Some further observations by way of context will perhaps not be go or come amiss.
First let us convey the gist of something we found in the book
Tonal Harmony, written for Northern American students by Stefan Kostka and a Lady. Dotted values, they tell us, present a problem where time-signatures are concerned. For example, if there are two beats per bar, and the beat note is a dotted crotchet, what would the time-signature be? 2/4½? 2/4+8? 2/8+8+8? There is no easy solution they say, and the method that survives to-day is the source of much confusion concerning compound beat. Simply stated, a compound time-signature informs the musician of the
number of divisions of the beat contained in a bar and of what the
division duration is. This means that the upper number of a compound time-signature will be 6, 9, or 12, because two beats times three divisions equals six, three beats times three divisions equals nine, and four beats times three divisions equals twelve. Here are some examples:
These compound time-signatures do
not they continue follow the erroneous rule so often learned by the student-musician that "
the upper number tells how many beats there are in a bar, and the lower number tells what note gets the beat." Of course there are some pieces in 6/8 for example that really do have six beats to the bar, but such a piece is not really in compound duple. A bar of 6/8 performed in six does not sound like compound duple; instead it sounds like two bars of simple triple, or 3/8. In compound duple the listener must hear two compound beats to the bar not six simple beats. In the same way a slow work notated in 2/4 might be conducted in four, which would seem to the listener to be simple quadruple. In both cases the usual division value has become the beat value.
Eaglefield Hull too is able to make an important contribution. The time he tells us seems at hand when the further requirements of the composer in the sub-divisions of the separate beats may well be deemed to require an improvement of the notation, or an altogether new system of signs, to indicate the various durations of sound. A distinct sign is certainly needed for the third part of a beat, since the use of compound time is often cumbersome, whilst the triplet is but a poor makeshift.
Half-beats against thirds, and quarters against sixths, are sufficiently commonplace he goes on to possess a less ambiguous notation than that at present in use. The composers' practice of marking irregular numbers over the beat hovers so indefinitely between the higher and lower powers that examiners are supplied with a never-failing source of confusing the candidate. The latter is asked to decide a question which the composers themselves have never solved - namely whether a quadruplet in compound time follows the rule of the quintuplet in simple time by being drawn from notes of higher power, as at (a), or lower power, as at (b), and how such practice is reconciled with the writing of the duplet, as at (c), and the Chopin values at (d):