A note on mathematical notation for musical intervals

by David C Keenan, 25-Nov-1999
last updated 2-Dec-1999


I think it is bad practice to describe intervals using fraction notation 2/1, 3/2 etc. I prefer to use the ratio notation 1:2, 2:3 etc., for the following reasons:

(a) Fraction notation is how individual pitches are notated in JI (Just Intonation). There's too much potential for confusing your readers if the same notation is used for intervals (the distance or relationship between two pitches).

(b) The ratio notation generalises to chords of more than two notes e.g. the major triad as 4:5:6, and allows utonalities to be expressed conveniently e.g. 1/6:1/5:1/4 for the minor triad. These are called extended ratios.

(c) While a 3/2 pitch is totally different from a 2/3 pitch, there is no difference between a 2:3 interval and a 3:2 interval. Although we have to write them in some order, the order carries no meaning. The colon is a left-right symmetrical symbol while the slash isn't.

(d) Although the order doesn't change the meaning, it is important to have a canonical order to facilitate human recognition. There is already a convention of ordering the notes by increasing pitch when using letter names e.g. C-G, C-E-G. This should also be followed for the ratios and extended ratios of intervals and chords e.g. 2:3, 4:5:6. In fact I think it makes good sense to use the colon with the letter names too, e.g. C:G, C:E:G, so the correspondence is made perfectly clear to the reader.

With regard to (d) above, I find it strange that many people obey this rule for chords but insist on reversing the order for intervals (2 note chords). The reason is probably historical, but perhaps it's time we eliminated this inconsistency.

With regard to (b) above, the notation for utonalities e.g. 1/6:1/5:1/4 still lends itself to the confusion between a generic chord, "the minor triad" and a one with particular pitches "the minor triad whose root is the pitch 1/6" in JI. An alternative is to express the generic utonality as 1/(6:5:4). Mixed-o-u-tonalities, such as the minor seventh chord, are more difficult.