by Christopher Schlegel

This table is derived from the ancient “Just” intonation ratios which consists of the following interval relationships:

1–9:8–2

2–10:9–M3

M3–16:15–4

4–9:8–5

5–10:9–M6

M6–9:8–M7

M7–16:15–8

It is a satisfactory place to start primarily because of the relative ease of use in computing general scale and chord ratio relationships. The flat 2, minor 3, flat 5, minor 6 and minor 7 intervals are derived from the major scale ratios also. If one multiplies all the ratios between the root and the octave the resultant ratio is of course 2:1. This multiplication of ratios is the proper mathematically manipulation to secure the resultant ratio across larger intervals.

“Equal temperament” intonation consists of splitting the octave up into exactly equal units. The foundation of developing any music system is the division of the octave because of the curvature of the cochlea in the inner ear. The root to octave ratio is 2:1. There are twelve notes in the chromatic scale to evenly distribute. Application of the multiplication of ratios principle results in the unit of division that forms a uniform half step throughout the octave being the twelfth root of 2. The advantage to this system is of course the ability to manufacture an instrument that is in tune with itself (and other likewise tuned instruments) across as large as range as is possible to build. Therefore it is possible to play over a many octave range, change root notes and thus key signatures as well as modulate within a key all while remaining properly intonated and thus in tune.