Overtone Music Network

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Three Tonal Systems

Cyan Blue: "12 TET" (twelve-tone equal temperament), which has been dominating our music culture for the last 100 years or so.
The even division (like hours of a clock) of the octave circle looks extremely pleasant to the eye but what is visually pleasant may not be harmonic to the ears - just think of the paper A format which is very harmonic to the eyes but in music the proportion 1: squareroot 2 is extremely dissonant.
The axises are found at 0-100-200-300-... etc. cent. The Point is that the reference of purity for the ears (and music theorists) and overtone freaks naturally are the intervals from the harmonic series (systemized in 'just tuning' -green) and the thirds and sixths of 12-TET are far from pure. We have actually made a musical convention which has sacrificed the experience of just thirds and sixths.

Magenta Red: Pythagorean tuning, generated by perfect fifth intervals (6 up and 5 down from the fundamantal). It is symmetrical (apart from the tritone) around the vertical line. The principle of using the perfect fifth (proportion 2:3) to generate a tonal system is the deepest background for the division of the octave circle into 12 half tones. Alas 12 fifths in succession is only approximately equivalent of 7 octaves:
1,5^12 ~ 2^7.
The derivation is known as the Pythagorean Comma.

Yellow: Just tuning. Intervals between elements from the harmonic series: 2 (octave), 3 (perf. fifth) and 5 (maj. just third) combined. Like Pythagorean Tuning it is not very suited for polyphonic music as it contains many different sizes of intervals.

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