New York meeting/ German Parade

I will be playing with the Mad Bavarian Brass Band Saturday 21/Sep/96
in New York, for a very big German-American parade. You can meet me in
the big tent at 2PM. The adress is 96, fifth avenue. I will arrive with
the parade, and will wear white socks, black shorts, white shirt. I will
be playing trumpet, some of you probably knew that already. You are all
invited to follow the band, bring your instrument if you think you can
fit in the group. The evening will be in a hotel, so you dont have to
worry about where you will sleep. I think it costs 5$ to get in the
tent, and if you want a hotel room, you will have to pay a part of the
price. This is our chance to meet other xenharmonists.

Talk to you soon,
-Kami

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Thanks for your comments, Paul (E). After four days I was beginning to
wonder if anybody gave a damn about my goofy ideas.

On Tue, 17 Sep 1996, PAULE wrote:
> The musical meaning of integer levels of
> consistency higher than 1, namely, the importance of matching just
> intonation in combinations of n-limit intervals, I find dubious. So level 2
> consistency seems like an arbitrary requirement, as level 2.1 consistency
> and level 1.9 consistency would lead to results which are slightly different
> but just as meaningful musically.

This is a good question, and I'm not sure if I have a good answer for
it. Probably the best answer I can give is this: the main thrust of my
theoretical wanderings in xenharmony is to discover tunings that are
analogous to common practice harmony, in the sense that they have
similar structures according to various hypothesized perceptual models.
Common practice 5-limit harmony did not find 5TET or 7TET sufficient for
its purposes, despite the fact that they are level 1 consistent at the
5-limit; it settled on 12TET, which is level 3 consistent. Therefore, I
tentatively conclude that there is some kind of advantage to higher
levels of consistency.

A possible explanation is this: an ET which is level 1 consistent at a
given limit may err from a just interval within that limit by up to
half a step. It seems to me that my ear, at least, would not readily
accept such an ET as an approximation to that interval when almost any
interval randomly chosen from the continuous gamut would be more closely
approximated by that ET. Since the average error of random intervals is
1/4 step, level 2 consistency at the N-limit ensures that N-limit
intervals will be better approximated than the average randomly chosen
interval. Thought of in this way, one can justify choosing level 2 as a
lower limit on consistency.

Does that make any sense?

--pH (manynote@library.wustl.edu or http://library.wustl.edu/~manynote)
O
/\ "Foul? What the hell for?"
-\-\-- o "Because you are chalking your cue with the 3-ball."

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Paul H.,

I believe your original definition leads to consistency for level
N>1.5. For 1will be consistent with one another.

I don't think the concept of consistency has much to do with whether a
particular JI interval is approximated well enough to be used as a harmonic
consonance. The absolute accuracy of the approximation is the determining
factor, not the accuracy relative to the size of the smallest steps in the
tuning. For example, the 12-tET major third sounds like a 5/4 even if 12-tET
is considered a subset of 72-tET, where the interval is off by an entire
step. I have derived an algorithm for determining the certainty with which
an arbitrary interval is perceived as one or another JI intervals, based on
J. L. Goldstein's 1973 paper, "An optimum processor theory for the central
formation of the pitch of complex tones." Manual Op de Coul has ingeniously
integrated this algorithm into Scala, under the Farey command. I leave it to
him to provide instructions. Using this algorithm, we find that for any
listener across the entire gamut of frequency resolution the most prominent
interpretation of the a 7-tET "third" will be 11/9, and the most prominent
interpretation 7-tET "tenth" will be as 17/7. Listeners at one extreme of
the gamut will perceive the 7-tET "sixth" as 5/3, but those at the other
extreme will hear it as 18/11. Thus the typically considered "imperfect
consonances" in triadic harmony are not well represented in 7-tET.

What consistency offers is a supplement to considerations of how good
the approximations to JI are. Essentially, composing with a consistent
tuning will be no more difficult than composing in JI, while in inconsistent
tunings, complications may arise if one attempts to always use the best
approximations to JI intervals.

- Paul E.


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On Thu, 19 Sep 1996, PAULE wrote:
> Paul H.,
>
> I believe your original definition leads to consistency for level
> N>1.5. For 1> will be consistent with one another.

Yes, I noted that in my "erratum" followup message. However, the
algorithm I just posted in response to John Chalmers is correct.

> I don't think the concept of consistency has much to do with whether a
> particular JI interval is approximated well enough to be used as a harmonic
> consonance.

I don't think that's exactly what I meant, or even what I said. But
anyway, see below.

