TUNING digest 801

> 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 0> 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.

> 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.

> >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.

> >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.

Interesting definition, but too narrow for me. I had nothing specific in
mind, although I was thinking of ETs, which behave differently from JI in
many ways, of course. We might tentatively talk about having one and only
one amount of P5s create a M3. We might even think of proper as having
only one cycle of P5s. Of course many tunings don't have either of these
characteristics. And many don't have their best M3 and m3 add up to a P5.

> You didn't address my point about
> fractional alterations

Didn't seem anything to address. I have no theory of fractional
alterations, especially when they should or should not be used.

> my request to clarify your criticism of
> Herf's and Sims' notations.

Another time, perhaps more detail will be possible. I am fairly busy but
will try to get to this.

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.,

There may be too many fractional parts or such a method may obscure
something else important if it is the only one used. They may be fine; I've
proposed some myself. See above comment.

> 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?

Perhaps that is a code for consistency again. (Acoustical functions?) I
agree that there might be a bizarre number of commas in various bizarre
orders, since everything is currently strange in such cases. All I tried
to do was derive a general system to see where it could be pushed. It is
likely that something else is better in some or even many cases. If so,
I look forward to seeing it.

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