2.3.5.7.13 subgroup - Revision history

FloraC: /* Regular temperaments */ correct factual error (cotoneum does keep the chain of fifths)

2026-06-14T11:52:55Z

Regular temperaments: correct factual error (cotoneum does keep the chain of fifths)

← Older revision		Revision as of 11:52, 14 June 2026
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	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, 7/4 at +25 gens, and 13/8 at −13 gens. [[Pontiac]] reaches 7/4 through +39 fifths, and 13/8 through −33 fifths, and it makes for a much better mapping, which is very well represented in [[171edo\|171]] and [[224edo]].		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, 7/4 at +25 gens, and 13/8 at −13 gens. [[Pontiac]] reaches 7/4 through +39 fifths, and 13/8 through −33 fifths, and it makes for a much better mapping, which is very well represented in [[171edo\|171]] and [[224edo]].

	For those searching higher accuracy temperaments, [[~~gariwizmic~~]] ~~also~~ keeps the chain of fifths~~, spliting the octave in half,~~ but does not temper out the schisma. It finds 5/4 at 39 fifths minus one [[semioctave]], 7/4 at −14 fifths, and 13/8 at −27 fifths plus a semioctave. This is a much worse mapping, but it ends at [[270edo]], which is known for its astounding accuracy in the 13-limit.		For those searching higher accuracy temperaments, one possibility is [[cotoneum]], which keeps the chain of fifths but does not temper out the schisma. It is well represented by [[217edo]], which inherits 31edo's [[2.5.7 subgroup\|2.5.7]] part and vastly improves upon 3 and 13, 13 itself being a semiconvergent. [[Gariwizmic]] also keeps the chain of fifths but splits the octave in halves. It finds 5/4 at 39 fifths minus one [[semioctave]], 7/4 at −14 fifths, and 13/8 at −27 fifths plus a semioctave. This is a much worse mapping, but it ends at [[270edo]], which is known for its astounding accuracy in the full 13-limit.

	Other non-chain-of-fifths temperaments that converge in 270edo, and are thus great candidates for the 2.3.5.7.13 subgroup are [[buzzard~~]], [[cotoneum~~]], [[newt]], and [[ennealimmal]]. ~~Cotoneum, well represented by [[217edo]], has 31edo's 2.5.7 and vastly improves upon 3 and 13; 13 itself being a semiconvergent. Ennealimmal~~ is ~~extremely~~ accurate and well represented, as it ~~can be naturally extended to the subgroup by adding the minisma, equating~~ the [[36/35]] generator to ~~the~~ [[1053/1024]]~~. The~~ pontigailimma is by extension tempered out too.		Other non-chain-of-fifths temperaments that converge in 270edo, and are thus great candidates for the 2.3.5.7.13 subgroup are [[buzzard]], [[newt]], and the [[ennealimmal]] extension that adds the minisma to the commas. The ennealimmal extension is very accurate and well represented, as it equates the [[36/35]] generator to [[1053/1024]]; the pontigailimma is by extension tempered out too.

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

FloraC: /* Regular temperaments */ neutrality (7-limit doesn't need to be singled out here)

2026-06-14T11:37:26Z

Regular temperaments: neutrality (7-limit doesn't need to be singled out here)

← Older revision		Revision as of 11:37, 14 June 2026
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	== Regular temperaments ==		== Regular temperaments ==
	=== Rank-1 temperaments (edos) ===		=== Rank-1 temperaments (edos) ===
	The 2.3.5.7.13 subgroup is relatively well approximated by the following edos (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs\| '''10''', 12f, 15, '''19''', 27, 31, 41, 46, '''53''', 72, 103, 111, 121, '''130''', 140, '''171''', 224, 243, '''270''', '''441''', … }}		The 2.3.5.7.13 subgroup is relatively well approximated by the following edos (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs\| '''10''', 12f, 15, '''19''', 27, 31, 41, 46, '''53''', 72, 103, 111, 121, '''130''', 140, '''171''', 224, 243, '''270''', '''441''', … }}

	~~<nowiki/>* Very accurate 7-limit but relatively innacurate prime 13.~~

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

FloraC: Improve wording & linking

2026-06-14T11:36:18Z

Improve wording & linking

← Older revision		Revision as of 11:36, 14 June 2026
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	The '''2.3.5.7.13 subgroup''' (a.k.a. ''yazatha'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], 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, 7 and 13; this makes it a rank-5 system. 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]], [[7/4]], [[13/8]], [[13/7]], [[13/10]], [[39/32]], and so on.		The '''2.3.5.7.13 subgroup''' (a.k.a. ''yazatha'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], 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, 7 and 13; this makes it a rank-5 system. 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]], [[7/4]], [[13/8]], [[13/7]], [[13/10]], [[39/32]], and so on.

	It can be thought out as an ~~extension~~ of the [[7-limit]] with a tridecimal xenharmonic touch, or as a retraction of the full 13-limit obtained by removing 11. It can be similar to the [[11-limit]], 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 [[expansion]] of the [[7-limit]] with a tridecimal xenharmonic touch, or as a [[retraction]] of the full [[13-limit]] obtained by removing 11. It can be similar to the [[11-limit]], 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]].

	The subgroup can be very easily rank-reduced into the 7-limit through the [[4096/4095\|minisma]], an unnoticeable comma which connects ratios of 35 to 13, such that for example {{nowrap\|[[36/35]]~[[1053/1024]]}}, or {{nowrap\|[[45/32]]~[[128/91]]}}. The same can be said with the [[pontigailimma]], an atomic comma which is harder to visualize but entails significantly more accuracy. See article for comma equivalences.		The subgroup can be very easily rank-reduced into the 7-limit through the [[4096/4095\|minisma]], an unnoticeable comma which connects ratios of 35 to 13, such that for example {{nowrap\|[[36/35]]~[[1053/1024]]}}, or {{nowrap\|[[45/32]]~[[128/91]]}}. The same can be said with the [[pontigailimma]], an atomic comma which is harder to visualize but entails significantly more accuracy. See article for comma equivalences.

Overthink: /* Rank-2 temperaments */ fifth count for consistency; better grammar

2026-05-01T00:29:28Z

Rank-2 temperaments: fifth count for consistency; better grammar

← Older revision		Revision as of 00:29, 1 May 2026
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	[[Catakleismic]] provides a low-badness approximation to the subgroup, using a slightly sharp ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, 7/4 at +22 gens and ~13/8 at +14 gens. It is coarsely represented by [[19edo]], and well represented by [[53edo]] and [[72edo]], with [[125edo]] and [[197edo]] making for much better approximations.		[[Catakleismic]] provides a low-badness approximation to the subgroup, using a slightly sharp ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, 7/4 at +22 gens and ~13/8 at +14 gens. It is coarsely represented by [[19edo]], and well represented by [[53edo]] and [[72edo]], with [[125edo]] and [[197edo]] making for much better approximations.

	No-11 [[cassandra]] provides a more complex temperament using a [[chain of fifths]], well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. It is also decent in [[147edo]], though inconsistent. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (−8 fifths) [[8192/6561]], the double-diminished octave 8388608/4782969 and the triple-augmented fourth (+20 fifths) 3486784401/2147483648 already sound very close to 5/4, 7/4, and 13/8 respectively. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 ~~and~~ 7 and 13 with −1.954{{c}} ~~and~~ +3.804{{c}} and +1.428{{c}} of error respectively.		No-11 [[cassandra]] provides a more complex temperament using a [[chain of fifths]], well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. It is also decent in [[147edo]], though inconsistent. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (−8 fifths) [[8192/6561]], the double-diminished octave (−14 fifths) 8388608/4782969, and the triple-augmented fourth (+20 fifths) 3486784401/2147483648 already sound very close to 5/4, 7/4, and 13/8 respectively. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5, 7, and 13 with −1.954{{c}}, +3.804{{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, 7/4 at +25 gens, and 13/8 at −13 gens. [[Pontiac]] reaches 7/4 through +39 fifths, and 13/8 through −33 fifths, and it makes for a much better mapping, which is very well represented in [[171edo\|171]] and [[224edo]].		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, 7/4 at +25 gens, and 13/8 at −13 gens. [[Pontiac]] reaches 7/4 through +39 fifths, and 13/8 through −33 fifths, and it makes for a much better mapping, which is very well represented in [[171edo\|171]] and [[224edo]].

