2.3.5.7.13 subgroup: Difference between revisions

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== 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): {{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 ===

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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]].

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 ===

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 == 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): {{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 ===
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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]].
-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 ===