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Physics 371 March 7, 2002

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. 1. Musical Interval.

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Physics 371 March 7, 2002

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  1. 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

  2. 1. Musical Interval • We perceive the intervals as being “the same” if the RATIO of the frequencies is the same: • example: 1:2 ratio (octave) 200Hz-> 400Hz • or 400Hz -> 800Hz • 2:3 ratio (“fifth”) 200Hz -> 300Hz • or 400Hz -> 600Hz • divide or multiply all frequencies by same number • -> same interval a musical interval is defined by frequency RATIO

  3. “roughness” fusionof tones within critical band two tones 1 1.12 fourth fifth 2 octave frequency ratio between tones 2. Consonance and Dissonance relative dissonance between pure tones: dissonant when both tones in same critical band beats

  4. sensory dissonance frequency ratio Dissonance Curve for Complex Tones Sethares, UW EEC

  5. Conclusion: simple number ratios between frequencies are consonant "simple" number ratios, e.g. 2:1 ("octave"); 3:2 ("fifth"); 5/4 ("major third") Physical basis (conjecture): no beats between overtones (overtones either agree exactly or not at all).

  6. 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 -> 300Hz -> 360Hz -> 480Hz

  7. Just Scale: white keys of the piano: THREE TRIADS multiply e.g. by 240 240 270 300 320 360 400 450 480Hz

  8. 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

  9. 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"

  10. Frequencies and frequency deviations measured in CENT octave: 12 semitones = 1200 cent proportion: examples:

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