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 which was Conlon Nancarrow's 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?


4 Answers 4


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:

At that time [of the composition of Study No. 41], I was looking for some irrational relationships. I had this book of engineering, and I looked up some relations that were roughly what I wanted. I didn’t want something that was so separated they didn’t even relate, or too close that you couldn’t hear it. I found that those particular numbers, transferred into simple numbers, gave the proportion more or less that I wanted. Not exact, but near enough. This was before I had written a note.

and the author's commentary:

Of course, by their very definition, irrational numbers cannot be specified; in order to have produced the Study Nancarrow had to approximate the proportions. Why not, then, simply use the rational equivalents? Part of the attraction must have been the gorgeous complexity of the original proportional structure. For a lover of numbers like Nancarrow, the proportion is a thing of beauty. And, of course, π means something even to the lay person: it is the ratio of the circumference of a circle to its diameter. Nancarrow gives no indication, however, that he had anything grander in mind than simply finding "some relations that were roughly what I wanted.”

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    Hey thanks! I didn't have the time or urgence to read the article, I think you did a great summary. Much obliged :) Commented Apr 21, 2014 at 21:11
  • Even if metronomes were only accurate to +/- 10%, the ratio between the tempos of two parts might be meaningfully expressed much more accurately. Although ratios like 3:2 or 2:1 would be more common, an irrational tempo ratio, if played accurately, would denote rather specific timing; a change of 0.03% in the tempo ratio would by the end of a five-minute piece represent a very audible (90ms) difference in timing. I think using two unrelated irrational numbers in the ratio was probably overly pretentious, but...
    – supercat
    Commented Oct 7, 2014 at 21:45
  • ...it's not hard to imagine how transcendental ratios, if played accurately, could lead to interesting patterns where one voice sometimes seems to lead the other and sometimes trail it.
    – supercat
    Commented Oct 7, 2014 at 21:46

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.

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    I would argue that this is not arrogant or adolescent. Einstein had to turn his back on the scientific conventions to come up with Relativity, no? How does one take the next step? To conform to the standards and conventions of today, is to merely take baby steps forward. Yes, all of these conventions exist for a reason but how will we ever know what is out there if no one ever takes the 'arrogant' step outside? Further, Schönberg may have been arrogant with his statement of German supremacy but the step to create tone rows was a brave and creative one, however unpopular serialism has been. Commented Mar 27, 2014 at 19:19
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    Einstein didn't try to refute Newtonian physics (nor did he), just to expand on it (which he of course did). Now, it isn't the entire body of 20th century music that is arrogant, nor is it Schönberg's entire body of music: he succeeded in spite of himself. But it is arrogant to say things like we can do whatever we want, because music can't be logically defined. We have, for example, John Cage's 4'33" presented as music, and also the inherent superficiality of serialism. These strike me as clever rather than profound, and it is that confusion of cleverness and profundity that I find arrogant.
    – BobRodes
    Commented Mar 27, 2014 at 19:48
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    You criticize Nancarrow for being arrogant and adolescent in his desire to revolutionize music and replace all that had gone before him. There's just one problem: he had no such desire; it's all in your head, not his. You're criticizing a strawman. He wasn't trying to revolutionize anything; he was just writing the kind of music he liked to write. He barely even showed it to the world because nobody seemed to be interested and he was living practically as a recluse. To accuse him of arrogance is quite breathtaking. He was quite close to disappearing without a trace. Commented Mar 27, 2014 at 22:57
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    David is correct about Nancarrow's reclusiveness. The comparison to make with Einstein is not that he had to "turn his back on convention" or anything of the sort, but that in order to make any sense of what he discovered, you needed a Ph.D. in theoretical physics--a layperson to this day may say the notion of curved spacetime is "arrogant", "nonsensical", or "juvenile". There may well be composers who spew out garbage that to the layperson is indistinguishable from Nancarrow, Berio, Schoenberg, etc... but it's not up to the layperson to make that judgement.
    – NReilingh
    Commented Mar 28, 2014 at 1:02
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    Just want to point out that there is nothing superficial about using tone rows to create music. If they are superficial, then so too are all other scales and systems of musical creation. Commented Mar 28, 2014 at 1:54

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.

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    I can think of a few names for this time signature, but none that I will mention here. Commented Mar 27, 2014 at 17:55
  • But how is it interpreted? It could be seen as somewhere between 5/8 and 6/8-ish, but what that information would give anyone is not clear. Commented Mar 27, 2014 at 18:03
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    Right! It would be good to see how many beats Nancarrow fits into each bar in the score. But, the piano roll itself could have any arbitrarily complex bar lengths and note lengths, as long as they are measurable when creating the piano roll. It doesn't really matter if this then makes perfect sense when presented in conventional notation. Commented Mar 27, 2014 at 18:10

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