Readable tupletsDiferent 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 dont prevent the user from creating tuplets that are ambiguous or even totally misleading, just as they dont prevent the user from bad notation in general the more fexible 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 ruleThe frst 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 defnition 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 isthat? Lets look at this example: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:It could again be noted that although the frst example is very misleading and would take some attention to fgure out, it would not result in any error or warning from LilyPond. And I dont 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) isnt allowed. Lets again look at an example:Here again it looks wrong; it seems very strange to put as many as fve 8ths in the duration of a quarter and we can just as easily as inthe previous example see the rationale for the rule. Here the duration of the fve 8ths should take up a half, as in this example:To conclude before moving to the next rule, this rule should be fairly easy to follow, and its actually what I would prefer and even recommend.The nearness ruleNow for the second rule. Lets begin with this example:This follows the frst rule. But in the same space as the seven tuplet 8ths we could ft eight 16ths and only 4 regular 8ths. Isnt it then more intuitive to use 16th notes for this instead?No, I dont really think so and this wouldnt be allowed by the frst 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 its important to note that this second rule is less defnite than the frst. It isnt hard to fnd examples where you couldnt 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 defnitely rule it out. On the other hand, even if this practice of 3/2 is not fully sanctioned by the second rule it is defnitely not contradictory to it.Maybe its too hard on this practice of seeking the nearness relation to require it to give a defnite answer in every instance. Maybe its even misleading to call it a rule. I guess its 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 fexible than to write music with pen and paper. But with tuplets at least I think the software invokes the idea that the fexibility which is allowed by the program is also a fexibility 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 isnt clear in detail on how to notate tuplets. Isnt it then more important to make the notation distinct and unambiguous (for example by clearly stating the ratio) than to follow these competing rules?Lets go back to the previously discussed ratio 3/4, heres the example again with explicit ratio and additionally the explicit note value of the denominator:Now this notation should be clear its three 8ths replacing four 8ths; although unusual, its now also unambiguous.If you have good reasons to create a non-standard tuplet its 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.Its equally easy to write 3/4 as 3/2 but the former will require a clarifcation and hence a greater efort 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 efort 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).