2.3.5.7.11.13.19.29 subgroup: Difference between revisions

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[[Category:Just intonation subgroups|#]]

[[Category:29-limit|#]]

=== Rank-2 temperaments ===

[[Hemififths]], through the 41 & 58h extension, provides a fairly simple but efficient way to approach the subgroup through a dicot structure, finding ~19/16 and ~29/16 to be a pythagorean comma above ~7/6 and ~16/9 respectively, 29/16 here is approximated to within a cent of accuracy.

[[Newt]] is arguably the most efficient rank-2 microtemperament of the subgroup despite its apparent great complexity, and much like hemififths, has a dicot structure. It has a strong extension of [[Garischismic clan#2.3.5.7.11.13.19 subgroup (neonewt)|neonewt]] (itself an extension of newt) for ~29/16. though its mapping is proportionally less accurate than the rest, being a cent off instead of tenths of a cent off.

[[Cassandra]] provides a chain-of-fifths framework for approaching the subgroup, and has a very strong extension for 19/16 as a minor third, but has three plausible extensions for ~29/16:

* 41 & 53 is the simplest, lowest-badness but least accurate, which simply equates it with ~9/5.

* 41 & 94 is more accurate and complex, which equates it with ~20/11.

* 53 & 94 is is the most accurate but most complex, which equates it with a stack of ~169/128 and ~11/8.

=== Rank-3 temperaments ===

[[Breed family#Freyr|Freyr]] detempers the hemififths extension and massively improves upon all primes in the subgroup, finding ~19/16 an aberschisma sharper except for ~29/16 which has the same mapping as hemififths but tuned less accurately because the fifth is closer to pure.

[[Cassaschismic]] detempers cassandra by readily including 19/16 as a minor third plus an aberschisma with incredible accuracy. Coincidentally like cassandra, it has three plausible extensions for ~29/16:

* 41 & 94 & 270 is the simplest, with the same exact mapping as 41 & 94 but less accurately tuned because the fifth is sharper.

* 41 & 53 & 270 is the lowest-badness, finding ~29/16 half a cent off as 2 aberschismas sharper than ~9/5 half a cent off pure.

* 53 & 94 & 217 is the worst extension with the most complex mapping of ~29/16 with close to no increase in accuracy from the others, but its mapping can be detempered further down to thousands of a cent of error precision with [[insatinismic]].

@@ Line 13: / Line 13: @@
 [[Category:Just intonation subgroups|#]]
 [[Category:29-limit|#]]
+=== Rank-2 temperaments ===
+[[Hemififths]], through the 41 & 58h extension, provides a fairly simple but efficient way to approach the subgroup through a dicot structure, finding ~19/16 and ~29/16 to be a pythagorean comma above ~7/6 and ~16/9 respectively, 29/16 here is approximated to within a cent of accuracy.
+[[Newt]] is arguably the most efficient rank-2 microtemperament of the subgroup despite its apparent great complexity, and much like hemififths, has a dicot structure. It has a strong extension of [[Garischismic clan#2.3.5.7.11.13.19 subgroup (neonewt)|neonewt]] (itself an extension of newt) for ~29/16. though its mapping is proportionally less accurate than the rest, being a cent off instead of tenths of a cent off.
+[[Cassandra]] provides a chain-of-fifths framework for approaching the subgroup, and has a very strong extension for 19/16 as a minor third, but has three plausible extensions for ~29/16:
+* 41 & 53 is the simplest, lowest-badness but least accurate, which simply equates it with ~9/5.
+* 41 & 94 is more accurate and complex, which equates it with ~20/11.
+* 53 & 94 is is the most accurate but most complex, which equates it with a stack of ~169/128 and ~11/8.
+=== Rank-3 temperaments ===
+[[Breed family#Freyr|Freyr]] detempers the hemififths extension and massively improves upon all primes in the subgroup, finding ~19/16 an aberschisma sharper except for ~29/16 which has the same mapping as hemififths but tuned less accurately because the fifth is closer to pure.
+[[Cassaschismic]] detempers cassandra by readily including 19/16 as a minor third plus an aberschisma with incredible accuracy. Coincidentally like cassandra, it has three plausible extensions for ~29/16:
+* 41 & 94 & 270 is the simplest, with the same exact mapping as 41 & 94 but less accurately tuned because the fifth is sharper.
+* 41 & 53 & 270 is the lowest-badness, finding ~29/16 half a cent off as 2 aberschismas sharper than ~9/5 half a cent off pure.
+* 53 & 94 & 217 is the worst extension with the most complex mapping of ~29/16 with close to no increase in accuracy from the others, but its mapping can be detempered further down to thousands of a cent of error precision with [[insatinismic]].
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