Schismic–commatic equivalence continuum - Revision history

FloraC: /* World calendar (5-limit) */ as outlined above, this is not a canonical restriction. Complete data

2026-09-03T10:13:26Z

World calendar (5-limit): as outlined above, this is not a canonical restriction. Complete data

← Older revision		Revision as of 10:13, 3 September 2026
Line 346:		Line 346:
	[[Badness]] (Sintel): 16.0		[[Badness]] (Sintel): 16.0

	== World calendar ~~(5-limit)~~ ==		== World calendar restriction ==
	: ''For extensions, see [[Septischismic clan #World calendar]].''		: ''For extensions, see [[Septischismic clan #World calendar]].''

Line 354:		Line 354:

	{{Mapping\|legend=1\| 4 1 44 \| 0 2 -13 }}		{{Mapping\|legend=1\| 4 1 44 \| 0 2 -13 }}
			: mapping generators: ~39858075/33554432, ~{{monzo\| -37 19 3 }}

	[[Optimal tuning]]s:		[[Optimal tuning]]s:
	* [[WE]]: ~39858075/33554432 = 299.~~989~~{{c}}, ~{{monzo\| -37 19 3 }} = 801.~~013~~{{c}}		* [[WE]]: ~39858075/33554432 = 299.9886{{c}}, ~{{monzo\| -37 19 3 }} = 801.0127{{c}}
	: [[error map]]: {{val\| -0.046, 0.059, 0.019 }}		: [[error map]]: {{val\| -0.046 +0.059 +0.019 }}
	* [[CWE]]: ~39858075/33554432 = 300.~~000~~{{c}}, ~{{monzo\| -37 19 3 }} = 801.~~045~~{{c}}		* [[CWE]]: ~39858075/33554432 = 300.0000{{c}}, ~{{monzo\| -37 19 3 }} = 801.0449{{c}}
	: error map: {{val\| 0.000, 0.135, 0.103 }}		: error map: {{val\| 0.000 +0.135 +0.103 }}

	~~[[Support]]ing [[ET]]s:~~ {{~~EDOs~~\|12, 340, 352, 364, 376, 388, ~~400~~, ~~412~~, ~~740~~, ~~764~~, ~~788, 1140, 1152~~, ~~1540~~}}		{{Optimal ET sequence\|legend=1\| 12, …, 340, 352, 364, 376, 388, 764, 1152, 1916, 3068, 8052bc, 11120bbcc }}

	Badness: 17.~~749~~		[[Badness]] (Sintel): 17.7

	[[Category:12edo]]		[[Category:12edo]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

Eliora: /* World calendar (5-limit) */

2026-09-01T18:42:43Z

World calendar (5-limit)

← Older revision		Revision as of 18:42, 1 September 2026
Line 362:		Line 362:

	[[Support]]ing [[ET]]s: {{EDOs\|12, 340, 352, 364, 376, 388, 400, 412, 740, 764, 788, 1140, 1152, 1540}}		[[Support]]ing [[ET]]s: {{EDOs\|12, 340, 352, 364, 376, 388, 400, 412, 740, 764, 788, 1140, 1152, 1540}}

			Badness: 17.749

	[[Category:12edo]]		[[Category:12edo]]
	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

Eliora: /* World calendar (5-limit) */

2026-09-01T18:40:39Z

World calendar (5-limit)

← Older revision		Revision as of 18:40, 1 September 2026
Line 347:		Line 347:

	== World calendar (5-limit) ==		== World calendar (5-limit) ==
	: ''For ~~7-limit and higher~~ extensions, see [[Septischismic clan #World calendar]].''		: ''For extensions, see [[Septischismic clan #World calendar]].''

