Schismic - Revision history

Overthink: /* Scales */ idiosyncratic terms

2026-04-29T19:19:11Z

Scales: idiosyncratic terms

← Older revision		Revision as of 19:19, 29 April 2026
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	== Scales ==		== Scales ==
			{{Idiosyncratic terms\|The later mos names are proposals that can be found on the page [[TAMNAMS Extension]].}}
	* [[5L 7s]] (p-chromatic)		* [[5L 7s]] (p-chromatic)
	* [[12L 5s]] (p-enharmonic)		* [[12L 5s]] (p-enharmonic)

FloraC: + category

2026-04-19T13:57:45Z

+ category

← Older revision		Revision as of 13:57, 19 April 2026
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	[[Category:Schismic\| ]] <!-- main article -->		[[Category:Schismic\| ]] <!-- main article -->
	[[Category:Rank-2 temperaments]]		[[Category:Rank-2 temperaments]]
			[[Category:Microtemperaments]]
	[[Category:Schismatic family]]		[[Category:Schismatic family]]

FloraC: Cleanup on hatnote & infobox

2026-02-09T09:55:28Z

Cleanup on hatnote & infobox

← Older revision		Revision as of 09:55, 9 February 2026
Line 1:		Line 1:
	~~: ''This page is about a~~ regular temperament sometimes known as "helmholtz"~~. For~~ the music theorist~~, see [[~~Hermann von Helmholtz~~]].''~~		{{About\|the regular temperament sometimes known as "helmholtz"\|the music theorist\|Hermann von Helmholtz}}
	{{Infobox regtemp		{{Infobox regtemp
	\| Title = Schismic		\| Title = Schismic
	\| Subgroups = 2.3.5		\| Subgroups = 2.3.5
	\| Comma basis = [[32805/32768]]		\| Comma basis = [[32805/32768]]
	\| ~~Generator~~ = 3/2		\| Edo join 1 = 12 \| Edo join 2 = 53
	\| Mapping = 1; 1 -8		\| Mapping = 1; 1 -8
			\| Generators = 3/2
			\| Generators tuning = 701.731
			\| Optimization method = CWE
			\| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]], [[12L 5s]]
	\| Pergen = (P8, P5)		\| Pergen = (P8, P5)
	\| Color name = Layoti		\| Color name = Layoti
	~~\| Edo join 1 = 12 \| Edo join 2 = 41~~
	~~\| Optimization method = CWE~~
	~~\| Generator tuning = 701.731~~
	~~\| MOS scales = [[2L 3s]], [[5L 2s]], [[5L 7s]], [[12L 5s]]~~
	\| Odd limit 1 = 5 \| Mistuning 1 = 0.217 \| Complexity 1 = 12		\| Odd limit 1 = 5 \| Mistuning 1 = 0.217 \| Complexity 1 = 12
	\| Odd limit 2 = (5-limit) 125 \| Mistuning 2 = 0.837 \| Complexity 2 = 29		\| Odd limit 2 = 5-limit 125 \| Mistuning 2 = 0.837 \| Complexity 2 = 29
	}}		}}
	'''Schismic''', '''schismatic''', or '''helmholtz''' is a [[5-limit]] [[regular temperament\|temperament]] which takes an almost just [[3/2\|perfect fifth]] and stacks it eightfold to reach [[8/5]], mapping [[5/4]] to the diminished fourth (e.g. C–F♭) and [[tempering out]] the schisma, [[32805/32768]].		'''Schismic''', '''schismatic''', or '''helmholtz''' is a [[5-limit]] [[regular temperament\|temperament]] which takes an almost just [[3/2\|perfect fifth]] and stacks it eightfold to reach [[8/5]], mapping [[5/4]] to the diminished fourth (e.g. C–F♭) and [[tempering out]] the schisma, [[32805/32768]].

