Tag Archives: music theory

Music theory for (ballet) dancers, the last word for now? Grant’s “Beating Time and Measuring Music in the Early Modern Era”


Roger Grant's book "Beating Time and Measuring Music"

I’ve just added Roger Grant’s Beating Time and Measuring Music in the Early Modern Era as my top choice for books on music theory for those interested in music-dance relationships (see my metre and rhythm page for a brief bibliography on  that topic). I don’t want to say too much, because it figures largely in a chapter in my PhD, and it’s too detailed and scholarly a book for me to summarize hastily. Suffice it to say, if you want to know what think about time signature and meter and movement, it’s all in this book. I’m glad I hadn’t read it when I was writing How Down is a Downbeat?, a journal article on music, ballet teaching and time signature that I wrote a few years ago; it would have tempted me to rewrite the whole thing. On the other hand, I wish I had read it when I first started teaching music for dance teachers back in 2000. However, some of the significant books and articles that Grant refers to in building his theory were published some years later than that. Is theory even the right word? I’m not sure: it’s history, but in order to understand the history, you have to change your ideas about what you thought was music theory. It’s amazing that in the 21st century, we’re still solving the problems unexamined or hidden by “rudimentary” music theory, e.g.—to name but one— why is a 6/8 called a compound time signature? What’s compound about it? 

The biggest problem with what is conventionally called “music theory” is that it presents as simple and straightforward (a matter of counting two or three) something which is exasperating in its complexity, not least because “time signature” as a subject leaves out the people who use it and the way they interpret it, but it is virtually meaningless without the (changing) practice in which it is embedded. I’ve hinted at this in many of my more recent postings on triple meter and Rothstein’s theory of  “Franco-Italian hypermeter.”   Grant discussed the way that the meaning of beat as movement has gradually disappeared, morphing into the concept of time as a endless stream of motionless, durationless ticks. This in fact was exactly how I used to teach music theory and meter, without realising the entailments or history of my own beliefs about what meter or musical time was. 

I am in awe of the way that Grant makes sense of such a complex assemblage of notation, musicians, practice, ideas, primers, teachers, and so on. It’s only when you’ve struggled to sort out some of these problems yourself that you realise how courageous and hard-working someone else has been at grappling with similar issues.  


And now for something completely sextuple


This variation by Mozart on “Ah, vous dirai-je maman”, K.265/300e has turned out to be a real life-saver in class for one of those ballet exercises where you need a 6/8 that gives you six quavers in a bar (click here to hear it). If you’re thinking “But that’s not a 6/8,” hold on, I’m coming to that, in this discussion of compound meter.

Compound meter in all but name in Mozart's "Ah, vous dirai-je maman"

Ah, vous dirai-je maman’, K.265/300e, Variation 3

The Mozart is useful for class, but it’s also an example of a particular kind of 6/8 that does what you’d think it would do, i.e. articulate six quavers that you can hear and count. Not that you’d want to count them, but they’re there, so you can hear why it’s called a six. Many pieces in 6/8 don’t go like that (they just jig along rumpty-tumpty fashion, so they sound barely distinguishable from a 2, so to see why it’s a six, you have to imagine the beats that you can’t hear.

What is compound about compound meter?

It’s things like this that make me dread trying to explain compound meters such as 6/8, coupled with the fact that the term “compound meter” (or “compound time signature”) does not convey anything useful or hearable in the “compound” part. The meaning that “compound” once had in this context is rarely taught in music theory  – that a 6/8 was at one time a way of writing two bars of 3/8 as compound bar, thus halving the number of barlines you had to draw.

Well, that’s part of the the story, at least. In Danuta Mirka’s Metric Manipulations in Haydn and Mozart,  she explains that the eighteenth century theorist Koch viewed 6/8 sometimes as a “compound metre” in this sense, and sometimes as a “mixed meter” or a simple meter of “tripled beats”. That is, some 6/8s are basically just 2/4s with triplets (and some 9/8s are just 3/4s with triplets), but for notational ease, you might sometimes write the “tripled” 2/4 as 6/8.

The difference is crucial – one is triple subdivision (tripled 2/4), the other (compound 6/8) is triple meter, even if they’re both notated as 6/8. Which brings us back to a well-worn topic on this site, truly triple meter. It’s truly triple rather than sextuple, because to Koch, it’s a compound of two 3/8 bars (with equal weight in both halves of the bar, and quavers establishing the meter. That in turn is one reason perhaps why Mozart didn’t write this as 6/8. It’s clearly duple, rocking between stronger and weak beats in each bar.

Is compound meter really just about subdividing beats into three?

This is why it’s so hard to teach about compound time signature as a concept to those (like dance teachers) who are trying to understand how it relates to hearing music. To recap a previous post: If you look at many music primers, they’ll tell you that compound time signatures are where the beats are divided into three, and simple are where they are divided into two. Nothing about the term “compound” suggests “divided into three”, and if you’re looking at a time signature like 6/8, unlike the simple meters, there is no visible beat to be divided, it’s already been divided as part of the time signature. It makes no sense, unless you  explain what I’ve explained above, which also explains what is simple about simple meters – not that the beat is divided into two, but that the bars are single units, not joined together as in compound signatures. But also, “compound time signature” only describes one concept of 6/8, and one which does not continue into the 19th century and beyond, where we describe it as if it were a duple metre with triple subdivision.

That is the why the Mozart piece is relatively unusual, and so useful. It is duple with triple subdivision, but it tips over into the realm of a truly triple meter because the movement that one hears, clearly on the musical surface, is of a continuous triple meter. It is hard to retain tripleness in the metrical slipstream of a piece which is duple at another level, but Mozart does it. Tears for Fears’ song “Everybody wants to rule the world” does it some of the time – there’s a constant, steady, truly triple 6/8 going on in a lot of the music, but the vocal line  exerts a strong duple pull.  Mozart’s advantage is that the tripleness is centre-stage in the melody, it’s not a support act. In “Everybody wants to rule the world”, it’s not exactly melody and accompaniment, they are simultaneous, equally salient layers of the music which both draw your attention (incidentally, the song’s time signature as published is notated in 12/8 with (4/4) in brackets).

6/8: two meters/time signatures masquerading as one

Disambiguating 6/8s into those which are characterised by triplet subdivision, and those which are truly triple meter seems to me to solve the problem, because it’s how people in the real world hear this music. You could argue that you should teach basic time signature before these more complex topics, but to my mind, teaching “compound time signature” by saying it “means” dividing a beat into 3, is oversimplifying the case to the point where it becomes difficult to understand because it doesn’t make sense. Koch’s theory isn’t simple, but it makes sense, and it reflects clearly the fact that 6/8 is not a single concept, but, echoing Justin London’s “many meters hypothesis”, it is a structure that has multiple expressions in real music.  A differentiation between two types of 6/8 is partially clarified in Labanotation and Benesh Notation, because you have to say what level of the beat you’re using as your pulse. However, the issue here is not about the pulse that you count or sense as being the main beat, but level of beat where the musical action happens. Music could be truly sextuple (i.e. triple x 2) compositionally, but whereas a dancer might not count it that way if it’s fast, the composer on the other still writes it that way, because that’s how the music moves.

It would be good if in ballet teaching we had words to describe different kinds of 6/8, at least at the point at which you learn about time signatures, so that you can account for the fact that some don’t sound like six at all, and some do. We need something like a “triplety-two” and “truly sextuple” and a “swingy two” for those things like 6/8 marches that barely reveal any of their sixy undergarments, and possibly a few more. Dance rhythms are handy – but only if both parties (teacher and musician) have the same shared vocabulary and understanding, and only up to a point. It would be nice to be able to have something that was like a 6/8 march metrically, but wasn’t a 6/8 march culturally (or is that impossible?).  Any ideas for some new terms?

Compound errors


Someone asked me last night what made me want to do a PhD in music in ballet training. I explained that one of the reasons came out of trying to teach time signature to trainee ballet teachers. It’s not that I think they ought to know about time signature, but I was filling in for someone who thought that they did. Trying to teach time signature did my head in, literally (you could say) to the extent that nothing but a PhD would really sort out the mess.

An example of 18/8 time signature by Alkan

Alkan’s ‘Barcarolette’ in 18/8. Don’t try asking for one of these in your ballet class

In the first class, I realised that the students knew more than I did about music theory, because they were studying for their ABRSM Grade 3 theory.  In decades since I did that exam, I’d forgotten about terms like ‘melodic and harmonic minor’ (someone later  asked me what ‘harmonic minor scales’ were for, and to this day, I don’t know). I’d also forgotten about simple and compound time signatures. As a musician, you look at a piece, see that it’s in 6/8, and play. There is no need to categorise it as ‘simple’ or ‘compound’.  I bluffed my way through the first class, and then went away to quickly mug up on all the theory I’d forgotten.

I was confused. What is a compound time signature? Why is it called ‘compound’? What is simple about a simple time signature?  Why does dividing a beat into three, rather than two, make it ‘compound’ rather than ‘simple’ (as most theory books tell you)? That leaves 3/8 in a rather odd position, because according to half of the theory, it should be compound, and to the other half, it should be simple. People argue about this on the internet, and one theorist says ‘Better to just remember that 3/8 is simple triple, for the exam’.

If (as other theorists tell you) compound time signatures are called compound, because they’re a compound of two or more bars of 3/8, then why logically, isn’t 4/4 also a compound time signature? It didn’t make sense to me, so how would I explain it to someone else?  Then in 2002, I bought the newly published Cambridge History of Music Theory, and read Caplin (2002, p.661), where I discovered that some theorists in the 18th century did claim that 4/4 was a compound time signature, for exactly the reason that I thought they should. The main thing is, they argued and disagreed about what ‘compound metre’ meant, and which metres were compound.

In the years I’ve spent dealing with scores and music notation, and working with dancers, metre and time signature has never gone away as a problem or an interest, because contrary to appearances, it’s culture, not arithmetic. My latest joyful discovery is two articles on metre in Communication in Eighteenth Century Music, one by Danuta Mirka, the other by William Rothstein. Mirka’s article is about the way that composers in the eighteenth century would manipulate the metre of music without changing the time signature (she differentiates between the ‘notated’ metre and the ‘composed’ metre – ‘notated’ is what’s on the page, ‘composed’ is what you hear, after the composer has played around with your sense of metrical accent).

Here’s the bit that I like a lot: Koch and Marpurg (18th century theorists) say that if you’re listening to something that has been notated in compound metre (remember, that’s either 6/8 or 4/4), you can tell whether it’s  ‘simple 4/4 or compound 4/4’ or ‘simple 6/8 or compound 6/8’ depending on whether caesuras (roughly speaking, the end of a short musical statement) come in the music. If the caesura comes in the middle of the bar, then it’s compound, if it comes at the beginning, it’s simple. Mirka’s article has several examples from Haydn that demonstrate this principle. One of them is the Adagio from Haydn’s string quartet in B flat major, Op. 50 No. 1 (arrangement for piano/cello from IMSLP here) where the first four bars are (in Koch/Marpurg’s terms) in simple 6/8 (because you only have one downbeat), and the last two are in compound 6/8 (because you’ve got two downbeats, and it’s effectively two bars of 3/8, notated as 6/8). The fascinating part here is that this can be construed as ‘balancing’ (in an aesthetic/theoretical sense) what appears to be an unbalanced (6 bar) phrase, because the first section is four bars of 6/8, the second is (effectively) four bars of 3/8. Whether you completely buy into that is a matter of theoretical position and interpretation – and Rothstein takes issue in his article in the same book (p. 114) with an analysis along the same lines by Maurer Zenck of a mid-bar cadence in Beethoven. Whatever position you take, the idea that there can be times when 6/8 or 4/4 are simple, and times when they’re compound, makes more sense than categorising 6/8 as “compound”, and 4/4 as “simple”.

