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WENDY’S XENHARMONIC KEYBOARD Continued Fractions” ([DM99]). This method will further demonstrate how the containment of the α and β scales JORDAN SCHETTLER inside the γ scale is analogous to the way in which the pen- tatonic scale (5 black keys) and heptatonic scale (7 white keys) fit into the familiar 12-tone scale found on a standard keyboard. Finally, we will illustrate this analysis by drawing 1. MOTIVATION Wendy’s γ scale keyboard. Wendy Carlos is an American composer and one of my favorite musical pioneers. She helped to popularize and 2. BACKGROUND improve the Moog synthesizer with the Grammy winning Before discussing the construction of any scales, we album Switched on Bach in 1968. I recall fondly when my first review some basic physics and functional analysis. grandfather gave me his copy of that record on vinyl... I didn’t know at first if I was playing it at the right speed! 2.1. Frequency and Pitch. Given an ideal taut string with Some of Carlos’ other best known works are the motion fixed endpoints (think of strings on a piano, violin, or gui- picture soundtracks for Stanley Kubrick’s A Clockwork Or- tar), vibrations lead to periodic compression waves in air ange in 1971, The Shining in 1980, and Walt Disney’s Tron which we perceive as sound. We define the fundamental in 1982. period T to be the number of seconds in one minimal cycle Here we are interested in Carlos’ 1986 original work of these periodic vibrations. The fundamental frequency Beauty in the Beast because of the experimentation with f = 1=T is the number of minimal cycles per second. The various so-called xenharmonic scales, i.e., scales other units of f here are measured in Hertz: than the familiar 12-tone scale on a standard keyboard. 1 This standard keyboard has equal spacing with 100 cents 1 Hz = 1 . sec between consecutive notes. In contrast, the title track on Beauty in the Beast uses a 9-tone and 11-tone scale with We will not write the units Hz, and we will simply regard f α = 77.995... and β = 63.814... cents, respectively, be- as a number in the interval of positive real numbers (0, 1). tween consecutive notes. The α and β scales are both For example, we literally define the frequency of Concert “contained” in a scale with γ = 35.097... cents between A (i.e., the A note above middle C on a piano) to be 440. consecutive notes. In her article “Tuning: At the Cross- Various physical properties of the string govern its fun- roads” ([Car87]), Wendy Carlos explains how she derived damental frequency as seen in the following formulation these scales by experimentation. In contrast, we will de- of Mersenne’s laws (see [Tip78]): if our given string has rive these scales using the theory of continued fractions length L, tension (aka pulling force) F , and linear density in the spirit of Dunne and McConnell’s article “Pianos and (aka thickness) µ, then pF /µ f = . 2L Either increasing the tension F , decreasing the density µ, or shortening the length L, has the affect of increasing the frequency f which we perceive as an increase in the pitch of the sound. For example, the L and µ for a given string on a guitar remain constant, but we can increase the pitch by adjusting the tuning peg to increase the tension F . As another example, the tensions and lengths of the strings on a given guitar should be similar, but the pitches sound different due to the variation in thickness µ of each indi- vidual string; the thinner strings (smaller µ) have a higher pitch. 2.2. Harmonics and Timbre. Given the above determi- nation of a pitch based upon only the length, tension, and density of a string, one might ask: Why does a Concert A sound different when played on piano versus a guitar or a synthesizer? The answer lies in the fact that our ideal FIGURE 1. Wendy Carlos string may vibrate at multiple frequencies simultaneously 1 2 JORDAN SCHETTLER not just the fundamental frequency. These other frequen- and these give rise to an orthonormal decomposition cies, which are positive integer multiplies of the fundamen- 1 X inx tal frequency f , are called harmonics: f , 2f , 3f , 4f , .... h(x) = cne One way to see how harmonics influence sound quality is n=−∞ in the context of the periodic functions of a real variable which means which model the periodic vibrations of the string. Perhaps N 2 the best known example of a periodic function is the sine X inx lim h(x) − cne = 0. function