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The Foundations of Scientific Musical Tuning by Jonathan Tennenbaum

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The Foundations of Scientific Musical Tuning by Jonathan Tennenbaum Click here for Full Issue of Fidelio Volume 1, Number 1, Winter 1992 The Foundations of Scientific Musical Tuning by Jonathan Tennenbaum want to demonstrate why, from a scientific stand­ teenth-century physicist and physiologist whose 1863 point, no musical tuning is acceptable which is not book, Die Lehre von den Tonempfindungen als physiolo­ I based on a pitch value fo r middle C of 256Hz (cycles gische Grundlage fu r die Theorie der Musik (The Theory per second), corresponding to A no higher than 432Hz. of the Sensations of Tone as a Foundation of Music Theory) In view of present scientificknowl edge, all other tunings became the standard reference work on the scientific including A =440 must be rejected as invalid and arbi­ bases of music, and remains so up to this very day. trary. Unfortunately, every essential assertion in Helmholtz's Those in fa vor of constantly raising the pitch typically book has been proven to be fa lse. argue, "What diffe rence does it make what basic pitch Helmholtz's basic fallacy-still taught in most music we choose, as long as the other notes are properly tuned conservatories and universities today-was to claim that relative to that pitch? After all, musical tones are just the scientificbasis of music is to be fo und in the properties frequencies, they are all essentially alike. So, why choose of vibrating, inert bodies, such as strings, tuning fo rks, one pitch rather than another?" To these people, musical pipes, and membranes. Helmholtz definedmusical tones tones are like paper money, whose value can be inflated merely as periodic vibrations of the air. The fundamental or deflated at the whim of whoever happens to be in musical tones, he claimed, are sine waves of various power. fr equencies. Every other tone is merely a superposition of This liberal philosophy of "free-floating pitch" owes added-up sine waves, called "overtones" or "harmonics." its present power and influence in large part to the The consonant musical intervals are determined by prop­ acoustical theories of Hermann Helmholtz, the nine- erties of the "overtone series" to be simple whole-number ratios of frequencies. Arguing from this standpoint, This article is based on a speech given by the author, Director Helmholtz demanded that musicians give up well-tem­ of the European Fusion Energy Foundation, at an April pering and return to a "natural tuning" of whole-number 1988 Schiller Institute conference on scientific tuning held ratios; he even attacked the music of I.S. Bach and in Milan, Italy. It appears also in the Institute's "Manual Beethoven fo r being "unnatural" on account of their on the Rudiments of Tuning and Registration." frequent modulations. 47 © 1992 Schiller Institute, Inc. All Rights Reserved. Reproduction in whole or in part without permission strictly prohibited. Helmholtz based his theory of human hearing on the geometrical configurations of molecules. In modern same fallacious assumptions. He claimed that the ear physics, this kind of self-organizing process is known as works as a passive resonator, analyzing each tone into a "soliton." Although much more detailed experimental its overtones by means of a system of tiny resonant work needs to be done, we know in principle that diffe r­ bodies. Moreover, he insisted that the musical tonalities ent frequencies of coherent solitons correspond to dis­ are all essentially identical, and that it makes no diffe r­ tinct geometries on the microscopic or quantum level of ence what fundamental pitch is chosen, except as an organization of the process. This was already indicated arbitrary convention or habit. by the work of Helmholtz's contemporary, Bernhard Riemann, who refuted most of the acoustic doctrines of ' Helmholtz's Theory: Helmholtz in his 1859 paper on acoustical shock waves. Helmholtz's theory of hearing also turned out to be Linear and Wrong fa llacious. The tiny resonators he postulated do not exist. Helmholtz's entire theory amounts to what we today , The human ear is intrinsically nonlinear in its fu nction, call in physics a "scalar," "linear," or at best, "quasi­ generating singularities at specific angles on the spiral linear" theory. Thus, Helmholtz assumed that all physi­ chamber, corresponding to the perceived tone. This is cal magnitudes, including musical tones, can at least an active process, akin to laser amplification, not just implicitly be measured and represented in the same way passive resonance. In fa ct, we know that the ear itself as lengths along a straight line. But, we know that every generates tones. important aspect of music, of the human voice, the hu­ Moreover, as every competent musician knows, the man mind, and our universe as a whole, is characteristi­ simple