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The other day I saw this question here:

I don't understand the bottom number in a time signature

I got really excited, as this has been bugging me for many years now. I started reading the answers, but they seem to answer a slightly different question than mine, and possibly different to that of the OP. Definitely a different question than I thought OP was asking. But the question was marked as answered, so I decided to write my own question instead of initiating a discussion in the comments.

The answers to @JessicaCunningham 's question make fine job explaining how time signatures like 3/4 and 3/8 are written differently. Some then go on to explain how 3/8 and 3/4 sound different, but they don't explain WHY they are different, except for "that's just the way it is"-type answers.

What I'm dying to know is: Is there some rule (or even a tendency or pattern) that tells you how stuff played in X/2, X/4, X/8 etc sound different, given a positive integer X?

NB: I don't care about the notation. I can read the different signatures, but this question is rather about the music than about the notation.

Following a convention defined on a case-by-case basis works great when playing "normal" music, but it only gets you so far. I know how 6/8, 3/4, 3/8, and even 7/8 sounds, but it would be great to be able to say anything at all about the sound (not the notation!) of 4/2, 3/16, 10/2, 10/4 etc, except for what I'm told by the numerator of the fraction.

I do suspect no such rule or pattern exists, except for higher denominators sometimes indicating faster pace.* I'm just griping for straws. In that case, all the information about the rhythm carried by the denominator could just as well be carried by a non-numerical name, like we do with Greek names of modes, but we just happened to choose numbers for historical reasons.

If this is how it is I'm totally fine with that; I'm not here to complain about tradition. I'm asking just in case there is something there. It would be so nice to actually understand what is happening and not just learn a bunch of cases by heart!

*I've seen What is the difference between 3/2 and 3/4 time signature and Is there any practical difference between 3/4 and 3/8 time? and What makes a composer decide between time signatures 3/4, 3/8, or 3/2?. They suggest larger denominators suggest quicker pace, but in my experience this isn't always the case. Also, we already have separate notation to tell us the number of beats per minute.

I maintain this is not a duplicate, as the kind of answer I'm looking for (what the rule is or a firm "there is no rule", in either case with some motivation) isn't given in any of the questions I've read. As everyone seem happy with the answers, I conclude they answer a different question then mine.

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  • 1
    The context in which the choice is made has changed over the centuries. If you want to infer something about a piece of music because it is written in 3/8 instead of 3/2, you probably need to know when and where it was written.
    – phoog
    Dec 13 '20 at 17:24
  • A knowledge of maths can be useful in music but it is easy to get carried away. Viewing time signatures as fractions is an example; it is tempting but not useful.
    – badjohn
    Dec 15 '20 at 16:25
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3/2, 3/4 and 3/8 are arithmetically identical. As are 2/2, 2/4 and 2/8. And 4/2, 4/4 and 4/8.

There are usage differences though, if not very clear-cut ones! 3/2 is unlikely to be a fast tempo, though a brisk march is quite likely to be notated in 2/2 (or in the equivalent 'cut time' with the slashed C time signature). A 'one in the bar' fast waltz is generally 3/4, a jig is 3/8. But Beethoven used 3/8 for Larghetto as well as for Andante con moto and Allegro con brio!

The point being, assuming speed from the lower number being 2, 4 or 8 involves historical perspective.

(Are you OK on why 3/4 is NOT the same as 6/8? Subdivisions - whether the bar splits into groups of 2 or of 3 notes) aren't an issue in the examples you asked about. They ARE an issue when extending the discussion to the difference between e.g. 3/4 and 6/8.)

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  • It seems you are confirming my suspicion that the choice of denominator is completely arbitrary, and only serves to convey information that really isn't numerical in nature. (And I'm totally OK with using different numerators! :) )
    – EdvinW
    Dec 12 '20 at 22:24
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There's no "sound of a time signature", just like there's no sound of a key signature. Each time signature corresponds to one or more classes of music that has been or could plausibly be written, using that time signature. But each such class can encompass a wide variety of different sounds, and the classes even overlap, meaning that the choice of time signature is subjective. Time signatures exist for writing and reading.

NB: I don't care about the notation. I can read the different signatures, but this question is rather about the music than about the notation.

Does each time signature correspond to a particular "sound", something that would be valid and identifiable in a culture with no tradition of written music, and music is only ever learned and passed on via listening and playing together, not reading and writing. I don't think so.

Do you know how African drummers traditionally learn rhythms? (This is based on a short drumming lesson with a Ghanaian drummer.) By playing together with other drummers. Do they see the rhythms as consisting of measures of 4 beats? No. Even if a Western musician would notate the rhythm using a 4/4 time signature. That's just the Western guy's fixation. "It is music, so it HAS to be broken down into measures?"

How do they traditionally start a song in Africa (at least in the area from which the teacher I mentioned comes from)? Someone plays this (as written in notation for a Western person to understand, because that's the only possible way here, audio clips are not supported):

African count-in

But that notation is not the sound. It is notation written in a language that suits you. But that doesn't mean that it is in any way a more "true" representation of what the music "is". It is just communication suited to you and your culture and the way of thinking.

In exactly what way is that rhythm pattern (also known as the 3-2 clave) "in 4/4"? Only the first and last notes coincide with the so-called "beats". And on the third beat there is a rest. If the purpose of playing that rhythm is to communicate the imaginary metric beats, then it feels like a rather ineffective way to do that.

Here's how a Western person starts a song.

Western count-in

They explicate, count in, the measurement units, the so-called beats, which are really just imaginary and wouldn't have a sound otherwise. This whole counting and measuring thing is a part of the Western culture. It's needed for that particular style of managing musical phenomena. But there are other styles.

If an actual repeating pattern could be written in 8/4 or 4/4 or 64/8 or whatever, which one do you choose? That decision is entirely based on the target culture and tradition.

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  • The African rhythm you cite isn't the only way to start a song, though (if you can use the word song). Another common rhythm has the last two notes in the opposite order, and there is also another class of rhythms that correspond to a European meter of 12/8 rather than 4/4. And I'm just speaking from my very limited knowledge of the music of Ghana; surely in all of Africa there are also options that are unused in West Africa.
    – phoog
    Dec 13 '20 at 17:28
  • @phoog There's 2-3 clave, but the 3-2 clave was taught to me as the rhythm to tap and then everyone understands that music starts. Note that it's not the actual rhythm of the song, it's only used as a count-in. It doesn't change the point though. Africa is a big continent! Dec 13 '20 at 17:30
  • My experience is otherwise. From which of Africa's thousands of musical traditions did your teacher come?
    – phoog
    Dec 13 '20 at 17:32
  • Do you think that the point I'm making is not valid - that time signatures are for a specific tradition and not a part of sound and music itself? I edited the text so now I'm not leading people to big conclusions about African music that might not be universally true. Dec 13 '20 at 17:33
  • No, the point is basically sound, but with some caveats. I'm thinking about adding an answer to discuss those.
    – phoog
    Dec 13 '20 at 17:39
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TL;DR

The upper number is more important in determining the feel of the music as there are default interpretations of how the pulses in a measure are emphasizes or de-emphasized, depending on how many pulses there are.

The lower number is an indicator of how the music will be notated, which does influence the interpretation, but in a generally more subtle way.


Explanation / Typical interpretations

Upper number

Based on the upper number in a time signature, here are the typical ways to play the music.

  • beat 1: In any time signature, the first beat is assumed to be the strongest emphasis, unless otherwise indicated.
  • final beat: The last beat of the measure is usually the weakest.
  • 2-time: If the top number is a multiple of 2, then the halfway point of the measure is the next strongest pulse. Where further halfway divisions are possible, they follow the same pattern, with the first half being stronger, and the second half being relatively weaker.
  • 3-time: If the top number is a multiple of 3, then both the second and third beats are relatively weak, with the stronger of the two being determined by the style of the music. For example, a waltz will tend to make beat two stronger than beat three, but a mazurka will tend to do the reverse.
  • 2-and-3-time: If the top number is divisible by both 2 and 3 (i.e.; divisible by 6), then the multiples of 2 take priority, followed by multiples of three.
  • not-2-or-3-time: If the top number is not divisible by 2 or 3, then the relative strength of pulses (other than beat 1) has to be indicated explicitly.
  • subdivisions: These tendencies hold for smaller subdivisions as well. For a single beat divided into three parts, the first part will be strongest, and so on.

