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The sequence (13/27, 1/27, 13/27) sum to a whole note, and it derives (presumably) from triplets, while the sequence (4/81, 259/1296, 1/1296) sum to a quarter, and may derive from triplets (the first one) and sextuplets (the last two). ('Derive' in the sense of being the result of such tuplet plus something else afterwards.)

While tuplets are typically exemplified by the same value repeated a number of times, like here:

enter image description here

the proposed sequences contain fairly different combinations.

  • Why would you want such durations? They look absolutely bizarre. You may struggle to get a note as short as 1/1296 in any normal notation, but if you can, it's possible to assemble all sorts of strange notes using a series of ties. – Simon B Apr 4 '16 at 20:09
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    @SimonB The point is not 'why', but 'how'. The question is tagged 'theory' for a good reason. – nightcod3r Apr 4 '16 at 20:53
  • Well, I can't quite get the math to work, so I'm calling it a night, without an answer... Sorry. The only thing I can say at this point has largely been said already by Dom, and the part I'm trying to figure out is how to explain using regular time to make this more legible. I'm having a hard time with the large denominators. High-rep users might be able to see my partial answer. – Josiah Apr 5 '16 at 0:21
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If that is what you want to do, you wouldn't use standard notation at all. It's ok to use other forms of representing music like graphic notation or even make your own when it doesn't fit in the system. You could theoretically represent it in standard notation, but I would be very hard to comprehend and write.

The way to represent it on standard notation wouldn't fit at least for the second one as you need to figure out where the lowest unit of the tuple falls between (16th and 32nd for the X/27 and 1024th and 2048th for the X/1269) and use ties to represent them.

I did a quick example of the 27th note example. First I took 27 32nd notes an marked them as a tuple and tied the notes to the correct value like so:

enter image description here

Then I started to reduce the ties by turning pairs of 32nd notes into 16th notes to give:

enter image description here

I kept doing this until there were no more pairs that could be grouped and ended up with what you see here:

enter image description here

The notation was not made for this which is why it's not easy to represent or comprehend in standard notion. You would never expect a musician to be able to sight read this as it's way too complicated and the 1269th notes are even more confusing and making them even more time consuming.

As a side note, the primes of basing everything off of a whole note is very faulty as it only fill a measure in 4/4. You would not use it in 5/4, 6/8, 7/8, 2/4 ect.

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    May be dotted-quarters to the right and left could make it look even simpler... – nightcod3r Apr 4 '16 at 22:36
  • @nightcod3r while it will increase readability, it will not reduce complexity which is the point of this post. A musician will never play this right the first time and some may never get it until they hear it. Don't forget the point of notation is to convey your music to others. It'd be much easier to generate this in a DAW or midi file and play it for someone who then can then replicate it. – Dom Apr 4 '16 at 22:45
  • For the 27th notes, I'd rebar the measure into 9/8, which should simplify things greatly. – cjm Apr 5 '16 at 6:08
  • Comments are not for extended discussion; this conversation has been moved to chat. – Matthew Read Apr 5 '16 at 21:11
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@Dom has showed you how to work out a solution in general. I want to add a bit of detail to his answer that seems to have gotten lost.

Namely: musical note durations don't occur in a vacuum. The human ear tends to naturally group them into a hierarchy of equally-spaced beats. And while the notation may be general enough to allow such arbitrary fractions, they will not be heard that way. Indeed, the ear is pretty bad about precisely measuring the lengths of times relative to each other, unless they have a simple relationship to each other. In fact, the usual subdivision at each level of this rhythmic hierarchy is 2-4 sub-beats per beat. Trying to fit in more usually causes the beat to mentally subdivide. Five beats is often heard as a longer group of 3 beats followed by a shorter group of 2, six beats is two equal groups of three, and seven is oft treated as a group of 4 followed by a group of 3.

In the case of your 27-lets, since 27 = 33 you have a three-fold division into threes: three groups of three-groups-of-three. If this is an essential feature of the music (breaking into 27 beats) then you'll probably want to use a nine-beat measure as a starting place (there are time signatures that have nine beats in them). For example, in a 9/8 time signature, you would only need a single set of triplets over 16th notes in order to express this figure, and I dare say it would almost be comprehensible to a normal musician.

If, on the other hand, the number 27 has no special significance, and the only idea that you are trying to convey is two equally long notes divided by a very quick note, there is a much cleaner solution, which involves using a grace note between two half notes. In this case, the actual sequence of durations would be (0.5 - δ, δ, 0.5) where you have an arbitrary short value δ < 1/16. I played both of these approaches in my notation editor, and, with no other context, they sounded identical to my ears.

enter image description here

To expand this briefly to 81, you have four levels of a hierarchy dividing into three at each level (81 = 34). Once again, we decide to use a measure with nine beats. Each of these beats then divides into a nontuplet.

Say "diddly-diddly-diddly" as fast as possible. That is one nontuplet, representing one beat of music. Now repeat that a total of nine times (tap out an even beat for every three diddlies). That is one measure of nine beats, each divided into nontuplets.

enter image description here

Note that using this notation, I am not dividing a whole note into 4 quarters, but rather dividing the entire measure into nine quarters. Because of this, the total of the sum that you created would not add to a quarter note, but would add to 9/4 = 2.25 quarter notes.

Dividing into 1296 = 34 * 24 would require breaking one of the nontuplets into sixteenths, which strains all credibility of meaningfulness. Instead of having four levels of rhythmic hierarchy (which is already a lot), you now have eight levels! That's too deep for comprehension, even aside from the fact that the note are too short to have that level of precision.

  • Brilliant. Just two questions: (1) why not using a time signature of 27/8? i.e., 9/8 has been used before, perhaps 27/8 hasn't, but theoretically it's a valid signature; and (2) any good reference on the perceptual capabilities of the human auditory cortex?, (I'm familiar with neuroscience, that wouldn't be a problem). – nightcod3r Apr 5 '16 at 13:53
  • (1) I wanted to stick with a relatively common musical case, although in theory, you could do that. For those kinds of corner cases, you may be interested in: en.wikipedia.org/wiki/… (2) Unfortunately, no[citation needed]. – Caleb Hines Apr 5 '16 at 14:16
  • My only concern about (2) is whether it is on the listening side (that we hierarchically group sounds), or may be it's on the visual/performance capabilities of the interpreter (that the sounds get grouped for the sake of execution). Collecting some citations on that point would make the answer just perfect. I'll raise the question if they don't show up here. – nightcod3r Apr 5 '16 at 14:33
  • I'd actually recommend asking it as a new question! (That way, someone more qualified to speak to music cognition than I can answer it) – Caleb Hines Apr 5 '16 at 14:46
  • More on this thread here. – nightcod3r Apr 5 '16 at 17:33

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