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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|rational intervals]] where [[2/1|2]], [[3/1|3]], [[5/1|5]], [[7/1|7]], and [[13/1|13]] are the only allowable [[Prime factor|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.
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 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 [[schismina]], an unnoticeable, tiny pife-like 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 '''way''' 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 ==
=== 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>*</nowiki>very accurate 7-limit but relatively innacurate prime 13.


=== 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.


[[Cassandra|No-11 cassandra]] provides a more complex temperament using a [[chain of fifths]], well represented with [[41edo|41-]] 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 doubly diminished octave 8388608/4782969 and the triple augmented fourth (+20 fifths) 3486784401/2147483648 already sound very close to 5/4 and 7/4 and 13/8 respectively, though 7/4 not so much. 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]].


For those searching higher accuracy temperaments, [[Wizmic microtemperaments#Gariwizmic|Gariwizmic]] also keeps the chain of fifths, spliting the octave in half, but does not temper the schisma. It finds 5/4 at 39 fifths - semioctave, 7/4 at -14 fifths, and 13/8 at -27 fifths + 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.


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.
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 ===
{4375/4374, 4096/4095} (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.


{[[1990656/1990625]], [[140625/140608]]}, the temperament that tempers 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.


=== Rank-4 temperaments ===
=== Rank-4 temperaments ===
{[[1990656/1990625]]}, the temperament that tempers 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.
 
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[[Category:Rank-5 temperaments|#]]
[[Category:13-limit|#]]
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