User:TallKite/The delta method: Difference between revisions

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WORK IN PROGRESS~~!!!~~

WORK IN PROGRESS

The delta method is a way to find the [[wikipedia:Stern–Brocot_tree|Stern-Brocot]] ancestors of a reasonably-sized ratio quickly and easily in one's head without using paper and pencil. It can be used to estimate the size in cents of a ratio. It also has applications in comparing edos.

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Consider two edos M-edo and N-edo, with M and N coprime. Every interval of M-edo (other than the unison and octave of course) will be approximated relatively better or worse by N-edo. By symmetry there are always two best approximations that are octave complements. They are called the '''nearest misses'''. The one(s) approximated the worst are called the '''farthest miss(es)'''. The farthest miss is always the [[Antipodes#Generalizations|generalized antipodes]] of M-edo with respect to either nearest miss. If M is even, then N is odd, and the farthest miss is the half-octave. But if M is odd, there are two such intervals, which are always octave complements. The smaller of the two is always exactly half of one of the nearest misses.

For example, consider 19edo as approximated by 12edo. The smaller nearest miss is found from the simpler stern-brocot ancestor of 19/12, which is 8/5. Pair 8 with 19 and 5 with 12. Thus 8\19 and its complement 11/19 are the two nearest misses, i.e. the two 19edo intervals closest to 12edo. Likewise 5\12 and 7\12 are the two 12edo intervals closest to 19edo. Since these intervals are all 4ths and 5ths, the generalized antipodes is the same as the standard circle-of-5ths antipodes, which for 19edo is half a 4th, and its complement a 5th higher. This is ~~4\19,~~ the aug 2nd or dim 3rd of 253¢, and ~~its complement~~ the aug 6th / dim 7th = 947¢. Thus if one wants 19edo to sound especially xenharmonic, one might feature these two intervals prominently, perhaps by using the temperament generated by them, [[Semaphore|Zozo/Semaphore]]. Conversely, to avoid offending ears accustomed to 12edo, one would avoid these intervals especially. (One might also avoid the 2nd farthest pair of misses, which are a 4th or 5th away from these.) Furthermore, if one wants to translate a 19edo piece to 12edo, the most difficult intervals to map will be these two antipodes.

For example, consider 19edo as approximated by 12edo. The smaller nearest miss is found from the simpler stern-brocot ancestor of 19:12, which is 8:5. Pair 8 with 19 and 5 with 12. Thus 8\19 and its complement 11\19 are the two nearest misses, i.e. the two 19edo intervals closest to 12edo. Likewise 5\12 and 7\12 are the two 12edo intervals closest to 19edo. Since these intervals are all 4ths and 5ths, the generalized antipodes is the same as the standard circle-of-5ths antipodes, which for 19edo is half a 4th, and its complement a 5th higher. This is the aug 2nd / dim 3rd of 253¢, and the aug 6th / dim 7th of 947¢. Thus if one wants 19edo to sound especially xenharmonic, one might feature these two intervals prominently, perhaps by using the temperament generated by them, [[Semaphore|Zozo/Semaphore]]. Conversely, to avoid offending ears accustomed to 12edo, one would avoid these intervals especially. (One might also avoid the 2nd farthest pair of misses, which are a 4th or 5th away from these.) Furthermore, if one wants to translate a 19edo piece to 12edo, the most difficult intervals to map will be these two antipodes.

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).

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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.

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.

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.

~~equal divisions of~~

All of the above generalizes to [[Edonoi|EDONOIs]].

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== Further notes ==

The delta method was invented by [[Kite Giedraitis]] in 2022. The ~~proof of~~ the ~~delta method is left to the reader.~~

The delta method was invented by [[Kite Giedraitis]] in 2022. The ratio approximations rely on the formula log [(a+b)/(c+d)] ≈ [a/(a+b)] * log [a/c] + [b/(a+b)] * log [b/d], where ad - bc = ±1

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-WORK IN PROGRESS!!!
+WORK IN PROGRESS
 The delta method is a way to find the [[wikipedia:Stern–Brocot_tree|Stern-Brocot]] ancestors of a reasonably-sized ratio quickly and easily in one's head without using paper and pencil. It can be used to estimate the size in cents of a ratio. It also has applications in comparing edos.
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 Consider two edos M-edo and N-edo, with M and N coprime. Every interval of M-edo (other than the unison and octave of course) will be approximated relatively better or worse by N-edo. By symmetry there are always two best approximations that are octave complements. They are called the '''nearest misses'''. The one(s) approximated the worst are called the '''farthest miss(es)'''. The farthest miss is always the [[Antipodes#Generalizations|generalized antipodes]] of M-edo with respect to either nearest miss. If M is even, then N is odd, and the farthest miss is the half-octave. But if M is odd, there are two such intervals, which are always octave complements. The smaller of the two is always exactly half of one of the nearest misses.
-For example, consider 19edo as approximated by 12edo. The smaller nearest miss is found from the simpler stern-brocot ancestor of 19/12, which is 8/5. Pair 8 with 19 and 5 with 12. Thus 8\19 and its complement 11/19 are the two nearest misses, i.e. the two 19edo intervals closest to 12edo. Likewise 5\12 and 7\12 are the two 12edo intervals closest to 19edo. Since these intervals are all 4ths and 5ths, the generalized antipodes is the same as the standard circle-of-5ths antipodes, which for 19edo is half a 4th, and its complement a 5th higher. This is 4\19, the aug 2nd or dim 3rd of 253¢, and its complement the aug 6th / dim 7th = 947¢. Thus if one wants 19edo to sound especially xenharmonic, one might feature these two intervals prominently, perhaps by using the temperament generated by them, [[Semaphore|Zozo/Semaphore]]. Conversely, to avoid offending ears accustomed to 12edo, one would avoid these intervals especially. (One might also avoid the 2nd farthest pair of misses, which are a 4th or 5th away from these.) Furthermore, if one wants to translate a 19edo piece to 12edo, the most difficult intervals to map will be these two antipodes.
+For example, consider 19edo as approximated by 12edo. The smaller nearest miss is found from the simpler stern-brocot ancestor of 19:12, which is 8:5. Pair 8 with 19 and 5 with 12. Thus 8\19 and its complement 11\19 are the two nearest misses, i.e. the two 19edo intervals closest to 12edo. Likewise 5\12 and 7\12 are the two 12edo intervals closest to 19edo. Since these intervals are all 4ths and 5ths, the generalized antipodes is the same as the standard circle-of-5ths antipodes, which for 19edo is half a 4th, and its complement a 5th higher. This is the aug 2nd / dim 3rd of 253¢, and the aug 6th / dim 7th of 947¢. Thus if one wants 19edo to sound especially xenharmonic, one might feature these two intervals prominently, perhaps by using the temperament generated by them, [[Semaphore|Zozo/Semaphore]]. Conversely, to avoid offending ears accustomed to 12edo, one would avoid these intervals especially. (One might also avoid the 2nd farthest pair of misses, which are a 4th or 5th away from these.) Furthermore, if one wants to translate a 19edo piece to 12edo, the most difficult intervals to map will be these two antipodes.
 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).
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 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.
-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.
+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.
-equal divisions of
 All of the above generalizes to [[Edonoi|EDONOIs]].
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 == Further notes ==
-The delta method was invented by [[Kite Giedraitis]] in 2022. The proof of the delta method is left to the reader.
+The delta method was invented by [[Kite Giedraitis]] in 2022. The ratio approximations rely on the formula log [(a+b)/(c+d)] ≈ [a/(a+b)] * log [a/c] + [b/(a+b)] * log [b/d], where ad - bc = ±1