Why is common time most... common?

What is "rhythmically stable"?

I would consider a rhythmically stable time signature to be composed of a pattern that is more intuitively obvious to the human brain to a degree where it is easy to follow. Listening to 4/4, one can "hear" the repeating pattern and "feel" the downbeat very very easily. It is comprised of the constant prime patter of 2+2+2 etc. The same is true of 3/4, three being prime. But with five, this pattern is reduced to 2+3 or 3+2, and therefor is harder to follow due to the possible ambiguity, and because it contains a combination of two prime patterns, each of which individually are intuitive, but together they create a pattern slightly harder to follow. 7 can be comprised of 2+2+3, 2+3+2, and 3+2+2, and so is more ambiguous to our ears and is comprised of a larger number of primes.

Each higher signature that is farther from the powers of two seems to get less and less stable due to the higher ambiguity and larger combinations of primes.

This is at least the way that I have come to view rhythmic "stability"

Please note that although one can likely define stability mathematically in relationship to the way we perceive rhythm (the same that has been done with consonance Relating Tuning and Timbre), environment is obviously a HUGE factor in what one considers "normal" or easy to follow, and the same way defining consonance and dissonance does not decide what sounds good, neither does a definition of rhythmic stability.

John M

What is "rhythmically stable"?

I would consider a rhythmically stable time signature to be composed of a pattern that is more intuitively obvious to the human brain to a degree where it is easy to follow. Listening to 4/4, one can "hear" the repeating pattern and "feel" the downbeat very very easily. It is comprised of the constant prime patter of 2+2+2 etc. The same is true of 3/4, three being prime. But with five, this pattern is reduced to 2+3 or 3+2, and therefor is harder to follow due to the possible ambiguity, and because it contains a combination of two prime patterns, each of which individually are intuitive, but together they create a pattern slightly harder to follow. 7 can be comprised of 2+2+3, 2+3+2, and 3+2+2, and so is more ambiguous to our ears and is comprised of a larger number of primes.

Each higher signature that is farther from the powers of two seems to get less and less stable due to the higher ambiguity and larger combinations of primes.

This is at least the way that I have come to view rhythmic "stability"

Please note that although one can likely define stability mathematically in relationship to the way we perceive rhythm (the same that has been done with consonance Relating Tuning and Timbre), environment is obviously a HUGE factor in what one considers "normal" or easy to follow, and the same way defining consonance and dissonance does not decide what sounds good, neither does a definition of rhythmic stability.

John M

So, any scientific research to back all that up? Cuz so far as I'm aware, that's all very much subjective, math ratios or not.

Hmm.

- 5 is a much a prime number as 2 and 3, so there's no reason to make a distinction based on that. The only difference in this context between those numbers is that some are larger than others (which again is the only reason we probably often split up 5/4 times, especially when conducting etc.).

- You can split up any positive number into a sum of smaller numbers if you want.

- Prime numbers per se are rather irrelevant when you're talking about SUMS of them. Primes are generally used as factors - they aren't very special if you simply add them together, so I don't see any relevance in using prime numbers as a basis here.

That's why, if we look at things this way, I prefer ratios, not simply prime numbers.

Maybe it was not clear that I consider equivalence, when I talk about ratios here (and as I said about 4/2 and 4/8, which can be the same as 4/4, they are simple in the end), just as we consider octave equivalence when we speak of musical intervals, scales and tunings and reduce intervals between 1 and 2 (octave, that is). If you write that 5/4 = 5/2 = 5/1 in a math class, you'll be likely kicked out. But here we look at it from a perceptual viewpoint... The pure major third is usually expressed as 5/4 instead of 5/2 or 5/1, which are still major thirds. However, in the world of musical intervals particularly, this could be oversimplification: an interval + octave(s) extension can more or less affect acoustical stability of various intervals due to the disposition of combinational tones.

I'm not entirely sure if you're saying that octave identity is valid in music or whether it is an invalid oversimplification. The last sentence suggests the latter, in which case I agree. Octave identity and treating intervals the same even when you add octaves to them is indeed a common thing in many traditional (and recent) music theories, but it is by no means an unquestionable fact. A 10th is something vastly different than a third to me (perceptively and theoretically) and not just because of combinational tones, but also because of pure intervallic "distance", because of a stronger timbral separation that different notes in different registers automatically produce, etc. When I consider the harmonic concepts of a piece I'm writing, the question whether I'm using a major second or a major ninth is essential to me and I would never randomly exchange them for each other. (And since Debussy/Grisey/Ligeti and many others we should have learned that harmony, timbre and register are strongly linked to each other and not clearly separable.)

Yes, you've got me right. Although octave identity can be useful (but leaky) abstraction for simplfying explanations, I don't see it entirely invalid - a tone or an interval in one octave sound both similar and different when compared to the same tone or interval in another octave. It's like a red colour, but darker or brighter, which never becomes green; or twins which only seem identical, but actually have different characters. A1 is different from A2, of course - they are unique different frequencies in the spectrum, and for me, too, there is a difference between a second and a ninth, for example.

Considering this, maybe a more accurate term would be octave similarity.

I'll start with the disclaimer that I all to often forget to include, that this view I'm relaying is a combination of how I think of things and of things I've had explained to me by fellow musicians with whom I've agreed.

- 5 is a much a prime number as 2 and 3, so there's no reason to make a distinction based on that.

Good point!

I think instead if in my post one were to replace where I refer to prime numbers as being so important with simply the numbers 2 and 3, things make more sense.

I should of said then:

Each higher signature that is farther from direct multiples of two and three seems to become less and less stable, and this I attribute to the higher ambiguity in the order of the combinations of two and three that our mind will naturally brake the signature into. (2,3,4,6,8,9,12,16,18,24 tend to be very stable and are easily broken down into regular patterns of 2 or 3)

I think that my definition of "rhythmic stability" is pretty accurate still:

I would consider a rhythmically stable time signature to be composed of a pattern that is more intuitively obvious to the human brain to a degree where it is easy to follow.

Another disclaimer:

Please note that although one can likely define rhythmic stability mathematically in relationship to the way we perceive rhythm (the same that has been done with consonance by Relating Tuning and Timbre ), environment is obviously a HUGE factor in what one considers "normal" or easy to follow, and the same way defining consonance and dissonance does not decide what sounds good, neither does a definition of rhythmic stability.

John M

And 7, 11, 13, 17, 19, ..., 61, ..., 101 and so on are all prime numbers...

But as I already mentioned, I don't like the expression rhythmic stability. I would prefer rhythmic complexity, for example. Rhythmic stability doesn't make sense to me as the latter does.

I have no idea how 4/4 became the standard for music, but some how or another it did and quite frankly I don't think there's much more to it. It just is what it is, no more, no less.