User:TallKite/The delta method: Difference between revisions

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*simplify by dividing both numerator and denominator by the delta to get the simpler ancestor

*subtract the simpler ancestor from the original ratio to get the more complex ancestor

*optional: confirm the answer by multiplying the numerator of one ancestor by the denominator of the other. The two products should differ by 1.

For example, 7/4 is delta-3. Bumping up, we get 8/5. But neither 8 nor 5 is divisible by 3. So instead we bump 7/4 down to 6/3. This simplifies to 2/1, which is the simpler ancestor. Subtract 2/1 from 7/4 to get 5/3 (because 7-2=5 and 4-1=3), which is the more complex ancestor. Optional confirmation: the two products are 2*3=6 and 7*1=7, which do indeed differ by 1.

But with delta-5 and higher, sometimes neither bumping up nor bumping down works. For example, with 12/7, neither 13/8 nor 11/6 are multiples of 5. When this happens, one must first '''unsimplify''' the ratio by doubling the numerator and the denominator. (12/7 = 24/14, bump to 25/15 = 5/3). If doubling doesn't work, try tripling, quadrupling, etc.

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Now consider 12edo as approximated by 19edo. The antipodes of any even edo is the half-octave, no matter what the generator is. Thus the 12edo tritone is the most difficult interval to translate to 19edo. Note that 12edo's nearest misses are quite close cents-wise to 19edo's. This is true for any two edos. But 12edo's farthest miss (the half-octave) is quite distant from 19edo's (the half-4th).

Now consider 11edo and 12edo. The nearest misses are 1\11 and 1\12. The farthest misses are 5\11, 6\11 and 6\12. This is an example of the farthest misses being fairly close to each other. This is also an example of an '''equidistant''' farthest miss, because 6\12 is midway between 5\11 and 6\11. As a result, mapping a 12edo piece to 11edo by simply selecting the closest approximation of each note fails in the sense that there is more than one way to do so.

Now consider 11edo and 12edo. The nearest misses are 1\11 and 1\12. The farthest misses are 5\11, 6\11 and 6\12. This is an example of the farthest misses being fairly close to each other. This is also an example of an '''equidistant''' farthest miss, because 6\12 is midway between 5\11 and 6\11. As a result, mapping a 12edo piece to 11edo by simply selecting the closest approximation of each note fails in the sense that there is more than one way to do so for 6\12.

If M and N are not coprime, they coincide exactly in places other than the unison and octave, in effect dividing the octave into several similar sections. For example, 12edo and 22edo coincide at the half-octave. We can think of 12edo as 6-'''edho''' (rhymes with "red toe", means equal division of the half octave), and 22-edo as 11-edho. Then we can proceed as with 6-edo and 11-edo, and afterwards simply shrink everything down to half size. The simpler ancestor of 11:6 is 2:1. Thus the nearest misses are 2\11 and 9\11 and 1\6 and 5\6. These convert to 2\22, 9\22, 1\12 and 5\12 (plus octave complements). 11edo's farthest misses from 6edo are 1\11 and 10\11. These convert to 1\22 and 10\22 (plus octave complements). Because 6 is even, 6edo's farthest miss from 11edo is the half-octave, 3\6. This converts to 3\12, the minor 3rd, which is equidistant from 5\22 and 6\22.

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==== Finding the natural generator ====

Every ~~non~~-~~perfect edo~~ has a "natural" heptatonic generator. For 13edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one edostep away from the octave. Thus the sharp means "sharpened by one edostep", major is one edostep wider than minor, and ups and downs aren't needed.

Every edo (except the [[Sharpness|sharp-0]] ones) has a "natural" heptatonic generator. For 13edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one edostep away from the octave. Thus the sharp means "sharpened by one edostep", major is one edostep wider than minor, and ups and downs aren't needed.

