User:TallKite/The delta method: Difference between revisions

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=== Comparing edos ===

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

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.

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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.

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

==== Spiral graphs ====

These spiral graphs for [https://en.xen.wiki/w/31edo#Relationship_to_12-edo 31edo], [https://en.xen.wiki/w/41edo#Relationship_to_12-edo 41edo] and [https://en.xen.wiki/w/53edo#Relationship_to_12edo 53edo] relate each of those edos to 12edo. Each chart has 12 "wheel-spokes". The larger edo's spiral of fifths is not really a spiral, it's a larger circle that you break into a chain and make a bunch of smaller 12-note loops with. Then add a few duplicates at each end of the chain, so that you can reconnect the ends mentally and get the original larger circle.

A 12-spoke spiral graph of fifths is only possible if the [[Sharpness|dodeca-sharpness]] (edosteps per pyth comma) of the larger edo is 1 or -1.

Such a spiral chart can be made for any two edos, as long as the are coprime. It's often a spiral of something other than fifths. In fact, it's a spiral of the nearest miss. For example, consider 8edo and 27edo. The near misses are 3\8 and 10\27. You get an 8-spoke spiral of 27edo major 3rds. This might be useful for someone researching octotonic scales in 27edo.

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

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 === Comparing edos ===
+All of the following generalizes to [[Edonoi|EDONOIs]].
 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.
@@ Line 94: / Line 96: @@
 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.
-All of the above generalizes to [[Edonoi|EDONOIs]].
+==== Spiral graphs ====
+These spiral graphs for [https://en.xen.wiki/w/31edo#Relationship_to_12-edo 31edo],  [https://en.xen.wiki/w/41edo#Relationship_to_12-edo 41edo] and [https://en.xen.wiki/w/53edo#Relationship_to_12edo 53edo] relate each of those edos to 12edo. Each chart has 12 "wheel-spokes". The larger edo's spiral of fifths is not really a spiral, it's a larger circle that you break into a chain and make a bunch of smaller 12-note loops with. Then add a few duplicates at each end of the chain, so that you can reconnect the ends mentally and get the original larger circle.
+A 12-spoke spiral graph of fifths is only possible if the [[Sharpness|dodeca-sharpness]] (edosteps per pyth comma) of the larger edo is 1 or -1.
+Such a spiral chart can be made for any two edos, as long as the are coprime. It's often a spiral of something other than fifths. In fact, it's a spiral of the nearest miss. For example, consider 8edo and 27edo. The near misses are 3\8 and 10\27. You get an 8-spoke spiral of 27edo major 3rds. This might be useful for someone researching octotonic scales in 27edo.
 ==== Finding the natural generator ====