User:Eufalesio/Ultimate - Revision history

Eufalesio: Changed how the term Ultimate is used

2026-08-15T18:00:04Z

Changed how the term Ultimate is used

← Older revision		Revision as of 18:00, 15 August 2026
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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.		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.		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...		* '''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.		* '''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]].		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]].

Eufalesio: /* The special place of 41edo, 94edo and 270edo */

2026-08-14T11:17:28Z

The special place of 41edo, 94edo and 270edo

← Older revision		Revision as of 11:17, 14 August 2026
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	The most error you'll get with this system resides in the chain of fifths (~+0.25c), having all other primes accurate to hundreds of a cent. This is in a sense is reminiscent of [[septimal meantone]], which can tune p5 and p7 near-pure by adding error to the fifth chain.		The most error you'll get with this system resides in the chain of fifths (~+0.25c), having all other primes accurate to hundreds of a cent. This is in a sense is reminiscent of [[septimal meantone]], which can tune p5 and p7 near-pure by adding error to the fifth chain.

	== The special place of 41edo, 94edo and 270edo ==		== The special place of 41edo, 53edo, 94edo and 270edo ==
	41edo is the coarsest cassandra edo, with a high ratio of accuracy to simplicity, and being the first ever edo to be distinctly consistent in the 9-odd-limit, making the most out of the next convergent chain of fifths. 94edo is arguably the best cassandra edo, making the most out of the chain of fifths; while more complex it can be extended to the 23-odd-limit; which could be useful to some. Not me.		41edo is the coarsest cassandra edo, with a high ratio of accuracy to simplicity, and being the first ever edo to be distinctly consistent in the 9-odd-limit, making the most out of the next convergent chain of fifths, and supplying

			53edo is the second coarsest, being better at some things 41edo is worse, and viceversa. It notably is extremely good in the yatha, with a fairly decent yazatha. It is also one of the best schismic temperaments, with nothing better than it except for 118edo and 171edo. Anything that doesn't use too many 11's and doesn't make much emphasis on 7's, can easily be done with 53edo.

			94edo is arguably the best cassandra edo, making the most out of the chain of fifths; while more complex it can be extended to the 23-odd-limit; which could be useful to some. Not me. When the whole 13-limit or 19-limit is desired, better to use 94edo to have everything decently well tuned.

	270edo is well known for its unbeatable 13-limit, for which, arguably, no other edo finer or coarser comes even close to its ratio of accuracy to "simplicity". It also technically has some useful interpretations for up to the [[53-limit]] which could be even more useful than that of 311edo, as seen by people like [[Godtone]].		270edo is well known for its unbeatable 13-limit, for which, arguably, no other edo finer or coarser comes even close to its ratio of accuracy to "simplicity". It also technically has some useful interpretations for up to the [[53-limit]] which could be even more useful than that of 311edo, as seen by people like [[Godtone]].

	41edo is particularly interesting because joining it with 270edo results in [[newt]], an extremely accurate rank 2 microtemperament of Ultimate that is practically indistinguishable from it. Instead of halving the poma, it halves the fifth, finding the MC "generator" at -41 generators, which firmly places this as a 41edo [[well temperament]].		41edo is particularly interesting because joining it with 270edo results in [[newt]], an extremely accurate rank 2 microtemperament of Ultimate that is practically indistinguishable from it. Instead of halving the poma, it halves the fifth, finding the MC "generator" at -41 generators, which firmly places this (including Ultimate) as a 41edo [[well temperament]].

	––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––		––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

Eufalesio: /* Easy tables */

2026-07-10T09:18:21Z

Easy tables

Eufalesio: blunder

2026-07-07T05:48:51Z

blunder

← Older revision		Revision as of 05:48, 7 July 2026
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	== Quick definition ==		== Quick definition ==
	Ultimate can be easily defined in the 13-limit as nullifying the [[2080/2079\|sinaisma]], [[minisma]], and [[eufalesma]]. This indirectly means that the [[Symbiotic comma\|salozo~~]], [[Nexus comma\|tribilo~~]], [[Wilschisma\|sathoyo]], [[Olympia\|salururu]], [[Garischisma\|sasaru]], [[Argyria\|lolotrizo]] commas, among an infinitude more, are all nullified too. In the 19-limit, it also tempers out the [[1225/1224\|subizoyoma]] and [[1216/1215\|sanoguma]].		Ultimate can be easily defined in the 13-limit as nullifying the [[2080/2079\|sinaisma]], [[minisma]], and [[eufalesma]]. This indirectly means that the [[Symbiotic comma\|salozo]], [[Wilschisma\|sathoyo]], [[Olympia\|salururu]], [[Garischisma\|sasaru]], [[Argyria\|lolotrizo]] commas, among an infinitude more, are all nullified too. In the 19-limit, it also tempers out the [[1225/1224\|subizoyoma]] and [[1216/1215\|sanoguma]].

	Ultimate has the following notable equations:		Ultimate has the following notable equations:

Eufalesio at 06:31, 28 June 2026

2026-06-28T06:31:13Z

← Older revision		Revision as of 06:31, 28 June 2026
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	=== Olympic - Metaolympic ===		=== Olympic - Metaolympic ===
	[[Olympic\|This]]. The best course of action for detempering Ultimate is to observe the ~~garischisma~~, resulting in rank-4 olympic, which is just {S64, S65}. Notationwise, this results in spliting the saruyoma ~~being observed~~. Unlike in JI, where the ~~schisma~~ is around half the ~~garischisma~~, here the ~~schisma~~ is ~1.6x LARGER, not smaller. A [[10241/10240\|tina (~10241/10240)]] is required to reach ~~prime~~ 17 and 19.		[[Olympic\|This]]. The best course of action for detempering Ultimate is to observe the sasaruma, resulting in rank-4 olympic, which is just {S64, S65}. Notationwise, this results in spliting the saruyoma in halves. Unlike in JI, where the layoma is around half the sasaruma, here the layoma is ~1.6x LARGER, not smaller. A heavily inflated [[10241/10240\|tina (~10241/10240)]] is one of these sasaruma halves; it and distinct layomas are required to reach primes 17 and 19.

	Olympic decouples 64/63 from the chain of fifths; 64/63 is now its own thing, and the poma is \\|↑. (\\| is a ~~garischisma~~ down)		Olympic decouples 64/63 from the chain of fifths; 64/63 is now its own thing, and the poma is \\|↑. (\\| is a sasaruma down)

	Everything else is the same:		Everything else is the same:
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	It's something I don't think I'll ever see myself doing because this accuracy is enough to represent highly fine edos such as 494, 764, or 935edo, which is already too much for me. Extensions to the 19-limit are bad: the strong one keeps 1729/1728 and 1216/1215, [[Catalog of rank-4 temperaments#Metaolympic\|'''Meta'''olympic]] works, and though it's a weak extension; it can still be written retrocompatibly.		It's something I don't think I'll ever see myself doing because this accuracy is enough to represent highly fine edos such as 494, 764, or 935edo, which is already too much for me. Extensions to the 19-limit are bad: the strong one keeps 1729/1728 and 1216/1215, [[Catalog of rank-4 temperaments#Metaolympic\|'''Meta'''olympic]] works, and though it's a weak extension; it can still be written retrocompatibly.

	Here, 19/16 is †\|\m3. 17/16 is 𐕣^\|\'''⇊⇊'''M2. (\|\ is a ~~schisma~~ up, †/𐕣 is a heavily inflated tina)		Here, 19/16 is †\|\m3. 17/16 is 𐕣^\|\'''⇊⇊'''M2. (\|\ is a layoma up, †/𐕣 is a heavily inflated tina)

	=== Insanismic - Insatinismic ===		=== Insanismic - Insatinismic ===
	Olympic STILL not enough? Split your losses and use [[5767168/5767125\|{S64/S65}]]. Now you observe the olympia and get a schismina accidental.		Olympic STILL not enough? Split your losses and use [[5767168/5767125\|{S64/S65}]]. Now you observe the olympia and get a schismina accidental.

	An olympia is 3 of these schisminas. ~~You could write~~ this as dots above or below the accidentals but this is possibly getting a tad crowded.		An olympia is 3 of these schisminas. Write this as dots above or below the accidentals but this is possibly getting a tad crowded.

	* ↑ is 64/63.		* ↑ is 64/63.
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	Insanismic is at a level of precision comparable to 8539edo and much, MUCH finer. The people at sagittal.org had already declared its own version of this notation to be of "Insane" precision. Personally, ''insane'' is way too nice of a descriptor. This is is beyond your grasp of perfection, trust me.		Insanismic is at a level of precision comparable to 8539edo and much, MUCH finer. The people at sagittal.org had already declared its own version of this notation to be of "Insane" precision. Personally, ''insane'' is way too nice of a descriptor. This is is beyond your grasp of perfection, trust me.

	With an average tuning error of -0.000234c, let's be frank, this is JI. Approachable, thanks to it being the end of the ULTRAOLYMPIC sequence and thus retrocompatible; obsessive, but approachable nontheless. You are quite frankly, batshit insane if you require this precision, but I still give you the tools to express that insanity.		With an average tuning error of -0.000234c, let's be frank, this is JI. Approachable, thanks to it being the end of the ULTRAOLYMPIC sequence and thus retrocompatible; obsessive, but approachable nontheless. You are quite frankly, batshit insane if you require this precision, but I still give you the tools to express that insanity. It looks funny.

