170
D. Rocchesso: Sound Processing
Musical scales
musical octave
chroma
place theory of hearing
virtual pitch
missing fundamental
temporal processing of
sound
timbre
scales that organize musical height. Musical scales are based on the subdivision
of the musical octave into a certain number of intervals. The musical octave is
usually defined as the frequency range having the higher bound that has twice
the value in Hertz of the first bound. On the other hand, the subjective scale
for pitch measures the subjective pitch relationship between two sounds, and it
is strictly connected with the spatial distribution of frequencies along the basi-
lar membrane. In musical reasoning, pitch is referred to as chroma, which is a
different thing from the tonal height that is captured by figure 6.
0
500
1000
1500
2000
2500
3000
100
1000
10000
Pitch in Mels
Frequency in Hz
Subjective pitch curve
pitch
Figure 6: Subjective frequency curve, mel vs. Hz.
So far, we have described pitch phenomena referring to the position of hair
cells that get excited along the basilar membrane. Indeed, the place theory of
hearing is not sufficient to explain the accuracy of pitch perception and some
intriguing effects such as the virtual pitch. In this effect, if a pure tone at fre-
quency f
1
is superimposed to a pure tone at frequency f
2
=
3
2
f
1
, the perceived
pitch matches the missing fundamental at f
0
= f
1
/2. If the reader, as an ex-
cercise, plots this superposition of waveforms, she may notice that the apparent
periodicity of the resulting waveform is 1/f
0
. This indicates that a temporal
processing of sound may occur at some stages of our perception. The hair cells
convey signals to the fibers of the acoustic nerve. These neural contacts fire at
a rate that depends on the transversal velocity of the basilar membrane and on
its lateral displacement. The rate gets higher for displacements that go from the
apex to the base, and this creates a periodicity in the firing rate that is multi-
ple of the waveform periodicity. Therefore, the statistical distribution of neural
spikes keep track of the temporal behavior of the acoustic signals, and this may
be useful at higher levels to extract periodicity information, for instance by
autocorrelation processes [86].
Even for pure tones, pitch perception is a complex business. For instance, it is
dependent on loudness and on the nature and quality of interfering sounds [42].
The pitch of complex tones is an overly complex topic to be discussed in this
appendix. It suffices to know that pitch perception of complex tones is linked to
the third (after loudness and pitch) and most elusive attribute of sound, that is
timbre.