User:BudjarnLambeth/12edo as a 2.3.5.17.19 tuning: Difference between revisions
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# If 3 is approximated within 15 cents, add 3 to the subgroup | # If 3 is approximated within 15 cents, add 3 to the subgroup | ||
## If it is not, then add the smallest multiple of 3, 60 or lower, it approximates within 15 cents (if any) | ## If it is not, then add the smallest multiple of 3, 60 or lower, it approximates within 15 cents (if any) | ||
## | ## Optionally, add the second-smallest multiple of 3, 60 or lower, it approximates within 15 cents, which is not a multiple of the previous one (if any) | ||
## Optionally, add the third-smallest multiple of 3, 60 or lower, it approximates within 15 cents, which is not a multiple of the previous ones (if any) | |||
# If 5 is approximated within 15 cents, add 5 to the subgroup | # If 5 is approximated within 15 cents, add 5 to the subgroup | ||
## If it is not, then add the smallest multiple of 5, 60 or lower, which it approximates within 15 cents (if any) | ## If it is not, then add the smallest multiple of 5, 60 or lower, which it approximates within 15 cents (if any) | ||
## If this is the same as a multiple of already added, keep that one | ## If this is the same as a multiple of already added, just keep that one | ||
# If 7 is approximated within 15 cents, add 7 to the subgroup | # If 7 is approximated within 15 cents, add 7 to the subgroup | ||
## If it is not, then add the smallest multiple of 7, 56 or lower, which it approximates within 15 cents (if any) | ## If it is not, then add the smallest multiple of 7, 56 or lower, which it approximates within 15 cents (if any) | ||
## If this is the same as a multiple of already added, keep that one | ## If this is the same as a multiple of already added, just keep that one | ||
# If 11 is approximated within 15 cents, add 11 to the subgroup | # If 11 is approximated within 15 cents, add 11 to the subgroup | ||
# If there are still spots free, and 13 is approximated within 7 cents, add 13 to the subgroup | # If there are still spots free, and 13 is approximated within 7 cents, add 13 to the subgroup | ||
# If there are still spots free, then add the smallest multiple of 11, 77 or lower, approximated within 15 cents to the subgroup (if any) | # If there are still spots free, then add the smallest multiple of 11, 77 or lower, approximated within 15 cents to the subgroup (if any) | ||
## If this is the same as a multiple of already added, keep that one | ## If this is the same as a multiple of already added, just keep that one | ||
# If any composite basis elements now in the subgroup share no common factors with any other element in the subgroup, remove them | |||
# If any primes 13, 17, 19 or 23 are approximated within 15 cents, include as many of those as there are basis element spots free (giving preference to harmonics with closer approximations first) | # If any primes 13, 17, 19 or 23 are approximated within 15 cents, include as many of those as there are basis element spots free (giving preference to harmonics with closer approximations first) | ||
# If there are aren't enough of those to fill all spots, fill the remaining spots with taxicab-2 intervals the edo approximates within 15 cents, giving preference to intervals with small primes | # If there are aren't enough of those to fill all spots, fill the remaining spots with taxicab-2 intervals the edo approximates within 15 cents, giving preference to intervals with small primes | ||
# Optionally, replace any one basis element with any composite harmonic 60 or smaller, that shares factors in common with at least 2 other basis elements, is approximated within 15 cents, and has not yet been added | |||
=== EDOs with 13 to 27 tones/octave === | === EDOs with 13 to 27 tones/octave === | ||
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# If 3 is approximated within 15 cents, add 3 to the subgroup | # If 3 is approximated within 15 cents, add 3 to the subgroup | ||
## If it is not, then add the smallest multiple of 3, 60 or lower, it approximates within 15 cents (if any) | ## If it is not, then add the smallest multiple of 3, 60 or lower, it approximates within 15 cents (if any) | ||
## | ## Optionally, add the second-smallest multiple of 3, 60 or lower, it approximates within 15 cents, which is not a multiple of the previous one (if any) | ||
## Optionally, add the third-smallest multiple of 3, 60 or lower, it approximates within 15 cents, which is not a multiple of the previous ones (if any) | |||
# If 5 is approximated within 15 cents, add 5 to the subgroup | # If 5 is approximated within 15 cents, add 5 to the subgroup | ||
## If it is not, then add the smallest multiple of 5, 60 or lower, which it approximates within 15 cents (if any) | ## If it is not, then add the smallest multiple of 5, 60 or lower, which it approximates within 15 cents (if any) | ||
## If this is the same as a multiple of already added, keep that one | ## If this is the same as a multiple of already added, just keep that one | ||
# If 7 is approximated within 15 cents, add 7 to the subgroup | # If 7 is approximated within 15 cents, add 7 to the subgroup | ||
## If it is not, then add the smallest multiple of 7, 56 or lower, which it approximates within 15 cents (if any) | ## If it is not, then add the smallest multiple of 7, 56 or lower, which it approximates within 15 cents (if any) | ||
## If this is the same as a multiple of already added, keep that one | ## If this is the same as a multiple of already added, just keep that one | ||
# If 11 is approximated within 15 cents, add 11 to the subgroup | # If 11 is approximated within 15 cents, add 11 to the subgroup | ||
# If there are still spots free, and 13 is approximated within 7 cents, add 13 to the subgroup | # If there are still spots free, and 13 is approximated within 7 cents, add 13 to the subgroup | ||
# If there are still spots free, then add the smallest multiple of 11, 77 or lower, approximated within 15 cents to the subgroup (if any) | # If there are still spots free, then add the smallest multiple of 11, 77 or lower, approximated within 15 cents to the subgroup (if any) | ||
## If this is the same as a multiple of already added, keep that one | ## If this is the same as a multiple of already added, just keep that one | ||
# If any composite basis elements now in the subgroup share no common factors with any other element in the subgroup, remove them | |||
# If any primes 13, 17, 19 or 23 are approximated within 15 cents, include as many of those as there are basis element spots free (giving preference to harmonics with closer approximations first) | # If any primes 13, 17, 19 or 23 are approximated within 15 cents, include as many of those as there are basis element spots free (giving preference to harmonics with closer approximations first) | ||
# If there are aren't enough of those to fill all spots, fill the remaining spots with taxicab-2 intervals the edo approximates within 15 cents, giving preference to intervals with small primes | # If there are aren't enough of those to fill all spots, fill the remaining spots with taxicab-2 intervals the edo approximates within 15 cents, giving preference to intervals with small primes | ||
# Optionally, replace any one basis element with any composite harmonic 60 or smaller, that shares factors in common with at least 2 other basis elements, is approximated within 15 cents, and has not yet been added | |||
=== EDOs with 28 to 52 tones/octave === | === EDOs with 28 to 52 tones/octave === | ||
# The subgroup should have | # The subgroup should have 6 basis elements | ||
# Primes 2, 3, 5, 7 and 11 must be added to the subgroup | # Primes 2, 3, 5, 7 and 11 must be added to the subgroup | ||
# If any primes 3, 5, 7 or 11 have more than 40% relative error, then they should be made a dual prime | # If any primes 3, 5, 7 or 11 have more than 40% relative error, then they should be made a dual prime | ||
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=== Carousel EDOs (20-34) === | === Carousel EDOs (20-34) === | ||
; 6 basis elements | ; 6 basis elements | ||
* [[20edo]]: 2 • 3 • 15 • 7 • 11 • 13 ('' | * [[20edo]]: 2 • 3 • 15 • 7 • 11 • 13 (''comp'') | ||
* [[21edo]]: 2 • 3 • 5 • 7 • 33 • 13 (''comp'') | * [[21edo]]: 2 • 3 • 5 • 7 • 33 • 13 (''comp'') | ||
* [[22edo]]: 2 • 3 • 5 • 7 • 11 • 17 (''no-n'') | * [[22edo]]: 2 • 3 • 5 • 7 • 11 • 17 (''no-n'') | ||
* [[23edo]]: 2 • 9 • 15 • 21 • 33 • 13 (''comp'') | * [[23edo]]: 2 • 9 • 15 • 21 • 33 • 13 (''comp'') | ||
* [[24edo]]: 2 • 3 • 5 • 7 • 11 • 13 (''lim'') | * [[24edo]]: 2 • 3 • 5 • 7 • 11 • 13 (''lim'') | ||
* [[25edo]]: 2 • 3 • 5 • 7 • 33 • 17 ('' | * [[25edo]]: 2 • 3 • 5 • 7 • 33 • 17 (''comp'') | ||
* [[26edo]]: 2 • 3 • 5 • 7 • 11 • 13 (''lim'') | * [[26edo]]: 2 • 3 • 5 • 7 • 11 • 13 (''lim'') | ||
* [[27edo]]: 2 • 3 • 5 • 7 • 11 • 13 (''lim'') | * [[27edo]]: 2 • 3 • 5 • 7 • 11 • 13 (''lim'') | ||