	Wouldn't I be also batshit insane by having researched it? No. I know I won't use it. But, knowing about it is important; how much '''JI'''uice I can squeeze out of 12edo... taking it to insanity? Well, '''here''' is the answer!		Wouldn't I be also batshit insane by having researched it? No. I know I won't use it. But, knowing about it is important; how much '''JI'''uice I can squeeze out of 12edo... taking it to insanity? Well, '''here''' is the answer!

Eufalesio: Reference to the rest of 12edo detemp sequences

2026-06-27T15:18:04Z

Reference to the rest of 12edo detemp sequences

← Older revision		Revision as of 15:18, 27 June 2026
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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.		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'' ''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. 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...		* '''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...

Eufalesio at 12:42, 22 June 2026

2026-06-22T12:42:38Z

Eufalesio: Final final

2026-06-22T11:52:49Z

Final final

← Older revision		Revision as of 11:52, 22 June 2026
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	The [[pergen]] is (P8, P5, ^1), where ^1 is the "minicomma" (from this point forward refered to as "MC"); a 3~5c interval that represents 385/384, 352/351, 5120/5103, 513/512, the layoma, etc. 4:5:6:7:9:11:13:15:17:19 octave reduced is notated as:		The [[pergen]] is (P8, P5, ^1), where ^1 is the "minicomma" (from this point forward refered to as "MC"); a 3~5c interval that represents 385/384, 352/351, 5120/5103, 513/512, the layoma, etc. 4:5:6:7:9:11:13:15:17:19 octave reduced is notated as:

	P1 – ^'''↓'''M3 – P5 – '''↓'''m7 – M2 – ⇈4 – v⇈m6 – ^'''↓'''M7 – ^^~~'''⇊⇊'''M2~~ – ^m3		P1 – ^'''↓'''M3 – P5 – '''↓'''m7 – M2 – ⇈4 – v⇈m6 – ^'''↓'''M7 – ^^⇊⇊M2 – ^m3

	Using pomas (pythagorean commas, [↑/'''↓''']) improves ~~this~~ notation for reasons that will be exposed later.		Using pomas (pythagorean commas, [↑/'''↓''']) improves [[User:Eufalesio/Ultimate#Nomenclature and notation\|the notation]] for reasons that will be exposed later.

	Some temperament properties of Ultimate and its subsets are exposed [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)\|here]].		Some temperament properties of Ultimate and its subsets are exposed [[User:Eufalesio/Important Tables#Temperament properties of Ultimate edos (I care about)\|here]].
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	It's something I don't think I'll ever see myself doing because this accuracy is enough to represent highly fine edos such as 494, 764, or 935edo, which is already too much for me. Extensions to the 19-limit are bad: the strong one keeps 1729/1728 and 1216/1215, [[Catalog of rank-4 temperaments#Metaolympic\|'''Meta'''olympic]] works, and though it's a weak extension; it can still be written retrocompatibly.		It's something I don't think I'll ever see myself doing because this accuracy is enough to represent highly fine edos such as 494, 764, or 935edo, which is already too much for me. Extensions to the 19-limit are bad: the strong one keeps 1729/1728 and 1216/1215, [[Catalog of rank-4 temperaments#Metaolympic\|'''Meta'''olympic]] works, and though it's a weak extension; it can still be written retrocompatibly.

	Here, 19/16 is †\|\m3. 17/16 is 𐕣^\|\'''⇊⇊'''M2. (\|\ is a schisma up, † is a heavily inflated tina)		Here, 19/16 is †\|\m3. 17/16 is 𐕣^\|\'''⇊⇊'''M2. (\|\ is a schisma up, †/𐕣 is a heavily inflated tina)

	=== Insanismic - Insatinismic ===		=== Insanismic - Insatinismic ===
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	––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––		––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

	Unlike Olympic, this one is undeniably '''WAY''' more accurate, at least 100 times more accurate. It can be weakly 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. 19/16 is †\|̈\m3. 17/16 is 𐕣^^~~'''⇊⇊̱'''M2~~.		Unlike Olympic, this one is undeniably '''WAY''' more accurate, at least 100 times more accurate. It can be weakly 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. 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! [expandable] Many thanks to Flora Canou for help with all the temp stuff... she's the real heroine here behind the numbers.		Look how goddamn accurate this temp is! Look, I even made a technical temp data section! [expandable] Many thanks to Flora Canou for help with all the temp stuff... she's the real heroine here behind the numbers.
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	These are all the accidentals you need to know to write in Ultimate, and even beyond it. For most cases, there is no need to go beyond two of anything, in the case of pomas however, you can end up using three or even more if you don't respell enharmonically with saquarumas, or go too far down the chain of fifths.		These are all the accidentals you need to know to write in Ultimate, and even beyond it. For most cases, there is no need to go beyond two of anything, in the case of pomas however, you can end up using three or even more if you don't respell enharmonically with saquarumas, or go too far down the chain of fifths.

