Monday, 29 June 2015

Adventures in … augmentation dots?

A couple of months ago, Aaron Cassidy gave an interesting lecture on rhythm. Aaron’s music is extremely complex—and in recent years, particularly since the Second String Quartet (2010) he has been interested in, if not simplifying, at least consolidating the vast amount of information his scores contain.

But there was one little moment in his talk (which is worth watching, by the way, particularly for his  exposition of the intricate rhythmic flow of elite cycling[1]) which sparked me. It was that Aaron had never found a convenient way to notate 9’s and 5’s. (See 1:06:00 in the talk.)

This got me thinking. Surely there has to be a convenient way of notating 9’s and 5’s—indeed, I’d felt I’d encountered it before.

It’s particularly odd that we haven’t retained a 9-length note, especially because of the tempus perfectum maior being the summit of mensural time measurement. Indeed, this frustrated the musicologist Willi Apel so much that he invented his own notation for a 9-length note. But more on that in a moment.

First, let’s see if we can tackle 5’s. Anyone who is familiar with Conlon Nancarrow’s brilliant pencil scores for his pieces will have seen his solution for notating 5’s. Consider the following excerpt from his Study no. 7. Is there a clearer way to notate this?

Nancarrow’s solution was to invent a new notehead to attach to the minims that would lengthen them by one quaver. The notehead looks something like a turn ornament. When this is done, the clarity of the line increases greatly.

While this is an elegant way of achieving a clear 5, it lacks extensibility—that is, if you want a length of 5 crotchets, you have to tie two of these together (unless you’re willing to use a stemless one like a semibreve, though Nancarrow never does this). Meanwhile, a length of five semiquavers is impossible to represent, without inventing some other notation.

Incidentally, George Crumb had another way of showing this length—by positioning another dot to the left of the note. The logic was probably that the note is double-dotted, but the second dot subtracts rather than adds that length. Thus a dotted note length 6, instead of being lengthened again to 7, is reduced to 5. I think this is a nice notation, but it suffers from lack of clarity if these dots collide with each other (if two Crumb-dotted notes are close together).

In a recent piece (a piano trio) I’d experimented with the ‘half-dot’, using a small circle—a la the harmonic marking—as an extendable way of achieving a length of five.

I was later very pleased to see that Ben Johnston does the same thing. See this excerpt from his Two Sonnets of Shakespeare (1978). The strings and winds divide the 8/8 bar into two notes, length 3 and 5, with the 5 showing the small clear circle after it.

The nice thing about this method is that it’s totally extendable. The Johnston half-dot can be appended to any note to add a quarter of its length.

Using this notation, let’s again look at that Nancarrow fragment. How would it appear using these Johnston half-dots?

Just using the half-dot, in combination with the other dots we already have, can yield some beautiful results. Notating ‘augmentation dot rubato’, as it might be called, is easy and can summarise enormously complicated rhythms with comparative notational efficiency. Consider this version of the song Old Man River.

With mensurstrich bars, the rhythm expands gracefully over the barlines, and while there is definite visible syncopation, when a human being plays this back, they would flex the rhythm according to feeling, rather than attempt the impossible task of actually calculating the underlying demisemiquavers. If this melody were notated ‘normally’, one would have to write something like this:

Here, one is trapped by the grid of the demisemiquavers, and further, it is much more difficult to see the melodic line, let alone flex with it.

Incidentally, one interesting feature of augmentation dots is that they can be used to subdivide bars of their length. A single-dotted note subdivides a bar of 12 (or 6). A half-dotted note subdivides a bar of 10 (of 5). And a double-dotted note subdivides a bar of 14 (or 7). In this way each of these lengths can be ‘converted’ into a bar of 4—in the same way that one could place a large 4-tuplet over the bar.

(From these notations, one can see that Sibelius struggles to notate the groupings in the upper, plain bar. In 14/8, the situation is particularly bad. I should also emphasise that Sibelius does not do well when notating these things—appending symbols to notes is fiddly, and if you realign anything in the bar, the whole arrangement can go out of alignment, leading to a time-consuming manual fix needed. I experimented with making these dots ‘custom articulations’, but even then problems arise. There is no way to make a custom augmentation dot, not even using a plug-in—it is a deeper part of the Sibelius architecture. The closest one could get would be to write a plug-in to detect notes of a given length and insert a symbol next to them. But even this would be subject to the same problems as doing it manually.)

So far, we’ve only considered the half-dot, as a notation for 5. But to notate a 9, we need another augmentation dot. Notating a 9 is like notating 4.5—we’re adding only an eighth of the length again. We need to invent a ‘quarter-dot’.

My proposal is to use, again, a clear dot, but with an oblique strike-through. I’m not particularly sure how clear this is—and another notation could plausibly be found.

This notation is interesting as it subdivides 9 into 4. This can produce some nice results—such as 5-4 ‘swing’. Consider this melody—from On the Sunny Side of the Street—the swing here is very light, close to the sort of swing rhythm that a jazz musician might actually use. It close to being straight quavers, but not quite.

The augmentation dots are interesting as they represent a ‘consolidation’ of the underlying grid. The 32nd or 64th notes that underlie the rhythm are present, but only ‘sort of’, as the rhythm is more ‘felt’ than calculated. As an example, consider this more complex ‘transcription’ of the waltz section from Strauss II’s Die Fledermaus overture, a piece which has many ‘conventional’ rhythmic nuances. Viennese waltzes by convention have an early onset second beat. I’ve also taken some liberties with the melody, to create ‘augmentation dot rubato’.

Here the augmentation dots really do nothing more than ‘suggest’ the push and pull of the rubato, as again, actually calculating the rhythms is extremely difficult. In computer playback, however, the extremity of the rubato is clearly audible.

In summary, then, here are all of the durations, from 8 to 16, summarised using these ‘custom’ augmentation dots. Some are common practice, others are invented.

The duration 18 presents some interesting difficulties. While it is possible to notate a semibreve with a quarter-dot, there is another notation, coming from Willi Apel, which is interesting and plausible.

This notation ‘dots’ a dotted note. It would, in other words, take a dotted note, and add half its length again. This, in 9/8, results in a dotted minim plus a dotted crotchet. Apel’s notation was to use a dotted minim with two dots, one positioned above the other. (This example comes from the appendix of his seminal book, The notation of polyphonic music.)

I therefore offer not one but two options for notating 9’s, depending on context. One could use the quarter-dot if you were equally subdividing the bar, into two halves of 4.5. Or one could use the Apel dotted-dot to imply threefold subdivision, as in tempus perfectum maior.

The 21/8 duration also presents some interesting opportunities. If one had divided the bar into three double-dotted crotchets, one could use a dotted double-dotted minim (a minim with two horizontal dots and one vertical dot) to show this. Or, one could use a double half-dot. Both these options appear above.

The only lengths my scheme omits are 17 and 19, but these are very uncommon. In any case, they can be shown as 17 = 12 + 5, and 19 = 10 + 9, or any other subdivision.

How, then, might we notate the rhythm that Aaron was interested in, at 1:06 of the video above?

How ideal is this solution? Could it be made clearer? Quite possibly. Different varieties of noteheads, that unlike Nancarrow’s, could be made ‘white’ as well as ‘black’, and hence extendable, might solve some of the clarity problem. One can see here that when the score is small, the quarter-dot is very similar in appearance to the half dot. Nevertheless, the rhythm on the right does have greater transparency than the one on the left.

The other thing these augmentation dots can do to aid us is give us greater precision in approximating a 7-tuplet rhythm like Aaron’s using 2-limit or 3-limit quantisation. Consider the following approximations.

The first, a) is very highly approximate, but gives a sense of the overall shape of the rhythm. On the other hand, I like b)—as it has much of the slightly lopsided quality of the 7-tuplets. The third approximation c), using its triple dots, is ungainly and not ideal (but almost identical to the 7-tuplet rhythm). However, I think my favourite is the fourth approximation d)—the advantage of this one is that it shows where the second half of the 4/8 bar occurs, something all the other rhythms do not.

Actually, not quite, as the second approximation does that in its own way—expanding out the first beat, and ‘contracting’ through use of the half-dot, the second beat. The question one is led to is ‘is the third note of b) “off the beat”?’ Indeed, is the third note of the original 7-tuplet rhythm “off the beat”? These are deep questions, to do with our notions of beat and rhythm—and when we use augmentation dots in this way, we can bend beats at will, it makes beats gluey, and while they retain their gravitational quality, they can fall away from the underlying grid.

In the end, these augmentation dots signal at least one way of explicitly (i.e. not merely through conventional performance practice) suggesting movement away from a rigid underlying grid. In Chopin performance, say, no one expects rhythms to be attacked robotically or without a healthy amount of license. But that kind of conventional performance approach cannot be ‘written in’ to a modern score particularly easily, especially when, for performers, the material may be very different from conventional notions of rubato pianism.

But with these augmentation dots, the rhythms are mostly ‘felt’; they elide the grid they sit on; they subvert it to a certain degree. Consider again that ‘transcription’ of Fledermaus. As a pianist, if you were presented with that excerpt, how would you play it? Would you try to ‘calculate out’ all the subtleties of the rhythms, or would you sight-read through and ‘feel out’ those subtleties?

I don’t have as much interest as Aaron does in re-inventing notation—as I feel that, given the limits placed on rehearsal time for new music, and the pressures on musicians performing it, more conventional notations are usually more efficient and transparent, and can still be used powerfully. But I do agree with him about how for a rhythm to be a rhythm (as opposed to a series of durations), it must have points of gravitation; it has pattern to it. It has 2-ness or 3-ness; it has upbeats or downbeats. To bend and stretch rhythms, so that they’re fluid and can be bent by the performers themselves, but still retain ‘up-ness’ and ‘down-ness’, ‘off-beat-ness’ and ‘on-beat-ness’, is a great priority.


Some of these ideas crop up in a piece I wrote for harpsichord, which was recorded recently. The score is available to download here, and can be listened to below.

[1] One can’t, or at least, I can’t, watch the section of the talk without thinking of Alan Partridge’s cycling commentary on The Day Today: ‘they look somehow like cattle, in a mad way, but cattle on bikes’.

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