Schisma - Revision history

Godtone: readd categorisation that was missed

2026-06-06T17:12:32Z

readd categorisation that was missed

← Older revision		Revision as of 17:12, 6 June 2026
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	[[Category:Schismic]]		[[Category:Schismic]]
			[[Category:Commas named for their regular temperament properties]]

Godtone: /* Garibaldi */ this notation looks silly and invalidates the other example

2026-06-06T17:03:22Z

Garibaldi: this notation looks silly and invalidates the other example

← Older revision		Revision as of 17:03, 6 June 2026
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	{{Main\| Garibaldi }}		{{Main\| Garibaldi }}

	Garibaldi tempers out [[225/224]] (S15) and [[5120/5103]] ([[64/63\|S8]]/[[81/80\|S9]]), and can be described as the 41 & 53 temperament in the 7-limit that equates the two aforementioned commas (S8 = (8/7)/(9/8) = 64/63 and S9 = (9/8)/(10/9) = 81/80) into a general purpose comma reached at 12 fifths via (9/8)<sup>6</sup> / (2/1). ~~<sup>~~This is derived as the schisma is also equal to [[225/224\|S15]]/([[5120/5103\|S8/S9]]).~~</sup>~~		Garibaldi tempers out [[225/224]] (S15) and [[5120/5103]] ([[64/63\|S8]]/[[81/80\|S9]]), and can be described as the 41 & 53 temperament in the 7-limit that equates the two aforementioned commas (S8 = (8/7)/(9/8) = 64/63 and S9 = (9/8)/(10/9) = 81/80) into a general purpose comma reached at 12 fifths via (9/8)<sup>6</sup> / (2/1). This is derived as the schisma is also equal to [[225/224\|S15]]/([[5120/5103\|S8/S9]]).

	==== 2.3.5.7.19 subgroup ====		==== 2.3.5.7.19 subgroup ====

Godtone: /* Garibaldi */ superscript derivation factoid

2026-06-06T17:02:47Z

Garibaldi: superscript derivation factoid

← Older revision		Revision as of 17:02, 6 June 2026
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	{{Main\| Garibaldi }}		{{Main\| Garibaldi }}

	Garibaldi tempers out [[225/224]] (S15) and [[5120/5103]] ([[64/63\|S8]]/[[81/80\|S9]]), and can be described as the 41 & 53 temperament in the 7-limit that equates the two aforementioned commas (S8 = (8/7)/(9/8) = 64/63 and S9 = (9/8)/(10/9) = 81/80) into a general purpose comma reached at 12 fifths via (9/8)<sup>6 / (2/1). This is derived ~~since~~ the schisma is also equal to [[225/224\|S15]]/([[5120/5103\|S8/S9]]).		Garibaldi tempers out [[225/224]] (S15) and [[5120/5103]] ([[64/63\|S8]]/[[81/80\|S9]]), and can be described as the 41 & 53 temperament in the 7-limit that equates the two aforementioned commas (S8 = (8/7)/(9/8) = 64/63 and S9 = (9/8)/(10/9) = 81/80) into a general purpose comma reached at 12 fifths via (9/8)<sup>6</sup> / (2/1). <sup>This is derived as the schisma is also equal to [[225/224\|S15]]/([[5120/5103\|S8/S9]]).</sup>

	==== 2.3.5.7.19 subgroup ====		==== 2.3.5.7.19 subgroup ====

Godtone: /* 2.3.5.7.17.19 12 & 118 & 171 (unnamed) */ fix error

2026-06-06T15:09:22Z

2.3.5.7.17.19 12 & 118 & 171 (unnamed): fix error

← Older revision		Revision as of 15:09, 6 June 2026
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	==== 2.3.5.7.17.19 12 & 118 & 171 (unnamed) ====		==== 2.3.5.7.17.19 12 & 118 & 171 (unnamed) ====
	By tempering [[1216/1215\|S16/S18]] we equate [[225/224\|S15]] with [[400/399\|S20]] (tempering the other comma of Nestoria) because of S15~S16~S18~S20 ~~(meaning the [[superparticular interval]]s in 14:15:16:17:18:19:20:21 are mapped equidistantly, so requiring a large edo tuning)~~, leading to:		By tempering [[1216/1215\|S16/S18]] we equate [[225/224\|S15]] with [[400/399\|S20]] (tempering the other comma of Nestoria) because of S15~S16~S18~S20, leading to:

