Bird's eye view of temperaments by accuracy: Difference between revisions
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=== [[Schismic]] === | === [[Schismic]] === | ||
Note count: 12 for {3, 5, 9, 15, 27, 45(, 81)} ([[5L 7s]] or [[12L 5s]]) | Note count: 12 for {3, 5, 9, 15, 27, 45(, 81)} ([[5L 7s]] or [[12L 5s]]) | ||
[[#Generator tunings|Generator tunings]]: 100\171, 131\224 | |||
Schismic is a very accurate and efficient [[5-limit]] temperament which is almost identical to [[Pythagorean tuning]] except that it tempers the perfect fifth very slightly flat so as to find [[8/5]] accurately at ([[9/8]])<sup>4</sup>, that is, as the [[Pythagorean augmented fifth]], or equivalently, finding [[5/4]] as the [[Pythagorean diminished fourth]]. Note that the smallest edo that validates its status as a microtemperament is [[118edo]], as [[53edo]], though a tone-efficient tuning, doesn't temper the fifth flat enough, being approximately the [[Pythagorean tuning]] of schismic. In schismic, (9/8)<sup>6</sup> overshoots the octave by [[~]][[81/80]] so that the syntonic comma and the [[Pythagorean comma]] are equated. | Schismic is a very accurate and efficient [[5-limit]] temperament which is almost identical to [[Pythagorean tuning]] except that it tempers the perfect fifth very slightly flat so as to find [[8/5]] accurately at ([[9/8]])<sup>4</sup>, that is, as the [[Pythagorean augmented fifth]], or equivalently, finding [[5/4]] as the [[Pythagorean diminished fourth]]. Note that the smallest edo that validates its status as a microtemperament is [[118edo]], as [[53edo]], though a tone-efficient tuning, doesn't temper the fifth flat enough, being approximately the [[Pythagorean tuning]] of schismic. In schismic, (9/8)<sup>6</sup> overshoots the octave by [[~]][[81/80]] so that the syntonic comma and the [[Pythagorean comma]] are equated. | ||
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Semaphore and bug are quite similar in that they have semi-twelfths (or semi-fourths). However, unlike bug, semaphore's semi-twelfths have a ratio of 12/7~7/4. | Semaphore and bug are quite similar in that they have semi-twelfths (or semi-fourths). However, unlike bug, semaphore's semi-twelfths have a ratio of 12/7~7/4. | ||
= Generator | = Generator tunings = | ||
Wherever you see "Generator tunings:" on this page, the tunings are given in the format ''a''\''b'', which means ''a'' steps of ''b'' [[edo]], which means the frequency ratio 2<sup>''a''/''b''</sup>. | Wherever you see "Generator tunings:" on this page, the tunings are given in the format ''a''\''b'', which means ''a'' steps of ''b'' [[edo]], which means the frequency ratio 2<sup>''a''/''b''</sup>. | ||