continued…

Following the premise of two notes separated by a specific interval distance converted into a ratio, the following is a list of three notes (forming basic triads) with two interval distances and thus a three-numbered ratio relationship. Keep in mind that some of these ratios are loose approximations.

Three note–ratio groupings:

Ratios —- Scale positions

1:2:3 —– 1,8,15

2:3:4 —– 1,5,8

3:4:5 —– 5,8,10

4:5:6 —– 1,3,5

5:6:7 —– 1,m3,b5 (diminished triad, very loose! Closer value is 26.67:32:45)

4:5:8 —– 1,M3,m6 (augmented triad)

6:7:8 —– 1,2,4 (very loose)

7:8:9 —– 1,2,M3 (very loose)

8:9:10 —– 1,2,m3 (very loose)

9:10:11 —– 1,2,m3 (very loose)

More exact ratios forming basic triads:

3:4:5 —– Major chord second inversion (closed)

4:5:6 —– Major chord root position (closed)

5:6:8 —– Major chord first inversion (closed)

10:12:15 —– Minor chord root position (closed)

12:15:20 —– Minor chord first inversion (closed)

15:20:24 —– Minor chord second inversion (closed)

4:5:8 —– Augmented chord (closed)

26.67:32:45 —– Diminished chord (closed)

Interestingly, the above chords acquire different ratios as the notes forming the triad are transposed into ranges outside closed position chords. However, even though these ratios are different they are simply multiples of the base chord ratio. These chords can be reduced or enlarged by factors of two (the octave effect) like this:

C1 (root) — E3 (third) — G2 (fifth) forms a ratio of 1:5:3.

Then transpose C1 up to C3 — 1 x 4 (2 octaves) = 4

Then transpose G2 up to G3 — 3 x 2 (1 octave) = 6

E3 is already in the correct octave (stays at 5)

So, the result is from 1:5:3 to the familiar 4:5:6

This aspect was pointed out to me in a very well constructed and informative website by Robert Asmussen called Tuning By Ratios. This site also includes an analysis of several Bach pieces by way of ratios and a computer program that uses ratios to realize pitches through the use of sine waves. It is a wonderful site and I suggest anyone interested in this topic should visit and study it. You can find it at this address:

http://www.leeds.ac.uk/music/studio/rproj_swss/tuning/httoc.htm