OK, So I'm a scientifically trained engineer with heavy math background and I just started learning a bit of music theory. It takes you 4 steps to get from C to E, so what do they call that? A third. Of course getting from C to Eb, which is actually three steps is also called a third. But here's where it gets really weird: the 7 steps from C to G is called a fifth and the 10 steps from C up to Bb is called a dominant seventh.
So, either musicians are nuts or mathematicians are nuts. Let's assume for the moment that it's the mathematicians who are misguided. What can they learn from the musicians' example. First lesson: You don't need all those number words like "seven", "five", and "nine". There's a more straightforward way to count with, for example, only four number words, "unit", "quad" and "dozen":
3. diminished quad
5. augmented quad
7. inverted augmented quad
8. inverted quad
9. inverted diminished quad
10. inverted semi-quad
11. inverted unit
With intermediate values such as:
3.25 diminished minor quad
3.5 minor quad
3.75 augmented minor quad
Second lesson: Number words don't have to actually be what they say they are. For example, just as 10 steps is a "seventh" in music, so we might say that "dozen" really means "nine" (or, more properly, "inverted diminished quad").
How much more sensible it would be to teach students that pi is "diminished quad point unit quad unit augmented quad inverted diminished quad", or to talk in history class about the "inverted augmented quad wonders of the ancient world."
On second thought, maybe it's the musicians who need revamp their descriptive terminology for intervals. Hmmm. Some how I don't think it's ever going to happen, but wouldn't it be nice if an interval of 10 steps was called an 10th instead of being called a "dominant seventh" so that actual mathematical analysis could be done on intervals?
Ah, but so it goes. I may some day understand music theory, if I can get my head around its irrational nomenclature.