There's a good mapping between the overtone series and how we perceive harmonic tension. I have no doubt that's tied to the mechanics of how we decode pitch.

If we take a frequency and divide it by ascending integers, we get the harmonic series:

f/1, f/2, f/3, f/4 ...

This yields the notes

C1, C2, G2, C3, E3, G3 ...

These map fairly well from least dissonant intervals to most dissonant intervals:

1, 5, 3, ...

We can add inversions (fourth, sixth) which gives a fairly complete scale:

1 3 4 5 6 8 ...

So in C major:

C E F G A C

It's not the same as the pentatonic scale, but more on that later. It also has some large gaps in it, between 1..3 and 6...8. There are various ways of filling in those gaps.

The other primary relation can be shown using the circle of fifths. For example, we can use the circle of fifths to derive the entire chromatic scale, ignoring the issue of the pythagorean comma:

C, G, D, A, E, B, F#/Gb, C#/Db, G#/Ab, D#/Eb, A#/Bb, F, C ...

We get the pitch set of any major scale by choosing one note to the left of the root, and five from the right. So the key of A major yields:

C, G, D, A, E, B, F#/Gb, C#/Db, G#/Ab, D#/Eb, A#/Bb, F

which rearranged gives:

A, B, C#, D, E, F#, G#, A

Back to the pentatonic scale. Given a major scale, we can reduce it by removing the half-steps, which are pretty universally heard as dissonances (of course, context changes this). So given a major scale, there are two scale degrees that are within a half-step: the 3-4 and the 7-8:

C, D, E, F, G, A, B, C

To get the most harmonically pleasing scale, we remove the clashing notes with the highest dissonance. That means we keep the 1, 3 and 5 (they are harmonically strongest, per the circle of fifths). So with the 3-4 we remove the 4, and 7-8 we remove the 7. This yields a major pentatonic scale, called a "gapped" scale to recognize that notes have been removed from it, leaving spaces:

C, D, E, G, A

These two principles - the overtone series and circle of fifths - provide a strong basis for explaining chords and scales. The circle of fifths also gives insight into harmonic movement, especially V->I movement.

-- David Cuny

My virtual singer development blog
Vocal control, you say. Never heard of it. Is that some kind of ProTools thing?

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