Physics 371 March 7, 2002

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Physics 371 March 7, 2002
Acoustics for Musicians

• Consonance /Dissonance
• Interval frequency ratio
• Consonance and Dissonance
• Dissonance curve
• The Just Scale
• major triad
• construction of just scale
• origin of the diatonic scale

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1. Musical Interval
We perceive the intervals as being the same if
the RATIO of the frequencies is the
same example 12 ratio (octave) 200Hz-gt
400Hz
or 400Hz -gt 800Hz 23 ratio
(fifth) 200Hz -gt 300Hz
or 400Hz -gt 600Hz divide or multiply
all frequencies by same number -gt same interval
a musical interval is defined by frequency RATIO
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2. Consonance and Dissonance
relative dissonance between pure tones dissonant
when both tones in same critical band
beats
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Dissonance Curve for Complex Tones
Sethares, UW EEC
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Conclusion simple number ratios between
frequencies are consonant
"simple" number ratios, e.g. 21
("octave") 32 ("fifth")
5/4 ("major third") Physical basis
(conjecture) no beats between overtones
(overtones either agree exactly or not at all).
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3. Construction of the Just Scale
freq. ratio of the THE TRIAD
4 5 6 (major
triad) or divide by 4 1 5/4
3/2 2 C
E G C call
first tone of scale C and assign to it a
frequency of 1unit (1 unit can be any number of
Hz) to get Hz, you could multiply by any number
you like, e.g. multiply by 60
240Hz -gt 300Hz -gt 360Hz -gt
480Hz
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Just Scale
white keys of the piano THREE TRIADS
multiply e.g. by 240 240 270 300
320 360 400 450 480Hz
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read off frequency ratios of intervals
name example ratio octave C - C 2 just
sixth C - A 5/3 fifth C - G 3/2 just
fourth C - F 4/3 just major third C-E
5/4 just minor third E - G 6/5
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if we call the first frequency (C) 1 the
frequencies become C D
E F G A B C
D 1 9/8 5/4
4/3 3/2 5/3 15/8 2 9/4
inter- val 9/8 10/9
16/15 9/8 10/9 9/8 16/15 incr
12.5 11.1 6.7 12.5 11.1 12.5
6.7 STEP 1 1 1/2 1
1 1 1/2
the "major just diatonic scale"
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Frequencies and frequency deviations measured in
CENT
octave 12 semitones 1200 cent proportion
examples