Ditto. I can also get an A bang in tune to 440 without a reference point, through nothing more than practice, having had to do it numerous times a day for many years. Can’t tune any other notes, though, because, well, A is the only one you need 
I don’t disagree.
Yeah, this learned skill is apparently is different (and a point I was trying to argue with the researcher, along with guitarists who always seem to know exactly where that E string should be tuned) than “true” absolute pitch.
It’s more variable than that, especially as (I’m sure my posts have made clear) I would argue that there’s no such thing as a pure scale!
String players do vary intonation slightly away from equal temperament, for example bringing leading notes ever so slightly closer to the tonic on which they are resolving, and pushing minor thirds just that little bit lower. The difference in timbres with inflexible instruments, namely the piano, helps avoid a clash, although the degree to which these alterations are made does depend on the ensemble in question. It’s one reason for enharmonic modulations sometimes not sitting comfortable, at least until they’ve been played through a few times so that the necessary compensations are made (sometimes an entirely subconcious process). I would suggest it’s also a reason that players’ early encounters with entirely atonal music, and in particular serialism, give some difficulties with intonation.
There’s also the related issue of the compromise between tuning resonant 3:2 fifths across strings, and adjusting towards equal temperament particularly to avoid a discordant interval between the violins’ E and the C of the violas & cellos.
Hey, Sapo, here is a virtual keyboard: Flash Piano - The Method Behind the Music
Click on each key, starting at one end, and working one key at a time to the other end. You’ll notice the same tonal distance – a semitone – between each key, whether it is a white key or a black key.
For example, if you start at F and work your way up through F#, G, G#, A, Bb, and B, you will be progressing up the keyboard one semitone at a time. To continue this sequence, just press the next key C – don’t worry that there is no black key there to press. As you can hear, no note is left out of the sequence despite there not being a black key between B and C.
Now keep on trucking along up the keyboard and you will see that the same thing happens between E and F, which are a semitone apart despite being two white notes. Every key, be it white or black, is a semitone apart from the key immediately beside it, be that key white or black.
If you want to get into the relationships between keys, google about on the term “music theory”. There are a lot of on-line music thory sites other than the one with the keyboard, so poke about until you find one that works for you.
A very useful link, thanks! I now have a good answer the next time I hear “But I don’t have a piano at home” from a kid! 
If you click on the “Click here to put the piano in a new window” line, it will open a keyboard popup that you can move about on your screen, and still use even when you go to other websites: http://numbera.com/musictheory/keyboardpop.html
Have a boo at the Railsback curve Piano acoustics - Wikipedia and how stretched tuning Pseudo-octave - Wikipedia is used to keep the overtones from conflicting they way they do if a piano is just tuned by the numbers Piano key frequencies - