All notation programs aim of course at being as powerful a tool as possible to help composers, arrangers and engravers rather than hold them back. I’d like to explore an area where I think you have to use these powers with care – I’m talking about tuplets.

This article is written from the perspective of writing music in LilyPond (this is after all the LilyPond blog) but my aim is that anyone interested in music notation should be able to follow this.

## Tuplets in LilyPond

Most readers of this blog would of course already know that as of LilyPond 2.17.x the syntax for writing a tuplet is this:

`\tuplet fraction { music }`

Those who aren’t familiar with LilyPond will note that it is text-based, and that ‘writing’ music here is more literal than in more graphical programs. Here’s an example with an ordinary triplet:

\tuplet 3/2 { g8 a b }

As a side note I personally think that tuplets are an area where the text-based model really is an advantage. I’d like to illustrate this by showing how easy it is to make a nested tuplet. Here is the last triplet quarter substituted with another triplet:

\tuplet 3/2 { g4 a \tuplet 3/2 { g8 a b } } g2

In my experience though the real advantage comes when you have to edit an already created tuplet.

## Readable tuplets?

Different notation programs may have small restrictions for what is possible but I think it can be assumed that the aim is to make notation as easy as writing with pen and paper, so in theory the only impediments are limited imagination and rhythmical miscalculations. But where this can make life easy for the person who writes the music it can make life hard on the person who is to perform it. When we are talking about nested tuplets that is of course very complex. But also single tuplets can be somewhat overpowering.

It should be noted that LilyPond and other notation programs don’t prevent the user from creating tuplets that are ambiguous or even totally misleading, just as they don’t prevent the user from bad notation in general – the more flexible and powerful the program, the more responsibility for the user. Hence you should always seek to follow notational practices to facilitate the interpretation of the score.

Unfortunately for tuplets the situation is even more complex because of there being two conventions regarding their notation that are somewhat competing:

## The mathematical rule

The first rule regarding tuplet notation is slightly mathematical. It says that if you calculate the tuplet fraction the result should always be between 1 and 2. (For those of you who would like an even more mathematical definition this could be stated as: `1 < Tf < 2`

where `Tf`

is the tuplet fraction.)

To further explain this rule and its rationale I would like to start with the fraction 3/4 (= 0.75 < 1), which is disallowed by the rule. Why is that? Let’s look at this example:

\tuplet 3/4 { g8 a b } d4 e

We have three 8ths in the space of four 8ths and then we have 2 ordinary quarters just like we prescribed. But this looks completely wrong. And that is of course because we have a very strong convention that three 8ths as a tuplet/triplet equals a quarter and not a half. Thus we normally *would* obey the mathematical rule, in the case of triplets at least. We would instead use the fraction 3/2, which is allowed by the rule (1 < 3/2 = 1.5 < 2). It would look like this:

\tuplet 3/2 { g4 a b } d4 e

It could again be noted that although the first example is very misleading and would take some attention to figure out, it would not result in any error or warning from LilyPond. And I don’t think it should, it should be up to the user to think of readability and proper notation. Furthermore a composer could have a very good reason to choose a non-standard notation and should be allowed to do so.

What happens on the other end, when the fraction is greater than 2? With this rule e.g. the fraction 5/2 (= 2.5 > 2) isn’t allowed. Let’s again look at an example:

\tuplet 5/2 { g8 a b c d } c2.

Here again it looks wrong; it seems very strange to put as many as five 8ths in the duration of a quarter and we can just as easily as in the previous example see the rationale for the rule. Here the duration of the five 8ths *should* take up a half, as in this example:

\tuplet 5/4 { g8 a b c d } c2

To conclude before moving to the next rule, this rule should be fairly easy to follow, and it’s actually what I would prefer and even recommend.

## The nearness rule

Now for the second rule. Let’s begin with this example:

\tuplet 7/4 { g8 a b c d e f } e2

This follows the first rule. But in the same space as the seven tuplet 8ths we could fit eight 16ths and only 4 regular 8ths. Isn’t it then more intuitive to use 16th notes for this instead?

\tuplet 7/8 { g16 a b c d e f } e2

No, I don’t really think so and this wouldn’t be allowed by the first rule, because the fraction is less than 1. But the supporters of the second rule would say that this is the way to do it. The second rule could roughly be stated as saying simply that tuplets should mimic the regular note values as closely as possible, thus I call it the “nearness rule”.

I think it’s important to note that this second rule is less definite than the first. It isn’t hard to find examples where you couldn’t absolutely settle what the rule would recommend. Take for instance the previous example of 3/4 — this is as close to regular note values as 3/2 but the general convention would definitely rule it out. On the other hand, even if this practice of 3/2 is not fully sanctioned by the second rule it is definitely not contradictory to it.

Maybe it’s too hard on this practice of seeking the nearness relation to require it to give a definite answer in every instance. Maybe it’s even misleading to call it a rule. I guess it’s on a more intuitive level when you choose the notation which is nearest to the standard division.

## Must I follow a rule?

To write music in a notational software is of course not more flexible than to write music with pen and paper. But with tuplets at least I think the software invokes the idea that the flexibility which is allowed by the program is also a flexibility I have as a creator. When writing with pen and paper you get a better sense of having to create the boundaries yourself. (This is my own experience anyway.)

I think I have shown that it is possible to construct examples which are fully possible to conceive and create but which run counter to our normal idea of how tuplets should be notated and interpreted. But I have also explained that the consensus isn’t clear in detail on how to notate tuplets. Isn’t it then more important to make the notation distinct and unambiguous (for example by clearly stating the ratio) than to follow these competing rules?

Let’s go back to the previously discussed ratio 3/4, here’s the example again with explicit ratio and additionally the explicit note value of the denominator:

\once \override TupletNumber.text = #(tuplet-number::append-note-wrapper tuplet-number::calc-fraction-text "8") \tuplet 3/4 { g8 a b } d4 e

Now this notation should be clear – it’s three 8ths replacing four 8ths; although unusual, it’s now also unambiguous.

If you have good reasons to create a non-standard tuplet it’s of course advisable to make it clear in this way. But I think the ability to make yourself clear should not be taken as an excuse for arbitrarily breaking the conventions that after all exist.

It’s equally easy to write 3/4 as 3/2 but the former will require a clarification and hence a greater effort from both you and your reader. So the choice should be obvious for most cases.

Again, this is not an argument against using the more explicit notation of the tuplet ratio to make it clearer. There are of course cases where it instead decreases the effort of the reader. It could be noted that Gould in *Behind Bars* recommends indicating ratios in full when using the nearness rule in a way that contradicts the mathematical rule (which she, like me, thinks is clearer).

## Conclusions

Creating tuplets in LilyPond and other notation software is very flexible and powerful, but you as a user should be careful not to misuse these powers. We have seen that currently there are two conventions on how to notate tuplets and that LilyPond in these examples isn’t taking a side between these two views, and as long as these two conventions exists side by side she should not do that. But you as a music writer should definitely take a side and only use one of the conventions at the time. Otherwise it could potentially be very confusing for the reader and performer of the music you write.

My main point here is not to restrict the use of tuplets (they are part of our musical language), but when they are used they should be notated carefully in order not to unnecessarily increase the complexity. Don’t trust the software to do the thinking for you — take those conventions that exists into account and if you want to go beyond then use the possibility of making the notation more explicit.

What do you think? Please share your thoughts in a comment!

## References:

Elaine Gould – Behind Bars (London, Faber Music Ltd, 2011)

Gardner Read – Modern Rhythmic Notation (London, Victor Gollancz Ltd, 1980)

Gilberto AgostinhoHi Peter, this was an excellent article, thanks for sharing it!

Peter BjuhrPost authorHi Gilberto, thanks!

karim haddadThanx for this very interesting article.

I have although some remarks regarding the mathematical rules. They come from where ? I looked up in the Read book and didn’t find any.

Well regarding these rules, there are examples that don’t satisfy the 1< tp < 2 .there are many tuplets in the music literature using for instance the 5:6 . It's easier to use 5:6 instead of 10:6 (or 5:3).

In the case of septolets, in the traditional theory (dubois) they use 7:4 instead of 7:8 because it is just before the 32nd notes (quarter division by eight). However 7:8 is logical depending on the musical context.

Best

Karim

Peter BjuhrPost authorThanks for your comment!

I know that many readers here have a copy of Behind Bars. You can find the topic of tuplet Note-values discussed on pp 203-204. I did reread Read as a research for this article, but I don’t have a copy and didn’t note the exact pages. I’ll check it up!

It should be noted that the names of the two rules are my invention and also that I describe them in my own words.

The examples you refer to with 5:6 and 7:8 seem to me to be examples of using the ‘nearness rule’ . As you also argue there are many examples of using the nearness rule in the music literature. If I gave the impression of the opposite it wasn’t intended!

Peter BjuhrPost authorIn Read – Modern Rhythmic Notation this is discussed in chapter 3.

Pierre-Luc GauthierThank you Peter for this great article.

Notation is indeed making the abstract as clear and intuitive as possible, whatever the rule.

Peter BjuhrPost authorThanks for your encouragement!

Urs LiskaJust recently I had published a post that can be taken as an example for multiple of your arguments: http://lilypondblog.org/2014/05/independent-meters/

I think this demonstrates how extremely powerful LilyPond (and its text input) is with regard to creating complex and nested tuplets. But it also shows that simply because it’s easily doable it doesn’t have to be a good idea to actually write it down. My exmples are completely mocked up to demonstrate the ease and perfection of LilyPond, but it could now be tempting to compose such complex music just because it’s easy to notate.

On the other hand I hope you didn’t mean to suggest that writing complex music is a dubious thing in itself? I know many composers who feel the need to express themselves in complex manners, and I don’t see any problem with that. Having powerful tools at hand is definitely a good thing for them.

I think it’s not a good idea to let your creativity as a composer be influenced too easily by what your tools offer you. But I find it much more disturbing that with the advent of the ubiquituous computer notation many composers let themselves rather be

restrictedby the functionality of their tools. I mean, with pen and paper you have significantly more options to invent creative notation – and music – than when immediately entering music into a notation program. I think if you have invented something and then struggle to convince the computer to give it an appropriate representation it’s awkward, but it’s OK. If you decidenotto write down an idea because your notation program doesn’t offer a convenient way to let you do it – this is what I’d consider an irritating development. I admit that I regret that a growing percentage of music composed nowadays tends to go backwards towards music that can be easily notated with computers.Peter BjuhrPost authorThanks for your comment, Urs!

About complexity, I think it would be easy to conclude that the more complex tuplets the more important to make them clear and readable. But I have an example where a simple triplet gets quite misleading; so I’m not sure where and if complexity will apply to this. Of course you may have a set of tuplets that you’re confident in using. But when you’re using a somewhat more non-standard tuplet I think you should always think twice. In my case as I tend to follow what I called the mathematical rule I will see if the ratio is allowed by that rule.

When speaking about complex music in general I think it should be remembered that music notation is a form of communication with the performers of the music who in turn will communicate the music to their audience. When the complexity of the music stretches this communication I think it can be a good thing. But when it breaks the communication I must conclude that it is a bad thing.

I know this isn’t very instructive for the more specific case, but sometimes you have to look at the wider picture.

About not getting too influenced by your tools I totally agree with you. But I fear it’s hard to avoid. I think I’ve always sought the straightest path from my imagination to the finished score.