2.3.5.7.13 subgroup: Difference between revisions

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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.

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=== 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.

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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.

~~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 ===

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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.
@@ Line 12: / Line 12: @@
 === 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.
@@ Line 20: / Line 20: @@
 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 ===