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<H2><A NAME="sec:Evaluation">Evaluation </H2>

<P>
This section describes the results of an experimental evaluation of
the system. Our evaluation tested the tolerance of the system with
respect to input errors, whether from mistakes in the user's
humming or from problems with the pitch-tracking.

<H3><A NAME="subsec:Robustness">Robustness </H3>

<P>
The effectiveness of this method is directly related to the accuracy
with which pitches that are hummed can be tracked and the accuracy of
the melodic information within the database. Under ideal circumstances,
we can achieve close to 100%accuracy tracking humming, where ideal
circumstances mean the user places a small amount of space between each
note and hits each note strongly. For this purpose, humming short notes
is encouraged. Even more ideal is for the user to aspirate the notes as
much as possible, perhaps going so far as to voice a vowel, as in
``haaa haaa haaa''. We have currently only experimented with male
voices.

<P>
The evaluation database currently contains a total of 183 songs. Each
song was converted from public domain General MIDI sources. Melodies
from different musical genres were included, including both classical
and popular music.  A few simple heuristics were used to cut down on
the amount of irrelevant information from the data, e.g. MIDI channel
10 was ignored as this is reserved for percussion in the General MIDI
standard.  However the database still contains a great deal of
information unrelated to the main theme of the melody. Even with this
limitation, we discovered that sequences of 10-12 pitch transitions
were sufficient to discriminate 90%of the songs.

<P>
As a consequence of using a fast approximate string matching algorithm,
search keys can be matched with any portion of the melody, rather than
just the beginning. As the size of the database grows larger, however,
this may not prove to be an advantage.

<H5><!WA43><!WA43><!WA43><!WA43><!WA43><!WA43><!WA43><!WA43><A HREF="#sec:Evaluation"><-- Evaluation</A></H5>

<H3><A NAME="subsec:Performance">Performance </H3>
<P>
The version of the pitch-tracker using a modified form of
autocorrelation (<!WA44><!WA44><!WA44><!WA44><!WA44><!WA44><!WA44><!WA44><A HREF=#note:2>**</A>) takes between 20 and 45
seconds on a Sparc 10 workstation to process typical sequences of hummed
notes. A brute-force search of the database unsurprisingly shows linear
growth with the size of the database, but remains below 4 seconds for
100 songs on a Sparc 2. Therefore the search time is currently
effectively limited by the efficiency of the pitch-tracker.

<P>
Contour representations for each song are currently stored in separate
files, so opening and closing files becomes a significant overhead.
Performance could be improved by packing all the songs into one file,
or by using a database manager. We plan to modularize our code to make
it independent of any particular database schema.

<H5><!WA45><!WA45><!WA45><!WA45><!WA45><!WA45><!WA45><!WA45><A HREF="#sec:Evaluation"><-- Evaluation</A></H5>

<H5><!WA46><!WA46><!WA46><!WA46><!WA46><!WA46><!WA46><!WA46><A HREF="#toc"><-- Table of Contents</A></H5>
<HR>
<H2><A NAME="sec:Future">Future directions and Related Work </A></H2>

<P>
We plan to improve the performance and speed and robustness of the
pitch-tracking algorithm by using a cubic-spline wavelet. The cubic
spline wavelet peaks at discontinuities in the signal (i.e. the air
impulses). One of the most significant features of the wavelet analysis
is that it can be computed in <i>O(n)</i> time.  Currently, the pitch
tracker is the slowest link in our system, so using wavelets for this
purpose has obvious advantages.

<P>
The pattern matching algorithm in its present form does not
discriminate the various forms of pattern matching errors discussed
earlier, but only accounts for them collectively. Some forms of errors
may be more common than others depending upon the way people casually
hum different tunes. For example drop-out errors reflected as dropped
notes in tunes are more common than transposition or duplication
errors. Tuning the key-search so that it is more tolerant to drop-out
errors, for example, may yield better results.

<P>
The melodic contours of the source songs are currently generated
automatically from MIDI data, which is convenient but not optimal. More
accuracy and less redundant information could be obtained by entering
the melodic themes for particular songs by hand. From a research
standpoint, an interesting question is how to extract melodies from
complex audio signals<!WA47><!WA47><!WA47><!WA47><!WA47><!WA47><!WA47><!WA47><A HREF=#ref:hawley>[4]</A>.

<P>
Finally, we would like to characterize the improvement gained by
increasing the resolution of the relative pitch differences by
considering query alphabets of three, five and more possible
relationships between adjacent pitches. Early experiments using an
alphabet of five relative pitch differences (same, higher, much higher,
lower, much lower) verified that changes of this sort are promising.
One drawback of introducing more resolution is that the user must be
somewhat more accurate in the intervals they actually hum. We will
explore the various tradeoffs involved. An important issue is precisely
where to draw the line between notes that are a little higher from the
previous note and those that are much higher.

