54edo: Difference between revisions

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It's a rare temperament which adds better approximations of the 11th and 15th harmonics from [[27edo]], which it doubles. 54edo contains an alternate (flat) mapping of the fifth and an "extreme bayati" 6 6 10 10 2 10 10 diatonic scale.

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]]. The 54cd val makes for an excellent tuning of 7-limit [[Augmented_family#Hexe|hexe temperament]], while the bdf val does higher limit [[Magic family #Muggles|muggles]] about as well as it can be tuned.

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 limits using the [[pajarous]] mapping and to the 13-limit using the f 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 [[Augmented_family#Hexe|hexe temperament]], while the bdf val does higher limit [[Magic family #Muggles|muggles]] about as well as it can be tuned. However, even these best temperament interpretations are quite high in badness compared to it's immediate neighbours 53 & 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.~~

~~The immediate close presence of [[53edo]] obscures 54edo and puts this temperament out of popular usage.~~

===Odd harmonics===

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 It's a rare temperament which adds better approximations of the 11th and 15th harmonics from [[27edo]], which it doubles. 54edo contains an alternate (flat) mapping of the fifth and an "extreme bayati" 6 6 10 10 2 10 10 diatonic scale.
-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]]. The 54cd val makes for an excellent tuning of 7-limit [[Augmented_family#Hexe|hexe temperament]], while the bdf val does higher limit [[Magic family #Muggles|muggles]] about as well as it can be tuned.
+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 limits using the [[pajarous]] mapping and to the 13-limit using the f 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 [[Augmented_family#Hexe|hexe temperament]], while the bdf val does higher limit [[Magic family #Muggles|muggles]] about as well as it can be tuned. However, even these best temperament interpretations are quite high in badness compared to it's immediate neighbours 53 & 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.
-The immediate close presence of [[53edo]] obscures 54edo and puts this temperament out of popular usage.
 ===Odd harmonics===
 {{harmonics in equal|54}}