Zhenlige: /* Tuning spectrum */

2026-01-28T09:09:15Z

Tuning spectrum

← Older revision		Revision as of 09:09, 28 January 2026
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	\| 69\118 \|\| \|\| 701.6949 \|\|		\| 69\118 \|\| \|\| 701.6949 \|\|
	\|-		\|-
	\| \|\| 5/4 \|\| 701.7108 \|\| 1/8-comma		\| \|\| 5/4 \|\| 701.7108 \|\| 1/8-comma, lower bound of 5-odd-limit diamond tradeoff
	\|-		\|-
	\| \|\| 25/24 \|\| 701.7252 \|\| 2/17-comma		\| \|\| 25/24 \|\| 701.7252 \|\| 2/17-comma
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	\| 31\53 \|\| \|\| 701.8868 \|\|		\| 31\53 \|\| \|\| 701.8868 \|\|
	\|-		\|-
	\| \|\| 3/2 \|\| 701.9550 \|\| Pythagorean tuning		\| \|\| 3/2 \|\| 701.9550 \|\| Pythagorean tuning, upper bound of 5-odd-limit diamond tradeoff
	\|-		\|-
	\| 24\41 \|\| \|\| 702.4390 \|\|		\| 24\41 \|\| \|\| 702.4390 \|\|

FloraC: Restore some deletions. Distinguish strong and weak extensions. Don't add "maqamschismic" until it's proven notable. - mysterious extra steps in the interval chain

2026-01-22T13:03:01Z

Restore some deletions. Distinguish strong and weak extensions. Don't add "maqamschismic" until it's proven notable. - mysterious extra steps in the interval chain

← Older revision		Revision as of 13:03, 22 January 2026
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	\| Odd limit 2 = (5-limit) 125 \| Mistuning 2 = 0.837 \| Complexity 2 = 29		\| Odd limit 2 = (5-limit) 125 \| Mistuning 2 = 0.837 \| Complexity 2 = 29
	}}		}}
	'''Schismic''', '''schismatic''', or '''helmholtz''' ~~(specifically in the~~ [[5-limit]]~~) is a~~[[regular temperament\|temperament]] which takes an almost just [[3/2\|perfect fifth]] and stacks it eightfold to reach [[8/5]], mapping 5/4 to the diminished fourth (e.g. C–F♭) and [[tempering out]] the [[~~schisma]],~~ 32805/32768~~. Put alternatively: 8/5 maps to the [[tetratone~~]].		'''Schismic''', '''schismatic''', or '''helmholtz''' is a [[5-limit]] [[regular temperament\|temperament]] which takes an almost just [[3/2\|perfect fifth]] and stacks it eightfold to reach [[8/5]], mapping [[5/4]] to the diminished fourth (e.g. C–F♭) and [[tempering out]] the schisma, [[32805/32768]].

	[[5/4]] maps equivalently to a major third minus one [[Pythagorean comma]], and thus, the Pythagorean and [[syntonic comma]]s are equated into one tempered comma, splitting octaves into two major thirds and one ~5/4.		[[5/4]] maps equivalently to a major third minus one [[Pythagorean comma]], and thus, the Pythagorean and [[syntonic comma]]s are equated into one tempered comma, splitting octaves into two diatonic major thirds and one downmajor third representing 5/4.

	It is one of the simplest [[microtemperament~~\|microtemperaments~~]], as the fifth generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the schisma is practically [[~~Unnoticeable~~ comma\|unnoticeable]]). Technically, the best tuning in the 5-limit is to flatten the fifth by a fraction of a cent, though tunings with sharper fifths (and worse 5-limit, like in 41 or 94edo) still work fine.		Schismic is one of the simplest [[microtemperament]]s, as the fifth generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the schisma is practically [[unnoticeable comma\|unnoticeable]]). Technically, the best tuning in the 5-limit is to flatten the fifth by a fraction of a cent, though tunings with sharper fifths (and worse 5-limit, like in [[41edo\|41-]] or [[94edo]]) still work fine.

	Extensions include:		Extensions of schismic include [[garibaldi]] and [[pontiac]]. Garibaldi equates the generalized comma further to [[64/63]] and [[50/49]] (tempering out [[225/224]] and [[5120/5103]]) to provide an efficient framework for [[7-limit]] harmony, though with worse 5-limit intonation since the tuning favors slightly sharp fifths; pontiac, which tempers out [[4375/4374]] to induce very little damage on schismic harmonies, at the cost of 7 being quite complex. Besides these, there is the 2.3.5.19-[[subgroup]] extension [[nestoria]], which equates the minor third to [[19/16]], major third to [[19/15]] and [[24/19]], and the minor second to [[19/18]] and [[20/19]] (tempering out [[513/512]] and [[361/360]]).

