2.3.5.7.13 subgroup: Difference between revisions

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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.

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 ~~doubly~~ 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 ~~relabelling~~ 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.

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 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.

=== Rank-3 temperaments ===

{4375/4374~~, 4096/4095~~} ({{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 schismina, 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 ===

{[[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.

[[Category:Just intonation subgroups]]

[[Category:Rank-5 temperaments]]

@@ Line 14: / Line 14: @@
 [[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 doubly 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 relabelling 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.
 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 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.
 === Rank-3 temperaments ===
-{4375/4374, 4096/4095} ({{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 schismina, 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 ===
-{[[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.
+[[Category:Just intonation subgroups]]
+[[Category:Rank-5 temperaments]]