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I’m studying Time Signature and I understand that the top number indicates the amount of beats per measure and the bottom note indicates the note value of those beats. But it got me thinking about this. Why don’t we use irrational time signatures that would contain 3,5,6,7,9,10,11 for a bottom note and how would we count those note values in comparison to 4 (quarter note beats), 8 (eighth note beats), 16 (16 note beats). Why are there no 3rd, 5th, 6th, 7th, 9th, 10th, 11th notes, etc? Is it simply because it is easier to sense the lengths of sounds if they are a power of two?

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    Does this answer your question? What is a rational rhythm?
    – Aaron
    Mar 24 at 15:41
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    I was afraid this question would arise from this one. There are many places in which someone might observe "This would be better/clearer/more efficient if we did it differently. Why do we do it this way?" The only answer is always "these practices are not usually invented in one day by one person with an end goal in mind. They evolve over time." And to "so why don't we change it now," the answer is "there's an inertia of practice that is hard to shift." Look at George Bernard Shaw's attempts to reform English phonetics and spelling. Mar 24 at 16:43
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    So your question relates more to the history of the notation and why it developed along a powers-of-two trajectory?
    – Aaron
    Mar 24 at 17:46
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    @Aaron umm, actually yes I guess that would be correct or more accurate then asking it the way I did
    – Emotion
    Mar 24 at 19:16
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    @gidds you could invent new symbols. For example, you could use triangles, or a different shape of flag, or what have you. Or you could use red ink to indicate division by a factor of three (not exactly a new concept, but it fell out of favor maybe 600 years ago).
    – phoog
    Mar 25 at 13:09

7 Answers 7

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Technically, third, fifth, sixth, etc. notes do exist - only they aren't usually called that. Here is what some third, sixth, twelfth and fifth notes look like:

A score in common time; first bar is a triplet of half notes, second bar is a triplet of quarter notes and two triplets of eighth notes; third bar os a quintuplet of quarter notes.

More commonly refered to with terms such as "a triplet of half notes" or "a quintuplet of quarter notes", but as you can see, they do have the length of 1/3, 1/6, 1/12 and 1/5 respectively.

What they don't appear as is the beat of a time signature.

The reasons why it's so lie, of course, in the long history of evolving notation. But the reason we don't miss such beats and don't feel the need to invent them is pretty much that the choice of the beat note is largely arbitrary. The only thing that really differs 3/4 from either 3/2 or 3/8 is the choice of which symbol we're going to use as the base one. A quarter note, or a half, or an eighth? If you adjust the tempo accordingly, they mean the same thing. So there's absolutely no reason to throw a 3/3 time into the mess. Let's just call it a 3/4 time at a slightly slower tempo, that's a much simpler frame of description.

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    Yes! Indeed, in many of Chopin's solo piano works, we have divisions by 7, 11, 13, and I think 17, etc. Maybe a 23. And quite a few bits of the transcriptions of Art Tatum's stuff has similar (not toooo surprising, since, apparently, he did study Chopin's music, whether from the score or from recordings, I have no idea... but evidently like the fluid sound). Mar 25 at 18:21
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When using time signatures in a straightforward way, it makes sense to choose ones that can be conveniently notated! We have quarter notes, so lower figure '4' is sensible. We have eight notes, so '8' is good. We don't often count a bar as a certain number of quintuplets!

When we DO use 'irrational' time signatures, it is to indicate a tempo relationship. '4/3' indicates a bar containing four notes if the length that were previously notated as triplets.

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Little-known factoid - you can take the reciprocal of the bottom number of a time signature and get the fraction of a whole note each component note of the time signature takes up.

For example:

  • 4/4 and 3/4 time involve 1/4 = quarter notes
  • 6/8 and 12/8 time involve 1/8 = 8th notes
  • Taking compound interpretations, 6/8 can be termed 2/(8/3) time, so it involves 3/8 = dotted quarter notes (i.e. 3/2 * 1/4 notes)

Pretty much only an implied tempo shift or tuple spam will therefore convince people to put odd numbers on the bottom of the time signature instead of powers of 2 or even fractions. Even then, a lot of people are going to want the tempo shift up front instead.

If you want odd numbers on the bottom of your time signature, then all of a sudden, you're going to want stuff like 1/3 notes or 1/5 notes:

  • 3/3 time involves 1/3 notes (component notes of a half-note triplet)
  • 4/5 time involves 1/5 notes (component notes of a quarter-note quintuplet)

This greatly increases the difficulty of your piece. To put this in perspective, I recall my clap-rhythms-by-sight-reading exercises to only introduce (8th-note) triplets and regular 8th notes in the same exercise in Grades 9-10. Trying to make people intuitively count out quintuplets or even large triplets is going to be heck, let alone septuplets.

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Why don’t we use irrational time signatures that would contain 3,5,6,7,9 for a bottom note and how would we count those note values in comparison to 4 (quarter note beats), 8 (eighth note beats), 16 (16 note beats)?

I feel compelled to note that only a musician (a species that is renowned for being unable to count higher than four) could call something like 4/7 "irrational."

It is also worthwhile to note that the constraint on the bottom number of time signatures is not simply that they be even, but that they be a power of two. The question recognizes this implicitly by including 6 in the list of unused time signature denominators.

There are two reasons for this. The most obvious is that all of the notes we have in our notational system are half as long as the next longer note and twice as long as the next shorter note. Yes, you can reduce the value of a note by using a tuplet, but that reduction applies to all note values within the tuplet, which continue to be related to one another only by factors of a power of two. It might make sense to have a time signature such as 5/7 if there were some note that by its form was intrinsically worth one seventh of a whole note, but there is no such note. In other words, implementing such time signatures would imply a much more radical overhaul of the metrical notation system, probably with greater cost in complexity than benefit in flexibility.

Second, and perhaps more importantly, what would be the point of such a system? The current system is already full of ambiguity. Waltzes are written in 3/4 by convention, but they can equally be written in 3/8, 3/16, or 3/2 (or even 6/8, 12/8, etc., etc.). A piece in duple time can be in 4/4, 2/2, 4/8, 4/2, and more. Meter nominally comes from dividing up time, but practically it comes from adding time units together. You can put three of them in a bar, or four, five, six, seven, etc., but it doesn't really make a difference what the note value is, the bottom number of the time signature. Because of conventions, people may find certain choices easier or harder to read, but mathematically it doesn't matter. In Monteverdi's time -- the dawn of metrical time signatures -- you often find a fast 3/1 that works well at dotted-semibreve=80 or so. You could copy it out with the note values divided by eight and mark it dotted-quarter=80; some modern musicians will find it easier to read, but no listener will be able to tell the difference. What purpose would it serve to add more options of things-we-can-put-three-of-into-a-measure? None, really, but it would make the system significantly more complicated.

Time signatures developed from mensural proportions, which worked a bit differently -- for example, they were true fractions, so there was no room for something such as 6/8. The main reason I mention this, however, is that in this system, each note duration could be divided into two or three of the next shorter duration. This was specified not by numbers but by other symbols: a circle meant three semibreves per breve, while a broken circle meant two (the forerunner of the C time signature for 4/4 time). Adding a dot in the center indicated that the semibreves was to be divided into three minims; without the dot, there were to be two. Putting a vertical stroke through the sign indicated that the basic unit of time was to be the breve rather than the semibreve, making everything twice as fast (the forerunner of both the term alla breve and the C-slash sign used to denote it).

One complication of this system is that even when the semibreve was to be divided into three minims, the composition of two minims was also represented by a semibreve. There were rules to determine whether a semibreve was longer or shorter. For example, in a series of alternating semibreves and minims, the semibreves would be worth two minims, but in a series of semibreves without minims, they would be worth three. In cases where the rules would imply that a semibreve should be two minims but it needed to be three, one would add a dot.

The numerical proportions began more as tempo marks. They couldn't be "time signatures" in the modern sense, specifying the organization of each bar, because there were no bars (bar lines were sometimes used to set off sections of a piece, rather like a modern double bar line, but we're not yet used to denote measures of regular length.) The slash for "twice as fast" wasn't enough. You start to see 3 and 3/2, which are not only faster but also organized in groups of 3, and I think this is where the idea that the number tells you about the metrical organization began. At the same time, the system whereby an undotted note could have the duration of three of the next smaller value fell out of use; people started using the dot always.

We're talking late 1500s here, and early 1600s. In the next decades, the modern meaning of the time signature began to assert itself as bar lines began to be used with increasing regularity. At this point, there was no interest in anything other than duple and triple divisions of time. The upper number in a time signature always had factors of only two and three: 2, 3, 4, 6, 8, 9, or 12. The lower number always had factors of two only: 2, 4, 8, or 16. In the 19th century, triplets and other tuplets, often with other prime factors, came into wider use. The larger tuplets were not necessarily strictly timed but were ornamental flourishes. They certainly were not a regular subdivision of a beat into 5 or 7 parts. The notation system represents them as exceptions because that is what they are.

As mentioned above, a system that has note divisions of other than two doesn't really get you much. One thing it does get you is that you would use the whole note differently in meters whether the top number is greater than or equal to the bottom number. For example, if you have a piece in 5/5, a whole note would last for an entire measure, whereas in 5/4 you need to tie a dotted half and half note or something like that. But it raises more questions: what symbol do you use to represent 1/5 of a measure? What about 2/5, 3/5, or 4/5?

I see that the question has been edited since I started writing this answer:

Why are there no notes called 3rd, 5th, 6th, 7th, 9th, 10th, 11th etc?

I think the above discussion still answers the new question, but this suggests other questions: What musical phenomenon would you use this system to describe? Would this really be the best way to describe it? Is there music that you want to play, or music that you've heard, that can't be rendered in western notation because of the absence of third, fifth, sixth, seventh notes, etc.?

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  • Thank you also for the historical notes! :) Mar 25 at 18:26
  • I learned those symbols as "common time" and "cut time". I had no idea those were just post hoc mnemonics. Although now I think of it, it makes sense
    – No Name
    Mar 26 at 21:02
  • @NoName indeed, especially since the words for "common" and "cut" don't start with "c" in most languages! But I think "common time" is as much a coincidence as a post hoc mnemonic; for example, in German it is known by the Italian term tempo ordinario.
    – phoog
    Mar 26 at 22:58
  • Mathematics is my day job and music just a hobby yet I do not object to "irrational" having a different meaning in music. It has a specific meaning within mathematics but I don't feel any need to impose that meaning onto other subjects.
    – badjohn
    Mar 27 at 12:47
  • Irrational is not the correct terminology. 5 and 7 time is irregular time signatures. I don't know where people got irrational from.
    – Neil Meyer
    Mar 27 at 17:59
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The system in common use has evolved over centuries not all in one go by any individual or committee. Could it be better - almost certainly yes. Is it likely to change significantly anytime soon - almost certainly no. So, if you want to be able to read and play existing music then just accept how it is and learn it.

If we were to design a new better and more logical system then I don't think that we would need to use a bottom number in the time signature at all. Being able to reset the unit of the beat does not have an obvious value. Pick one symbol, eg. the crotchet, to be the beat and stick to it. I don't plan to do this redesigning.

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That would be the best way for the simple reason that it is very easy to play a fastest tap rate and then count down from that. You could have a 64th note, then 2 64th notes (a 32nd), then 3 (a dotted 32nd), then 4 (a 16th) then 5 (a doted 64th tied to a 32nd) then 6 (a dotted 16th) then 7 and so on, as here: enter image description here

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The composer Henry Cowell describes a system using different noteheads for 1/3 and 1/5 notes etc. in his 1930 book "New Musical Resources". This was never taken up by other composers.

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