I recently found this article on wikipedia about lists of musical works in unusual time signatures and the first unusual time signature is (1/√π)/√⅔. I looked up the piece that was listed as having that time signature (Conlon Nancarrow - Study for Player Piano 41a) and listened to it and could not begin to count it at all. So is the time signature actually useful and if it is, is there a specific name for time signatures like this?
I think the author of that Wikipedia page has rather misinterpreted Nancarrow's title page for the Study (linked on Roland Bouman's comment to the question). (1/√π)/√⅔ refers to a tempo ratio between two voices, not a time signature.
Nancarrow was rather obsessed with canons. The canon is a form where multiple voices each play the same music at some time offset (i.e. the second voice enter a bar after the first). Nancarrow wrote tempo canons where the voices are at various tempo ratios growing more and more complex over his career.
The Study for Player Piano No. 41 is structured in 3 movements, 41A and 41B are two voice canons and 41C is both 41A and 41B played together. 41A has a tempo ratio of (1/√π)/√⅔, which is what the Wikipedia page mentions. This does not refer to a time signature, a regular grouping of beat stressing, but to the ratio between the two voices in the canon. So, for example, if the first voice was at ♩=100, the second would be at ♩= 100 * (1/√π)/√⅔ ≈ 69.098829894267098
41B is in a similarly ridiculous ratio of (1/(π^1/3)) / ((13/16)^1/3) and the final movement Nancarrow notates as having a ratio of 41B/41A = [(1/(π^1/3)) / ((13/16)^1/3)] / [(1/√π)/√⅔]
The article Roland Bouman mentions has much more detail and analysis of what Nancarrow actually intended by these numbers, and how precise he was actually intending to be. The most interesting section, especially for those noting how pretentious such notation is (which I think is an accurate observation) is a quote from Nancarrow about how he picked the ratio:
and the author's commentary:
I would say that the specific name is "experimental." My feeling is that it comes from the school of thought that attempts to turn the back on musical tradition and come up with something new. There's a certain arrogance to it in my opinion (famously, Schönberg said upon coming up with his rather superficial tone-row concept that he had assured the supremacy of German music for the next hundred years); musical architecture is based on a great deal of trial and error and it's unlikely that some inspiration would turn it all on its head and replace it just because someone wants that to happen.
For example, there's John Cage working with randomness. He would do things like drop a string to determine the shape of a musical line, and often used the I Ching as a means of composing music. (Please note that I take wisdom where I find it, and there's plenty in the I Ching IMHO.) The thinking behind using randomness is to get oneself out of one's own way, so to speak, and allow some deeper influence to contribute to the composition. However, people have been doing that in some way ever since music began; more common is the idea of standing aside and inviting "the Muse" to enter. Then there's Bach, with his "I play the notes as they are written, but it is God who makes the music."
I guess my feeling is that there came a point where we decided that "anything goes" and wound up having this rather adolescent departure from the core of music (whatever that is), and we're hopefully getting back to work.
This is either compositional wanking or a composer having a joke at literalists' expense. Since any metronome (or human) can only approximate any time period to some precision, the beat will always be a rational part of a second. For that matter, the repeatability of the beat will only be exact to some rational limit. So claiming you want the beat to be, say, the first real root of an N-th order polynomial divided by the value of the j-th Bessel function of the second kind evaluated at sqrt(37) , is pointless.
Personally, I view it as a cute joke, and would go on to play the piece at whatever speed felt appropriate.
Wow, what a great question!
Sorry, I don't know of a specific name for this kind of time signature. Judging whether this is a useful time signature is almost a philosophical question. If the main function of a time signature is to provide performance information to the performer, then no, this isn't very useful. But, a time signature can also be used to describe an aspect of the rhythmic organisation of a piece, independently of any performance considerations. I would argue that this is the sole purpose of the time signature in this case. After all, the score originally served little purpose as a performance guide, as the piece is written for player piano - no score, and so time signature, was originally needed for performance of the piece, with the piano roll supplying the information necessary for performance.
Instead, the time signature here is merely descriptive, giving information about this piece.
And, of course, because no performer has to use the time signature, there is no need for the time signature to actually reflect rhythms actually found in the music. Equally, the actual rhythms of this piece could be forced to fit into another time signature, or be presented in a score with no time signature.
I love the idea of trying to count along with a piece in this time signature! I haven't listened to Nancarrow pieces for a while, so can't remember this one, but presumably one of the most interesting aspects of a piece like this is the rhythmic complexity allowed by mechanical performance; the ear is presented with something which we can't easily count along with. And so, if there is no easily discernible pulse or simple rhythmic organisation, the time signature becomes essentially meaningless; any time signature would do in this case, so why not one as outlandish as this? I can't help thinking that this is supposed to be humorous, like one of Satie's performance markings, even if it does clearly relate to specific temporal relationships in the music.