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5

Eliora: + 5-limit world calendar

2026-09-01T18:40:19Z

+ 5-limit world calendar

@@ Line 345: / Line 345: @@
 [[Badness]] (Sintel): 16.0
 [[Category:12edo]]
 [[Category:Equivalence continua]]

FloraC: /* Heptacot */ update

2026-05-20T11:03:18Z

Heptacot: update

← Older revision		Revision as of 11:03, 20 May 2026
Line 325:		Line 325:
	[[Badness]] (Sintel): 8.02		[[Badness]] (Sintel): 8.02

	== Heptacot ==		== Heptacot (5-limit) ==
	Heptacot tempers out the [[heptacot comma]] and divides the [[3/2\|perfect fifth]] into seven equal parts, the most notable example being [[12edo]] (7\12). It corresponds to {{nowrap\| ''n'' {{=}} 7 }}, meaning the Pythagorean comma is equated with a stack of seven schismas. It was named by [[Tristan Bay]] in 2024, for it splits the perfect fifth into seven.		: ''For extensions, see [[Garischismic clan #Heptacot]].''

			The 5-limit version of heptacot tempers out the [[heptacot comma]] and divides the [[3/2\|perfect fifth]] into seven equal parts, the most notable example being [[12edo]] (7\12). It corresponds to {{nowrap\| ''n'' {{=}} 7 }}, meaning the Pythagorean comma is equated with a stack of seven schismas. It was named by [[Tristan Bay]] in 2024, for it splits the perfect fifth into seven.

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5

FloraC: /* Sextile (5-limit) */ update a link

2026-05-16T10:16:16Z

Sextile (5-limit): update a link

← Older revision		Revision as of 10:16, 16 May 2026
Line 283:		Line 283:

	== Sextile (5-limit) ==		== Sextile (5-limit) ==
	{{See also\| ~~Landscape microtemperaments~~ #Sextile }}		{{See also\| Garischismic clan #Sextile }}

	The 5-limit version of sextile reaches the [[interval class]] of [[5/1\|5]] by −6 [[3/2\|perfect fifths]] (i.e. a diminished fifth) minus a period of 1/6-octave. It corresponds to {{nowrap\| ''n'' {{=}} 6 }}, meaning the Pythagorean comma is equated with a stack of six schismas.		The 5-limit version of sextile reaches the [[interval class]] of [[5/1\|5]] by −6 [[3/2\|perfect fifths]] (i.e. a diminished fifth) minus a period of 1/6-octave. It corresponds to {{nowrap\| ''n'' {{=}} 6 }}, meaning the Pythagorean comma is equated with a stack of six schismas.

FloraC: Move discussion on the k-continuum below m-continuum as they are more closely related.

2026-04-24T12:04:48Z

Move discussion on the *k*-continuum below *m*-continuum as they are more closely related.

← Older revision		Revision as of 12:04, 24 April 2026
Line 11:		Line 11:
	* [[Quindromeda]] ({{nowrap\| ''n'' {{=}} 5 }}) does not split the octave but splits the fourth in five, as 5 is coprime with 12.		* [[Quindromeda]] ({{nowrap\| ''n'' {{=}} 5 }}) does not split the octave but splits the fourth in five, as 5 is coprime with 12.

	Alternatively, because the the 5-limit otonal detemperament of 12edo is a 4×3 rectangle (known as the [[duodene]]), we may be interested in expressing the continuum in terms of the boundary commas of this detemper, that is, as {{nowrap\| ([[81/80]])<sup>''k''</sup> ~ ([[128/125]]) }}. This corresponds to these commas' structural significance via 128/125 being entirely in the 2.5 subgroup while 81/80 explains 5 in the simplest way relative to the 3-limit. This choice of coordinates has the advantage of finding all temperaments discussed in a relatively intuitive and simple way so that less accurate but structurally simpler temperaments are found at integer points while microtemperaments converging to atomic are found as successive mediants towards the JIP. Specifically, its JIP is at 1.90915584… which is approximated very closely by the microtemperament atomic at {{nowrap\| 21/11 {{=}} 1.90909… }} so that the main continuum can be seen as taking successive mediants towards 2/1 (schismic) starting from 1/1 (diaschismic). It is noted for its nontrivial relation to the other better-motivated (in terms of mapping) coordinates discussed.		{\| class="wikitable center-1"

	{\| class="wikitable center-1 ~~center-2~~"
	\|+ style="font-size: 105%;" \| Temperaments with integer ''n''		\|+ style="font-size: 105%;" \| Temperaments with integer ''n''
	\|-		\|-
	! rowspan="2" \| ''n''		! rowspan="2" \| ''n''
	~~! rowspan="2" \| ''k''~~
	! rowspan="2" \| Temperament		! rowspan="2" \| Temperament
	! colspan="2" \| Comma		! colspan="2" \| Comma
Line 25:		Line 22:
	\|-		\|-
	\| -3		\| -3
	~~\| 9/4~~
	\| [[Triscordial]]		\| [[Triscordial]]
	\| (40 digits)		\| (40 digits)
Line 31:		Line 27:
	\|-		\|-
	\| -2		\| -2
	~~\| 7/3~~
	\| [[Biscordial]]		\| [[Biscordial]]
	\| (30 digits)		\| (30 digits)
Line 37:		Line 32:
	\|-		\|-
	\| -1		\| -1
	~~\| 5/2~~
	\| [[Gracecordial]]		\| [[Gracecordial]]
	\| (22 digits)		\| (22 digits)
Line 43:		Line 37:
	\|-		\|-
	\| 0		\| 0
	~~\| 3~~
	\| [[Compton]]		\| [[Compton]]
	\| [[531441/524288]]		\| [[531441/524288]]
Line 49:		Line 42:
	\|-		\|-
	\| 1		\| 1
	~~\| ∞~~
	\| [[Meantone]]		\| [[Meantone]]
	\| [[81/80]]		\| [[81/80]]
Line 55:		Line 47:
	\|-		\|-
	\| 2		\| 2
	~~\| 1~~
	\| [[Diaschismic]]		\| [[Diaschismic]]
	\| [[2048/2025]]		\| [[2048/2025]]
Line 61:		Line 52:
	\|-		\|-
	\| 3		\| 3
	~~\| 3/2~~
	\| [[Misty]]		\| [[Misty]]
	\| [[67108864/66430125]]		\| [[67108864/66430125]]
Line 67:		Line 57:
	\|-		\|-
	\| 4		\| 4
	~~\| 5/3~~
	\| [[Undim]]		\| [[Undim]]
	\| (26 digits)		\| (26 digits)
Line 73:		Line 62:
	\|-		\|-
	\| 5		\| 5
	~~\| 7/4~~
	\| [[Quindromeda]]		\| [[Quindromeda]]
	\| (34 digits)		\| (34 digits)
Line 79:		Line 67:
	\|-		\|-
	\| 6		\| 6
	~~\| 9/5~~
	\| [[Sextile]]		\| [[Sextile]]
	\| (44 digits)		\| (44 digits)
Line 85:		Line 72:
	\|-		\|-
	\| 7		\| 7
	~~\| 11/6~~
	\| [[Heptacot]]		\| [[Heptacot]]
	\| (52 digits)		\| (52 digits)
Line 91:		Line 77:
	\|-		\|-
	\| 8		\| 8
	~~\| 13/7~~
	\| [[World calendar]] restriction		\| [[World calendar]] restriction
	\| (62 digits)		\| (62 digits)
Line 97:		Line 82:
	\|-		\|-
	\| 9		\| 9
	~~\| 15/8~~
	\| Quinbisa-tritrigu (12 & 441)		\| Quinbisa-tritrigu (12 & 441)
	\| (70 digits)		\| (70 digits)
Line 103:		Line 87:
	\|-		\|-
	\| 10		\| 10
	~~\| 17/9~~
	\| Lesa-quinbigu (12 & 494)		\| Lesa-quinbigu (12 & 494)
	\| (80 digits)		\| (80 digits)
Line 109:		Line 92:
	\|-		\|-
	\| 11		\| 11
	~~\| 19/10~~
	\| Quadtrisa-legu (12 & 559)		\| Quadtrisa-legu (12 & 559)
	\| (88 digits)		\| (88 digits)
Line 115:		Line 97:
	\|-		\|-
	\| 12		\| 12
	~~\| 21/11~~
	\| [[Atomic]]		\| [[Atomic]]
	\| (98 digits)		\| (98 digits)
Line 121:		Line 102:
	\|-		\|-
	\| 13		\| 13
	~~\| 23/12~~
	\| Quintrila-theyo (12 & 677)		\| Quintrila-theyo (12 & 677)
	\| (106 digits)		\| (106 digits)
	\| {{Monzo\| -176 92 13 }}		\| {{Monzo\| -176 92 13 }}
	\|-		\|-
	~~\| …~~
	\| …		\| …
	\| …		\| …
Line 133:		Line 112:
	\|-		\|-
	\| ∞		\| ∞
	~~\| 2~~
	\| [[Schismic]]		\| [[Schismic]]
	\| [[32805/32768]]		\| [[32805/32768]]
Line 140:		Line 118:

	We may invert the continuum by setting ''m'' such that {{nowrap\| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''syntonic–commatic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.0908441588…. The [[syntonic comma]] is way larger but much simpler than the schisma. As such, this continuum does not contain as many [[microtemperament]]s, but has more useful lower-complexity temperaments.		We may invert the continuum by setting ''m'' such that {{nowrap\| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''syntonic–commatic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.0908441588…. The [[syntonic comma]] is way larger but much simpler than the schisma. As such, this continuum does not contain as many [[microtemperament]]s, but has more useful lower-complexity temperaments.

			Alternatively, because the the 5-limit otonal detemperament of 12edo is a 4×3 rectangle (known as the [[duodene]]), we may be interested in expressing the continuum in terms of the boundary commas of this detemper, that is, as {{nowrap\| ([[81/80]])<sup>''k''</sup> ~ ([[128/125]]) }}. This corresponds to these commas' structural significance via 128/125 being entirely in the [[2.5 subgroup]] while 81/80 explains 5 in the simplest way relative to the 3-limit. This choice of coordinates is a flip of the ''m''-continuum such that microtemperaments converging to atomic are found as successive mediants towards the [[JIP]]. Specifically, its JIP is at 1.90915584…, which is approximated very closely by the microtemperament atomic at {{nowrap\| 21/11 {{=}} 1.90909… }} so that the main ''n''-continuum can be seen as taking successive mediants towards 2/1 (schismic) starting from 1/1 (diaschismic). It is noted for its nontrivial relation to the other better-motivated (in terms of mapping) coordinates discussed.

	{\| class="wikitable center-1 center-2"		{\| class="wikitable center-1 center-2"

FloraC: Correction

2026-04-23T04:29:24Z

Correction

← Older revision		Revision as of 04:29, 23 April 2026
Line 346:		Line 346:

	== Heptacot ==		== Heptacot ==
	Heptacot tempers out the [[heptacot comma]] and divides the [[3/2\|perfect fifth]] into seven equal parts, the most notable example being [[12edo]] (7\12). It corresponds to {{nowrap\| ''n'' {{=}} 7 }}, meaning the Pythagorean comma is equated with a stack of seven schismas. It was named by [[~~Akselai~~]] in 2024, for it splits the perfect fifth into seven.		Heptacot tempers out the [[heptacot comma]] and divides the [[3/2\|perfect fifth]] into seven equal parts, the most notable example being [[12edo]] (7\12). It corresponds to {{nowrap\| ''n'' {{=}} 7 }}, meaning the Pythagorean comma is equated with a stack of seven schismas. It was named by [[Tristan Bay]] in 2024, for it splits the perfect fifth into seven.

	[[Subgroup]]: 2.3.5		[[Subgroup]]: 2.3.5

FloraC: Page move followup

2026-04-23T04:27:18Z

Page move followup

← Older revision		Revision as of 04:27, 23 April 2026
Line 1:		Line 1:
	{{Technical data page}}		{{Technical data page}}
	The '''~~schismic–Pythagorean~~ equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[5-limit]] [[regular temperament\|temperaments]] which equate a number of [[32805/32768\|schismas (32805/32768)]] with [[Pythagorean comma\|Pythagorean comma ({{monzo\| -19 12 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[12edo]].		The '''schismic–commatic equivalence continuum''' is a [[equivalence continuum\|continuum]] of [[5-limit]] [[regular temperament\|temperaments]] which equate a number of [[32805/32768\|schismas (32805/32768)]] with [[Pythagorean comma\|Pythagorean comma ({{monzo\| -19 12 }})]]. This continuum is theoretically interesting in that these are all 5-limit temperaments [[support]]ed by [[12edo]].