	[[Subgroup]]: 2.3.5.7.17.19		[[Subgroup]]: 2.3.5.7.17.19

Godtone: /* Temperaments */ improve

2026-06-06T14:52:54Z

Temperaments: improve

← Older revision		Revision as of 14:52, 6 June 2026
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	== Other intervals ==		== Other intervals ==
	Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the [[Pythagorean comma]] and [[septimal comma]] is called the [[septimal schisma]]. Other examples are [[undevicesimal schisma]], [[Alpharabian schisma]] and [[tridecaschisma]].		Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the [[Pythagorean comma]] and [[septimal comma]] is called the [[septimal schisma]]. Other examples are [[undevicesimal schisma]], [[Alpharabian schisma]] and [[tridecaschisma]].


	== Temperaments ==		== Temperaments ==
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	{{See also\| No-sevens subgroup temperaments #Nestoria }}		{{See also\| No-sevens subgroup temperaments #Nestoria }}

	Nestoria tempers out [[361/360]] (S19) and [[513/512]] (S15/S20), and can be described as the 12 & 53 temperament in the 2.3.5.19 subgroup. This is derived since the schisma is expressible as [[361/360\|S19]]/([[1216/1215\|S16/S18]])<sup>2</sup> and ([[513/512\|S15/S20]])/([[1216/1215\|S16/S18]]).		Nestoria tempers out [[361/360]] (S19) and [[513/512]] (S15/S20), and can be described as the 12 & 53 temperament in the 2.3.5.19 subgroup. This is derived since the schisma is expressible as [[361/360\|S19]]/([[1216/1215\|S16/S18]])<sup>2</sup> and ([[513/512\|S15/S20]])/([[1216/1215\|S16/S18]]). This corresponds to making 19/16 the minor third and 24/19~19/15 the major third; a good tuning for this is [[65edo]], or if you prefer a more accurate [[19/16]], [[77edo]].

	=== Garibaldi ===		=== Garibaldi ===
	{{Main\| Garibaldi }}		{{Main\| Garibaldi }}

	Garibaldi tempers out [[225/224]] (S15) and [[5120/5103]] (S8/S9), and can be described as the 41 & 53 temperament in the 7-limit. This is derived since the schisma is also equal to [[225/224\|S15]]/([[5120/5103\|S8/S9]]).		Garibaldi tempers out [[225/224]] (S15) and [[5120/5103]] ([[64/63\|S8]]/[[81/80\|S9]]), and can be described as the 41 & 53 temperament in the 7-limit that equates the two aforementioned commas (S8 = (8/7)/(9/8) = 64/63 and S9 = (9/8)/(10/9) = 81/80) into a general purpose comma reached at 12 fifths via (9/8)<sup>6 / (2/1). This is derived since the schisma is also equal to [[225/224\|S15]]/([[5120/5103\|S8/S9]]).

	==== 2.3.5.7.19 subgroup ====		==== 2.3.5.7.19 subgroup ====
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	==== 2.3.5.7.17.19 12 & 118 & 171 (unnamed) ====		==== 2.3.5.7.17.19 12 & 118 & 171 (unnamed) ====
	By tempering [[1216/1215\|S16/S18]] we equate [[225/224\|S15]] with [[400/399\|S20]] (tempering the other comma of Nestoria) because of S15~S16~S18~S20, leading to:		By tempering [[1216/1215\|S16/S18]] we equate [[225/224\|S15]] with [[400/399\|S20]] (tempering the other comma of Nestoria) because of S15~S16~S18~S20 (meaning the [[superparticular interval]]s in 14:15:16:17:18:19:20:21 are mapped equidistantly, so requiring a large edo tuning), leading to:

