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I have learnt all the complicated mathematics subjects in my school, but that bottom number, I can't understand. Ok, the notation will be different, and it will say what kind of note gets the beat, but why do I need to know that if it won't change what I'll play? Can someone show me the difference if I play something in 5/4 or 5/8, or 7/4 and 7/8?

There was a time when I thought I understood, I thought that since it was a quaver note in the bottom I would count 1 and 2 and 3 and 4...; if it was 16th note in the bottom I would count 1 e and a 2 e and a 3 e and a 4; but I asked on reddit and they said I'm completely wrong.

After all, if it's only going to make a small difference in how I read and no difference in how I play the song or count, then what's the point of the bottom number? It's just there to complicate things?

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    Not sure if it will help but in case it does, knowing math is no benefit to understanding time signatures. They are musical symbols not based on mathematics. – Todd Wilcox Apr 25 at 17:23
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    @ToddWilcox - there is still an element of maths in time signatures. Which bears a relationship with fractions which to a degree they actually are. Numerator tells how many, denominator tells what each is. – Tim Apr 25 at 17:40
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    Does this answer your question? I don't understand the bottom number in a time signature – Neal Apr 25 at 20:45
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    @Neal I've updated the title in an attempt to clarify the distinction between this question and the proposed duplicate. – Aaron Apr 25 at 22:00
  • @ToddWilcox time signatures are musical symbols that are very much based on mathematics. – phoog Apr 26 at 5:09
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I believe there are fundamentally 2 main reasons I can think of:

  1. historically, the tempo was not strict as it has been considered in the last 2-3 centuries, and it was usually decided based on the time signature; so, for instance, 3/8 was considerably faster than 3/4, and that's usually true even for today's music: if the fraction has a larger denominator, it's usually played fastly;
  2. readability;

Tempo indications based on signature

Before tempo markings (and, obviously, the introduction of metronome), the signature was used to indicate how fast a piece should have been played.

See the following diagram, taken from this answer:

Tempo markings

As you can see, the suggestions (created to relate the signatures to modern metronome marks) are very different.

Readability

Readability is a very important aspect in music writing: the musician must focus only on playing, not decyphring what's written (just like an actor must focus on delivering the feeling of the lines, not how they're written). Using simpler tempo markings could cause serious issues, especially for slower music that has phrasings based on small subdivisions. This aspect is obviously also considered for writing: many notation aspects are "tricks" composers used to write "less" and faster - and, I believe, to use less ink and reduce risk of errors).

Consider the following excerpt from the 8th Prelude of the Well-Tempered Clavier, first book, which is written in 3/2:

Bach excerpt

Now, let's see how it would look if it were written in 3/4:

Terrible!

While it is "mathematically" correct, it's simply awful to read.


Let's take the third Prelude instead:

Third prelude

You could say that, if written in 3/4, it could have been more easy to read. But then we have to recall the first rule (remember, it was written in the first half of the 1700).

Finally, the only problem could rise for equivalent fractions, but that's easily answered: in music, equal mathematical values don't always mean the same musical result: 6/8 is not 3/4; 8/16 is not 4/8 (nor 2/4), and it usually is considered a "tempo misto" (sorry, I don't know the English term), often as 3+3+2.

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  • Unusual key sig. at the end - but at least all the # are within the stave. – Tim Apr 25 at 17:46
  • @Tim yeah, I didn't realize that. But I don't think it's about making them fit the stave, as the lower part has that B above it, so I assume it was just to put them "in linear order". – musicamante Apr 25 at 17:50
  • Somehow, though, it's a tidier version than the existing one. I've always questioned why the G# needed to be out of the stave although maybe that puts a question mark over the bass clef? – Tim Apr 25 at 18:19
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    @J... no, maybe I finally found out, I believe they're called Additive rhythms, which include not only odd meters, but also even meters that use odd subdivisions (8 as 3+3+2, etc). – musicamante Apr 26 at 19:15
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After all, if it's only going to make a small difference in how I read and no difference in how I play the song or count, then what's the point of the bottom number?

