User:Eufalesio/12edo detemperament sequences: Difference between revisions

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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.

12edo is an undisputed paragon of tonality among music theory of old and young. In the world of xenharmonic music, it is thus extremely desirable, if not necessary, to have a temperament supported by 12edo so that a simple workflow and notation are used. At least, [[RTT]] speaking. I will be neglecting Ringer scales and JI well temperaments as I don't care about them.

12edo is an undisputed paragon of tonality among music theory of old and young. In the world of xenharmonic music, it is thus extremely desirable, if not necessary, to have a temperament supported by 12edo so that a simple workflow and notation are used. At least, [[RTT]] speaking. I will be neglecting [[Ringer scale|Ringer scales]] and JI [[Well temperament|well temperaments]] as I don't care about them.

The easiest way to detemper 12edo is to observe (untemper) one or many more of its nullified commas; though observing a number of similar sized commas and tempering out their difference also works. This produces a rank-2 temperament. A rank-2 temperament can be "retempered" into a finer edo by adding another comma that 12edo doesn't temper out. All that I've explained here can be done with any rank temperament; rank-2s can be detempered into rank-3s, retempered into much finer edos.

The easiest way to detemper 12edo is to observe (untemper) one or many more of its nullified commas, though observing a number of similar sized commas and tempering out their difference also works. This produces a rank-2 temperament. A rank-2 temperament can be "retempered" into a finer edo by adding another comma that 12edo doesn't temper out. All that I've explained here can be done with any rank temperament; rank-2s can be detempered into rank-3s, retempered into much finer edos.

The temperaments I'll be showing always have an whole or split octave period, and a whole or split fifth generator, and in any case, notation-wise from a D, the perfect fifth or fourth is NEVER spelt with an accidental.

The temperaments I'll be showing always have an whole or split octave period, and a whole or split fifth generator, and in any case, notation-wise from a D, the perfect fifth or fourth shall be <big>'''NEVER'''</big> spelt with an accidental, that is to say... it is a [[Nominal-accidental chain|nominal]].

== 7-limit ==

Let us take a look at some 7-limit temperaments through the [[Patent val|patent]] [[Val|valuation]] (direct approximation primes 2 through 7); which can be analyzed as tempering out 81/80, 64/63, 225/224; and by extension~~: 531441/524288~~, 126/125, 2048/2025, 36/35, 5120/5103... etc.

Let us take a look at some 7-limit temperaments through the [[Patent val|patent]] [[Val|valuation]] (direct approximation primes 2 through 7); which can be analyzed as tempering out 81/80, 64/63, 225/224; and by extension a long list of commas such as the [[pythagorean comma]], [[126/125]], [[2048/2025]], [[36/35]], [[5120/5103]], [[32805/32768]]... etc.

=== 12 ===

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* With septimal meantone:

** Adding [[126/125]] or [[1029/1024]] or [[1728/1715]] gives 31edo.

** Adding [[126/125]] or [[1029/1024]] or [[1728/1715]] gives [[31edo]].

** Adding [[15625/15552]] or [[3125/3072]] or [[49/48]] gives 19edo.

** Adding [[15625/15552]] or [[3125/3072]] or [[49/48]] gives [[19edo]].

** Adding [[16807/16384]] gives [[50edo]].

* With pajara:

** Adding [[250/243]] gives 22edo.

** Adding [[250/243]] or [[875/864]] gives [[22edo]].

** Adding [[4375/4374]] gives 34edo, through the [[Wart notation|34d]] [[val]].

** Adding [[4375/4374]] gives [[34edo]], through the [[Wart notation|34d]] [[val]].

** Adding [[686/675]] gives 56edo.

** Adding [[686/675]] gives [[56edo]].

* With garibaldi:

** Adding 3125/3072 or [[245/243]] or [[2401/2400]] gives [[41edo]].

** Adding [[3125/3072]] or [[245/243]] or [[2401/2400]] gives [[41edo]].

** Adding 15625/15552 or 4375/4374 gives [[53edo]].

** Adding [[15625/15552]] or [[4375/4374]] gives [[53edo]].

** Adding the [[wizma]] or [[stearnsma]] gives [[94edo]].

* With septimal diaschismic:

** Adding 245/243 or 4375/4374 gives 46edo.

** Adding [[245/243]] or [[4375/4374]] gives [[46edo]].

** Adding 1728/1715 gives 58edo.

** Adding [[1728/1715]] gives [[58edo]].

* With compton:

** Adding 15625/15552 or 4375/4374 or 2401/2400 gives 72edo.

** Adding [[15625/15552]] or [[4375/4374]] or [[2401/2400]] gives [[72edo]].

Any of these rank-2s can be detempered into a rank-3 temperament which only tempers out one comma. The ones I deem most notable are:

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=== 12d ===

Taking the next best approximation of 7/4, a major sixth, however stupid that might sound... another family of temperaments can be derived. 12d can be analyzed as tempering out 81/80, ~~4375~~/~~4374~~, [[~~2048~~/~~2025~~]]. This is a bad exotemperament because it implies 21/20, 28/27 are tempered out; but so are 32805/32768, ~~and~~ [[~~875~~/~~864~~]].

Taking the next best approximation of 7/4, a major sixth, <small>however stupid that might sound</small>... another family of temperaments can be derived. 12d can be analyzed as tempering out [[81/80]], [[2048/2025]], [[875/864]]. This is a bad exotemperament because it implies 21/20, 28/27 are tempered out; but so are [[32805/32768]], [[4375/4374]].

* Observing 2048/~~2045~~ gives [[flattone]].

* Observing [[2048/2025]] gives [[flattone]].

* Observing 81/80 gives [[srutal]].

* Observing [[81/80]] gives [[srutal]].

* Observing and tempering 2048/~~2405~~ and 81/80 together whilst tempering out 4375/4374 gives [[pontiac]].

* Observing and tempering [[2048/2025]] and [[81/80]] together whilst tempering out [[4375/4374]] gives [[pontiac]].

With flattone:

* Adding [[50/49]] gives 26edo.

* Adding [[50/49]] gives [[26edo]].

* Adding 3125/3087 gives [[45edo]].

* Adding [[3125/3087]] gives [[45edo]].

With srutal:

* Adding 126/125 or 245/243 gives 46edo.

* Adding [[126/125]] or [[245/243]] gives [[46edo]].

* Adding 1728/1715 or [[4000/3969]] gives [[80edo]].

* Adding [[1728/1715]] or [[4000/3969]] gives [[80edo]].

With pontiac:

* Adding 225/224 or 5120/5103 or 15625/15552 gives 53edo.

* Adding [[225/224]] or [[5120/5103]] or [[15625/15552]] gives [[53edo]].

* Adding 1029/1024 or 3136/3125 gives [[118edo]].

* Adding [[1029/1024]] or [[3136/3125]] gives [[118edo]].

* Adding 2401/2400 or wizma gives [[171edo]].

* Adding [[2401/2400]] or [[wizma]] gives [[171edo]].

* Adding 16875/16807 gives [[224edo]].

* Adding [[16875/16807]] gives [[224edo]].

Through this val, I deem the following rank-3 temperaments notable:

* Ragismic {4375/4374}

* [[Ragismic]] {4375/4374}

* Orthoschismic {32805/32768}

* [[Orthoschismic]] {32805/32768}

== 13-limit ==

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Maps 11/8 to a fourth, 13/8 to a minor sixth. It is not the best val temperament wise, but since the corrections of p11 and p13 are in opposite directions to 5/4 and 7/4; but for this, the garibaldi extension [[cassandra]] is arguably the best pick, which is just a relabeling of the [[3-limit]], and can be easily detempered further into what I believe to be the [[Cassaschismic|BEST temperament (Ultimate)]]... see [[User:Eufalesio/Ultimate|here]] for more info!

The path through cassandra into Ultimate is the easiest, as it keeps the chain of fifths, and a pythagorean-septimal comma ([[gary]]) with a hyper accurate [[zala]], but adds an independent generator for 5 and 13, a "minicomma" that works as aberschisma ~ major minthma. Thus, the mappings of 5 and 13 can be thought as the same as cassandra, but altered by +1 / -1 minicomma. It can be detempered further into rank-4 [[olympic]] and finally, the goddamn-fucking-insanely accurate rank-5 temp [[5767168/5767125#Insanismic|Batshit AKA ''Insanismic'' for the prudes.]]

The path through cassandra into Ultimate is the easiest, as it keeps the chain of fifths, and a pythagorean-septimal comma ([[gary]]) with a hyper accurate [[zala]], but adds an independent generator for 5 and 13, a "minicomma" that works as aberschisma ~ major minthma. Thus, the mappings of 5 and 13 can be thought as the same as cassandra, but altered by +1 / -1 minicomma. It can be detempered further into rank-4 [[olympic]] and finally, the goddamn-fucking-insanely accurate rank-5 temp [[5767168/5767125#Insanismic|Batshit AKA ''Insanismic for the prudes.'']]

