168/5767125#Insanismic|insanismic]] and a detemperament of [[ | Insatinismic is a [[Catalog of rank-5 temperaments|rank-5 temperament]] for the [[19-limit]] and no-23 [[31-limit]] that is a [[weak extension]] of [[5767168/5767125#Insanismic|insanismic]] and a detemperament of [[septischismic]]. It is not a member of any known temperament family or clan. | ||
Data below is given because infobox regtemp doesn't work with rank-5 temperaments:{{Databox|Temperament data|2=[[Subgroup]]: 2.3.5.7.11.13.17.19 | Data below is given because infobox regtemp doesn't work with rank-5 temperaments:{{Databox|Temperament data|2=[[Subgroup]]: 2.3.5.7.11.13.17.19 | ||
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Optimal ET sequence: {{EDOs|41g, 53, 94, 176g, 217, 270, 311, 487, 581, 1115g, 1385, 1696, 1966, 2597, 2814, 3084, 3178, 3395, 5144, 6573, 7958, 8269, 8539, 11934, 16808, 20203}}. | Optimal ET sequence: {{EDOs|41g, 53, 94, 176g, 217, 270, 311, 487, 581, 1115g, 1385, 1696, 1966, 2597, 2814, 3084, 3178, 3395, 5144, 6573, 7958, 8269, 8539, 11934, 16808, 20203}}. | ||
Badness (Sintel): 0.279}}Ploidacots, pergens, MOS scales and odd-limit detunings are unapplicable or too convoluted to be descriptive. | Badness (Sintel): 0.279<br> | ||
[[Subgroup]]: 2.3.5.7.11.13.17.19.29<br>[[Comma list]]: 5767168/5767125, [[11413376/11413325]], [[633556/633555]], [[176176/176175]] | |||
[[Mapping]]: [⟨ 1 0 0 0 5 8 -9 0 19 ], | |||
⟨ 0 1 0 0 0 0 -2 1 -3 ], | |||
⟨ 0 0 1 0 0 -1 -2 1 -3 ], | |||
⟨ 0 0 0 1 1 0 0 2 3 ], | |||
<nowiki>⟨ 0 0 0 0 -9 -3 8 -8 -21 ]]</nowiki> | |||
Mapping generators: ~2, ~3, ~5, ~7, ~608/385 | |||
[[Optimal tuning|Optimal tunings]] | |||
: [[WE]]: ~2 = 1199.999454¢, ~3/2 = 701.955619¢, ~5/4 = 386.314591¢, ~7/4 = 968.825991¢, ~385/608 = -791.051281¢ | |||
: error map: -0.000546 0.000618 0.000877 0.000084 -0.000591 -0.000463 0.00017 -0.001072 0.001239 | |||
Badness (Sintel): 0.354 | |||
[[Subgroup]]: 2.3.5.7.11.13.17.19.29.31<br>[[Comma list]]: 5767168/5767125, [[11413376/11413325]], [[633556/633555]], [[176176/176175]], [[178126/178125]] | |||
[[Mapping]]: [⟨ 1 0 0 0 5 8 -9 0 19 -8 ], | |||
⟨ 0 1 0 0 0 0 -2 1 -3 0 ], | |||
⟨ 0 0 1 0 0 -1 -2 1 -3 6 ], | |||
⟨ 0 0 0 1 1 0 0 2 3 2 ], | |||
<nowiki>⟨ 0 0 0 0 -9 -3 8 -8 -21 -10 ]]</nowiki> | |||
Mapping generators: ~2, ~3, ~5, ~7, ~608/385 | |||
[[Optimal tuning|Optimal tunings]] | |||
: [[WE]]: ~2 = TBA¢, ~3/2 = TBA¢, ~5/4 = TBA¢, ~7/4 = TBA¢, ~385/608 = TBA¢ | |||
: error map: TBA | |||
Badness (Sintel): 0.777}}The optimal patent val is much finer than 20203, but for all intents and purposes, 20203 marks is practically identical to insatinismic. | |||
Ploidacots, pergens, MOS scales and odd-limit detunings are unapplicable or too convoluted to be descriptive. | |||
Edo join: 53 & 94 & 270 & 3395 & 8539 (19-limit); 53k & 217k & 311k & 3395k & 8539 (31-limit) | |||
As a detemperament of cassaschismic, it does the following: | As a detemperament of cassaschismic, it does the following: | ||
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As a weak extension of insanismic, it does the following: | As a weak extension of insanismic, it does the following: | ||
* The fifth generator is trisected for primes 11 and | * The fifth generator is trisected for primes 11, 13 and 29, and can be instead taken to be ~1540/1539, or a redundant pair of 4096/4095~4225/4224 and 10241/10240. 10241/10240 is used to reached 17/16, 19/16 and 31/16. | ||
=== Add-29,31 extension === | |||
Insatinismic is much worse at extending to the no-23 31-limit despite the great accuracy, since it requires detempering from the suboptimal septischismic add-29,31 basis 53k & 217k & 311k, instead of 217 & 270 & 311 or 12e & 270 & 311. This adds the commas 176176/176175 and 178126/178125 into the list, where prime 29 has the most error out of all, around ~3x as much - still, just 0.003 cents. Prime 31 fits much better into the temperament, though it is the worst to write. | |||
