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On Tuning Theory and False Reification by David B. Doty is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 United States License.

On Tuning Theory and False Reification

This essay originally appeared in 1/1 11:2, Spring 2002.

The tendency has always been strong to believe that whatever received a name must be an entity or being, having an independent existence of its own. And if no real entity answering to the name could be found, men did not for that reason suppose that none existed, but imagined that it was something peculiarly abstruse and mysterious. ——John Stuart Mill1

To reify (from Latin res: a thing) is to treat an idea or abstraction as if it had concrete or material ­existence. False reification, then, is the error of assuming that a pattern of data represents a “thing” in the real world, when there is no evidence for the existence of the thing other than the pattern. One of the oldest and best know examples is the Zodiac. In ancient times, people in various cultures identified patterns of stars along the ecliptic with figures of animals, mythical beings, and the like (with different cultures identifying different patterns). Then, for reasons beyond the scope of this essay, people developed the belief, still held by many today, that the positions of the sun, moon, and planets relative to these figures at the time of a person’s birth determine his or her character, personality, and fate, in a way that reflects the attributes of the animals or mythical beings identified with the star patterns and the planets.2 Yet the patterns themselves exist only relative to the position of an observer on earth—some stars in a constellation may be a few dozen light years away, others millions—an observer elsewhere in the universe would not see the same patterns. And the identities of the constellations are clearly creations of the human imagination, and some rather vivid imaginations at that.

I first encountered the concept of false reification in Stephen Jay Gould’s excellent book, The Mismeasure of Man.3, 4 The reification error that most concerns Gould is that scores from IQ tests represent a real-world phenomenon called “general intelligence” (and, further, that this commodity is heritable). However, I find the concept equally applicable to tuning theory and, by extension, to music theory in general. And, although no misconception in music theory is likely to result in such serious social consequences as those produced by the misapplication of intelligence testing, it can still lead to the expenditure of much time and energy on the formulation, presentation, and debating of theories that ultimately have little bearing on how any actual music might be composed or heard. Typically, these theories are based on mathematical models or constructive methods abstracted from scales or other musical structures.

Of course, everyone who has worked with Just Intonation is aware that numbers can represent real, significant musical phenomena. The very basis of Just Intonation is that musical intervals that can be represented by small integer frequency ratios are a distinct class, readily distinguished from (and, arguably, more beautiful than) those represented by more complex ratios. Clearly, this distinction extends to chords (triads, tetrads, pentads, and beyond) that can be represented as low-numbered subsets of a harmonic series. Given these undoubted facts, it would seem reasonable to suppose that the numerical properties of larger musical structures—scales, gamuts, melodies, harmonic progressions, and so on—would be similarly descriptive or predictive of their perceivable musical properties. To a certain extent, this is undoubtedly true. A scale or gamut comprising a network of closely connected consonant intervals based on low prime-limit ratios will certainly sound different and have different musical properties than one comprising large number and/or high prime-limit ratios (or irrational intervals); a scale with equal tetrachords will have different melodic properties than one with unequal tetrachords, and so on. These properties are obvious to the ear and do not require sophisticated mathematical analysis to discover.

Given that some characteristics of a scale, tuning system, or composition can be represented mathematically, it might seem a reasonable conclusion, or at least a reasonable speculation, that all significant features can be so represented. Perhaps so, although the truth of this statement is far from self-evident. From this arises the corollary that all features of a tuning system (or of a musical composition) that can be represented mathematically are musically significant and that there is a rather precise correspondence between mathematical and musical properties. Continuing down this path, we arrive at the idea that structures that are mathematically interesting, elegant, beautiful, or sophisticated necessarily have similar musical properties, or that the more sophisticated the mathematics required to describe or generate a tuning, the better will be the resulting music. Here we are deep in the territory of false reification. The computation, manipulation, and examination of the mathematical representations of tunings and other musical structures easily becomes an end in itself, far removed from the creation of actual music; in other words, it becomes a kind of recreational mathematics.

Part of the problem is the inherent difficulty of proving the truth of any proposition of music theory (or, by extension, of aesthetics) whatever. Because they are working with tables of numbers, geometric lattices, equations, and the like, tuning theorists may conclude that their theories can be proven on paper (or on a computer screen) like any other mathematical theorem. But to the extent that the numbers are meant to represent musical (that is, aesthetic), as opposed to merely acoustic phenomena, this cannot be so, at least not with our current level of understanding of human perception and emotional response. The proof of music must be in the hearing. Thus, to test any prescriptive theory, actual music based on the theory must be composed and heard. But this results in further difficulties. Different composers, given the same material, will produce vastly different results; indeed the same composer, working with the same material at different stages in his or her career (or even on successive days), may produce vastly different results. And, of course, different listeners may have very different reactions to the same composition and performance. How, then, can we determine to what extent the aesthetic impression created by a musical composition results from the theoretical properties of the raw material, the musicality, creativity, or stylistic preferences of the composer, or the musical tastes of the listeners?

Supposing that much of what passes for tuning theory is, in fact, recreational mathematics, is there any harm in this? Recreational mathematics is, no doubt, an enjoyable, blameless hobby that can occasionally result in mathematical discoveries of real value. And it is certainly possible that composers, in experimenting with the structures devised by mathematicians, may discover new musical possibilities that might otherwise go unnoticed. The harm, if any, is of two sorts. The first is that talented composers (or potential composers) may be seduced by the beauty of the numbers and spend too much time in mathematical exercises, to the neglect of composing, or be led to the false conclusion that music can be understood from the manipulation of numbers, without the intervention of the ears. A second and worse harm would result if it were believed that a high degree of mathematical sophistication was a prerequisite for composing in Just Intonation, where in fact no more than an understanding of elementary arithmetic is required.

Should the mathematical study of tuning be abandoned, then? I do not propose this, nor do I think that such a proposal would be taken very seriously by those engaged in this activity. What I do suggest, however, its that those who are interested in composing music resist the temptation to be seduced into recreational mathematics and instead devote more time to the aural study of tunings. Spend more time with your instruments and voice and less with computers and calculators, and better music will almost certainly be the result. The ear is and must ever be the best and final judge of music. A modest amount of math is necessary for the understanding of Just Intonation, but more is not necessarily better. 


1. A System of Logic, Ratiocinative and Inductive, Being a Connected View of the Principles of Evidence and the Methods of Scientific Investigation, 2 volumes. London, Parker: 1843.

2. In an odd twist, most Western astrologers are not even concerned with the real positions of the constellations as they appear in the sky today, but with fixed “signs” based on their positions in the second century c.e., established by our old friend Claudius Ptolemy.

3. New York and London, W.W. Norton & Company: 1981; revised and expanded, 1996.

4. Gould died on May 20, 2002, as I was finishing this essay. As a paleontologist, he made important contributions to the theory of evolution. He was a skilled and prolific writer, known to many for his essays in the magazine Natural History, and for books such as Ontogeny and Phylogeny (1977), Wonderful Life (1989), and Full House (1996), in addition to The Mismeasure of Man, cited above. I will miss him.