Yahoo Tuning Groups Ultimate Backup tuning representation, fuzzy logic, optimization, adaptive tuning

I've recently placed a lot of my work on the web.
Much of it is tuning-related and might be of interest
to members. The tuning-related material deals mainly
with fuzzy logic, optimization, and the representation
of tunings and scales in general. In particular, there
are detailed algorithms for tuning definition,
optimization and adaptive (unstable) tuning, as well as
treatments of other topics, e.g., temperament.
Most of the tuning-related material is in section 5.5 of
the thesis (The Underlying Physical Forms: Pitch and
Pitch Organization); the short paper gives an overview
of the work as a whole, but does not deal with tuning
directly. Since members' interests in music are, no
doubt, wide ranging, many might find the rest of the
thesis interesting as well. everything is at:
http://members.fortunecity.com/odradek5/pp/rustyprogress.html
Please let me know what you think---I'm eager to hear
your opinions.

-m

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Hi Michael!

Though I'd like at some point to look at your paper in more detail, a
quick skim shows that we are in agreement on many point which I often
find myself debating on this list. I have to admit, though, that I
pretty much skipped to one section of interest. So please forgive
this knee-jerk reaction to the section. I'm playing devil's advocate
here (though I really believe what I say below).

As I see it, the three compositional paradigms you present
(Pythagorean, Equal-Temperament, and JI) are not the three most
important ones, even by the criteria you name. In fact there is
another paradigm that I was surprised to see omitted not only because
of its importance, but because in many ways it's a more
inherently "fuzzy" tuning than the ones you name. I am speaking, of
course, of the meantone paradigm.

The meantone paradigm was employed by composers beginning in the 15th
century, supplanding the Pythagorean paradigm that had exclusive
importance for centuries prior. The meantone paradigm had about 99.9%
importance in Western music of the 16th and 17th centuries, declining
in the 18th century, and the Equal Temperament paradigm began to
overtake it in the 19th century.

In the meantone paradigm, notation is the same as in the Pythagorean
paradigm, and the meaning of fifths and fourths in ratio terms is the
same (fuzzily speaking), but there's an important difference. While
thirds and sixths were not considered stable consonant sonorities in
the Pythagorean era, in the meantone era they were, and they
approximate the following ratios:

major sixth -- 5:3
major third -- 5:4
minor third -- 6:5
minor sixth -- 8:5

Now the inherent fuzziness of this paradigm is clear from the
following calculation. Starting from A as 1/1, the perfect fourth up
is D at 4/3, and the perfect fourth up from D is G at 4/3 * 4/3 =
16/9. But starting again from A as 1/1, the minor third up is C at
6/5, and the perfect fifth up from C is G at 6/5 * 3/2 = 9/5. The two
pitches for G (9/5 and 16/9) differ by the ratio of 81/80
(the "syntonic comma" or "comma") Composers in the meantone paradigm,
however, did not distinguish the two versions of the note G, or the
two and more versions of every pitch that would arise from such a
construction. Rather, they clearly operated on a fuzzy notion of what
these ratios were, allowing a few cents deviation (especially in
melodic intervals) from the ideal consonant ratios. In particular,
the perfect fourth was typically _widened_ by an indefinite amount,
the perfect fifth _narrowed_, and the minor third often but not
always _narrowed_, so that the two versions of the note G above would
come out to the same pitch (fuzzily speaking). Note that none of the
consonances need to be altered by more than a quarter of a comma if a
fixed-pitch instrument is used; if flexible-pitch instruments are
used, none of the consonances need to be altered at all unless all
four notes (A, C, D, G) are played simultaneously, and a
melodic "fuzziness" of about a quarter of a comma in each pitch can
accomodate the paradigm instead.

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. In particular, the diatonic scale in fixed meantone
temperament will have more consonant thirds, and consonant triads,
than any fixed tuning of the diatonic scale in JI. Furthermore, if
the tuning is not fixed, a meantone paradigm will allow for the same
degree of consonance (that is, perfection) as a JI paradigm but with
a much smaller degree of fuzziness in the size of notated intervals
and pitches.

I'll stop now for surely you know most or all of the above, and you
can help me focus in on what aspect of these issues might be relevant
for your project, if any. Also, feel free to change the subject and
ask about any other aspect of tuning theory/practice that you might
be interested in learning about/getting more feedback on.

Congratulations on the great work and looking forward to many
stimulating discussions to come!

-Paul

representation, fuzzy logic, optimization, adaptive tuning

🔗Michael Saunders <michaelsaunders7@hotmail.com>

🔗paul@stretch-music.com

🔗Michael Saunders <michaelsaunders7@hotmail.com>

🔗monz <joemonz@yahoo.com>

🔗paul@stretch-music.com