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Pythagorean tuning

One way of forming the diatonic scale using pure fifths is shown below (see notation example):

1st bar: C is the starting point. Half notes are used to express the tone of the scale to be tuned.
2nd bar: C to G', i.e. a rising pure fifth; the frequency ratio between the tones is 3 (G') to 2 (C).
3rd to 6th bar:A rising fifth from G' to D'' followed by a falling octave in order to fit the tones within the same octave range. The pointer of the ratio is tripled and the nominator doubled, except when performing an octave transfer (without which the ratio would be 9:4).
7th bar: The F is achieved with a falling fifth. The ratio results from the fact that in a harmonic F series, F is the fourth and C the third harmonic.

pure scale

What results is the seven tones of the diatonic scale (C, G, F, D, A, E and B), which comprise a major scale in Pythagorean tuning. Comparing the scale with an equal-tempered scale, one notices that the difference between the tone positions increases by two cents at each fifth. If the entire chroma (12 consecutive fifths) was tuned using only fifths, the difference would become 1/8 step (approximately 24 C), which would be disturbing even to an untrained ear. This discrepancy is called the Pythagorean comma, and it was one the primary reasons why other tuning systems (temperaments) were studied in the medieval ages.