The point is, you have to write SOMETHING there, or otherwise the top number doesn't mean ANYTHING.

It's about culture, conventions, tradition and language. Not natural science, laboratory measurements and math. If you ask yourself "should I write this in 7/4 or 7/8" and cannot tell the difference, then you lack cultural, not theoretical knowledge. To me, 7/4 means something slower than 7/8 and so that I don't think about dividing the bigger ticks. But it's not a very commonly used time signature, so I might look at and listen to more pieces written in 7/4 to study how people traditionally understand it.

What does 4/4 mean? It won't tell you the intended pulse of the song. The time signature doesn't specify if you should be thinking One-e-and-a Two-e-and-a Three-e-and-a Four-e-and-a, or One-and Two-and Three-and Four-and. In other words, is it an 1/8 beat or 1/16th beat feel. Or if it should be played with swing. The time signature gives you a class or category of rhythmic feelings that have traditionally been written with that time signature. To know the intended pulse and feel of the song, you'll have to look at other hints, such as what is the smallest time value in the melody, is there 1/16 syncopation, what is the tempo marking, does it say "swing", or is there perhaps a name of a rhythm style such as tango.

Why aren't 8th beat songs written in 8/8, and 16th beat songs in 16/16? Why do people say "How do you do?" How do you do what? What's the point, if the answer is going to be "How do you do"? You can't calculate it, you need to get involved in the culture to understand it.

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The other important information that the "denominator" of a time signature conveys is the beat in relation to other sections of the music. The meter of a piece does not have to remain constant, and as an example, it is possible to go from 4/4 to 3/8. In that case, the new time signature uses the eighth note as its beat for a triple meter, and the old time signature was using the quarter note as its beat in quadruple time. Thus, one of the reasons the beat value is significant is that the beat value can change.

Theoretically, it makes no difference whether the above example was 4/4 to 3/8 or whether it was written as 4/2 to 3/4. However, musicians have expectations about what types of musical ideas should be expressed in what place in the meter. In a lot of rock music, the snare will hit on the 2nd and 4th beats of a measure, and it is most common that these represent the 2nd and 4th quarter notes in 4/4. In most pop songs, the quickest syncopations fall on the 16th-note subdivisions. In most music that swings, the swing will be expressed as uneven 8th notes in the notation. These are not hard-and-fast rules, but following the note-values that musicians expect is part of the practice of writing easily understandable sheet music.

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    "In a lot of rock music, the snare will hit on the 3rd beat of a measure" Do you mean in general or as an common occurrence? Because, in the first case, rock (and 4-beat-pop in general, but it actually is valid even from roots of jazz music), the snare hits on 2 and 4. In the second, that's referred to as "half-time" or "half-tempo". – musicamante Apr 26 at 19:04
  • @musicamante Good catch, and good point. – user45266 Apr 26 at 21:17
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When counting with the system you're using, the beat is always the whole number.

  • X/2 time: minim = 1 2 3 ...; crotchet = 1 & 2 & ...; quaver = 1 e & a 2 e & a ....
  • X/4 time: crotchet = 1 2 3 ...; quavers = 1 & 2 & ...; semiquavers = 1 e & a 2 e & a ....
  • X/8 time: quaver = 1 2 3 ...; semiquaver = 1 & 2 & ...; demisemiquaver = 1 e & a 2 e & a ....
  • X/16 time: semiquaver = 1 2 3 ...; etc.

For the answer to the rest of the question, see Is there any real-world difference between time signatures such as 4/4 and 8/8?

