User:TallKite/The delta method

WORK IN PROGRESS

The delta method is a way to find the 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.

The usual way of finding a Stern-Brocot ancestor is to use the extended Euclidean algorithm, which requires a computer. But there's times when computer usage isn't appropriate. You might be at a company retreat, and your boss's boss is telling a long anecdote. You lose interest, and start to wonder, which interval in 31edo is the farthest from 41edo? Or you're attending a performance of your friend's 17-minute-long minimalist 23-limit piece, and you start to wonder, roughly how many cents is 23/13 anyway? The delta method allows you to solve such problems in your head.

Background and terminology

The delta of a ratio is simply the numerator minus the denominator. All superparticular ratios are delta-1. Both 5/3 and 7/5 are delta-2.

Every ratio occurs only once in the Stern-Brocot tree. Every ratio has two ancestors and two children. Both ancestors will have a smaller integer limit, and one will always be smaller than the other. Thus there is a simpler ancestor and a more complex ancestor.

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The stern-brocot tree is also used for edo fractions. In this form it's called the scale tree.

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The process

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 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

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.

The correct factor to multiply by is always less than half of the delta, and is always coprime with the delta.

delta	1	2	3	4	5	6	7	8	9	10	11	12
possible unsimplifications	(none)				double	(none)	double or triple	triple	double or quadruple	triple	double, triple, quadruple or quintuple	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.

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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¢).

Comparing edos

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 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 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, 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).

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.

All of the above generalizes to EDONOIs.

Examples of approximating a ratio

Delta-1 ratios

The simpler ancestor is always 1/1. Bump the ratio down to get the more complex ancestor.

For example, 5/4 yields 4/3 and 1/1. Since 4/3 is four times as complex as 1/1, 5/4 falls about four times closer to it.

Delta-2 ratios

Bump down and simplify to get the simpler ancestor. Bump up and simplify to get the more complex ancestor.

9/7: Bump up/down to get 10/8 and 8/6, which yields 5/4 and 4/3. Since 5/4 is slightly more complex than 4/3, 9/7 lies slightly flat of the midpoint between the two.

19/17: Bump to get 20/18 and 18/16, which yields 10/9 and 9/8. 19/17 lies almost exactly midway between the two. Thus it's very close to the quarter-comma meantone major 2nd.

Delta-3 ratios

Bump either up or down, whichever gives multiples of three, then simplify. Subtract to get the more complex ancestor.

• 10/7: Bump down to get 9/6, which simplifies to 3/2. Subtract 3/2 from 10/7 to get 7/5.
• 11/8: Bump up to get 12/9 = 4/3. Subtract to get 7/5.

Delta-4 ratios

Bump either up or down, whichever gives multiples of four, then simplify. Subtract to get the more complex ancestor.

• 13/9: Bump down to get 12/8 = 3/2. Subtract to get 10/7.
• 15/11: Bump up to get 16/12 = 4/3. Subtract to get 11/8.

Delta-5 ratios

If the numerator mod 5 is 1 or 4, bump it. (16/11: Bump down to get 15/10 = 3/2. Subtract to get 13/9)

If not, double the ratio before bumping.

• 13/8: Double to get 26/16. Bump down to get 25/15 = 5/3. Subtract from 13/8 to get 8/5.
• 17/12: Double to get 34/24. Bump up to get 35/25 = 7/5. Subtract from 17/12 to get 10/7.

Delta-6 ratios

Bump it. (17/11 --> 18/12 --> 3/2 and14/9)

Delta-7 ratios

• If the numerator mod 7 is 1 or 6, bump it.
• If the numerator mod 7 is 2 or 5, triple the ratio before bumping. (16/9 --> 48/27 --> 49/28 --> 7/4 and 9/5)
• If the numerator mod 7 is 3 or 4, double the ratio before bumping. (17/10 --> 34/20 --> 35/21 --> 5/3 and 12/7)

Delta-8 ratios

• If the numerator mod 8 is 1 or 7, bump it.
• If the numerator mod 8 is 3 or 6, triple the ratio before bumping. (19/11 --> 57/33 --> 56/32 --> 7/4 and 12/7)

Delta-9 ratios

• If the numerator mod 9 is 1 or 8, bump it.
• If the numerator mod 9 is 2 or 7, quadruple the ratio before bumping. (20/11 --> 80/44 --> 81/45 --> 9/5 and 11/6)
• If the numerator mod 9 is 4 or 5, double the ratio before bumping. (22/13 --> 44/26 --> 45/27 --> 5/3 and 17/10)

Delta-10 ratios

• If the numerator mod 10 is 1 or 9, bump it.
• If the numerator mod 10 is 3 or 7, triple the ratio before bumping. (23/13 --> 69/39 --> 70/40 --> 7/4 and 16/9)

Further notes

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