Kleismic family - Revision history

Godtone: bracketing issue

2026-08-28T13:29:43Z

bracketing issue

← Older revision		Revision as of 13:29, 28 August 2026
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	==== Other subgroup extensions ====		==== Other subgroup extensions ====
	The main extension of note is a very remarkable extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]]), [[676/675\|S13/S15 = S26]]} }}.		The main extension of note is a very remarkable extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]],) [[676/675\|S13/S15 = S26]]} }}.

	See [[#Subgroup extensions]].		See [[#Subgroup extensions]].
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	=== Kleismic (2.3.5.13) a.k.a. cata ===		=== Kleismic (2.3.5.13) a.k.a. cata ===
	The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]]), [[676/675\|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] S13/S15 = ([[4/3\|16/12]])/([[15/13]])<sup>2</sup>. From here, we notice that (26/15)/(13/9) = 6/5, so all that remains is dividing 13/9 into two 6/5's via the semiparticular S10/S12 = (13/9)/([[6/5\|12/10]])<sup>2</sup>, hence explaining the mapping of the entire 2.3.5.13 subgroup.		The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]],) [[676/675\|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] S13/S15 = ([[4/3\|16/12]])/([[15/13]])<sup>2</sup>. From here, we notice that (26/15)/(13/9) = 6/5, so all that remains is dividing 13/9 into two 6/5's via the semiparticular S10/S12 = (13/9)/([[6/5\|12/10]])<sup>2</sup>, hence explaining the mapping of the entire 2.3.5.13 subgroup.

	Subgroup: 2.3.5.13		Subgroup: 2.3.5.13

Godtone: /* Kleismic (2.3.5.13) a.k.a. cata */ not happy with the lack of structure explanation in this

2026-08-28T13:26:13Z

Kleismic (2.3.5.13) a.k.a. cata: not happy with the lack of structure explanation in this

← Older revision		Revision as of 13:26, 28 August 2026
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	=== Kleismic (2.3.5.13) a.k.a. cata ===		=== Kleismic (2.3.5.13) a.k.a. cata ===
			The structure of the temperament as dividing 3/1 into 6 equal parts can be deduced completely from its [[S-expression]]-based comma list of {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]]), [[676/675\|S13/S15 = S26]]} }}. Specifically, dividing 3/1 into two halves of ~26/15 is equivalent to dividing 4/3 into two halves of ~15/13, hence the [[semiparticular]] S13/S15 = ([[4/3\|16/12]])/([[15/13]])<sup>2</sup>. From here, we notice that (26/15)/(13/9) = 6/5, so all that remains is dividing 13/9 into two 6/5's via the semiparticular S10/S12 = (13/9)/([[6/5\|12/10]])<sup>2</sup>, hence explaining the mapping of the entire 2.3.5.13 subgroup.

	Subgroup: 2.3.5.13		Subgroup: 2.3.5.13

Godtone: /* Subgroup extensions */ add note in a place that feels more natural

2026-08-26T23:22:38Z

Subgroup extensions: add note in a place that feels more natural

← Older revision		Revision as of 23:22, 26 August 2026
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	== Subgroup extensions ==		== Subgroup extensions ==
			For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].

	=== Kleismic (2.3.5.13) a.k.a. cata ===		=== Kleismic (2.3.5.13) a.k.a. cata ===
	Subgroup: 2.3.5.13		Subgroup: 2.3.5.13

Godtone: /* Subgroup extensions */ fix link

2026-08-26T23:22:21Z

Subgroup extensions: fix link

← Older revision		Revision as of 23:22, 26 August 2026
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	Temperaments involving larger splits include [[#Sqrtphi\|sqrtphi]], [[#Quartkeenlig\|quartkeenlig]], [[#Novemkleismic\|novemkleismic]]. Those split the kleismic structure into five to nine parts.		Temperaments involving larger splits include [[#Sqrtphi\|sqrtphi]], [[#Quartkeenlig\|quartkeenlig]], [[#Novemkleismic\|novemkleismic]]. Those split the kleismic structure into five to nine parts.

	==== ~~Subgroup~~ extensions ====		==== Other subgroup extensions ====
	The main extension of note is a very remarkable extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]]), [[676/675\|S13/S15 = S26]]} }}.		The main extension of note is a very remarkable extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]]), [[676/675\|S13/S15 = S26]]} }}.

Godtone: /* Subgroup extensions */ nowrap

2026-08-26T23:21:24Z

Subgroup extensions: nowrap

← Older revision		Revision as of 23:21, 26 August 2026
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	==== Subgroup extensions ====		==== Subgroup extensions ====
	The main extension of note is a very remarkable extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]]), [[676/675\|S13/S15 = S26]]}.		The main extension of note is a very remarkable extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {{nowrap\| {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]]), [[676/675\|S13/S15 = S26]]} }}.

	See [[#Subgroup extensions]].		See [[#Subgroup extensions]].