== Notation == | == Notation == | ||
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=== Minimal Insatinismic notation{{Idiosyncratic}} === | === Minimal Insatinismic notation{{Idiosyncratic}} === | ||
This notation is a direct mapping retrieved from [[Sintel]]'s temperament evaluator [https://sintel.pythonanywhere.com/result?subgroup=19&reduce=spine&weights= | This notation is a direct mapping retrieved from [[Sintel]]'s temperament evaluator [https://sintel.pythonanywhere.com/result?subgroup=2.3.5.7.11.13.17.19.29&reduce=spine&weights=weil&target=&edos=53%2C94%2C217%2C3395%2C8539&submit_edo=submit&commas= here], shuffling the generators to have them be in decreasing order of size. Either way, it requires 5 pairs of accidentals to fully notate on a score. | ||
* Sharps/Flats | * Sharps/Flats | ||
* Pythagorean commas | * Pythagorean commas | ||
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* 17/16: '''major''' second - 4 pythagorean commas + 2 schismas - 8 kevolismas | * 17/16: '''major''' second - 4 pythagorean commas + 2 schismas - 8 kevolismas | ||
* 19/16: minor third + schisma - 2 garischismas + 8 kevolismas | * 19/16: minor third + schisma - 2 garischismas + 8 kevolismas | ||
* 29/16: '''major sixth''' + 5 pythagorean commas - 3 schismas - 3 garischismas + 21 kevolismas | |||
* 31/16: major seventh + TBA | |||
The notation can be equivalently and naively described as using: | The notation can be equivalently and naively described as using: | ||
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Any of the above expressions can be rewritten by converting syntonic comma = pythagorean comma - schisma; septimal comma = pythagorean comma + garischisma. | Any of the above expressions can be rewritten by converting syntonic comma = pythagorean comma - schisma; septimal comma = pythagorean comma + garischisma. | ||
A redundant accidental, the [[mercator comma]], can be introduced to keep the amount of pythagorean commas needed less than two, since it is the difference between 4 pythagorean commas and one limma. 17/16 is a type of '''major''' second. Using a mercator comma allows it to be respelled as a minor second + pythagorean comma - mercator comma... etc. | A redundant accidental, the [[mercator comma]], can be introduced to keep the amount of pythagorean commas needed less than two, since it is the difference between 4 pythagorean commas and one limma. 17/16 is a type of '''major''' second as in [[Septischismic family#Septinoellismic|septinoellismic]]. Using a mercator comma allows it to be respelled as a minor second + pythagorean comma - mercator comma... etc. Same with 29/16, which instead of being spelt a major sixth can be spelt a minor seventh + pythagorean comma + mercator comma... etc. | ||
=== Eufalesio's ULTRAOLYMPIC [sic] notation === | === Eufalesio's ULTRAOLYMPIC [sic] notation === | ||
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* 19/16: minor third + schisma + 3 minas + tina | * 19/16: minor third + schisma + 3 minas + tina | ||
And any interval between them can be found by subtracting each accidental independently. | And any interval between them can be found by subtracting each accidental independently, keeping in mind that an excess of septimal commas can be converted into buzzardmas. | ||
Not shown in the image are 29/16 and 31/16, which are extremely complex: | |||
* 29/16: minor seventh + septimal comma + buzzardma - aberschisma - 2 schismas - 7 minas <br>OR minor seventh + septimal comma + buzzardma - 3 aberschismas + 10 minas - 3 tinas | |||
* 31/16: major seventh + septimal comma - buzzardma + 5 aberschismas + schisma - 9 minas - tina | |||
As such, it is recommended to use a [[3969/3968|yunzee comma]] to notate 31/16 and 29/16, if strictly sticking to this system, producing: | |||
* 29/16: minor seventh + buzzardma + 2 aberschismas + schisma + two tinas + yunzee | |||
* 31/16: octave - 2 septimal commas - yunzee (by definition). | |||