PHYS 103lecture #11

Musical Scales

Properties of a useful scale

• An octave is divided into a set number of notes• Agreed-upon intervals within an octave– not necessary for consecutive notes to have the same

interval– Examples: diatonic, pentatonic, blues, Indian

• Most intervals should be consonant (pleasing)– exact frequency ratios (e.g. 3:2 or 4:3) are preferred

• Intervals should be consistent– Frequency ratios are the same for a given interval– Example: C-G (fifth) is equivalent to D-A (fifth)

Pythagorean Scale

Construction of a diatonic scale based on the interval of a fifth

frequency ratio of a perfect fifth is 3/2 (going up) or 2/3 (going down)

Pythagoras (ca. 500 BC) supposedly observed that consonant intervals produced by two vibrating strings occurred when the string lengths had simple ratios.

L=2 units

L=3 units

Pythagorean Scale

1. Start with some pitch, called the tonic, which is the foundation of the scale. Any frequency will do. Let’s call this note C. fC = 400 Hz.

2. Determine the pitch that is a fifth above the tonic: 600 Hz. (G)

3. The next note is a fifth above G: 900 Hz. But notice that this note is more than an octave above C. So we drop down an octave by dividing by 2. Call this note D: fD = 450 Hz.

4. What is the interval between the tonic and this new note?

5. Repeat this process (multiply the previous frequency by 3/2 and divide by 2 if needed to stay within the octave) until you have a total of 6 notes.

6. The seventh (final) note of our scale is obtained by going down a fifth from the tonic, then multiplying by 2 to get back to the correct octave.

Construction of a diatonic scale based on the interval of a fifth

frequency ratio of a perfect fifth is 3/2 (going up) or 2/3 (going down)

€

f =3

2fC =

€

f =3

2fG =

3

2×3

2× fC =

Circle of fifthsnote Pythagorean recipe overall

ratioreduced ratio

interval

C 1 1 1 unison

G fifth

D octave + major second

A octave + major sixth

two octaves + major third€

32

€

32 ×

32 ×

32

€

94

€

32

€

32 ×

32

€

278

€

2 × 2716

€

2 × 98

€

32

€

32 ×

32 ×

32 ×

32

€

8116

€

4 × 8164

Pythagorean Problems

*

**

major third is slightly sharp (frequency is a little too high)

€

816454=81

80

€

271653=81

80major sixth is slightly sharp by the same amount

€

243128

158

=81

80major seventh is also slightly sharp by the same amount

Chromatic Just Scalenote span

interval frequency ratio

C-C# semitone 16/15C-D whole step 9/8C-D# minor third 6/5C-E major third 5/4C-F perfect fourth 4/3C-F# augmented fourth 45/32C-G perfect fifth 3/2C-G# minor sixth 8/5C-A major sixth 5/3C-A# minor seventh 16/9C-B major seventh 15/8C-C octave 2/1

Ideal intervals from C, but others not so good.

F#-C# should be a perfect fifth (3/2), but is actually 1024/675 = 1.52

Half-steps come in three different sizes!

F#-C#

half step ratio

16/15

135/128

16/15

25/24

16/15

135/128

16/15

16/15

25/24

16/15

135/128

16/15

Equal tempered scale• Every half-step must be identical in a chromatic scale• This means the ratio of each half step is a constant• 12 half-steps = 1 octave

half-step + half-step + half-step + half-step + half-step + half-step + half-step + half-step+ half-step + half-step + half-step

€

r = 212 =1.05946

€

r12 = 2

€

r × r × r × r × r × r × r × r × r × r × r × r = 2

This guarantees that every interval is the same, regardless of which note you start from.

Comparing Scalesnote span

interval just ratio equal tempered

C-C# semitone 1.0667 1.0595C-D whole step 1.125 1.121C-D# minor third 1.200 1.188C-E major third 1.250 1.259C-F perfect fourth 1.333 1.333C-F# augmented fourth 1.406 1.413C-G perfect fifth 1.500 1.497C-G# minor sixth 1.600 1.586C-A major sixth 1.667 1.681C-A# minor seventh 1.778 1.781C-B major seventh 1.875 1.887C-C octave 2.000 2.000