Subject: Re: representation, fuzzy logic, optimal tuning, adaptive tuning

>> >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?

With them (and with the two others I described, overtone and acoustic),
it's easy to generate intervals with a few parameters without reference
to the construction of an entire scale (which comes a few pages later).
My list of intervallic tuning paradigms was never meant to be exhaustive.
I represent intervals in a very general way, as fuzzy quantities of pitch
distance. The classes in the section you're looking at are just examples
of handy ways to produce interval objects by parameter.

>>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....
>All meantone intervals are then just a certain number of meantone fifths >plus a certain number of octaves

But is there a certain fifth that you think of as a "meantone fifth"
(not just a perfect fifth) that has its own quality and that you have
a formula for generating?

The section on temperaments describes this in detail---how rusty's
usual method of tuning is the reverse of the historical method.
That section also has a more general version of the algorithm you
suggest, finding cycles and sequences of target intervals, calculating
"commas" and assigning crisp tempers---but that's getting ahead of
the story!

>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.

Maybe I'm not familiar enough with the method you're using; if it's
really different from the algorithm I described in the temperament
section, I should probably include it. Maybe I should include it even
if it isn't, but my general idea was interval definitions first, optimization
second.

>I've done many similar optimizations.

yes---I think those are fasicnating. Do you use dissonance curves?
I thought that my curves should be user-definable---matters of opinion.

For the record---though the date on that version of the thesis says 2001,
I actually wrote it all 3--6 years ago (I submitted in 9/98). After all
that time, I still don't know if the correct spelling is intervallic or
intervalic.

-m
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--- In tuning@y..., "Michael Saunders" <michaelsaunders7@h...> wrote:
>>
> >Hmm . . . so what is it about Pythagorean, 12-tET, and JI that put
them in
> >the former category?
>
> With them (and with the two others I described, overtone and
acoustic),
> it's easy to generate intervals with a few parameters without
reference
> to the construction of an entire scale (which comes a few pages
later).

I don't see how meantone is any different.

> My list of intervallic tuning paradigms was never meant to be
exhaustive.
> I represent intervals in a very general way, as fuzzy quantities of
pitch
> distance. The classes in the section you're looking at are just
examples
> of handy ways to produce interval objects by parameter.

So perhaps meantone would fall under the Pythagorean umbrella?
>
> >>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....
> >All meantone intervals are then just a certain number of meantone
fifths
> >plus a certain number of octaves
>
> But is there a certain fifth that you think of as a "meantone fifth"
> (not just a perfect fifth) that has its own quality and that you
have
> a formula for generating?

Sure -- there are a number of possibilities. For example, one can run
a least-squares optimization on the three errors: the deviation of
the fifth itself from 3:2, the deviation of the major sixth generated
by three such fifths (minus an octave) from 5:3, and the deviation of
the major third generated by four such fifths (minus two octaves)
from 5:4. The result is 7/26-comma meantone temperament (Woolhouse
1835), i.e. (3/2)*(80/81)^(7/26).
>
> The section on temperaments describes this in detail---how rusty's
> usual method of tuning is the reverse of the historical method.
> That section also has a more general version of the algorithm you
> suggest, finding cycles and sequences of target intervals,
calculating
> "commas" and assigning crisp tempers---but that's getting ahead of
> the story!

I need to catch up!
>
> >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.
>
> Maybe I'm not familiar enough with the method you're using; if it's
> really different from the algorithm I described in the temperament
> section, I should probably include it. Maybe I should include it
even
> if it isn't, but my general idea was interval definitions first,
> optimization
> second.

That's my idea too, if I'm understanding you correctly.
>
> >I've done many similar optimizations.
>
> yes---I think those are fasicnating. Do you use dissonance curves?
> I thought that my curves should be user-definable---matters of
opinion.

I have a whole list on the dissonance curves I use. It's
harmonic_entropy@yahoogroups.com.

>
> For the record---though the date on that version of the thesis says
2001,
> I actually wrote it all 3--6 years ago (I submitted in 9/98).

Cool. I see you make a lot of points that I often argue in favor of
on this list.

BTW, you may be fascinated by John deLaubenfels' adaptive tuning
methods: www.adaptune.com.