Bird's eye view of temperaments by accuracy: Difference between revisions
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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. | ||
=== [[Orwell]] === | === [[Orwell|Undecimal Orwell]] === | ||
[[#Note counts|Note count]]: 22 for {3, 5, 7, 9, 11, 15, 21, 25, 33, 35, 45, 55, 75, 77} ([[9L 13s]]) | [[#Note counts|Note count]]: 22 for {3, 5, 7, 9, 11, 15, 21, 25, 33, 35, 45, 55, 75, 77} ([[9L 13s]]) | ||
[[#Generator tunings|Generator tunings]]: 5\22, 7\31, 12\53, 19\84 | [[#Generator tunings|Generator tunings]]: 5\22, 7\31, 12\53, 19\84 | ||
Orwell is | Orwell is a tempermant generated by a slightly sharp subminor third representing [[7/6]], seven of which reach [[6/1]]. Three subminor thirds reach [[8/5]], and two reach [[11/8]]. This temperament is arguably one of the simplest 11-limit temperaments with decent accuracy, and is supported by the highly notable edos {{edos|22, 31, and 53.}} This temperament is more accurate in lower limits, very nearly being a microtemperament in the 5-limit (5-odd-limit minimax error 1.006{{c}}!), and quite good accuracy in the 7-limit, with the least accurate prime being 11, so this temperament can only barely be considered "low accuracy". However, the mapping for 11 is very simple, so this temperament really comes into its own in the 11-limit. Important commas tempered out by orwell include [[99/98]], [[121/120]], [[176/175]], [[225/224]], [[385/384]], [[540/539]], and [[1728/1715]]. | ||
== ~17-limit focus == | == ~17-limit focus == | ||