[bigsnip]
> What consistency offers is a supplement to considerations of how good
> the approximations to JI are. Essentially, composing with a consistent
> tuning will be no more difficult than composing in JI, while in inconsistent
> tunings, complications may arise if one attempts to always use the best
> approximations to JI intervals.

Here's where we part company, I'm afraid. I find level 1 consistency to
be far too low a standard. Very weird things can happen in level 1
consistent tunings. A simple example: 5TET is consistent at the
5-limit. A 16/15, represented as a 4/3 less a 5/4, becomes 2-2=0 steps,
or a unison. However, a 25/24, represented as a 5/4 less a 6/5, becomes
2-1=1 step. In other words, a larger interval becomes a unison while a
smaller interval does not. Maybe this doesn't bother you, but for me it
causes cognitive dissonance.

--pH (manynote@library.wustl.edu or http://library.wustl.edu/~manynote)
O
/\ "Foul? What the hell for?"
-\-\-- o "Because you are chalking your cue with the 3-ball."

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On Thu, 19 Sep 1996, PAULE wrote:
> >the average error of random intervals is
> >1/4 step [given] level 2 consistency . . .
>
> Paul H., surely the average error is less than 1/4 step, since that is the
> maximum error. [snip]

Er, I think you've misunderstood me. (Gotta watch those comma splices.)
What I meant in the above, up to the word "step", was just what you say
below:

> Of course, the mean absolute error of random intervals, without
> assuming any consistency at all, is .25 step, since all values from 0 to .5
> step are taken on with equal probability.

In other words, I _meant_ "random intervals, without assuming any
consistency at all". The bit about level 2 consistency belonged to a
different clause.

--pH (manynote@library.wustl.edu or http://library.wustl.edu/~manynote)
O
/\ "Foul? What the hell for?"
-\-\-- o "Because you are chalking your cue with the 3-ball."

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On Fri, 20 Sep 1996, it was written:
> > In particular, no tuning can fail to be level
> > 1 consistent according to your algorithm, and some tunings would be level 2
> > consistent according to this algorithm that wouldn't be according to your
> > original definition.
>
> Er, pardon me but it _is_ correct; at least, it is cribbed directly from
> the code that generated the table at
> , and I haven't
> discovered any errors in that table yet.

That, of course, was not the most convincing answer in the world. Here
then, a formal proof:

Define Approx(N, P) on positive odd P and positive N as

Approx(N, P) = round(N * log2(P))

Define Error(N, P) as

Error(N, P) = N * log2(P) - Approx(N, P)

My algorithm searches for the values of P within a given limit having
the least and greatest values of Error for a given N[-TET]. If the
different between them is greater than 1/(2 * L), it declares N-TET to
be level L inconsistent.

It is fairly obvious that anything the algorithm rejects as level 1
inconsistent is so. Assume that Error(N, P1) - Error(P2) is greater
than 1/2. Now consider the triad formed by the root, P1, and P2.

round(N * (log2(P1) - log2(P2)))
=
round((Approx(N, P1) + Error(N, P1)) -
(Approx(N, P2) + Error(N, P2)))
=
round((Approx(N, P1) - Approx(N, P2)) -
(Error(N, P1) - Error(N, P2)))

Approx is always integer. Any integer plus a value greater than 1/2
will not round to the original integer. Thus, this triad is
inconsistently represented.

So, all the algorithm's rejects are inconsistent, but might some that it
accepts also be inconsistent? Let us examine an arbitrary triad P1, P2,
P3 in a tuning which is accepted at level 1 by the algorithm. If the
algorithm accepted the tuning, then by the above,

abs(Error(N, P1) - Error(N, P2)) < 1/2
abs(Error(N, P1) - Error(N, P3)) < 1/2
abs(Error(N, P2) - Error(N, P3)) < 1/2

Therefore, examining the interval P3/P1 = P2/P1 * P3/P2,

round((Approx(N, P3) + Error(N, P3)) -
(Approx(N, P2) + Error(N, P2))
+
round((Approx(N, P2) + Error(N, P2)) -
(Approx(N, P1) + Error(N, P1))

=

(Approx(N, P3) - Approx(N, P2)) +
(Approx(N, P2) - Approx(N, P1))

=

Approx(N, P3) - Approx(N, P1)

=

round((Approx(N, P3) + Error(N, P3)) -
(Approx(N, P1) + Error(N, P1))

therefore the triad P1, P2, P3 is consistent.

The same reasoning can be extended to higher consistency levels.

--pH (manynote@library.wustl.edu or http://library.wustl.edu/~manynote)
O
/\ "Foul? What the hell for?"
-\-\-- o "Because you are chalking your cue with the 3-ball."

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