source file: mills2.txt Date: Tue, 9 Jul 1996 12:13:58 -0700 Subject: Another post from McLaren From: John Chalmers 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 Received: from eartha.mills.edu [144.91.3.20] by vbv40.ezh.nl with SMTP-OpenVMS via TCP/IP; Tue, 9 Jul 1996 22:30 +0100 Received: from by eartha.mills.edu via SMTP (940816.SGI.8.6.9/930416.SGI) for id NAA14751; Tue, 9 Jul 1996 13:30:35 -0700 Date: Tue, 9 Jul 1996 13:30:35 -0700 Message-Id: <960709202707_101655.321_IHN235-1@CompuServe.COM> Errors-To: madole@ella.mills.edu Reply-To: tuning@eartha.mills.edu Originator: tuning@eartha.mills.edu Sender: tuning@eartha.mills.edu