Draft. To appear in Plack, C. and Oxenham. A. (eds) Pitch. New York: SpringerVerlag (2004).6Pitch perception modelsAlain de Cheveigné1 IntroductionThis chapter discusses models of pitch, old and recent. The aim is to charttheir common points – many are variations on a theme – and differences,and build a catalog of ideas for use in understanding pitch perception. Thebusy reader might read just the next section, a crash course in pitch theorythat explains why some obvious ideas don’t work and what are currently thebest answers. The brave reader will read on as we delve more deeply intothe origin of concepts, and the intricate and ingenious ideas behind themodels and metaphors that we use to make progress in understanding pitch.2 Pitch Theory in a NutshellPitch-evoking stimuli usually are periodic, and the pitch usually is related tothe period. Accordingly, a pitch perception mechanism must estimate theperiod T (or its inverse, the fundamental frequency F0) of the stimulus.There are two approaches to do so. One involves the spectrum and the otherthe waveform. The two are illustrated with examples of stimuli that evokepitch, such as pure and complex tones.2. 1 SpectrumThe spectral approach is based upon Fourier analysis. The spectrum of apure tone is illustrated in Figure 1A. An algorithm to measure its period(inverse of its frequency) is to look for the spectral peak and use its positionas a cue to pitch. This works for a pure tone, but consider now the soundillustrated in Figure 1B, that evokes the same pitch. There are several peaksin the spectrum, but the previous algorithm was designed to expect onlyone. A reasonable modification is to take the largest peak, but consider nowthe sound illustrated in Figure 1C. The largest spectral peak is at a higherharmonic, yet the pitch is still the same. A reasonable modification is toreplace the largest peak by the peak of lowest frequency, but consider nowthe sound illustrated in Figure 1D. The lowest peak is at a higher harmonic,yet the pitch is still the same. A reasonable modification is to use thespacing between partials as a measure of period. That is all the more22reasonable as it often determines the frequency of the temporal envelope ofthe sound, as well as the frequency of possible difference tones (distortionproducts) due to nonlinear interaction between adjacent partials. However,consider now the sound illustrated in Figure 1E. None of the inter-partialintervals corresponds to its pitch, which (for some listeners) is the same asthat of the other tones.This brings us to a final algorithm. Build a histogram in the followingway: for each partial, find its subharmonics by dividing the frequency of thepartial by successive small integers. For each subharmonic, increment thecorresponding histogram bin. Applied to the spectrum in Figure 1E, thisproduces the histogram illustrated in Figure 1F. Among the bins, some arelarger than the rest. The rightmost of the (infinite) set of largest bins is thecue to pitch. This algorithm works for all the spectra shown. It illustrates theprinciple of pattern matching models of pitch perception.2. 2 WaveformThe waveform approach operates directly on the stimulus waveform.Consider again our pure tone, illustrated in the time domain in Figure 2A.Its periodic nature is obvious as a regular repetition of the waveform. A wayto measure its period is to find landmarks such as peaks (shown as arrows)and measure the interval between them. This works for a pure tone, butconsider now the sound in Figure 2B that evokes the same pitch. It has twopeaks within each period, whereas our algorithm expects only one. A trivialmodification is to use the most prominent peak of each period, but considernow the sound in Figure 2C. Two peaks are equally prominent. A tentativemodification is to use zero-crossings (e.g. negative-to-positive) rather thanpeaks, but then consider the sound in Figure 2D, which has the same pitchbut several zero-crossings per period. Landmarks are an awkward basis forperiod estimation: it is hard to find a marking rule that works in every case.The waveform in Figure 2D has a clearly defined temporal envelope with aperiod that matches its pitch, but consider now the sound illustrated inFigure 2E. Its pitch does not match the period of its envelope (as long as theratio of carrier to modulation frequencies is less than about 10, see Plackand Oxenham, Chapter 2).This brings us to a final algorithm that uses, as it were, every sample as a“landmark”. Each sample is compared to every other in turn, and a count iskept of the inter-sample intervals for which the match is good. Comparisonis done by taking the product, which tends to be large if samples x(t) andx(t-τ) are similar, as when τ is equal to the period T. Mathematically:r(τ) = x(t)x(t −τ)dt∫(1)defines the autocorrelation function, illustrated in Figure 2F. For a periodicsound, the function is maximum at τ=0, at the period, and at all itsmultiples. The first of these maxima with a strictly positive abscissa can beused as a cue to the period. This algorithm is the basis of what is known asthe autocorrelation (AC) model of pitch. Autocorrelation and patternmatching are both adequate to measure periods as required by a pitch model,and they form the basis of modern theories of pitch perception.Created 29/01/04 17:18 IrcamLast saved 31/01/04 8:55 This copy printed 31/01/04 8:5533We reviewed a number of principles, of which some worked and othersnot. All have been used in one pitch model or another. Those that use aflawed principle can (once the flaw is recognized) be ruled out. It is harderto know what to do with the models that remain. The rest of this chaptertries to chart out their similarities and differences. The approach is in parthistorical, but the focus is on the future more than on the past: in whatdirection should we take our next step to improve our understanding ofpitch?2. 3 What is a model?An important source of disagreement between pitch models, often notexplicit, is what to expect of a model. The word is used with variousmeanings. A very broad definition is: a thing that represents another thingin some way that is useful.