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From: mclaren
Subject: Another book review
--
In 1993 Martin Vogel's "Orpheus" publishing
house released a book titled "Mathematical
Models of Musical Scales: A New Approach" by
Mark Lindley and Ronald Turner-Smith.
Orpheus is the same publishing house that released
Leigh Gerdine's superlative english translation of
Adriaan Fokker's "New Music With 31 Tones," and
since Lindley/Turner-Smith's book was also in english, I
read it with great anticipation.
Let me preface my remarks by pointing out that Mark
Lindley is one of the most knowledgeable and respected
scholars of historical intonation. He has written articles
on 16th- and 17th-century intonation which qualify as
definitive, and he's written a book which stands alone
as a reference on the tuning of lutes & other antique
fretted instruments.
Alas, this book is a disappointment in several respects,
though it has much to recommend it.
The problems with "Mathematical Models of Musical
Scales" are many and varied.
To summarize:
[1] The book starts with a distinctly polemical slant
and never lets up.
[2] Instead of using the standard measurement of cents,
Lindley and Turner-Smith inexplicably choose to use
millioctaves (1/1000 of an octave). Worse, they
introduce confusing and bizarre temrinology: "flog,"
set membership symbols, Boolean logic symbols, etc.
Such symbology and terrinology would prove dense
but reasonable reading for a mathematics paper--
but for a music paper, this kind of mathematical
arcana is damned hard to read and extract any
musical meaning from.
In particular, it's almost impossible to shift gears
from sentences like "A mapping which is both 1-1 *and*
onto is called a bijection or a 1-1 correspondence. A
bijection of a set X *to itself* is called a
permutation of X." to sentences like "In 18th century
French keyboard tuning, one or two major thirds (Db-F,
Ab-C) were larger than pure by a little more than 2% of
an octave, although people at the time said these
particular 3rd were harsh and in fact meantone
temperaments had originally come into use because 15th-
century organists considered unacceptable a major 3rd
(comprising two 9:8 whole tones) which is larger than
pure by a little *less* than 2% of an octave."
[3] Throughout the book, Lindley and Turner-Smith
consistently disparage and belittle non-12
divisions of the octave. Typical is the following:
"However in the 1550s a talented French composer,
Costeley, wrote a song for the 19 divisions, and in
1577 a Spanish theorist, Salinas, showed how to
tune it on a keyboard instrument by using the pure
major 6ths and minor 3rd of 1/3-comma meantone
temperament. Experimentation went on in the 17th
century and since, but posterity has judged the
system unsatisfactory." This will come as a
surprise to many of the members of this tuning
forum, who are at present engaged in composing
and performing music in the 19-tone equal
temperament. In fact 19-TET is the most common,
almost the closest in sound to 12-TET of the "teen"
temperaments (22-TET is closer), and perhaps the
easiest to use of the < 20 TET equal temperaments.
[4] Despite the title, the overwhelming thrust
of this book is a consideration of general mathematical
models of the musical scales used from the 15th through
the 18th centuries. Lindley and Turner-Smith do not,
despite the implication of their title, consider
the musical ramifications of systems such as
Harry Partch's extended just intonation, Enrique
Moreno's non-octave scales, Mandelbaum's multiple
divisions of the octave, or my own non-just non-
equal-tempered tunings.
To dilate on these points:
[1] The polemical slant. The reader gets hit between
the eyes with the first sentence of the introduction:
"We reject the ancient Pythagorean idea that music
somehow 'is' number, and we show how to design
mathematical models for musical scales and
systems according to some more modern principles."
[Lindley, Mark, and Ronald Turner-Smith, "Mathematical
Models of Musical Scales," Orpheus, Bonn 1993,
Page 7]
This passage contains two statements which are
open to serious debate. First, the authors reject
categorically the Pythagorean idea of "numerus
sonorus" in favor of an emprical Aristoxenian
definition of intervals and of music.
But the debate twixt Pythagorean and Aristoxenian
views on the nature of music has raged since the
beginning of Western music and no one has *ever*
succeeded in resolving the question.
Despite Lindley/Turner-Smith's efforts to paint
the Pythagorean viewpoint as "mystical" and
"unscientific," the fact remains that the Pythagorean
view is the basis of Western theoretical science.
In fact the Pytahgorean view ought to be called
the "theoretical science" view, since it encapsulates
and perfectly expresses all the ideals held by
scientific pure theorists: in the view of the pure
theorist, an infallible test of whether a mathematical
theory accurately models the real world is the
degree of mathematical beauty and "elegance" which
the theory exhibits. Pure theorists typically reject
a scientific model of reality whose mathematics are
ugly and messy yet whose predictions are in good
accord with experiment, in favor of a scientific model
of reality whose mathematics are elegant and gorgeous
yet whose predicitions have not been tested yet, or
don't fall quite as close to experimental numbers.
There is good precedent for this. The final elaboration
of the Potelamic epicycle model of the universe in
the 16th century produced excellent agreement with
astronomical observations of the movements of the
planets. But scientists rejected the model of
Ptolemaic epicycles in favor of the Copernican
model largely on the basis of the mathematical
ugliness and complexity of the Ptolemaic system,
versus the simplicity and elegance of the
Copernican.
In modern times the best expression of the pure
theoretical (AKA Pythagorean) viewpoint is
given by Einstein: he was once asked what he would
have done if a phsyical experiment had contradicted
his theory of special relativity, and he answered
by saying that he would have felt sorry for the
dear Lord.
"Time and again the passion for understanding has
led to the illusion that man is able to comprehend
the objective world rationally, by pure thought,
without any empirical foundations--in short, by
metaphysics. I believe that every true theorist is a
kind of tamed metaphysicist, no matter how pure
a 'positivist' he may fancy himself. The metaphysicist
believes that the logically simple is also the real.
The tamed metaphysicist believes that not all that
is logically simple is embodied in experienced
relatiy, but tha the totality of all sensory experience
can be 'comprehended' on the basis of a conceptual
system built on premises of great simplicity.
The skeptic will say that this is a 'miracle creed.'
Admittedly so, but it is a miracle creed which has
been borne out to an amazing extent by the
development of science." [Einstein, Albert,
"Ideas and Opinions, Pg. 333: originally from
"Scientific American," Vol. 182, No. 4, April, 1950]
This strain of Western thought is so fundamental
to rational inquiry that it is untenable to dismiss
the Pythagorean viewpoint as cavalierly as Lindley
and Turner-Smith have, just as it untenable to
cavalierly define *all* of music in terms of
the Pythagorean viewpoint that intervals
are numbers, thus numbers are simple ratios and
thus the simplest ratios are the most audibly
consonant, quod erat demonstrandum (yet dead
wrong).
As I've pointed out in prior posts, the strict
extreme hard-line Pythagorean viewpoint
fails when it encounters reality, since
the 3:5:7 triad demonstrably sounds less
consonant than the 4:5:6 triad--yet the
integers of the 3:5:7 triad are obviously
smaller. A cadence which ends with a 4/3
in the bass is universally proscribed in all
Western harmony texts as a dissonance
to be avoided in favor of a 4:5:6 triad in
root position as the preferred final chord--
yet the 4/3 is clearly a smaller interval
than the 3/2. And so on.
The Pythagorean hard-line view on musical
consonance and auditory perception also
fails the test of experiment, since 130
years of psychoacoustic tests have shown
no evidence of small-integer-ratio detectors
in the human auditory system, and systematic
evidence that intervals not described in
terms of small whole number ratios are
heard as "pure" and "perfect consonances,"
while the purportedly "true" and "natural"
small-integer ratio intervals are universally
heard as "impure," "too narrow," "out of tune,"
and "flat."
