Mathematical Means from the Timaeus

I. The Arithmetic Mean

The arithmetic mean c of numbers a and b is their average, defined by c = . Its definition may be expressed as c exceeds a by the same as b exceeds c or c-a = b-c.

II. The Geometric Mean

The geometric mean of a and b is c = . If c= then we have a:c = c:b, or a/c=c/b.

III. The Harmonic Mean

The harmonic mean of a and b is . Plato defines it as the number c for which the ratio of the amount by which c exceeds a to the amount by which b exceeds c is equal to the ratio of a to b; in other words .

IV. Examples

1. The arithmetic mean of 1 and 2 is .

2. The harmonic mean of 1 and 2 is .

3. The harmonic mean of 2 and 4 is .

4. The arithmetic mean of 2 and 4 is .

Musical Ratios in the Timaeus

The following basic ratios are equivalent to the given intervals:

2:1 - the octave, C to C

3:2 - the fifth, C to G

4:3 - the fourth, C to F

The interval from F to G is a whole tone or two semitones. The basic ratios are multiplied or divided to find the sum or difference of intervals. Thus the tone which is the difference of G and F is given by (3:2)¸(4:3)=9:8.

The semitone can be computed by taking the difference of F and C (5 semitones) and subtracting from that two whole tones (4 semitones). (4:3)¸(9:8)¸(9:8)=256:243.