Pythagoras (approx. 580-500 BC) laid the foundation for our tone system. He used a monochord to establish the relationship between string length and pitch.
Dividing string length by 2 will bring a tone one octave higher and the string vibrates twice as quickly. Dividing by 3 causes a tone tone which is an octave plus a fifth higher and the string vibrates 3 times as quickly. So: inversely proportional. For just going one fifth higher (so not octave plus fifth) this means: string length x 2/3, frequency x3/2.
Let's continue. Dividing string length by 4: the double octave, frequency x4. In fact it is not a new tone that is created through multiplying by 2. Neither through multiplying by 4: octave plus octave is the result.
And how about dividing string length by 5? Now a 'new' tone is created: two octaves plus a major third. Pythagoras did not use this new tone in his system. He established the intervals within the octave by just using the numbers 2 and 3.
He reasoned from the 'mesos' (μέσος = middle tone) and went a fifth higher. By going one octave lower from there (string length x2, frequency x 1/2) he ended up one fourth below the 'mesos'. So the original string length is now multiplied by 4/3 and the frequency by 3/4.
The tone one tone distance above is reached by getting two fifths up and one octave down: 3/2 x 3/2 x 1/2 = 9/8. With that the whole octave can be built. So, thinking from the middle tone Pythagoras produces this table:
| E | F | G | A | B | C | D | E | |
| string length | 4/3 | 81/64 | 9/8 | 1 | 8/9 | 27/32 | 3/4 | 2/3 |
| frequency ratio |
3/4 | 64/81 | 8/9 | 1 | 9/8 | 32/27 | 4/3 | 3/2 |
Read more about this subject in: temperaments.
So far we divided string length by 2, 3, 4 and 5. With 2, 3 and 5 a new tone emerges (octave, fifth, major third). With 4 not really (double octave). Either with 6, because divided by 6 can be understood as divided by 2 and followed by division by 3. That brings us again to the fifth (plus two octaves). 7 gives indeed a new tone (the septh), 8 does not (3 octaves). So the prime numbers 5 and 7 provide a new tone because they have no other factor than itself and 1. This is just a theoretical side road without much practical significance, because dividing by 5 already provides problems for the music system, let alone dividing by 7 or 11. Not for nothing Pythagoras used only 2 and 3. His major third has the ratio 81/64, while the 'pure' major third corresponds with the ratio 5/4, i.e. 80/64. The "difference" (81/80 in ratio) is known as Didymos' comma or syntonic comma.
A string can vibrate in different ways. A plucked string shows this:
Putting a finger extactly halfway the string while plucking (making a 'node' in the vibration) makes the string vibrate like this: As mentioned above, then you will hear the octave.
Making the node on 1/3 string length then the other node (on 2/3) emerges automatically and it vibrates in this way. The fifth plus octave will be heard.
Of course things are a bit different in real life. A combination of the vibrations described above appear when a string is plucked. The sound consists of the keynote and its overtones: So the vibration is rather complicated.
To resume: the overtones wil always be heard when a string is plucked, i.e. the pure intervals major third, fifth, octave and higher.
The overtones provide the 'colour' of the tone. Emphasis on the high overtones gives a sharp tone (guitarist plucks at the bridge, at the end of the string), while more lower overtones make the tone rounder, darker. (guitarist plucks the middle of the string).
back Questions or remarks?
