2.3.5.13 subgroup - Revision history

Overthink: /* Rank-2 temperaments */ is this word needed

2026-07-19T20:59:58Z

Rank-2 temperaments: is this word needed

← Older revision		Revision as of 20:59, 19 July 2026
Line 14:		Line 14:
	[[Tridecaschismic]], which adds prime 13 to schismic via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] and especially 53edo. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and -1.428{{c}} of error respectively.		[[Tridecaschismic]], which adds prime 13 to schismic via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] and especially 53edo. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and -1.428{{c}} of error respectively.

	Cata and tridecaschismic unite ~~both~~ in 53edo.		Cata and tridecaschismic unite in 53edo.

	Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.		Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.

Eufalesio: /* Regular temperaments */

2026-07-19T20:08:58Z

Regular temperaments

← Older revision		Revision as of 20:08, 19 July 2026
Line 12:		Line 12:
	The 2.3.5.13 version of [[kleismic]] (sometimes called ''cata'') provides a fairly low complexity approximation to the subgroup, using a slightly sharp [[~]]6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. Two generators reach ~[[13/9]], [[tempering out]] the marveltwin comma [[325/324]]. Then ~[[26/15]] is found at three generators, with two such intervals reaching ~3/1, tempering out [[676/675]]. The interval at +4 generators is a third of a [[9/8]] whole tone, representing all of [[25/24]], [[26/25]], and [[27/26]]. Good tunings of cata include [[34edo]] and especially [[53edo]], with other tunings such as [[87edo]] and [[140edo]] being usable as well.		The 2.3.5.13 version of [[kleismic]] (sometimes called ''cata'') provides a fairly low complexity approximation to the subgroup, using a slightly sharp [[~]]6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. Two generators reach ~[[13/9]], [[tempering out]] the marveltwin comma [[325/324]]. Then ~[[26/15]] is found at three generators, with two such intervals reaching ~3/1, tempering out [[676/675]]. The interval at +4 generators is a third of a [[9/8]] whole tone, representing all of [[25/24]], [[26/25]], and [[27/26]]. Good tunings of cata include [[34edo]] and especially [[53edo]], with other tunings such as [[87edo]] and [[140edo]] being usable as well.

	~~The~~ [[~~schismic~~]] ~~extension that~~ adds prime 13 via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] and 53edo~~, though [[94edo]] is more optimized and can extend to other subgroups~~. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and -1.428{{c}} of error respectively.		[[Tridecaschismic]], which adds prime 13 to schismic via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] and especially 53edo. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and -1.428{{c}} of error respectively.

			Cata and tridecaschismic unite both in 53edo.

	Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.		Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.

Overthink: /* Rank-2 temperaments */ +20 fifths octave reduced is actually 839.1 cents

2026-05-09T00:55:51Z

Rank-2 temperaments: +20 fifths octave reduced is actually 839.1 cents

← Older revision		Revision as of 00:55, 9 May 2026
Line 12:		Line 12:
	The 2.3.5.13 version of [[kleismic]] (sometimes called ''cata'') provides a fairly low complexity approximation to the subgroup, using a slightly sharp [[~]]6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. Two generators reach ~[[13/9]], [[tempering out]] the marveltwin comma [[325/324]]. Then ~[[26/15]] is found at three generators, with two such intervals reaching ~3/1, tempering out [[676/675]]. The interval at +4 generators is a third of a [[9/8]] whole tone, representing all of [[25/24]], [[26/25]], and [[27/26]]. Good tunings of cata include [[34edo]] and especially [[53edo]], with other tunings such as [[87edo]] and [[140edo]] being usable as well.		The 2.3.5.13 version of [[kleismic]] (sometimes called ''cata'') provides a fairly low complexity approximation to the subgroup, using a slightly sharp [[~]]6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. Two generators reach ~[[13/9]], [[tempering out]] the marveltwin comma [[325/324]]. Then ~[[26/15]] is found at three generators, with two such intervals reaching ~3/1, tempering out [[676/675]]. The interval at +4 generators is a third of a [[9/8]] whole tone, representing all of [[25/24]], [[26/25]], and [[27/26]]. Good tunings of cata include [[34edo]] and especially [[53edo]], with other tunings such as [[87edo]] and [[140edo]] being usable as well.