	* [[Garibaldi]], which equates the generalized comma further to [[64/63]] and [[50/49]] (tempering out [[225/224]]) to provide an efficient framework for [[7-limit]] harmony, though with worse 5-limit intonation since the tuning uses slightly sharp fifths.		A notable example of a [[weak extension]] is [[sesquiquartififths]], which tempers out [[2401/2400]] and splits the fifth in fourths, inducing very little damage with a less complex mapping of 7 at the cost of quadrupling the complexity of 3 and 5.
	* [[Pontiac]], which tempers out [[4375/4374]] ~~to induce very little damage on schismic harmonies, at the cost of 7 being quite complex.~~
	* [[~~Sesquiquartififths~~]], which tempers out [[2401/2400]] and splits the fifth in fourths, inducing very little damage with a less complex mapping of 7 at the cost of quadrupling the complexity of 3 and 5.
	* The 2.3.5.13 [[subgroup]] extension [[Schismatic family#Maqamschismic (2.3.5.13)\|maqamschismic]], (tempering out the [[325/324\|marveltwin comma]]) and finds [[13/8]] at the dupminor sixth (^^Ab from C). See [[2.3.5.13 subgroup]] for more details.
	* The 2.3.5.19 [[subgroup]] extension [[nestoria]], which equates the minor third to [[19/16]], major third to [[19/15]] and [[24/19]], and the minor second to [[19/18]] and [[20/19]] (tempering out [[513/512]] and [[361/360]]).

	This page, however, focuses on the basic 5-limit temperament.		This page, however, focuses on the basic 5-limit temperament.
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	\| 20.77		\| 20.77
	\| 81/80		\| 81/80
	\|-
	~~\|13~~
	~~\|722.49~~
	~~\|243/160~~
	\|-
	~~\|14~~
	~~\|224.22~~
	~~\|256/225~~
	\|}		\|}
	<nowiki/>* In 5-limit CWE tuning		<nowiki/>* In 5-limit CWE tuning

	== Notation ==		== Notation ==
	Using schismic can be a challenge because it defies the tradition of diatonic {{w\|tertian harmony}} in [[chain-of-fifths notation]]; The just major ~~third~~ is not C–E–G like in [[meantone]], but rather, C–F♭–G. To address that, an additional module of accidentals such as arrows to represent the comma step may be adopted, allowing the user to write the chord above as C–vE–G.		Using schismic can be a challenge because it defies the tradition of diatonic {{w\|tertian harmony}} in [[chain-of-fifths notation]]; The just major triad on C is not C–E–G like in [[meantone]], but rather C–F♭–G. To address that, an additional module of accidentals such as arrows to represent the comma step may be adopted, allowing the user to write the chord above as C–vE–G.

	== Scales ==		== Scales ==
			* [[5L 7s]] (p-chromatic)
	~~=== MOS scales ===~~		* [[12L 5s]] (p-enharmonic)
			* [[12L 17s]] (pythagotonic)
	* [[5L 7s]] (p-chromatic) ~~[17edo]~~		* [[12L 29s]] (pythamystonic)
			* [[12L 41s]] (antipythomerc)
	* [[12L 5s]] (p-enharmonic) ~~[29edo]~~		* [[53L 12s]] (m-chro antipythomerc)
	* [[12L 17s]] (pythagotonic) ~~[41edo]~~
	* [[12L 29s]] (pythamystonic) ~~[53edo]~~
	* [[12L 41s]] (antipythomerc) ~~[65edo]~~
	* [[53L 12s]] (m-chro antipythomerc) ~~[118edo]~~

	~~EDO in brackets represents basic step ratio.~~

	=== Scala files ===		=== Scala files ===

Eufalesio: Added and reworded to make more succint

2026-01-22T12:26:39Z

Added and reworded to make more succint

← Older revision		Revision as of 12:26, 22 January 2026
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	\| Odd limit 2 = (5-limit) 125 \| Mistuning 2 = 0.837 \| Complexity 2 = 29		\| Odd limit 2 = (5-limit) 125 \| Mistuning 2 = 0.837 \| Complexity 2 = 29
	}}		}}
	'''Schismic''', '''schismatic''', or '''helmholtz''' ~~is a~~ [[5-limit]] [[regular temperament\|temperament]] which takes ~~a roughly justly tuned~~ [[3/2\|perfect fifth]] and stacks it ~~eight times~~ to reach [[8/5]], ~~thus finding the 5th harmonic at~~ the diminished fourth (e.g. C–F♭) and [[tempering out]] the [[schisma]], 32805/32768. ~~[[5~~/4]] can be respelled as a major third flattened by one [[Pythagorean comma]], and thus, the Pythagorean and [[syntonic comma]]s are equated into a generalized "comma", and the octave can be split into two diatonic major thirds and one downmajor third representing 5~~/4. It is one of~~ the ~~most basic examples of a~~ [[~~microtemperament~~]], as the fifth generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the difference between [[8192/6561]] and 5/4, the schisma being tempered out, is approximately 2 cents, which is [[just-noticeable difference\|unnoticeable]] to most people). Technically, the best tuning in the 5-limit is to flatten the fifth by a fraction of a cent, though tunings on both sides of the just interval work fine.		'''Schismic''', '''schismatic''', or '''helmholtz''' (specifically in the [[5-limit]]) is a[[regular temperament\|temperament]] which takes an almost just [[3/2\|perfect fifth]] and stacks it eightfold to reach [[8/5]], mapping 5/4 to the diminished fourth (e.g. C–F♭) and [[tempering out]] the [[schisma]], 32805/32768. Put alternatively: 8/5 maps to the [[tetratone]].

	Extensions ~~to schismic~~ include [[~~garibaldi~~]], which equates the generalized comma further to [[64/63]] and [[50/49]] (tempering out [[225/224~~]] and [[5120/5103~~]]) to provide an efficient framework for [[7-limit]] harmony, ~~and unlike~~ 5-limit ~~schismic performs best with a fifth tuned~~ slightly sharp ~~of just;~~ [[~~pontiac~~]], which tempers out [[4375/4374]] to induce very little damage on schismic harmonies, at the cost of 7 being quite complex; and the 2.3.5.19 [[subgroup]] extension [[nestoria]], which equates the minor third to [[19/16]], major third to [[19/15]] and [[24/19]], and the minor second to [[19/18]] and [[20/19]] (tempering out [[513/512]] and [[361/360]]). This page, however, focuses on the basic 5-limit temperament.		[[5/4]] maps equivalently to a major third minus one [[Pythagorean comma]], and thus, the Pythagorean and [[syntonic comma]]s are equated into one tempered comma, splitting octaves into two major thirds and one ~5/4.

			It is one of the simplest [[microtemperament\|microtemperaments]], as the fifth generator can be detuned by a fraction of a cent from just, or left untouched entirely (as the schisma is practically [[Unnoticeable comma\|unnoticeable]]). Technically, the best tuning in the 5-limit is to flatten the fifth by a fraction of a cent, though tunings with sharper fifths (and worse 5-limit, like in 41 or 94edo) still work fine.

			Extensions include:

			* [[Garibaldi]], which equates the generalized comma further to [[64/63]] and [[50/49]] (tempering out [[225/224]]) to provide an efficient framework for [[7-limit]] harmony, though with worse 5-limit intonation since the tuning uses slightly sharp fifths.
			* [[Pontiac]], which tempers out [[4375/4374]] to induce very little damage on schismic harmonies, at the cost of 7 being quite complex.
			* [[Sesquiquartififths]], which tempers out [[2401/2400]] and splits the fifth in fourths, inducing very little damage with a less complex mapping of 7 at the cost of quadrupling the complexity of 3 and 5.
			* The 2.3.5.13 [[subgroup]] extension [[Schismatic family#Maqamschismic (2.3.5.13)\|maqamschismic]], (tempering out the [[325/324\|marveltwin comma]]) and finds [[13/8]] at the dupminor sixth (^^Ab from C). See [[2.3.5.13 subgroup]] for more details.
			* The 2.3.5.19 [[subgroup]] extension [[nestoria]], which equates the minor third to [[19/16]], major third to [[19/15]] and [[24/19]], and the minor second to [[19/18]] and [[20/19]] (tempering out [[513/512]] and [[361/360]]).

			This page, however, focuses on the basic 5-limit temperament.

