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VI. Scales & Consonance
Dr. Bill Pezzaglia
Incomplete Rough Draft
Updated
April 23, 2012
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Outline
A.
B.
C.
D.
Musical Scales & Intervals
Consonance & Dissonance
The Harmonic Series
References
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A. Musical Scales
1. Three Basic Scales
2. Labeling Notes
3. Musical Intervals
1. Three Basic Scales
(a) Chromatic Scale: Octave is divided into 12 semitone
steps (halfsteps)
(b) Diatonic Scale (white keys)
7 notes
(c) Pentatonic Scale (black keys)
5 notes
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1b Diatonic Scales
You can build more (7 note) scales by starting on any of
the white keys. Classic “greek” names:
Soldiers should only listed to Dorian or Phrygian scales (Plato)
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2. Labeling the Notes
(a) Boethian Notation: white keys are labeled “A”, “B”,
“C” through “G”.
(b) Sharps & Flats: black keys are notated
by # or “b”
(c) Octave Numbering: C2 is octave above C1
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2b Sharps and Flats
• In modern tuning, D#=Eb, they are “enharmonic keys”,
i.e. equivalent sounds.
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2. The Piano
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88 Keys (36 black, 52 white)
Start at A0 (27.5 Hz), end at C8 (4186 Hz)
Range: 7 octaves (plus 3 notes)
3. Musical Intervals
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Label White Keys 1 through 7
1 2 3 4 5 6 7
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CD
CE
CF
CG
CA
CB
CC
M2
M3
P4
P5
M6
M7
P8
Major 2nd (Dissonant)
Major 3rd
Perfect 4th
Perfect 5th
Major 6th
Major 7th
Octave (Consonant)
Demonstration Link:
http://www.music.sc.edu/fs/bain/atmi02/partch/index.html
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C.1. Harmonic Modes
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• Daniel Bernoulli (1728?) shows string can vibrate in
different modes, which are multiples of fundamental
frequency (called “Harmonics” by Sauveur)
n=1
f1
n=2
f2=2f1
n=3
f3=3f1
n=4
f4=4f1
n=5
f5=5f1
C.2. Harmonic Series
The musical notes of harmonic series
Reference: http://www.music.sc.edu/fs/bain/atmi02/hs/index-audio.html
Sound: http://www.music.sc.edu/fs/bain/atmi02/hs/playback/partials/hs1-12-c.mov
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C.3. Pythagorean Ratios
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Musical Intervals can be expressed as pure
mathematical ratios.
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Lower numbers sound more consonant
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Bigger numbers sound more dissonant
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CD
CE
CF
CG
CA
CB
CC
M2
M3
P4
P5
M6
M7
P8
Major 2nd
Major 3rd
Perfect 4th
Perfect 5th
Major 6th
Major 7th
Octave
8:9
4:5
3:4
2:3
3:5
8:15
1:2
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D. References
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http://en.wikipedia.org/wiki/Consonance_and_dissonance
http://www.tunesmithy.connectfree.co.uk/musical_note_intervals.htm
Bugle Demo of harmonics: http://www.philtulga.com/harmonics.html