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Pythagoras


582-500 BC

Discovered the relationship between harmony and numbers.

More famous for his geometry equation for right-angled triangles, Pythagoras has also been credited with discovering the relationship between harmony and numbers.

Using a monochord, which has one string and a moveable bridge, he could produce two different tones at the same time by plucking the string on either side of the bridge. Marking the position of the bridge gave him the measurements to support his discoveries that:

  • A musical octave occurs when the length of one string is exactly twice or one-half the length of the other string. The ratio is 2:1 or 1:2, and the fractions are 2/1 and 1/2. (An octave is the same note at a higher or lower pitch. A scale is made up of the notes between two octaves. It is called an octave because it is the eighth note in a seven-note scale.)
  • The most pleasing harmony occurs when the length of the second string is 3/2 or 3/4 that of the first string. We call this note a fifth because it is the fifth note of the modern western scale.
  • The next most harmonious notes occur with the second string being 4/3 or 2/3 of the first string. This is now known as the fourth note in our modern western scale.

Pythagoras built a seven-note harmonic musical scale using only three ratios:

  • 1/2 for an octave
  • 3/2 for a perfect fifth
  • 4/3 for a perfect fourth

For example, he created the "second", a 9/8 ratio, by increasing a fifth (3/2) by another fifth (3/2), and then, to keep it in the same octave, he applied the octave 1/2 rule. The result was 3/2 x 3/2 x 1/2 = 9/8.

In order of the familiar scale, do-re-mi-fa-so-la-ti-do, here are the calculations for his scale:

  • 1 do - 1/1 - key or tonic = 1.000
  • 2 re - 3/2 x 3/2 = 9/4 x 1/2 = 9/8 = 1.1250
  • 3 me - 9/8 x 9/8 = 81/64 = 1.26525
  • 4 fa - 4/3 - as discovered = 1.3333
  • 5 so - 3/2 - as discovered = 1.5000
  • 6 la - 9/8 x 3/2 = 27/16 = 1.6875
  • 7 ti - 3/2 x 9/8 x 9/8 = 243/128 = 1.8984375
  • 8 do - 2/1 - the octave = 2.0000

Great metaphysical significance was given by him to the fact that all harmonies came from the numbers one, two, three and four.

When tones of a pure harmonuc scale are sounded simultaneously, another harmonic tone is created that Helmholtz called a combinational tone. Denny Doherty of the Mamas and Papas called it the "fifth voice".

However, because the tuning is based on ratios of the first note, changing the key or root in a song means changing the tuning of any fixed-tone instrument, such as a piano or guitar.

A more basic problem, called the Pythagorean comma is described as follows:

  • After cycling through the fifth note of a scale twelve times, we should arrive back at the first note seven octaves higher. However, 3/2 multiplied by itself 12 times is higher than 2 multiplied by itself 7 times. The difference is about one-quarter of a semitone per octave.

Over the centuries, musicians have dealt with this problem in various ways using meantone temperament, well temperament, equal temperament and now by usings a just intonation tuning cube that tunes electronic musical instruments as they are being played.

Justonic Pitch Palette software, for Windows and Mac OS9 , calculates and sends the MIDI data necessary to keep your synthesizer tuned to pure harmony in every Root of every Key in any Scale. It also includes a collection of tools for any musician interested in experimenting with adding new tuning to their performance.

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