It's taken a year or so, but I use (don't ask why)
the opening note of Coronation Street (a British soap) that is a C note , the major 3 of Ab.
Every time I walk past a piano - several times a day, I hum that note, then play it to check (much to my wife's amusement and disgust).
Just done it now, and got in the crack to one side of the correct note.
Getting better but not perfect yet.
So, yes, it can be learnt.
Obviously.
Some have perfect/absolute pitch, but we mere mortals put up with relative pitch, where we get a note and can pitch another from it.
Again, a skill to be practised. With tuners as small as they are,or using a mobile, you could carry one around, and try a note every so often.
It may take time, but it's a good way to bore your friends.
50 yrs ago - pre tuners, during caravan holidays, I used to drop a house key on a hard surface to keep guitars in 'concert pitch'.
Daft, but it worked !
music.stackexchange.com/users/2808/Tim
I had to learn Bach's Minuet for a piano class.
I played it so much that I remember the D
I started on every time.
I also have a small vocal part in a song that I had to practice getting perfectly correct even if I can't hear myself at all, that starts on a D.
Now I'm pretty good at just singing, hearing a D and can sometimes identify other notes from this D.
It takes some real training but just practicing different modes of making or hearing the note repetitively to really drill it into your head.
It's still not always accurate of course, but it'll get better and if you don't have perfect pitch, it's something!!
music.stackexchange.com/users/12118/Eichhörnchen
"Perfect pitch" (the ability to sing and/or recognize a particular note) can be developed by some people, to some degree.
Not everyone, and generally not as accurately as a tuner.
(The folks who are precise report that it may be more a handicap than a help, since they may be annoyed by any instrument that's tuned slightly differently.)
I know that I naturally tend to start some songs in D-minor, just because that's where my voice naturally falls.
That sort of thing, tapping "muscle memory" and the resonances of your own body, might be adequate for solo performance.
As noted above, it isn't likely to be good enough for performing with others where you have to agree on a tuning.
Electronic or mechanical tuners can be tiny and cheap.
Tuner apps exist for most of the popular smartphones and tablets.
I suspect that if you care enough to want to be on the right note, you want to be on the RIGHT note, and are better off using mechanical assistance unless you want to spend lots of time practicing this skill that could be spent practicing the instrument.
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You're asking how to remember a pitch
.It's more like remembering one pound by hand, than notes in music.
Therefore you have to repeat the pitch as others answered: listen to a pitch, recall it by heart, listen, recall, ...
It's simple rote learning.
http://en.wikipedia.org/wiki/Rote_learning
For that matter if you tune an instrument every day, you would be aware of Hz difference rather than note difference although I can't say you will have one herz accuracy.
music.stackexchange.com/users/11047/lamusique
Use a spectrum analyser (there's an app), that will allow you to correlate your natural tone.
mine is 146hz..
whats that, a D?
so i can read this value off on the analyser, and repeat recall until its lodged deep inside my neural pathway.
music.stackexchange.com/users/50340/user50340
The sensitivity to the relative pitch differences may depend on the absolute value of the pitch.
Very low or very high notes are more difficult to recognize and tell apart, even if in still hear-able sound range.
As you have an instrument, try to experiment where approximately your individual ear is the most sensitive to the pitch differences, and then learn to remember the absolute frequency in this range.
Simply play, record into mobile phone and use later as a ringtone and wake up signal in the morning.
music.stackexchange.com/users/8737/h22
I'm wondering what ways there are to remember a note. For example, as a guitarist it'd come in handy if one can hum an E note so you can tune your E string according to it.  Are there common techniques to remember a note? Or is it just practice and feeling?
Techniques for remembering a pitch without reference
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music.stackexchange.com/questions/20302
music.stackexchange.com/users/9626/strudelkopf
It's simple rote learning.  It takes some real training but just practicing different modes of making or hearing the note repetitively to really drill it into your head. One trick is to simply play the note, record it into mobile phone and use later as a ringtone and wake up signal in the morning. The most common technique to achieve this is probably just to practise. However, you could also consider recording a note as part of your ringtone to help you memorise it.
The most common technique is probably just to practise repeatedly. You could record a note as your ringtone on your phone.