Advanced dotting...

In my notation seminar, we were talking last Friday about "Crumb dots". These are George Crumb's idiosyncratic notation where dots are placed on either side of a note. Such as . ♩ . which is worth the length of a quarter note tied to a sixteenth note. I think the logic is that it's a double-dotted quarter note where the second dot instead subtracts instead of adds.  In any case, it's worth 1.25 quarter notes in length.

We also talked in the seminar earlier in the semester about medieval "dot groups."  These are not symbols found in actual medieval music, but things that are really useful for transcribing medieval music, where 9/8, 9/16, 9/4, 9/2, etc. are commonly implied meters.  There is no single note that can fill up a measure of 9/X where X is a power of two.  So we tend to use things such as a dotted quarter note tied to a dotted eighth note ( ♩. ⁀♪. ) for 9/16, and so on. But consider that that figure comprises two notes, the second of which is half the length of the first, and they are tied together.  That is the basic definition of a dotted note: a dotted half note is a half note tied to the note half the length of a half note, or a quarter.  So what ♩. ⁀♪. needs is a way of "dotting" a dotted quarter note, or  ( ♩. ) . ––note that this note has a different length than a double-dotted quarter note (which is worth 7/16, not 9/16) and this "dotted-dotted-quarter note" is worth more than a half note.  When it is used, it is usually written with two dots vertically aligned: ♩:

Formulas and Extensions to negative numbers

Working out the length of notes with multiple dots can be hard work.  In this screen capture from the "Reimagined" Battlestar Galactica, Kara "Starbuck" Thrace doesn't get it right even though the fate of humanity rests on her deciphering the secrets of a mysterious melody:

The third line should have two 32nd notes, not 16ths, and the fourth line should have triplet 64th notes.

So, let's get back to the basics. How long is a note with d dots? Let's see.  If a quarter note gets the beat then:

♩ = 1 beat
♩. = 1.5 beats
♩.. = 1.75 beats
♩... = 1.875 beats

and so on to

lim (d _⃗ ∞ ) = 2

So, what's the pattern?

1
1 + 1/2
1 + 3/4
1 + 7/8
1 + (n – 1)/n

Where n = 2^d

Given that, we can work out that –1 dots is:  1 + (2^–1– 1) / 2^–1 or 1 + (–½) / ½ or 1 – 1 or 0. So a note with negative 1 dots has no length.

For d = –2 we get –2 beats.  I don't know what that would mean.  I suppose play the note backwards beginning two beats before it should start?  (David Lewin has written about the uselessness of negative note lengths).

Just to round it all out:

d = –3, implies duration –6 beats.
d = –4, implies duration –14 beats.

d = –5, implies duration –30 beats.

or for any negative d, duration = – (2^|d| – 2).

[This section added 15 August 2015]:
Note also that the fraction of the preceding duration added by each dot is as follows:

One dot: 3:2
Two dots: 7:6
Three dots: 15:14
Four dots: 31:30

So the numerators follow the series of Mersenne numbers while the denominators are always one less.  The general formula this is:

(2^{numDots + 1}– 1) : (2^{numDots + 1}– 2)

Look what happens though if d (numDots) is zero:

(2¹– 1) : (2¹– 2)  = 1 : 0

Thus a note with no dots is 1:0 or infinitely longer than what preceded it in the dot series. One way of looking at this is that a plain note receives all of its duration not from its shape or number of flags, but from the fact that it has no dots; if it lacked no dots then it would have no duration. :-)  I told this theory to Noam Elkies and he said that I was dotty.

Anyhow, back to other weird numbers of dots.

Non-integer dots

Fractional dots create irrational durations, a more useful concept than negative durations. Conlon Nancarrow^[*]  has used these durations in some of his pieces, such as one where two canonic lines are set against each other with tempi in the ratio of √2 to 2.  After the first note of the piece, no two notes will ever coincide no matter how long the piece lasts.   There's no way to notate anything like this in normal notation, but with fractional dots, we can do this.  If n = 2^d and beats = 1 + (n – 1)/n then a note with 1/2 dot has n =  2½ or √2, so a half-dotted quarter lasts 1 + (√2 - 1)/√2 beats, or approximately 1.29 beats.  With this knowledge we can create many irrational lengths for notes just by giving fractional dots. (Though not all irrational lengths.  For instance Nancarrow's "Transcendental Etude" puts two canonic lines against each other with tempi in the ratio of e to π.  Fractional dots could not notate this).

But before trying to figure those out (an exercise for the reader...), there are still lots of rational number lengths that cannot be notated with fractional (i.e. rational number) dots.  Is there a way to get any other rational length instead of just the ones based on powers of 2 that are standard? Can we, for instance, notate a triplet only with partial or negative or partial negative dots? Yes, we can, by making the number of dots a logarithm!  For instance, to get a note that is 5/3 the basic length, we set the number of dots to log₂(3). That's a triplet half note. To get a triplet quarter, we subtract two dots from that, or (log₂(3) – 2) dots. Or for a quintuplet, use (log₂(5) – log₂(3) – 1) dots.

Trying to come up with a way of notating this makes me understand why someone decided that a triplet mark was simpler. But since it is not too hard to prove that any nested tuplet can be written as a single tuplet, we find that any rational duration can be notated as a single note with different numbers of, possibly irrational, dots.  Something for the new complexity school to pursue next.  Or hopefully not.

[*] an earlier version of this post referred to "Colon Nancarrow" who is neither a piece of punctuation nor a part of the digestive tract.  He is also not the same as Loren Nancarrow, a really great weatherman in San Diego, that I've always wanted an excuse to make a shoutout to in a Conlon Nancarrow post.