PITCH RECOGNITION
Pitch Recognition with Wavelets
1.130 Wavelets and Filter BanksMay 15, 2003Project by:
Stephen Geiger
[email protected]
This report investigates the use of wavelets for pitch
recognition. A method is developed using the Continuous Wavelet
Transform at various scales to identify individual notes.
Successful results were obtained for computer generated polyphonic
piano music that included octave intervals. The current method
requires the training of the system before recognition is possible
and may only work on some instruments. However, it seems possible
that the method could be extended to recognize real polyphonic
piano music.
Outline Introduction
Problem Description
Existing Methods
Developed Method and Results
Conclusions
References
Appendix A Matlab Code
Appendix B Additional Results
Introduction
Pitch recognition, the ability to identify notes contained in an
audio signal, is a task some humans are quite proficient at. Given
the sound of a dropped metal trash can lid, (or perhaps preferably
a violin) they can respond with the name of a corresponding musical
note. This ability is typically referred to in the music world as
perfect pitch. Not all humans seem to have this capability, and
there has been somewhat limited success in creating computerized
systems capable of pitch recognition. Research in this area has
been approached with different motivating factors from several
fields. Perhaps the most obvious application is in automatic
transcription of music [1][2][3]. There is also interest in pitch
recognition for analyzing models of musical instruments [4], speech
analysis [5], and from the perspective of perceptual computing [6].
The aim of this work was to explore the use of wavelets [7] for
computer based pitch recognition.
Problem Description
Pitch is one of the properties of sound. It is perhaps most
simply described as how high or low a sound is (not loud and soft,
but high and low). Pitch also refers to what musical note a sound
can be described as. In more technical terms, pitch relates to the
fundamental frequency of a sound. Each musical note has a unique
fundamental frequency. However, a sound or note typically does not
consist of one pure frequency. This is shown in the following
graph: SHAPE \* MERGEFORMAT
Relative Frequency Content of a Computer Generated Piano
Sound
The graph displays the frequencies present in a Middle C (C4)
with fundamental frequency 262 Hz. It can be seen that there is a
large frequency component at the fundamental frequency, and that
there are frequency components at integer multiples of this
frequency (harmonics). The fundamental frequency is not always the
largest component as shown here:
Relative Frequency Content of a Computer Generated Oboe SoundIn
the case of the Oboe sound, the fundamental frequency is again 262
Hz, and is present with its harmonics; but, one can notice that the
most prominent frequency component is the 4th harmonic.
What may not be obvious is that to the human ear this sound will
be heard as having the same pitch as sinusoidal wave at the
fundamental frequency of 262 Hz. This is despite the fact the
strength of the fundamental frequency component in the signal is
relatively small. In fact, there can be cases where the fundamental
frequency of a sound is not even present in the signal. It is
worthwhile to note that the varying distribution of strengths of
frequency components in a note is what determines its musical
property called timbre. This is the property that makes an oboe
sound like an oboe, and a piano sound like a piano, and a trumpet
sound like a trumpet, etc.Two more relevant terms to mention are
monophonic and polyphonic. A monophonic sound is one where there is
only one pitch present at any given time. Some examples would be
one person singing or a single trumpet. Polyphonic sounds are ones
that contain multiple notes simultaneously such as an orchestra, or
barbershop quartet.
There are several existing methods for monophonic pitch
recognition and these have had some success. Polyphonic pitch
recognition has proven significantly more difficult. This is
partially because the combined of frequency spectrum of various
notes, is more difficult to analyze, and is especially so in the
case of identifying two pitches related by an interval of one
octave (for example a middle C and the next highest C played
together). This is because all of the frequency components found in
the higher note in an octave will also be present in the lower note
[8].
