To Daniel Wolf, re Heinrich Schenker

I believe that I'm going to disappoint you. But I have to confess that I was
certainly not carried by Schenkerian spirit when I did my scale
constructions. The reason is simple: I didn't know about his work. I was
simply led by the idea that a harmonic scale results from the attempt to fill
the framework of a large consonant interval (for instance the octave)
with a chosen triad (for instance the Major) and its sub-intervals. The
smallest intervals created in this process are then used to fill the
voids and achieve approximate equidistance. This infill has practicality
for musical use as its reason. Alright, that sounds rather mechanistic, but
doesn't already the choice of the triad introduce tonality into this
process?

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Dear me, dear me. I've been lurking on this list for a few weeks,
waiting to get a feel for it, but I must rise up now in defense of
mathematics. Yo, Brian! Get yourself a good book on Number Theory.
Hardy & Wright is a good choice. (Yes, that's *Number* Theory, not
*Group* Theory. The integers modulo N do form a group under addition,
but that is not relevant to your post. Issues of compositeness and
relative primality are pure Number Theory.) In said book you will
discover that 7 mod 12 -5, not 5, and that yes-indeedy-do there are 12
(count 'em: 12 !) distinct equivalence classes modulo 12.

Faulty whole-number mathematics on the tuning list? For shame!

Gordon Collins
Yet another harpsichordist who has forever abandoned the harsh world of
12TET


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> The integers modulo N do form a group under addition,

Hmmm... They are certainly closed over addition, and there certainly
seem to be identity and inverse elements. But I can't recall what else is
required for them to constitute a group.

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