Rothstein’s article “National metrical types in music of the eighteenth and early nineteenth centuries” is probably one of the most fascinating and enlightening articles about metre and time signature I’ve ever read. His theory is that barring music is a matter, to some degree, of what he calls ‘national metrical types’. ‘Broadly speaking’, he says:

Italian and French composers were more likely in the nineteenth century to place cadences on the first beat of a bar, whereas German composers often placed them later. Conversely, phrases in German music were less likely to begin in mid-bar, beginning instead on the the downbeat or with a short anacrusis, one-third of a bar or less in length. (Rothstein 2008, p.113)

I’ll try to summarise as briefly as possible, with inevitable loss of detail and accuracy – better to read the article yourself, don’t take my word for it: In compound metres, he distinguishes between French, Italian and ‘neutral’ barring. French compound metre is where you get half-bar anacruses, and cadences on a downbeat, and it’s rare in the late 18th century onwards. Bach’s Badinerie from the Suite in B minor is written in 2/4, with a half bar anacrusis, but he could have written it as a French compound 4/4, i.e. so that you have a 3-beat anacrusis – because that’s effectively what the music does. German compound metre  is where you get a short or no anacrusis, and cadences are on the second beat of the bar. ‘Italian‘ is where you might just as well have written it in 3/8 or 2/4 (what Mirka calls compound 6/8 or compound 4/4, in reference to Marpurg & Koch).

This has finally solved a mystery for me that bugged me all the time I was preparing the scores for the RAD’s new Grades 4-5 syllabus. There were two pieces, one by Verdi (E03b, Canzone Greca from the ballet music from Otello), and one by Bizet (E5a, prelude to l’Arlèsienne) where the melody begins on the half bar, where I nearly rebarred the music so that the downbeats fell on ‘one’, because it sounds like ‘one’. But then I realised that this made the cadence land in the middle of the bar, and that looked wrong. So I ended up having to just write a big ‘1’ underneath the half bar in the music, so that there wouldn’t be any arguments in the studio. I can’t say for sure what the answer is, but in the case of l’Arlèsienneit seems to me that this is a clear case of French compound metre  – the point is to get that final cadence on a downbeat (just as many French words are end-accented), and not make such a big deal about the metrical accents in between: it’s long jump, rather than hurdles.  Otello, I’m inclined to think is somewhere between French and Italian compound, in Rothstein’s terms, because you could rewrite the beginning in 2/4, but when the long legato melody that comes in in the middle, it begins with almost an entire bar anacrusis, and it cadences on the downbeat. If you rewrote this in 2/4, you’d lose those long lines. It’s precisely the ambiguity of this music that I suspect makes it so effective for fondus, because you never got a strong sense of either up or down, it’s in a constant state of fluid tension.

So, 15 years after I started teaching, and now that I have stopped teaching, I finally know something about compound metre that makes sense. Unfortunately, I don’t think we’ll ever escape the tendency on teaching courses to reduce knowledge about time signature to a catechism of partial truths, like the notion that there is a fixed, categorical difference between 2/4 and 4/4, or that 6/8 always sounds different to 3/4.  The answer to the question “What’s the difference between a 2/4 and a 4/4” for the moment will have to remain “What does your teacher say you have to say it is to pass the exam?”

If you’re a music theorist (like Mirka, Rothstein or Caplin) and you’re reading this thinking “Why on earth is someone who doesn’t understand all this, trying to teach music to ballet teachers?” then you’ve got a good point. Hands up, people like me shouldn’t be trying to teach time signature when they don’t understand it themselves, but they do.  Or maybe we shouldn’t be trying to teach about time signature at all, if in fact it’s so darn complicated that you need a PhD to understand it properly.  Or maybe we should just stick to basic ‘facts’, and not get into this kind of detail. But you can’t do that with dancers, because they’ll ask the kind of awkward questions that lead you straight back into complex theory of metre. And that, roughly speaking, is one of the things that got me into this PhD.


Caplin, W.E. 2002 Theories of musical rhythm in the eighteenth and nineteenth centuries, In T. Christensen (ed.) Cambridge History of Music Theory. Cambridge: Cambridge University Press, 657–694.

Mirka, D. 2008 Metre, phrase structure and manipulations of musical beginnings In D. Mirka & K. Agawu (eds.) Communication in eighteenth-century music. Cambridge: Cambridge University Press, 83–111.

Rothstein, W. 2008 National metrical types in music of the eighteenth and early nineteenth centuries In D. Mirka & K. Agawu (eds.) Communication in eighteenth-century music. Cambridge UK; New York: Cambridge University Press, 112–159.

See also:

Rothstein, W. 2011. Metrical Theory and Verdi’s Midcentury Operas. Dutch Journal of Music Theory, Vol. 16 No. 2, pp 93-111.  Available online at