Why does it not matter what octave you're tuning to? If you wind a string to two different octaves of the same note, surely it's going to have a big effect on the sound, no?
Why does it not matter what octave you're tuning to?
If you want to set a string to a certain pitch, of course it does matter what octave you adjust the string to.
Setting a string to A3 (220Hz) is not the same as setting it to A4 (440 Hz). Not only will the sound be different, but you might make the string very hard to play if it is too slack, or break your string (or your instrument!) if it is too tight.
However, "tuning to" can refer to the reference you are listening to while you tune. And because of the phenomenon of octave equivalence, it is possible to listen to A3 as a reference while adjusting an instrument to play A4, as long as you have some way of not getting confused about which octave you are actually adjusting the string to.
This is normally quite easy, because usually an instrument is approximately in tune - the strings might be a semitone out-of-tune at the most. So if you need to tune the A string on your guitar, you can play any A on the piano*, and tune your guitar to the correct A, because you know that you only need to adjust the guitar slightly.
*Notes near the extremes of the piano might not be good choices, because the strings behave in a somewhat 'less-ideal' way.
True enough, sonically. You'll also probably end up with two different guitar parts too. 'Why does it not matter?' - it does.
I think you may be referring to tuning an A string (for example) to an A note - but maybe an A note in a different octave. If so, the frequencies of each note are double or haqlf to arrive at another A. It will be an 'octave copy' of what you need, but for a lot of folk, it's a good reference point. Not as good as the actual note you need to match, but good enough. We do it all the time when singing. If a man and a woman were singing the same melody, they'd be an octave or two apart, without thinking about it. Neither would be trying to reach the other's actual notes, only octave copies thereof.
That apart, it's impossible to 'wind a string to two different octaves'. There's only one string, therefore only one note available. The strings on any instrument are the correct gauge for the notes they're going to be tuned to. Go down an octave and they're too floppy; go up and they're way too tight. Neither move is sensible.
@topomorto provided a very nice answer so I won't repeat what he already wrote.
But... FWIW, it does matter to me in which octave the reference pitch is.
1 (E) 329.63 Hz E4 2 (B) 246.94 Hz B3 3 (G) 196.00 Hz G3 4 (D) 146.83 Hz D3 5 (A) 110.00 Hz A2 6 (E) 82.41 Hz E2
When I tune the low E2 string, I have trouble tuning it to the high E4 string as a reference pitch.
I usually fret string 4 at the 2nd fret for an E3 reference pitch, and also fret the 6 string at the 5th fret for the unison with the open string 5 (A2.)
In this sense the octave does matter to me for tuning. If the reference pitch is 2 octaves away, I can't hear the "beating" between the two pitches as clearly.
Just to add the boring mathematical basis for the other excellent answers: the reason we have the "octave equivalence effect" is because tones an octave apart, a 2/1 ratio in frequency, are the closest to being the same tone mathematically, a 1/1 ratio, and we hear this similarity.