User:Eufalesio/Ultimate: Difference between revisions

Line 1:

~~iis~~ 53 & 94 & 217, with mapping {{Mapping|1 0 0 25 -33 -13|0 1 0 -14 23 12|0 0 1 0 0 -1}}. It's otherwise known by in the wiki as ''[[cassaschismic]]'' (tad less rambly), also [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here]]; but we will simply call it '''Ultimate'''. Our reasoning of this will become clear. Or at least, we expect you to understand why it's clear in our mind.

is 53 & 94 & 217, with mapping {{Mapping|1 0 0 25 -33 -13|0 1 0 -14 23 12|0 0 1 0 0 -1}}. It's otherwise known by in the wiki as ''[[cassaschismic]]'' (tad less rambly), also [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here (data dump)]]; but we will simply call it '''Ultimate'''. Our reasoning of this will become clear. Or at least, we expect you to understand why it's clear in our mind.

Special thanks for [[Kite Giedraitis]] for feedback and edits.

Line 28:

=== Interval list ===

Here is a quick compressed cheat sheet of octave-reduced intervals. This is a ~~MASSIvE~~ simplification with many (infinitely many) intervals left out for the sake of brevity. For every entry here, ratios here represent pitch-classes and their pitch class inverses; so for instance 8/5 pitch class is mapped to +8 fifths -1 MC, being the octave inverse of 5/4 pitch class negates the mappings so it is found at -8 fifths + 1 MC. There are no octave reduced primes or prime inverses with positive fifth-span and MC-span.

Here is a quick compressed cheat sheet of octave-reduced intervals. This is a MASSIVE simplification with many (infinitely many) intervals left out for the sake of brevity. For every entry here, ratios here represent pitch-classes and their pitch class inverses; so for instance 8/5 pitch class is mapped to +8 fifths -1 MC, being the octave inverse of 5/4 pitch class negates the mappings so it is found at -8 fifths + 1 MC. There are no octave reduced primes or prime inverses with positive fifth-span and MC-span.

{| class="wikitable"

!

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== Justification ==

The chain of fifths is a very important framework historically. It's been in Western music THE way to think about everything all the way from plainchant to Renaissance meantone temperaments to the modern day; where the 12-pitch-class circle of fifths is taught; 12edo, a massively over-represented tuning. It has a bit of a bad reputation in the xen circles, but the more I researched, the more I realized it is a '''paragon''', and that its position nowadays is very much well earned.

The chain of fifths is a very important framework historically. It's been in Western music THE way to think about everything all the way from plainchant to Renaissance meantone temperaments to the modern day – where the 12-pitch-class circle of fifths is taught; 12edo remains a massively over-represented tuning. It has a bit of a bad reputation in the xen circles, but the more I researched, the more I realized it is a '''paragon''', and that its position nowadays is very much well earned.

My main aim is to expand tonality with JI, and there is no better way to do so than to also extend the fundamental tuning framework to its logical conclusion. Behold, the ''sequence''.

My main aim is to expand tonality with JI, and there is no better way to do so than to also extend the fundamental tuning framework to its logical conclusion. Behold, ''the'' '''''ultimate''''' ''sequence''.

* '''12edo''' introduces the [[compton]] framework, which closes the chain of fifths with 12 flat, but proportionally very good fifths. Compton ''sensu stricto'' uses an independent generator to each all the different primes, and is generally a very good temperament. However, if the fifths are tuned sharper to become closer to just, the chain goes on for longer...

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Unlike Olympic, this one is undeniably '''WAY''' more accurate, at least 100 times more accurate. It can be extended to the whole 19-limit by adding S76/S77 and S2431 (or the [[devicisma]]) to the comma list, however, a tina is required to reach prime 17 and 19, which is close, but not equal to a third of a schismina. The tina is 10241/10240. 19/16 is †|̈\m3. 17/16 is 𐕣^^'''⇊⇊̱'''M2.

Look how goddamn accurate this temp is! Look, I even made a technical temp data section!

Look how goddamn accurate this temp is! Look, I even made a technical temp data section! [expandable]

=== Batshit AKA Batshit-Insanic or simply call it Insanic if profanity is not allowed ===

[[Subgroup]]: 2.3.5.7.11.13

{{Databox|1=[[Subgroup]]: 2.3.5.7.11.13

[[Comma list]]: [[5767168/5767125|S64/S65]]

[[Mapping]]:

[[Mapping]]: [⟨ 1 0 0 0 2 7 ],

~~{| class="right-all"~~

⟨ 0 1 0 0 0 0 ],

|[⟨

⟨ 0 0 1 0 0 -1 ],

|1

⟨ 0 0 0 1 1 0 ],

|0

|0

Mapping generators: ~2, ~3, ~5, ~7, ~128/65

|2

|7

|],

|-

|⟨

|0

|1

|0

|],

|-

|⟨

|0

|1

|0

| -1

|],

|-

|⟨

|0

|1

|0

|],

|-

|⟨

|0

| -3

| -1

|]]

|}Mapping generators: ~2, ~3, ~5, ~7, ~128/65

[[Optimal tuning|Optimal tunings]]? (the calculator isn't very helpful here...)

