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Recently I came across this List of musical works in unusual time signatures on Wikipedia. Some of the time signatures mentioned have fractional numbers on the top. e.g. two and a half over four, three fifths over four, four thirds over two, etc.
What is the correct way to interpret such time signatures?
These time signatures are very interesting in nature and have a very specific purpose just like most things in music.
A measure of (3/5)/4 would mean each measure contains 3/5 of a quarter note or 3 eighth note quintuplets. A simpler example is 2(1/2)/4 where where would be 2 and a half quarter notes or 5 eighth notes.
As Matthew Read pointed out in the comments a time signature of 2(1/2)/4 can be represented as 5/8, but there is much more going on. The obvious difference between the two is that in 5/8 the eighth note gets the beat and in 2(1/2)/4 the quarter note gets the beat. This may seem insignificant, but this is a huge difference. In 2(1/2)/4 a full measure does not contain a full amount of beats. It's a half beat short. This gives a very distinct feel that may be thought of as rushed or unbalanced that would not be observed in 5/8.
One good example where this type of time signature can be applied is The Ocean by Led Zepplin. The intro and transitions are typically notated as alternating measures of 4/4 and 7/8. While that notationally is correct, it does not accurately represent the feel of this section. The quarter note is always felt as the beat to and when you start back on beat one on the measure of 4/4 it feels rushed and it feels as though you're missing half of the beat. The two measures also work together very well and act as one unit thus it is valid and in some ways better to look at these two bars as one single measure of 7(1/2)/4 instead of 4/4 and 7/8.
Listen and you'll understand why this makes sense to notate it this way:
Another example on this topic is Schism by Tool. There are many different time signatures used in this song, but one section of the song the interlude can be looked at as 6(1/2)/8. The reason for this is the combination is how a combination of measures 6/8 and 7/8 are used. The measure of 6/8 does not complete the idea by itself and is only complete with the 7/8, but the measure of 7/8 cannot stand alone and is independent. It could also be looked at as 13/8, but the two measures are still somewhat independent, thus the idea of 6.5/8.
Here's an article about it from Wikipedia:
"Schism" is renowned for its use of uncommon time signatures and the frequency of its meter changes. In one analysis of the song, the song alters meter 47 times. The song begins with two bars of 5/4, followed by one bar of 4/4, followed by bars of alternating 5/8 and 7/8, until the first interlude, which consists of alternating bars of 6/8 and 7/8.
The following verse exhibits a similar pattern to the first, alternating bars of 5/8 and 7/8. The next section is bars of 6/4 followed by one bar of 11/8. This takes the song back into alternating 5/8 and 7/8. Another 6/8 and 7/8 section follows, and after this the song goes into repeating 7/8 bars.
The middle section is subsequently introduced, consisting of three bars of 6/8, one bar of 3/8, and one bar of 3/4 repeating several times. At one point it interrupts with two bars of 6/8 followed by a bar of 4/8, twice. A single bar of 4/8 is played before the meter switches back to a set of 6/8 for two bars and 2/4 for one bar. This repeats, setting up another section: two bars of 9/8 followed by a bar of 10/8, that pattern again, and then a single bar of 9/8 followed by a bar of 6/8 and then a bar of 7/8. Next is a set of two bars of 6/8 followed by a bar of 2/8 repeated four times then a single bar of 6/8. The outro has alternating bars of 5/8 and 7/8, ending with alternating 6/8, 2/8 that one could interpret as pulsing with a 4/4 feel.
The band has referred to the time signature as 6.5/8. Although many composers would use 13/16 instead, 6.5/8 is still a valid fractional time signature.
I also suggest listening to it to get a good idea:
I'm going to make a literal and mathematical assessment here, not because I'm not sure how to interpret why someone would write that way but because I really don't think it matters HOW it's written. It matters how the piece is heard, not about some strange writing on a page.
So if I play in 3/4 time, that has a waltz feel, and if I play in 4/4 time, that has the common time sound. I could also play in 3.5/4 time, at an awkward place between the two. But what would that sound like? It would sound like 3/4 time, but with an extra eighth. It would sound like 4/4 time, but with an eighth missing. It wouldn't really sound correct, would it?
So it seems to me that any music written in such a notation accounts for this awkward, irregular feeling. It was made that way for that reason. So the "correct way" to play the music would be to do what the composer heard it as.
Yet all music is open to interpretation. You might have noticed that 3.5/4 time is also 7/8, and that is a much more easy way to read (and notate!) the music. It would still be the same amount of beats per measure, because of the math of how it translates. Ahh, so can't i just drop the denominator then? Of course! It would be played exactly the same way, and written with much more ease.
As for "what is the meaning," I would say it is just a fraction of a measure, and do what you will to understand how to play it so it sounds nice.
As for "What is the correct way to interpret" such a meter, I would say that depends on whether you want to follow what the composer intended, or play it as an interpretation of the work. Either way you should make an assessment on that based on the piece.
Better to think larger sometimes. 3.5/4 is like seven beats over two measures. Doesn’t make sense to think in one measure increments if it’s meant to be felt over two.
It’s definitely situation specific, but I’d say be on the lookout for the possibility that it makes more sense than it looks like.
As has been already pointed out in other answers, a fractional time signature can also be written as an integer time signature over a larger denominator - e.g. 3½/4 can be written as 7/8. This is a compound time signature and in compound time signatures the beat is always some multiple of the lower number, not the lower number itself. For instance, 6/8 time is counted with two beats to the bar.
You can argue about which notation is more useful. I just wanted to point out here that if the intended groupings are not obvious then the composer usually annotates the groupings above the time signature. For example, 7½/4, mentioned in another answer, can also be written as 15/8. The intended feel could be conveyed quite precisely with the annotation (2+2+2+2)+(2+2+2+1). I'm not sure if that is clearer or not than 7½/4.
The examples I've seen have all been where there are various groupings of 3 and 2, for example 13/8 annotated with 3+3+2+2+3.