FloraC: Style

2026-04-19T10:31:20Z

Style

← Older revision		Revision as of 10:31, 19 April 2026
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	The '''2.3.5.7.13 subgroup''' (~~AKA~~ ''yazatha'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], 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, 7 and 13; this makes it a rank-5 system. 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]], [[7/4]], [[13/8]], [[13/7]], [[13/10]], [[39/32]], and so on.		The '''2.3.5.7.13 subgroup''' (a.k.a. ''yazatha'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], 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, 7 and 13; this makes it a rank-5 system. 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]], [[7/4]], [[13/8]], [[13/7]], [[13/10]], [[39/32]], and so on.

	It can be thought out as an extension of the [[7-limit]] with a tridecimal xenharmonic touch, or as a retraction of the full 13-limit obtained by removing 11. It can be similar to the [[11-limit]], specially considering neutral interval pairs such as ~~{{nowrap\|~~39/32~11/9}} and ~~{{nowrap\|~~16/13~27/22}}, which are connected by the small comma of [[352/351]].		It can be thought out as an extension of the [[7-limit]] with a tridecimal xenharmonic touch, or as a retraction of the full 13-limit obtained by removing 11. It can be similar to the [[11-limit]], 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]].

	The subgroup can be very easily rank-reduced into the 7-limit through the [[4096/4095\|minisma]], an unnoticeable comma which connects ratios of 35 to 13, such that for example {{nowrap\|[[36/35]]~[[1053/1024]]}}, or {{nowrap\|[[45/32]]~[[128/91]]}}. The same can be said with the [[pontigailimma]], an atomic comma which is harder to visualize but entails significantly more accuracy. See article for comma equivalences.		The subgroup can be very easily rank-reduced into the 7-limit through the [[4096/4095\|minisma]], an unnoticeable comma which connects ratios of 35 to 13, such that for example {{nowrap\|[[36/35]]~[[1053/1024]]}}, or {{nowrap\|[[45/32]]~[[128/91]]}}. The same can be said with the [[pontigailimma]], an atomic comma which is harder to visualize but entails significantly more accuracy. See article for comma equivalences.
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	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	[[Catakleismic]] provides a low badness approximation to the subgroup, using a slightly sharp ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, 7/4 at +22 gens and ~13/8 at +14 gens. It is coarsely represented by [[19edo]], and well represented by [[53edo]] and [[72edo]], with [[125edo]] and [[197edo]] making for much better approximations.		[[Catakleismic]] provides a low-badness approximation to the subgroup, using a slightly sharp ~6/5 as a generator, finding ~5/4 at +5 gens, ~3/2 at +6 gens, 7/4 at +22 gens and ~13/8 at +14 gens. It is coarsely represented by [[19edo]], and well represented by [[53edo]] and [[72edo]], with [[125edo]] and [[197edo]] making for much better approximations.