	[[Subgroup]]: 2.3.5.7.17.19		[[Subgroup]]: 2.3.5.7.17.19
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	{{Optimal ET sequence\|legend=1\| 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh }}		{{Optimal ET sequence\|legend=1\| 12, 29, 41, 53, 106d, 118, 171, 289h, 460hh }}

	~~{{Todo\| improve readability }}~~

	=== 2.3.5.41 53 & 65 (unnamed) ===		=== 2.3.5.41 53 & 65 (unnamed) ===
	The schisma can additionally split into two superparticular commas in the 41-limit: 32805/32768 = ([[1025/1024]])*([[6561/6560]]). Tempering both of these out provides a ~~natural~~ mapping for prime 41~~, if~~ a ~~little less practical than those~~ for ~~19 or 7~~.		The schisma can additionally split into two superparticular commas in the 41-limit: 32805/32768 = [[1025/1024]] * [[6561/6560]]. Tempering both of these out provides a microtemperament-accuracy mapping for prime 41 via tempering out [[6561/6560\|S81]] = (81/80)/(82/81) (the second of the aforementioned commas) s.t any accurate schismic tuning (one with a very slightly flat 81/80) will have a good tuning for an otonal supermajor third [[41/32]] and a flat supermajor second (41/32)/(9/8) = [[41/36]].

	== History and etymology ==		== History and etymology ==

Godtone: when i talked to fumica, i couldnt get to the bottom of what was wrong with the page per se, so im going to revert and try to improve some issues i see to fit with Xenharmonic_Wiki:Notability_guidelines

2026-06-06T14:44:47Z

when i talked to fumica, i couldnt get to the bottom of what was wrong with the page per se, so im going to revert and try to improve some issues i see to fit with Xenharmonic_Wiki:Notability_guidelines

@@ Line 13: / Line 13: @@
 {{Wikipedia| Schisma }}
-The '''schisma''', '''32805/32768''', is a small interval about 2 [[cent]]s. It arises as the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]).
+The '''schisma''', '''32805/32768''', is the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]).
 == Temperaments ==
 Tempering out this comma gives a [[5-limit]] microtemperament called [[schismic|schismatic, schismic or helmholtz]], which if extended to larger [[subgroup]]s leads to the [[schismatic family]] of temperaments.
-== History and etymology ==
+=== Nestoria ===
 ''Schisma'' is a borrowing of Ancient Greek, meaning "split". The term was first used by [[Boethius]] (6th century), in his ''De institutione musica'', using it to refer to half of the [[Pythagorean comma]]. The modern sense was introduced by [[Helmholtz]]' ''On the Sensations of Tone'', in particular the translation by [[Alexander Ellis]], where it is spelled ''skhisma''. Since it is extremely close to the [[superparticular]] ratio 887/886 {{nowrap|(2<sup>-1</sup>⋅443<sup>-1</sup>⋅887)}}, it is used interchangably with this interval in some of Helmholtz' writing.
 == Trivia ==
@@ Line 34: / Line 84: @@
 [[Category:Schismic]]

Sintel: Undo revision 231381 by Godtone (talk)

2026-06-01T01:16:35Z

Undo revision 231381 by Godtone (talk)

