Don’t Blame Me, It Was Henry Cowell’s Idea



How’s it look? This is how far I’ve succeeded in notating meters like 2/3, 7/12, and 5/6 in Sibelius. I have to put in a false meter, input the notes, delete the meter, insert the real meter via “Text > Special Text > Time Signatures,” and then add the brackets manually. I can’t decide whether it’s clearer to put a “3” over two or four quarter-note triplets or “3:2.” And I wish I could put a space in that bracket for the number. Suggestions welcome.

Those examples are from my I’itoi Variations of 1985. I admit that since I started using notation software in the mid-90s I shy away from meters like that – though Sibelius has opened up a myriad other possibilities in multitempo music for Disklavier, so it’s a trade-off. I notice that Michael Gordon, the other composer I know of to use unconventional tuplet groupings the way I do, always seems to expend considerable ingenuity in aligning his tuplets so that they’ll fit within a conventional meter – at least in pieces I have scores of, like Yo Shakespeare:


and Trance:


If I’m going to use this idea at all, I find this solution too limiting. I want 7/5 meter, and 25/13, and the whole megillah.

[UPDATE:] Not only is it too limiting, ultimately I think it’s too difficult to perform. Michael writes these kind of rhythms for new-music groups, and they negotiate them well as a chamber-music kind of thing, but I’ve never seen him try it with orchestra. For conducting purposes, I think it would be easier to notate the passages of his above as alternations of 2/4 and 2/3 meter, which would also allow the music to expand outside the 4/4 framework. For instance, I can’t imagine maintaining a 4/4 beat through the Trance example above, but I think my examples could be conducted.

I think that humans are capable of learning to switch to a tempo 2/3, or 4/3, or 3/4 as fast as a preceding tempo – and that, in fact, the composers of my son’s generation are pushing us toward that capacity.


  1. mclaren says

    Outstanding! You really ought to give Finale a try, though. Finale makes it trivial to do this kind of thing — it even puts in the brackets automatically if you check an optional box. Finale doesn’t care whether the X in the “X in the time of Y” is larger or smaller than Y — you just enter whatever you want, no hassle. So 3:2 is as easy as 2:3, or for that matter 11:23 is as easy as 23:11.

    Even better, you can copy an entire block of these polymeters and then paste it in across barlines, and Finale will automatically adjust the tuplets to the hideous-looking crap required to split, say, a 5:4 or 7:5 across a barline. (Sibelius 3 won’t let me do that! It gives me the error message “passage contains tuplets.” Yeah? So??? Shut up, Mr. Computer, ‘n DO WHAT I GODDAMN TELL YA!)

    The only thing I’ve had trouble with in Finale is fragmented tuplets of the kind Gordon writes above. To handle those, I’ve had to add an extra staff, then put in rests in the main staff where the fragmentary tuplet should be, then insert the broken tuplets in the additional staff with rests where the regular non-tuplet notes should be. A pain in the butt, but it works.

    Caveat emptor: Finale remains hideously unintuitive. You must memorize all the weird arcane procedures for doing stuff. It’s not possible to figure out how to do most stuff in Finale, you must look it up in the manual and memorize the procedure step by %#*$ing step. I’ve used Sibelius 3, and Sibelius makes it much easier to do simple stuff…but! But…in Sibelius it’s just so much harder to do exotic things like insert a 7:9 tuplet across a barline that I gave up. And since life is already complicated enough when you’re writing microtonal music, that’s that for that. YMMV.

    I have still not figured out how to insert a broken tuplet that doesn’t continue directly across barlines, or notate a Nancarrowesque gradual tempo speed-up using any existing notation program. Cakewalk makes it trivial to create gradual systematic accelerandi or decelerandi, and you can overlap ’em in multiple simultaneous tempo streams…but notating em? Ai caramba. We’ve hit the outer edge of common practice notation at that point, methinks.

