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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?

Brian THOMAS
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    Seems like this belongs on Math.SE. I think it’s a geometric series that sums to 2, yes – Todd Wilcox Oct 28 '21 at 12:32
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    I know this is music **theory**, but is that taking it too far..? – Tim Oct 28 '21 at 13:07
  • It's a bit like the tortoise & the hare problem… it would approach 2, but never actually reach it. – Tetsujin Oct 28 '21 at 13:45
  • This sounds like the most brilliant ploy ever for a composer to delay a commission. "It will be ready when it's ready, ok? I'm on the last note!" "That's what you said 10 years ago!" – Andy Bonner Oct 28 '21 at 13:50
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    I’m voting to close this question because it is a hypothetical thought exercise ultimately unrelated to music. – Andy Bonner Oct 28 '21 at 13:51
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    Ah, Achilles & Tortoise, not tortoise & hare - one of Zeno's paradoxes https://en.wikipedia.org/wiki/Zeno%27s_paradoxes#Achilles_and_the_tortoise – Tetsujin Oct 28 '21 at 14:09
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    I'm voting to reopen. Yes, indeed this question involves math, but just high school math, and it also has strong connection with music theory. I'm not convinced we should close a question just because it's interdisciplinary. Maybe it should be migrated to math, but then OP would need to explain what a dot is, which here most of us understands. I can also imagine someone else searching for a similar question, even if I don't see an immediate practical application. – user1079505 Oct 29 '21 at 00:53
  • I've voted to keep closed. To reopen, I think OP needs to make a case for why the question is useful. Ordinarily, a lack of usefulness would just be a downvote reason, but since the question is already closed, I think, for now, it should stay that way. – Aaron Oct 29 '21 at 01:20
  • While I also tend to like theoretical extremes to demonstrate concepts, I think this one is a little outside what most would consider a working extreme that show the concept. Even a note with 10 dots would stretch the bounds extremes with this concept. I do however think there is a way to ask this question that makes more sense as a whole, but with the focus on at what point the subdivisions become too small to be distinguishable notation wise. At that point, X dots on a one beat note vs the two beat note doesn't matter and the answer would be well before infinity. – Dom Oct 29 '21 at 02:11
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    ∑(n=0,∞)1/2^n=2 – Elements in Space Oct 29 '21 at 04:54
  • One way in which i found the question genuinely useful: I actually didn't know - or at least had forgotten - that you could put multiple dots after a note to change the duration in the way suggested... – Нет войне Oct 29 '21 at 07:14
  • As currently written, this question is a duplicate of [What is the smallest difference in note lengths that an average listener can still perceive?](https://music.stackexchange.com/q/73170/70803). An answer in terms of dots, however, would be meaningless, since it would be dependent on tempo. – Aaron Oct 29 '21 at 08:32
  • As posed the question may not belong here, although it has an answer in the comments. But there are some interesting side issues here in my view. For example has anyone ever used more than three dots after a note rather than using a different notation and why. If that was the question I would be happy to vote to reopen. – JimM Oct 29 '21 at 08:55
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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.) – Andy Bonner Oct 29 '21 at 17:29
  • Yeah these are two completely different questions please pick one to focus on. There's already one answer that only focuses on only one of them. – Dom Oct 29 '21 at 21:08

3 Answers3

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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.

phoog
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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 '21 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. – Scott Wallace Oct 29 '21 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 '21 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 '21 at 14:10
  • @phoog - Sorry, by multi-hit, I mean multi-dot. – Dekkadeci Oct 29 '21 at 14:34
  • @phoog - Indeed. But dots alone would need to be further qualified to cover all exigencies. Not that such exigencies are musically important. – Scott Wallace Oct 29 '21 at 15:21
  • @Dekkadeci aha, that makes more sense. Double-dotted notation is not that common, but I never found it particularly difficult to read, perhaps excepting the first time I encountered it. I suppose it would be difficult to decipher outside of the usual context of exaggerated "normal" long-short dotted rhythm, but don't remember ever encountering a double dot in any other context. – phoog Oct 30 '21 at 12:07
  • @phoog - I just remember reading a recommendation some time ago to split double-dotted notation into 2 notes tied together (possibly in a more pop music context). – Dekkadeci Oct 30 '21 at 13:54
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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.

user1079505
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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.

Kevin
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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 '21 at 12:00