@@ Line 13: / Line 13: @@
 {{Wikipedia| Schisma }}
-The '''schisma''', '''32805/32768''', is the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]).
+The '''schisma''', '''32805/32768''', is a small interval about 2 [[cent]]s. It arises as the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]).
 == Temperaments ==
 Tempering out this comma gives a [[5-limit]] microtemperament called [[schismic|schismatic, schismic or helmholtz]], which if extended to larger [[subgroup]]s leads to the [[schismatic family]] of temperaments.
-=== Nestoria ===
+== History and etymology ==
-{{See also| No-sevens subgroup temperaments #Nestoria }}
+''Schisma'' is a borrowing of Ancient Greek, meaning "split". The term was first used by [[Boethius]] (6th century), in his ''De institutione musica'', using it to refer to half of the [[Pythagorean comma]]. The modern sense was introduced by [[Helmholtz]]' ''On the Sensations of Tone'', in particular the translation by [[Alexander Ellis]], where it is spelled ''skhisma''. Since it is extremely close to the [[superparticular]] ratio 887/886 {{nowrap|(2<sup>-1</sup>⋅443<sup>-1</sup>⋅887)}}, it is used interchangably with this interval in some of Helmholtz' writing.
-Nestoria tempers out [[361/360]] (S19) and [[513/512]] (S15/S20), and can be described as the 12 & 53 temperament in the 2.3.5.19 subgroup. This is derived since the schisma is expressible as [[361/360|S19]]/([[1216/1215|S16/S18]])<sup>2</sup> and ([[513/512|S15/S20]])/([[1216/1215|S16/S18]]).
+== Other intervals ==
+Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the [[Pythagorean comma]] and [[septimal comma]] is called the [[septimal schisma]]. Other examples are [[undevicesimal schisma]] and [[Alpharabian schisma]].
 == Trivia ==
@@ Line 84: / Line 34: @@
 [[Category:Schismic]]

Godtone: all the temps mentioned here except one (which is undocumented AFAIK!) are both non-duplicate (are presented to link and refer to other existing temps and only give an overview) and are here because they are simple and efficient ways of expressing the schisma. i thought it's agreed not to delete info and make the wiki less navigable

2026-05-31T13:47:46Z

all the temps mentioned here except one (which is undocumented AFAIK!) are both non-duplicate (are presented to link and refer to other existing temps and only give an overview) and are here because they are simple and efficient ways of expressing the schisma. i thought it's agreed not to delete info and make the wiki less navigable

@@ Line 13: / Line 13: @@
 {{Wikipedia| Schisma }}
-The '''schisma''', '''32805/32768''', is a small interval about 2 [[cent]]s. It arises as the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]).
+The '''schisma''', '''32805/32768''', is the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]).
 == Temperaments ==
 Tempering out this comma gives a [[5-limit]] microtemperament called [[schismic|schismatic, schismic or helmholtz]], which if extended to larger [[subgroup]]s leads to the [[schismatic family]] of temperaments.
-== History and etymology ==
+=== Nestoria ===
 ''Schisma'' is a borrowing of Ancient Greek, meaning "split". The term was first used by [[Boethius]] (6th century), in his ''De institutione musica'', using it to refer to half of the [[Pythagorean comma]]. The modern sense was introduced by [[Helmholtz]]' ''On the Sensations of Tone'', in particular the translation by [[Alexander Ellis]], where it is spelled ''skhisma''. Since it is extremely close to the [[superparticular]] ratio 887/886 {{nowrap|(2<sup>-1</sup>⋅443<sup>-1</sup>⋅887)}}, it is used interchangably with this interval in some of Helmholtz' writing.
 == Trivia ==
@@ Line 34: / Line 84: @@
 [[Category:Schismic]]

FloraC: Move history and etymology below temps

2026-05-02T18:54:01Z

Move history and etymology below temps

← Older revision		Revision as of 18:54, 2 May 2026
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	{{~~interwiki~~		{{Interwiki
			\| en = 32805/32768
	\| de = 32805/32768		\| de = 32805/32768
	~~\| en = 32805/32768~~
	\| es =		\| es =
	\| ja =		\| ja =
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	The '''schisma''', '''32805/32768''', is a small interval about 2 [[cent]]s. It arises as the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]).		The '''schisma''', '''32805/32768''', is a small interval about 2 [[cent]]s. It arises as the difference between the [[Pythagorean comma]] and the [[syntonic comma]]. It is equal to ([[9/8]])<sup>4</sup>/([[8/5]]) and to ([[135/128]])/([[256/243]]) and also to ([[9/8]])<sup>3</sup>/([[64/45]]).

	~~== History and etymology ==~~
	''Schisma'' is a borrowing of Ancient Greek, meaning "split". The term was first used by [[Boethius]] (6th century), in his ''De institutione musica'', using it to refer to half of the [[Pythagorean comma]]. The modern sense was introduced by [[Helmholtz]]' ''On the Sensations of Tone'', in particular the translation by [[Alexander Ellis]], where it is spelled ''skhisma''. Since it is extremely close to the [[superparticular]] ratio [[887/886]] {{nowrap\|(2<sup>-1</sup> 443<sup>-1</sup> 887)}}, it is used interchangably with this interval in some of Helmholtz' writing.

	== Temperaments ==		== Temperaments ==
	~~{{main\|Schismatic family}}~~
	Tempering out this comma gives a [[5-limit]] microtemperament called [[schismic\|schismatic, schismic or helmholtz]], which if extended to larger [[subgroup]]s leads to the [[schismatic family]] of temperaments.		Tempering out this comma gives a [[5-limit]] microtemperament called [[schismic\|schismatic, schismic or helmholtz]], which if extended to larger [[subgroup]]s leads to the [[schismatic family]] of temperaments.

			== History and etymology ==
			''Schisma'' is a borrowing of Ancient Greek, meaning "split". The term was first used by [[Boethius]] (6th century), in his ''De institutione musica'', using it to refer to half of the [[Pythagorean comma]]. The modern sense was introduced by [[Helmholtz]]' ''On the Sensations of Tone'', in particular the translation by [[Alexander Ellis]], where it is spelled ''skhisma''. Since it is extremely close to the [[superparticular]] ratio 887/886 {{nowrap\|(2<sup>-1</sup>⋅443<sup>-1</sup>⋅887)}}, it is used interchangably with this interval in some of Helmholtz' writing.

	== Other intervals ==		== Other intervals ==

	Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the [[Pythagorean comma]] and [[septimal comma]] is called the [[septimal schisma]]. Other examples are [[undevicesimal schisma]] and [[Alpharabian schisma]].		Commas arising from the difference between a stack of Pythagorean intervals and other primes may also be called schismas. The difference between the [[Pythagorean comma]] and [[septimal comma]] is called the [[septimal schisma]]. Other examples are [[undevicesimal schisma]] and [[Alpharabian schisma]].

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	The schisma explains how the greatly composite numbers 1048576 (2<sup>20</sup>) and 104976 (18<sup>4</sup>) look alike in decimal. The largest common power of two between these numbers is 2<sup>5</sup>, (when 1049760 is written to equalize) and when reduced by that, 1049760/1048576 becomes 32805/32768.		The schisma explains how the greatly composite numbers 1048576 (2<sup>20</sup>) and 104976 (18<sup>4</sup>) look alike in decimal. The largest common power of two between these numbers is 2<sup>5</sup>, (when 1049760 is written to equalize) and when reduced by that, 1049760/1048576 becomes 32805/32768.

	It's also very close in size—about 0.~~0013¢~~ off—from the difference between 3/2 and 7\12, which is about 1.~~9550009¢~~. Tempering out this difference instead results in [[atomic]], an extremely high accuracy temperament.		It is also very close in size—about 0.0013{{c}} off—from the difference between 3/2 and 7\12, which is about 1.9550009{{c}}. Tempering out this difference instead results in [[atomic]], an extremely high accuracy temperament.

	== See also ==		== See also ==

TallKite: updated the comma color name

2026-04-29T04:09:52Z

updated the comma color name

← Older revision		Revision as of 04:09, 29 April 2026
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	\| Ratio = 32805/32768		\| Ratio = 32805/32768
	\| Name = schisma		\| Name = schisma
	\| Color name = ~~Ly-2~~, ~~Layo comma~~		\| Color name = LyM, layoma
	\| Comma = yes		\| Comma = yes
	}}		}}