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### 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

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

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)

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)

- 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.
- Determine the pitch that is a fifth above the tonic: 600 Hz. (G)
- 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.
- What is the interval between the tonic and this new note?
- 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.
- 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.

Pythagorean Problems

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

major sixth is slightly sharp by the same amount

major seventh is also slightly sharp by the same amount

Chromatic Just Scale

Half-steps come in three different sizes!

Ideal intervals from C, but others not so good.

F#-C#

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

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

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

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