Don Page comma

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By a Don Page comma is meant a comma computed from two other intervals by the method suggested by the Don Page paper, Why the Kirnberger Kernel Is So Small. If a and b are two rational numbers > 1, define r = ((a-1)(b+1)) / ((b-1)(a+1)). Suppose r reduced to lowest terms is p/q, and a and b are written in monzo form as u and v. Then the Don Page comma is defined as DPC(a, b) = qu - pv, or else minus that if the size in cents is less than zero. The reason for introducing monzos is purely numerical; if monzos are not used the numerators and denominators of the Don Page comma quickly become so large they cannot easily be handled. In ratio form, the Don Page comma can be written a^q / b^p, or the reciprocal of that if that is less than 1.

Bimodular approximants

If x is near to 1, then ln(x)/2 is approximated by bim(x) = (x-1)/(x+1), the bimodular approximant function, which is the Padé approximant of order (1, 1) to ln(x)/2 near 1. The bimodular approximant function is a Möbius transformation and hence has an inverse, which we denote mib(x) = (1+x)/(1-x), which is the (1, 1) Padé approximant around 0 for exp(2x). Then bim(exp(2x)) = tanh(x), and therefore ln(mib(x))/2 = artanh(x) = x + x^3/x + x^5/5 + ..., from which it is apparent that bim(x) approximates ln(x)/2, and mib(x) approximates exp(2x), to the second order; we may draw the same conclusion by directly comparing the series for exp(2x) = 1 + 2x + 2x^2 + O(x^3) with mib(x) = 1 + 2x + 2x^2 + O(x^3) and ln(x)/2 = (x-1)/2 - (x-1)^2/4 + O(x^3), which is the same to the second order as bim(x). Using mib, we may also define BMC(a, b) = DPC(mib(a), mib(b)), where BMC is an acronym for "bimodular comma".

If r is as above we have that r = bim(a)/bim(b), and depending on common factors the corresponding Don Page comma is equal to an nth power of a^bim(b) / b^bim(a) = mib(u)^v/mib(v)^u for some n. If we set a = 1+x, b = 1+y, then r = r(x, y) is an analytic function of two complex variables with a power series expansion around x=0, y=0. This expansion begins as r(x, y) = 1 - (xy^3 - x^3y)/24 + (3xy^4 + x^2y^3 - x^3y^2 - 3x^4y)/48 + ..., with its first nonconstant term of total degree four, and so when x and y are small, r(x, y) will be close to 1. The nth power will still be close to 1, and if there are common factors in the denominators of the exponents, the resulting comma will be less complex (lower in height), and this is the really interesting case. For example, if a = 7/6 and b = 27/25, we obtain (7/6)^(1/26) / (27/25)^(1/13), the lcm of 13 and 26 is 26, and taking the 26th power gives us 4375/4374. The common factor of 13 leads to the resulting comma (4375/4374) being relatively simple.

What is going on here becomes clearer if we shift to BMC rather than DPC. If bim(x) was an exact logarithmic function rather than an approximation, then the Don Page commas would all be 1. They measure the deviation between an approximate relationship between intervals and an exact one. For example. bim(11/9) = 1/10 and bim(3/2) = 1/5, and two 11/9 intervals fall short of 3/2 by (3/2)/(11/9)^2 = BMC(1/10, 1/5) = 243/242. Not all relationships between intervals of this sort arise from bimodular approximation. The syntonic comma, 81/80, is how much two 9/8 intervals exceed 5/4, and how much two 10/9 intervals fall short of it. But bim(10/9) = 1/19 and bim(9/8) = 1/17, neither of which will add up to bim(5/4) = 1/9. Instead mib(1/18) = 19/17 will give BMC(1/18, 1/9) = 1445/1444, a whole other deal. To get 81/80, note that bim(4/3) = 1/7 and bim(9/5) = 2/7, and BMC(1/7, 2/7) = 81/80.

