Music and the Chromatic Scale

Octave: a doubling of pitch -- a doubling of the frequency of a musical tone.

Equal temperament: the chromatic scale divides an octave into 12 semitones: each halfstep (to the next higher semitone) increases the frequency by a factor of 21/12 = 1.0594631. The division of the octave into 12 semitones is partly a cultural artifact and partly due to human physiology. Some of the powers of 21/12 are nearly simple ratios; pleasing intervals are heard when tones are sounded together:

n 0 1 2 3 4 5 6 7 8 9 10 11 12
2n/12 1 1.059 1.122 1.189 1.260 1.335 1.414 1.498 1.587 1.682 1.782 1.888 2.000
nearly 1 - 9/8 - 5/4 4/3 - 3/2 - 5/3 7/4 15/8 2
scale 1 - 2 - 3 4 - 5 - 6 - 7 1
C key C C# D D# E F F# G G# A A# B C

An instrument tuned with equal temperament can be played in different keys and sound good without needing to be retuned.

The guitar demonstrates some of these features of our musical scales, by its very construction -- frets are placed to set the length of the vibrating portion of a string. Tuning a Guitar can be done using the harmonic relationships discussed above.

Tuning a guitar


Harmonics are struck by plucking the string while lightly touching it near the fret indicated, to force a node in its vibrations at that place. Matches or near matches of harmonics (shown by the colors) are useful for tuning: adjust the string tension until the beats go away.

fundamental
harmonic 4x
harmonic 3x
guitar string
frequency
5th fret = 1/4
7th fret = 1/3
E (6)
329.628
1318.512
988.884
B (5)
246.942
987.768
740.826
G (4)
195.998
783.992
587.994
D (3)
146.832
587.328
440.496
A (2)
110.000
440.000
330.000
E (1)
82.407
329.628
247.221