2.3.5.7.13 subgroup: Difference between revisions
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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''' (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 | 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 [[ | 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 == | ||
=== 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 | 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''', … }} | ||
=== 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 | 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]]. | ||
For those searching higher accuracy temperaments, [[ | 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]], [[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 === | ||
{[[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 | {[[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. | ||
Latest revision as of 11:52, 14 June 2026
The 2.3.5.7.13 subgroup (a.k.a. yazatha in color notation) is a just intonation subgroup consisting of rational intervals where 2, 3, 5, 7, and 13 are the only allowable prime factors, 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 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 minisma, an unnoticeable comma which connects ratios of 35 to 13, such that for example 36/35~1053/1024, or 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
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): 10, 12f, 15, 19, 27, 31, 41, 46, 53, 72, 103, 111, 121, 130, 140, 171, 224, 243, 270, 441, …
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.
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 ¢, +3.804 ¢, and +1.428 ¢ 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 171 and 224edo.
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 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, 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
{4096/4095, 4375/4374} (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.
Rank-4 temperaments
{1990656/1990625}, the temperament that tempers out the pointigailimma alone is an unfathomably accurate nanotemperament, due to the extremely tiny size of the pontigailimma.