2.3.5.7.13 subgroup: Difference between revisions

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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]].

The subgroup can be very easily rank-reduced into the 7-limit through the [[~~schismina~~]], 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.

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 ==

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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]]. 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.

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 schismina, equating the [[36/35]] generator to the [[1053/1024]]. The pontigailimma is by extension tempered out too.

=== 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 ~~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.

{[[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.

@@ Line 3: / Line 3: @@
 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]].
-The subgroup can be very easily rank-reduced into the 7-limit through the [[schismina]], 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.
+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 ==
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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]]. 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.
+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 schismina, equating the [[36/35]] generator to the [[1053/1024]]. The pontigailimma is by extension tempered out too.
 === 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 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.
 {[[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.