For n>1 BMC(1/n, 1/(2n)) goes 27/25, 50/49, 245/243, 243/242, 847/845, 676/675, 2025/2023, 1445/1444, 3971/3969, 2646/2645, 6877/6875, 4375/4374, 10935/10933, 6728/6727, 16337/16335, 9801/9800, 23275/23273, 13690/13689, 31941/31939..., with BMC(1/13, 1/26) being our example 4375/4374. Similarly, BMC(1/n, 1/(3n)) goes 375/343, 128/125, 6655/6591, 1029/1024, 34391/34295, 4000/3993, 109503/109375, 10985/10976, 268279/268119, 24576/24565... and BMC(2/n, 3/n) goes 49/27, 432/343, 9/8, 3125/2916, 3267/3125, 1372/1331, 1352/1323, 35721/35152, 3125/3087, 85184/84375, 7803/7744, 19773/19652, 123823/123201, 337500/336091, 3136/3125... .

We might also note that successive superparticular ratios such as 10/9 and 11/10 or 12/11 and 13/12 exhibit a some tendency to produce strong (in the sense of low badness figures for the corresponding temperament) Don Page commas, not surprising when we note that in this case the first non-constant term of the one-variable expansion is of order five.

Examples

5-limit commas

DPC(5/3, 3) = BMC(1/2, 1/4) = 27/25

DPC(4/3, 5/2) = BMC(1/7, 3/7) = 135/128

DPC(5/3, 2) = BMC(1/2, 1/3) = 648/625

DPC(4/3, 9/5) = BMC(1/7, 2/7) = 81/80

DPC(5/4, 2) = BMC(1/9, 1/5) = 128/125

DPC(4/3, 5/3) = BMC(1/7, 1/4) = 16875/16384, negri

DPC(3/2, 5/3) = BMC(1/5, 1/4) = 20000/19683, tetracot

DPC(10/9, 32/25) = BMC(1/19, 7/57) = |8 14 -13>, parakleisma

DPC(5/4, 4/3) = BMC(1/9, 1/7) = |32 -7 -9>, escapade

DPC(6/5, 5/4) = BMC(1/11, 1/9) = |-29 -11 20>, gammic

DPC(10/9, 9/8) = BMC(1/19, 1/17) = |-70 72 -19>

DPC(81/80, 25/24) = BMC(1/161, 1/49) = |71 -99 37>, raider

DPC(81/80, 128/125) = BMC(1/161, 3/253) = |161 -84 -12>, the atom

7-limit commas

DPC(7/5, 2) = BMC(1/6, 1/3) = 50/49

DPC(6/5, 7/4) = BMC(1/11, 3/11) = 875/864

DPC(7/5, 5/3) = BMC(1/6, 1/4) = 3125/3087

DPC(9/7, 5/3) = BMC(1/8, 1/4) = 245/243

DPC(7/6, 8/5) = BMC(1/13, 3/13) = 1728/1715

DPC(8/7, 3/2) = BMC(1/15, 1/5) = 1029/1024

DPC(5/4, 7/5) = BMC(1/9, 1/6) = 3136/3125

DPC(9/8, 10/7) = BMC(1/17, 3/17) = 5120/5103

DPC(27/25, 7/6) = BMC(1/26, 1/13) = 4375/4374

11-limit commas

DPC(11/10, 4/3) = BMC(1/21, 1/7) = 4000/3993

DPC(10/9, 11/8) = BMC(1/19, 3/19) = 8019/8000

DPC(11/9, 3/2) = BMC(1/10, 1/5) = 243/242

DPC(5/4, 11/7) = BMC(1/9, 2/9) = 176/175

DPC(8/7, 11/9) = BMC(1/15, 1/10) = 41503/41472

13-limit commas

DPC(15/14, 16/13) = 43904/43875

DPC(14/13, 5/4) = 10985/10976

DPC(11/10, 15/13) = 225000/224939

DPC(15/13, 4/3) = 676/675

DPC(13/11, 7/5) = 847/845

DPC(6/5, 13/9) = 325/324

Here are some complex Don Page commas derived from other commas:

DPC(525/512, 245/243) = |-153 277 -18 -87>

DPC(49/48, 50/49) = |-487 -97 -198 392>

DPC(10/9, 11/10) = |40 -38 40 0 -21>

DPC(11/10, 12/11) = |-67 -23 -21 0 44>

DPC(77/75, 245/243) = |0 286 -99 -103 19>

DPC(55/54, 56/55) = |-442 -327 220 -111 220>

DPC(176/175, 540/539) = |-58 -249 -137 139 110>