Lower number

The purpose of the lower number is only to indicate which note type is used to indicate the basic pulse. However, there are some interpretive indications of the choice. Choosing a small number tends to look and feel slower; whereas choosing a large number tends to look and feel faster. A composer might strategically choose the bottom number of the time signature to send a message to the performer.


Examples

In these examples, the numbers above are the beat numbers; the numbers below are each beat's relative strength: 1 = strongest; 2 = 2nd strongest; etc.


     Beat number
Time signature
     "Default" relative strength (1 = strongest; 2 = next strongest; etc.)

     1 2 | 1 2 | 1 2 |
2/X
     1 2 | 1 2 | 1 2 |



     1 2 3 4 | 1 2 3 4 |
4/X
     1 3 2 4 | 1 3 2 4 |



     1 2 3 4 5 6 7 8 | 1 2 3 4 5 6 7 8 |
8/X
     1 5 3 7 2 6 4 8 | 1 5 3 7 2 6 4 8 |


     1 2 3 | 1 2 3 | 1 2 3 |
3/X
     1 2 3 | 1 2 3 | 1 2 3 | or |1 3 2 | 1 3 2 | 1 3 2 | depending on style


     1 2 3 4 5 6 | 1 2 3 4 5 6 |
6/X
     1 3 5 2 4 6 | 1 3 5 2 4 6 |


     1 2 3 4 5 6  7 8 9  10 11 12 |
12/X
     1 5 9 3 7 11 2 6 10 4  8  12 |

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  • I'd like it notated more explicitly that the top number being a multiple of 2 or 4 takes priority over the top number being a multiple of 3 when it comes to which pulses are emphasized. 6/8, 12/8, 12/16, and 6/16 are all compound meters and are all more common than all 8/X meters combined, I've found.
    – Dekkadeci
    Dec 12 '20 at 20:11
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    I have to say I find the "Examples" section totally baffling...
    – Judy N.
    Dec 12 '20 at 20:57
  • @JudyN.Happy to revise, but need a better feel for your bafflement.
    – Aaron
    Dec 12 '20 at 21:14
  • Your "default" relative strengths for 6/X and 12/X look like expanded versions of 3/X time instead of the compound meters they really are. Try "1 4 2 5 3 6" for 6/X time and "1 7 4 10 2 8 5 11 3 9 5 12" for 12/X time instead.
    – Dekkadeci
    Dec 13 '20 at 14:28
  • @Dekkadeci I'm trying to convey that for 6-time, it's typically two groups of 3, making beat 4 the second strongest. Were I to change as you suggest, that would make it three groups of two, with beat 3 being the second strongest. Similarly for 12 -- I'm looking to convey four groups of three, with the primary stresses ordered like 4/4.
    – Aaron
    Dec 13 '20 at 17:48
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Assuming you halve or double the note values accordingly, there is no difference between say 3/2 3/4 and 3/8. Depending on the style of the piece, one of these might look more appropriate in a given situation. It's up to the composer to decide which one they think is better. Tradition is important: a Viennese Waltz would sound the same if it was written in 3/2 or 3/8, but it would look really silly in anything other than 3/4. Tangos are traditionally notated in 4/8, but they're often easier to read re-notated in 4/4. A majestic adagio might look better in 3/2, a light scherzo might look better in 3/8.

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  • This sounds good, and it's actually really nice to hear! Whenever I've brought this up with a musician they've always told me that of course there is a logical system to interpret denominators. They've then tried to explain about subdivisions of the bar, tempos etc, and in the end left me totally confused. This makes sense.
    – EdvinW
    Dec 12 '20 at 22:40
  • Subdivisions aren't an issue in the examples you asked about. They ARE an issue when extending the discussion to the difference between 3/4 and 6/8. Dec 13 '20 at 16:07
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I know only 2 reasons for notating music in 2/2 than in 2/4:

a) deciphering the rhythm (the shorter values) in 2/2 might be easier to read for beginners.

b) hand writing half notes in 2/2 might need less ink and writing quavers instead of quarters need less time.

All arguments and speculations that the performance and the impression between the the two kind of notating were different are nonsense.

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