The natural heptatonic generator of N-edo is simply the nearest miss of that edo and 7edo. The natural generator is always one of these:

@@ Line 32: / Line 32: @@
 *<u>simplify</u> by dividing both numerator and denominator by the delta to get the simpler ancestor
 *<u>subtract</u> the simpler ancestor from the original ratio to get the more complex ancestor
+*optional: <u>confirm</u> the answer by multiplying the numerator of one ancestor by the denominator of the other. The two products should differ by 1.
+For example, 7/4 is delta-3. Bumping up, we get 8/5. But neither 8 nor 5 is divisible by 3. So instead we bump 7/4 down to 6/3. This simplifies to 2/1, which is the simpler ancestor. Subtract 2/1 from 7/4 to get 5/3 (because 7-2=5 and 4-1=3), which is the more complex ancestor. Optional confirmation: the two products are 2*3=6 and 7*1=7, which do indeed differ by 1.
 But with delta-5 and higher, sometimes neither bumping up nor bumping down works. For example, with 12/7, neither 13/8 nor 11/6 are multiples of 5. When this happens, one must first '''unsimplify''' the ratio by doubling the numerator and the denominator. (12/7 = 24/14, bump to 25/15 = 5/3). If doubling doesn't work, try tripling, quadrupling, etc.
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 Now consider 12edo as approximated by 19edo. The antipodes of any even edo is the half-octave, no matter what the generator is. Thus the 12edo tritone is the most difficult interval to translate to 19edo. Note that 12edo's nearest misses are quite close cents-wise to 19edo's. This is true for any two edos. But 12edo's farthest miss (the half-octave) is quite distant from 19edo's (the half-4th).
-Now consider 11edo and 12edo. The nearest misses are 1\11 and 1\12. The farthest misses are 5\11, 6\11 and 6\12. This is an example of the farthest misses being fairly close to each other. This is also an example of an '''equidistant''' farthest miss, because 6\12 is midway between 5\11 and 6\11. As a result, mapping a 12edo piece to 11edo by simply selecting the closest approximation of each note fails in the sense that there is more than one way to do so.
+Now consider 11edo and 12edo. The nearest misses are 1\11 and 1\12. The farthest misses are 5\11, 6\11 and 6\12. This is an example of the farthest misses being fairly close to each other. This is also an example of an '''equidistant''' farthest miss, because 6\12 is midway between 5\11 and 6\11. As a result, mapping a 12edo piece to 11edo by simply selecting the closest approximation of each note fails in the sense that there is more than one way to do so for 6\12.
 If M and N are not coprime, they coincide exactly in places other than the unison and octave, in effect dividing the octave into several similar sections. For example, 12edo and 22edo coincide at the half-octave. We can think of 12edo as 6-'''edho''' (rhymes with "red toe", means equal division of the half octave), and 22-edo as 11-edho. Then we can proceed as with 6-edo and 11-edo, and afterwards simply shrink everything down to half size. The simpler ancestor of 11:6 is 2:1. Thus the nearest misses are 2\11 and 9\11 and 1\6 and 5\6. These convert to 2\22, 9\22, 1\12 and 5\12 (plus octave complements). 11edo's farthest misses from 6edo are 1\11 and 10\11. These convert to 1\22 and 10\22 (plus octave complements). Because 6 is even, 6edo's farthest miss from 11edo is the half-octave, 3\6. This converts to 3\12, the minor 3rd, which is equidistant from 5\22 and 6\22.
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 ==== Finding the natural generator ====
-Every non-perfect edo has a "natural" heptatonic generator. For 13edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one edostep away from the octave. Thus the sharp means "sharpened by one edostep", major is one edostep wider than minor, and ups and downs aren't needed.
+Every edo (except the [[Sharpness|sharp-0]] ones) has a "natural" heptatonic generator. For 13edo, it's a 2\13 2nd (and its octave inverse of course), because seven 2\13's falls only one edostep away from the octave. Thus the sharp means "sharpened by one edostep", major is one edostep wider than minor, and ups and downs aren't needed.
 The natural heptatonic generator of N-edo is simply the nearest miss of that edo and 7edo. The natural generator is always one of these: