continued…

The octave is the cohesive factor in comparing just and equal temperament systems. It is always 2:1 in both because this corresponds to human hearing physiology. The curvature of the cochlea (the snail shell shaped part of the ear) is the reason humans can identify two pitches at a ratio of 2:1 as a strong unison sound and thus called the octave and furthermore the reason that both pitches have the same letter name in scales. This is a crucial fact in the intervallic structure of music. If the cochlea was a straight tube we couldn’t identify the octave as a unison sound and the musical alphabet would have to be organized differently (possibly with no repetition of letters).

While the half step intervals (and thus whole steps also) are exactly uniform in equal temperament, the flaws of the just intonation system become apparent under analysis. The available whole steps of the major scale must shift in value to fit the pattern. The just scale will not be in tune with itself far past an octave and certainly not in tune should a modulation or transposition be attempted. Consider the following ratios in trying to form a just intonation chromatic scale:

M3–16:15–4 is the half step ratio

Therefore two half steps could be 16/15 x 16/15 = 256/225

However we know from the established rules of the just system that

1–9:8–2 and 2–10:9–M3 are the two different whole step ratios

256:225 is approx. 1.377 (repeating 7)

9:8 is approx. 1.125

10:9 is approx. 1.11 (repeating 1)

Furthermore, the ratio of 1–flat 2 could then be 16:15 (standard half step) or

3:2.82842 . . . (square root of 9 over the square root of 8) or 3.16227 . . .:3 (square root of 10 over square root of 9)