Deconstructing the Rhythm Dot – The Mathematics of Dotted Notes

Updated on  August 9, 2018 to add an addendum on tupletting pairs of tied notes.

I recently needed to figure out ways to represent dotted notes so they would appear without a dot, and in the process, I found a number of ways to break down dotted notes into smaller notes. You may never need to do what I had to do, but someday one of these techniques may turn out to be useful.

DOTTED NOTES


A rhythm dot added to a note adds half the duration of the original note, and for multiple dots, each dot adds half the value of the previous dot.

(Sibelius allows up to triple-dotted notes, the maximum number of rhythm dots allowed in Finale is determined by location of the note relative to the barline, and Dorico has separate controls to allow the maximum number of rhythm dots to be different on simple beats and compound beats.)

Using a quarter note / crotchet as an example:

  • the duration of a single-dotted quarter is a quarter note plus an eighth note/quaver, or 1 + ½ = 1½ times the original duration.
  • the duration of a double-dotted quarter is a quarter note plus an eighth note/quaver, plus a 16th note/semiquaver, or 1 + ½ + ¼ = 1¾ times the original duration.
  • the duration of a triple-dotted quarter is a quarter note plus an eighth note/quaver, plus a 16th note/semiquaver, plus a 32nd note/demisemiquaver, or 1 + ½ + ¼ + ⅛ = 1⅞ times the original duration.

As we go along it will be convenient to represent the fractions as a ratio, so we will have:

  • Single = 1 + ½ = 1½ = 3/2
  • Double  =  1 + ½ + ¼ = 1¾ = 7/4
  • Triple = 1+ ½ + ¼ + ⅛ = 1⅞ = 15/8

BREAKING DOWN DOTTED NOTES


Here are several ways to break dotted notes into smaller units. In the following examples I will refer to the value of the note without a rhythm dot as the base duration, and the value of the dotted portion as the dot duration. For a double-dotted half note / minim:

  • The base duration is a half note.
  • The dot duration is the value of the 2 dots, a quarter note plus an 8th note, or a dotted quarter.

I could not find any names for these processes, so I have made up these names:

  • Unfolding a dotted note will break it down into a number of tied notes, all with the same duration.
  • Deconstructing a dotted note breaks the note unto a series of tied notes, each of which has the duration represented by the note or rhythm dot at its position, so that in the end none of the notes are dotted
  • Splitting breaks the note into 2 tied notes, one with the base duration, and one with the dot duration. Any dotted note can be split in this way.
  • Tupletting a dotted note replaces the dotted note with a tuplet containing one or more undotted notes. In some simple cases, like replacing a single-dotted quarter note with a duplet of 2 quarter notes, this might be something that you would present to a performer.

I developed this technique because I needed to hide a tied dotted note. It was very useful to be able to make a tuplet of undotted notes and combine those notes into a single undotted note, which I could then hide without making its tie disappear. This is pretty much of a special case. It is very unlikely that you would want a performer to have to figure out the meaning of an 8:15 tuplet with a single half note in it, played at the duration of a triple-dotted half note!

The examples should make this description intelligible.

In Sibelius, all of these these processes except tupletting can be produced using the Divide Durations plugin, and I used that plugin to produce these examples.

EXAMPLES OF DECONSTRUCTING DOTTED NOTES


Unfolding, Deconstructing, and Splitting:


These are relatively straightforward once you know the appropriate ratios.

When unfolding a single-dotted note, the ratio 3/2 tells you there will be 3 notes whose durations are each ½ of the base duration, so a dotted half note becomes 3 quarter notes. A double-dotted note has 7/4 times the duration of the undotted note, and will become 7 notes, each with the duration of the undotted note divided by 4.  A triple-dotted note is 15/8 times the base duration, and you get what you might expect if you are this far down the rabbit hole.

It is also useful to understand that you can always split a dotted note into one note of the base duration and one of the dot duration, and that the dot duration can always be represented as a single note, as opposed to something like a half note tied to an 8th note, which cannot be represented as a single note outside of a tuplet.

Splitting was my first approach to hiding a tied dotted note. If you split a dotted note and then reverse the order of the split notes you can hide the dotted portion. The tie from that note will also be hidden, but that is what I wanted in this case, and the second note will still display its tie. This gave me a stable tie from nowhere, but it used up a lot of space that was tricky to deal with.

