Bird's eye view of temperaments by accuracy: Difference between revisions
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== ~17-limit focus == | == ~17-limit focus == | ||
=== [[Buzzard]] === | |||
Note counts: | |||
* 31 for {3, 5, ''7'', 9, 13, 15, ''21'', 27, ''35''(, 81)} ([[5L 28s]]) | |||
* 43 for {3, 5, ''7'', 9, 11, 13, 15, ''21'', 27, 33, ''35'', 39(, 81)} (5L 38s) | |||
Bound-violating intervals: [[7/4]], [[11/7]] and various intervals made with compound intervals of 7 (corresponding odds italicized), which are the simplest (see [[#2.3.7 Buzzard]]) | |||
Its [[S-expression]]-based comma list is {[[1728/1715|S6/S7]], [[5120/5103|S8/S9]]}, with the structure of its 7-limit implied by these equivalences combined with the nontrivial [[JI]] equivalence [[36/35|S6]] = [[64/63|S8]] × [[81/80|S9]]. Tempering out S8/S9 leverages this by splitting [[36/35|S6]] into two syntonic~septimal commas, so buzzard naturally finds an interval between [[6/5]] and [[7/6]] which in the 7-limit is [[32/27]] and in the 13-limit is [[13/11]], while tempering out S6/S7 implies that [[49/48|S7]] is also split into two so that the system also finds an interval between 7/6 and 8/7 which in the 7-limit is 7/6 inflected down by a comma or 8/7 inflected up by a comma, and in the 13-limit is [[15/13]], so that it is clear this system naturally wants to be extended to and interpreted in at least the full 13-limit. Because of the sharp 3 and 5 (in an optimized tuning), [[~]][[16/15]] is tuned quite flat so that a very natural extension to prime 17 exists by equating it with a sharp [[~]][[17/16]] (tempering out [[256/255|S16]]). | |||
[[53edo]] and [[58edo]] are good tunings for the 13-limit; though 58edo is more accurate, 53edo supports a variety of structures that might be preferred over the ones 58edo supports, such as [[#Cata]] (and its best extension to prime 7, [[#Catakleismic]], as well as another more complex extension to prime 7 called [[countercata]]), [[#Schismic]] and especially [[marvel]] (so that it supports [[#Garibaldi]]); by contrast, 58edo may be preferred for supporting [[#Echidna]] and [[#Diaschismic]]. [[111edo]] is a very elegant tuning for higher limits, combining it with [[#Sensible]] and [[#Sendai]] (two extensions of [[#Sensipent]]. | |||
== Higher-limit focus == | == Higher-limit focus == | ||
=== [[Sendai]] === | === [[Sendai]] === | ||