A First Course in Physics · Ch. XVIII, §§ 475–486 · Chapter 18: Properties of Musical Sounds
56. Strings, overtones, quality and harmony
Goal: Your student can state the laws of vibrating strings (length and tension), explain fundamentals and overtones, explain what gives a sound its quality, and explain discord as a matter of beats.
You'll need
- The book, open to these pages
- A guitar, violin or ukulele if you have one; otherwise a rubber band stretched around an open box, with a pencil as a movable bridge
- A piano (for the sympathetic-vibration test), if available
- A notebook and pencil
- A calculator (optional, for the problems)
What's in the book
The sonometer; vibration numbers inversely as length and as the square root of the tension; nodes and loops in strings; fundamentals and overtones (harmonics); a string plucked at different points; quality set by the overtones present; manometric flames; Helmholtz's resonators; sympathetic vibrations; harmony and discord explained by beats. Nine problems follow, with a portrait of Helmholtz.
The lesson, step by step
Hook question
SayA piano and a trumpet play the same note. Why do they sound different?
Read together
Read §§ 475–486. The cause of quality:
the characteristic of a tone which we call its quality is determined simply by the number and prominence of the overtones which are present in the tone.
Try it: the law of lengths
Pluck the open string (or rubber band). Press it exactly at the middle and pluck one half: an octave higher (twice the vibrations). Press at one third: the sol above that (three times). Tighten the string: the pitch rises (§ 476).
Try it: harmonics
On a guitar or a long rubber band, touch the string very lightly (don't press) exactly at the middle and pluck near the end: a clear, high, bell-like note, the first overtone (§ 479). Touch lightly at one third: the second overtone. Then pluck the open string near the end and in the middle: same pitch, different quality (§ 481).
Explain the idea
A string vibrates as a whole and at the same time in halves, thirds, quarters and so on. The whole gives the fundamental; the parts give overtones with 2, 3, 4… times the vibrations. Which overtones are strong decides the quality. Discord is beats: two notes (or their overtones) differing by about 30 vibrations a second sound worst (§ 486).
What we know now: today a computer can show a sound's overtones on a screen (a spectrum), doing electronically what Helmholtz did with his resonators.
Try it: sympathetic vibration
If you have a piano, hold down the right pedal and sing a loud note into it, then stop: the strings sing it back (§ 483–484).
Say it back
Your student explains why piano hammers are covered with felt (§ 481).
Problems
Work problems 1–9 on book pages 379–380. Use the scale table on page 368.
Afterwards
Listen to the same note played on two different instruments (or two voices) and describe the difference in quality.
Answers to the book's problems
1. One fourth of the string's length from the end (C′ is 4/3 the vibrations of G, so ¾ of the string must vibrate). 2. They are the same (half as many from doubling the length, twice as many from four times the tension). 3. B″, two octaves above B (5 × the vibrations of G). 4. Fourth overtone E″; fifth overtone G″. 5. 9 kg (vibrations in the ratio 3 : 2 need tensions in the ratio 9 : 4). 6. 2000 vibrations per second; mi two octaves above (mi″). 7. About 4147 (27 × 128 for seven octaves, × 6/5 for the two notes A to C). 8. Lowest note about 42 ft.; highest about 0.27 ft. (about 3.3 in.). 9. Bowing near the end brings out many overtones and a rich tone; bowing at the middle kills the even overtones and gives a hollow sound.
Worked out for this site from the scan. If a number in your copy differs, trust the book.