Using dots, ties and tuplets of standard note values (whole, half, quarter, etc.), are there values that cannot be represented?
Examples:
what rational numbers are not in this list?
Update:
Said in a general form: what n/m cannot be represented?
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:
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:
Ties allow you to add any place values, regardless of if they are adjacent:
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:
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).
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.
One can approximate irrational note values rather easily in standard notation. For example, a note with the value Sqrt(2)/2 can be approximated by a fraction like 12/17 (larger numbers in the fraction make the approximation better and smaller ones make it less accurate.) The theory of continued fractions explains how it works. The error for this approximation is less than 1/289. It's always less than 1/denominator squared.
One then writes a 17-plet (17 eighths in the space of a half note for example) then groups 12 of them together (dotted half would do). Of course, another irrational value would be necessary to fill out the measure (should the piece be written with barlines.)