	One instance is writing 5/4 above a 16/11, which is 20/11. Above a C, this is '''⇊'''G - ^'''↓⇊'''B.		One instance is writing 5/4 above a 16/11, which is 20/11. Above a C, this is '''⇊'''G - ^'''↓'''⇊B.

	A phonetic coding proposal and the alterations for written text are:		A phonetic coding proposal and the alterations for written text are:
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	\|+		\|+
	! rowspan="2" \|Main accidentals		! rowspan="2" \|Main accidentals
	~~(ASCII)~~
	! colspan="9" \|Ultimate		! colspan="9" \|Ultimate
	! colspan="6" \|ULTRAOLYMPIC		! colspan="6" \|ULTRAOLYMPIC
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	! colspan="2" \|Tina		! colspan="2" \|Tina
	\|-		\|-
	\| rowspan="2" \|~~Symbols~~		\| rowspan="2" \|ASCII-ish symbols
	\| rowspan="2" \|h		\| rowspan="2" \|h
	\|#		\|#
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	\| rowspan="2" \|M		\| rowspan="2" \|M
	\| rowspan="2" \|m		\| rowspan="2" \|m
	\| rowspan="2" \|^		\| rowspan="4" \|^
	\| rowspan="2" \|v		\| rowspan="4" \|v
	\| rowspan="2" \|<nowiki>Y \| X</nowiki>		\| rowspan="2" \|<nowiki>Y \| X</nowiki>
	\| rowspan="2" \|<nowiki>y \| x</nowiki>		\| rowspan="2" \|<nowiki>y \| x</nowiki>
	\| rowspan="2" \|' " "'		\| rowspan="2" \|' " "'
	\| rowspan="2" \|. .. …		\| rowspan="2" \|. .. …
	\| rowspan="2" \| †		\| rowspan="4" \| †
	\| rowspan="2" \| ‡		\| rowspan="2" \| ‡ / t
	\|-		\|-
	\|##		\|##
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	\|H		\|H
	\|h		\|h
			\|-
			\| rowspan="2" \|Unicode symbols
			\| rowspan="2" \|♮
			\|♯
			\|♭
			\|↑
			\|⇈
			\| rowspan="2" \|Γ
			\| rowspan="2" \|L
			\| rowspan="2" \|<nowiki>/\| \| \|\</nowiki>
			\| rowspan="2" \|<nowiki>\\| \| \|/</nowiki>
			\| rowspan="2" \|̇ ̈ ̄
			\| rowspan="2" \|̣ ̤ ̱
			\| rowspan="2" \|𐕣
			\|-
			\|𝄪
			\|𝄫
			\|'''↓'''
			\|⇊
	\|-		\|-
	\| rowspan="2" \|Spoken		\| rowspan="2" \|Spoken
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	\|sharp		\|sharp
	\|flat		\|flat
	\|~~super~~		\|(su)per
	\|sub		\|sub
	\| rowspan="2" \|me		\| rowspan="2" \|me
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	\|sesquidim		\|sesquidim
	\|}		\|}
	<nowiki>*</nowiki>Not strictly necessary, but it does simplify notation immensely. The saquaruma equals the mercator comma in Ultimate (also being tempered out in 53edo), but not beyond it. It is useful for ultra-long chains of fifths and notating p17, which requires at least four rumas. It follows the following equation: ⇈⇈P1 = ~~mm2~~, ⇈⇈m2 = m'''↓'''M2; 17/16 is then \|\^~~μ↑m2~~.		<nowiki>*</nowiki>Not strictly necessary, but it does simplify notation immensely. The saquaruma equals the mercator comma in Ultimate (also being tempered out in 53edo), but not beyond it. It is useful for ultra-long chains of fifths and notating p17, which requires at least four rumas. It follows the following equation: ⇈⇈P1 = Lm2, ⇈⇈m2 = L'''↓'''M2; 17/16 is then ^^L↑m2 in Ultimate; or 𐕣\|\^L↑̱m2 in Batshittin.

	<big>Of course, you can use whatever phonetic coding you want, as long as it's not ''too bespoke''. IFYKYK.</big>		<big>Of course, you can use whatever phonetic coding you want, as long as it's not ''too bespoke''. IFYKYK.</big>

	=== Syntax ===		=== Syntax ===
	For an accidental: (tina) + (~~schismina~~) + (sasaruma/layoma) + [~~saruyoma~~] + [~~poma~~] + [apotome]		For an accidental: (tina) + (minas) + (sasaruma/layoma) + [saruyomas] + (saquaruma) + [pomas] + [apotome]

	* v# = downsharp		* v# = downsharp
	* ^^~~'''⇊'''~~ = duphypo		* ^^h = duphypo

	For intervals: [accidental] + [5L 7s 6\|5 nominal]		For intervals: [accidental] + [5L 7s 6\|5 nominal]

	* ~~'''↓'''M2~~ = submajor second		* sM2 = submajor second
	* ^~~'''↓'''M3~~ = upsubmajor third		* ^sM3 = upsubmajor third
	* ~~v⇈m6~~ = downhyperminor sixth		* vHm6 = downhyperminor sixth

	For pitches: [5L 7s 6\|5 nominal] + [accidental] + [octave number] <small>(scientific please, A4=440Hz)</small>		For pitches: [5L 7s 6\|5 nominal] + [accidental] + [octave number] <small>(scientific please, A4=440Hz)</small>

	* D4 = dee four		* D4 = dee four
	* ~~E⇈b4~~ = e hyperflat four		* EHb4 = e hyperflat four
	* F^~~'''↓'''~~#5 = ef upsubsharp five		* F^s#5 = ef upsubsharp five
	I left out chords on purpose. Kite has made much better systems for naming them; see [[Kite's color notation#Chord names]] and [[Hi-lo notation]].		* E^^MSb6 ef dupmeperflat six
			I left out chords on purpose. Kite has made much better systems for naming them; see [[Kite's color notation#Chord names\|Color notation (chords)]] and [[Hi-lo notation]].

Eufalesio: Final edits, this is going to be seen by a wider public!

2026-06-22T09:53:41Z

Final edits, this is going to be seen by a wider public!

Show changes

Eufalesio: Forced everything to be 3line paragraphs for readmaxxing

2026-06-19T21:07:42Z

Forced everything to be 3line paragraphs for readmaxxing

Show changes

← Older revision		Revision as of 09:18, 10 July 2026
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	et cetera... Ultimate is much, much better only in the [[2.3.5.7.11.13.19 subgroup\|yathalazana]], but it still manages to make a decent, albeit complex mapping.		et cetera... Ultimate is much, much better only in the [[2.3.5.7.11.13.19 subgroup\|yathalazana]], but it still manages to make a decent, albeit complex mapping.

	The [[pergen]] is (P8, P5, ^1), where ^1 is the "minicomma" (~~from this point forward refered to~~ as "MC"); a 3~5c interval that represents 385/384, 352/351, 5120/5103, 513/512, the layoma, etc. 4:5:6:7:9:11:13:15:17:19 octave reduced is notated as:		The [[pergen]] is (P8, P5, ^1), where ^1 is the "minicomma" (hereafter noted as MC); a ~4{{C}} interval that represents 385/384, 352/351, 5120/5103, 513/512, the layoma, etc. 4:5:6:7:9:11:13:15:17:19 octave reduced is notated as:

	P1 – ^'''↓'''M3 – P5 – '''↓'''m7 – M2 – ⇈4 – v⇈m6 – ^'''↓'''M7 – ^^⇊⇊M2 – ^m3		P1 – ^'''↓'''M3 – P5 – '''↓'''m7 – M2 – ⇈4 – v⇈m6 – ^'''↓'''M7 – ^^⇊⇊M2 – ^m3
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	== Beyond Ultimate - the ULTRAOLYMPIC sequence ==		== Beyond Ultimate - the ULTRAOLYMPIC sequence ==
	There's a reason why I deem Ultimate ultimate. You're supposed to end there and go no further, because Ultimate is right at the limit of practicality. If you're stubborn enough to ignore my warnings and venture into the land of ~~impossible~~... I still know how you can continue the into the ULTRAOLYMPIC sequence and add extra pairs of accidentals that are fully retrocompatible with Ultimate.		There's a reason why I deem Ultimate ultimate. You're supposed to end there and go no further, because Ultimate is right at the limit of practicality. If you're stubborn enough to ignore my warnings and venture into the land of insanity... I still know how you can continue the into the ULTRAOLYMPIC sequence and add extra pairs of accidentals that are fully retrocompatible with Ultimate.

