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.