	The [[schismic]] extension that adds prime 13 via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and +1.428{{c}} of error respectively.		The [[schismic]] extension that adds prime 13 via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and -1.428{{c}} of error respectively.

	Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.		Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.

Overthink: /* Rank-2 temperaments */ use "kleismic" as the main name

2026-05-05T21:51:34Z

Rank-2 temperaments: use "kleismic" as the main name

← Older revision		Revision as of 21:51, 5 May 2026
Line 10:		Line 10:

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	[[~~Cata~~]] provides a fairly low complexity approximation to the subgroup, using a slightly sharp [[~]]6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. Two generators reach ~[[13/9]], [[tempering out]] the marveltwin comma [[325/324]]. Then ~[[26/15]] is found at three generators, with two such intervals reaching ~3/1, tempering out [[676/675]]. The interval at +4 generators is a third of a [[9/8]] whole tone, representing all of [[25/24]], [[26/25]], and [[27/26]]. Good tunings of cata include [[34edo]] and especially [[53edo]], with other tunings such as [[87edo]] and [[140edo]] being usable as well.		The 2.3.5.13 version of [[kleismic]] (sometimes called ''cata'') provides a fairly low complexity approximation to the subgroup, using a slightly sharp [[~]]6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. Two generators reach ~[[13/9]], [[tempering out]] the marveltwin comma [[325/324]]. Then ~[[26/15]] is found at three generators, with two such intervals reaching ~3/1, tempering out [[676/675]]. The interval at +4 generators is a third of a [[9/8]] whole tone, representing all of [[25/24]], [[26/25]], and [[27/26]]. Good tunings of cata include [[34edo]] and especially [[53edo]], with other tunings such as [[87edo]] and [[140edo]] being usable as well.

	The [[schismic]] extension that adds prime 13 via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and +1.428{{c}} of error respectively.		The [[schismic]] extension that adds prime 13 via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and +1.428{{c}} of error respectively.

Overthink: interseptimals; explain cata; cassandra restriction

2026-05-05T21:50:48Z

interseptimals; explain cata; cassandra restriction

← Older revision		Revision as of 21:50, 5 May 2026
Line 2:		Line 2:

	It can be thought out as an extension of the familiar [[5-limit]] with a tridecimal xenharmonic touch, or as a retraction of the 13-limit obtained by removing [[7/1\|7]] and [[11/1\|11]]. It shares some qualities with the [[2.3.5.11 subgroup]], specifically considering [[neutral (interval quality)\|neutral]] interval pairs such as [[39/32]]~[[11/9]] and [[16/13]]~[[27/22]], which differ by the small comma of [[352/351]].		It can be thought out as an extension of the familiar [[5-limit]] with a tridecimal xenharmonic touch, or as a retraction of the 13-limit obtained by removing [[7/1\|7]] and [[11/1\|11]]. It shares some qualities with the [[2.3.5.11 subgroup]], specifically considering [[neutral (interval quality)\|neutral]] interval pairs such as [[39/32]]~[[11/9]] and [[16/13]]~[[27/22]], which differ by the small comma of [[352/351]].

			This subgroup is notable for containing the simplest JI representations of [[interseptimal]] intervals, which are halfway between two interval categories, with [[15/13]] being an ultramajor second/inframinor third, [[13/10]] being an ultramajor third/infrafourth, [[20/13]] being an ultrafifth/inframinor sixth, and [[26/15]] being an ultramajor sixth/inframinor seventh. Importantly, 15/13 is close to half of the [[4/3\|perfect fourth]], and 26/15 is close to half of the [[3/1\|perfect twelfth]], with two intervals of 15/13 falling short of 4/3 by [[676/675]], the island comma.

	== Regular temperaments ==		== Regular temperaments ==
Line 8:		Line 10:

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	[[Cata]] provides a fairly low complexity approximation to the subgroup, using a slightly sharp ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. It is ~~well represented by~~ [[34edo~~\|34-~~]] and [[53edo]], with [[87edo]] being ~~an almost perfect approximation~~.		[[Cata]] provides a fairly low complexity approximation to the subgroup, using a slightly sharp [[~]]6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. Two generators reach ~[[13/9]], [[tempering out]] the marveltwin comma [[325/324]]. Then ~[[26/15]] is found at three generators, with two such intervals reaching ~3/1, tempering out [[676/675]]. The interval at +4 generators is a third of a [[9/8]] whole tone, representing all of [[25/24]], [[26/25]], and [[27/26]]. Good tunings of cata include [[34edo]] and especially [[53edo]], with other tunings such as [[87edo]] and [[140edo]] being usable as well.

