Octaves and Fifths on a String

In the picture

1. 1Full string: the starting note, at the base frequency
2. 2Two-thirds of the string: 3/2 the frequency, a perfect fifth higher
3. 3Half of the string: twice the frequency, one octave higher
4. 4Movable bridge: shortening the vibrating part raises the pitch; repeating the 3:2 step and folding back into one octave gives the Pythagorean scale

Examples

• Pressing a guitar string at the 12th fret halves its length and plays the same note one octave higher.
• Violin strings are tuned a fifth apart, G, D, A, E, each one vibrating 3/2 as fast as the one below.

Quiz: Octaves and Fifths on a String (Age 16)

1. 1

   If you make the vibrating part of a string half as long, what happens to the note?

   • It sounds an octave higher, at twice the frequency
   • It sounds a fifth higher, at 3/2 the frequency
   • It sounds an octave lower, at half the frequency
2. 2

   A string shortened to two-thirds of its length plays a perfect fifth above the full string. What is the frequency ratio of that fifth?

   • 4:3
   • 3:2
   • 2:1
3. 3

   How did Pythagoras build his musical scale?

   • By dividing the octave into equal steps that all have the same ratio
   • By multiplying by 3/2 again and again, and halving whenever a note went past the octave
   • By multiplying by 2 again and again, and taking two-thirds whenever a note went past the octave

The answers, each with its explanation, are in the app. This lesson has left the free week, so its quiz is part of the subscription.