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. 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.

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.

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== Notation ==

=== Ups and downs notation ===

~~Using~~ [[~~Helmholtz–Ellis~~]] ~~accidentals~~, ~~54edo~~ can also be notated ~~using~~ [[~~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).

It can also be notated by borrowing [[Helmholtz–Ellis]] accidentals:

Here, a sharp raises by eight steps, and a flat lowers by eight steps, so single, double, and triple arrows along with Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals can be used to fill in the gap.

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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. 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.
+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.
@@ Line 308: / Line 308: @@
 == Notation ==
 === Ups and downs notation ===
-Using [[Helmholtz–Ellis]] accidentals, 54edo can also be notated using [[ups and downs notation]]:
+edo 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}}
+It can also be notated by borrowing [[Helmholtz–Ellis]] accidentals:
 {{Sharpness-sharp8}}
 Here, a sharp raises by eight steps, and a flat lowers by eight steps, so single, double, and triple arrows along with Stein–Zimmerman [[24edo#Notation|quarter-tone]] accidentals can be used to fill in the gap.