	The [[schismic]] extension that adds prime 13 via [[tempering out~~]] the~~ [[~~marveltwin comma~~]] provides a more complex temperament, well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and +1.428{{c}} of error respectively.		The [[schismic]] extension that adds prime 13 via tempering out [[325/324]] provides a more complex temperament, well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This mapping for 13 is a [[restriction]] of the full 13-limit [[cassandra]] mapping. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and +1.428{{c}} of error respectively.

	Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.		Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.

FloraC: Style

2026-04-19T10:26:55Z

Style

← Older revision		Revision as of 10:26, 19 April 2026
Line 1:		Line 1:
	The '''2.3.5.13 subgroup''' (~~AKA~~ ''yatha'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[13/1\|13]] are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 13. This is an infinite set, and is still infinite even if we [[Octave reduction\|octave-reduce]] every interval in it. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[13/8]], [[13/10]], [[39/32]] and so on.		The '''2.3.5.13 subgroup''' (a.k.a. ''yatha'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[13/1\|13]] are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 13. This is an infinite set, and is still infinite even if we [[Octave reduction\|octave-reduce]] every interval in it. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[13/8]], [[13/10]], [[39/32]] and so on.

	It can be thought out as an extension of the familiar [[5-limit]] with a tridecimal xenharmonic touch, or as a retraction of the 13-limit obtained by removing [[7/1\|7]] and [[11/1\|11]]. It shares some qualities with the [[2.3.5.11 subgroup]], specifically considering [[~~Neutral~~ (interval quality)\|neutral]] interval pairs such as [[39/32]]~[[11/9]] and [[16/13]]~[[27/22]], which differ by the small comma of [[352/351]].		It can be thought out as an extension of the familiar [[5-limit]] with a tridecimal xenharmonic touch, or as a retraction of the 13-limit obtained by removing [[7/1\|7]] and [[11/1\|11]]. It shares some qualities with the [[2.3.5.11 subgroup]], specifically considering [[neutral (interval quality)\|neutral]] interval pairs such as [[39/32]]~[[11/9]] and [[16/13]]~[[27/22]], which differ by the small comma of [[352/351]].

	== Regular temperaments ==		== Regular temperaments ==
Line 14:		Line 14:
	Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.		Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.

	For those searching [[~~Very~~ high accuracy temperaments\|very high-accuracy temperaments]], the 2.3.5.13 extension of [[egads]] ({{nowrap\| 19 & 422 }}) provides an extremely complex, though insanely accurate representation of the subgroup, with lower badness than cata and with an almost just ~6/5 as a generator, finding 5/4 at -51 gens, 3/2 at -52 gens, and 13/8 at -138 gens, of which [[1342edo]] offers practically perfect approximations.		For those searching [[very high accuracy temperaments\|very high-accuracy temperaments]], the 2.3.5.13 extension of [[egads]] ({{nowrap\| 19 & 422 }}) provides an extremely complex, though insanely accurate representation of the subgroup, with lower badness than cata and with an almost just ~6/5 as a generator, finding 5/4 at -51 gens, 3/2 at -52 gens, and 13/8 at -138 gens, of which [[1342edo]] offers practically perfect approximations.

	=== Rank-3 temperaments ===		=== Rank-3 temperaments ===
	[[Marveltwin]] offers a very low-complexity approximation to the subgroup, reaching [[16/13]] through ([[10/9]])<sup>2</sup>, and condensing the subgroup into a 5-limit [[planar temperament]].		[[Marveltwin]] offers a very low-complexity approximation to the subgroup, reaching [[16/13]] through ([[10/9]])<sup>2</sup>, and condensing the subgroup into a 5-limit [[planar temperament]].

	[[Catasma\|{140625/140608}]], the temperament that tempers the catasma alone, is an extremely accurate temperament, which also appears in the same Egads extension (catabolic). Non-cata edos at the boundary of usability are, [[407edo]], [[441edo]], [[494edo]], [[901edo]], and of course [[1342edo]].		{[[Catasma\|140625/140608]]}, the temperament that tempers the catasma alone, is an extremely accurate temperament, which also appears in the same Egads extension (catabolic). Non-cata edos at the boundary of usability are, [[407edo]], [[441edo]], [[494edo]], [[901edo]], and of course [[1342edo]].

	[[Category:Just intonation subgroups\|#]]		[[Category:Just intonation subgroups\|#]]
	[[Category:Rank-4 temperaments\|#]]		[[Category:Rank-4 temperaments\|#]]
	[[Category:13-limit\|#]]		[[Category:13-limit\|#]]

Eufalesio: color notation

2026-04-19T09:52:17Z

color notation

← Older revision		Revision as of 09:52, 19 April 2026
Line 1:		Line 1:
	The '''2.3.5.13 subgroup''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[13/1\|13]] are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 13. This is an infinite set, and is still infinite even if we [[Octave reduction\|octave-reduce]] every interval in it. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[13/8]], [[13/10]], [[39/32]] and so on.		The '''2.3.5.13 subgroup''' (AKA ''yatha'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[13/1\|13]] are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 13. This is an infinite set, and is still infinite even if we [[Octave reduction\|octave-reduce]] every interval in it. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[13/8]], [[13/10]], [[39/32]] and so on.

	It can be thought out as an extension of the familiar [[5-limit]] with a tridecimal xenharmonic touch, or as a retraction of the 13-limit obtained by removing [[7/1\|7]] and [[11/1\|11]]. It shares some qualities with the [[2.3.5.11 subgroup]], specifically considering [[Neutral (interval quality)\|neutral]] interval pairs such as [[39/32]]~[[11/9]] and [[16/13]]~[[27/22]], which differ by the small comma of [[352/351]].		It can be thought out as an extension of the familiar [[5-limit]] with a tridecimal xenharmonic touch, or as a retraction of the 13-limit obtained by removing [[7/1\|7]] and [[11/1\|11]]. It shares some qualities with the [[2.3.5.11 subgroup]], specifically considering [[Neutral (interval quality)\|neutral]] interval pairs such as [[39/32]]~[[11/9]] and [[16/13]]~[[27/22]], which differ by the small comma of [[352/351]].

Overthink: links; change wordings

2026-03-16T05:15:45Z

links; change wordings

← Older revision		Revision as of 05:15, 16 March 2026
Line 1:		Line 1:
	The '''2.3.5.13 subgroup''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[13/1\|13]] are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 13. This is an infinite set and still infinite even if we ~~restrict consideration to a single~~ octave. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[13/8]], [[13/10]], [[39/32]] and so on.		The '''2.3.5.13 subgroup''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], and [[13/1\|13]] are the only allowable [[prime factor]]s, so that every such interval may be written as a ratio of integers which are products of 2, 3, 5 and 13. This is an infinite set, and is still infinite even if we [[Octave reduction\|octave-reduce]] every interval in it. Some examples within the [[octave]] include [[5/4]], [[3/2]], [[13/8]], [[13/10]], [[39/32]] and so on.

	It can be thought out as an extension of the familiar [[5-limit]] with a tridecimal xenharmonic touch, or as a retraction of the 13-limit obtained by removing 7 and 11. It ~~can be similar to~~ the [[2.3.5.11 subgroup]], ~~specially~~ considering neutral interval pairs such as 39/32~11/9 and 16/13~27/22, which ~~are connected~~ by the small comma of [[352/351]].		It can be thought out as an extension of the familiar [[5-limit]] with a tridecimal xenharmonic touch, or as a retraction of the 13-limit obtained by removing [[7/1\|7]] and [[11/1\|11]]. It shares some qualities with the [[2.3.5.11 subgroup]], specifically considering [[Neutral (interval quality)\|neutral]] interval pairs such as [[39/32]]~[[11/9]] and [[16/13]]~[[27/22]], which differ by the small comma of [[352/351]].

