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Using dots, ties and tuplets of standard note values (whole, half, quarter, etc.), are there values that cannot be represented?

Examples:

  • 1/8 is an eighth
  • 7/16 is a two-dotted-quarter
  • 15/8 is a three-dotted whole
  • 1/3 results from a triplet of a whole
  • 11/24 is a tie of 1/3 and 1/8
    ...

what rational numbers are not in this list?

Update:
Said in a general form: what n/m cannot be represented?

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    I'm not sure I really understand what is going on here. Can you maybe do something to make this question more clear. – Neil Meyer Apr 4 '16 at 9:27
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    Irrational portions of a beat or measure cannot be represented, but I don't see how they could reasonably be accurately played. I.e., the cardinality of the set of possible note values is countably infinite. – Todd Wilcox Apr 4 '16 at 10:14
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    @ToddWilcox Aah, you can do anything with enough practice :) – topo morto Apr 4 '16 at 10:21
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    @topomorto too bad you don't have an infinite amount of time to practise... – leftaroundabout Apr 4 '16 at 11:03
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    Is this really some kind of math question in disguise? I believe the answer depends on how general tuplets you allow. For example, if you want a note with the mathematical duration 1/97 (one ninety-seventh), is it allowed to create a tuplet embracing 97 notes, and write the number "97" above it? – Jeppe Stig Nielsen Apr 4 '16 at 12:49
24

Math Alert! Also, I will be very much discussing what is theoretically possible, not necessarily what is convenient for the poor musician.

The notation for musical rhythm is more or less equivalent to writing a fractional number in binary (e.g. using a radix point). Each note type represents a different place value. For example:

  • Whole note = 1.02
  • Half note = 0.12
  • Quarter note = 0.012
  • Eighth note = 0.0012
  • and so forth...

Since you can (theoretically) add an arbitrary number of flags to a note head, you can carry this out as long as you please.

Dotted notes (single or double) represent adding two (or three) adjacent places:

  • Dotted half note = 0.102 + 0.012 = 0.112

Ties allow you to add any place values, regardless of if they are adjacent:

  • Half note tied to eighth note = 0.1002 + 0.0012 = 0.1012

Since, given enough place values, any numerical value can be represented in binary (i.e. a sum of the value of each place), the answer to your question is technically, no, there are no values that cannot be represented. However, this is not practical, since many numbers would require an infinite series of digits to represent. These can be broken into two classes of numbers (as you allude to in your question) -- rational and irrational.

Tuplets can be used to handle the rational case. If ties represent addition, then an n-tuplet represents division of a time unit into n equally-sized portions. Any rational number can be written as a fraction, m/n, which has the same value as 1/n added to itself m times. This number can be represented musically by tying m copies of an n-tuplet. For example:

  • 31/87 can be represented by tying together 31 notes that are 87-lets.

However, in the irrational case, the value cannot be represented as a ratio or fraction, so tuplets don't work. This leaves you back to writing out an infinite series of tied notes, which cannot be improved upon because irrational numbers require an infinite, non-repeating representation.


The final question that must be considered is the precision required. While theoretically, you would require an infinite number of notes, with an increasing number of flags on the staff, the human ear is limited in how precisely it can detect rhythms. At a certain point, rounding off becomes inevitable. For example, even NASA requires only 15 digits of pi (in base ten) in order to calculate the positions of interplanetary probes. Digits beyond this are insignificant enough as to not matter. Similarly, any number inside a computer must be rounded off to some finite approximation, simply because of limited memory.

So if you limit yourself to a basic unit of precision (perhaps a 128th note?), then you can represent any multiple of that basic unit (using ties), as well as any rational number (using tuplets).

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    I didn't use "floating point" because that implies to me an additional exponential scaling factor that doesn't come into play here. However, I also forgot that decimal implied base ten. I've reworded my answer to remove all instances of that word. Thanks for keeping my terminology honest! – Caleb Hines Apr 4 '16 at 14:36
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    Note that floating point is more properly a computer science term, not a math term. It does get used a lot in many different contexts, but not at all in pure mathematics classes or texts (in my experience). – Todd Wilcox Apr 4 '16 at 16:13
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    @ToddWilcox true, 'cos as Caleb pointed out it relates to an implementation detail... I'm still fishing for a more general term for a number representation that involves a radix point... – topo morto Apr 4 '16 at 16:53
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    @topomorto: "Real"? – Lightness Races in Orbit Apr 4 '16 at 20:34
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    @BarryTheHatchet Indeed. Or, for the non-integer portion, "fractional part": "For a positive number written in a conventional positional numeral system (such as binary or decimal), the fractional part equals the digits appearing after the radix point." -- en.wikipedia.org/wiki/Fractional_part – Caleb Hines Apr 4 '16 at 20:40
4

Notation as we know it has been around for many centuries, and although it was quite vague initially, it's sorted. Using ties, dotted notes and tuplets, as you say, means that any note duration in any time signature can be written. O.k. as Bob says, some could end up ugly and difficult to read, but if something had been deemed unwriteable, a new configuration would surely have been introduced by now.

  • You're right. I just needed a sort of proof for that assertion. – nightcod3r Apr 4 '16 at 16:49

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