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 Perfect 4ths & 5ths generate Pythagorean scale


Starting from any note set as the tonic of an octave of eight notes, the seven notes above it can be generated by ascending and descending intervals of perfect fifths & fourths. There are two ways of creating the octave of notes without repeating a note:




The diagram illustrates the first alternative. Alternating jumps of a perfect fifth and a perfect fourth generate the second alternative.

9/8 and the "leimma" of 256/243 (so-called because the left-over (Greek: leimma) interval between the penultimate note with tone ratio 128/243 and the octave with tone ratio 2 is 256/243. The leimma is the equivalent of the semi-tone in the equal-tempered, musical scale through which most Western music is played today.



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