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 [[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 | The 54cdd 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. | 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. | ||
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=== Ups and downs notation === | === Ups and downs notation === | ||
54edo can be notated with [[ups and downs]], spoken as up, dup, trup, quup (or downquip), dudsharp, downsharp, sharp, upsharp, etc. and down, dud, trud, quud (or upquid), dupflat, etc. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and quupflat (quadruple-up flat) is equivalent to quid (quintuple-down). | 54edo can be notated with [[ups and downs]], spoken as up, dup, trup, quup (or downquip), dudsharp, downsharp, sharp, upsharp, etc. and down, dud, trud, quud (or upquid), dupflat, etc. Note that quudsharp (quadruple-down sharp) is equivalent to quip (quintuple-up) and quupflat (quadruple-up flat) is equivalent to quid (quintuple-down). | ||
{{Ups and downs sharpness | {{Ups and downs sharpness}} | ||
It can also be notated by borrowing [[Helmholtz–Ellis]] accidentals: | It can also be notated by borrowing [[Helmholtz–Ellis]] accidentals: | ||