Existing Methods
A brief overview of some of the methods that have been tried for
pitch detection is presented here. Monophonic transcription
techniques include time domain techniques based on zero crossings
and auto-correlation, and frequency domain techniques based on
discrete Fourier transform and cepstrum methods, see references in
[8]. The estimation of local maxima to find the pitch period (which
can be switched easily to frequency) with the incorporation of
wavelets is described in [1][9]. Another technique using wavelets
to estimate pitch period and a comparison to auto-correlation
methods is presented in [4].The use of models of human pitch
perception is also described in [8], as well as the concept of
blackboard systems. This approach incorporates various sources of
knowledge and these sources could include music theory or
statistical and probabilistic knowledge [2][6]. Lastly, it is worth
noting that one approach to dealing with the problem of
distinguishing octaves is to incorporate instrument models.
Developed Method and Results
Taking a different approach, the method developed in this work
makes use of the Continuous Wavelet Transform (CWT), and uses a 2nd
Order Gaussian Wavelet. The Continuous Wavelet Transform is defined
as follows:
And the 2nd Order Gaussian Mother Wavelet has the following
appearance:
When the scaling parameter, a, in the wavelet transform is
varied it has the effect of stretching or compressing the mother
wavelet.
The implementation of the CWT found in the Matlab Wavelet
Toolbox was used, and further explanation of the CWT, and the 2nd
Order Gaussian wavelet can be found in the Wavelet Toolbox Users
Guide [10].
The idea for this method is based on an observation made by
Jeremy Todd [11]. In his work he found that taking the CWT of a
recording of a piano using a certain CWT scale parameter and a 2nd
Order Gaussian wavelet function the onset of a specific note (a G4)
could be easily identified. This observation is shown in the
following illustration:
SHAPE \* MERGEFORMAT
Furthermore, Todd observed that the same result would occur in
situations with polyphony as well. This was particularly
interesting.I started my work by running a number of continuous
wavelet transforms of varying scale on some test signals (computer
generated piano sounds), and observing the results. After looking
at a number of results it was possible to identify CWT scale
coefficients to respond to all the notes in the musical scale
starting at C4. (Note: in the previous sentence the term scale is
used with two different meaning; the former instance using its
wavelet definition, and the latter its musical definition). These
results are shown here:
The CWT with each one of the selected scaling factor had large
values at the occurrence of a specific note, and comparatively
small value during the rest of the signal. Next, we can observe the
results of the CWTs in the presence of some polyphony:
and:
In both cases this method worked, even with the presence of
polyphony. Furthermore, in the second example we see that both the
C and the G are not affected by the presence of other octaves.
(Note: the three areas of large response on the first line [Scale
=594] of the second example are correct. The second two occurrences
of the C are found in the bass clef).
One of the next steps to take was testing whether notes played
with a different instrument (i.e. having a different timbre) would
work or not. This test was run using a computer generated brass
sound, and the results clearly show that it did not work.
This result was somewhat expected, and it suggests that the CWT
is acting as an instrument model of sorts. When the scale parameter
of the CWT is adjusted it affects frequency response. So at certain
scale parameters it appears that this frequency response is
tailored such that it responds to one pitch more than others.
Based on the encouraging results so far, investigation was
continued to see how effective this method could be on a larger
scale. At this point a training algorithm was written to aid in the
identification of appropriate scale factors corresponding to
various pitches. The computer was programmed to train itself to
find applicable scale factors.The algorithm was implemented in
Matlab and works as follows:
Different sound files were created for each note in a range of
desired notes. The CWT of each sound file was taken
The maximum results of the CWTs from each sound file were
compared. If the maximum CWT coefficient from one file was at least
twice the value of those in all other files it was considered a
result.
For all the results the following were recorded: the scale
factor, pitch of the sound file, and the factor its max value
exceeded all others by.
This process was repeated over a range of CWT scale factors in
hopes of finding results for every pitch in the desired range of
notes. At the end the scale factor of the best result for each
pitch was collected. (The code for this algorithm, as well as some
of the other work for this project, is included in Appendix A).