	See [[Schismatic family #Schismic, schismatic, a.k.a. helmholtz]] for technical data.		See [[Schismatic family #Schismic, schismatic, a.k.a. helmholtz]] for technical data.
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	\| 20.77		\| 20.77
	\| 81/80		\| 81/80
			\|-
			\|13
			\|722.49
			\|243/160
			\|-
			\|14
			\|224.22
			\|256/225
	\|}		\|}
	<nowiki/>* In 5-limit CWE tuning		<nowiki/>* In 5-limit CWE tuning

	== Notation ==		== Notation ==
	Using schismic can be a challenge because it defies the tradition of {{w\|tertian harmony}} in [[chain-of-fifths notation]]. The just major ~~triad on C~~ is C–F♭–G~~, for example~~. To address that, an additional module of accidentals such as arrows to represent the comma step may be adopted, allowing the user to write the chord above as C–vE–G.		Using schismic can be a challenge because it defies the tradition of diatonic {{w\|tertian harmony}} in [[chain-of-fifths notation]]; The just major third is not C–E–G like in [[meantone]], but rather, C–F♭–G. To address that, an additional module of accidentals such as arrows to represent the comma step may be adopted, allowing the user to write the chord above as C–vE–G.

	== Scales ==		== Scales ==

			=== MOS scales ===

			* [[5L 7s]] (p-chromatic) [17edo]

			* [[12L 5s]] (p-enharmonic) [29edo]
			* [[12L 17s]] (pythagotonic) [41edo]
			* [[12L 29s]] (pythamystonic) [53edo]
			* [[12L 41s]] (antipythomerc) [65edo]
			* [[53L 12s]] (m-chro antipythomerc) [118edo]

			EDO in brackets represents basic step ratio.

	=== Scala files ===		=== Scala files ===
	* [[Clipper32805]] – in a 1–3–5 equal-beating tuning		* [[Clipper32805]] – in a 1–3–5 equal-beating tuning

FloraC at 09:13, 7 January 2026

2026-01-07T09:13:06Z

← Older revision		Revision as of 09:13, 7 January 2026
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	* [https://x31eq.com/schismic.htm ''Schismic Temperaments''] by [[Graham Breed]]		* [https://x31eq.com/schismic.htm ''Schismic Temperaments''] by [[Graham Breed]]

	[[Category:~~Schismatic~~\| ]] <!-- main article -->		[[Category:Schismic\| ]] <!-- main article -->
	[[Category:Rank-2 temperaments]]		[[Category:Rank-2 temperaments]]
	[[Category:Schismatic family]]		[[Category:Schismatic family]]

FloraC: /* Tunings */ ++

2026-01-07T09:07:44Z

Tunings: ++

← Older revision		Revision as of 09:07, 7 January 2026
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	\|-		\|-
	\| 38\65 \|\| \|\| 701.5385 \|\|		\| 38\65 \|\| \|\| 701.5385 \|\|
			\|-
			\| \|\| 45/32 \|\| 701.6294 \|\| 1/6-comma
	\|-		\|-
	\| \|\| 15/8 \|\| 701.6759 \|\| 1/7-comma		\| \|\| 15/8 \|\| 701.6759 \|\| 1/7-comma
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	\|-		\|-
	\| \|\| 9/5 \|\| 701.7596 \|\| 1/10-comma		\| \|\| 9/5 \|\| 701.7596 \|\| 1/10-comma
			\|-
			\| \|\| 81/80 \|\| 701.7922 \|\| 1/12-comma
	\|-		\|-
	\| 31\53 \|\| \|\| 701.8868 \|\|		\| 31\53 \|\| \|\| 701.8868 \|\|

FloraC: + more tunings and a link to a Scala file

2026-01-07T09:03:44Z

+ more tunings and a link to a Scala file

@@ Line 86: / Line 86: @@
 == Notation ==
 Using schismic can be a challenge because it defies the tradition of {{w|tertian harmony}} in [[chain-of-fifths notation]]. The just major triad on C is C–F♭–G, for example. To address that, an additional module of accidentals such as arrows to represent the comma step may be adopted, allowing the user to write the chord above as C–vE–G.
 == Tunings ==
 === Target tunings ===
 {| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
-|+ style="white-space: nowrap;" | Target tunings
+|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings
 ! rowspan="2" | Target
 ! colspan="2" | Minimax

FloraC: /* Tunings */ + a tuning spectrum

2025-12-16T13:36:24Z

Tunings: + a tuning spectrum

@@ Line 106: / Line 106: @@
 | {{Monzo| 0 -10 17 }}
 |}
 === Tuning spectrum ===
-{{Todo|inline=1|expand}}
+{| class="wikitable center-all left-4"
 == External links ==