notation of proportions

[Gagnon, TD 91:]
> I would like to know how you do calculate
> such an interval?
>
> a pure 6:8:9
>
> I know 9:8... but I don't know what is meaning
> an interval with three numbers.

An interval is two notes, thus two numbers.
Three numbers indicate a triad, or three notes.
The numbers show the proportions between all
three notes as a unit.

In other words, your example of 6:8:9 means:

1) the interval between the middle note and the bottom
note would be 6:8 = 3:4,

2) the interval between the top note and the middle
note would be 8:9,

3) the interval between the top note and the bottom
note would be 6:9 = 2:3

For some reason, the ratios of intervals (2 notes)
are usually spoken of in the reverse fashion,
with the higher number first, i.e., 4:3, 9:8, and 3:2,
but both ways are in common use.

In proportions (more than 2 notes) we usually start
with the smaller number on the left.

- Monzo
http://www.ixpres.com/interval/monzo/homepage.html

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