 The Fingering Forum
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Author: ~Heather ~
Date: 2004-03-02 21:46
I still can't grasp the concept of the bottom number in the time signature. I mean in 3/4 time how in the world does the 4 mean that every quater note gets one beat? Then in 6/8 every half note gets the beat. Can somebody please explain how they came up with this!!!!??? Thanks a bunch! :D
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Author: TorusTubarius
Date: 2004-03-02 22:50
Ever hear somebody say that math and music are intricately related? It is true, more true than I ever even dreamed when I was a kid and just starting my first piano lesson.
Another thing I have learned is that when you're striving to achieve a more complete understanding of something, you often have to leave behind certain maxims or rules-of-thumb that may have been useful in practice, but are actually inconsistent with theory. I think you have reached this point with the rule "the four means every quarter gets one beat." Forget it, it doesn't mean that at all. They just tell you that as a quick and dirty explanation. Here's what time signatures really tell you:
It's easiest to understand the true meaning of a time signature when considering whole notes, so that is what we'll do. Basically, a time signature tells you where each measure is drawn in relation to the whole notes. Each one tells you what fraction of the whole note is contained within each measure.
So take 4/4. 4/4 = 1 arithmetically speaking. So this tells you that there will be 1 whole note in each measure. This is the largest value note you can have per measure.
Take 2/4. 2/4 = 1/2. Here we see there is one-half a whole note in each bar. So now the whole note is split in between two measures, and now the biggest note value you can have is half notes in each measure.
Take 3/4. 3/4 cannot be reduced down any further, so you can just read this directly and say that now there will be only three-quarters of a whole note in each measure. Thus the largest note value you can have per measure is the dotted half-note (one half plus one quarter,
1/2 + 1/4 = 3/4)
Now take 6/8. If we reduce this fraction, you get this:
6/8 = 3/4. So again, you only get three-quarters of a whole note in each measure. What's the big idea? Why not just write everything as 3/4? Well the reason we express the fraction as 6/8 is so that instead of counting quarters we now count eights in each measure. Think of it like this:
We saw above that 3/4 time is literally three-quarter time, that is three-quarters of a whole note per measure. And by convention we count each quarter in the measure as one of those three quarters.
6/8 time is likewise literally sixth-eighths of a whole note per measure. And we count each eighth as one of those six-eighths of a whole note. This makes it easier to write music and read music where you have lots of three eighth-note groupings. Instead of having to split up the three quarter notes in 3/4, we now have the whole note split up again into eighths which can be grouped anyway we like or counted individually as 1-2-3-4-5-6, 1-2-3-4-5-6, etc.
Now what I said above, <i>by convention</i> is important. Because when you think about it, there is no mathematical reason why we couldn't in fact go back and write out every 6/8 song as a 3/4 song or every 3/4 song as a 6/8 song. But how would you express groups of three eighth notes like you get in 6/8 songs in a 3/4 time signature? You would have to split up the three quarter notes unevenly, which defeats the purpose of calling it a 3/4 piece in the first place. What a composer is really saying when he puts a piece in 3/4 is "hey musician, pay attention to the fact that each measure here contains three-quarters of a whole note."
This doesn't necessarily mean that we should give every quarter note the beat, indeed there are many a scherzo movement written in 3/4 which are so fast you only get one beat per measure. But what it does mean is that we should conceive of the music as being some division of the three quarters which comprise each measure as opposed to some grouping of the 6 eighths. That's why you don't see 3/4 song with two groupings of 3 eighth notes in each measure like you do in 6/8; to have written it as such would be to have unevenly split up the three quarters to which our attention is drawn by the 3/4 time signature. So if you think about it, a time signature tells us much more than just the rhythm of the piece or the mathematical division of the whole note among the measures. It also tells us something about the construction of the melody, the way in which the music will move from chord to chord, and the overall "feel" of the piece.
Now consider a time signature which when reduced is greater than 1. This would be something like 6/4, 12/8, or 9/8.
6/4 = 3/2 = 1.5 Thus we can see that there will be one and one-half whole notes per measure. Therefore the largest note value in the measure can be a dotted whole note (a whole note plus a half).
12/8 = 1.5 also. And the relationship between 6/4 and 12/8 is analogous to the relationship I discussed above between 3/4 and 6/8. In 6/4 you'll get rhythms which have the six quarters as their core such as 1&-2&-3&-4&-5&-6&, 1&-2&-3&-4&-5&-6& etc. Where as in 12/8, since now all the eighth notes have been designated, you'll get other types of groupings such as groups of threes, 1&u-2&u-3&u-4&u, 1&u-2&u-3&u-4&u etc.
9/8 doesn't reduce any further, but you could think of it as being 8/8 + 1/8. Thus you get one whole note plus one more eighth of a whole note per measure. And because each eighth note is made explicit in the time signature, we expect to see groupings of each of the nine eighths in each measure, and in fact we do almost inevitably.
Still another way to think about it would be to take a theoretical piece of music which is 4 measures of 4/4 time. The piece would have a total of 4 whole notes. If you removed all the measure lines, what time signature would you have? Removing all the measure lines would give you one big measure that encompasses the whole piece. Since in that measure there would be 4 whole notes, you get a time signature of 4/1! You'll never see anything like that of course, but theoretically it could exist.
Anyway, I hope that helps.
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Author: Dee
Date: 2004-03-02 23:13
Well Torus explanation is nice but probably a little bit more than you are ready for.
In 6/8 time, an 8th gets one beat with 6 beats to the measure. However, many pieces play much better if it is conducted in two beats to a measure and then you mentally subdivide each of those beats into 3 parts. It is a matter of convenience. Afterall you don't want the conductor waving his arm so fast that it falls off!!
So in counting in your head for 6/8, you think:
ONE two three FOUR five six
The conductor will show the beat on counts one and four.
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