Re: Complexity formulae; Monzo's lattice diagrams

I wrote,

>According to [Dave Keenan], that sum models critical-band roughness
very well up to ratios of 17

Re-reading Dave's e-mail to me, I see that the sum only models
critical-band roughness well if the _sum_ is 17 or less. So intervals
like 11/8 and 13/8 are already too complex for this type of formula, as
they are too close to other ratios of similar complexity to be clearly
distinguished from them. Note that my harmonic entropy model, to the
accuracy that I've evaluated it so far, did not show local minima at
11/8 or 13/8, but there was a very tiny one at 11/6.

Joe Monzo noted that he could not fit 11/6 in with the other minima in a
3-d lattice. To the entire enterprise of creating that lattice, I would
reply that the minima are not really a scale, except that they represent
the points you might stop at if tuning one note against a fixed pitch.
They are not conceived in relation to one another, although many of them
happen to be quite consonant with one another. Lattice diagrams are
useful when trying to grasp _all_ the interrelationships within a scale.
The harmonic entropy concept can be applied to all those
interrelationships, not just the ones formed with a single tonic. So if
you were really going to derive a scale (for a type of music in which a
lattice is relevant) from the harmonic entropy concept, you would not
simply take the minima and construct them all upwards from a single
tonic.