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I try to understand odd time by listening to the underlying pulse and the duration of notes, but it doesn't always work when the rhythm gets complicated.
What I mean is visualising the pulse as the duration of an eighth note etc. and to make sure the beats I'm counting stop exactly as the bar stops and move on to the next bar.
Often times, it's not easy to decide whether the song is a 13/8 or 13/16 because sometimes the change on tempo counfuse me.
It's unusual for a whole song to be in 13/something - unless it's Greek, from experience. But x/8 and x/16 are rather different in that one is twice as fast, given the same bpm.
Do you mean there might be a 13/x bar somewhere in a piece? If so, it's pretty rare, bu if it is, it'll be counted something along the lines of 1231231231234. Even the humble 5/4 is split into 3+2, or 2+3. There will inevitably be divisions of 2,3 or 4, in a regular pattern.
As an aside, the piece you mentioned initially may be 4/4, 4/4, 7/4 from a brief hearing...
The distinction between 13/8 and 13/16 or 13/4 is probably not easy to distinguish, somewhat of a preference. You should definitely be able to tell 12/8 from 13/8. The song has to express the time signature in the melody, in the phrasing, if not in an obvious rhythmic pattern.
If you're trying to transcribe the piece what works for me is trying to find shortest division of time, fastest "beat" then try to count how many of that beat I tap before a phrase or riff repeats. But whether you call that beat an 8th note or 16 note is really up to you, there is some room for choice here. Different musicians may write the same piece in 8ths and some in 16ths. Once you get the fast beat fitting try dividing in half and see if you still end on the end of a phrase. With practice you learn to hear entire phrases. If there is a tempo change then you need to divide the song in pieces (as if it were two songs) and apply the procedure to each piece then put it together. I'm guessing my the 13/8 you are not looking at a simple waltz. Keep in mind that the artist may be playing notes faster than the beat definition. So you may tap 26 at first (which is an even number) but if you can track the phrases in 13 you have a measure if 13/something (you are free to call that something and 8th or a quarter note if you like). Many time signatures like 5/4 and 7/8 are divided up into shorter groves (at Tim indicated in his answer). If you listen to a lot of Rush you get good at identifying 7/8. And they will sometime toggle between 7 and 6 which adds to 13.