Tupletting

I found it useful to look at the unfolded state for each dotted note type when working out the details of tupletting.

I had been looking for a way to represent a dotted note without the dot so I could hide notes using a notehead style. If a dotted note is hidden with a notehead style the dot is still visible. I wanted to represent the note as a tuplet that could contain a single note, and then I could hide the tuplet and the notehead and stem.

For a single-dotted note, I knew from experience that a 2:3 tuplet using half the original base duration (which is the duration of each unfolded note) would work. For a dotted half note, I could make a 2:3 duplet of quarter notes, since 2 quarter notes in the time of 3 gives the same duration as a dotted half note. I could then combine the 2 quarter notes in the duplet into a single half note, which does not have a dot and uses up less space than 2 separate notes.

Double and triple-dotted notes were trickier, but looking at the unfolding, I saw that a double-dotted note generates 7 notes, each of whose duration was ¼ of the base duration. This suggested a tuplet of the form n:7. When I realized that I wanted n to be a power of 2, so the notes could be combined without needing a dot, I tried 4:7. For the double-dotted half note, I got 4-8th notes in the tuplet, and I could then combine the 8th notes into a single half note.

I then realized that a triple-dotted note needed the tuplet to be n:15. The nearest power of 2 less than 15 is 8, so for the dotted half note I got 8-16th notes in the tuplet, and then I could combine those notes into a single half note.

Each form of the dotted note can thus be represented as a tuplet with a single note that has the base duration by choosing appropriate tuplet ratios. You can also keep as many of the notes separate as you like.

So:

  • a single-dotted note can be represented as a 2:3 tuplet of notes that are 1/2 the duration of the undotted note
  • a double-dotted note can be represented as a 4:7 tuplet of notes that are 1/4 the duration of the undotted note
  • a triple-dotted note can be represented as an 8:15 tuplet of notes that are 1/8 the duration of the undotted note

which happens to match the ratios discussed above. This is unlikely to be a coincidence.

Here is an example of using tupletting to hide a dotted note with a tie or slide and keep the tie visible.

Now you know what I know about dotted notes. I hope you will find some of this useful.

Addendum: Tupletting pairs of tied notes

 I occasionally encounter pairs of notes that cannot be represented as a single note. These are undotted notes, the second of which is no larger than ¼ the duration of the first note. Here are some examples, using a quarter note as the first note, though any undotted note can be used.Like dotted notes, these can also be represented as a single note in a tuplet. Because there are many such pairs, I will express the tuplet ratio by a calculation procedure rather than enumerating all the possibilities. Let’s call the larger note B (for Base) and the smaller note R (for Remainder). We need to calculate the ratio of the durations, and the American naming system makes that pretty easy. Let’s represent a whole note as the number 1, a half note as ½, a quarter note and ¼, and so on.

  1. Get the duration of the first note B, and the second note R as fractions, expressing B in terms of R. For example, a Whole note plus an 8th note would be 1 + 1/8, and 1 is eight 8th notes, or 8/8, so we have B + R = T(otal), or 8/8 + 1/8 = 9/8.
  2. The top number (numerator) of the fraction representing B will be the numerator for the tuplet ratio
  3. The numerator for the fraction representing T will be the bottom number (denominator) for the tuplet ratio.
  4. In the example of 8/8 + 1/8 = 9/8, the tuplet will be 8:9
  5. The unit size for the tuplet will be the second note R .

Here are some examples in addition to those in the figure above.  A Sibelius plugin would use internal Sibelius units of 256 units to a quarter note for the durations, though any consistent unit could be used:

Units: Q(uarter, E(ighth, W(hole), 64th, 128th

Q+64th  (16/64 + 1/64 = 17/64). Tuplet is 16:17, unit is 64th

E+128th (16/128 +1/128 = 17/128). Tuplet is 16:17, unit is 128th

W+8th (8/8 + 1/8 = 9/8). Tuplet is 8:9, unit is 8th

Experimentally, in Sibelius, tuplet ratios larger than 128:129  (are not accepted, so for half notes and above, some combinations will not be allowed, and will be rejected. An example of a 128:129 ratio is W + 128th, so these are not too likely to be encountered.

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