	No-11 [[cassandra]] provides a more complex temperament using a [[chain of fifths]], well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. It is also decent in [[147edo]], though inconsistent. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (−8 fifths) [[8192/6561]], the double-diminished octave 8388608/4782969 and the triple-augmented fourth (+20 fifths) 3486784401/2147483648 already sound very close to 5/4, 7/4, and 13/8 respectively. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 7 and 13 with −1.954{{c}} and +3.804{{c}} and +1.428{{c}} of error respectively.		No-11 [[cassandra]] provides a more complex temperament using a [[chain of fifths]], well represented with [[41edo]] and 53edo, though [[94edo]] is more optimized and can extend to other subgroups. It is also decent in [[147edo]], though inconsistent. [[Pythagorean tuning]] also works surprisingly well, where the diminished fourth (−8 fifths) [[8192/6561]], the double-diminished octave 8388608/4782969 and the triple-augmented fourth (+20 fifths) 3486784401/2147483648 already sound very close to 5/4, 7/4, and 13/8 respectively. This is not so much a temperament as it is a relabeling of the 3-limit, which offers 5 and 7 and 13 with −1.954{{c}} and +3.804{{c}} and +1.428{{c}} of error respectively.
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	For those searching higher accuracy temperaments, [[gariwizmic]] also keeps the chain of fifths, spliting the octave in half, but does not temper out the schisma. It finds 5/4 at 39 fifths minus one [[semioctave]], 7/4 at −14 fifths, and 13/8 at −27 fifths plus a semioctave. This is a much worse mapping, but it ends at [[270edo]], which is known for its astounding accuracy in the 13-limit.		For those searching higher accuracy temperaments, [[gariwizmic]] also keeps the chain of fifths, spliting the octave in half, but does not temper out the schisma. It finds 5/4 at 39 fifths minus one [[semioctave]], 7/4 at −14 fifths, and 13/8 at −27 fifths plus a semioctave. This is a much worse mapping, but it ends at [[270edo]], which is known for its astounding accuracy in the 13-limit.

	~~Another~~ non-chain-of-fifths temperaments that converge in 270edo, and are thus great candidates for the 2.3.5.7.13 subgroup are [[buzzard]], [[cotoneum]], [[newt]], and [[ennealimmal]]. Cotoneum, well represented by [[217edo]], has 31edo's 2.5.7 and vastly improves upon 3 and 13; 13 itself being a semiconvergent. Ennealimmal is extremely accurate and well represented, as it can be naturally extended to the subgroup by adding the minisma, equating the [[36/35]] generator to the [[1053/1024]]. The pontigailimma is by extension tempered out too.		Other non-chain-of-fifths temperaments that converge in 270edo, and are thus great candidates for the 2.3.5.7.13 subgroup are [[buzzard]], [[cotoneum]], [[newt]], and [[ennealimmal]]. Cotoneum, well represented by [[217edo]], has 31edo's 2.5.7 and vastly improves upon 3 and 13; 13 itself being a semiconvergent. Ennealimmal is extremely accurate and well represented, as it can be naturally extended to the subgroup by adding the minisma, equating the [[36/35]] generator to the [[1053/1024]]. The pontigailimma is by extension tempered out too.

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

Eufalesio: color notation

2026-04-19T09:49:39Z

color notation

← Older revision		Revision as of 09:49, 19 April 2026
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	The '''2.3.5.7.13 subgroup''' is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], 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, 7 and 13; this makes it a rank-5 system. 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]], [[7/4]], [[13/8]], [[13/7]], [[13/10]], [[39/32]], and so on.		The '''2.3.5.7.13 subgroup''' (AKA ''yazatha'' in [[color notation]]) is a [[just intonation subgroup]] consisting of [[rational interval]]s where [[2/1\|2]], [[3/1\|3]], [[5/1\|5]], [[7/1\|7]], 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, 7 and 13; this makes it a rank-5 system. 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]], [[7/4]], [[13/8]], [[13/7]], [[13/10]], [[39/32]], and so on.

	It can be thought out as an extension of the [[7-limit]] with a tridecimal xenharmonic touch, or as a retraction of the full 13-limit obtained by removing 11. It can be similar to the [[11-limit]], specially considering neutral interval pairs such as {{nowrap\|39/32~11/9}} and {{nowrap\|16/13~27/22}}, which are connected by the small comma of [[352/351]].		It can be thought out as an extension of the [[7-limit]] with a tridecimal xenharmonic touch, or as a retraction of the full 13-limit obtained by removing 11. It can be similar to the [[11-limit]], specially considering neutral interval pairs such as {{nowrap\|39/32~11/9}} and {{nowrap\|16/13~27/22}}, which are connected by the small comma of [[352/351]].

