RE: TUNING digest 802

Paul R,

What symbol did I use multiply?

>> Here's a definition of consistency: Given an odd number n, an
octave-based
>> equal temperament is consistent within the n-limit if, for any odd
numbers
>> a, b, and c such that 0approximates
>> a:b plus the number of steps that best approximates b:c is equal to the
>> number of steps that best approximates a:c.

>Multiple uses of the same symbol make this confusing or actually
>impossible to figure out. Thanks, though, for the earlier references.

I'd be happy to clear this up, but I don't see what your complaint is
directed at.

>> Exactly. Here's the implicit derivation of your strict mathematical
>> definition: the syntonic comma is (3:2)^4/(5:4) (ignoring octaves);
>> therefore the best syntonic comma is 4 times the best perfect fifth minus

>> the best major third, mod the number of notes per octave. Another, just
as
>> musically relevant and just as mathematically strict, definition would
be:
>> the syntonic comma is (3:2)^3/(5:3) (again ignoring octaves); therefore
the
>> best syntonic comma is 3 times the best perfect fifth minus the best
major
>> sixth, mod the number of notes per octave. If the two definitions lead to
a
>> conflict, I see no reason one should take precedence. Therefore, in such
>> cases, there is no "best" syntonic comma.

>Perhaps someone else would like to explain the precedence of 1:5 over
>3:5. It's fairly clear to me, which does not invalidate attempts to base
>a system on 3:5. I don't know whether something could be concocted to
>include both in all cases.

That's precisely the point. In a case, like 20tet, where the two approaches
lead to different results, then the tuning is simply not compatible with a

just-intonation view. Defining a "best" size for the syntotic comma is
futile in such systems. The precedence of 1:5 over 3:5 seems consistent with
an Euler-Fokker, or square lattice, philosophy but not an Erv Wilson, or
triangle lattice, philosophy. I have investigated the relationship between
the two approaches in terms of lattices and made some geometrical
discoveries in the process. Both approaches extend to any number of
dimensions -- perhaps I'll write a paper on this someday. Paul, I guess your
notation system succeeds within the Euler-Fokker philosophy, but I lean
towards a more egalitarian one.

> >The method . . . . also allows for an improvement in Blackwood's
notations
>> in a few cases.

>Blackwood's notation for 16, 18, and possibly a few others isn't
>consistent in potential use on all steps of the tuning. (I haven't his
>work at hand as I write this.) Examining the scales he writes out at the
>beginning of each piece in the 12 etudes should show what I am getting
>at.

Ah, I'll bet this derives from his desire to preserve standard notation of
diminished scales. See his PNM article.

-Paul E.


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