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If a chord can be said to have a Quality which relates to whether it is major, minor, augmented or diminished then by what name do we call the property of a single note's flatness or sharpness?
Note: I'm aware of the brevity of this question but think it is rather clear. Also fretted-note is the only tag I can seem to tie to the post.
Based on your question and the comment to left to Neil's answer, the term that seems to fit your needs most closely is accidental. That said, I'm not sure it's exactly what you want, and I'm not sure what you want actually exists. This is because the notion of "flatness" or "sharpness" are not absolute qualities the way major and minor are, rather they are relative to some reference key.
An accidental is a note which is not one of the eight notes defined by the (current) key signature of the piece (Neil's definition is close, but accidentals are based on key signature, not on the current scale one is playing). So for example, if the key signature indicates C-Major, then both D# and Db are accidentals. However, if the key signature is E-Major, then D# is not an accidental, and D-natural is (because D# is in the key of E and D-natural isn't).
In fact, there exists such things as double-sharps and double-flats, in case one wants to indicate a note which is sharper/flatter than a note which already has a sharp/flat. For example, again suppose the piece is in the key of E-Major, in which D# is not an accidental. Now suppose the composer wants to notate the note that is a half-step sharp of D#. He can write Dx (the symbol used isn't really an "x", but it's close). A Dx (D-double sharp) in enharmonically equivalent to an E, but theoretically distinct, and many composers know and care about the distinction. If you're curious about this distinction, leave a comment and I'll expand.
This truly fills your criteria, but I think you were intending to speak on a slightly higher level. For any given pitch in the set of 12 we use in modern harmony (that is, including C, F#, Eb, etc.), the degree to which that note is in or out of tune in either a flat or sharp direction is called intonation.
This might be closer to what you're really looking for, but as @Alex Basson mentioned, you should always consider that different keys have different sets of "home notes". When you see an accidental (naturals included) in a line that is adjusting the note to one that is outside of the written key, you can give it any number of names: accidental, chromaticism, color note, secondary tonicization, alteration.
I don't like accidental because it refers to a symbol, chromaticism is a bit too generic, and color note/secondary tonicization are too specific to common practice. Alteration, on the other hand, is a term that easily applies to this situation, describes the property of such a note, and applies to situations elsewhere. For example, an altered dominant chord has tones that are moved up or down a half step using accidentals; when you describe the difference between major and mixolydian you must alter the 7th scale degree. I suppose you could say that G major is just C major, whose 4th has been altered up half a step, and whose tonic has been moved to G.
"Sharp" means a note that is half a step higher, and "flat" means half a step lower. The way your question is currently written, you're looking for a name for something that is both half a step higher and half a step lower, which makes no sense.
Perhaps you're looking for the term accidental? An accidental is a note not part of the current scale. For example, an e-flat would be accidental in the key of C.
forename which is a property we share, our names differ and having two forenames would be confusing. So I can call a D note D, a sharp D note D#, but what might we say in such a case: 'the something of the note determines whether it is sharp or flat'. Where something could be replaced with offset on the fretboard from the perfect note. But there must be a phrase for that, a single word, like Quality for a property of chords as described in my Q.
Mar 19 '11 at 14:43
This is probably the best reference for the concept you're looking for:
http://en.wikipedia.org/wiki/Interval_(music)#Interval_number_and_quality
Your basic three-note chord contains two intervals: from the root to the third note above it in the scale, and from the root to the fifth above it. Both intervals have a quality, as do all of the intervals in the 12-note western system. If you go through the major scale and construct each of the 1-3-5 chords, one for each note in the scale (the so-called harmonic structure of the scale), you find out a couple important facts.
The first is that all seven chords have a "perfect" fifth interval except for the seven chord. That one has a diminished fifth interval, but the seven chord is usually left out of songs or modified, so in practice the fifth is almost always perfect.
That leaves the third interval, which is much more interesting. In three of the chords, it's major and the other four are minor. So there's more going on here. Because the fifth isn't very interesting, the quality of the chord as a whole is taken from that interesting third interval.
Why the explanation? Because your question about the "single" notes quality is one and the same with the chord's quality, namely the chord quality is just a shorthand reference the the third's quality. So they're the same notion.
The only nit I'd pick with your question is that there isn't any idea of quality without the comparison of two notes to one another, so a single note such as A has no quality. The third note of a scale, however, does have a quality because it is compared to the root note of the scale. For the twelve possible notes in the western system, the intervals (with their qualities) go:
Minor second, major second, minor third, major third, perfect fourth, augmented fourth/diminished fifth, perfect fifth, minor sixth, major sixth, minor seventh, major seventh
Technically, any perfect can then be further augmented or diminished (making it no longer perfect), which means I missed some overlapping possibilities in that list, such as the major third/diminished fourth overlap. I left those out to keep the list legible.
Likewise minors can be further diminished and majors can be further augmented, but not vice-versa.
Intervals are really the fundamental unit of music and emotion, so it's a good question and it doesn't hurt to spend some time getting to know the language of them so you have a way of communicating about them aside from just playing them.
If you really want to bend your mind, check out just intonation on wikipedia and learn why the western system isn't perfect (the Pythagorean comma, for example) and why there exist other systems than the 12-note system.
Notes are not inherantly sharp or flat. the only thing that determines if we hear them that way is the CURRENT MUSICAL FRAMEWORK, by which I mean the RELATIONSHIPS between notes we just heard, are hearing, or are about to hear. This is what is meant by a particular "Key". Chord "qualities" are all about the relationships of 3 or more pitches, not about which particular pitches are used. This is why a C major triad has the same "quality" (in this case, major) as a B major triad.
breaking music down to the level of single pitches in this way, means that there is no music. isolated pitches are not music, and listeners don't hear pitches that way anyway. We usually hear and pay more attention to the relationships between the pitches.
It might be helpful to distinguish flat/sharp when just diatonic notes of the scale from flat/sharp where you are actually raising or lowering a note to make it non-diatonic.
In the former case, there isn't really a term as far as I'm aware as there's nothing special about, say, F♯ over F other than a notational (and keyboard) bias towards C-major and the corresponding modes.
However, in the latter case, you could describe the note as non-diatonic or chromatic. For example, in C major, F♯ would be considered a chromatic note (although not that in G major, D major, A major, etc it would not). Note also that in keys like G major, D major, A major, etc it is F♮ that would be considered chromatic.
The term accidental generally only refers to the symbols ♭,♯ ,♮ in the specific context of a score so doesn't seem relevant to your specific question.
