A Metric Notation Issue…

Okay, here’s a notation issue.

Consider a musical passage that begins with a simple meter in which the quarter note gets the beat, then alternates regularly with a compound meter in which the subdivided beat is equivalent to the triplet eighth of the prevailing simple meter and in which extra subdivided beats are occasionally added or subtracted. Or to put it another way, consider the alternation between a prevailing simple meter in which the quarter note gets the beat and a compound meter, with occasional asymmetrical meters (e.g. 7/8, 5/8) thrown in, in which the eighth note is equivalent to the triplet eighth of the prevailing simple meter.

Here’s a possible example. (Click on the image to enlarge)

Audio (Forgive the poor midi!)

What is the best way to notate such a passage so that it is most easily read? It seems to me that there are at least four different possibilities, not all of which are sensible:

1. The TEMPO CHANGE method

I suppose one could notate such a passage by indicating a tempo change at each shift. After all, that is essentially what is going on here—the shift between two different musics of two different tempi.

But surely this won’t do. In addition to it simply being extremely unwieldy, it fails to adequately show the correspondence between the triplet eighth of the prevailing simple meter and the eighth of the new compound meter.


One could indicate the correspondence of note values just mentioned by showing the equivalence each time a shift occurs.

But this, too, proves to be unwieldy and awkward. And if the two musics came to  resemble one another on the page more closely, reading the duration equivalency equations become even more difficult to read. There must be a better method.

3. The BOULEZ Method

In Le Marteau sans maître, Boulez used fractional time signatures (such as 4/3 over 2) which, perhaps after a few moments of utter confusion, make perfect sense.

The “Boulez” method certainly makes sense, but is the awkwardness of reading fractional time signatures simply too much trouble?


I first came across the “Ferneyhough” method when looking at a piano piece by Thomas Adès called Traced Overhead (1995-96). I later realized that Ferneyhough used this method before Adès in his Etudes Transcendantales from the early 1980s. In this method, unusual “denominators” of the time signature such as 12 or 20 are used. Just as the lower “4” of a 4/4 meter indicates that the quarter note gets the beat since it is 1/4 of a whole note, the denominator “12” would indicate that that note value which is 1/12 of a whole note is what gets the beat. That note value would be the triplet eighth. So a 5/12 time signature means that 5 beats (or beat divisions, more accurately) fill the measure and the beat is the triplet-eighth. A 9/20 time signature would mean that 9 quintuplet-sixteenths fill the measure. Like the “Boulez” method, the “Ferneyhough” method makes perfect sense once you get your head around it. Here is what it would look like:

Of the four, methods 3 and 4 are better, and while it is extremely odd to use “Ferneyough” and “clarity” in  the same sentence, it does appear that Method 4 is the neatest of the four, though it perhaps requires the most “figuring out” at the beginning. Furthermore, it seems to me that with some practice, one could become quite fluent at reading time signatures with “12,” “20” and “24” denominators.

So what do you say? What would you prefer to see as a performer? Are there any better “methods”? Do you know of other composers who have tackled this issue?


About Luke Dahn

Composer and music theorist Luke Dahn is Visiting Assistant Professor of Music Theory at the University of Utah. He is also co-founder and artistic co-director of Ensemble Périphérie, and lives in Salt Lake City with his wife Yu Jueng and daughter Mae. http://www.lukedahn.net.
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10 Responses to A Metric Notation Issue…

  1. Steve Layton says:

    I suppose the #3 “Boulez” version appeals strongest to me. It only takes an extra second’s thought and counting to feel that.

    But — and of course I don’t know what else follows this passage — I’m still trying to figure out why you’re trying to stay mostly with quarter-note denominators in all the other measures, when eighths and 16ths instead seem so much clearer and representative of what’s happening outside the tuplet shifts.

  2. Great post! We all face these challenges, but you are right in wanting to make it as clear and ‘sight-readable’ as possible. I would probably use a combination of 1 and 2 using the tempo markings from 1 and putting the equation of 2 in parenthesis. I would also make the 2nd measure be 3/4 since every other instance of this bass clef figure is a pickup to a downbeat. It just conducts better to me.

