RE: [tuning] Re: MOS's, generators

I wrote,

>>I don't think that criterion could help you decide whether n for a
>>Pythagorean diatonic scale should be 12 or 17 (or 29, . . .)

Jason wrote,

> That's not what the criterion is for. It's a criterion for
determining
>what are good fits, of which there are always more than one. The decision
>of how to represent the Pythagorean diatonic could simply be, take the
>lowest n which gives a good fit (since this is a translation between one
>mathematical description and another, we have to give a tolerance in this
>case. For translations from heard music to a representation, other factors
>such as cultural indoctrination may also play a role, but I presume that we
>will be able to specify by some other criteria which intervals must be the
>same, and which different).

So you're saying that _all_ scales are perceptually forced into the lowest
equal temperament which gives a good fit? That kind of assumption may sit
well with theorists like Balzano and Clough-Carey-Clampitt, who've
"explained" a lot of things on these sorts of bases, but I don't buy it, not
one bit -- there's always a better, more acoustical explanation, with fewer
arbitrary assumptions.

Equal temperament is a contrivance that offers many great benefits, but I
don't believe it is of perceptual origin (though in the _particular_ case of
the Pythagorean scale the fit to 12 is so good that it is hard to avoid,
even perceptually).

>Yes, and this fact gives us a useful method of finding the generator of
>the scale at any step of the process. But here's where I cash in on the
>observation above that each choice of a previous number of places to go
>back to find the number to add to the last, given the weak criterion, gives
>a unique l:s in the following scale: to explain the value of the strong
>criteria. If the choice is always 1, then l:s is always 2:1.

??? Actually, it approaches 1:phi, unless you're dealing with a Yasser-type
situation where you use an equal temperament which is chosen by using the
next term in the sequence (i.e., using an additional "choice of 1").