1. PHYS 103 lecture #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 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

4. 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. f C = 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: f D = 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)

5. Circle of fifths notePythagorean recipeoverall ratio reduced ratio interval C111 unison G fifth D octave + major second A octave + major sixth two octaves + major third

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 note span intervalfrequency ratio C-C#semitone16/15 C-Dwhole step9/8 C-D#minor third6/5 C-Emajor third5/4 C-Fperfect fourth4/3 C-F#augmented fourth45/32 C-Gperfect fifth3/2 C-G#minor sixth8/5 C-Amajor sixth5/3 C-A#minor seventh16/9 C-Bmajor seventh15/8 C-Coctave2/1 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 25/24 16/15 135/128 16/15

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 note span intervaljust ratioequal tempered C-C#semitone1.06671.0595 C-Dwhole step1.1251.121 C-D#minor third1.2001.188 C-Emajor third1.2501.259 C-Fperfect fourth1.333 C-F#augmented fourth1.4061.413 C-Gperfect fifth1.5001.497 C-G#minor sixth1.6001.586 C-Amajor sixth1.6671.681 C-A#minor seventh1.7781.781 C-Bmajor seventh1.8751.887 C-Coctave2.000