Do we perceive octaves as precisely 2:1 or is that an approximation? i.e. is it really 2.041256:1 or similar.
Is this problem then because we start with integer ratios? Is this problem solvable if we started a "true(?)" irrational ratio?
Is this problem then because we start with integer ratios? Is this problem solvable if we started a "true(?)" irrational ratio?
You can try by putting 250 and 510 into the boxes here: http://onlinetonegenerator.com/binauralbeats.html
and revert to a nice-sounding octave with 250 and 500.
This frequency relationship is approximately true for many, but not all, acoustic instruments.
http://sethares.engr.wisc.edu/html/soundexamples.html
The mp3 at the above website called "Challenging the octave" gives an example of a bell that sounds more in tune when the "octave" is a frequency ratio of 2.1 vs the usual 2.
If you have any interest in music theory, even if you've already studied traditional western music theory, read Dr. Sethares' work on the subject: http://sethares.engr.wisc.edu/ttss.html (He was mentioned elsewhere in the comments, but he's too awesome to risk missing.)
In the perfect world you have e.g.:
+ C major scale: C D E F G A B C, where frequencies of all notes depend on frequency of note C (like, “fifth” from C is G and it's exactly 3/2 of C)
+ and D major: D E* F♯* G* A* B* C♯* … D (and depend likewise). But now G is not necessary equal to G*, but they are close. So here comes the idea of equal temperament where octaves are strictly 2:1 as they suppose to be, but all notes between are equally scattered (on log scale) in between.
So TL;DR: nowadays it's all approximation. You can do it perfectly, but only for one root.