The debate twixt Pythagorean (pure theorist)
and Aristoxenian (applied experimental science)
viewpoints is ongoing. It represents a fundamental
dichotomy at the heart of Western culture. It
cannot be resolved.
For Lindley and Turner-Smith to dismiss this
fundamental dichotomy at the heart of Western
culture with a wave of their hands is astonishing.
One would have thought a scholar of Lindley's
range and depth would recognize that
this ongoing debate twixt two schools of thought
is essential to the health of Western music.
Over the centuries the debate has shifted from
one side to the other, then back again, as new
evidence and new mathematics come to the fore.
At present the pendulum is swinging toward the
Aristoxenian extreme with the discovery that
even the most linear physical laws exhibit
unpredictably chaotic behavior. However, recently
Ed Witten and others succeeded in demonstrating
that all of the five competing models of superstring
theory are subsumed under one overarching model.
If Witten and company can find a way to generate
actual numbers from superstring theory, the
pendulum may swing back to the Pyhagorean
extremum.
And so it goes in music. With the collapse of
Helmholtz's theory and the findings of Ward,
Corso, Pikler, Sundberg and many others, grave
doubts have been cast on the advisability of
viewing music in terms of small whole numbers
as an adequate explanation of how we hear and
how we compose. However, future experimental
evidence or future mathematical results might
shift the burden of proof back onto the Fetis-Ward-
Burns-Corso contingent who view harmony as
mostly a matter of brainwashing and "consonance"
and "dissonance" are largely learned habits.
The point is that the musical and cultural debate
is ongoing and never-ending, and it *cannot* be
prestidigitated away as Lindley and Turner-Smith
have sought to do.
The second problem with Lindley's and Turner-
Smith's argument is the implication that "modern"
methods of analyzing and generating musical scales
somehow equals "better." There is no evidence
for this. Newer musical methods are not necessarily
"better" than older music. Moreover,
the entire notion of the musical utility of *this* or
*that* mathematical method is cyclic. Just intonation
and equal temperament periodically come into
and go out of fashion. This has happened throughout
the history of Western music, and will doubtless
happen again.
[2] John Chalmers has already expressed doubts
about the wisdom of using non-standard units of
musical measurement such as the millioctave.
For my part, let me point out that it is as confusing
as encountering speed limit signs which give
numbers of furlongs per fortnight when
your speedometer is calibrated in miles per hour.
It is simply incomprehensible to me why Lindley
and Turner-Smith would choose to use a non-
standard unit of musical interval measurement.
[3] Lindley's and Turner-Smith's negative attitude
toward musical intonations with more than 12
pitch classes must be understood in context.
They are probably simply saying that there is
little historical precedent for such usage; but
this is mainly a question of technology, rather
than aesthetics or musicality. In the era of
wooden machines (viz., the piano, the harpsichord)
it would have been impossibly difficult &
expensive to build a 5-octave instrument with
31 equal tones to the octave. If such an instrument
could have been built, its keys would have been
too narrow to be fingered; and the instrument
itself would have been too mechanically complex
and too fragile to survive an actual performance.
Today, however, with digital tehcnology, other
intonations than 12 equal tones to the octave
can easily be had.
Since Lindley and Turner-Smith's book is largely
an historical study (despite the title), its
dismissive attitude toward > 12 pitch classes
constitutes a judgment of practicality rather
than intrinsic worth, and so ought not to be taken
as the final word musical composition in
non-12 tunings--especially for *modern*
composers with access to retunable
digital instruments.
[4] The title of the book is probably overly
ambitious. It would be better if the book had
been called "Mathematical Models of Musical
Scales from the Common Practice Period of
Western Music." Within that context, the book
works very well. In particular, the footnotes
and musical examples are superb. Lindley
reveals the true range of his scholarship in
the footnotes and appendices--which make
more interesting reading than the rest of
the text.
All in all, this is a book recommended for those
interested in historical intonation with a
heavily partisan bias (toward emprical
science, away from pure theory).
--mclaren

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Catler and such

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