[with] -- the foundations of rhythm
Western music notates rhythms in time signatures. A time signature like 3/4 means that the piece is felt in 3 groups of quarter notes. Similarly, 6/8 denotes 6 groups of eighth notes. With this flexible system, there are no limits to the number of time signatures that can be dreamed up: pi/4, 3/10, or 29/16, are all technically permissible. But what is the cognitive reality behind these formalisms?
When trying to understand a phenomenon, it is helpful to go back to its first or most primitive manifestation. As discussed in another post, Gregorian chant is the bedrock of Western music. In Gregorian chant, neumes are organized into groups of 1, 2, or 3. Aside from being theologically significant, combinations of 1, 2, and 3 encompass the three primitive types of rhythm: unity, binary, and ternary. All other rhythms can be decomposed into these “metarhythms”.
For example, 4/4 is really two groups of 2. Likewise, 6/8 is not 6, but 3 + 3. Rather than saying that a song is “felt in 7”, it is more realistic to say that it is felt in 2 + 2 + 2 + 1, 2 + 2 + 3, or 3 + 3 + 1. Different people will count it in different ways. This sort of flexibility is essential to all cognitive models.
The human mind is myopic. It sees no farther than it has to. Mathematical abstractions are powerful, but they can be misleading if they are not tempered by cognitive reality.