It can instead be detempered into [[hecate]], which either cassandra or compton through the shit extension 12e & 72 can do. But I do not recommend going into hecate. It's decent, but it decouples the generic comma of cassandra from the chain of fifths and thus the garischisma is observed, and can't be detempered further into hyper-accurate temperaments.

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=== 12e ===

Cassandra and Ultimate are both easy 19-limit temperaments. By tempering out 1225/1224 and 1216/1215, the temps can be strongly extended to include the two new primes. Ultimate is notable as arguably the best [[yazalathana]], altering the minor third of cassandra by +1 minicomma with great accuracy and negligible extra error.

Cassandra and Ultimate are both easy 19-limit temperaments. By tempering out [[1225/1224]] and [[1216/1215]], the temps can be strongly extended to include the two new primes. Ultimate is notable as arguably the best [[yazalathana]], altering the minor third of cassandra by +1 minicomma with great accuracy and negligible extra error.

17 is more complex and less accurately tuned, but still within reasonable use and accuracy... though, it needs the 12e'''g''' val. This can mitigated if you use [[Buzzardsma|~~buzzardmas~~]] as accidentals... but still, kinda sorta not the best.

17 is more complex and less accurately tuned, but still within reasonable use and accuracy... though, it needs the 12e'''g''' val. This can mitigated if you use [[Buzzardsma|saquarumas]] as accidentals... but still, kinda sorta not the best.

In Olympic, you '''must''' use the weak extension [[Catalog of rank-4 temperaments#Metaolympic|Metaolympic]], in which your garischisma is split in two heavily inflated [[10241/10240]] halves, and you use one of these halves to reach 17/16 and 19/16, -1 and +1 respectively. Through [[User:Eufalesio/Ultimate#Insatinismic AKA Batshittin|Batshittin]] (a weak extension of Batshit), you use the same 10241/10240 and [[Mina|minas]]; 4096/4095 ~ 4225/4224.