  2. says

    mclaren, i can always tell it’s you by your rampant use of boldface. cracks me up every time.
    anyway, i didn’t even realize finale made irrational time signatures that easy. guess i can go back and re-edit my irrational times signature piece from last year. but i didn’t use tuplet brackets, i just used the time signature (4/6, 5/6, etc.). i personally find the additional tuplet confusing. others?
    one thing that i still don’t know how to do without scads of editing is have different parts in different time signatures. finale will stretch out the 3/8 bar to take the same space as a 4/4 bar instead of lining up with just the first three eighths. do you know if they’ve made this any easier? maybe i’ll just look it up myself after this month of craziness is over…

  3. says

    Unfortunately, I believe Mr McLaren is writing only of complex ‘tuplets,’ not time signatures. Finale will let you put some pleasant things in a time signature, e.g., 4.2/4, but I believe there is no (easy) way to have a time signature of 4/3. I’ve given up on convincing Finale to do incomplete tuplets, (except at the end of the bar!) so I’ve just given up writing them. I’m very impressed by the stamina that you all have in fighting against the unnecessary limitations of these programs. I simply don’t have the patience. I just write what the program tells me to write and leave it at that. It’s very very sad.

  4. says

    Andrea, they haven’t. However, I agree 100% with what mclaren stated above. I have used Finale since the early 90’s and while it’s not as elegant an application as I thought it was when it was back at version 3.5.2, it can still do a lot of things many other notation programs just can’t. Given that my music is pretty repetitive at times, the fact that I can have Finale repeat a measure dozens of times if I wanted to, but Sibelius can only handle a few repeat loops, makes Sibelius not a tenable option for me.

    That said, I do wish it had better handling of some elements of time signatures, and you note above. I have some music that has different time signatures in different staves, and Finale can’t have them not line up, which his weird. I’d also love for Finale to handle large cross-staff time signatures that I used to do all the time with my hand notation. But that’s just something that affects the look of the score, rather than function.

  5. mclaren says

    Erling Wold is correct that you can’t get irrational time signatures in Finale (or Sibelius either), at least, not as far as I can tell. But you can get exactly the same effect with either tuplets, or a compound time signature.
    To get the same things as 4/3, just use (3+3+3)/8, since what you want is 3 dotted quarter notes. Alternatively, you could just write down 4/4 with a 3:4 tuplet inside.
    Giving each staff a different time signature in Finale is trivial, but, alas, grotesquely non-inuitive.
    1. click on the staves icon in the menu bar.
    2. Where it says allow independent properties (or something like that, it’s in the lower left hand corner) choose allow independent time signature.
    3. Apply your different time signatures to each staff by clicking on the control point for each staff and entering a different time signature in each staff.
    Erling, I did explain above how to enter broken tuplets in Finale. Basically, you have to split up each staff with broken tuplets into two staves, with the tuplets on one staff + rests where the missing elements of the tuplets should be, and the rest of the measure on the original staff with rests where the broken tuplets should be. You can do this stuff. You don’t have to limit yourself to what the computer wants you to do.
    KG replies: McLaren, I do appreciate all your advice, but 4/3 isn’t three dotted-quarter notes, it’s four triplet quarter-notes. ???

  6. says

    Sibelius’ bracket menu has half-brackets that allow you to leave the space the way you want. You have to place one, then the other, then write the the tuplet’s numbers, and then adjust the thing. A bit of ant’s work.

  7. mclaren says

    Then I guess I’m completely confused. Thought I knew what Henry Cowell was getting at, but apparently not.
    4/3 would seem to be 100% completely totally impossible according to the basic laws of mathematics.
    Look, let’s take a simple example: 4:3 quarter notes in the time of 4/4. Each of the 4:3 quarter notes has 3/4 of the length of a normal quarter note, so you get 4 * 3/4 = 3 quarter notes and then you need an extra quarter note to make up the total 4 beats in the measure. Fine, no probelm.
    But now if we write down a time signature of 4/3, that suggests from what you say, Kyle, that each note has to have 4/3 the length of a normal quarter note. That’s impossible, since you’ve got 4 quarter notes per measure. 4 * 4/3 = 16/3, which is more than 4 beats per measure. It’s impossible. It can’t be done. The mathematics blows up. It doesn’t make any sense.
    KG lol: Well, your logic is impeccable up to a point. That’s where Henry Cowell comes in. He theorized that if 1/4th of a whole note is a quarter-note, then 1/3rd of a whole note is a third-note, and 1/5 is a fifth-note, and a triplet 8th-note is a 12th-note. And you don’t have to use them in groups of 12, there can be 7 or 11. In *New Musical Resources* he invented a notation using different noteheads to distinguish third-notes, fifth-notes, and so on, so brackets weren’t necessary. Unfortunately, he didn’t live long enough to design notation software, or the world would be a much less limited place.

  8. Juhani Nuorvala says

    Brian Ferneyhough uses 5/12, 4/3 and such time signatures, and leaves out the tuplet brackets, so that a time signature change from, say, 5/4 to 5/6 is, in effect, a tempo modulation where quarternotes in the new meter are triplet quarternotes of the old meter. But I believe Kyle wants to preserve Cowell’s original idea – Cowell had actual sixth-notes, fifth-notes etc. that each looked different: each had a different kind of notehead, triangular, square etc. Now, this way you can do more than with Ferneyhough’s notation (and Galen’s suggestion?). Using split or “over-long” tuplet brackets á la Gordon or Gann is a way of indicating independent third-notes, fifth-notes etc. so that it isn’t necessary for the performers to learn and memorize new symbols.
    – Incidentally, Feldman uses split triplets in his piece Piano (1977).

  9. Jonathan Mayhew says

    So for a time signature of 9/8, you would put 9/9 instead, with the logic that there are the 8th note triplets are each 1/9th of the total measure?
    KG replies: A 9th-note is a 9th of a whole note, not necessarily a 9th of a measure. I would use 9/8 if I wanted three groups of three 8th-notes. Nine 8th-notes is longer than a whole note. If I wanted a whole-note-long measure with a 9 tuplet, I’d use 4/4. If I wanted 10/9, or 7/9, or 8/9, I’d use that.

  10. Galen H. Brown says

    The more I think about this the more it makes sense and the more comfortable I get with having odd numbers in the denominator. And I certainly don’t have any better ideas. I don’t think there’s any need for the triplet bracket to say 3:2 — once you can get your head around the concept it doesn’t really add to comprehension, and the more the notes look like regular triplets in unusual groupings the faster I think people would be able to understand, although in a sense it’s true that leaving out the “:2” is somewhat less accurate, and I may be persisting in my bias toward accomodating current notational practices rather than finding the best solutions to notational problems that exceed current practice anyway. In my own defense, though, my accomodations for “the way people already think” extends only to writing maximally interpretable notation, not limiting the kinds of rhythms that are acceptable to write.
    In the example of Trance, would you consider instead of alternating measures of 2/4 and 2/3 having one compound meter of 2/4+2/3?
    KG replies: 2/4 + 2/3? That’s 14/12 – one of my favorite meters!

  11. says

    I’m sorry, but I don’t understand why you’ve got the extra brackets on measure 275. Am I right that in Piano I you want four quarter-note triplets? If so, just use 4/6 meter; four quarter notes in piano I; and dotted-quarter, dotted-quarter, quarter in piano II. I’ve got similar issues with your 7/12 measure.

  12. DJA says

    one thing that i still don’t know how to do without scads of editing is have different parts in different time signatures. finale will stretch out the 3/8 bar to take the same space as a 4/4 bar instead of lining up with just the first three eighths.
    Figure out where the first place in the music is that of all the time signatures you are using will have a common barline — for instance, if you just had simultaneous 4/4 and 3/8, the first common barline happens after the 12th quarter note — that’s after 3 bars of 4/4 and 8 bars of 3/8. So you would make the actual meter 12/4. Then place the in-between barlines on each staff graphically (easy to do with shape expressions + metatools). Then, with each staff set to enable independent time sigs, edit the “Show As” to display the correct 4/4 and 3/8 time sigs on the appropriate staves.
    I find the Michael Gordon notation much easier to read, perform, and conduct than time sigs like 2/3, etc. I think I’d even prefer something like 2.66/4. But perhaps that’s just unfamiliarity.
    As far as orchestral works go, I don’t have the score, but doesn’t Decasia have some of these kinds of rhythms?

  13. DJA says

    Also, I figured out how to create the “Trance” example in Finale, including correct playback:
    First enter the figure as regular quarter notes and eighth notes.
    Next, for the bars with partial tuplets, create a “4 quarters in the space of 6 quarters” tuplet over the entire bar. This turns all quarter notes in the measures into quarter note triplets. Hide this tuplet.
    Next, we need to change the eighth notes into non-tuplet eighth notes. This can be done by applying a nested tuplet to the eighth notes: “four eighth notes in the space of sixth eighth notes.” Hide this tuplet as well.
    Finally, add the “3”s graphically. (Finale also makes it easy to add incomplete triplet brackets if that’s what you prefer.)

  14. says

    So, I kind of went nuts on this. Here is a post where I explain in greater detail how to do the incomplete tuplets in Finale. I also made two audio files of the bassline from “Trance” — one with a straight quarter note click, and one with a 2/4 + 2/3 click. I was curious which would be easier to feel. The answer was not what I was expecting. (At least, so long as we are only talking about the bassline. Once you add in all the other parts, all bets are off.)

  15. PostClassic says

    KG replies: Todd – 4/6 equals 2/3, does it not? And why put two dotted quarters against three triplet quarter notes? It doesn’t matter what your issues are – the math works out.

  16. Frederick Schneider says

    Juicy topic.
    Just an FYI, in the shareware Harmony Assistant (and probably its cheaper cousin Melody Assistant), one can input time signatures with irrational denominators. It’s quite intuitive in that respect. There’s also provisions for microtonal music in the software if anyone wants to check it out.

  17. says

    It could be that I’m not sure what you really want in m.275, but it seems to me that you either use the brackets or the 2/3 meter, but not both.
    As for 2/3 or 4/6, it doesn’t really matter, it would depend on what you wanted to be normative.
    My reasoning: as a measure of 2/3 it’s a grouping of two half note triplets in a measure. You want the first piano to play four quarter note triplets and the second piano to playing “full” quarter notes followed by 2/3 of a quarter note. As such, the bracket on piano I is unnecessary – its implied in the meter; on piano two for you to notate it as you have it, you’d need to eliminate the 2/3 meter and give it a meter of 2.666/4. Within the 2/3 meter, what you’ve got is something not equal to a full measure of 2/3.
    I suggested the two dotted-quarters for piano 2 becuase in 2/3 a dotted quarter is equal to a quarter at x/4. It would also allow you to demonstrate the polyrhythm with Piano I more clearly.
    Fun and games however you look at it.

  18. todd says

    Now that I’ve done the dishes, I think I understand where you’re coming from.
    If the quarter note is normative in 3/4 we have three quarter notes. If the 3rd note is normative in 2/3 we need 2 third-notes. The only way we can understand a third note given our system is with a bracket, thus a measure of 2/3 should have two half-notes with broken tuplet brackets over them. Thus, Cowell creates new noteheads to get around this problem. By this logic, quarter notes in this system should be grouped in groups of two and be bracketed with broken tuplet brackets.

    Where I’m coming from is to think of the 2/3 as articulating the big bracket already, by virtue of the meter – effectively the meter standins for a metric modulation. By this logic to articulate two impulses in 2/3 we need only 2 half notes.

  19. says

    Not for nothing, Thomas Ades frequently uses these “irrational” meters in orchestra works (cf Asyla, third movement) all over the place. And the Berlin Philharmonic does it fine (how the Boise Philharmonic will handle it is another question…)
    (I do seem to keep posting comments on Kyle’s posts two months too late.)

  20. Mr.Joe says

    hey guys, i’m wondering if you can help me understand how these “irrational” time signatures are possible. i can’t find any books or anything to help me understand. . . i understand that 4/3 could basically be a dotted half note gets 1 beat? then how could n/5 or n/7, 11, etc be possible. and how would you exactly play them i were to sit at my piano or guitar? could someone try to explain?
    KG replies: In 4/3, a triplet half-note gets a beat, and there are four of them. The only book is Henry Cowell’s New Musical Resources. The entire rhythm section is fantastic, and you can read the whole thing in an hour.

  21. Ged says

    Thanks to Kyle and everyone for an enlightening discussion. I am so glad this webpage exists. I’ve been wondering how to notate incomplete tuplets recently.
    The idea of ‘irrational’ time signatures, with odd denominators or multiples thereof, does seem appealing. So, if you have, say, a bar with two quarter notes followed by two quarter-note triplets, a time signature of 5/6 would fit that snugly.
    OK, so my question is: given that the ‘6th-note’ has been declared the ‘norm’ in the context of this bar, wouldn’t it be redundant to bracket the two quarter note triplets with a ‘3’? Conversely, shouldn’t I bracket the preceding two quarter notes together with a ‘2’, or ‘2:3’, as these are the ‘exception’ to the ‘rule’ set by the new time signature?
    Then, of course, there’s the question of how best to do this in software – oh, heck! – that’s a can of worms for sure. I think I’ll just scribble some ideas on a bit of paper for now :)
    Kyle, I think I will buy a copy of Henry Cowell’s book, actually. I’ve heard a lot of good things about it.

  22. Sami says

    If I may contribute a few ideas on the topic:
    1) First of all, the term “irrational” is completely false. Irrationals are numbers that cannot be written as fractions at all, like the square root of root (or any non-square integer for that matter). A time signature like 4:3 is very rational, with the difference that it doesn’t have a power of 2 at the denominator. That surely keeps time signatures like that rational, only non-standard. Now, if we think of something as simple as 5/8 as non-standard, I think we kind of defeat the purpose. So, I would propose we don’t accept a name that is (mathematically, at least) false from the beggining. Try something else, like “non-binary” (binary meaning a power of two).
    2)Second, a time signature like 4:3 simply means that we have extended a 4/4 measure by one half-note triplet. In other words, just think of 3 half-note triplets which make up the 4/4 measure (or 3/3 measure if you will, both of these end up giving 1, which is the default duration of the whole note anyway.), plus an additional 1/3, which one can easily visualize as the very first one of the three half-note triplets of the next measure. Put the first 4/4 measure, add this half-note triplet note right at the end and that gives you a 4:3 time signature.
    Want another way? Take 8th note triplets in groups of 4 – tetrads of 8th-note triplets. Instead of doing three tetrads in one 4/4 measure, try four. You have played one 4:3 measure. If you try this with a metronome playing 4/4 and you play the 4:3 over it, you will see that the “1” of the next 4:3 measure is write on the 2nd triplet of the 2nd qurter note of the second 4/4 measure.
    But then, one may say that this whole thing – four tetrads, I repeat – can actually be written as 16th notes and be done with. Sure, if it is alone. But if the next measure is trully a 4/4 (as opposed to the 4:3ds we play), then it is completely different, and very justifiable. For example:
    4 tetrads of 8th-note 3-plets (4 groups of 8th-note triplets in groups of four, comprising a 4:3 measure), followed by 4 tetrads of 16th notes (again, 4 groups of 16th notes in groups of four, comprising a very tired 4/4 measure). It will sound like you have suddenly shifted gears! You have a 4:3 ratio of increase in speed of the notes, with the same rhythmic underpinning – 4 slow tetrads (in triplets) followed by 4 fast tetrads (in 16th notes). Traditionally, the only way to write this is using rhythmic modulation (8th-note triplet of first measure equals three 16th notes of second measure.)
    3) I personally love this system, because it trully gives you the possibility to play any rhythmic ration right away. Now, the question is: is there a situation written using this system, that the traditional system of meters even with the help of rhythmic modulations cannot represent? The answer is a most definite yes: It is when we use THIS system with rhythm modulations – the possibilities are endless. Try moving from 4:3 measure to a 7:5ths measure, using an additional time modulation of values not in the two time signatures (for example, 7 8th notes of the first measure equal 7 8th-note septuplets from the second measure), and you have something non representable by the traditional notation – or, in the best of cases, impossibly hard to notate. With the non-binary rhythms though, these are not simpler to write, they are also simpler to play. One only has to really understand the various durations the same way as they know the powers of two. I think it is all a matter of education.