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MÜLLER et al.: SIGNAL PROCESSING FOR MUSIC ANALYSIS 1091
Fig. 3. Calculation of a log-frequency spectrogram as a columnwise linear
mapping of bins from a conventional (linear-frequency) spectrogram. The im-
ages may be interpreted as matrices, but note the bottom-to-top sense of the
frequency (row) axis.
Although conceptually simple, such a mapping often gives
unsatisfactory results: in the figure, the logarithmic frequency
axis uses
(one bin per semitone), starting at
Hz (A2). At this point, the log-frequency bins have centers
only 6.5 Hz apart; to have these centered on distinct STFT bins
would require a window of 153 ms, or almost 7000 points at
Hz. Using a 64-ms window, as in the figure, causes
blurring of the low-frequency bins. Yet, by the same token, the
highest bins shown—five octaves above the lowest—involve av-
eraging together many STFT bins.
The long time window required to achieve semitone resolu-
tion at low frequencies has serious implications for the temporal
resolution of any analysis. Since human perception of rhythm
can often discriminate changes of 10 ms or less [4], an anal-
ysis window of 100 ms or more can lose important temporal
structure. One popular alternative to a single STFT analysis is to
construct a bank of individual bandpass filters, for instance one
per semitone, each tuned the appropriate bandwidth and with
minimal temporal support [6, Sec. 3.1]. Although this loses the
famed computational efficiency of the fast Fourier transform,
some of this may be regained by processing the highest octave
with an STFT-based method, downsampling by a factor of 2,
then repeating for as many octaves as are desired [7], [8]. This
results in different sampling rates for each octave of the analysis,
raising further computational issues. A toolkit for such analysis
has been created by Schorkhuber and Klapuri [9].
E. Time-Chroma Representations
Some applications are primarily concerned with the chroma
of the notes present, but less with the octave. Foremost among
these is chord transcription—the annotation of the current chord
as it changes through a song. Chords are a joint property of
all the notes sounding at or near a particular point in time, for
instance the C Major chord of Fig. 2, which is the unambiguous
label of the three notes C, E, and G. Chords are generally defined
by three or four notes, but the precise octave in which those notes
occur is of secondary importance. Thus, for chord recognition, a
representation that describes the chroma present but “folds” the
octaves together seems ideal. This is the intention of chroma
representations, first introduced as Pitch Class Profiles in [10];
the description as chroma was introduced in [11].
Fig. 4. Three representations of a chromatic scale comprising every note on
the piano from lowest to highest. Top pane: conventional spectrogram (93-ms
window). Middle pane: log-frequency spectrogram (186-ms window). Bottom
pane: chromagram (based on 186-ms window).
A typical chroma representation consists of a 12-bin vector
for each time step, one for each chroma class from C to B. Given
a log-frequency spectrogram representation with semitone res-
olution from the preceding section, one way to create chroma
vectors is simply to add together all the bins corresponding to
each distinct chroma [6, Ch. 3]. More involved approaches may
include efforts to include energy only from strong sinusoidal
components in the audio, and exclude non-tonal energy such as
percussion and other noise. Estimating the precise frequency of
tones in the lower frequency range may be important if the fre-
quency binning of underlying transform is not precisely aligned
to the musical scale [12].
Fig. 4 shows a chromatic scale, consisting of all 88 piano
keys played one a second in an ascending sequence. The top
pane shows the conventional, linear-frequency spectrogram, and
the middle pane shows a log-frequency spectrogram calculated
as in Fig. 3. Notice how the constant ratio between the funda-
mental frequencies of successive notes appears as an exponen-
tial growth on a linear axis, but becomes a straight line on a
logarithmic axis. The bottom pane shows a 12-bin chroma rep-
resentation (a “chromagram”) of the same data. Even though
there is only one note sounding at each time, notice that very
few notes result in a chroma vector with energy in only a single
bin. This is because although the fundamental may be mapped
neatly into the appropriate chroma bin, as will the harmonics at
, etc. (all related to the fundamental by octaves),
the other harmonics will map onto other chroma bins. The har-
monic at
, for instance, corresponds to an octave plus 7 semi-
tones
, thus for the C4 sounding at 40 s, we
see the second most intense chroma bin after C is the G seven
steps higher. Other harmonics fall in other bins, giving the more
complex pattern. Many musical notes have the highest energy in
the fundamental harmonic, and even with a weak fundamental,
the root chroma is the bin into which the greatest number of
low-order harmonics fall, but for a note with energy across a
large number of harmonics—such as the lowest notes in the
figure—the chroma vector can become quite cluttered.
One might think that attempting to attenuate higher har-
monics would give better chroma representations by reducing
these alias terms. In fact, many applications are improved by
whitening the spectrum—i.e., boosting weaker bands to make
the energy approximately constant across the spectrum. This