54edo: Difference between revisions
Prime table, a little more theory. |
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The immediate close presence of [[53edo]] obscures 54edo and puts this temperament out of popular usage. | The immediate close presence of [[53edo]] obscures 54edo and puts this temperament out of popular usage. | ||
{{primes in edo|54}} | |||
== Theory == | == Theory == | ||
54edo is suitable for usage with [[dual-fifth tuning]] systems, or alternately, no-fifth tuning systems. | 54edo is suitable for usage with [[dual-fifth tuning]] systems, or alternately, no-fifth tuning systems. | ||
It's a rare temperament which adds approximations of the 11th and 15th harmonics. | 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. 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. | |||
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. The 54cd val makes for an excellent tuning of 7-limit [[Augmented_family#Hexe|hexe temperament]]. | |||
{| class="wikitable" | {| class="wikitable" | ||
|+Table of intervals | |+Table of intervals | ||
Revision as of 16:08, 24 November 2021
54-EDO is an equal temperament that divides the octave into 54 equal parts, each 22.2222 cents in size.
The immediate close presence of 53edo obscures 54edo and puts this temperament out of popular usage.
Script error: No such module "primes_in_edo".
Theory
54edo is suitable for usage with dual-fifth tuning systems, or alternately, no-fifth tuning systems.
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. The 54cd val makes for an excellent tuning of 7-limit hexe temperament, while the bdf val does higher limit muggles about as well as it can be tuned.
| Degree | Name | Cents | Approximate Ratios |
|---|---|---|---|
| 0 | Natural Unison | 0.000 | |
| 1 | Ninth-tone | 22.222 | |
| 2 | Extreme bayati quarter-tone | 44.444 | |
| 3 | Third-tone | 66.666 | |
| 4 | 88.888 | ||
| 5 | 111.111 | ||
| 6 | Extreme bayati neutral second | 133.333 | |
| 7 | 155.555 | ||
| 8 | Minor whole tone | 177.777 | 10/9 |
| 9 | Symmetric whole tone | 200.000 | 9/8 |
| 10 | Extreme bayati whole tone | 222.222 | |
| 12 | Septimal submajor third | 266.666 | 7/6 |
| 17 | Classical major third | 377.777 | 5/4 |
| 18 | Symmetric major third | 400.000 | 29/23, 19/16 |
| 25 | Undecimal superfourth | 555.555 | 11/8 |
| 26 | Septimal minor tritone | 577.777 | 7/5 |
| 27 | Symmetric tritone | 600.000 | |
| 28 | Septimal major tritone | 633.333 | 10/7 |
| 36 | Symmetric augmented fifth | 800.000 | |
| 44 | Harmonic seventh | 977.777 | 7/4 |
| 54 | Octave | 1200.000 | Exact 2/1 |