User:TallKite/The delta method: Difference between revisions

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Adding 1 to both numerator and denominator is called '''bumping up'''. Subtracting 1 from both is called '''bumping down'''. Note that bumping up increases the integer limit, but ''decreases'' the size in cents. The basic process is:

* possibly octave-reduce, see below

* possibly unsimplify, see below

*bump the ratio up or down to get a new ratio in which both the numerator and the denominator are multiples of the delta

*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.

*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.

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For very large deltas, one might want to use the octave complement. For example, 27/16 is delta-11, but 32/27 is only delta-5. The ancestors of 32/27 are 13/11 and 19/16. Thus 27/16 is between 22/13 and 32/19. ~~Its~~ actual ancestors ~~are~~ 5/3 ~~and~~ 22/13.

For very large deltas, one might want to use the octave complement. For example, 27/16 is delta-11, but 32/27 is only delta-5. The ancestors of 32/27 are 13/11 and 19/16. Thus 27/16 is between 22/13 and 32/19. But 32/19 is more complex than 27/16, not simpler. To find the actual ancestors, "subtract" one from another: 32/19 "minus" 22/13 = 10/6 = 5/3. indeed, the actual ancestors of 27/16 are 22/13 and 5/3.

...

The two numbers in the ratio can be thought of as two edos (see "comparing edos" below). This suggests another way of dealing with very large deltas. To solve 27/16, we think of it as a 16edo interval 27\16 and '''octave-reduce''', getting 11\16. Since that's larger than a half-octave, we can further reduce by finding the octave complement 5\16. The question becomes, how many 5\16 steps equals ±1 edostep of 16edo when octave-reduced? Three steps is 15\16, one short of an octave, so the answer is 3. We multiply the larger edo (27) by 3 and get 81. We divide 81 by the smaller edo (16) and get about 5. Thus the simpler stern-brocot ancestor of 27/16 is 5/3.

Another example: 72/41 --> 72\41 --> 31\41 --> 10\41 --> 4*(10\41)=40\41 --> 4*72=288 --> round(288/41)=7 --> 7/4

== Applications ==

=== Approximating ratios ===

Two ratios can be combined to make a 3rd ratio via the [[mediant]] ~~or "~~freshman sum". The 3rd ratio is always intermediate in cents between the other two. For example 8/5 "plus" 15/8 equals 23/13. One can work backwards and decompose any ratio into two simpler ratios, one larger and one smaller. In this example, knowing that 23/13 lies between 8/5 and 15/8 isn't very useful. Far better to find the two stern-brocot ancestors. The delta method gives 7/4 and 16/9, telling us that 23/13 sounds like a slightly flat minor 7th. Furthermore, because 16/9's integer limit is about double that of 7/4, 23/13 is about twice as close to 16/9 than 7/4. If one knows that 7/4 = 969¢ and 16/9 = 996¢, one can estimate 23/13 to be about 985¢ (actual size is 988¢).

Two ratios can be combined to make a 3rd ratio via the [[mediant]] aka freshman sum. The 3rd ratio is always intermediate in cents between the other two. For example 8/5 "plus" 15/8 equals 23/13. One can work backwards and decompose any ratio into two simpler ratios, one larger and one smaller. In this example, knowing that 23/13 lies between 8/5 and 15/8 isn't very useful. Far better to find the two stern-brocot ancestors. The delta method gives 7/4 and 16/9, telling us that 23/13 sounds like a slightly flat minor 7th. Furthermore, because 16/9's integer limit is about double that of 7/4, 23/13 is about twice as close to 16/9 than 7/4. If one knows that 7/4 = 969¢ and 16/9 = 996¢, one can estimate 23/13 to be about 985¢ (actual size is 988¢).

=== 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 ====

@@ Line 28: / Line 28: @@
 Adding 1 to both numerator and denominator is called '''bumping up'''. Subtracting 1 from both is called '''bumping down'''. Note that bumping up increases the integer limit, but ''decreases'' the size in cents. The basic process is:
+* possibly <u>octave-reduce</u>, see below
 * possibly <u>unsimplify</u>, see below
 *<u>bump</u> the ratio up or down to get a new ratio in which both the numerator and the denominator are multiples of the delta
 *<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.
+*<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.
@@ Line 71: / Line 72: @@
 |quintuple
 |}
-For very large deltas, one might want to use the octave complement. For example, 27/16 is delta-11, but 32/27 is only delta-5. The ancestors of 32/27 are 13/11 and 19/16. Thus 27/16 is between 22/13 and 32/19. Its actual ancestors are 5/3 and 22/13.
+For very large deltas, one might want to use the octave complement. For example, 27/16 is delta-11, but 32/27 is only delta-5. The ancestors of 32/27 are 13/11 and 19/16. Thus 27/16 is between 22/13 and 32/19. But 32/19 is more complex than 27/16, not simpler. To find the actual ancestors, "subtract" one from another: 32/19 "minus" 22/13 = 10/6 = 5/3. indeed, the actual ancestors of 27/16 are 22/13 and 5/3.
-...
+The two numbers in the ratio can be thought of as two edos (see "comparing edos" below). This suggests another way of dealing with very large deltas. To solve 27/16, we think of it as a 16edo interval 27\16 and '''octave-reduce''', getting 11\16. Since that's larger than a half-octave, we can further reduce by finding the octave complement 5\16. The question becomes, how many 5\16 steps equals ±1 edostep of 16edo when octave-reduced? Three steps is 15\16, one short of an octave, so the answer is 3. We multiply the larger edo (27) by 3 and get 81. We divide 81 by the smaller edo (16) and get about 5. Thus the simpler stern-brocot ancestor of 27/16 is 5/3.
+Another example: 72/41 --> 72\41 --> 31\41 --> 10\41 --> 4*(10\41)=40\41 --> 4*72=288 --> round(288/41)=7 --> 7/4
 == Applications ==
 === Approximating ratios ===
-Two ratios can be combined to make a 3rd ratio via the [[mediant]] or "freshman sum". The 3rd ratio is always intermediate in cents between the other two. For example 8/5 "plus" 15/8 equals 23/13. One can work backwards and decompose any ratio into two simpler ratios, one larger and one smaller. In this example, knowing that 23/13 lies between 8/5 and 15/8 isn't very useful. Far better to find the two stern-brocot ancestors. The delta method gives 7/4 and 16/9, telling us that 23/13 sounds like a slightly flat minor 7th. Furthermore, because 16/9's integer limit is about double that of 7/4, 23/13 is about twice as close to 16/9 than 7/4. If one knows that 7/4 = 969¢ and 16/9 = 996¢, one can estimate 23/13 to be about 985¢ (actual size is 988¢).
+Two ratios can be combined to make a 3rd ratio via the [[mediant]] aka freshman sum. The 3rd ratio is always intermediate in cents between the other two. For example 8/5 "plus" 15/8 equals 23/13. One can work backwards and decompose any ratio into two simpler ratios, one larger and one smaller. In this example, knowing that 23/13 lies between 8/5 and 15/8 isn't very useful. Far better to find the two stern-brocot ancestors. The delta method gives 7/4 and 16/9, telling us that 23/13 sounds like a slightly flat minor 7th. Furthermore, because 16/9's integer limit is about double that of 7/4, 23/13 is about twice as close to 16/9 than 7/4. If one knows that 7/4 = 969¢ and 16/9 = 996¢, one can estimate 23/13 to be about 985¢ (actual size is 988¢).
 === 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 91: / 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 ====