the overtone series for strings

The equal-tempered scale has 12 intervals (half-steps) making up an octave (factor of two). The ratio, r, of frequencies for a half-step therefore satisfies r^12 = 2, which means r = 1.0595. The scale, notated with names commonly used in jazz harmony, then corresponds to frequency multiples of:

tonic    maj2nd    maj3rd  4th       5th       maj6th    maj7th  octave
1.000    1.1225    1.2599  1.3348    1.4983    1.6818    1.8877  2.0000
    min2nd    min3rd            b 5th     # 5th     dom7th
    1.0595    1.1892            1.4142    1.5874    1.7818

A string, fixed at both ends, vibrates naturally with the length equal to integer multiples of a half wavelength. Since frequency is inversely proportional to wavelength, this results in the overtone series for string harmonics:

  f    f/f_1 freq/tonic    approx. interval
  ---  ----- ----------    ----------------
  f_1    1    1  = 1.0          tonic
  f_2    2    2  = 2.0          tonic
  f_3    3   3/2 = 1.5           5th
  f_4    4    2  = 2.0          tonic
  f_5    5   5/4 = 1.25        maj 3rd
  f_6    6   6/4 = 1.5           5th
  f_7    7   7/4 = 1.75        dom 7th
  f_8    8    2  = 2.0          tonic
  f_9    9   9/8 = 1.125    maj 2nd (or 9th)
  f_10  10  10/8 = 1.25        maj 3rd
  f_11  11  11/8 = 1.375  between 4th and b 5th
The first ten harmonics fall close to the intervals of the equal-tempered scale (and define the perfect-tempered scale) with eight being notes of the major triad. The eleventh harmonic is off-key, as is the thirteenth and many more.

To strike these harmonics, one touches the string at 1/2, 1/3, 1/4, 1/5, 1/6, etc. of the length from one end to damp out the lower harmonics and then lets the string vibrate freely.

The timber of a stringed instrument is determined by the relative amplitude of each of these overtones. This is governed by the type of string, feedback between the string and sounding board, the sounding board itself, and the way the string is plucked.

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Sam Hokin
Last modified: Mon Aug 16 00:06:01 GMT 1999