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# The subgroup should have 3 [[basis element]]s
# The subgroup should have 3 [[basis element]]s
# If the EDO approximates 3 or more [[prime]]s 11 or lower within 15 [[cents]], then choose the best 3 and use those as its subgroup
# If the EDO approximates 3 or more [[prime]]s 11 or lower within 15 [[cents]], then choose the best 3 and use those as its subgroup
# If it approximates less than 3 such primes, then include all the ones it does approximate, and fill the remaining spots with odd harmonics smaller than 40 that it approximates within 15 cents (giving preference to the lowest harmonics first)  
# If it approximates less than 3 such primes, then include all the ones it does approximate, and fill the remaining spots with [[11-limit]] composite harmonics smaller than 60 that it approximates within 15 cents (giving preference to harmonics with lower prime factors first and excluding powers of two)
# If there are aren't enough of those to fill all 3 spots, fill the remaining spots with [[taxicab distance|taxicab-2]] intervals the edo approximates within 15 cents, giving preference to intervals with small primes and intervals that are approximated more precisely (use direction to decide which to value more on an edo-by-edo basis)
# If there are aren't enough of those to fill all 3 spots, fill the remaining spots with [[taxicab distance|taxicab-2]] intervals the edo approximates within 15 cents, giving preference to intervals with small primes
# If there are still spots left open, fill them with the smallest composite harmonics of any size that are approximated within 15 cents


=== EDOs with 7 to 12 tones/octave ===
=== EDOs with 7 to 12 tones/octave ===
# The subgroup should have 5 basis elements
# The subgroup should have 5 [[basis element]]s
# If the EDO approximates any primes 11 or lower within 15 cents, then add all of those to its subgroup
# Add prime 2 to the subgroup
# If there are still spots left over, and the EDO does not approximate prime 3, then check if it approximates any harmonics 6, 9, 12 or 15 within 15 cents. If so, add them all to the subgroup starting with the smallest until all spots are filled or all harmonics have been added.
# If 3 is approximated within 15 cents, add 3 to the subgroup
# If there are still spots left over, if the EDO does not approximate one or both of 5 or 3, check if it approximates 5/3 within 15 cents and if so, add that as a basis element
## If it is not, then add the smallest multiple of 3 with up to 2 digits it approximates within 15 cents
# If there are still spots left over, if the EDO does not approximate one or both of 7 or 3, check if it approximates 7/3 within 15 cents and if so, add that as a basis element
## If no such multiple exists, move on to the next step
# If there are still spots left over, if the EDO does not approximate one or both of 7 or 5, check if it approximates 7/5 within 15 cents and if so, add that as a basis element
# If 5 is approximated within 15 cents, add 5 to the subgroup
# Do the same as above for 11/3, then 11/5, then 11/7
## If it is not, then add the smallest multiple of 5 with up to 2 digits it approximates within 15 cents
# If there are still spots left over, check if the EDO approximates any primes 13, 17, 19 or 23 within 15 cents, if so, then add all of those to its subgroup giving preference to the lowest ones and to the ones approximated most closely (use discretion to choose which of those things to weight more heavily) until all spots have been filled or all primes have been added
## If no such multiple exists, move on to the next step
# If there are still spots left open, choose either (A) or (B) at your discretion on an edo-by-edo basis, or combine both for edos where that makes sense:
# If 7 is approximated within 15 cents, add 7 to the subgroup
## (A) fill the remaining spots with odd harmonics smaller than 40 that it the edo approximates within 15 cents (giving preference to harmonics that are multiples of simple harmonics like 21 or 33)  
## If it is not, then add the smallest multiple of 7 with up to 2 digits it approximates within 15 cents
## (B) fill the remaining spots with taxicab-2 intervals the edo approximates within 15 cents, giving preference to intervals with small primes and intervals that are approximated more precisely (use direction to decide which to value more on an edo-by-edo basis)
## If no such multiple exists, move on to the next step
# If there are still spots left open, fill them with the smallest remaining integer harmonics of any size that are approximated within 15 cents
# If 11 is approximated within 15 cents, add 11 to the subgroup
## If it is not, then add the smallest multiple of 11 with up to 2 digits it approximates within 15 cents
## If no such multiple exists, move on to the next step
# If it approximates primes 13, 17, 19 or 23 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


=== EDOs with 13 to 27 tones/octave ===
=== EDOs with 13 to 27 tones/octave ===
# The subgroup should have 6 basis elements
# The subgroup should have 6 [[basis element]]s
# If the EDO approximates any primes 13 or lower within 15 cents, then add all of those to its subgroup
# Add prime 2 to the subgroup
# If there are still spots left over, and the EDO does not approximate prime 3, then check if it approximates any harmonics 6, 9, 12 or 15 within 15 cents. If so, add them all to the subgroup starting with the smallest until all spots are filled or all harmonics have been added.
# If 3 is approximated within 15 cents, add 3 to the subgroup
# If there are still spots left over, if the EDO does not approximate one or both of 5 or 3, check if it approximates 5/3 within 15 cents and if so, add that as a basis element
## If it is not, then add the smallest multiple of 3 with up to 2 digits it approximates within 15 cents
# If there are still spots left over, if the EDO does not approximate one or both of 7 or 3, check if it approximates 7/3 within 15 cents and if so, add that as a basis element
## If no such multiple exists, move on to the next step
# If there are still spots left over, if the EDO does not approximate one or both of 7 or 5, check if it approximates 7/5 within 15 cents and if so, add that as a basis element
# If 5 is approximated within 15 cents, add 5 to the subgroup
# Do the same as above for 11/3, then 11/5, then 11/7, then 13/3, then 13/5, then 13/7, then 13/11
## If it is not, then add the smallest multiple of 5 with up to 2 digits it approximates within 15 cents
# If there are still spots left over, check if the EDO approximates any primes 17, 19 or 23 within 15 cents, if so, then add all of those to its subgroup giving preference to the lowest ones and to the ones approximated most closely (use discretion to choose which of those things to weight more heavily) until all spots have been filled or all primes have been added
## If no such multiple exists, move on to the next step
# If there are still spots left open, choose either (A) or (B) at your discretion on an edo-by-edo basis, or combine both for edos where that makes sense:
# If 7 is approximated within 15 cents, add 7 to the subgroup
## (A) fill the remaining spots with odd harmonics smaller than 40 that it the edo approximates within 15 cents (giving preference to harmonics that are multiples of simple harmonics like 21 or 33)  
## If it is not, then add the smallest multiple of 7 with up to 2 digits it approximates within 15 cents
## (B) fill the remaining spots with taxicab-2 intervals the edo approximates within 15 cents, giving preference to intervals with small primes and intervals that are approximated more precisely (use direction to decide which to value more on an edo-by-edo basis)
## If no such multiple exists, move on to the next step
# If there are still spots left open, fill them with the smallest remaining integer harmonics of any size that are approximated within 15 cents
# If 11 is approximated within 15 cents, add 11 to the subgroup
## If it is not, then add the smallest multiple of 11 with up to 2 digits it approximates within 15 cents
## If no such multiple exists, move on to the next step
# If 13 is approximated within 15 cents, add 14 to the subgroup
## If it is not, move on to the next step
# If it approximates primes 17, 19 or 23 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


=== EDOs with 28 to 52 tones/octave ===
=== EDOs with 28 to 52 tones/octave ===
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