Structure metric: Difference between revisions

The [https://en.wikipedia.org/wiki/Gromov_product Gromov product] is a construction in the theory of metric spaces, which depends on a choice of base point. For our purposes that choice won't matter, and we may assume it is the 1/1 of the scale. If x is the base point, and y and z are any points, then the Gromov product is defined to be (y, z)_x = ('''d'''(x, y) + '''d'''(x, z) - '''d'''(y, z))/2. Assuming x is 1, this becomes (y, z) = ('''d'''(1, y) + '''d'''(1, z) - '''d'''(y, z))/2. The Gromov product matrix is then G = ((i, j)) for all points x_i other than 1 (or other than 0, using logarithmic measures such as cents) taken in some order.

If d is a metric, the pth power of d for p ≥ 0 will at least be a distance function, though for some choices of p it might violate the triangle inequality. The pth power transform of the metric d leads to the p distance matrix Dp = ('''d'''(i, j)^p). This is an N dimensional symmetric square matrix, where N is the cardinality of the scale within a single period. Corresponding to it is an N-1 dimensional symmetric square matrix Gp = ((i, j)^p), the p Gromov product matrix. If Gp is positive semidefinite, then the metric space is said to have p-negative type. If it is positive definite, the space is of strict p-negative type. The space is embeddable in a Euclidean space if and only if it is of 2-negative type; and if and only if it is embeddable in a Euclidean space of N-1 dimensions but in no lesser number of dimensions, it is of strict 2-negative type. It follows that if Gp is positive semidefinite, the p/2-th power transform d^(p/2) of the metric embeds in Euclidean space, and if it is positive definite, such an embedding requires N-1 dimensions. If the space is of p-negative type, it is of strict q-negative type for any q<p. The supremum of all the exponents q where the space is of strict q-negative type is an exponent p which is of negative type but not strict negative type. This exponent is called the supremal p-negative type (and also the maximal generalized roundness.) A space (and hence for us, a scale) with a higher supremal p-negative type is "rounder", and with a lower one "flatter". Below is a listing of some scales (either JI or in some edo) by increasing roundness.

p = 1.1135814 [[duodene|duodene]], [[novadene|novadene]], [[marveldene|marveldene]]; these are not isometric

p = 1.1366768 [[domdimpajinjmean|miller7]], [[wilson_class|wilson_class]], [[dekany-cs|dekany-cs]]; these are isometric

 

p = 1.2651510 [[zeus8tri|zeus8tri]], [[star|star]], [[nova|nova]]; these are not isometric

 

p = 1.3404363 [[thirteendene|thirteendene]]

 

p = 1.3563125 [[wilson17|wilson17]]

 

p = 1.3652790 [[centaur|centaur]]

 

p = 1.5709365 [[zarlino|zarlino]]

 

p = 1.5865859 Cps([2,3,5,7,9,11], 3), the eikosany

 

p = 1.6426289 [[mandala|mandala]], the stellated hexany.

 

p = 1.8225500 [[zeus7tri|zeus7tri]], [[diamond5|diamond5]] the 5-limit tonality diamond; these are not isometric

 

 

p = 1.9855771 [[blue-ji|blue]]

 

p = 2 exactly all MOS scales, also [[diamond7|diamond7]] the 7-limit tonality diamond

 

p = 2.1918973 [[shell5_3|shell5_3]]

 

p = 2.4079115 [[shell5-2|shell5-2]]

 

p = 2.7580875 Cps([2,3,5,7,11], 2) and Cps([2,3,5,7,11], 3), the 2)5 and 3)5 dekanys; these are isometric

 

p = 3.1062837 [[Hexany|hexany]], [[hexagon|hexagon]], isometric

p = 4.4843144 otonal and utonal pentad; isometric

 

p = 6.9477267 otonal and utonal heptad; isometric

 

p = ∞ otonal and utonal tetrad; this implies the space is ultrametric

 

If D is the distance matrix of a finite metric space of n points, let S be the sum of elements of D. S can also be described as twice the sum of all the distances in the metric since these are counted twice in D. Then, the ''average distance'' in the space is S/(2(n-1)(n-2)), and the ''sparsity'' of the space is S/(2(n-1)^2(n-2)). The sparsity is 1 when all points are at the same distance, but otherwise less.  

Tempering will often shrink distances and so increase density. For example, the duodene has a sparsity of 0.3686. Tempering by [[Srutal|srutal]], where 2048/2025 is tempered out reduces that to 0.2860, and tempering by meantone to 0.2364. Tempering both gives 12et, and the sparsity becomes 0. To give another example, [[pentadekany2|pentadekany2]], which is Cps([2,3,5,7,9,11], 3), has a sparsity of 0.4796; tempering out 3025/3024 lowers that to 0.4772; tempering further to portent (which tempers out 385/384, 441/440 and 1029/1024 as well as 3025/3024) lowers that to 0.4521, miracle tempering brings it down to 0.4286, and 72et brings that down to 0.4282. If A and B are two metric matricies for the same set of points, then A ''dominates'' B if A - B has all coefficients greater than or equal to zero. If A dominates B and is not identical with B, then the sparsity of A is greater than that of B. In the situation above, we may regard the notes in the tempered versions as the same points as in the untempered version, and we have chains of domination, where for instance the metric matrix for pentadekany2 dominates its tempering by 3025/3024, which dominates the tempering by portent, and so forth.

An invariant related to sparsity is ''spread''. If n is a point, define the spread polynomial of n to be the sum sp(n) = ∑ t^'''d'''(n, i) over all points i, where t is an indeterminate. Then the spread is the rational function sprea'''d'''(t) = ∑1/sp(n) over all points n. Spread as a function decreases between 0 and 1, with sprea'''d'''(0) = '''P''', the number of notes in the scale and therefore points in the space, and sprea'''d'''(1) = 1. We can think of t = 0 as the highest magnification, with each of the points showing clearly, and t = 1 as the lowest, where all points have merged together. In between, at t = 1/2 or (a traditional choice, for some reason) t = exp(-1), we have a sparsity measure. Spread could use more study as it applies to scales; one notable fact for example is that most scales seem to have a spread inflection point between 0 and 1, a place where the second derivative has a local minimum. However, MOS scales do not give a spread function with such an inflection point, and it is easy to construct non-scale metric spaces where spread is not inflected. This inflection is related to the fact that spread for scales tends to be relatively large--notes tend to be far apart. Especially for larger scales, the tendency of spread to stick close to the maximum value for most values of t in the range 0 to 1 is striking.

In most instances, spread is a rational function of complicated appearance, but in a few special cases it is quite simple. We have, for instance, sprea'''d'''(Euler(3*5)) = 4/(t^3 + 2t^2 +1), sprea'''d'''(Euler(3*5*7)) = 8/(t^7 + 3t^6 + 3t^4 + 1), sprea'''d'''(hexany) = 6/(t^6 + 4t^4 + 1), sprea'''d'''(dekany) = 10/(3t^9 + 6t^7 + 1), sprea'''d'''(pentadekany) = 16/(6t^14 + 8t^11 + 1), sprea'''d'''(eikosany) = 20/(t^19 + 9t^18 + 9t^14 + 1).