are 2 enharmonic notes really the same note?

Question

Because we use equal temperament tuning as opposed to pythagorean tuning or just tuning are two enharmonic notes aka G1 and G2 or C5 and C4 really the same note? or is one them off by a few hertz or more than a few hertz? And would that be why I can't hear an octave? and why that two notes such as C4 and C5 or E4 and E2 sound like garbage to me when played together?

Answer 1

'Enharmonic' is not the word you are looking for. From the ever-reliable Wikipedia:

[An] enharmonic equivalent is a note, interval, or key signature that is equivalent to some other note, interval, or key signature but "spelled", or named differently

Octaves are not enharmonic. They are not the same note. However, they do have a special relationship; a jump of an octave represents a doubling (or halving) of frequency.

I assume you probably know this already, just clearing up some terminology.

To your actual question; Equal temperament is defined to preserve the octave relationship. It's not a universal property of every tuning system, but it's pretty common.

So no, if the octaves are in tune, then they should have the correct relationship, and not be off by any hertz. This may not be the case in the real world, particularly if junior concert bands or bagpipes are involved.

You may not enjoy the sound of pure octaves; it's fairly sterile. But they should not be out of tune.

Now, all of that being said, pianos (and other similar instruments) are tuned using something called Stretched Tuning. As a result, the octaves are not exact doublings of frequency. But I don't think you'd say they are out of tune. In fact, this is intended to make the piano sound more in tune. Some digital pianos emulate stretched tuning; some do not.

Answer 2

The hearing physiology is completely different from the sound generation acoustics. And it is less well understood. But some discrepancies between acoustics in instruments and the psychoacoustics have been described. One of the most obvious things is that the spectrum of the sound has a greater influency on musical harmony than the harmonic series from mathematics. Both are weakly linked together. Steady nonmoving fusing sounds tend to have a harmonic spectrum, while timbres with inharmonic spectra tend to have some instability or vibrating component in the timbre.

So when your octave sounds bad, it may be that the octave is not really an octave, or it may be that your timbre has a component which dissonates with the octave of another component. This is often the case with instruments that have a multidimensional geometry like Glockenspiel, Drums, Xylophones etc. Tubes and Strings are mainly one-dimensional while they do not necessarily have a perfect harmonic spectrum due to friction, stiffness or because the different waves extend more or less over the instrument geometry.

Actually, without knowing your particular sound, nobody can really anwer your question.

Answer 3

Octaves are the only interval that is perfectly tune in equal temperament.

When you say an octave "sounds like garbage" to you, it's unclear what you mean. If you mean that it doesn't sound particularly interesting then I agree, that's the whole point, octaves are so harmonious that they disappear, two notes played together in octaves simply blend together. If you mean that it sounds unpleasant, then whatever instrument you're using is probably out of tune which is why you're hearing a clash.

Here is a short audio recording demonstration of what octaves sound like together, and then contrasting with an out of tune octave:

https://vocaroo.com/i/s1jNwyc6Yrmw