54edo: Difference between revisions

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== Theory ==

54edo is suitable for usage ~~with~~ [[dual-fifth tuning]] ~~systems~~, or ~~alternately~~, no-fifth tuning ~~systems~~. ~~It has an ultrahard diatonic scale using~~ the sharp fifth of [[27edo]] and an ultrasoft diatonic ~~using the flat fifth~~. ~~The soft diatonic~~ scale is so soft, with L/s = 8/7, that it stops sounding like [[meantone]] or even [[flattone]], but just sounds like a [[circulating temperament]] of [[7edo]].

54edo is suitable for usage as a [[dual-fifth tuning]] system, or alternatively, a no-fifth tuning system. Using the sharp fifth, it can be viewed as two [[ring number|rings]] of [[27edo]], which adds better approximations of the [[11/1|11th]] and [[15/1|15th harmonics]]. Using the flat fifth, it generates an ultrasoft diatonic scale. This scale is so soft, with L/s = 8/7, that it stops sounding like [[meantone]] or even [[flattone]], but just sounds like a [[circulating temperament]] of [[7edo]].

~~It is a rarely used temperament which adds better approximations of the~~ [[~~11/1|11th~~]] ~~and [[15/1|15th harmonics]] from [[~~27edo]], ~~which it doubles~~. It ~~contains an alternate (flat) mapping of the fifth and an "extreme bayati" 6 6 10 10 2 10 10 diatonic scale.~~

The [[patent val]] of this edo takes the same fifth as 27edo, but the mapping for harmonic 5 is different. It tempers out [[2048/2025]] in the 5-limit, making it a [[diaschismic]] system. It is the highest edo in which the best mappings of the major 3rd ([[5/4]]) and harmonic 7th ([[7/4]]), 17\54 and 44\54, are exactly 600 cents apart, making them suitable for harmonies using tritone substitutions. In other words, this is the last edo tempering out [[50/49]]. This means it extends quite simply to the 7- and 11-limit using the [[pajarous]] mapping and to the 13-limit using the 54f val, falling neatly between the 7- and 13-limit [[Target tuning #Minimax tuning|minimax tunings]]. The 54cd val makes for an excellent tuning of 7-limit [[hexe]] temperament, while the 54bdf val does higher limit [[muggles]] about as well as it can be tuned. However, even these best temperament interpretations are quite high in badness compared to its immediate neighbours 53- and 55edo, both of which are historically significant for different reasons, leaving it mostly unexplored so far.

~~Using the patent val, 54edo~~ tempers out [[2048/2025]] in the 5-limit, making it a [[diaschismic]] system. It is the highest [[edo]] in which the best mappings of the major 3rd ([[5/4]]) and harmonic 7th ([[7/4]]), 17\54 and 44\54, are exactly 600 cents apart, making them suitable for harmonies using tritone substitutions. In other words, this is the last edo tempering out [[50/49]]. This means it extends quite simply to the 7- and 11-limit using the [[pajarous]] mapping and to the 13-limit using the 54f val, falling neatly between the 7- and 13-limit [[Target tuning #Minimax tuning|minimax tunings]]. The 54cd val makes for an excellent tuning of 7-limit [[hexe]] temperament, while the 54bdf val does higher limit [[muggles]] about as well as it can be tuned. However, even these best temperament interpretations are quite high in badness compared to its immediate neighbours 53- and 55edo, both of which are historically significant for different reasons, leaving it mostly unexplored so far.

=== Odd harmonics ===

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 == Theory ==
-edo is suitable for usage with [[dual-fifth tuning]] systems, or alternately, no-fifth tuning systems. It has an ultrahard diatonic scale using the sharp fifth of [[27edo]] and an ultrasoft diatonic using the flat fifth. The soft diatonic scale is so soft, with L/s = 8/7, that it stops sounding like [[meantone]] or even [[flattone]], but just sounds like a [[circulating temperament]] of [[7edo]].
+edo is suitable for usage as a [[dual-fifth tuning]] system, or alternatively, a no-fifth tuning system. Using the sharp fifth, it can be viewed as two [[ring number|rings]] of [[27edo]], which adds better approximations of the [[11/1|11th]] and [[15/1|15th harmonics]]. Using the flat fifth, it generates an ultrasoft diatonic scale. This scale is so soft, with L/s = 8/7, that it stops sounding like [[meantone]] or even [[flattone]], but just sounds like a [[circulating temperament]] of [[7edo]].
-It is a rarely used temperament which adds better approximations of the [[11/1|11th]] and [[15/1|15th harmonics]] from [[27edo]], which it doubles. It contains an alternate (flat) mapping of the fifth and an "extreme bayati" 6 6 10 10 2 10 10 diatonic scale.
+The [[patent val]] of this edo takes the same fifth as 27edo, but the mapping for harmonic 5 is different. It tempers out [[2048/2025]] in the 5-limit, making it a [[diaschismic]] system. It is the highest edo in which the best mappings of the major 3rd ([[5/4]]) and harmonic 7th ([[7/4]]), 17\54 and 44\54, are exactly 600 cents apart, making them suitable for harmonies using tritone substitutions. In other words, this is the last edo tempering out [[50/49]]. This means it extends quite simply to the 7- and 11-limit using the [[pajarous]] mapping and to the 13-limit using the 54f val, falling neatly between the 7- and 13-limit [[Target tuning #Minimax tuning|minimax tunings]]. The 54cd val makes for an excellent tuning of 7-limit [[hexe]] temperament, while the 54bdf val does higher limit [[muggles]] about as well as it can be tuned. However, even these best temperament interpretations are quite high in badness compared to its immediate neighbours 53- and 55edo, both of which are historically significant for different reasons, leaving it mostly unexplored so far.
-Using the patent val, 54edo tempers out [[2048/2025]] in the 5-limit, making it a [[diaschismic]] system. It is the highest [[edo]] in which the best mappings of the major 3rd ([[5/4]]) and harmonic 7th ([[7/4]]), 17\54 and 44\54, are exactly 600 cents apart, making them suitable for harmonies using tritone substitutions. In other words, this is the last edo tempering out [[50/49]]. This means it extends quite simply to the 7- and 11-limit using the [[pajarous]] mapping and to the 13-limit using the 54f val, falling neatly between the 7- and 13-limit [[Target tuning #Minimax tuning|minimax tunings]]. The 54cd val makes for an excellent tuning of 7-limit [[hexe]] temperament, while the 54bdf val does higher limit [[muggles]] about as well as it can be tuned. However, even these best temperament interpretations are quite high in badness compared to its immediate neighbours 53- and 55edo, both of which are historically significant for different reasons, leaving it mostly unexplored so far.
 === Odd harmonics ===