1. Physics 371 March 7, 2002 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. 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 1. Musical Interval a musical interval is defined by frequency RATIO

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

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

5. "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). Conclusion: simple number ratios between frequencies are consonant

6. 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 3. Construction of the Just Scale

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

8. read off frequency ratios of intervals: nameexampleratio octaveC - C 2 just sixthC - A 5/3 fifthC - G 3/2 just fourthC - 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: