You don't need a new universe. You just need a species that hears sounds a little differently. We won't like listening to their music, but it'll be really great for them.
3^12 ~= 2^19
Then you can cut string B into thirds, and it too will sound different.
If you pluck both of your new strings at the same time, you'll find they sound quite nice together (the difference between these two is called a "fifth").
And after 12 rounds of cutting string B into thirds, and 19 rounds of cutting string A in half, you'll happen to have a string from each group that are almost identical in length and pitch.
But it won't line up exactly! They'll be about 1.4% different in length, which roughly works out to a quarter-semitone difference in pitch (i.e. 1/4th the distance from one piano key to the next).
But my comment was in reply to this statement specifically:
> Somewhere there is a perfect universe where 12 fifths form an octave.
For this to be true, I think you'd indeed need a new universe where 3^12 = 2^x, where X is a whole integer.
And I agree with you. While I can imagine a culture that hadn't "discovered" the octave, it's difficult to imagine people who can't even perceive octaves when presented with them, and quite easy to imagine other creatures that cannot perceive them.
ABCABCABCABCABCABC
Just to be clear: the magical number here is 2 (not 12 or 8 or 7) since "octave equivalence" refers to the fact that you can multiply a frequency by 2 and get the same note.
Unless the sound is a perfect sine wave, there isn't a particular frequency associated with it due to these alternative interpretations.
When you overlay two waveforms that are related x:1, the zero points of the waveforms align. The wave resets at the same instant. If instead the waveforms are related not exactly, then you get a change in where the reset point is that drifts and causes the wave to exhibit beating (changes in volume).
If you have a 2:1 (octave) relationship between 2 waveforms, then you won't hear the beating as the beat frequency exactly overlaps the frequency of the lower frequency.
If you have say two notes that are not quite in the octave ratio, then you hear beating at the difference in frequency. E.g. say 440Hz and 888Hz (instead of 880Hz), you have beating occurring at 448Hz, which you'll hear as an 8Hz (448Hz-440Hz) wobble in the sound volume of the combined wave.
If we take a look into a human ear we'll see there an apparatus which physically splits sound into a spectrum of frequencies (a long narrowing tube, sound comes from one end and it creates resonances at different places) and a lot of receptors which placed in such a way that allows them to specialize on different frequencies.
I know nothing that would point to an innate ability of this apparatus to feel octave as something special compared with a mix of two random frequencies. So the peculiarity we hear pops up on later stages of sound processing. But why it pops up?
Moreover there is an evidence (I have no link, sorry) that the peculiarity of an octave is a cultural thing, not a genetic one. European music teaches us to feel octave as something special. There are tribes not exposed to European music who doesn't feel consonance and dissonance like we do.
My hypothesis that a mind picks correlation between a frequency and a double frequency. I mean it is not because of some funny math comparing two sine waves, it is because sounds essentially are not single sines. Two sounds forming an octave are both sums of many sine waves, and there is a huge overlap between sets of frequencies, so they sound similar. And then, when mind trains on this data, it becomes conditioned on a similarity of double frequencies, so it starts think of two sines with frequency ratio of 2:1 as of similar. The similarity is just a correlation.
And people who didn't tried to make a music with strings cannot grasp the idea, because natural sounds mostly much more complex than just sum of a several sines with frequencies that are multiples of some base frequency.
The higher tone also excites the sensors for the lower tone.
Wait, does it? I might be spouting bullshit, sorry.
Thinking about it, I'm not so sure it doesn't. I'm not a physicist to know it for sure.
The octave rule definitely has some physical/biological founding, it’s not purely cultural. Although culture also plays a role in its preponderance.
In nature, sounds produce harmonics i.e. when two objects collide they usually create waves of frequency f, 2f, 3f, 4f... in various (usually exponentially decreasing) weights. It's very rare to find pure sounds (i.e. only f frequency) in nature. The interval between f and 2f is an octave apart (1:2 ratio); the interval between 2f and 3f is a perfect fifth (2:3 ratio). So, when you actually hear a sound, you actually hear an octave and a fifth too, and how dominant this octave and fifth changes the "timbre" of the sound. This way, you know the source of the sound independent of the frequency. For example, both a violin and a piano can produce the note A4 at 440Hz, but anyone can easily determine if it's a piano or violin. The reason is, when a piano produces A4, it sounds not only just 440Hz but also 880Hz and 1320Hz etc... too and the relative volume of 880Hz and 1320Hz will be different than that of violin. Your brain automatically interprets these volume weights as "timbre" and the fundamental frequency 440Hz as "pitch".
Consequently, in order for your brain to be able to process the timbre of a sound it needs to find octaves and fifths between each fundamental note it hears. This means there might be something universal about octave and fifth (and other decreasingly consonant intervals such as major third etc...). Maybe we "understand" music because our brain is hard-wired to search for octaves and fifths in all sounds, in order to analyze timbre and in order to process spoken language. If this hypothesis is true, maybe an alien species could have octave/fifth/major third based music too! (if they have music at all, of course)
There is! At least for the kinds of instruments that are conventionally used in Western music. The harmonic series arises naturally from the physical properties of a string or wind instrument (e.g. violins, guitars, pianos, flutes, brass, organs, etc). As a very rough description of the physical phenomena, the tones we hear arise from a full spectrum, atonal excitation (like a pluck or a reed flapping) bouncing back and forth along the length of string or tube, which is basically a one-dimensional "waveguide". Frequencies that are aligned with the harmonic series naturally reinforce themselves, in the same way that putting energy at the top of the arc of a playground swing has more of an effect than in the middle.
Notably, musical instruments that are not strings or tubes, or more general sound-producing bodies, have more complicated patterns of sound waves dispersing through them, and don't typically follow the harmonic series. Pitched percussion, drum heads, or bells have more complicated harmonic spectra than the standard harmonic series (as they are generally thought of as 2D or 3D waveguides where cancellation/reinforcement patterns are less straightforward), as do less musically conventional sounds like knocking two rocks together or striking an arbitrary surface.