This algorithm was applied to several different types of
training signals. It was tried on computer generated piano sound
for a range of three octaves, for a real guitar sound (albeit
electric guitar, 70s Ibanez Les Paul), for a set of pure sinusoidal
waves, and lastly for a training set of all 88 keys from the
computer generated piano.
The training on the three octave range was able to find results
for all pitches except the bottom two notes. This is likely due to
the fact that a limited set of CWT scales was searched, and it is
hypothesized that given a large range these values would have been
found as well. The results are shown here.
The training on the real guitar sound met with limited success.
Only 5 out of 8 notes were identified in the training process
(again for a somewhat limited set of scales); however, the results
were not completely successful in identifying the corresponding
notes in a test file. It wasnt a complete failure, and could merit
a more thorough try, but the guitar is expected to be a more
difficult case than a basic computer generated sound, or even a
real piano.
The results for the sinusoidal wave form were found as a step to
help gain a better understanding about the relationship between
scale and frequency. It can be observed that changing the CWT scale
shifts the frequency response of the transform. It also can be
observed that some interesting relationships exist between which
scales yield results for which notes as seen in the following two
graphs.
Successful Results from the Training Algorithm For 8 Sinusoidal
Pitches in a C Scale
Successful Results from the Training Algorithm For 3 Octaves of
a Computer Generated Piano Sound
In the first graph with pure sinusoidal sounds, the scale
frequency relationship seems a little more straightforward then in
the second case. There also could be some patterns in the second
graph as well, though they are less apparent.
The tests with all 88 notes were abandoned after considering the
time they required to run, and the amount of time left to complete
this work. Its worth noting here that running CWTs for a number of
test files at a number scales, could take a number of hours. This
could possibly be sped up noticeably with shorter test files or a
lower sampling rate, but this was not investigated. The initial
results from the training for 88 notes showed some interesting
results, picking out notes 70-88. The notes appeared to show up in
more clearly defined regions than in the three scale test case. It
seems possible that a training run of the three scale test case at
higher CWT scales might yield similar results.
Lastly, a fragment of the right hand part of Chopins Prelude in
C, Op. 28 No. 1 was tested, and the results were output into more
music like format for comparison:
A Test Fragment by Chopin
A comparison of the musical score and the graph reveals that the
method successfully identified all the notes contained in this
polyphonic fragment. This is noteworthy, as the method was
successful, even in situations with polyphony and octaves.
Conclusions
An application of the Continuous Wavelet Transform to pitch
recognition was explored, and some interesting results were found.
The method demonstrated the ability to recognize a reasonably
complex polyphonic fragment and octaves, which means it compares
favorably with some of the other results in literature I came
across. Two significant drawbacks of the method are that it
required training on an instrument sounds, and it is possible that
it might be effective only on some instruments. The most obvious
next step would be to try and apply the current technique to a real
piano and observe how well it worked. One issue that might need to
be dealt with is variation in the volumes of notes played, as this
might interfere with the simple maximum method used for identifying
results. Possibly some type of compression or normalization could
be applied. Another issue would be the identification of the
beginning and ends of notes. If this was handled successfully the
system would be well on its way to handling basic music
transcription. Maybe similar techniques to those used in wavelet
edge detection could be applied to this problem?
References[1] Kevin Chan, Supaporn Erjongmanee, Choon Hong Tay,
Real Time Automated Transcription of Live Music into Sheet Music
using Common Music Notation, 18-551 Final Project (Carnegie
Mellon), May 2000.
[2] Martin, K. D. (1996). A Blackboard System for Automatic
Transcription of Simple Polyphonic Music. M.I.T. Media Lab
Perceptual Computing Technical Report #385, July 1996.
[3] Michelle Kruvczuk, Ernest Pusateri, Alison Covell, Music
Transcription for the Lazy Musician, 18-551 Final Project (Carnegie
Mellon), May 2000.
[4] John Fitch, Wafaa Shabana, A Wavelet-Based Pitch Detector
for Musical Signals.
