Insatinismic: Difference between revisions
Public birth, now to make it less idiosyncratic, more neutral! |
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: | ||
* It decouples the septimal comma from the chain of fifths (observing the garischisma), | * It decouples the septimal comma from the chain of fifths (observing the garischisma), | ||
* It observes 4096/4095 and 4225/4224 but tempers them together (observing the olympia and | * It observes [[4096/4095]] and [[4225/4224]] but tempers them together (observing the [[olympia]] and [[2080/2079|ibnsinma]] too), as seen in insanismic | ||
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 | 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 === | ||
[[Eufalesio]] had devised a somewhat accessible notation of differently sized accidentals can be devised by reusing the suit of accidentals that is used for 12e, cassandra, | [[Eufalesio]] had devised a somewhat accessible notation of differently sized accidentals can be devised by reusing the suit of accidentals that is used for 12e, cassandra, cassaschismic, and expanding upon them to accomodate the minuscule alterations. This has the benefit of being a natural representation of [[olympic]], keeping the representations of primes 2-13,19 as simple as possible. | ||
[[File:ULTRAOLYMPIANsymbolset.jpg|center|thumb|960x960px]] | [[File:ULTRAOLYMPIANsymbolset.jpg|center|thumb|960x960px]] | ||
Buzzardmas are to ULTRAOLYMPIC notation what mercator commas are to minimal. They are the difference between 4 arrow-unison and a minor second, only that the arrow here is a septimal comma, not a pythagorean one. | Buzzardmas are to ULTRAOLYMPIC notation what mercator commas are to minimal. They are the difference between 4 arrow-unison and a minor second, only that the arrow here is a '''septimal''' comma, not a pythagorean one. | ||
Thus, as seen in the notation, octave reduced harmonics can be described as follows: | Thus, as seen in the notation, octave reduced harmonics can be described as follows: | ||
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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). | |||