	=== Olympic - Metaolympic ===		=== Olympic - Metaolympic ===
	[[Olympic\|This]]. The best course of action for detempering Ultimate is to observe the sasaruma, resulting in rank-4 olympic, which is just {S64, S65}. Notationwise, this results in spliting the saruyoma in halves. Unlike in JI, where the layoma is around half the sasaruma, here the layoma is ~1.6x LARGER, not smaller. A heavily inflated [[10241/10240\|tina (~10241/10240)]] is one of these sasaruma halves; it and distinct layomas are required to reach primes 17 and 19.		[[Olympic\|This]]. The best course of action for detempering Ultimate is to observe the sasaruma, resulting in rank-4 olympic, which is just {S64, S65}. Notationwise, this results in spliting the saruyoma in halves with no meaningful notation changes. Unlike in JI, where the layoma is around half the sasaruma, here the layoma is ~1.6x LARGER, not smaller. A heavily inflated [[10241/10240\|tina (~10241/10240)]] is one of these sasaruma halves; it and distinct layomas are required to reach primes 17 and 19.

	Olympic decouples 64/63 from the chain of fifths; 64/63 is now its own thing, and the poma is \\|↑. (\\| is a sasaruma down)		Olympic decouples 64/63 from the chain of fifths; 64/63 is now its own thing, and the poma is \\|↑. (\\| is a sasaruma down)
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	=== Insanismic - Insatinismic ===		=== Insanismic - Insatinismic ===
	Olympic STILL not enough? Split your losses and use [[5767168/5767125\|~~{S64/S65}~~]]. Now you observe the olympia and get a schismina accidental.		Olympic STILL not enough? Split your losses and use [[5767168/5767125#Insanismic\|Insanismic]]. Now you observe the olympia and get a schismina accidental.

	An olympia is 3 of these schisminas. Write this as dots above or below the accidentals but this is possibly getting a tad crowded.		An olympia is 3 of these schisminas. Write this as dots above or below the accidentals but this is possibly getting a tad crowded.
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	With an average tuning error of -0.000234c, let's be frank, this is JI. Approachable, thanks to it being the end of the ULTRAOLYMPIC sequence and thus retrocompatible; obsessive, but approachable nontheless. You are quite frankly, batshit insane if you require this precision, but I still give you the tools to express that insanity. It looks funny.		With an average tuning error of -0.000234c, let's be frank, this is JI. Approachable, thanks to it being the end of the ULTRAOLYMPIC sequence and thus retrocompatible; obsessive, but approachable nontheless. You are quite frankly, batshit insane if you require this precision, but I still give you the tools to express that insanity. It looks funny.

	Wouldn't I be also batshit insane by having researched it? No. ~~I know I won't use it. But, knowing~~ about it is important; how much '''JI'''uice I can squeeze out of 12edo... taking it to insanity? Well, '''here''' is the answer!		Wouldn't I be also batshit insane by having researched it? No. Knowing about it is important; how much '''JI'''uice I can squeeze out of 12edo... taking it to insanity? Well, '''here''' is the answer!

	––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––		––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––
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	Look how goddamn accurate this temp is! Look, I even made a technical temp data section! [expandable] Many thanks to Flora Canou for help with all the temp stuff... she's the real heroine here behind the numbers.		Look how goddamn accurate this temp is! Look, I even made a technical temp data section! [expandable] Many thanks to Flora Canou for help with all the temp stuff... she's the real heroine here behind the numbers.

	=== Insanismic AKA Batshit (temperament) ===		=== Insanismic AKA Batshit (the temperament) ===
	I'm aware temps can't have NSFW names. "Insanismic" is the SFW, clean term. The one for the front page. The ''correct'' mask.		I'm aware temps can't have NSFW names. "Insanismic" is the SFW, clean term. The one for the front page. The ''correct'' mask.

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	== Nomenclature and notation ==		== Nomenclature and notation ==
	This notation can be easily spoken as well as written, adapting Kite's color notation and ups and downs into a nice collage. At least, the part of Ultimate. For the things beyond Ultimate I <s>stole</s> borrowed some Sagittal nomenclature of the Magrathean symbol set that had the same comma functions. Behold!		This notation can be easily spoken as well as written, adapting Kite's color notation and ups and downs into a nice collage. At least, the part of Ultimate. For the things beyond Ultimate I <s>stole</s> borrowed some Sagittal nomenclature of the Magrathean symbol set that had the same comma functions. Behold!
	[[File:ULTRAOLYMPIANsymbolset.jpg\|center\|thumb\|960x960px\|The name of this notation is henceforth the ULTRAOLYMPIC notation, and the main 4 systems. Metaolympic would go in the middle of Ultimate and Batshit, with tinas but without mina accidentals. Not included since it ''kinda'' breaks the sequence.]]		[[File:ULTRAOLYMPIANsymbolset.jpg\|center\|thumb\|960x960px\|The name of this notation is henceforth the ULTRAOLYMPIC notation, and the main 4 systems. ALL CAPS, and pronounced loudly. Metaolympic would go in the middle of Ultimate and Batshit, with tinas but without mina accidentals. Not included since it ''kinda'' breaks the sequence.]]

