Bird's eye view of temperaments by accuracy: Difference between revisions
m add generator tunings for 2.3.7 buzzard |
add generator tunings for superpyth and mohaha |
||
| Line 330: | Line 330: | ||
* 12 for {3, 5, 7, 9} ([[5L 7s]] and [[5L 12s]]) | * 12 for {3, 5, 7, 9} ([[5L 7s]] and [[5L 12s]]) | ||
Bound-violating intervals: [[9/8]] (and [[8/7]] in flatter tunings like 22edo) | Bound-violating intervals: [[9/8]] (and [[8/7]] in flatter tunings like 22edo) | ||
[[#Generator tunings|Generator tunings]]: 13\22, 16\27, 29\49 | |||
Superpyth is the natural "opposite" of [[#Septimal meantone]] in a surprisingly large number of surprisingly exact senses; the main ones of note are that the [[3/2|fifth]] is mistuned in opposite directions (sharp in superpyth), and that superpyth makes prime 7 most immediately accessible on the chain of fifths with prime 5 requiring more complex movements (when measured in number of fifths), while septimal meantone does the opposite. Superpyth makes the major third [[~]][[9/7]], the minor third [[~]][[7/6]], the major second a blend between a sharp [[~]][[9/8]] and a flat [[~]][[8/7]] and the minor second [[~]][[28/27]], which is tuned very flat so that it becomes a quarter-tone in any good tuning of superpyth. Superpyth finds [[~]][[5/4]] as the augmented second and [[~]][[6/5]] correspondingly as the diminished fourth. | Superpyth is the natural "opposite" of [[#Septimal meantone]] in a surprisingly large number of surprisingly exact senses; the main ones of note are that the [[3/2|fifth]] is mistuned in opposite directions (sharp in superpyth), and that superpyth makes prime 7 most immediately accessible on the chain of fifths with prime 5 requiring more complex movements (when measured in number of fifths), while septimal meantone does the opposite. Superpyth makes the major third [[~]][[9/7]], the minor third [[~]][[7/6]], the major second a blend between a sharp [[~]][[9/8]] and a flat [[~]][[8/7]] and the minor second [[~]][[28/27]], which is tuned very flat so that it becomes a quarter-tone in any good tuning of superpyth. Superpyth finds [[~]][[5/4]] as the augmented second and [[~]][[6/5]] correspondingly as the diminished fourth. | ||
| Line 340: | Line 342: | ||
* 9 for {3, 5, 9, 11(, 33, 35)} ([[7L 3s]], note odd 35 comes from the [[mohajira]] mapping of 7 specifically) | * 9 for {3, 5, 9, 11(, 33, 35)} ([[7L 3s]], note odd 35 comes from the [[mohajira]] mapping of 7 specifically) | ||
* 20 or 23 for adding {7, 15, 21(, 35)} ([[7L 10s]] and [[7L 17s]]) | * 20 or 23 for adding {7, 15, 21(, 35)} ([[7L 10s]] and [[7L 17s]]) | ||
[[#Generator tunings|Generator tunings]]: 9\31, 16\55 | |||
Mohaha is a 2.3.5.11 (no-7's [[11-limit]]) "hemi-meantone" temperament that splits [[#Meantone]]'s fifth into two [[~]][[11/9]]'s by tempering out [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], which as [[81/80|S9 = (9/8)/(10/9)]] is tempered implies tempering out [[121/120|S11 = (11/10)/(12/11) = (11/8)/(15/11)]] as well. It has two main extensions to the full 11-limit; if you accept the [[#Septimal meantone]] mapping of 7 you get [[migration]], but maybe more natural is if you instead equate the flat [[~]][[33/32]] interval with S6 = [[36/35]] = ([[6/5]])/([[7/6]]), which results in [[mohajira]], which finds 7 at a negative number of gens so that composite harmonics of 7 are simpler to find (as primes 3, 5 and 11 are all found at a positive number of gens). Because of this, mohajira is usually the preferred extension as it is more note-efficient, but both extensions merge in [[31edo]], which is a good tuning for both. | Mohaha is a 2.3.5.11 (no-7's [[11-limit]]) "hemi-meantone" temperament that splits [[#Meantone]]'s fifth into two [[~]][[11/9]]'s by tempering out [[243/242|S9/S11 = (12/8)/(11/9)<sup>2</sup> = (3/2)/(11/9)<sup>2</sup>]], which as [[81/80|S9 = (9/8)/(10/9)]] is tempered implies tempering out [[121/120|S11 = (11/10)/(12/11) = (11/8)/(15/11)]] as well. It has two main extensions to the full 11-limit; if you accept the [[#Septimal meantone]] mapping of 7 you get [[migration]], but maybe more natural is if you instead equate the flat [[~]][[33/32]] interval with S6 = [[36/35]] = ([[6/5]])/([[7/6]]), which results in [[mohajira]], which finds 7 at a negative number of gens so that composite harmonics of 7 are simpler to find (as primes 3, 5 and 11 are all found at a positive number of gens). Because of this, mohajira is usually the preferred extension as it is more note-efficient, but both extensions merge in [[31edo]], which is a good tuning for both. | ||
| Line 349: | Line 353: | ||
== No-3's focus == | == No-3's focus == | ||
== No-5's focus == | == No-5's focus == | ||
= '''Very low accuracy (<~18c)''' = | = '''Very low accuracy (<~18c)''' = | ||