	All temperaments in the continuum satisfy {{nowrap\|(32805/32768)<sup>''n''</sup> ~ {{monzo\| -19 12 }}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 12edo due to it being the unique equal temperament that [[tempering out\|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 12.0078623975…, and temperaments having ''n'' near this value tend to be the most accurate ones – indeed, the fact that this number is so close to 12 reflects how small [[Kirnberger's atom]] (the difference between 12 schismas and the Pythagorean comma) is.		All temperaments in the continuum satisfy {{nowrap\|(32805/32768)<sup>''n''</sup> ~ {{monzo\| -19 12 }}}}. Varying ''n'' results in different temperaments listed in the table below. It converges to [[schismic]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all 5-limit temperaments supported by 12edo due to it being the unique equal temperament that [[tempering out\|tempers out]] both commas and thus tempers out all combinations of them. The just value of ''n'' is approximately 12.0078623975…, and temperaments having ''n'' near this value tend to be the most accurate ones – indeed, the fact that this number is so close to 12 reflects how small [[Kirnberger's atom]] (the difference between 12 schismas and the Pythagorean comma) is.
Line 139:		Line 139:
	\|}		\|}

	We may invert the continuum by setting ''m'' such that {{nowrap\| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''~~syntonic–Pythagorean~~ equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.0908441588…. The [[syntonic comma]] is way larger but much simpler than the schisma. As such, this continuum does not contain as many [[microtemperament]]s, but has more useful lower-complexity temperaments.		We may invert the continuum by setting ''m'' such that {{nowrap\| 1/''m'' + 1/''n'' {{=}} 1 }}. This may be called the ''syntonic–commatic equivalence continuum'', which is essentially the same thing. The just value of ''m'' is 1.0908441588…. The [[syntonic comma]] is way larger but much simpler than the schisma. As such, this continuum does not contain as many [[microtemperament]]s, but has more useful lower-complexity temperaments.

	{\| class="wikitable center-1 center-2"		{\| class="wikitable center-1 center-2"

FloraC: FloraC moved page Schismic–Pythagorean equivalence continuum to Schismic–commatic equivalence continuum: Pythagorean is assumed so it conveys nothing. This is the continuum of the Pythagorean comma, so commatic is the word to use here.

2026-04-23T04:22:57Z

FloraC moved page Schismic–Pythagorean equivalence continuum to Schismic–commatic equivalence continuum: Pythagorean is assumed so it conveys nothing. This is the continuum of the Pythagorean comma, so *commatic* is the word to use here.

← Older revision	Revision as of 04:22, 23 April 2026
(No difference)

Schismic–commatic equivalence continuum - Revision history

FloraC: /* World calendar (5-limit) */ as outlined above, this is not a canonical restriction. Complete data

Eliora: /* World calendar (5-limit) */

Eliora: /* World calendar (5-limit) */

Eliora: + 5-limit world calendar

FloraC: /* Heptacot */ update

FloraC: /* Sextile (5-limit) */ update a link

FloraC: Move discussion on the *k*-continuum below *m*-continuum as they are more closely related.

FloraC: Correction

FloraC: Page move followup

FloraC: FloraC moved page Schismic–Pythagorean equivalence continuum to Schismic–commatic equivalence continuum: Pythagorean is assumed so it conveys nothing. This is the continuum of the Pythagorean comma, so *commatic* is the word to use here.

FloraC: Move discussion on the k-continuum below m-continuum as they are more closely related.

FloraC: FloraC moved page Schismic–Pythagorean equivalence continuum to Schismic–commatic equivalence continuum: Pythagorean is assumed so it conveys nothing. This is the continuum of the Pythagorean comma, so commatic is the word to use here.