RE: [tuning] metrics: tenney, barlow, etc...

Paul Fly wrote,

>but i am at a loss to understand; what is the 'minima of all
>dimensions' in a lattice?

Scholz means the point in the lower left corner (in 2-d; the generalization
to higher dimensions is obvious) of the smallest rectangle enclosing the
pitch set -- in other words, the greatest common divisor (GCD) of the notes
when expressed in integer form.

See also http://www.ixpres.com/interval/dict/harmonicdistance.htm and
http://www.ixpres.com/interval/dict/complex.htm ("indigestibility" is,
unfortunately (or perhaps fortunately!), offline).

A justification for Tenney's harmonic distance for dyads (assuming
octave-equivalence is _not_ invoked):

It's the only complexity function I know of for which the following holds:
(a) Take all ratios with less than a certain complexity
(b) Calculate the mediant-to-mediant "width" for each ratio (i.e., the size
of its "domain")
(c) The width of the ratio can be predicted to within a factor of 2 (and
much better for the relatively simple ratios), as a function of the
complexity of the ratio (using the _same_ definition of complexity).

For most complexity functions, this property is fulfilled except that a
_different_ complexity function predicts the width. For example, if
complexity of the ratio n/d in lowest terms is defined as n (as we've seen
-- "the Farey series"), then width is predicted by 1/d. If complexity is
defined as n^2 + d^2, then width is predicted by sqrt(1/n^2 + 1/d^2).

But for the Tenney harmonic distance n*d, the width is predicted by
1/sqrt(n*d), which is a function of the Tenney harmonic distance! In
particular, if one takes all ratios n/d such that n*d<T (harmonic distance
is less than T), then the width of each ratio will be between 1/sqrt(n*d*T)
and 2/(sqrt(n*d*T), with the widths of the relatively simple ratios tending
to be very close to the latter.

So in a sense, the Tenney harmonic distance (or any monotonic function
thereof) is the only self-consistent complexity measure for dyads (that I
know of). In fact, the factor of 2 above could be increased to any finite
number and still none of the other complexity functions I know of would
qualify.