Long answer: they aren't. They are enharmonic equivalents in the vast majority of music that uses any of the conventional Western systems of tuning (as the author sort of goes out of their way to demonstrate in the article), and if you use or invent a different system then what you call the notes is kind of up to you since it's your system.
Not on a piano, but for all the other instruments with variable pitch (e.g. fretless strings, voice, trombones) they are. The enharmonic is useful information and we'll know whether to place that note just a little under or above its usual pitch to make the chord more in tune. We don't need to invent a new system to do this, we do it every day within the 12-tet system we already have.
Most of the time, even if you're playing a variable pitch instrument you're going to be tuning to fixed pitch because you'll have at least one fixed pitch instrument (eg a piano) and if you don't you'll just sound out of tune.
In cases (eg a consort group or string quartet or something) where you're all variable pitch, you'll be tuning to each other and to the scale/key as appropriate to the music and whatever sounds good. You may well sweeten the thirds or widen the fifths a bit etc but that doesn't apply to this question here because you're really not going to see the enharmonic equivalents in the same piece the absolute vast majority of the time for stylistic reasons and if you ever did you would just be tuning to each other to make the vertical incidences sound good rather than thinking consciously of tuning a d-sharp one way and an e-flat another way.
Source: Have a degree and postgrad in music, used to be a professional double bass player[1], spouse has a degree and postgrad in music and teaches at 2 conservatoires in London as well as performing professionally, mostly early music in small consort groups where this sort of tuning thing comes up a lot.
[1] So yeah you can make the standard joke about what do double bass players know about tuning.
For me the enharmonic equivalent is usually just a totally different harmony, so that is what I tune to. As a result they are quite different notes. I try to do that consciously - also while playing with fixed pitch instruments when possible (like the grandparent comment explained).
Even if playing with a fixed-pitch instrument, it only really sounds out of tune if they're playing the same notes. Which in the styles I play isn't an issue.
So I guess how often this happens in practice varies between styles and eras of music, which would make sense to me. I haven't ever done early music and know nothing about it (other than trombones used to be designed terribly and we now know how to make better ones ;)
Source: Also have a postgrad in music, probably from one of the conservatories your spouse teaches at, and still play trombone professionally.
I don’t think that’s correct -- they are identical in 12-TET, but all the other tuning systems either treat them as different notes, or attempt to compromise between the alternatives in a way that favours certain keys over others.
Maybe this is isn’t critical info for a lot of people, but it is important foundational knowledge if you’re a music student, or just interested in music theory.
This is not true in general. It is only true for instruments that use the well tempered tuning, e.g. a piano. But for example, the violin and cello do not.
I actually did read the article and even prior to that do know about deriving pitches from harmonics. What I posted was still correct in spite of the downvotes.
Trying to sing a D# in a B major chord the same way you would a Eb in a C minor won't be a great experience.
Most of the adjustments will happen automatically if you listen to your fellow singers and have experience. But they do happen.
Expansion: in non-tempered, natural tuning, such as often used when playing a violin, there are differences between some sharps and flats built from different notes, because natural harmonic intervals, based on frequency ratios like 3:2, do not split the frequency range in a completely log-linear way. This is why, say, G# and Ab may be not the same for purposes of pure natural harmony [1].
Equal temperation was invented to overcome this. J.S.Bach wrote a great showcase for it, Woll-Tepmeriertee Klavier, which involves harmonies and chord progressions that are hard or impossible to achieve with natural tuning without producing weird dissonances.
Now I'm wondering if I was told wrong 50 years ago, or if this is new research?
https://en.m.wikipedia.org/wiki/The_Well-Tempered_Clavier#We...
The “Bach standardized the world on 12TET” trope is old and enjoyable enough to make a good story that unspecialized music teachers have parroted along for generations.
We've got better access to information now. I've corrected music teachers on this specific topic in the past. Some gratefully accept. Some pull out the “but the teacher here is me card”, so I'm sure a few more generations are going to be needed.