Incredibly, the sounds produced by elegant mathematical harmonies also sound pleasant to our ears - as if the ears are a sense into the underlying world of vibration and resonance. Let's continue.
Taking the ratio of 3:2 means shortening the monochord by a third. The frequency (or pitch) of the note produced by the shortened monochord is 50 percent higher. This note also sounds good paired with the corresponding root note.
When notes sound good together they are said to be consonant and when they sound terrible they are said to be dissonant. This pair of notes are consonant - nearly as consonant as they come (except for the octave). This note is called the perfect fifth. The reasons why it is called that and not something like the perfect third will come later.
You can find these notes on your guitar by looking at the 7th fret. The note played at the 7th fret is the perfect fifth of the open string note. Again, see for yourself how the distance from the nut to the 7th fret is exactly one third of the length of the string from nut to bridge.
We are making good progress in harmoniously dividing the string. So lets continue with another simple ratio.
Another simple fundamental ratio is to divide the string in the ratio of 4:3. That is, shorten the string by a quarter. The frequency of this note is one third higher than the root note. This note is called the perfect fourth.
You can find these notes on your guitar by looking at the 5th fret. The note at the 5th fret is the perfect fourth of the open string root note. You will notice that the distance from the nut to the 5th fret is one quarter the length of the string from nut to bridge.
We have made really good progress here, because so far we have got a scale of notes starting to form. We have a scale of octaves and a couple of notes dividing up the octave - the perfect fourth and perfect fifth, resulting in the whole tone. This is an important interval, and will be crucial for further divisions of an octave, to follow.