a) Excerpt of waveform. Note that the waveform repeats every 2.27 ms, which is the period. b) Spectrum. Note the peaks at integer multiples of the F0, characteristic of a periodic sound. In this case the F0 is physically present, but the second, third, and fourth harmonics actually have higher amplitudes. c) Autocorrelation. The correlation coefficient is always 1 at a lag of 0 ms, but because the waveform is periodic, correlations close to 1 are also found at integer multiples of the period (2.27, 4.55, 6.82, and 9.09 ms in this example). Note that whereas evaluating periodicity in the waveform requires examining every point, the autocorrelation makes periodicity more explicit – it is signified by high correlations at regularly spaced intervals.

a) Scales. In Western music the octave is divided up into 12 semitones, equally spaced on a logarithmic frequency scale. All musical scales are drawn from this set of notes, which are given letter names. Three common scales are depicted, in the key of C: the major scale (outline circles); the harmonic minor scale (solid circles); a pentatonic scale (circles with lines). The solfege note names for the major scale are also included on the left. Other musical systems use different scales, but usually share some properties of Western scales - notes are typically not separated by less than a semitone, and the steps separating notes are usually not all equal. b) Musical intervals derived from the chromatic scale. In the equal-tempered scale that is common nowadays, the ratios in the rightmost column are only approximate. c) Tonal hierarchy. The figure plots results of probe-tone experiments from Krumhansl and colleagues. Listeners were played a melody in the key of C major or C minor, after which they were asked to rate the appropriateness of different probe tones. Notes outside the scale (white circles) are rated as inappropriate. The tonic (C) is most appropriate, followed by the other notes of the major triad (E and G for the major key; D# and G for the minor key), followed by the other notes of the key. The particular key structure shown here (i.e. the pattern of intervals between the notes) is specific to Western music, but qualitatively similar structures appear to be used and perceived in many other musical systems as well.

a) Transposition. Figure depicts two five-note melodies. The second melody is shifted upwards in pitch. In this case both the contour and the intervals are preserved. b) Contour and intervals of “Somewhere Over the Rainbow”.

a) Beating. The red and blue waveforms differ in frequency by 10%, and so shift in and out of phase over time. The first dashed line denotes a point where they are in phase; the second line denotes a point where they are out of phase. The black waveform is the sum of the red and blue, and exhibits beats – amplitude fluctuations that result from the changing phase relations of the two components. b) Spectrum for a perfect fifth harmonic interval (a consonant interval in Western music). Blue and red circles indicate the frequency components of the two notes of each interval. Note that for the perfect fifth, the components of the two notes either exactly coincide, or are far apart. c) Spectrum for a minor second harmonic interval, considered dissonant in Western music. Note that the frequency components tend to be close in frequency but not exactly overlapping. d) Waveform of perfect fifth. e) Waveform of minor second. Note amplitude modulation, produced by pairs of components that beat. f) Autocorrelation of waveform of perfect fifth. Note the regular peaks near 1 – the signature of periodicity. g) Autocorrelation of waveform of minor second. Note the absence of regular peaks.