User:Eufalesio/Ultimate: Difference between revisions

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is ~~53 & 94 & 217, with mapping {{Mapping|1 0 0 25 -33 -13 47 -6|0 1 0 -14 23 12 -30 5|0 0 1 0 0 -1 2 1}} in the~~ [[~~19-limit~~]]~~. 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'''~~. The choice of name will become clear. Or at least, I expect you to understand why it's clear in my mind~~.

is a special denominatinon I give to [[septischismic]] (also [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here (data dump)]]). The real name is ''septischismic, 12e Ultimate'', ; but we will simply call it '''Ultimate''' here.

Special thanks for [[Kite Giedraitis]] for feedback and edits, for [[Godtone]] (osmium) for teaching me [[S-expression|S-expressions]], and for [[Flora Canou]] for helping me document and put the temperaments up to standards.

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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.

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. Among all paths I've considered [[User:Eufalesio/12edo detemperament sequences|(here)]], this, is what I consider to be the best. Behold, ''the'' ''~~Ultimate~~'' ''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. Among all paths I've considered [[User:Eufalesio/12edo detemperament sequences|(here)]], this, is what I consider to be the best for me. Behold, ''the'' ''12e'' ''detemperament'' ''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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* '''Ultimate''' is not just an extension of the concept, but what I believe to be the '''end''' of that extension. Ultimate adds an independent "minicomma" generator to reach p5 and p13, which acts as 385/384, ~352/351, ~5120/5103, layoma, ... etc, being around 4.4c. It doesn't begin to be fully useful up until 217edo, but 270edo and 311edo are arguably the best tunings.

* 217edo is ~~alright~~, but not the best, which would make it a waste of time if it weren't a multiple of 31edo. [[176edo]] also supports it earlier at full structure, but it blows as an equal tuning because the MC is too wide.

* 217edo is good, but not the best, which would make it a waste of time if it weren't a multiple of 31edo. [[176edo]] also supports it earlier at full structure, but it blows as an equal tuning because the MC is too wide.

'''The key reasons''' ~~on why~~ Ultimate ~~is ultimate~~:

'''The key reasons''' for which to settle with Ultimate and its temperaments:

* 270edo and 311edo are inside the supported equal tunings.

* 270edo and 311edo are inside the supported equal tunings, two widely known incredibly good edos.

* ~~Going any further would make~~ the ~~edosteps~~ not ~~discernible~~.

* It is a temperament of [[olympic]], an extremely good 13-limit temperament.

* ~~No other edo in their vicinity~~ is ~~as good as them~~.

* It provides one of the easiest, if not the easiest way to achieve near-perfect JI with a '''whole''' 5-limit basis. That is to say, no prime is split in equal parts. Primes beyond 5 are reached with a linear combination of 2, 3, 5.

* It has the chain of fifths as a primordial spine, with minimal alterations of the third generator to reach the rest of primes.

* It is extremely easy to notate.

Beyond that, I see no reason to use edos. The sequence still continues beyond Ultimate, which I dub the [[User:Eufalesio/Ultimate#Beyond Ultimate - the ULTRAOLYMPIC sequence|ULTRAOLYMPIC sequence]].

@@ Line 1: / Line 1: @@
-is 53 & 94 & 217, with mapping {{Mapping|1 0 0 25 -33 -13 47 -6|0 1 0 -14 23 12 -30 5|0 0 1 0 0 -1 2 1}} in the [[19-limit]]. 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'''. The choice of name will become clear. Or at least, I expect you to understand why it's clear in my mind.
+is a special denominatinon I give to [[septischismic]] (also [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)|here (data dump)]]). The real name is ''septischismic, 12e Ultimate'', ; but we will simply call it '''Ultimate''' here.
 Special thanks for [[Kite Giedraitis]] for feedback and edits, for [[Godtone]] (osmium) for teaching me [[S-expression|S-expressions]], and for [[Flora Canou]] for helping me document and put the temperaments up to standards.
@@ Line 141: / Line 141: @@
 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. Among all paths I've considered [[User:Eufalesio/12edo detemperament sequences|(here)]], this, is what I consider to be the best. Behold, ''the'' ''Ultimate'' ''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. Among all paths I've considered [[User:Eufalesio/12edo detemperament sequences|(here)]], this, is what I consider to be the best for me. Behold, ''the'' ''12e'' ''detemperament'' ''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 151: / Line 151: @@
 * '''Ultimate''' is not just an extension of the concept, but what I believe to be the '''end''' of that extension. Ultimate adds an independent "minicomma" generator to reach p5 and p13, which acts as 385/384, ~352/351, ~5120/5103, layoma, ... etc, being around 4.4c. It doesn't begin to be fully useful up until 217edo, but 270edo and 311edo are arguably the best tunings.
-* 217edo is alright, but not the best, which would make it a waste of time if it weren't a multiple of 31edo. [[176edo]] also supports it earlier at full structure, but it  blows as an equal tuning because the MC is too wide.
+* 217edo is good, but not the best, which would make it a waste of time if it weren't a multiple of 31edo. [[176edo]] also supports it earlier at full structure, but it blows as an equal tuning because the MC is too wide.
-'''The key reasons''' on why Ultimate is ultimate:
+'''The key reasons''' for which to settle with Ultimate and its temperaments:
-* 270edo and 311edo are inside the supported equal tunings.
+* 270edo and 311edo are inside the supported equal tunings, two widely known incredibly good edos.
-* Going any further would make the edosteps not discernible.
+* It is a temperament of [[olympic]], an extremely good 13-limit temperament.
-* No other edo in their vicinity is as good as them.
+* It provides one of the easiest, if not the easiest way to achieve near-perfect JI with a '''whole''' 5-limit basis. That is to say, no prime is split in equal parts. Primes beyond 5 are reached with a linear combination of 2, 3, 5.
+* It has the chain of fifths as a primordial spine, with minimal alterations of the third generator to reach the rest of primes.
+* It is extremely easy to notate.
 Beyond that, I see no reason to use edos. The sequence still continues beyond Ultimate, which I dub the [[User:Eufalesio/Ultimate#Beyond Ultimate - the ULTRAOLYMPIC sequence|ULTRAOLYMPIC sequence]].