m-excellent pairs

Let $\nu$ a positive irrational number and $m$ a positive integer.

A pair of integers $(a,b)$ is said m-good if: $a\times \lceil \nu \times b\rceil -b\times \lfloor \nu \times a\rfloor =m$
($\lceil x\rceil$ and $\lfloor x\rfloor$ are respectively the smallest integer $\ge x$ and the largest integer $\le x$)

A pair of positive integers $(a,b)$ is said m-excellent if:
$(a,b)$ is m-good and neither $(a-b,b)$ nor $(a,b-a)$ are m-good.

How many m-excellent pairs ($m=123456789101112$) are there if $\nu =0.5772156649015314\dots$?