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

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

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

Dissonance Curve for Complex Tones

Sethares, UW EEC

frequency ratio

sen

sory

dis

son

ance

"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 ratiosbetween frequencies are consonant

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

white keys of the piano: THREE TRIADS

multiply e.g. by 240 240 270 300 320 360 400 450 480Hz

Just Scale:

read off frequency ratios of intervals:

name example ratiooctave C - C 2just sixth C - A 5/3fifth C - G 3/2just fourth C - F 4/3just major third C-E 5/4just minor third E - G 6/5

if we call the first frequency (C) =1the 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"

Frequencies and frequency deviations measured in CENT

octave: 12 semitones = 1200 cent

proportion:

cent1200

=log(f1/f2)log2

examples: