1
Spring 2007 – Week 10 PHY 131 30
Consonance/Dissonance & Critical Band
Spring 2007 – Week 10 PHY 131 31
Consonance vs. Dissonance
• Complex tones– More consonant intervals have more
harmonics which coincide.– More dissonant intervals have more
harmonics which mismatch within a criticalband.
– Explains why small integer ratios areconsonant.
2
Spring 2007 – Week 10 PHY 131 32
Consonance vs. Dissonance
2093 2060 R2093 2198 R2093 2180 R2093
1831 1766 R1831 1884 R1831 1744 R1831
1570 1472 R1570 157015701570 1570
13081308 1256 R1308 13081308
1047 1177 R1047 942 R10471047 1177 R
785 883 R785785 872 R785 785
523 589 R523 628 R523523
262 294262 314262 436262 392
Fifth 3:2 Maj. 6th 5:3 Min. third 6:5 2nd 9:8
Spring 2007 – Week 10 PHY 131 33
Consonance vs. frequency
3
Spring 2007 – Week 10 PHY 131 36
Examples of scales
• Scale -- a succession of discrete frequencies(pitches) used in music
• Note -- an individual frequency in the scale
• Examples– Pentatonic: C-D-F-G-A– Diatonic: C-D-E-F-G-A-B– Church modes
Spring 2007 – Week 10 PHY 131 37
Scales
• A musical scale is a set of pitches withinan octave which we use to play music.
• What scales do we use? Why thosescales?
• Geographically and culturally distantgroups of humans have settled on thesame scales. Can we understand this?
4
Spring 2007 – Week 10 PHY 131 38
Scales 2
• Build a scale from consonant intervals.
• Earliest music was monophonic andpresumably vocal -- any set of pitcheswould be allowed.
• However, successive pitches are linkedby reverberation and memory sointervals occur even in monophonicmusic.
Spring 2007 – Week 10 PHY 131 39
Pentatonic scale• Use octave and fifth five times:
• In order, CDFGA, appears in many cultures– China, Polynesia, Africa, American Indians, Celts, Scots, Led Zeppelin
f =32f0 f =
3232f0
⎛⎝⎜
⎞⎠⎟=94f0 f =
32
⎛⎝⎜
⎞⎠⎟3
f0 =278f0f0
262 Hz 393 Hz 294 Hz 442 Hz 349 Hz
C FG D A
f = fo32
=23fo
5
Spring 2007 – Week 10 PHY 131 40
Diatonic scale
• Keep going, up two more fifths.
f =32
⎛⎝⎜
⎞⎠⎟4
f0 =8116
f0 f =32
⎛⎝⎜
⎞⎠⎟5
=24332
f0
f =148116
f0 =8164
f0 f =1424332
f0 =243128
f0
332 Hz 497 Hz
E B
Spring 2007 – Week 10 PHY 131 41
Diatonic scale
C D E F G BA C
f098f08164
f043f0
32f0
243128
f02716
f0 2 f098
98
256243
98
98
98
256243
• Keep going, up five more fifths to get twelve noteswithin an octave. This is called a chromatic scale.
• Problems:– Circle of fifths doesn’t close:– The major third C-E is sharp. 3
2⎛⎝⎜
⎞⎠⎟12
≈ 27
6
Spring 2007 – Week 10 PHY 131 42
Circle of fifths
Spring 2007 – Week 10 PHY 131 43
Temperament
• Now that we have our diatonic scale weneed to decide how to adjust (temper)the intervals.
• Infinite number of ways to do this.We’ll look at three.– Meantone temperament– Just temperament– Equal temperament
7
Spring 2007 – Week 10 PHY 131 44
Meantone temperament• Get exactly a 5:4 major third:
– Squeeze the fifth:
– Now circle of fifths falls short.– Fix by making one of the fifths (A flat-E flat) too wide.
Called the wolf fifth.
• Key of C sounds good. Keys get worse as you addsharps and flats.
• Demo: wolf fifth, 5:4 third vs. equal tempered third.
x4
4=54⇒ x = 54 ≈ 1.49527
Spring 2007 – Week 10 PHY 131 45
Just intonation (three chords and the truth)
• There are three major triads in thediatonic scale.– I or tonic: C E G– IV or subdominant: F A C– V or dominant: G B D
• Adjust the tuning so that all three havePythagorean intervals in the ratio 4:5:6
8
Spring 2007 – Week 10 PHY 131 46
Just intonation
C D E F G BA C
f098f054f0
43f0
32f0
53f0
158f0 2 f0
98
109
1615
98
109
98
1615
• Problems:– 5th from D-A is imperfect (40:27 not 3:2)– Different size whole steps: retune for each
key.– Enharmonic notes (G sharp and A flat)
don’t match.
