RE: P. Erlich's comments

Well, maybe we won't agree to disagree just yet.

>"Best" has a strict mathematical, as opposed to musical definition.
>Consequently, the most desirable musical result may conflict.

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.

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

I'd be interested in hearing more about this.

>There is no more inherent reason to expect the m3 and M3 to "behave
>properly" than there is for cycles of P5 or for the M3 and P5 together.

To behave properly is to behave as in just intonation. In just intonation,
cycles of P5 don't close or form just M3s, so there is no inherent reason to
expect equal temperaments to behave this way. In just intonation, a M3 plus
a m3 is a P5, so this is a reasonable expectation of any equal temperament
in which we hope to use 5-limit harmony. If this expectation is not
fulfilled, don't use 5-limit symbols to notate that equal temperament. Or if
you do, at least don't call them "best."

>Some tunings, like 25 and marginally like 17, may have two minor thirds.
>Tunings with large numbers of units in the octave may have several of
>everything. I don't find this a fault of notation in general or of my
>proposal.

Neither do I. In fact, I think your method achieves the goals of being both
"general" and "standard." Using new symbols for various alterations is fine,
but naming them after various commas is useful only when the relevant
structural characteristics of the tuning mirror those of just intonation.
Otherwise it is just misleading. You didn't address my point about
fractional alterations, which seem just as necessary, and often at least as
meaningful, as the "commas," and my request to clarify your criticism of
Herf's and Sims' notations. What wrong with using lots of fractional
alterations, whose ordering is intuitive, combined with an understanding of
how they represent the syntonic comma, septimal comma, etc., as opposed to
using combinations of 12-based commas, with possibly bizarre orderings,
combined with an understanding of how these commas may fail to fulfill their
acoustical functions?


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