sinusoidal signals produced by electronic circuits cally nonlinear. Every physical or aesthetic theory based (such as the Hammond electronic organ) do not constitute on the assumption of only linear or scalar magnitudes, musical tones. Prior to Helmholtz, it was generally un­ is bound to be fa lse. derstood that the human singing voice, and more specifi­ A simple illustration should help clarify this point. cally, the properly trained bel canto voice, is the standard Compare the measurement of lengths on a straight line of all musical tone. Historically, all musical instruments with that of arcs on the circumference of a circle. A were designed and developed to imitate the human voice straight line has no intrinsic measure; before we can as closely as possible in its nonlinear characteristics. measure length, we must first choose some unit, some The bel canto human voice is fo r sound what a laser interval with which to compare any given segment. The is fo r light: The voice is an acoustical laser, generating choice of the unit of measurement, however, is purely the maximum density of electromagnetic singularities arbitrary. per unit action. It is this property which gives the bel The circle, on the contrary, possesses by its very nature canto voice its special penetrating characteristic, but also an intrinsic, absolute measure, namely one complete cycle determines it as uniquely beautiful and uniquely musical. of rotation. Each arc has an absolute value as an angle, By contrast, electronic instruments typic�lly produce and the regular self-divisions of the circle definecertain Helmholtzian sine-wave tones, which are ugly, "dead," specificangles and arcs in a lawful fa shion (e.g., a right and unmusical exactly to the extent that they are incoher­ angle, or the 1200 angle subtended by the side of an ent and inefficient as electromagnetic processes. equilateral triangle inscribed in the circle). Just as the process of rotation, which creates the circle, Tuning Is Based on the Voice imposes an absolute metric upon the circle, so also the process of creation of our universe determines an abso­ The human voice defines the basis fo r musical tuning lute value fo r every existence in the universe, including and, indeed, fo r all music. This was clearly understood musical tones. Helmholtz refused to recognize the fact long before Helmholtz, by the scientific current associ­ that our universe possesses a special kind of curvature, ated with Plato and St. Augustine, and including Nico­ such that all magnitudes have absolute, geometrically­ laus of Cusa, Leonardo da Vinci and his teacher Luca determined values. This is why Helmholtz's theories are Pacioli, and Johannes Kepler. In fa ct, Helmholtz's book systematically wrong, not merely wrong by accident or was a direct attack on the method of Leonardo da Vinci. through isolated errors. Straight-line measures are If Helmholtz's theories are wrong, and those of Plato intrinsically fa llacious in our universe. through Kepler and Riemann have been proven cor­ For example, sound is not a vibration of the air. A rect-at least as fa r as these went-then what conclu­ sound wave, we know today, is an electromagnetic pro­ sions fo llow fo r the determination of musical pitch to­ cess involving the rapid assembly and disassembly of day? Let us briefly outline the compelling reasons fo r 48 FIGURE I. The Golden Section arises as the ratio of the side to the diagonal of a pentagon. C=256Hz as the only acceptable sci­ entific tuning, which have emerged from a review of the classical work of Kepler et al. as well as modern scientific research. The human voice, the basic in­ strument in music, is also a living process. Leonardo and Luca Pacioli demonstrated that all living pro­ cesses are characterized by a very FIGURE 2. Leonardo da Vinci's drawing of the hu specific internal geometry, whose man body inscribed in a circle demonstrates Golden Section proportions. most direct visible manifestation is the morphological proportion of the Golden Section. In elementary ge- ometry, the Golden Section arises as the ratio between the fo llowing two series of tones, whose musical signifi­ the side and the diagonal of a regular pentagon (see cance should be evident to any musician: C-B-G-C, Figure 1). The Golden Section naturally fo rms what we and C-E-F#-G. In the firstseries, the diffe rences of the call a self-similar geometric series-a growth process in frequencies between the successive tones fo rm a self­ which each stage fo rms a Golden Section ratio with similar series in the proportion of the Golden Section. the preceding one. Already before Leonardo da Vinci, The frequency differences of the second series decrease Leonardo Pisano (also called Fibonacci) demonstrated according to the Golden Section ratio (see Figure 3). that the growth of populations of living organisms al­ ways fo llows a series derived from the Golden Section.
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