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The Foundations of Scientific Musical Tuning by Jonathan Tennenbaum

want to demonstrate why, from 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 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 , 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 "." 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 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-#-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 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. In extensive morphological studies, Leonardo da Vinci The Golden Section showed that the Golden Section is the essential character­ To understand the well-tempered system better, we must istic of construction of all living fo rms. For example, firstexamine the reason why certain specificproportions, Figure 2 illustrates the simplest Golden Section propor­ especially the Golden Section, predominate in our uni­ tions of the human body. . verse, whereas others do not. Since music is the product of the human voice and There is nothing mysterious or mystical about the human mind-i.e., of living processes-therefore, ev­ appearance of the Golden Section as an "absolute value" erything in music must be coherent with the Golden fo r living processes. Space itself-that is, the visual space Section. This was emphatically the case fo r the develop­ in which we perceive things-has a specific "shape" ment of Western music from the Italian Renaissance up coherent with the Golden Section. For, space does not through Bach, Mozart, and Beethoven. exist as an abstract entity independent of the physical The Classical well-tempered system is itself based on universe, but is itself created. The geometry of space the Golden Section. This is very clearly illustrated with reflects the characteristic curvature underlying the pro-

49 A A A A C E-flat G C' FIGURE 3. The frequency values L- 1 /cp2 ----lL...1 --- lei> ___ -'I IL ______1 of these two basic series of musical tones are ordered A A A A according to the Golden Section. C E F-sharp G ______-'1 1 '-1 --- 1 lei> ----,1'-1Icp2 ----l

cess of generation of the universe as a whole. We know eter. By fo lding again, we obtain a point, the center of that space has a specificshape, because only fivetypes of the circle, as the intersection of two diameters, as in regular solids can be constructed in space: the tetrahe­ Figure 6. This alone creates fo r us the basic "elements" dron, cube, octahedron, dodecahedron, and icosahedron of plane geometry. Also, by rotating a circle we obtain (see Figure 4). the sphere (see Figure 7). These fivesolids are uniquely determined characteris­ Further constructions, using circular action alone, tics of space. They are absolute values fo r all of physics, generate the regular polygons-the equilateral triangle, biology, and music. Indeed, Luca Pacioli emphasized square, and pentagon-which fo rm the fa ces of the five that all the solids are derived fr om a single one, the regular solids. From these uniquely determined poly­ dodecahedron, and that the latter is uniquely based upon gons, Kepler derived the fu ndamental musical intervals the Golden Section. Hence, the Golden Section is the of the fifth, fo urth, and , without any reference principal visual characteristic of the process of creation to overtones. These polygons embody the principle of of the universe. self-division of circular action by 3, 4, and 5. The , In his Mysterium Cosmographicum, Kepler provided or division by 2, we already obtained as the very first fu rther, decisive proof fo r Leonardo and Pacioli's result of fo lding the circle against itself. From division method. He demonstrated that the morphology of the by 2, 3, 4, and 5 we obtain, fo llowing Kepler, the fo llow­ solar system, including the proportions of the planetary ing values fo r the basic musical intervals: octave, 1 :2; orbits, is derived from the five regular solids and the fifth, 2:3; fo urth, 3:4; major third, 4:5. Golden Section. Figure 5 shows Kepler's fa mous con­ Division by seven is invalid, Kepler argued, because struction of the planetary orbits through a nested series the heptagon is not constructible from circular action of concentric spheres, whose spacing is determined by alone, nor does it occur in any regular solid. Since inscribed regular solids. Therefore, the solar system has Kepler's musical ratios are uniquely coherent with the the same morphological characteristics as living or­ regular solids, they are uniquely coherent with the gamsms. Golden Section underlying those solids. Kepler located the underlying reason fo r these mor­ Kepler went on to demonstrate that the angular veloc­ phological characteristics in the generating process of the ities of the planets as they move in their elliptical orbits universe itself, and this he attempted to identify with the around the sun, are themselves proportioned according help of what is called the isoperimetric theorem. This to the same ratios as the fu ndamental musical intervals theorem states that among all closed curves having a (see Table I, page 56). Since Kepler's time, similar rela­ given parameter, the circle is the unique curve which tions have been demonstrated in the system of moons of encloses the greatest area. Circular action is the maxi­ various planets, and provisionally also even in the motion mally efficientfo rm of action in visible space, and there­ of spiral galaxies. fo re coheres uniquely with the bel canto musical tone and the beam generated by a laser. Kepler reasoned that c= 256 As a 'Keplerian Interval' if circular action reflects uniquely the creative process of the universe, then the fo rm of everything which exists­ C=256 has a uniquely defined astronomical value, as a of atoms and molecules, of the solar system, and the Keplerian interval in the solar system. The period of one musical system-must be constructible using nothing cycle of C=256 (1/2 56 of a second) can be constructed as but circular action. fo llows. Take the period of one rotation of the Earth. By this procedure, called "synthetic geometry," we Divide this period by 24 (=2X3X4), to get one hour. generate from the circle, by fo lding it upon itself (i.e., Divide this by 60 (= 3 X 4 X 5) to get a minute, and again circular action applied to itself),a straight line, the diam- by 60 to obtain one second. Finally, divide that second