	== Regular temperaments ==		== Regular temperaments ==

Overthink: /* Rank-2 temperaments */ Improve sentence; link

2026-03-16T05:10:29Z

Rank-2 temperaments: Improve sentence; link

← Older revision		Revision as of 05:10, 16 March 2026
Line 10:		Line 10:
	[[Cata]] provides a fairly low complexity approximation to the subgroup, using a slightly sharp ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. It is well represented by [[34edo\|34-]] and [[53edo]], with [[87edo]] being an almost perfect approximation.		[[Cata]] provides a fairly low complexity approximation to the subgroup, using a slightly sharp ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, and ~13/8 at +14 gens. It is well represented by [[34edo\|34-]] and [[53edo]], with [[87edo]] being an almost perfect approximation.

	The [[schismic]] extension via [[tempering out]] ~~the [[schisma]] and~~ the [[marveltwin comma]] provides a more complex temperament, well represented with [[41edo~~\|41-~~]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and +1.428{{c}} of error respectively.		The [[schismic]] extension that adds prime 13 via [[tempering out]] the [[marveltwin comma]] provides a more complex temperament, well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (-8 fifths) [[8192/6561]] and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound extremely close to 5/4 and 13/8 respectively. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 13 with -1.954{{c}} and +1.428{{c}} of error respectively.

	Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.		Other approximations of [[schismic]] reach prime 13 through other means, such as [[hemischis]], dividing prime 3 in 2 and finding 3/2 at +2 gens, 5/4 at -16 gens, and 13/8 at -13 gens. [[Helenus]] reaches 13/8 through -33 fifths, but it is a worse mapping.

	For those searching very high-accuracy temperaments, the 2.3.5.13 extension of [[egads]] ({{nowrap\| 19 & 422 }}) provides an extremely complex, though insanely accurate representation of the subgroup, with lower badness than cata and with an almost just ~6/5 as a generator, finding 5/4 at -51 gens, 3/2 at -52 gens, and 13/8 at -138 gens, of which [[1342edo]] offers practically perfect approximations.		For those searching [[Very high accuracy temperaments\|very high-accuracy temperaments]], the 2.3.5.13 extension of [[egads]] ({{nowrap\| 19 & 422 }}) provides an extremely complex, though insanely accurate representation of the subgroup, with lower badness than cata and with an almost just ~6/5 as a generator, finding 5/4 at -51 gens, 3/2 at -52 gens, and 13/8 at -138 gens, of which [[1342edo]] offers practically perfect approximations.

	=== Rank-3 temperaments ===		=== Rank-3 temperaments ===

FloraC: /* Regular temperaments */ review list

2026-02-13T09:48:23Z

Regular temperaments: review list

← Older revision		Revision as of 09:48, 13 February 2026
Line 5:		Line 5:
	== Regular temperaments ==		== Regular temperaments ==
	=== Rank-1 temperaments (edos) ===		=== Rank-1 temperaments (edos) ===
	It is relatively well approximated by the following edos (bold ones do particularly well in this subgroup): ~~[[7edo~~\|7~~]], [[15edo\|15]], [[19edo\|19]]~~, ~~[[24edo\|24]]~~, ~~[[27edo\|27]]~~, ~~[[31edo\|31]]~~, '''~~[[34edo\|34]]~~'''~~, [[46edo\|46]], [[50edo\|50]]~~, '''~~[[53edo\|53]]~~'''~~, [[62edo\|62]], [[68edo\|68]], [[77edo\|77]], [[80edo\|80]]~~, '''~~[[87edo\|87]]~~''', ~~[[94edo\|94]]~~, ~~[[96edo\|96]]~~, ~~[[106edo\|106]]~~, ~~[[130edo\|130]]~~, ~~[[137edo\|137]]~~, ~~[[140edo\|140]]~~, ~~[[159edo\|159]]~~, ~~'''[[171edo\|171]]'''~~, ~~[[190edo\|190]]~~, '''~~[[217edo\|217]]~~'''~~, [[224edo\|224]], '''[[270edo\|270]]''', [[311edo\|311]]~~, …		The 2.3.5.13 subgroup is relatively well approximated by the following edos (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs\| '''7''', 10, 12, 15, '''19''', '''34''', '''53''', 130, 140, 164, 183, 217, 270, 354, 388, 407, '''441''', … }}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===