Approaching TD 1000!

Hi Tuning List,

As those of you that have been around a while know, every once in a
while (it has been a long time now!) I post a short statistical summary
of the Tuning list. It seems that as the list approaches Digest 1000
(just three more to go!) such a posting would be appropriate, with
perhaps a bit of historical stuff.

The list was started on February 2 1994 by Greg Higgs and myself, and
announced on USENET. By the Feb 3 we were congratulating ourselves
because we had 25 subscribers! In less than a year, the list had grown
and brought the machine it was originally implemented on (a NeXT "Turbo
slab") to a virtual standstill, so it was moved to its current home
(eartha@mills.edu).

As of Feb 23rd, 1997, the list had 350 subscribers, distributed around the
globe/nets as follows. (List membership normally oscillates between a
high of about 360 and a low of about 325. That is, it is about 90%
stable.)

Austria 3, Australia 18, Canada 13, Switzerland 2, Commercial Providers
100, Germany 12, US Education 101, Spain 3, Finland 1, France 4, US
Government 2, Greece 2, Hong Kong 2, India 1, Italy 2, Japan 4, Korea
1, US Military 1, Mexico 1, "Network (.net)" 30, the Netherlands 10,
Norway 1, Non-profit Organizations 9, Poland 1, Sweden 3, Singapore 1,
United Kingdom 20, "United States" 1, South Africa 1.

(Note: there is no way of geographically locating the "Commercial"
subscribers, although a majority seem to be in English speaking
countries.)

I would like to comment here that, although the US Education subscribers
have the barest of pluralities(!), the shift toward commercial and
international subscribers over the last couple of years has been
dramatic. It seems that for the first year or so, about 75-80% of the
subscribers were US Education (.edu). Now they are about 28%.

There have been 5695 messages posted to the list, and the archive now
contains 15.3 Megabytes of data (including mail headers, etc.).

The list physically resides at Mills College in Oakland, California, USA,
currently on an SGI Indy. Oakland is across the Bay from San Francisco.

Dave Madole
Technical Director, Center for Contemporary Music
Listserv Administrator

Mills College
Oakland, CA 94613
510-430-2336

madole@mills.edu


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From: PAULE

>>The whole point is that it doesn't matter what the exact form dissonance
>>function is! The formulas that result from this assumption is, as I tried to
>>make clear in words,
>>minimize (error of 3:1)^2+(error of 5:1)^2+(error of 5:3)^2 and
>>minimize (3*error of 3:1)^2+(5*error of 5:1)^2+(5*error of 5:3)^2.

>>The results of applying these to meantone tunings I have already posted in
>>both cents and exact formulae. If you want me to work you through the
>>calculus please try to reach me offline, I don't think we need to scare any
>>more musicians off this list.

>I, for one, would certainly not be scared off the list by such a
discussion.

I think this is overkill, since any meantone tuning from 1/3 comma to 1/5
comma is fine by me, all sounding wonderfully smooth. But one is naturally
curious. Since a meantone tuning is generated by the (flattened) perfect
fifth, which we will represent by v, express these intervals in terms of
that fifth. Since were are assuming a tuning with perfect octaves, it
doesn't matter which octave inversion or extension of the consonant
intervals we examine. Expressing everything in units of 1octave:

(1) minimize
(v-log(3/2)/log(2))^2+(4*v-log(5)/log(2))^2+(3*v-log(10/3)/log(2))^2
(2) minimize
(3*(v-log(3/2)/log(2)))^2+(5*(4*v-log(5)/log(2)))^2+(5*((3*v-log(10/3)/
log(2)))^2

Calculus lesson 1: To find the value of a variable that maximizes or
minimizes a function, differentiate the function with respect to the
variable, set the derivative equal to zero, and solve for the variable.

If you don't know how to differentiate a function, or why the above is true,
please consult a calculus textbook. Anyway, the resulting derivatives are

(1) 52*v-2*log(3/2)/log(2)-8*log(5)/log(2)-6*log(10/3)/log(2)
which simplifies to
-2*(-26*v*log(2)-2*log(3)+2*log(2)+7*log(5))/log(2)
which simplifies to
52*v-4+(4*log(3)-14*log(5))/log(2)

and

(2) 1268*v-18*log(3/2)/log(2)-200*log(5)/log(2)-150*log(10/3)/log(2)
which simplifies to
2*(634*v*log(2)+66*log(3)+66*log(2)-175*log(5))/log(2)
which simplifies to
1268*v-132+(132*log(3)-350*log(5))/log(2)

Setting these equal to zero and solving for v gives

(1) v (2-(2*log(3)+7*log(5))/log(2))/26
or
v .58013737 octaves
or
v 696.1648 cents

and

(2) v (66-(66*log(3)+175*log(5))/log(2))/634
or
v .58001560 octaves [.58 octaves is of course 29/50 octaves or the 50-tET
perfect fifth]
or
v 696.0187 cents.

>I am particularly interested in the ideas you call "virtual pitch, residue
>pitch, complex tone pitch, fundamental tracking, and possibly periodicity
>pitch, I have been working on this for some time" and I'd love to see some of
>these ideas posted to the list.

I'm sure someone else can give you a fuller bibliography on these ideas than
I. A good start might be Juan Roederer's "Introduction to the Psychophysics
of Music." I posted some of my musings, complete with mathematical
derivations, a few months ago. This procedure I extended from intervals to
chords with a kludge. I would now like to make this more rigorous, but I
need to drop the assumption that ratios are always expressed in lowest
terms. (This was a nice assumption because it allowed be to use the theory
of rational approximations. There is no theory of rational approximations
for ratios of three of more numbers. Graphing x/y vs. x/z for all
combinations of x, y, and z integers less than N (N fixed) will give you
some idea why no such theory exists. There is evidence (including personal
experience) that even a rich complex tone, played quietly and in the
presence of noise, can sometimes be interpreted as harmonics 2, 4, 6, 8, . .
., or even 3, 6, 9, 12, . . . of the consciously perceived fundamental. The
lowest-terms restriction would not allow for these phenomena.) Anyway, I'm
sort of not working on this now, but J. Kukula has suggested that wavelet
theory might be useful.

Brian McLaren has made the observation that "scientific" models of
perception and cognition are continously revised to reflect the mathematics
and computer science in vogue at the time. Rather than take this as evidence
for a Kuhnian attack on these models, I would suggest that they represent
successively better approximations to the truth, just as Newtonian physics
remains a good approximation though relativity and quantum mechanics are far
more accurate. So, when I cite the results of Goldstein, J. L. 1973. "An
optimum processor theory for the central formation of the pitch of complex
tones." J. Acoust. Soc. Amer. Vol. 54 p. 1499, and I often do, I realize
that there may be improvements to the model, but as a first-order
approximation, it fit the data quite well.

Where has Brian been lately? I sort of miss him in a weird way. I kind of
wanted to respond to his assertion that neutral thirds are more consonant
than tritones by saying that, for tones with strong third partials and weak
fifth partials, he is certainly right, if a roughness-type definition of
consonance (based on Plomp and Levelt's critical bands) is used. The reason
many objected is that when gradually changing the size of the interval, the
_local_ minimum of dissonance achieved around the 7:5 is much more distinct
than that achieved at the 11:9. In an absolute sense, however, the 7:5 may
be more dissonant.

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