FloraC: Name change followup

2026-03-03T11:04:43Z

Name change followup

← Older revision		Revision as of 11:04, 3 March 2026
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	For those searching higher accuracy temperaments, [[gariwizmic]] also keeps the chain of fifths, spliting the octave in half, but does not temper out the schisma. It finds 5/4 at 39 fifths minus one [[semioctave]], 7/4 at −14 fifths, and 13/8 at −27 fifths plus a semioctave. This is a much worse mapping, but it ends at [[270edo]], which is known for its astounding accuracy in the 13-limit.		For those searching higher accuracy temperaments, [[gariwizmic]] also keeps the chain of fifths, spliting the octave in half, but does not temper out the schisma. It finds 5/4 at 39 fifths minus one [[semioctave]], 7/4 at −14 fifths, and 13/8 at −27 fifths plus a semioctave. This is a much worse mapping, but it ends at [[270edo]], which is known for its astounding accuracy in the 13-limit.

	Another non-chain-of-fifths temperaments that converge in 270edo, and are thus great candidates for the 2.3.5.7.13 subgroup are [[buzzard]], [[cotoneum]], [[newt]], and [[ennealimmal]]. Cotoneum, well represented by [[217edo]], has 31edo's 2.5.7 and vastly improves upon 3 and 13; 13 itself being a semiconvergent. Ennealimmal is extremely accurate and well represented, as it can be naturally extended to the subgroup by adding the ~~schismina~~, equating the [[36/35]] generator to the [[1053/1024]]. The pontigailimma is by extension tempered out too.		Another non-chain-of-fifths temperaments that converge in 270edo, and are thus great candidates for the 2.3.5.7.13 subgroup are [[buzzard]], [[cotoneum]], [[newt]], and [[ennealimmal]]. Cotoneum, well represented by [[217edo]], has 31edo's 2.5.7 and vastly improves upon 3 and 13; 13 itself being a semiconvergent. Ennealimmal is extremely accurate and well represented, as it can be naturally extended to the subgroup by adding the minisma, equating the [[36/35]] generator to the [[1053/1024]]. The pontigailimma is by extension tempered out too.

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

Eufalesio: Changed Schismina to Minisma among other thi

2026-03-03T09:09:33Z

Changed Schismina to Minisma among other thi

← Older revision		Revision as of 09:09, 3 March 2026
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	It can be thought out as an extension of the [[7-limit]] with a tridecimal xenharmonic touch, or as a retraction of the full 13-limit obtained by removing 11. It can be similar to the [[11-limit]], specially considering neutral interval pairs such as {{nowrap\|39/32~11/9}} and {{nowrap\|16/13~27/22}}, which are connected by the small comma of [[352/351]].		It can be thought out as an extension of the [[7-limit]] with a tridecimal xenharmonic touch, or as a retraction of the full 13-limit obtained by removing 11. It can be similar to the [[11-limit]], specially considering neutral interval pairs such as {{nowrap\|39/32~11/9}} and {{nowrap\|16/13~27/22}}, which are connected by the small comma of [[352/351]].

	The subgroup can be very easily rank-reduced into the 7-limit through the [[~~schismina~~]], an unnoticeable comma which connects ratios of 35 to 13, such that for example {{nowrap\|[[36/35]]~[[1053/1024]]}}, or {{nowrap\|[[45/32]]~[[128/91]]}}. The same can be said with the [[pontigailimma]], an atomic comma which is harder to visualize but entails significantly more accuracy. See article for comma equivalences.		The subgroup can be very easily rank-reduced into the 7-limit through the [[4096/4095\|minisma]], an unnoticeable comma which connects ratios of 35 to 13, such that for example {{nowrap\|[[36/35]]~[[1053/1024]]}}, or {{nowrap\|[[45/32]]~[[128/91]]}}. The same can be said with the [[pontigailimma]], an atomic comma which is harder to visualize but entails significantly more accuracy. See article for comma equivalences.

	== Regular temperaments ==		== Regular temperaments ==
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	For those searching higher accuracy temperaments, [[gariwizmic]] also keeps the chain of fifths, spliting the octave in half, but does not temper out the schisma. It finds 5/4 at 39 fifths minus one [[semioctave]], 7/4 at −14 fifths, and 13/8 at −27 fifths plus a semioctave. This is a much worse mapping, but it ends at [[270edo]], which is known for its astounding accuracy in the 13-limit.		For those searching higher accuracy temperaments, [[gariwizmic]] also keeps the chain of fifths, spliting the octave in half, but does not temper out the schisma. It finds 5/4 at 39 fifths minus one [[semioctave]], 7/4 at −14 fifths, and 13/8 at −27 fifths plus a semioctave. This is a much worse mapping, but it ends at [[270edo]], which is known for its astounding accuracy in the 13-limit.

