http://people.csail.mit.edu/jaffer/Music/Overtones
	Why Musical Scales Work

Although there are many resources tabulating the distance between natural overtones and the notes in musical scales, a diagram showing the position of (string) frets and the string lengths corresponding to overtone ratios should be more accessible to those who are geometrically inclined.

The blue ticks are the fret positions for a 36.cm string supported by a bridge at the left end (0.cm). Each magenta and grey tick is positioned at its manifest fraction of the full length (36.cm).

The (magenta) fractions with 1 in the numerator are pure overtones. But we perceive tones produced by doubling or halving a length as the same pitch in different octaves. The fractions with 2 or 4 in their numerators are respectively one and two octaves lower in pitch than their pure overtones.

Each up-interval should be paired with a down-interval to complete a scale. The gray ticks with power-of-2 denominators (4, 8, 16, 32, 64) are these reciprocals. 9/16 pairs with 8/9; 5/8 with 4/5; and 3/4 with 2/3.

The blue syllables identify a minor scale starting at the far right "do". Six of its seven notes are close to overtones.

A pentatonic scale, which was the oldest music scale known to many human cultures, is composed of the five notes with the best matches. Their syllables are imprinted on magenta circles in the diagram.