  3. lukedahn says:

    Ralph, Right. I didn’t mention the possibility of combining methods 1 & 2. I agree that would probably be better than either of those two methods on their own. As for making measure 2 a 3/4 measure, I see your point. I think it depends on what the composer is after. While every other instance of the bass clef figure is a pickup to a downbeat as you say, every other instance of that figure enters on a downbeat as well. So what should eb emphasized: the downbeats on which they enter, or the downbeats to which they lead?

    Steve, good point about the meter not reflecting what’s actually happening in the music. It’s the “written meter” vs. “perceived meter” question. I have always generally leaned more toward using a simpler notated meters (which do not always reflect the perceived meter) and writing the syncopations for performers to create the perceived meter. So for measures 3-5, instead of using perhaps 5/8, 5/16, 3/8, 5/16, 2/4, I would generally prefer to use the simpler 4/4, 3/4, 3/8. Granted, I don’t doubt that the final aural result changes depending on which way it is notated. Will an accent that is meant to be played as a “perceived” downbeat really played as such if it appears on the final sixteenth of a 4/4 measure?

  4. Scott says:

    Luke, this is great. Reminds me of the conversations about irrational note durations we used to have back in the WMU days (remember us theorizing about attaching a number to the stem to indicate partial tuplets?). To some extent, especially in the example you have here, this would be easier to deal with by leaving out time signatures and barlines entirely, or by beaming the triplet across the barline (sans time signatures). In a larger ensemble setting that would probably be impossible, of course.

    In a solo or small chamber context, though, this strikes me as something of a self-imposed problem stemming largely from our notation software. Finale and Sibelius require measures and meters, even if they can be hidden. Written on paper (and, again, omitting time signatures), the whole problem sort of disappears, doesn’t it?

    Fantastic post. I’ve been following you since you started, but am only now breaking the comment ice. Keep it up!

  5. lukedahn says:

    Scott, I do remember theorizing about this in our WMU days! I agree that for a solo work, it may be best to do without time signatures and even perhaps barlines. Then one could simply use incomplete triplets as I did in the very first image. A large ensemble work is a different matter. It would be interesting to develop a system of attaching numbers to stems/beams/flags. Why don’t you get to that in your spare time.
    Thanks for commenting. I will probably see you on your visit to Iowa City in a couple of weeks.

  6. Evan says:

    Luke, great job laying out the 4 possibilities! While I had my own composerly opinions, I asked a friend who plays a lot of new music, including Adès. She said the fourth notation, the “Ferneyhough”, was the easiest to read–5/12. Now if we could only commit to this notation, get used to it, and teach it to performers.

  7. lukedahn says:

    Evan, I think I agree with your friend. But you can’t say “I had my own opinion” and then not give it! That’s dirty pool! :o)

  8. Pedro Dias says:

    Hi. I was curious about which software you used for the examples and how you did the “ferneyhough” one (which btw is the one i prefer).

  9. lukedahn says:

    Hi Pedro. I used Finale for these examples. To achieve the “5/12” measure, I hid the original time signature for the measure, added extra space at the beginning of the measure (about .3 inches in this case), then typed in the “5” and “12” as text blocks using the Maestro font. There may be other ways of achieving this, but this seemed simplest to me. Let me know if any of this needs more explanation.

  10. Jacob says:

    There is another solution, which works in some cases (though not so much in this one with all it’s short note durations), and that would be to expand all note-durations such that the triplets become normal-length notes. In this case that’d mean you’d have to notate everything else as quadruplets, and the triplets as straight quavers in bars of 9/8 – 6/8 – 12/8 – 9/8 – 8/16 – 5/8 – 6/8 – 7/8 – 9/8.

    Obviously, that method’s pretty hideous in this case, but there are cases where it provides much neater notation, for example, I was working on a peace in which quavers were grouped 2+3+2+2+2+3, and underneath that were 3 groups of 7 notes each with a value of a member of a triplet. Rather than notating those broken triplets in groups of 7, it made a lot more sense to expand out the notation such that each crotchet of the non-tuplet melody became a dotted quaver (so 2+3+2+2+2+3/8 becomes 6+9+6+6+6+9/16), and have the groups of 7 become standard quavers.

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