A First Course in Physics · Ch. XVIII, §§ 471–474 · Chapter 18: Properties of Musical Sounds
55. The physics of the musical scale
Goal: Your student can explain that harmonious intervals have simple vibration ratios, build the major scale from three major chords, and find the vibration number of any note of the scale.
You'll need
- The book, open to these pages
- A piano, keyboard, or a free keyboard or tuner app
- Eight identical glass bottles or glasses and water (a bottle xylophone)
- A notebook and pencil
- A calculator (optional, for the problems)
What's in the book
A siren disk with rows of 24, 30, 36 and 48 holes plays do, mi, sol, do'; the octave (1 : 2), fifth (2 : 3), fourth (3 : 4) and major third (4 : 5); the major chord (4 : 5 : 6); the major diatonic scale built from three major chords, with a table of relative and absolute vibration numbers (C = 256); standard pitches. Six problems follow.
The lesson, step by step
Hook question
SayWhy do some pairs of notes sound sweet together and others clash?
Read together
Read §§ 471–474 and study the table on book page 368. The book's discovery:
harmonious musical intervals correspond to very simple vibration ratios, and the more harmonious the combination the simpler the ratio.
Explain the idea
do : mi : sol : do' = 4 : 5 : 6 : 8. The octave doubles the vibrations; sol is 3/2 of do. The scale is built from three major chords: do-mi-sol, fa-la-do', sol-si-re (§ 473). With C = 256: D 288, E 320, F 341, G 384, A 427, B 480, C' 512.
Try it: hear the ratios
On a keyboard or app, play C and the C above together (octave), then C and G (a fifth), C and E (a third), then C and C-sharp (a clash). Play the chord C–E–G. A tuner app will show the vibration numbers (in hertz).
Try it: a bottle scale
Fill eight bottles with different amounts of water and tap them (or blow across them) to tune a scale, checking against the keyboard. Which needs more water for a higher note when tapped? When blown? (Tapping sounds the glass and water; blowing sounds the air column.)
What we know now
Today's standard pitch is A = 440 vibrations per second (fixed in 1939), a little higher than the 435 of the book's "international pitch." Pianos are now tuned in "equal temperament," which makes every half step the same ratio, so most intervals are very slightly off the book's simple ratios.
Say it back
Your student explains how the three major chords give all eight notes of the scale.
Problems
Work problems 1–6 on book page 369.
Afterwards
Pick out simple tunes on the bottle scale or a keyboard and listen for octaves and fifths.
Answers to the book's problems
1. G′, the sol of the octave above (768). 2. G″, two octaves above that G (1536). 3. E″, two octaves above E (1280). 4. 4.5 ft. 5. 2560 vibrations per second (an E, E‴). 6. F 400, A 500.
Worked out for this site from the scan. If a number in your copy differs, trust the book.