Insatinismic: Difference between revisions

(One intermediate revision by the same user not shown)

Line 1:

Insatinismic is a [[rank-5 temperament]] for the ~~add~~-29-19-limit that is a [[weak extension]] of [[5767168/5767125#Insanismic|insanismic]] and a detemperament of [[~~cassaschismic~~]]. It is not a member of any known temperament family or clan.

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

Line 39:

: 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 ],

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:

Line 48:

Line 69:

As a weak extension of insanismic, it does the following:

* 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 and 19/16.

* 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 ==

Line 71:

Line 95:

* 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:

Line 81:

Line 106:

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 ===

Line 98:

Line 123:

* 17/16: minor second + septimal comma - buzzardma + aberschisma + 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

* 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:

~~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~~.

* 29/16: minor seventh + buzzardma + 2 aberschismas + schisma + two tinas + yunzee

* 31/16: octave - 2 septimal commas - yunzee (by definition).

@@ Line 1: / Line 1: @@
-Insatinismic is a [[rank-5 temperament]] for the add-29-19-limit that is a [[weak extension]] of [[5767168/5767125#Insanismic|insanismic]] and a detemperament of [[cassaschismic]]. It is not a member of any known temperament family or clan.
+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
@@ Line 39: / Line 39: @@
 : 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:
@@ Line 48: / Line 69: @@
 As a weak extension of insanismic, it does the following:
-* 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 and 19/16.
+* 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 ==
@@ Line 71: / Line 95: @@
 * 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:
@@ Line 81: / Line 106: @@
 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 ===
@@ Line 98: / Line 123: @@
 * 17/16: minor second + septimal comma - buzzardma + aberschisma + 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
+* 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:
-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.
+* 29/16: minor seventh + buzzardma + 2 aberschismas + schisma + two tinas + yunzee
+* 31/16: octave - 2 septimal commas - yunzee (by definition).

Insatinismic: Difference between revisions

Eufalesio (talk | contribs)