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  • I don't understand the '1 2 3...' in each example. Help! – Tim Apr 25 at 17:42
  • The whole numbers correspond to the basic pulse of the music — the beats. The top number in a time signature (X in these examples) indicates the number of beats. The "&" and "e/a" represent divisions of beats: "&" marks the 1/2-beat; "e" and "a" mark 1/4 and 3/4 parts of a beat. – Aaron Apr 25 at 19:41
  • I've been aware of that for many decades!!! Some things I don't understand. Why 1 2 3...??? I'm not a complete idiot, but working on it. Just help them out when they need some support. – Tim Apr 25 at 20:20
  • The idea behind 1 2 3 is to keep track of which beat within the measure you're on. Some (piano) method books, however, have begun moving away from that, especially in the earliest levels. Instead, they count the length of the notes. So, crotchet, crotchet, minim would be 1 1 1-2. – Aaron Apr 25 at 20:28
  • @Tim why not? Being able to count out loud while playing (assuming the instrument allows it) should be a basic skill. Or am I not understanding your question? – musicamante Apr 25 at 23:09
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Well, if an English text is preceded with "[to be read in a heavy Yorkshire accent]" this does not really have an effect on the sequence of words following as such (assuming they don't employ special terms or phonetic spelling) but are still an important performance indicator. Syllable stress will be totally different than when preceded with "[to be read in a heavy French accent]".

The meter specifies how the material is structured, where the accents in the music lie and how they are connected. In some respects, it tells you what kind of dances will work with the music.

Now the most important number certainly is the top number. However the choice of bottom number is still somewhat relevant by convention rather than logic: 3/1 tends to be faster than 3/4 (since 3/1 is a baroque triplet rhythm), 3/8 has a somewhat more rigid substructure than 3/4 (which may have a somewhat elongated 4 in execution) because of different dances they are associated with, 4/4 is a standard structure while 4/8 again tends to be a bit faster and with less of an accent on the 3rd beat and so on and so on.

Basically you are best off not trying too much to understand the underlying logic but just taking in examples as you go and eventually developing some sort of feeling for it.

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I don't think the confusion is your basic understanding of meters and time signatures, but probably how you are counting aloud with syllables.

In simple time signatures - those that subdivide the beat by 2 - it's the note value that gets the beat. Ex...

4 (four beats per measure)
4 (quarter note gets the beat)

In compound time - those that subdivide the beat by 3 - it's the value of the subdivisions. Ex...

6 (divide this by three, 6/3 = 2, there are two beats per measure)
8 (each of the 2 beat is subdivided by three eighth notes)

Time signatures are not fractions.

Time signatures are not really math.

Time signatures are short hand signs for meters. You could replace them with any other sign. Case in point: the C for common time or 4/4.

After learning how to read a time signature, it's probably more practical to just recognize the commonest by memory: 2/4/, 3/4, 4/4, 3/8, 6/8, 9/8, 12/8`.

Additive time signatures like 7/8, 5/4, etc. are sort of "advanced" but shouldn't be too hard to deal with provided you understand simple and compound time signature.

...it will say what kind of note gets the beat, but why do I need to know that if it won't change what I'll play? Can someone show me the difference if I play something in 5/4 or 5/8...I thought that since it was a quaver note in the bottom I would count 1 and 2 and 3 and 4...; if it was 16th note in the bottom I would count 1 e and a 2 e and a 3 e and a 4...

Technically, the person at Reddit was right to say your counting aloud is wrong. If the 16th note was on the bottom, like 4/16, then the beat count should still be "1,2,3,4" on each sixteenth note.

I think the think to keep in mind is: beats, subdivisions, beats per measure.

When using the rhythm syllables, like "1 e & a", be careful of what is a beat and what is a subdivision. There are different systems of syllables, but I think it doesn't matter much what you say as long as you keep to numbers on the beats, which is what the person a Reddit must have taken issue with in your example. Here is how I think you would count aloud for some examples...

enter image description here

Notice that with 5/4 the beats are subdivided by 2 so it a kind of simple meter, but in 5/8 you get beamed groups of 3 eighth notes subdividing a beat, which is like compound meters, mixed with groups of 2 eighth note subdivisions. In odd or additive meters like these the bottom number can signal the difference between simple to compound feel.

4/16 looks strange... because it is, but if you wrote it, that is how it would be counted. It's essentially common time, but you change which note values are assigned to a full beat. There probably isn't a practical reason to do this.

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