About metrics

It seems there is confusion about metrics.

A metric measure something. One have to reflect clearly what is supposed to be measured and for that, to distinguish first the pertinent entities.

At microtonal level, i.e. independently of any interactional system, one have to reflect, the most appropriately possible, what corresponds to the perception.
It is well known there exist two irreducible manner to order a set of pitch height intervals. One could call that
the ordinary or melodic order and the harmonic order.
There exist in this way two distinct distances to be measured, one on each ordered structure. One could call that
the melodic distance and the harmonic distance.
At macrotonal level, i.e. when rules are added on how interact the elements, one have to reflect the geometry of the created global entity. In pure JI, one can simply use the Euclidean distance, once is determined the quadric or hyperquadric defining the scalar product and the orthogonality on that space, created by choice.

Microtonal level

There is a unique representation of a rational interval X in an appropriate primal basis <pi>.
X = <pi> (xi) = (p0^x0) (p1^x1) ... (pN^xN)
I argue that the simplest appropriate distances between the intervals X and Y are
Ordinary or Melodic Distance
| sum ( ( yi - xi ) log pi ) |
Tenney or Harmonic Distance
sum ( | yi - xi | log pi )
I remark also that the harmonic distance concept has no sense in the real segment of pitch heights.

Macrotonal level

Suppose one choose to use heptatonic modes in prime 5-limit. One knows that the simplest val is [7 11 16], which is simply
Round ( 7 [log 2 log 3 log 5] / log 2 )
Using X = (w,x,y) rather than (x0,x1,x2), that means each interval has now a class or degree D(X) = 7w + 11x + 16y, or simply 4x + 2y (mod 7) in the octave.That means also the set of unison vectors, i.e. all the intervals having the degree 0 in the octave, is a sublattice of the <3 5> ZxZ lattice.

Is it sufficient to define a system ? No.

Defining a system implies equally the choice of the minimal unison vectors around the unison which determines the fundamental domain of that system. In a plane, like here, that implies the choice of the six " nearest " unison vectors giving an hexagonal shape in which the unison, at the center, is the unique interval having the degree 0.

Nearest here implies an undefined macrotonal metric. One have to define a macrotonal distance permitting to say which among such vectors are the nearest of the unison. There are different possibilities. Each choice determine a distinct system.

Suppose one choose to get the " simplest ".

Can one only observe the skewness of possible hexagons to determine that? No.

Simplest can only have a microtonal sense since the macrotonal metric is not yet chosen. It's where the harmonic distance has to be used. Comparing the harmonic norm (i.e. harmonic distance from unison) for all intervals of each hexagons, it appears clearly here that the simplest case corresponds to unison vector U = 81/80, V = 25/24, U+V and the three inverses. What is well-known to correspond at the Zarlino system with modes using 16/15, 10/9 and 9/8 as steps.

An Euclidean space is defined as a vector space with a scalar product, which in turn is defined by the choice of a bilinear form corresponding to a quadric or hyperquadric. The appropriate choice to reflect the geometric properties of such tonal systems is to fit the quadric on the chosen minimal unison vectors.

The ellipse
3 x^2 + 13 y^2 + 3 xy = 49
passes on the six chosen unison vectors. Using then the matrix M = [(3, 3/2) (3/2, 13)], the scalar product
(X|Y) = X M Y
define an appropriate structure of Euclidean space with the correlative Euclidean distance, so there exist an orthonormal basis in which the ellipse is a circle of radius 7, and the distance of (x,y) from unison, for instance, is defined as sqrt (x^2 + y^2).

Pierre