[5] Inge Gavat, Matei Zirra, Valentin Enescu, Pitch Detection of
Speech by Dyadic Wavelet Transform.
http://www.icspat.com/papers/181mfi.pdf[6] Martin, K. D. and
Scheirer, E. D. (1997). Automatic Transcription of Simple
Polyphonic Music: Integrating Musical Knowledge. Presented at SMPC,
August 1997.
[7] Robi Polikar, The Wavelet Tutorial,
http://engineering.rowan.edu/~polikar/WAVELETS/WTtutorial.html.
[8] Martin, K. D. (1996). Automatic Transcription of Simple
Polyphonic Music: Robust Front End Processing. M.I.T. Media Lab
Perceptual Computing Technical Report #399, November 1996,
presented at the Third Joint Meeting of the Acoustical Societies of
America and Japan, December, 1996.
[9] Tristan Jehan, Musical Signal Parameter Estimation, Thesis
CNMAT, 1997.
http://cnmat.cnmat.berkeley.edu/~tristan/Report/Report.html[10]
Wavelet Toolbox Users Guide (MATLAB), Mathworks, 1997.[11] Jeremy
Todd, A Comparison of Fourier and Wavelet Approaches to Musical
Transcription, 18.327 Final Project (MIT).Appendix A Matlab
CodeAppendix B Additional ResultsFrequency, Hz
Frequency, Hz
EMBED Equation.3
CWT @
Specific
Scale
Original
Signal
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Original
Scale: 594
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Original
Scale: 594
530
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Original
Scale: 594
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NOTE NUMBER
SCALE
EMBED Excel.Sheet.8
NOTE NUMBER
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EMBED Excel.Sheet.8
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3ScalesResults3
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Note Number
Scale
GuitarResults2
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Sinusoidal Chart
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3ScalesChart
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499482.5565.5
500485.5567.5
507486.5589.5
510496.5591.5
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831985.5
838992.5
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3ScalesResults1
212142.177643
193182.212915
203202.395037
213212.084562
123943.035565
123952.926503
123962.550727
123982.054979
124272.520421
114422.578317
114432.423267
114442.383787
114452.866099
114463.594242
114472.055408
124642.550141
104723.201081
104752.32026
104763.476367
114792.681666
204972.164351
204992.983465
205002.030092
105072.274153
105102.491557
105112.26256
225192.752277
225232.98961
95292.149883
95302.05013
95322.471369
165383.465797
165393.259343
165404.606992
165414.408437
165422.436256
125643.510069
125654.068228
125662.056952
125672.792566
85903.092185
85913.120598
85922.37176
85943.997286
85952.886781
85962.323083
85972.418848
156012.068142
156022.865551
156032.990665
156064.071281
156073.884132
66182.058132
96202.380116
66222.119798
146382.322108
146422.305401
146432.248866
176642.239396
106712.939709
106724.023252
106733.102698
106742.161128
106762.014186
86952.009291
56982.186252
127042.136699
127052.068791
127102.205146
127112.076163
137162.796206
137173.108981
137182.446398
137192.278683
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137234.750081
37282.163997
37352.077926
147512.022979
167562.079683
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128143.415824
78182.166499
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38262.40999
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108452.163603
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119132.969486
119142.699302
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79973.338417
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79992.029896
710002.025474
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1310082.132004
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510302.006307
2210422.356103
810552.312996
810563.564305
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2111042.15011
711212.13836
711222.818514
711232.100654
711292.285881
711302.453265
