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:
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).
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