representation, fuzzy logic, optimal tuning, adaptive tuning

thank you paul, it's good to hear from you!

>...there is another paradigm that I was surprised to see omitted ... the >meantone paradigm.

there are two reasons for that---
1. the section you're reading is concerned with the representation
of intervals themselves, not with the representations of scales and
tunings (that comes a bit later). i don't know of any systematic
way to generate meantone intervals, as such, outside the context
of the complete scale (but cf. with the later section of the thesis
called "temperaments").
2. the paradigms that i outlined were only intended to be examples
of handy classes for generating intervals according to familiar
paradigms; i could add others later.

>In the meantone paradigm, ... the meaning of fifths and fourths in ratio >terms is the same (fuzzily speaking)...
>Now the inherent fuzziness of this paradigm ...
>they clearly operated on a fuzzy notion of what these ratios were

yes, exactly. this is where it gets interesting. in rusty, i represent
intervals (and all other quantities) fuzzily---as functions (valued
on [0,1]) of goodness (or "appropriateness" or "suitability")
vs. pitch-distance. so, a perfect fifth, say, might be represented
by a fairly broad curve peaking at 3/2. in the later sections, you'll
see how one can define a tuning as a network of fuzzy intervals
and let my labrynthine optimization algorithm make all those
compromises in ratio automatically, according to the slack offered
by the breadth of the curves---if you get my meaning---doing what
the old tuners used to do by ear and trial-and-error.

>In the beginning of your section on Just Intontation, you write,
>"Since it is the tuning which is most often consonant".

>Now there are some problems with this statement. I assume that
>consonance is to be a fuzzy characteristic, so that small deviations
>of a fraction of a comma will only slightly reduce the truth-value of
>whether one of the consonant ratios mentioned above is still
>consonant.

yes. i meant that in two senses:
1. a (necessarily) simple ji scale won't have to make many compromises,
2. ji is often consonant because instruments often have harmonic spectra.

also, i guess i was reacting against the 12tet world a bit!

thanks for your comments and thanks for reading about my work---
i hope you enjoy it as much as i did!

---m

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--- In tuning@y..., "Michael Saunders" <michaelsaunders7@h...> wrote:
>
> thank you paul, it's good to hear from you!
>
>
> >...there is another paradigm that I was surprised to see
omitted ... the
> >meantone paradigm.
>
> there are two reasons for that---
> 1. the section you're reading is concerned with the representation
> of intervals themselves, not with the representations of scales and
> tunings (that comes a bit later).

Hmm . . . so what is it about Pythagorean, 12-tET, and JI that put
them in the former category?

> i don't know of any systematic
> way to generate meantone intervals, as such, outside the context
> of the complete scale

Hmm . . . I may be able to help you there. As you know, I've done a
lot of meantone optimizations and such, and never had to refer to a
complete scale . . . just a chain of four fifths at most. All
meantone intervals are then just a certain number of meantone fifths
plus a certain number of octaves (assuming that's what you mean
by "generate")

> (but cf. with the later section of the thesis
> called "temperaments").

I'll have to look at that and see what I can grok.

> 2. the paradigms that i outlined were only intended to be examples
> of handy classes for generating intervals according to familiar
> paradigms; i could add others later.

OK -- I think the meantone paradigm _still_ has great importance for
most Western classical musicians, who continue to distinguish between
augmented sixths and minor sevenths, for example.
>
> yes, exactly. this is where it gets interesting. in rusty, i
represent
> intervals (and all other quantities) fuzzily---as functions (valued
> on [0,1]) of goodness (or "appropriateness" or "suitability")
> vs. pitch-distance. so, a perfect fifth, say, might be represented
> by a fairly broad curve peaking at 3/2. in the later sections,
you'll
> see how one can define a tuning as a network of fuzzy intervals
> and let my labrynthine optimization algorithm make all those
> compromises in ratio automatically, according to the slack offered
> by the breadth of the curves---if you get my meaning---doing what
> the old tuners used to do by ear and trial-and-error.

I've done many similar optimizations. For example, I've found that if
you start with a 7-tone-equal-temperament scale, and roll down the 42-
dimensional dissonance surface, you'll end up with a diatonic scale
in a quasi-meantone temperament (with the central fifths tempered the
most and the outer fifths tempered the least).