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 = 1.8501138 [[raven|raven]]

 

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