Godtone: /* Subgroup extensions */ paste

2026-08-26T23:20:45Z

Subgroup extensions: paste

← Older revision		Revision as of 23:20, 26 August 2026
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	==== Subgroup extensions ====		==== Subgroup extensions ====
	The ~~kleismic family boasts~~ a very remarkable extension to the [[2.3.5.13 subgroup]], ~~which has further extensions with higher primes~~. ~~These are listed at~~ the ~~bottom of this page~~, in [[#Subgroup extensions]].		The main extension of note is a very remarkable extension to the [[2.3.5.13 subgroup]], as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]]), [[676/675\|S13/S15 = S26]]}.

			See [[#Subgroup extensions]].

	== Catakleismic ==		== Catakleismic ==

Godtone: /* Kleismic (2.3.5.13) a.k.a. cata */ cut

2026-08-26T23:19:33Z

Kleismic (2.3.5.13) a.k.a. cata: cut

← Older revision		Revision as of 23:19, 26 August 2026
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	== Subgroup extensions ==		== Subgroup extensions ==
	=== Kleismic (2.3.5.13) a.k.a. cata ===		=== Kleismic (2.3.5.13) a.k.a. cata ===
	Hanson lends itself nicely to this extension in the 2.3.5.13 subgroup, as the hemitwelfth, reached by three generator steps, can be interpreted as [[26/15]]. Notice 15625/15552 = ([[325/324]])⋅([[625/624]]) and 325/324 = (625/624)⋅([[676/675]]). The [[S-expression]]-based comma list of the temperament is {[[325/324\|S10/S12 = S25⋅S26]], ([[625/624\|S25]]), [[676/675\|S13/S15 = S26]]}. For the high-limit version of cata with a 1\5 period, see [[thunderclysmic]].

	Subgroup: 2.3.5.13		Subgroup: 2.3.5.13

Godtone: /* Overview to extensions */ move

2026-08-26T23:19:03Z

Overview to extensions: move

← Older revision		Revision as of 23:19, 26 August 2026
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	Temperaments involving larger splits include [[#Sqrtphi\|sqrtphi]], [[#Quartkeenlig\|quartkeenlig]], [[#Novemkleismic\|novemkleismic]]. Those split the kleismic structure into five to nine parts.		Temperaments involving larger splits include [[#Sqrtphi\|sqrtphi]], [[#Quartkeenlig\|quartkeenlig]], [[#Novemkleismic\|novemkleismic]]. Those split the kleismic structure into five to nine parts.

			==== Subgroup extensions ====
	The kleismic family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], which has further extensions with higher primes. These are listed at the bottom of this page, in [[#Subgroup extensions]].		The kleismic family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], which has further extensions with higher primes. These are listed at the bottom of this page, in [[#Subgroup extensions]].

	~~==== Subgroup extensions ====~~

	== Catakleismic ==		== Catakleismic ==

Godtone: /* Overview to extensions */ readd cata here.

2026-08-26T23:08:57Z

Overview to extensions: readd cata here.

← Older revision		Revision as of 23:08, 26 August 2026
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	=== Overview to extensions ===		=== Overview to extensions ===
			==== Full 7-limit extensions ====
	The second comma of the [[normal forms #Normal forms for commas\|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], hemifamity, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan.		The second comma of the [[normal forms #Normal forms for commas\|normal comma list]] defines which [[7-limit]] family member we are looking at. [[4375/4374]], the ragisma, gives catakleismic. [[875/864]], the keemic comma, gives keemun. [[5120/5103]], hemifamity, gives countercata. [[179200/177147]], the tolerant comma, gives metakleismic. [[64/63]], the archytas comma, gives catalan. Catakleismic, keemun, countercata, metakleismic, and catalan all have octave period and use the minor third as a generator; catakleismic, countercata, and metakleismic define the 7/4 more complexly but more accurately than keemun and catalan.

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	Temperaments involving larger splits include [[#Sqrtphi\|sqrtphi]], [[#Quartkeenlig\|quartkeenlig]], [[#Novemkleismic\|novemkleismic]]. Those split the kleismic structure into five to nine parts.		Temperaments involving larger splits include [[#Sqrtphi\|sqrtphi]], [[#Quartkeenlig\|quartkeenlig]], [[#Novemkleismic\|novemkleismic]]. Those split the kleismic structure into five to nine parts.

	The kleismic family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], which has further extensions with higher primes. These are listed at the bottom of this page, in [[#Subgroup extensions]].		The kleismic family boasts a very remarkable extension to the [[2.3.5.13 subgroup]], which has further extensions with higher primes. These are listed at the bottom of this page, in [[#Subgroup extensions]].

			==== Subgroup extensions ====

	== Catakleismic ==		== Catakleismic ==

FloraC: - interwiki (the German article corresponds to Kleismic)

2026-08-14T12:03:47Z

- interwiki (the German article corresponds to Kleismic)

← Older revision		Revision as of 12:03, 14 August 2026
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	~~{{interwiki~~
	~~\| en = Kleismic family~~
	~~\| de = Hanson-Kleismisch~~
	~~\| es =~~
	~~\| ja =~~
	}}
	{{Technical data page}}		{{Technical data page}}
	The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5\|classical minor third]]s falls short of the [[3/1\|3rd]] [[harmonic]].		The [[5-limit]] parent comma for the '''kleismic family''' is [[15625/15552]], the kleisma, which is the amount by which a stack of six [[6/5\|classical minor third]]s falls short of the [[3/1\|3rd]] [[harmonic]].