1. VI. Scales & Consonance Dr. Bill Pezzaglia1
VI. Scales & Consonance Dr. Bill Pezzaglia
Incomplete Rough Draft
Updated April 23, 2012

2. Outline Musical Scales & Intervals Consonance & Dissonance2
Outline
Musical Scales & Intervals
Consonance & Dissonance
The Harmonic Series
References

3. A. Musical Scales Three Basic Scales Labeling Notes Musical Intervals3
A. Musical Scales
Three Basic Scales
Labeling Notes
Musical Intervals

4. 1. Three Basic Scales 4
Chromatic Scale: Octave is divided into 12 semitone steps (halfsteps)
Diatonic Scale (white keys) 7 notes
Pentatonic Scale (black keys) 5 notes

5. 1b Diatonic Scales 5
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)

6. 2. Labeling the Notes 6
Boethian Notation: white keys are labeled “A”, “B”, “C” through “G”.
Sharps & Flats: black keys are notated by # or “b”
Octave Numbering: C2 is octave above C1

7. 2b Sharps and Flats 7
In modern tuning, D#=Eb, they are “enharmonic keys”, i.e. equivalent sounds.

8. 2. The Piano 88 Keys (36 black, 52 white)
Start at A0 (27.5 Hz), end at C8 (4186 Hz)
Range: 7 octaves (plus 3 notes)

9. 3. Musical Intervals 9 Label White Keys 1 through 7
CD M2 Major 2nd (Dissonant)
CE M3 Major 3rd
CF P4 Perfect 4th
CG P5 Perfect 5th
CA M6 Major 6th
CB M7 Major 7th
CC P8 Octave (Consonant)
Demonstration Link:

10. C.1. Harmonic Modes 10
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

11. C.2. Harmonic Series The musical notes of harmonic series 11
Reference:
Sound:

12. C.3. Pythagorean Ratios 12
Musical Intervals can be expressed as pure mathematical ratios.
Lower numbers sound more consonant
Bigger numbers sound more dissonant
CD M2 Major 2nd 8:9
CE M3 Major 3rd 4:5
CF P4 Perfect 4th 3:4
CG P5 Perfect 5th 2:3
CA M6 Major 6th 3:5
CB M7 Major 7th 8:15
CC P8 Octave 1:2

13. 13
D. References
Bugle Demo of harmonics: