User:BudjarnLambeth/12edo as a 2.3.5.17.19 tuning: Difference between revisions
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=== Why different sized EDOs have different procedures === | === Why different sized EDOs have different procedures === | ||
As EDOs get bigger and their step size gets smaller, their step size gets closer and closer to the [[just-noticeable difference]] | As EDOs get bigger and their step size gets smaller, their step size gets closer and closer to the [[just-noticeable difference]]. | ||
Also, as EDOs get bigger, more notes per octave need to be labelled with a JI approximation, so more | This means that if a smaller EDO has high [[relative error]] on a [[prime]], it will sound like the prime is not there at all (no-no), but if a larger EDO has high relative error on a prime, especially a small prime, it will sound like there are two versions of the prime (dual). | ||
Different approaches are needed for different EDO sizes to reflect this. | |||
Also, as EDOs get bigger, more notes per [[octave]] need to be labelled with a [[JI]] approximation, so more [[basis element]]s are needed to produce those labels. Whereas, as EDOs get smaller, too many basis elements just make it needlessly complicated to navigate them, and fewer basis elements are better. So this is another reason for differing approaches at different EDO sizes. | |||
=== EDOs with 1 to 6 tones/octave === | === EDOs with 1 to 6 tones/octave === | ||
# The subgroup should have 3 | # The subgroup should have 3 basis elements | ||
# If the EDO approximates 3 or more | # If the EDO approximates 3 or more primes 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 [[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 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 11-limit [[taxicab distance|taxicab-2]] intervals the edo approximates within 15 cents, giving preference to intervals with lower primes | # If there are aren't enough of those to fill all 3 spots, fill the remaining spots with 11-limit [[taxicab distance|taxicab-2]] intervals the edo approximates within 15 cents, giving preference to intervals with lower primes | ||