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Pretty sure the answer is two beats. Am I right?

But that's a side issue, interesting though it is.

What are the benefits of using dot notation to specify a note length over notating the note length using several tied noteheads?

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    Seems like this belongs on Math.SE. I think it’s a geometric series that sums to 2, yes Oct 28, 2021 at 12:32
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    I know this is music theory, but is that taking it too far..?
    – Tim
    Oct 28, 2021 at 13:07
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    I’m voting to close this question because it is a hypothetical thought exercise ultimately unrelated to music. Oct 28, 2021 at 13:51
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    Ah, Achilles & Tortoise, not tortoise & hare - one of Zeno's paradoxes en.wikipedia.org/wiki/…
    – Tetsujin
    Oct 28, 2021 at 14:09
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    Since I voted to close, there have been several edits. The latest, "What are the benefits ...", is a perfectly valid question. But seems to me, a completely separate question from "how long is an infinite-dots note." I would suggest editing to remove the original question and fully convert to the new one. (One of my objections to the original is it's a "how many angels can dance on the head of a pin" problem: it's impossible to write infinite dots within our universe. It's a "how long would a performance of 4'33" last at the speed of light" sort of question.) Oct 29, 2021 at 17:29

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What are the benefits of using dot notation to specify a note length over notating the note length using several tied noteheads?

It's easier to write: fewer noteheads, stems, and possibly flags, and no ties.

It's easier to read: no need to differentiate between a tie and a slur.

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  • I've actually read that one drawback is that multi-hit notation is harder to read by virtue of being rarer notation, so it's less likely to be understood.
    – Dekkadeci
    Oct 29, 2021 at 12:28
  • Since this is all theoretical and unlikely to have any musical significance in practice anyway, I'd like to point out (as I'm sure you know, phoog) that neither dots nor several tied noteheads can cover all possible note lengths: for instance, a note length that is 1/3 of a beat longer. Oct 29, 2021 at 13:11
  • @Dekkadeci by multi-hit do you mean ties? I am sure that the unfamiliarity is a big part of it, but I suppose it's not the only reason.
    – phoog
    Oct 29, 2021 at 14:06
  • @ScottWallace for that you need either compound meter or the triplet sign, possibly in addition to tied notes. Interestingly, the dot arose in the context of triple subdivisions, indicating that the note decorated with the dot is "perfect," consisting of three subdivisions, when one might otherwise read it as "imperfect," having two subdivisions and 2/3 the length of a perfect note.
    – phoog
    Oct 29, 2021 at 14:10
  • @phoog - Sorry, by multi-hit, I mean multi-dot.
    – Dekkadeci
    Oct 29, 2021 at 14:34
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I'm going to challenge the frame of your question. Music notation is, above all else, a tool used to communicate musical ideas. Math notation is similar, but math has an idea of rigor that isn't present is music. Correct notation is easy to read first and foremost, even if it doesn't perfectly reflect what is being played. It's better to indicate that a piece is swung, leave it at that, and write eighth notes everywhere, for example, than it is to add triplets that show precisely how long each note should be played.

Given this, a note with an infinite number of dots wouldn't ever be written down. You could notate it if you really wanted to with something like a note followed by a bunch of dots and an infinity sign. But that would always be much harder to read than a half note, so it's not a useful communication tool and fails the most important function of notation.

Likewise, that's why we use dotted notes instead of tied noteheads in some situations. Depending on the context, dotted notes can simply be easier to read.

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    "triplets that show precisely how long each note should be played": even triplets don't capture swing precisely.
    – phoog
    Oct 30, 2021 at 12:00
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Pretty sure the answer is two beats. Am I right?

Yes. The length of the notes with increasing number of dots is described by geometric series. https://en.wikipedia.org/wiki/Geometric_series In the page we find formula showing that for infinite series:

1 + 1/2 + 1/4 + ... = 1 + r + r² + ... = (for -1 < r < 1, which is fulfilled by r=1/2) 1/(1-r) = 2.

Of course in practice writing infinite number of dots seems non-realistic, but also there are limits of hearing, as well as performer capabilities, imposing how many dots would make a note indistinguishable from a twice longer note; it would happen with a finite number of dots.

What are the benefits of using dot notation to specify a note length over notating the note length using several tied noteheads?

Dots were used to indicate a dotted rhythm (i.e. pairing of a longer and a shorter note, like dotted eight note and a sixteen note). That's a very characteristic and significant rhythm, which could be compared to swing rhythm.

Just like swing is not always performed with popular triplet division, dotted rhythms are also sometimes not performed with mathematical precision, except while swing rhythm is rather softened, dotted rhythm is sometimes played sharper than written. Also, unlike swing rhythm, dotted rhythm very often coexists with straight rhythm.

It seems therefore reasonable to have a dedicated notation for dotted rhythm, to distinguish it from any other arbitrary rhythmic structure.

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    I think dotted notation is also much more readable than ties.
    – Theodore
    Oct 29, 2021 at 19:57

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