sin(x) which has period 2π and corresponds to N!1 n=−N a sound we would recognize as a dial tone. In general, inx we use the word timbre to indicate the quality of a sound The functions e are analogous to the unit vectors e1 = independent of the pitch. For example, sin(2πfx) corre- (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1) in three-dimensional 3 sponds to a dial tone with arbitrary frequency f ; larger space R , and the above orthonormal decomposition is values of f correspond to higher pitched dial tones. The analogous to writing a vector cosine function of frequency f is just a horizontal shift of v = (c1, c2, c3) = c1e1 + c2e2 + c3e3. the sine function, namely cos(2πfx) = sin(2πfx + π=2), so In fact, the three-dimensional Pythagorean theorem this does not correspond to a different timbre. However, it 2 2 2 2 turns out that we can use sines and cosines to describe c1 + c2 + c3 = jjvjj every sufficiently nice periodic function. Hence any kind also holds in our infinite dimensional Hilbert space of sound made from the vibrations of our idealized string 1 X 2 2 can be built up from combining various dial tones. There is (2) jcnj = jjh(x)jj . a very elegant interpretation of this which begins with Eu- n=−∞ ler’s formula e2πifx = cos(2πfx) + i sin(2πfx) where i is a 2 Those functions h(x) which describe timbres of vibrat- fixed imaginary unit: i = −1. By allowing complex valued ing strings are real valued, so the complex conjugate sat- functions, we can capture all the harmonics with powers of isfies h(x) = h(x), which forces cn = c−n; this means our this exponential function. For instance, the nth harmonic amplitudes are real-valued and for n ≥ 1 satisfy is contained in the expression an = −2 · Im(cn) bn = 2 · Re(cn) (e2πifx )±n = cos(2π(nf )x) ± i sin(2π(nf )x). where cn = Re(cn) + iIm(cn) with Re(cn), Im(cn) 2 R. If The physical properties of the string dictate the extent to we further assume h(x) is odd, i.e., h(−x) = −h(x), then which various harmonics appear. In other words, the am- bn = 0 = c0 for all n ≥ 1, which implies that we can write plitude (aka volume) of certain harmonics will be greater our orthonormal decomposition as than others depending upon the instrument, and the sum 1 X of all these harmonics at various amplitudes determine the h(x) = an sin(nx) timbre. The language of Hilbert spaces neatly packages n=1 this viewpoint. For simplicity, we assume for the moment where an is the amplitude of the nth harmonic. For exam- that our period is 2π = 1=f , so all of our possible timbres ple, we define the “sawtooth” function s(x) on the interval are of the form [0, 2π) by −i2x −ix ix i2x ··· + c−2e + c−1e + c0 + c1e + c2e + ··· ( 0 if x = 0 s(x) = π − x where our coefficients cn give the amplitudes of the nth if 0 < x < 2π, 2 harmonic with n ≥ 1: an = i(cn − c−n) for the sines and and then we extend s(x) to by 2π-periodicity. See Figure bn = cn + c−n for the cosines. How do we interpret this R infinite sum? We define the Hilbert space L2([0, 2π]) to 2. The sawtooth function is real-valued, odd, and has a be the classes of all complex valued, 2π-periodic functions particularly nice orthonormal decomposition 1 h(x) with finite norm squared X 1 s(x) = sin(nx). 1 Z 2π n (1) jjh(x)jj2 = jh(x)j2 dx n=1 2π 0 The sound corresponding to the sawtooth does not come and where another such function g(x) is in the same class from some classical stringed instrument, rather is has a as h(x) when jjg(x) − h(x)jj2 = 0. Here the j · j2 in the “synthesizer” sound. Indeed, some analog synthesizers, integrand denotes the usual complex modulus squared: like the Moogs played by Wendy Carlos, use the sawtooth 2 2 2 ja + ibj = a + b for a, b 2 R. For any h(x) representing function in subtractive synthesis. This means that the syn- a class in L2([0, 2π]), we define Fourier coefficients thesizer has a built-in approximation to the sawtooth func- tion and can filter out or amplify various harmonics to pro- Z 2π 1 −inx duce a wide range of sounds from this one function. In cn = h(x)e dx 2π 0 contrast, additive synthesis would take various sine/cosine WENDY’S XENHARMONIC KEYBOARD 3 the A note in the sense that this class is literally the set of frequencies which determine an A note. We define the note space N2 := f[f ]: f 2 (0, 1)g where the subscript 2 indicates that two frequencies are related precisely when their quotient is an integer power of 2.

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