: [[CWE]]: ~2 = 1200.000 ¢, ~3/2 = 701.956 ¢, ~5/4 = 386.315 ¢, ~7/4 = 968.827 ¢, ~~~608~~/~~385~~ = 1173.154¢

: [[CWE]]: ~2 = 1200.000 ¢, ~3/2 = 701.956 ¢, ~5/4 = 386.315 ¢, ~7/4 = 968.827 ¢, ~128/65 = 1173.154¢

: error map: -0.000, 0.001, 0.001, 0.001, 0.001, 0.003

Optimal ET sequence: {{EDOs|7, 10e, 12e, 30b, 31e, 34, 36, 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1609, 1696, 1920, 2190, 2684, 3395, 5144, 5585, 6079, 8269, 8539, 11664, 14124, 14348, 14618, 20203}}

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[[Comma list]]: S64/S65, [[11413376/11413325|S76/S77]], [[633556/633555]]

[[Mapping]]:

[[Mapping]]: [⟨ 1 0 0 0 5 8 -9 0 ],

~~{| class="right-all"~~

⟨ 0 1 0 0 0 0 -2 1 ],

|[⟨

⟨ 0 0 1 0 0 -1 -2 1 ],

|1

⟨ 0 0 0 1 1 0 0 2 ],

|0

|0

|5

|8

| -9

|0

|],

|-

|⟨

|0

|1

|0

| -2

|1

|],

|-

|⟨

|0

|1

|0

| -1

| -2

|1

|],

|-

|⟨

|0

|1

|0

|2

|],

|-

|⟨

|0

| -9

| -3

|8

| -8

|]]

|}

Mapping generators: ~2, ~3, ~5, ~7, ~608/385

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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}}.