	Another non-chain-of-fifths temperaments that converge in 270edo, and are thus great candidates for the 2.3.5.7.13 subgroup are [[buzzard]], [[cotoneum]], [[newt]], and [[ennealimmal]]. Ennealimmal is extremely accurate and well represented, as it can be naturally extended to the subgroup by adding the schismina, equating the [[36/35]] generator to the [[1053/1024]]. The pontigailimma is by extension tempered out too.		Another non-chain-of-fifths temperaments that converge in 270edo, and are thus great candidates for the 2.3.5.7.13 subgroup are [[buzzard]], [[cotoneum]], [[newt]], and [[ennealimmal]]. Cotoneum, well represented by [[217edo]], has 31edo's 2.5.7 and vastly improves upon 3 and 13; 13 itself being a semiconvergent. Ennealimmal is extremely accurate and well represented, as it can be naturally extended to the subgroup by adding the schismina, equating the [[36/35]] generator to the [[1053/1024]]. The pontigailimma is by extension tempered out too.

	=== Rank-3 temperaments ===		=== Rank-3 temperaments ===
	{[[4096/4095]], [[4375/4374]]} ({{nowrap\| 270 & 441 & 935 }}) is very accurate and has very low badness. As the pontigailimma is the difference between the ragisma and ~~schismina~~, it is tempered out too.		{[[4096/4095]], [[4375/4374]]} ({{nowrap\| 270 & 441 & 935 }}) is very accurate and has very low badness. As the pontigailimma is the difference between the ragisma and the minisma, it is tempered out too.

	{[[140625/140608]], [[1990656/1990625]]}, the temperament that tempers out the pontigailimma and the catasma, is also extremely accurate, orders of magnitude more than the last one.		{[[140625/140608]], [[1990656/1990625]]}, the temperament that tempers out the pontigailimma and the catasma, is also extremely accurate, orders of magnitude more than the last one.

FloraC at 17:48, 2 March 2026

2026-03-02T17:48:36Z

← Older revision		Revision as of 17:48, 2 March 2026
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	{[[1990656/1990625]]}, the temperament that tempers out the pointigailimma alone is an unfathomably accurate nanotemperament, due to the extremely tiny size of the pontigailimma.		{[[1990656/1990625]]}, the temperament that tempers out the pointigailimma alone is an unfathomably accurate nanotemperament, due to the extremely tiny size of the pontigailimma.

	[[Category:Just intonation subgroups]]		[[Category:Just intonation subgroups\|#]] <!-- 1-digit number -->
	[[Category:Rank-5 temperaments]]		[[Category:Rank-5 temperaments\|#]]
			[[Category:13-limit\|#]]

FloraC: /* Regular temperaments */ review edo list

2026-03-02T17:46:55Z

Regular temperaments: review edo list

← Older revision		Revision as of 17:46, 2 March 2026
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	== Regular temperaments ==		== Regular temperaments ==
	=== Rank-1 temperaments (edos) ===		=== Rank-1 temperaments (edos) ===
	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, '''19''', '''53''', 72, 130, 140, 171, '''~~224~~''', 243, '''270''', '''441''', ~~494...~~}}		The 2.3.5.7.13 subgroup is relatively well approximated by the following edos (decreasing [[TE error]], bold ones do particularly well in this subgroup): {{EDOs\| '''10''', 12f, 15, '''19''', 27, 31, 41, 46, '''53''', 72, 103, 111, 121, '''130''', 140, '''171''', 224, 243, '''270''', '''441''', … }}

	<nowiki/>* Very accurate 7-limit but relatively innacurate prime 13.		<nowiki/>* Very accurate 7-limit but relatively innacurate prime 13.