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So, I read somewhere that there was an ocarina with "only 4 holes" that could play "all 8 notes of the scale" because "all the holes were different sizes.

This raised many questions. Why shouldn't it be able to play 16 different notes? 2^4=16, after all. Is the normal way to make wind instruments to have all the holes be the same size? That just seems inefficient. And, after looking into it a little more, there was also the question of what the crap is up with musical scales.

Okay, so to clear things up for anyone who is as confused as I was, here's what I dug up. It's 4 holes of 3 different sizes, so there are 12 meaningfully different combinations. Also, once through the musical scale is 12 notes, with sharps and flats included. So that all works out.

It turns out that ocarinas have in fact traditionally had far more than 4 holes, with little improvement in the number of notes they can play. Apparently, the reason for this is that musical instruments have traditionally been designed by musicians, not mathematicians. And as it turns out, musicians don't immediately take the logarithm of their desired number of notes when faced with the task of figuring out how many holes/whatever will be necessary.

As for what the crap is up with musical scales, for some reason, the notes aren't all the same distance apart. Lord only knows why. Because they just go and add sharps and flats to the wider intervals. So why assign symbols in such an ass-backward fashion?

Description

So, I read somewhere that there was an ocarina with "only 4 holes" that could play "all 8 notes of the scale" because "all the holes were different sizes.

This raised many questions. Why shouldn't it be able to play 16 different notes? 2^4=16, after all. Is the normal way to make wind instruments to have all the holes be the same size? That just seems inefficient. And, after looking into it a little more, there was also the question of what the crap is up with musical scales.

Okay, so to clear things up for anyone who is as confused as I was, here's what I dug up. It's 4 holes of 3 different sizes, so there are 12 meaningfully different combinations. Also, once through the musical scale is 12 notes, with sharps and flats included. So that all works out.

It turns out that ocarinas have in fact traditionally had far more than 4 holes, with little improvement in the number of notes they can play. Apparently, the reason for this is that musical instruments have traditionally been designed by musicians, not mathematicians. And as it turns out, musicians don't immediately take the logarithm of their desired number of notes when faced with the task of figuring out how many holes/whatever will be necessary.

As for what the crap is up with musical scales, for some reason, the notes aren't all the same distance apart. Lord only knows why. Because they just go and add sharps and flats to the wider intervals. So why assign symbols in such an ass-backward fashion?