@@ Line 1: / Line 1: @@
 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.
-edo is an undisputed paragon of tonality among music theory of old and young. In the world of xenharmonic music, it is thus extremely desirable, if not necessary, to have a temperament supported by 12edo so that a simple workflow and notation are used. At least, [[RTT]] speaking. I will be neglecting Ringer scales and JI well temperaments as I don't care about them.
+edo is an undisputed paragon of tonality among music theory of old and young. In the world of xenharmonic music, it is thus extremely desirable, if not necessary, to have a temperament supported by 12edo so that a simple workflow and notation are used. At least, [[RTT]] speaking. I will be neglecting [[Ringer scale|Ringer scales]] and JI [[Well temperament|well temperaments]] as I don't care about them.
-The easiest way to detemper 12edo is to observe (untemper) one or many more of its nullified commas; though observing a number of similar sized commas and tempering out their difference also works. This produces a rank-2 temperament. A rank-2 temperament can be "retempered" into a finer edo by adding another comma that 12edo doesn't temper out. All that I've explained here can be done with any rank temperament; rank-2s can be detempered into rank-3s, retempered into much finer edos.
+The easiest way to detemper 12edo is to observe (untemper) one or many more of its nullified commas, though observing a number of similar sized commas and tempering out their difference also works. This produces a rank-2 temperament. A rank-2 temperament can be "retempered" into a finer edo by adding another comma that 12edo doesn't temper out. All that I've explained here can be done with any rank temperament; rank-2s can be detempered into rank-3s, retempered into much finer edos.
-The temperaments I'll be showing always have an whole or split octave period, and a whole or split fifth generator, and in any case, notation-wise from a D, the perfect fifth or fourth is NEVER spelt with an accidental.
+The temperaments I'll be showing always have an whole or split octave period, and a whole or split fifth generator, and in any case, notation-wise from a D, the perfect fifth or fourth shall be <big>'''NEVER'''</big> spelt with an accidental, that is to say... it is a [[Nominal-accidental chain|nominal]].
 == 7-limit ==
-Let us take a look at some 7-limit temperaments through the [[Patent val|patent]] [[Val|valuation]] (direct approximation primes 2 through 7); which can be analyzed as tempering out 81/80, 64/63, 225/224; and by extension: 531441/524288, 126/125, 2048/2025, 36/35, 5120/5103... etc.
+Let us take a look at some 7-limit temperaments through the [[Patent val|patent]] [[Val|valuation]] (direct approximation primes 2 through 7); which can be analyzed as tempering out 81/80, 64/63, 225/224; and by extension a long list of commas such as the [[pythagorean comma]], [[126/125]], [[2048/2025]], [[36/35]], [[5120/5103]], [[32805/32768]]... etc.
 === 12 ===
@@ Line 22: / Line 22: @@
 * With septimal meantone:
-** Adding [[126/125]] or [[1029/1024]] or [[1728/1715]] gives 31edo.
+** Adding [[126/125]] or [[1029/1024]] or [[1728/1715]] gives [[31edo]].
-** Adding [[15625/15552]] or [[3125/3072]] or [[49/48]] gives 19edo.
+** Adding [[15625/15552]] or [[3125/3072]] or [[49/48]] gives [[19edo]].
 ** Adding [[16807/16384]] gives [[50edo]].
 * With pajara:
-** Adding [[250/243]] gives 22edo.
+** Adding [[250/243]] or [[875/864]] gives [[22edo]].
-** Adding [[4375/4374]] gives 34edo, through the [[Wart notation|34d]] [[val]].
+** Adding [[4375/4374]] gives [[34edo]], through the [[Wart notation|34d]] [[val]].
-** Adding [[686/675]] gives 56edo.
+** Adding [[686/675]] gives [[56edo]].
 * With garibaldi:
-** Adding 3125/3072 or [[245/243]] or [[2401/2400]] gives [[41edo]].
+** Adding [[3125/3072]] or [[245/243]] or [[2401/2400]] gives [[41edo]].
-** Adding 15625/15552 or 4375/4374 gives [[53edo]].
+** Adding [[15625/15552]] or [[4375/4374]] gives [[53edo]].
 ** Adding the [[wizma]] or [[stearnsma]] gives [[94edo]].
 * With septimal diaschismic:
-** Adding 245/243 or 4375/4374 gives 46edo.
+** Adding [[245/243]] or [[4375/4374]] gives [[46edo]].
-** Adding 1728/1715 gives 58edo.
+** Adding [[1728/1715]] gives [[58edo]].
 * With compton:
-** Adding 15625/15552 or 4375/4374 or 2401/2400 gives 72edo.
+** Adding [[15625/15552]] or [[4375/4374]] or [[2401/2400]] gives [[72edo]].
 Any of these rank-2s can be detempered into a rank-3 temperament which only tempers out one comma. The ones I deem most notable are:
@@ Line 48: / Line 48: @@
 === 12d ===
-Taking the next best approximation of 7/4, a major sixth, however stupid that might sound... another family of temperaments can be derived. 12d can be analyzed as tempering out 81/80, 4375/4374, [[2048/2025]]. This is a bad exotemperament because it implies 21/20, 28/27 are tempered out; but so are 32805/32768, and [[875/864]].
+Taking the next best approximation of 7/4, a major sixth, <small>however stupid that might sound</small>... another family of temperaments can be derived. 12d can be analyzed as tempering out [[81/80]], [[2048/2025]], [[875/864]]. This is a bad exotemperament because it implies 21/20, 28/27 are tempered out; but so are [[32805/32768]], [[4375/4374]].
