Abstract
Introduction
Results & Discussion
Conclusion
Ashly Kastenschmidt and Sarah Luman � Mathematics� University of Wisconsin-Eau Claire
Dr. James S. Walker
Ashly Kastenschmidt and Sarah Luman � Mathematics� University of Wisconsin-Eau Claire
Dr. James S. Walker
Perception of Loudness in Dissonance
and Harmonic Tones
Perception of Loudness in Dissonance
and Harmonic Tones
Methods and Materials� A Fletcher-Munson curve (left) was estimated to make
the interpolation curves (right) to be used in MATLAB.
This will allow the program to convert to phons which
are used to measure loudness.
We thank the Office of Research and Sponsored Programs for supporting this research, and Learning & Technology Services for printing this poster.
Acknowledgements
and References� We would like to thank William A. Sethares in his work
done creating the roughness curve.
(1) H. Fletcher and W. Munson. (1933). Loudness, its
definition, measurement and calculation, J. of the
Acoust. Soc. Am.,5:82– 108.
(2) W. Sethares. (1994). Local consanance and the
relationship between timbre and scale, J. Acoust. Soc.
Am., 94(3):1218–1228.
(3) MATLAB 8.0 and Statistics Toolbox 8.1, The
MathWorks, Inc., Natick, Massachusetts, United States.
Two different models for measuring auditory
roughness were created to see how humans perceive
loudness in dissonance and harmonic tones. Estimated
Fletcher-Munson curves that show the relation between
frequency of pitches and their loudness was incorporated
into a MATLAB code. This MATLAB code used
roughness measures by Sethares. Then dissonance graphs
were produced from the MATLAB code that also
compared Sethares original dissonance curve with the
new dissonance curve that incorporated the Fletcher-
Munson curves. Then two different models were created
with constant amplitude harmonics and exponential
decreasing harmonics. The constant amplitude harmonics
curve showed that there was significance in perception of
loudness in dissonance curves compared to Sethares
dissonance curve. However, the exponential decreasing
harmonics curve was very similar to the Sethares
dissonance curve.
�A roughness curve, created by Sethares, was coded
into MATLAB and used to compare the new
dissonance curves. These new curves were created
using the estimated Fletcher-Munson curve.
There were two different models created in
MATLAB. The first model created used exponential
decreasing harmonics whereas the other model had
constant amplitude harmonics. Each dissonance curves
has Sethares curve to compare to, which is in blue. The inspiration for this research was based on the
work done by Sethares. For the past 20 years he has been
analyzing sensory dissonance and its relation to musical
analysis and composition. Most of his work was based
on amplitude and the roughness in the harmonics. The
main purpose in this research is to incorporate phons into
dissonance curves which allows us to measure the
loudness for a certain frequency.
The figure above shows, in red, the exponential
decreasing harmonics incorporated with the Fletcher-
Munson curve is very similar to Sethares roughness
curve. There is a slight difference before and after the
musical interval 2 that shows there is more dissonance
around those intervals than the Sethares roughness curve
expects. Overall the largest peaks right after the musical
interval 0 and the peaks at intervals 1.5 and 2 have the
same dissonance as Sethares’s curve. Both curves show
a good representation of the dissonance perceived by the
human ear.
The red line in the figure below shows the constant
amplitude harmonic curve with the Fletcher-Munson
curve. This curve is very different from Sethares’s
roughness curve. It shows similar trends in the
dissonance level right after the musical interval 0, and at
interval 2. The interval at 1.5 has a greater dissonance
whereas there is less dissonance right before 1.5. Right
between 1.5 and 2 the dissonance is greater compared to
Sethares’s curve. This shows that when tones are played
at constant amplitudes, the dissonance in certain tones are
more noticeable to the human ear.
Once the Fletcher-Munson curve was estimated and
coded into MATLAB we were able to compare Sethares
roughness curve to the two new models. We found that
when the exponential decreasing harmonics curve was
very similar to Sethares’s curve, whereas the constant
amplitude harmonics curve showed greater dissonance in
most of the graph.