54edo: Difference between revisions

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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{{c}} 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 of 54edo are quite high in [[badness]] compared to its immediate neighbours [[53edo|53-]] and [[55edo]], both of which are [[Historical temperaments|historically significant]] for different reasons, leaving it mostly unexplored so far.

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. The overall best val of 54edo in the 17-limit is probably 54c, which preserves the 2.3.5.7.13 mapping of 27edo and corrects the 11th and 17th harmonics with a consistently sharp tendency.

However, even these best temperament interpretations of 54edo are quite high in [[badness]] compared to its immediate neighbours [[53edo|53-]] and [[55edo]], both of which are [[Historical temperaments|historically significant]] for different reasons, leaving it mostly unexplored so far.

=== Odd harmonics ===

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 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{{c}} 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 of 54edo are quite high in [[badness]] compared to its immediate neighbours [[53edo|53-]] and [[55edo]], both of which are [[Historical temperaments|historically significant]] for different reasons, leaving it mostly unexplored so far.
+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. The overall best val of 54edo in the 17-limit is probably 54c, which preserves the 2.3.5.7.13 mapping of 27edo and corrects the 11th and 17th harmonics with a consistently sharp tendency.
+However, even these best temperament interpretations of 54edo are quite high in [[badness]] compared to its immediate neighbours [[53edo|53-]] and [[55edo]], both of which are [[Historical temperaments|historically significant]] for different reasons, leaving it mostly unexplored so far.
 === Odd harmonics ===