50 FIGURE 5. Kepler's construction of the planetary c,', orbits, nested according to a series of inscribed regular ({ffrOctahedron solids.

Icosahedron

Dodecahedron

FIGURE 4. Th e five Platonic solids.

by 256 (=2X2X2X2X2X2X2X2). These divisions are all Keplerian divisions derived by circular action alone. It is easy to verify, by fo llowing through the indicated series of divisions, that the rotation of the Earth is a "G," twenty-four lower than C=256. Similarly, o Q - CD C=256 has a determinate value in terms of the complete system of planetary motions. By contrast, A =440 is a purely arbitrary value, having no physical-geometrical justification.A =440 is an insane o - ED tuning in the rigorous sense that it bears no coherent � relationship with the universe, with reality. FIGURE 6. Using "synthetic geometry, " we generate a Today, we can add some essential points to this. straight line and a point fr om the circle, by fo lding it Kepler's solution was absolutely rigorous, as fa r as it upon itself. went; however, circular action is only an incomplete representation of creative action in the universe. The next great step was taken by Carl Friedrich Gauss at the beginning of the nineteenth century. Gauss introduced conical spiral action, instead of mere circular action, as '"�''''' '''''' the basis fo r synthetic geometry. Spiral action combines the isoperimetric principle of the circle with the principle of growth expressed by the Golden Section. (' -.rcf-» Let us demonstrate conical spiral action in the bel ,/ \,' ...... -..,...... canto voice. Have a soprano sing a scale upward, starting a) circular action b) double self-reflexive c) triply self-reflexive at middle C (=256). As the frequency increases, so does circular action circular action the intensity of the sound produced. The more precise FIGURE 7. The sphere is generated by triply-connected term fo r this intensity is "energy flux density." But this circular action. increase is not merely linear extension, not merely in-

51 A + B = � = G = 1. 5 = arithmetic mean 2 2

...fAB = -[2= F' = 1.41 4 = geometric mean A = 1 2AB 4 = C256 -- = - = Fq = 1.333 = mean --t----lI:r===�1E:::3� A+B 3 \\ i , , i ' , , i ," \ ,' ! , \, I. " , FIGURE 8. The octave corresponds to one fu ll revolution 'I' · · ...... of spiral action, while the fift h, diminished fift h, and 0 •• •• .••. , • i fo urth are, respectively, the arithmetic, geometric, and iz harmonic means.

crease in scalar magnitude. As our singer sings upward, construction identifiesthe frequency regions and angular two important events occur. First, she must make a displacements within which the well-tempered values register shift, at F#,in order to maintain the "isoperime­ are to be defined.) tric," least-action fo rm of bel canto tone. We shall return Most important, the halfway point of the fu ll cycle to the register shift in a moment. The second event starting at C, is F#, the diminished fifth from C, or the occurs upon arrival at the octave, C=512. We hear very interval once known as the "devil's interval." In terms of clearly, that one cycle of action has been completed, like geometrical proportion, this F# is located as the geometric a 360° rotation. This proves that there is a rotational mean of C=256 and its octave, C=512. component of action to increase the frequency or energy If we carry out synthetic geometry constructions with flux density. Again, Helmholtzian straight-line action conical spiral action, just as Kepler did with circular does not exist. action, we discover wonderful things. For example, con­ The true geometry of the singer's action is therefore struct the characteristic of the conical volume bounded most simply represented by spiral action upward on a cone. In Figure 8, the cone's axis represents frequency. Arithmetic mean Geometric mean Each circular cross-section of the cone represents a bel radial length radial length halfway canto musical tone. The spiral makes one complete rota­ around the scale tion in passing from C=256 to C=512, and one more cycle would bring it from C=512 to the next higher octave, C= 1024. Thus, the interval of an octave corre­ .-.- F-,/ sponds to one complete 360° cycle of conical spiral action. Not only the octave, but all musical intervals, corre­ spond to specific angles on conical-spiral action. This is most clearly seen if we project our conical spiral onto a A plane perpendicular to the axis (see Figure 9). If we divide a fu ll 360° rotation into twelve equal angles, then each such (30°) displacement corresponds to a interval in frequency. The radial lengths defined by the spiral at the indicated twelve angles are exactly propor- . tional to the frequencies of the equal-tempered musical scale. The interval of a fifth corresponds to rotation through 7/1 2 of the circle, or 210°. The interval of a minor C' third. corresponds to a right angle, and so fo rth. FIGURE 9. Th e self-similar conical spiral projected (The equal-tempered system is only an approximation onto a plane, showing the intervals of the equal­ of a rigorous well-tempered system whose details have tempered scale. yet to be fully elaborated. Nevertheless, the indicated

52 a is the radius at perihelion b is the radius at aphelion 2ab/(a + b) is the harmonic mean, which occurs at the latus rectum FIGURE 10. (a b)/2 is the semi-major axis Projection onto a + plane of the ellipse -{iFis the semi-minor axis fo rmed by slicing diagonally between the circularcuts representing C=256 and C=512, showing the important division points of the octave.

by the circles at C=256 and C=5l2, by slicing the cone actly at the geometrical-mean or halfway point in the diagonally across those two circles. The result is an cycle of conical spiral action. The same process repeats ellipse. Project this ellipse onto the plane. The principal in the next-higher octave, where the shift from second parameters of the resulting plane ellipse define exactly to third register of the soprano comes once again at H, the frequency ratios fo r the most important division­ the geometric mean. points of the octave (see Figure 10): The bel canto shift is a physical event of fu ndamental importance, and not merely a technical question fo r the C=256 corresponds to the periheJ'ion of the ellipse voice. In physical terms, the register shift constitutes a C=512 corresponds to the aphelion singularity, a nonlinear phase change comparable to the F corresponds to the semi-latus rectum transformation from ice to water or water to steam. An FI corresponds to the semi-minor axis G corresponds to the semi-major axis even better comparison is to the biological process of cell division (mitosis). In every case, we see that in C=256 At the same time, F, H, and G correspond to the har­ tuning, the region of this singularity coincides with the monic, geometric, and arithmetic means, respectively, of principal geometrical division of conical spiral action. the octave. These three means fo rmed the basis of classi­ (Here we take the soprano voice, fo r musical and devel­ cal Greek theories of architecture, perspective, and mu­ opmental reasons, as the fundamental reference fo r the sic. The same notes F, FI, and G mark the principal human voice in general.) division of the basic C-major scale. This scale consists of Our solar system also makes a "register shift." It has two congruent : C-D-E-F and G-A-8-C. long been noted that the inner planets (Mercury, Venus, The dividing-tone is Fl . Earth, and Mars) all share such common fe atures as relatively small size, solid silico-metallic surface, fe w Physical Significance moons, and no rings. The outer planets (Jupiter, Saturn, Uranus, and Neptune) share a second, contrasting set of Of the Register Shift characteristics: large size, gaseous composition, many Let us now return to our soprano. She makes the first moons, and rings. The dividing-point between these two register shift, from firstto second register, exactly at this sharply contrasting "registers" is the asteroid belt, a ring­ point of division. The firsttetra chord, C-D-E-F, is sung like system of tens of thousands of fragmentary bodies in the first register, while G-A-8-C are sung in the believed to have arisen from an exploded planet. second register. The register shift divides the scale ex- It is easy to verify that the solar-system register shift

