On a violin, cello etc, during normal play (drawing the bowhair

I like the genuine inquiry into the math behind a simple, non-mathematical task.

Bump

> music theorists can't into sensible notation
That's why if you want to analyze something mathematically, you need a mathematical notation.

Suppose you have a
> predictable pattern of notes that doesn't change depending on direction. Any scale, any arpeggio, etc.
Arrange all the notes in order from lowest to highest pitch.
Label the starting note 0, the note with the next pitch up as 1, then 2, 3, ... etc.
Likewise, label the note with the next lowest pitch compared to the starting note as -1, then -2, -3, ... etc.

(Jargon alert: what I am doing is constructing a homomorphism from your pattern of notes to the set of integers.)

Then if you start on a downbow at 0 and begin playing, alternating between downbows and upbows, the downbows will occur on the even numbers and the upbows on odd numbers.

If you have four different states and cycle through them regularly, they'll occur on the numbers (read: notes) that are multiples of 4, 1 greater than a multiple of 4, 2 greater than a multiple of 4, and 3 greater than a multiple of 4 respectively. The cycling through bow states corresponds naturally (the technical term is "isomorphism") to cycling through remainders of numbers when divided by 4.

>>8559917
Oh, yeah, that sounds exactly right. So it takes two steps after the initial downbow at 0 to get to a downbow again, and the position of that next downbow is -2, 0 or 2.

Homomorphism and isomorphism are two words I have sorely needed throughout my life.

>>8560959
ur mom is a homomorphism

TL;DR

Im sorry OP. Even though im a musician i had no patience to read your unnecessarily long post. You should practice your writing skills before posting.

It's because you have two "states" and two directions in which you can take the scale (up or down). Say you have a linearly-ordered, finite set of squares (like a file on a chess board), and you label these squares according to the ordered set of notes you are playing (a scale, arpeggio, et cetera). Then any way you play that doesn't skip notes, as you mentioned, corresponds to what mathematicians call a walk through the line of squares, moving one square to the left or right at each unit of time. Imagine colouring the squares assigned to upbow notes white, and downbow notes black. Then, the colour of this walk along the squares will alternate at each step, since you cannot stay still and each of a squares neighbors have a different colour than it.

File: Chessboard_2.jpg (3MB, 3704x2864px) Image search: [Google]

3MB, 3704x2864px

>>8561047
It's probably you being a musician that caused you to have no patience to read it, since I tried to write this for the hypothetical mathematician who has never encountered the cello in their life. I do agree my writing is shit though

>>8561152
Yeah, that makes sense.

Hm, and if you had the entire chessboard and moved like a king piece on it (stepwise), it would still be upbows assigned to one color and downbows assigned to the other. And it would do so for any set of notes you assign it.

This sounds like shitty graphic score music in the making. [spoiler]I kind of want to write it now.[/spoiler]

>>8561522
Yes, it works in all dimensions. That's both cool and possibly useful!

>>8559855
It's because there are 8 notes in a scale and 2 | 8. Literally that simple.

>>8561645
But there are 15 notes in a two-octave scale and 29 notes in a four-octave scale.

>>8561526
It also works for three. Does it work no matter how many dimensions?

>>8562209
Yes, this can be shown a number of ways, but the simplest is by induction over "shells" about a starting square. The relevance of this is that you can label your various axes with different note sets, and then any path through this space will correspond precisely to a sequence of chords, or something more general.