Insatinismic: Difference between revisions
Add-29,31 are a pain to use, acknowledge, WIP |
29 and 31 sting like sons of a bitch, declare the pain despite the low badness and extremely small error |
||
| Line 39: | Line 39: | ||
: error map: -0.000546 0.000618 0.000877 0.000084 -0.000591 -0.000463 0.00017 -0.001072 0.001239 | : error map: -0.000546 0.000618 0.000877 0.000084 -0.000591 -0.000463 0.00017 -0.001072 0.001239 | ||
Badness (Sintel): 0.354}}Ploidacots, pergens, MOS scales and odd-limit detunings are unapplicable or too convoluted to be descriptive. | 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: | ||
| Line 51: | Line 72: | ||
=== Add-29,31 extension === | === Add-29,31 extension === | ||
Insatinismic is worse at extending to the no-23 31-limit, since it requires detempering from the suboptimal septischismic add-29,31 basis 53k & 217k & 311k, instead of 217 & 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. | 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 == | ||
| Line 74: | Line 95: | ||
* 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 | * 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: | ||
| Line 84: | Line 106: | ||
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. Same with 29/16, which instead of being spelt a major sixth can be spelt a minor seventh + 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 === | ||
| Line 101: | Line 123: | ||
* 17/16: minor second + septimal comma - buzzardma + aberschisma + schisma - 3 minas - tina | * 17/16: minor second + septimal comma - buzzardma + aberschisma + schisma - 3 minas - tina | ||
* 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, 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 | * 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). | |||