-* Observing 2048/2045 gives [[flattone]].
+* Observing [[2048/2025]] gives [[flattone]].
-* Observing 81/80 gives [[srutal]].
+* Observing [[81/80]] gives [[srutal]].
-* Observing and tempering 2048/2405 and 81/80 together whilst tempering out 4375/4374 gives [[pontiac]].
+* Observing and tempering [[2048/2025]] and [[81/80]] together whilst tempering out [[4375/4374]] gives [[pontiac]].
 With flattone:
-* Adding [[50/49]] gives 26edo.
+* Adding [[50/49]] gives [[26edo]].
-* Adding 3125/3087 gives [[45edo]].
+* Adding [[3125/3087]] gives [[45edo]].
 With srutal:
-* Adding 126/125 or 245/243 gives 46edo.
+* Adding [[126/125]] or [[245/243]] gives [[46edo]].
-* Adding 1728/1715 or [[4000/3969]] gives [[80edo]].
+* Adding [[1728/1715]] or [[4000/3969]] gives [[80edo]].
 With pontiac:
-* Adding 225/224 or 5120/5103 or 15625/15552 gives 53edo.
+* Adding [[225/224]] or [[5120/5103]] or [[15625/15552]] gives [[53edo]].
-* Adding 1029/1024 or 3136/3125 gives [[118edo]].
+* Adding [[1029/1024]] or [[3136/3125]] gives [[118edo]].
-* Adding 2401/2400 or wizma gives [[171edo]].
+* Adding [[2401/2400]] or [[wizma]] gives [[171edo]].
-* Adding 16875/16807 gives [[224edo]].
+* Adding [[16875/16807]] gives [[224edo]].
 Through this val, I deem the following rank-3 temperaments notable:
-* Ragismic {4375/4374}
+* [[Ragismic]] {4375/4374}
-* Orthoschismic {32805/32768}
+* [[Orthoschismic]] {32805/32768}
 == 13-limit ==
@@ Line 83: / Line 83: @@
 Maps 11/8 to a fourth, 13/8 to a minor sixth. It is not the best val temperament wise, but since the corrections of p11 and p13 are in opposite directions to 5/4 and 7/4; but for this, the garibaldi extension [[cassandra]] is arguably the best pick, which is just a relabeling of the [[3-limit]], and can be easily detempered further into what I believe to be the [[Cassaschismic|BEST temperament (Ultimate)]]... see [[User:Eufalesio/Ultimate|here]] for more info!
-The path through cassandra into Ultimate is the easiest, as it keeps the chain of fifths, and a pythagorean-septimal comma ([[gary]]) with a hyper accurate [[zala]], but adds an independent generator for 5 and 13, a "minicomma" that works as aberschisma ~ major minthma. Thus, the mappings of 5 and 13 can be thought as the same as cassandra, but altered by +1 / -1 minicomma. It can be detempered further into rank-4 [[olympic]] and finally, the goddamn-fucking-insanely accurate rank-5 temp [[5767168/5767125#Insanismic|Batshit AKA ''Insanismic'' for the prudes.]]
+The path through cassandra into Ultimate is the easiest, as it keeps the chain of fifths, and a pythagorean-septimal comma ([[gary]]) with a hyper accurate [[zala]], but adds an independent generator for 5 and 13, a "minicomma" that works as aberschisma ~ major minthma. Thus, the mappings of 5 and 13 can be thought as the same as cassandra, but altered by +1 / -1 minicomma. It can be detempered further into rank-4 [[olympic]] and finally, the goddamn-fucking-insanely accurate rank-5 temp [[5767168/5767125#Insanismic|Batshit AKA ''Insanismic for the prudes.'']]
 It can instead be detempered into [[hecate]], which either cassandra or compton through the shit extension 12e & 72 can do. But I do not recommend going into hecate. It's decent, but it decouples the generic comma of cassandra from the chain of fifths and thus the garischisma is observed, and can't be detempered further into hyper-accurate temperaments.
@@ Line 101: / Line 101: @@
 === 12e ===
-Cassandra and Ultimate are both easy 19-limit temperaments. By tempering out 1225/1224 and 1216/1215, the temps can be strongly extended to include the two new primes. Ultimate is notable as arguably the best [[yazalathana]], altering the minor third of cassandra by +1 minicomma with great accuracy and negligible extra error.
+Cassandra and Ultimate are both easy 19-limit temperaments. By tempering out [[1225/1224]] and [[1216/1215]], the temps can be strongly extended to include the two new primes. Ultimate is notable as arguably the best [[yazalathana]], altering the minor third of cassandra by +1 minicomma with great accuracy and negligible extra error.
-is more complex and less accurately tuned, but still within reasonable use and accuracy... though, it needs the 12e'''g''' val. This can mitigated if you use [[Buzzardsma|buzzardmas]] as accidentals... but still, kinda sorta not the best.
+is more complex and less accurately tuned, but still within reasonable use and accuracy... though, it needs the 12e'''g''' val. This can mitigated if you use [[Buzzardsma|saquarumas]] as accidentals... but still, kinda sorta not the best.
 In Olympic, you '''must''' use the weak extension [[Catalog of rank-4 temperaments#Metaolympic|Metaolympic]], in which your garischisma is split in two heavily inflated [[10241/10240]] halves, and you use one of these halves to reach 17/16 and 19/16, -1 and +1 respectively. Through [[User:Eufalesio/Ultimate#Insatinismic AKA Batshittin|Batshittin]] (a weak extension of Batshit), you use the same 10241/10240 and [[Mina|minas]]; 4096/4095 ~ 4225/4224.