53 stated in dozens of interviews and record jackets, the A Brief pragmatic reason: German instruments of the period 1780-1 827, and even replicas of those instruments, can f only be tuned at A=430. History o Tuning The demand by Czar Alexander, at the 1815 Congress of Vienna, for a "brighter" sound, began the demand fo r he firstexplicit reference to the tuning of middle a higher pitch from all the crowned heads of Europe. C at 256 oscillations per second was probably While Classical musicians resisted, the Romantic school, Tmade by a contemporary of J.S. Bach. It was at led by Friedrich Liszt and his son-in-law Richard that time that precise technical methods were developed, Wagner, championed the higher pitch during the 1830's making it possible to determine the exact pitch of a given and 1840's. Wagner even had the bassoon and many note in cycles per second. The first person said to have other instruments redesigned so as to be able to play only accomplished this was Joseph Sauveur (1653-1716), at A = 440 and above. By 1850, chaos reigned, with major called the fa ther of musical . He measured the European theatres at pitches varying fr om A =420 to pitches of organ pipes and vibrating strings, and defined A = 460, and even higher at Venice. the "ut" (nowadays known as "do") of the musical scale In the late 1850's, the French government, under the at 256 cycles per second. influence of a committee of led by bel canto J.S. Bach, as is well known, was an expert in organ proponent Giacomo Rossini, called fo r the first standard­ construction and master of acoustics, and was in constant ization of the pitch in modern times. France conse­ contact with instrument builders, scientists, and musi­ quently passed a law in 1859 establishing A at 435, the cians all over Europe. So we can safely assume that he lowest of the ranges of pitches (from A=434 to A=456) was fa miliar with Sauveur's work. In Beethoven's time, then in common use in France, and the highest possible the leading acoustician was Ernst Chladni (1756-1 827), pitch at which the soprano register shifts may be main­ whose textbook on the theory of music explicitly defined tained close to their disposition at C = 256. It was this C=256 as the scientific tuning. Up through the middle French A to which Verdi later referred, in objecting to of the present century, C=256 was widely recognized as higher tunings then prevalent in Italy, under which the standard "scientific" or "physical" pitch. circumstance "we call A in Rome, what is Bb in Paris." In fa ct, A=440 has never been the international stan­ dard pitch, and the firstint ernational conference to im­ pose A =440, which fa iled, was organized by Nazi Pro­ 19. WHAT ARE THE QUALITIES OF MOSlC? \ paganda Minister Joseph Goebbels in 1939. Strike Dote middle C OD aDy . What iJ standard pitch ? th� . 1 Throughout the seventeeth, eighteenth, and nine­ average. well tuned and it gives 256 vibrations per second. ; teenth centuries, and in fact into the 1940's, all standard Likewise the middle forks that ar w.d in all physical i laboratories are all tuned to 256 vibration.e per second. Thil! : ... U.S. and European textbooks on physics, sound, and gives the nr.te .\.7 vibn.tioOJ! :per ��ond. · .... \&.e Fig; 19... 1 .) ; music took as a given the "physical pitch" or "scientific Theother notes of the Bcale. vibrate liccordingt.o a fixed ratio, like th.t eho"llll in the dikgram: In . which.is DQW little pitch" ofC=256, including Helmholtz's own texts them­ used,midd' � "" t.. -- �1 _..:L_,.. .: __...... _ ... _.,J l_,'_�,�.,:._;.,"" t,.",·.'::� selves. Figure 13 shows pages from two standard modern l':XPI.ANATlON 0))" so�m n:RMS American textbooks, a 1931 standard phonetics text,

and the official 1944 physics manual of the U.S. War Standard of Pitch. 10 this book wbe,,: other authQrs aro £j'lOtt-d, the Department, which begin with the standard definition �. COl $t.aHd��rd of rnuskal pilCh 'lJ�ed by thctn i� oftt'n -retain.ed. On the other ' 0. •• hand: where it rompariso!1 Qf two au thors l