<P>
Previous work on efficiently searching a database of melodies by
humming seems to be limited. Mike Hawley <!WA48><!WA48><!WA48><!WA48><!WA48><!WA48><!WA48><!WA48><A HREF=#ref:hawley>[4]</A>
briefly discusses a method of querying a collection of melodic themes
by searching for exact matches of sequences of relative pitches input
by a MIDI keyboard. We have incorporated approximate pattern matching,
implementing an actual audio database (of MIDI songs) and most
significantly by allowing queries to be hummed. Kageyama and Takashima
<!WA49><!WA49><!WA49><!WA49><!WA49><!WA49><!WA49><!WA49><A HREF=#ref:kageyama>[8]</A> published a paper on retrieving melodies
using a hummed melody in a Japanese journal, but we were unable to
locate a translated version.
<H5><!WA50><!WA50><!WA50><!WA50><!WA50><!WA50><!WA50><!WA50><A HREF="#toc"><-- Table of Contents</A></H5>

<HR>
<H2><A NAME="sec:Footnotes">Footnotes </A></H2>

<DL>
<DT> <A NAME=note:1>*</A>
<DD> The terms vocal folds and vocal chords are more or less used as
synonyms in the literature.

<DT> <A NAME=note:2>**</A>
<DD> The modifications include low-pass filtering and
center-clipping (as described in Sondhi's paper
<!WA51><!WA51><!WA51><!WA51><!WA51><!WA51><!WA51><!WA51><A HREF=#ref:sondhi>[13]</A>) which help
eliminate the formant structure that generally causes difficulty for
autocorrelation based pitch detectors.

</DL>

<H5><!WA52><!WA52><!WA52><!WA52><!WA52><!WA52><!WA52><!WA52><A HREF="#toc"><-- Table of Contents</A></H5>
<HR>
<H2><A NAME="sec:References">References </A></H2>
<DL>
<DT><A NAME=ref:asifa><B>1</B></A><DD>
Ricardo A. Baesa-Yates and Chris H. Perleberg.
Fast and practical approximate string matching.
<em>Combinatorial Pattern Matching, Third Annual Symposium</em>, pages
  185-192, 1992.

<DT><A NAME=ref:asifd><B>2</B></A><DD>
Ricardo Baesa-Yates and G.H. Gonnet.
Fast string matching with mismatches.
<em>Information and Computation</em>, 1992.

<DT><A NAME=ref:handel><B>3</B></A><DD>
Stephen Handel.
<em>Listening: An Introduction to the Perception of Auditory
  Events</em>.
The MIT Press, 1989.

<DT><A NAME=ref:hawley><B>4</B></A><DD>
Michael Jerome Hawley.
<em>Structure out of Sound</em>.
PhD thesis, MIT, September 1993.

<DT><A NAME=ref:hess><B>5</B></A><DD>
Wolfgang Hess.
<em>Pitch Determination of Speech Signals</em>.
Springer-Verlag, Berlin Heidelberg, 1983.

<DT><A NAME=ref:hirano><B>6</B></A><DD>
M. Hirano.
Structure and vibratory behavior of the vocal folds.
In M. Sawashima and F.S. Cooper, editors, <em>Dynamic aspects of
speech production</em>, pages 13-27. University of Tokyo Press, 1976.

<DT><A NAME=ref:autocor><B>7</B></A><DD>
L.R. Rabiner J.J. Dubnowski and R.W. Schafer.
Real-time digital hardware pitch detector.
<em>IEEE Transactions on Acoustics, Speech and Signal Processing</em>,
  ASSP-24(1):2-8, Feb 1976.

<DT><A NAME=ref:kageyama><B>8</B></A><DD>
T. Kageyama and Y. Takashima.
A melody retrieval method with hummed melody (language: Japanese).
<em>Transactions of the Institute of Electronics, Information and
  Communication Engineers D-II</em>, J77D-II(8):1543-1551, August 1994.

<DT><A NAME=ref:asifb><B>9</B></A><DD>
G. Landau and U. Vishkin.
Efficient string matching with k mismatches.
<em>Theoretical Computer Science</em>, 43:239-249, 1986.

<DT><A NAME=ref:opp><B>10</B></A><DD>
A. V. Oppenheim.
A speech analysis-synthesis system based on homomorphic filtering.
<em>J. Acoustical Society of America</em>, 45:458-465, February 1969.

<DT><A NAME=ref:dtsp><B>11</B></A><DD>
Alan V. Oppenheim and Ronald W. Schafer.
<em>Discrete-time Signal Processing</em>.
Prentice Hall, Englewood Cliffs, NJ, 1989.

<DT><A NAME=ref:plomp><B>12</B></A><DD>
R. Plomp.
<em>Aspects of tone sensation</em>.
Academic Press, London, 1976.

<DT><A NAME=ref:sondhi><B>13</B></A><DD>
M. M. Sondhi.
New methods of pitch extraction.
<em>IEEE Trans. Audio Electroacoust. (Special Issue on Speech
  Communication and Processing-Part II</em>, AU-16:262-266, June 1968.

<DT><A NAME=ref:mlh><B>14</B></A><DD>
James D. Wise, James R. Caprio, and Thomas W. Parks.
Maximum likelihood pitch estimation.
<em>IEEE Trans. Acoustics, Speech, Signal Processing</em>, 24(5):418-423,
  October 1976.

</DL>

<H5><!WA53><!WA53><!WA53><!WA53><!WA53><!WA53><!WA53><!WA53><A HREF="#toc"><-- Table of Contents</A></H5>
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