	=== Easy tables ===		=== Easy tables ===
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	\|}		\|}
	<nowiki>*</nowiki>Not strictly necessary, but it does simplify notation immensely. The saquaruma equals the mercator comma in Ultimate (also being tempered out in 53edo), but not beyond it. It is useful for ultra-long chains of fifths and notating p17, which requires at least four rumas. It follows the following equation: ⇈⇈P1 = Lm2, ⇈⇈m2 = L'''↓'''M2; 17/16 is then ^^L↑m2 in Ultimate; or 𐕣\|\^L↑̱m2 in Batshittin.		<nowiki>*</nowiki>Not strictly necessary, but it does simplify notation immensely. The saquaruma equals the mercator comma in Ultimate (also being tempered out in 53edo), but not beyond it. It is useful for ultra-long chains of fifths and notating p17, which requires at least four rumas. It follows the following equation: ⇈⇈P1 = Lm2, ⇈⇈m2 = L'''↓'''M2; 17/16 is then ^^L↑m2 in Ultimate; or 𐕣\|\^L↑̱m2 in Batshittin.

			It is also unnecessary in 12edo and 41edo, since it equals a flat and a poma up respectively in those systems, and it is a way to skip the bad 19-limit mappings of 12eg and 41g.

	<big>Of course, you can use whatever phonetic coding you want, as long as it's not ''too bespoke''. IFYKYK.</big>		<big>Of course, you can use whatever phonetic coding you want, as long as it's not ''too bespoke''. IFYKYK.</big>

← Older revision		Revision as of 12:42, 22 June 2026
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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.		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'' ''~~'''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. 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...		* '''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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	* No other edo in their vicinity is as good as them.		* No other edo in their vicinity is as good as them.

	Beyond that, I see no reason to use edos. The sequence still continues beyond Ultimate~~... in~~ which ~~a more appropiate name would be~~ the ULTRAOLYMPIC sequence.		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]].

	270edo and 311edo inherit a chain of fifths that is consistent with cassandra, which itself is an extension of the circle of fifths. The only addition is a single edostep, and respectively, the entire 13-limit is tuned to unfathomable precision, and the 41-limit is fully accessible and very well tuned. However, I prefer sticking to the 13-limit, so 270edo is the best equal tuning.		270edo and 311edo inherit a chain of fifths that is consistent with cassandra, which itself is an extension of the circle of fifths. The only addition is a single edostep, and respectively, the entire 13-limit is tuned to unfathomable precision, and the 41-limit is fully accessible and very well tuned. However, I prefer sticking to the 13-limit, so 270edo is the best equal tuning.
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	==== Ultimate ''sensu stricto'' ====		==== Ultimate ''sensu stricto'' ====
	It is possible to forgo edos altogether and use Ultimate as is, providing the absolute best tuning possible. However, it's a rank-3 ~~system~~. It has no extra unisons unlike the equal tunings. This is reserved for when 270edo is not just enough, and beating is something to avoid as much as possible.		It is possible to forgo edos altogether and use Ultimate as is, providing the absolute best tuning possible. However, it's a rank-3 temperament. It has no extra unisons unlike the equal tunings. This is reserved for when 270edo is not just enough, and beating is something to avoid as much as possible.

	The most error you'll get with this system resides in the chain of fifths (~+0.25c), having all other primes accurate to hundreds of a cent. This is in a sense is reminiscent of [[septimal meantone]], which can tune p5 and p7 near-pure by adding error to the fifth chain.		The most error you'll get with this system resides in the chain of fifths (~+0.25c), having all other primes accurate to hundreds of a cent. This is in a sense is reminiscent of [[septimal meantone]], which can tune p5 and p7 near-pure by adding error to the fifth chain.

	== The special place of 41edo, 94edo and 270edo ==		== The special place of 41edo, 94edo and 270edo ==
	41edo is the coarsest cassandra edo, with a high ratio of accuracy to simplicity, and being the first ever edo to be distinctly consistent in the 9-odd-limit, making the most out of the next convergent chain of fifths. 94edo is arguably the best cassandra edo, making the most out of the chain of fifths~~, which though~~ more complex can be extended to the ~~entire~~ 23-odd-limit; which could be useful to some.		41edo is the coarsest cassandra edo, with a high ratio of accuracy to simplicity, and being the first ever edo to be distinctly consistent in the 9-odd-limit, making the most out of the next convergent chain of fifths. 94edo is arguably the best cassandra edo, making the most out of the chain of fifths; while more complex it can be extended to the 23-odd-limit; which could be useful to some. Not me.

