As other commenters have said, approximating integer ratios is important.
1:2 is the octave
2:3 is the perfect fifth
3:4 is the perfect fourth
4:5 is the major third
5:6 is the minor third
and it just so happens that these ratios are close to powers of the 12th root of 2.
2^(12/12) is the octave
2^(7/12) is the perfect fifth
2^(5/12) is the perfect fourth
2^(4/12) is the major third
2^(3/12) is the minor third
You can do the math and verify those numbers are relatively close.
It’s important to recognize that this correspondence is relatively recently discovered; it was developed independently in china in 1584 and in europe in 1605, and coexisted with other schemes for finding approximations of those ratios for hundreds of years, and there are still people who think that this system sucks and we should use a different one, because of minor differences in pitch. (Also, the Chinese system actually used 24th roots of 2, not 12th roots.) This system is called “Equal Temprament”, and there any many other tuning systems that make slightly different choices.
Why not just use the exact integer ratios instead of the approximate ones? Well, if you’re playing on a violin or singing, you can use exact integer ratios. But if you’re using a fixed-note instrument, like a guitar (with frets) or a piano, then you have to deal with the issue that if you go up 1 octave and 1 minor third from a note, and also go up three perfect fourths, you get two notes that are almost identical, but different enough to go out of tune. (This is called the Syntonic comma.) So which one do you put on the piano? If you choose one, the other will sound a little wrong. Or, you could choose the average, and they’ll both sound a little wrong.
As other commenters have said, approximating integer ratios is important.
1:2 is the octave
2:3 is the perfect fifth
3:4 is the perfect fourth
4:5 is the major third
5:6 is the minor third
and it just so happens that these ratios are close to powers of the 12th root of 2.
2^(12/12) is the octave
2^(7/12) is the perfect fifth
2^(5/12) is the perfect fourth
2^(4/12) is the major third
2^(3/12) is the minor third
You can do the math and verify those numbers are relatively close.
It’s important to recognize that this correspondence is relatively recently discovered; it was developed independently in china in 1584 and in europe in 1605, and coexisted with other schemes for finding approximations of those ratios for hundreds of years, and there are still people who think that this system sucks and we should use a different one, because of minor differences in pitch. (Also, the Chinese system actually used 24th roots of 2, not 12th roots.) This system is called “Equal Temprament”, and there any many other tuning systems that make slightly different choices.
Why not just use the exact integer ratios instead of the approximate ones? Well, if you’re playing on a violin or singing, you can use exact integer ratios. But if you’re using a fixed-note instrument, like a guitar (with frets) or a piano, then you have to deal with the issue that if you go up 1 octave and 1 minor third from a note, and also go up three perfect fourths, you get two notes that are almost identical, but different enough to go out of tune. (This is called the Syntonic comma.) So which one do you put on the piano? If you choose one, the other will sound a little wrong. Or, you could choose the average, and they’ll both sound a little wrong.