Reply to Gary Morrison; Meeting in NYC

>One of Ivor Darreg's observations about 17TET is that it reverses one of
>the
>usual premises of harmony. In particular, the major third is such a
dreadful
>dissonance that it must be resolved outward to fourths. That then makes it
a
>good candidate for quartal and quintal harmony (Ivor didn't say that
>specifically though). So it's not necessarily a case of putting any kind
of
>harmony behind a 17TET melody sounding "YUCK!", but triadic harmony in
>particular.

Very true! Or you could resolve the major third outward to a fifth, as was
done by Medieval polyphonists working in Pythagorean tuning.

>> I just happened to note that theorists who derive the diatonic scale from

>> three triads are perpetrating a historico-geographic fallacy.

>Is that the topic of your up-coming Xenharmonikon paper? It sounds
>interesting, since that's one of two ways that come to mind immediately as
the
>basis for the construction of the traditional major scale. The other of
>course
>is tetrachord. Since there are big dissimilarities between the scales
>constructed by those means, and since the scale itself came about before
>triadic
>harmony, I can imagine that there is fertile ground for a paper there.

I just added this to my paper as an aside, but it is a good reflection of
the philosophy of the paper in general. Of course the major mode was not one
of the favorite modes of the diatonic scale until triadic harmony had been
in use for some time -- the tritone resolves to the tonic triad only in the
major and minor modes. My paper is an attempt to find a set of pitches
which, like the diatonic scale, stands up on a melodic basis alone, but
leads as naturally to 7-limit harmony and 7-limit tonality, as the diatonic
scale leads to 5-limit harmony and tonality, and (if you believe Yasser was
on the right track) as the pentatonic scale leads to 3-limit harmony and
tonality. The attempt succeeds even though I look no farther than the first
34 equal temperaments (the solution is found in 22-equal).

This brings me to recount the events of this weekend. Johnny Reinhard,
Richard Kassell, Adam Silverman, and I met in Manhattan on Saturday for a
Chinese-Latin lunch. Much enlightening discussion ensued. I spent much of
the rest of the day with Johnny. His wife is a world-class microtonalist in
her own right; we brought in her defective Kurzweil K2000 only to find out
that the model is no longer in production -- it has been superseded by, if I
recall correctly, the VX-24, which has additional expansion ports or
something. Anyway, Johnny practically drowned me in bliss with his
collection of music and written materials. Our dinner discussion over sushi
found him confirming my view that the melodic properties of the diatonic
scale are best reflected by Pythagorean intonation, its harmonic properties
by just intonation, and the integration of the two by temperament (this
could mean meantone temperament, not necessarily 12-equal). He associates
these aspects with the left-brain, right-brain, and whole-brain functions,
respectively. (I may have reversed the first two.)

Gary: I believe you are right in pointing to the tetrachord as the basis of
the diatonic scale. The reason this favors Pythagorean tuning is that only
in this tuning does each and every octave species contain two identical
tetrachords. (The chromatic and enharmonic tetrachords, of course, fail to
produce a scale with this property). The pentatonic scale and my new scale
have this property as well. It is worth pointing out, however, that many
theorists have instead found the basis in a property one might call "maximal
evenness," or spacing two unequally-sized steps as uniformly as possible
around the octave. Although this can lead to the diatonic and pentatonic
scales if 12-tet is presumed (another historical fallacy), it leads to a
scale in 22-tet that is slightly different from the one that is constructed
according to "tetrachords." Both scales are so interesting harmonically that
I have allowed for either derivation in my paper. I do think the
"tetrachordal" one is superior melodically, though.


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