711312.109204
1211362.146678
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1211402.558493
1211412.373317
1211423.75586
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1211523.185047
812192.784557
812202.559068
2012662.235563
2012672.108195
2012742.070478
2012752.184847
2012762.077282
1112832.294904
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1112852.03324
1013452.245692
1013512.183885
1013522.483285
1013532.964855
1013542.896236
1013553.071583
1013564.210997
1013575.7284
1013586.311492
1013595.490148
1013608.212197
1013616.405734
1013623.824978
1013634.031316
1013644.024063
1013653.367693
1013662.8847
1013672.059401
1013682.204621
1013692.187233
1914202.016631
1914212.098074
1914222.190312
1914302.1536
1914312.12057
1914322.029217
1415052.176181
1415062.42536
1415072.180954
915252.315788
915262.620054
915272.907879
915282.754995
915292.522607
1815942.459006
1815952.495861
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1816022.604942
1816032.858121
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1816053.034099
1816062.997259
1816072.598434
1816082.264764
1716902.043466
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1717032.015216
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817122.243798
817132.385439
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817152.346288
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418282.054677
418292.197262
418302.354005
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2219602.144258
2219622.287787
2219632.454804
2219642.538682
2219652.410123
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2219692.07843
3ScalesResults2
20213.52.237525
22292.52.015675
16315.52.152406
19318.52.459157
20319.52.29491
10332.52.24193
12396.53.221832
11415.52.036767
11442.53.01314
11443.52.029879
11444.52.147233
11446.52.341882
12463.52.760469
12464.52.086933
10469.52.289853
10471.52.246745
10472.52.690418
10474.52.168868
10475.52.670309
11478.52.009287
11481.52.42831
11482.52.273234
7485.52.371171
7486.52.165277
20496.52.187615
20497.53.029671
20499.52.286733
3ScalesResults3
10506.52.084705
10507.52.235192
10510.52.896844
22516.52.039151
22517.52.110993
22518.52.143455
22522.53.817154
22523.52.856244
9529.52.895495
9530.53.353053
9531.52.646454
16536.52.966781
16538.53.29103
16539.54.684091
16540.55.276969
16541.54.183998
16542.52.175177
6545.52.473525
21554.52.088811
12563.52.242728
12564.53.527681
12565.53.179989
12567.52.174375
8589.52.289029
8591.52.699494
8593.52.705104
8596.52.803951
15601.52.382
15602.53.592575
15605.53.877179
15606.55.370706
15607.52.152265
9619.52.491008
6621.52.406058
9623.52.016377
11634.52.204935
17663.52.569585
10670.52.298315
10671.54.073235
10672.53.736069
10673.52.023352
10674.53.351876
21685.52.016702
4686.52.028145
22701.52.021406
12704.52.170082
12705.52.16512
12709.52.141011
12710.52.139792
13716.53.307506
13717.53.218774
13718.53.838929
13721.53.715156
13722.54.226244
13723.52.676037
3728.52.29823
3734.52.241034
14751.52.144459
16756.52.140589
16757.52.206811
16758.52.150385
16759.52.774111
16760.52.535803
16761.52.147802
16762.52.29336
16763.52.390304
4784.52.360903
11790.52.467306
11791.53.912961
11792.53.063538
11796.53.115746
11797.53.943577
11798.52.659171
5801.52.152254
5802.52.679552
12806.52.240367
12807.53.399004
12808.52.630413
12809.52.280543
12813.52.84327
12814.52.078805
7817.52.231381
7818.52.23428
3823.52.548049
3824.52.844683
3825.52.728131
3826.52.223328
3829.52.700978
3830.52.881632
10837.52.446532
10838.52.26018
10843.52.082547
10844.52.217708
15853.53.227568
15854.52.324903
15855.52.143762
15856.52.238491
5877.52.126164
11886.52.017565
7895.52.117832
14903.52.707813
14904.52.681601
11912.52.328501
11913.52.947694
11914.52.272725
9939.52.088466
9940.53.214567
9941.53.42054
9942.52.435257
9943.52.012856
10965.52.288062
10968.52.629413
7975.52.006197
4984.52.207811
4985.52.31063
7992.52.346587
7993.52.189582
7994.52.443476
7995.53.672563
7996.54.071258
7997.52.754862