	270edo is well known for its unbeatable 13-limit, for which, arguably, no other edo finer or coarser comes even close to its ratio of accuracy to "simplicity". It also technically has some useful interpretations for up to the [[53-limit]] which could be even more useful than that of 311edo, as seen by people like [[Godtone]].		270edo is well known for its unbeatable 13-limit, for which, arguably, no other edo finer or coarser comes even close to its ratio of accuracy to "simplicity". It also technically has some useful interpretations for up to the [[53-limit]] which could be even more useful than that of 311edo, as seen by people like [[Godtone]].

	41edo is particularly interesting because joining it with 270edo results in [[newt]], an extremely accurate rank 2 ~~subset temperament~~ of Ultimate that is practically indistinguishable from it. Instead of halving the poma, it halves the fifth, finding the MC "generator" at -41 ~~gens~~, which firmly places this as a 41edo [[well temperament]].		41edo is particularly interesting because joining it with 270edo results in [[newt]], an extremely accurate rank 2 microtemperament of Ultimate that is practically indistinguishable from it. Instead of halving the poma, it halves the fifth, finding the MC "generator" at -41 generators, which firmly places this as a 41edo [[well temperament]].

	––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––		––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––––

	94edo and 270edo ~~have the key property of being~~ even, tempering out the [[kalisma]] and allowing the poma to be halved. Using them this way is reminiscent of [[~~Gariwizmic~~]], a very similar subset of Ultimate, but with the MC found deep in the generator chain, not independent. This is useful for easier navigation within a DAW, though optimization ends here.		94edo and 270edo are even, tempering out the [[kalisma]] and allowing the poma to be halved. Using them this way is reminiscent of [[gariwizmic]], a very similar subset of Ultimate, but with the MC found deep in the generator chain, not independent. This is useful for easier navigation within a DAW, though optimization ends here.

	It's possible to use ~~Gariwizmic~~ wholesale, though it only slightly improves 270edo in precision, Newt is a much better choice for accuracy's sake, though 94edo does ''not'' support it. Gariwizmic provides structure, not necessarily the tuning.		It's possible to use gariwizmic wholesale, though it only slightly improves 270edo in precision, Newt is a much better choice for accuracy's sake, though 94edo does ''not'' support it. Gariwizmic provides good structure, not necessarily the best tuning.

	== Beyond Ultimate - the ULTRAOLYMPIC sequence ==		== Beyond Ultimate - the ULTRAOLYMPIC sequence ==
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	I'm aware temps can't have NSFW names. "Insanismic" is the SFW, clean term. The one for the front page. The ''correct'' mask.		I'm aware temps can't have NSFW names. "Insanismic" is the SFW, clean term. The one for the front page. The ''correct'' mask.

	But know: the ''real'' name is '''batshit.''' Because that's what this temperament is. Batshit insane! Yes, I like crude, evocative names. The English language has a beautiful expression for this special obsession that I will not '''not''' use. The unfathomable precision it bestows. The madness that ensues upon the search of perfection. ~~Here~~ is perfection! GO CRAZY!{{Databox\|1=Temperament data\|2=[[Subgroup]]: 2.3.5.7.11.13		But know: the ''real'' name is '''batshit.''' Because that's what this temperament is. Batshit insane! Yes, I like crude, evocative names. The English language has a beautiful expression for this special obsession that I will not '''not''' use. The unfathomable precision it bestows. The madness that ensues upon the search of perfection. HERE is perfection! GO CRAZY!{{Databox\|1=Temperament data\|2=[[Subgroup]]: 2.3.5.7.11.13

	[[Comma list]]: [[5767168/5767125]]		[[Comma list]]: [[5767168/5767125]]
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	==== Insatinismic AKA Batshittin ====		==== Insatinismic AKA Batshittin ====
	The "Tinismic" part comes from it being a weak extension that notation-wise results in the addition of a tina accidental.{{Databox\|1=Temperament data\|2=[[Subgroup]]: 2.3.5.7.11.13.17.19		The "Tinismic" part comes from it being a weak extension that notation-wise results in the addition of a tina accidental. {{Databox\|1=Temperament data\|2=[[Subgroup]]: 2.3.5.7.11.13.17.19
	[[Comma list]]: 5767168/5767125, [[11413376/11413325]], [[633556/633555]]		[[Comma list]]: 5767168/5767125, [[11413376/11413325]], [[633556/633555]]