81edo: Difference between revisions

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{{Infobox ET
{{Infobox ET}}
| Prime factorization = 3<sup>4</sup>
| Step size = 14.815¢
| Fifth = 47\81 = 696.30¢
| Major 2nd = 13\81 = 192.59¢
| Semitones = 5:8 (74.1¢:118.5¢)
| Consistency = 7
}}


[[File:81 EDO structure-accidentals-notation-colour notation.png|thumb|
[[File:81 EDO structure-accidentals-notation-colour notation.png|thumb|

Revision as of 19:12, 4 October 2022

← 80edo 81edo 82edo →
Prime factorization 34
Step size 14.8148 ¢ 
Fifth 47\81 (696.296 ¢)
Semitones (A1:m2) 5:8 (74.07 ¢ : 118.5 ¢)
Dual sharp fifth 48\81 (711.111 ¢) (→ 16\27)
Dual flat fifth 47\81 (696.296 ¢)
Dual major 2nd 14\81 (207.407 ¢)
Consistency limit 7
Distinct consistency limit 7
By Tom Winspear, utilizing the Accidents shown below. Left: Chain of 4ths/5ths , Right: Chromatic view. Black font represents the '6 accidentals deep' notation that covers the chromatic scale with enharmonics only across EF & BC. White text displays deep enharmonics in the ambiguous infrared & ultraviolet area of the colour notation.

81edo divides the octave into 81 steps of 14.815 cents each. It is notable as a tuning for meantone and related temperaments and is the optimal patent val for a number of them. In particular it is the optimal patent val for 5-limit meantone, 7-limit meantone, 11-limit meanpop, 13-limit meanpop, and the rank three temperament erato. The electronic music pioneer Daphne Oram was interested in 81edo.

As a step in the Golden meantone series of EDOs, 81 EDO marks the point at which the series ceases to display audible changes to meantone temperament, and is also the EDO with the lowest average and most evenly spread Just-error across the scale (though 31 EDO does have the best harmonic 7th).

81 EDO Accidentals created and used by Tom Winspear. Based on those provided in Scala though with a logic correction. The innermost accidentals represent the 15 cent EDOstep, followed by two, then the bracket representing three. Conventional sharp/doublesharp/flat/doubleflat accidentals are reached in steps of five and the pattern repeats itself on them. The chromatic scale can be notated utilizing only six accidentals in either direction - the rest are for enharmonics.

Script error: No such module "primes_in_edo".

Comma list

  • 5-limit commas: 81/80, [-48 1 20⟩
  • 7-limit commas: 81/80, 126/125, [-24 1 0 8⟩
  • 11-limit commas: 81/80, 126/125, 385/384, 12005/11979
  • 13-limit commas: 81/80, 105/104, 144/143, 196/195, 6655/6591

Scales

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