User:BudjarnLambeth/Sandbox2: Difference between revisions
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== Octave stretch or compression == | == Octave stretch or compression == | ||
{{main|23edo and octave stretching}} | |||
; [[zpi| | 23edo is not typically taken seriously as a tuning except by those interested in extreme [[xenharmony]]. Its fifths are significantly flat, and is neighbors [[22edo]] and [[24edo]] generally get more attention. | ||
* Step size: | |||
However, when using a slightly [[stretched tuning|stretched octave]] of around 1216 [[cents]], 23edo looks much better, and it approximates the [[perfect fifth]] (and various other [[interval]]s involving the 5th, 7th, 11th, and 13th [[harmonic]]s) to within 18 cents or so. If we can tolerate errors around this size in [[12edo]], we can probably tolerate them in stretched-23 as well. | |||
Stretched 23edo is one of the best tunings to use for exploring the [[antidiatonic]] scale (and its 9-note extension, the [[superantidiatonic]] scale), since its fifth is more [[consonant]] and less "[[Wolf interval|wolfish]]" than fifths in other [[pelogic family]] temperaments. | |||
What follows is a comparison of stretched- and compressed-octave 23edo tunings. | |||
; [[zpi|86zpi]] | |||
* Step size: 51.653{{c}}, octave size: NNN{{c}} | |||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this. | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this. | ||
{{Harmonics in cet| | {{Harmonics in cet|51.653|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}} | ||
{{Harmonics in cet| | {{Harmonics in cet|51.653|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}} | ||
; [[ | ; [[60ed6]] | ||
* Step size: NNN{{c}}, octave size: NNN{{c}} | * Step size: NNN{{c}}, octave size: NNN{{c}} | ||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this. | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this. | ||
{{Harmonics in equal| | {{Harmonics in equal|60|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} | ||
{{Harmonics in equal| | {{Harmonics in equal|60|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}} | ||
; [[ | ; [[zpi|85zpi]] | ||
* Step size: | * Step size: 52.114{{c}}, octave size: NNN{{c}} | ||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this. | ||
{{Harmonics in cet| | {{Harmonics in cet|52.114|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}} | ||
{{Harmonics in cet| | {{Harmonics in cet|52.114|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}} | ||
; | ; 23edo | ||
* Step size: NNN{{c}}, octave size: NNN{{c}} | * Step size: NNN{{c}}, octave size: NNN{{c}} | ||
Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}. | Pure-octaves EDONAME approximates all harmonics up to 16 within NNN{{c}}. | ||
{{Harmonics in equal| | {{Harmonics in equal|23|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONAME}} | ||
{{Harmonics in equal| | {{Harmonics in equal|23|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONAME (continued)}} | ||
; [[WE|ETNAME, SUBGROUP WE tuning]] | ; [[WE|23et, 13-limit WE tuning]] | ||
* Step size: | * Step size: 53.237{{c}}, octave size: NNN{{c}} | ||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this. | |||
{{Harmonics in cet|53.237|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}} | |||
{{Harmonics in cet|53.237|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}} | |||
; [[WE|23et, 2.3.5.13 WE tuning]] | |||
* Step size: 53.447{{c}}, octave size: NNN{{c}} | |||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this. | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. Its SUBGROUP WE tuning and SUBGROUP [[TE]] tuning both do this. | ||
{{Harmonics in cet| | {{Harmonics in cet|53.447|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning}} | ||
{{Harmonics in cet| | {{Harmonics in cet|53.447|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ETNAME, SUBGROUP WE tuning (continued)}} | ||
; [[ | ; [[59ed6]] | ||
* Step size: NNN{{c}}, octave size: NNN{{c}} | * Step size: NNN{{c}}, octave size: NNN{{c}} | ||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this. | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this. | ||
{{Harmonics in equal| | {{Harmonics in equal|59|6|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} | ||
{{Harmonics in equal| | {{Harmonics in equal|59|6|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}} | ||
; [[zpi|ZPINAME]] | ; [[zpi|84zpi]] | ||
* Step size: 52.615{{c}}, octave size: NNN{{c}} | |||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning ZPINAME does this. | |||
{{Harmonics in cet|52.615|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in ZPINAME}} | |||
{{Harmonics in cet|52.615|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in ZPINAME (continued)}} | |||
; [[36edt]] | |||
* Step size: NNN{{c}}, octave size: NNN{{c}} | * Step size: NNN{{c}}, octave size: NNN{{c}} | ||
_ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning | _ing the octave of EDONAME by around NNN{{c}} results in improved primes NNN, but worse primes NNN. This approximates all harmonics up to 16 within NNN{{c}}. The tuning EDONOI does this. | ||
{{Harmonics in | {{Harmonics in equal|36|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in EDONOI}} | ||
{{Harmonics in | {{Harmonics in equal|36|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in EDONOI (continued)}} | ||
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