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PHYS 103 lecture #11

PHYS 103 lecture #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

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PHYS 103 lecture #11

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  1. PHYS 103lecture #11 Musical Scales

  2. 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)

  3. 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)

  4. 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.

  5. Circle of fifths

  6. Pythagorean Problems * * * 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

  7. 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

  8. 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.

  9. Comparing Scales

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