For those who think that math deliver the most precise answers to everything…
I would perhaps say that in music, math creates one structure of music, at least in notated music in any style in Western music, since fairly simple fractions make the rhythmic order as seen on the notes.
When you play a piece of written music you need to have this order sort of fastened in your mind to the extent that you can play the rhythm correctly, but you also need to be so sure of it that you can keep the rhythm and at the same time let go of it. If you play it mathematically correct it will most of the times sound to rigid, in this style it will be not with a swing.
You can also analyse music according to a flock of other ideas, put in your own story or stories to a certain extent and see it as symbols of practically anything. In short, you are expected to put your own personality into it when you play it, with whatever ideas that will work to make something interesting and nice out of it.
To actually do this with your fingers and mouth, not only think it, you need that same freedom in relation to the rhythm, so that you can improvise within the given pattern, the given notes. It can become a rubber band which you can’t totally leave, but you can make your own pattern within the music, within the notes.
The result can, if you’re lucky, be something completely new.
Now, if you try to measure the performed rhythm with a technical instrument of some kind, which on the sheet music is written perfect and plain, you will most probably discover a lot of mathematical inaccuracies which is due to the performers’ own way of forming the music.
So what’s left of the concepts of eights, quarter notes etc are perhaps ideas which you can hear and follow very well when you listen, but the accuracy of the sounds, the actual played music, you will need some other concepts than pure numbers to catch.
Every musician and every genre will have their own words for the things that are going on, and no one will ever make a definitive version of a piece because the chore of those concepts, even if the concepts exist (you can hear the rhythm and everything else, right?) – even if they exist, it is not possible to define them completely, or if you manage to do so, they will possible die in your mind and leave you with a disability to play or listen.
Life is not supposed to stop, and the representation of life, the pictures that we make of life, in short…art, are also not supposed to lose their ability to interest us and give us new versions of the truths hidden inside them.
In music that is not notated, the same things go on, only the ideas maybe spread out more and happen in different notes every time it is brought to life through the live playing of music.
The idea itself remains alive and present, but not completely finished in the sense that it cannot be told in another way the next time someone plays.
Especially in Count Basie’s playing with his band you can experience this combination of laziness and energy in attitude and in the relationship to the notes, and you need the security both of your instrument and of the notes to half leave them as you go along, just as if you walked down a path which you know very well, but which you see with fresh eyes as you move forward, and with small detours on the way.