Badness (Sintel): 0.279

Badness (Sintel): 0.279}}

––––––––––––

@@ Line 1: / Line 1: @@
-iis 53 & 94 & 217, with mapping {{Mapping|1 0 0 25 -33 -13|0 1 0 -14 23 12|0 0 1 0 0 -1}}. It's otherwise known by in the wiki as ''[[cassaschismic]]'' (tad less rambly), also [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here]]; but we will simply call it '''Ultimate'''. Our reasoning of this will become clear. Or at least, we expect you to understand why it's clear in our mind.
+is 53 & 94 & 217, with mapping {{Mapping|1 0 0 25 -33 -13|0 1 0 -14 23 12|0 0 1 0 0 -1}}. It's otherwise known by in the wiki as ''[[cassaschismic]]'' (tad less rambly), also [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here (data dump)]]; but we will simply call it '''Ultimate'''. Our reasoning of this will become clear. Or at least, we expect you to understand why it's clear in our mind.
 Special thanks for [[Kite Giedraitis]] for feedback and edits.
@@ Line 28: / Line 28: @@
 === Interval list ===
-Here is a quick compressed cheat sheet of octave-reduced intervals. This is a MASSIvE simplification with many (infinitely many) intervals left out for the sake of brevity. For every entry here, ratios here represent pitch-classes and their pitch class inverses; so for instance 8/5 pitch class is mapped to +8 fifths -1 MC, being the octave inverse of 5/4 pitch class negates the mappings so it is found at -8 fifths + 1 MC. There are no octave reduced primes or prime inverses with positive fifth-span and MC-span.
+Here is a quick compressed cheat sheet of octave-reduced intervals. This is a MASSIVE simplification with many (infinitely many) intervals left out for the sake of brevity. For every entry here, ratios here represent pitch-classes and their pitch class inverses; so for instance 8/5 pitch class is mapped to +8 fifths -1 MC, being the octave inverse of 5/4 pitch class negates the mappings so it is found at -8 fifths + 1 MC. There are no octave reduced primes or prime inverses with positive fifth-span and MC-span.
 {| class="wikitable"
 !
@@ Line 139: / Line 139: @@
 == Justification ==
-The chain of fifths is a very important framework historically. It's been in Western music THE way to think about everything all the way from plainchant to Renaissance meantone temperaments to the modern day; where the 12-pitch-class circle of fifths is taught; 12edo, a massively over-represented tuning. It has a bit of a bad reputation in the xen circles, but the more I researched, the more I realized it is a '''paragon''', and that its position nowadays is very much well earned.
+The chain of fifths is a very important framework historically. It's been in Western music THE way to think about everything all the way from plainchant to Renaissance meantone temperaments to the modern day – where the 12-pitch-class circle of fifths is taught; 12edo remains a massively over-represented tuning. It has a bit of a bad reputation in the xen circles, but the more I researched, the more I realized it is a '''paragon''', and that its position nowadays is very much well earned.
-My main aim is to expand tonality with JI, and there is no better way to do so than to also extend the fundamental tuning framework to its logical conclusion. Behold, the ''sequence''.
+My main aim is to expand tonality with JI, and there is no better way to do so than to also extend the fundamental tuning framework to its logical conclusion. Behold, ''the'' '''''ultimate''''' ''sequence''.
 * '''12edo''' introduces the [[compton]] framework, which closes the chain of fifths with 12 flat, but proportionally very good fifths. Compton ''sensu stricto'' uses an independent generator to each all the different primes, and is generally a very good temperament. However, if the fifths are tuned sharper to become closer to just, the chain goes on for longer...
@@ Line 236: / Line 236: @@
 Unlike Olympic, this one is undeniably '''WAY''' more accurate, at least 100 times more accurate. It can be extended to the whole 19-limit by adding S76/S77 and S2431 (or the [[devicisma]]) to the comma list, however, a tina is required to reach prime 17 and 19, which is close, but not equal to a third of a schismina. The tina is 10241/10240. 19/16 is †|̈\m3. 17/16 is 𐕣^^'''⇊⇊̱'''M2.
-Look how goddamn accurate this temp is! Look, I even made a technical temp data section!
+Look how goddamn accurate this temp is! Look, I even made a technical temp data section! [expandable]
 === Batshit AKA Batshit-Insanic <small><small><small><small><small>or simply call it</small></small></small></small></small> Insanic<small><small><small><small><small> if profanity is not allowed</small></small></small></small></small> ===
-[[Subgroup]]: 2.3.5.7.11.13
+{{Databox|1=[[Subgroup]]: 2.3.5.7.11.13
 [[Comma list]]: [[5767168/5767125|S64/S65]]
-[[Mapping]]:
+[[Mapping]]: [⟨	1	0	0	0	2	7	],
-{| class="right-all"
+⟨	0	1	0	0	0	0	],
-|[⟨
+⟨	0	0	1	0	0	-1	],
-|1
+⟨	0	0	0	1	1	0	],
-|0
+<nowiki>⟨	0	0	0	0	-3	-1	]]</nowiki>
-|0
-|0
+Mapping generators: ~2, ~3, ~5, ~7, ~128/65
-|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, ~128/65
 [[Optimal tuning|Optimal tunings]]? (the calculator isn't very helpful here...)
-: [[CWE]]: ~2 = 1200.000 ¢, ~3/2 = 701.956 ¢, ~5/4 = 386.315 ¢, ~7/4 = 968.827 ¢, ~608/385 = 1173.154¢
+: [[CWE]]: ~2 = 1200.000 ¢, ~3/2 = 701.956 ¢, ~5/4 = 386.315 ¢, ~7/4 = 968.827 ¢, ~128/65 = 1173.154¢
 : error map: -0.000, 0.001, 0.001, 0.001, 0.001, 0.003
 Optimal ET sequence: {{EDOs|7, 10e, 12e, 30b, 31e, 34, 36, 41, 46, 53, 84, 87, 130, 183, 217, 224, 270, 494, 764, 935, 1075, 1205, 1609, 1696, 1920, 2190, 2684, 3395, 5144, 5585, 6079, 8269, 8539, 11664, 14124, 14348, 14618, 20203}}
@@ Line 301: / Line 261: @@
 [[Comma list]]: S64/S65, [[11413376/11413325|S76/S77]], [[633556/633555]]
-[[Mapping]]:
+[[Mapping]]: [⟨	1	0	0	0	5	8	-9	0	],
-{| class="right-all"
+⟨	0	1	0	0	0	0	-2	1	],
-|[⟨
+⟨	0	0	1	0	0	-1	-2	1	],
-|1
+⟨	0	0	0	1	1	0	0	2	],
-|0
+<nowiki>⟨	0	0	0	0	-9	-3	8	-8	]]</nowiki>
-|0
-|0
-|5
-|8
-| -9
-|0
-|],
-|-
-|⟨
-|0
-|1
-|0
-|0
-|0
-|0
-| -2
-|1
-|],
-|-
-|⟨
-|0
-|0
-|1
-|0
-|0
-| -1
-| -2
-|1
-|],
-|-
-|⟨
-|0
-|0
-|0
-|1
-|1
-|0
-|0
-|2
-|],
-|-
-|⟨
-|0
-|0
-|0
-|0
-| -9
-| -3
-|8
-| -8
-|]]
-|}
 Mapping generators: ~2, ~3, ~5, ~7, ~608/385
@@ Line 367: / Line 276: @@
 '''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
+Badness (Sintel): 0.279}}
 ––––––––––––