Like most of the Tool fan base, I want to new music from the guys, so any click-bait about caught me up so far...

Recently, we've mentioned something about trying to write in 5/7 time signature.
I'm not so strong with math (I get through with basic calculations so YAY ME!):
But it got me wondering about dividing seconds and beats in 7 instead of 1, 2, 4, 8, 16 etc...

Any mathematical geniuses out there?
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No such thing as a 5/7 time sig, because there is no such thing as a 1/7 note.
What's your source? What's the track?
Standard rhythmic theory does not allow the note value (bottom number of time sig) to be anything other than 2, 4, 8, 16 (32). This is just trying to find a convenient way of measuring the flow of time, so other longer and short durations can be placed against these.

But the bigger numbers are for notational convenience ... you're not going to tap your foot to a1/16th or 1/32 note.

So,let's say you do want a system using a 7th. Let's say you can tap your foot to it. You've now got a nightmare working out where any of the even numbered note values would fall.

The interesting part of the time signature is the top number (5 in your case) ... the bottom number just indicates how fast these notes flow by.

The top figure only becomes real, feel it in the music, by adding accents of some sort, which nearly always means breaking into groups of 2s or 3s (or multiples) by using accents. ONE two ONE two three ...repeat, ONE two three ONE two ... ONE two Three four ONE ... etc. Without the stress, its just a number on a chart. Or try playing 3 even duration notes that occupy the same duration as the 5 ... that gets hard.

If you want to experiment, do that with the top number. The bottom is effectively a time scale, and not much else.
A 7th note = a quarter note septuplet (divide a whole note into 7 equally long notes and you get 7 quarter note septuplet notes).

I don't know how 5/7 would sound any different from 5/4. You can't really hear the difference between quarter notes and quarter note septuplet notes out of context. I mean, it doesn't matter whether the basic value is a quarter note septuplet note or a quarter note.

I guess "5/7" would be possible if you for example wrote a piece with a melody that moves in normal quarter notes and a drum beat that uses quarter note septuplets and repeats after 5 notes. But then again, that would be just a weird polyrhythm, not sure if I would call it "5/7 time signature". But if you didn't use any polyrhythms, 5/7 would sound exactly the same as 5/4.
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Western music defines standard note durations by fractions of powers of 2 (whole (1/1 = 1/(2^0)) note, half (1/2) note, quarter (1/4) note, eighth, sixteenth... note: these are based on n/n time for n a power of 2, so a whole note would get n beats' duration, thus completing the measure. A half note thus gets 2 beats' duration in 4/4, a whole measure in 2/4, 2/3 of a measure in 6/8, and so forth).

Triplets (3 notes within a beat) aren't written as 1/3 notes (if a beat were a measure); they're written as one of these 2^n fractions with a 3 around said grouping:

The guy is playing 3 (eighth) notes in the place of 2, but the note value is defined as an eighth note triplet. (idk about the next two lines, just used Google )

The same extends to other tuplets.

I'd suggest linking the song for further analysis, btw. You can definitely have 2 polyrhythms on top of each other (making a 35-beat loop), but the overriding rhythm would most likely be 5/n or 7/n, n a power of 2.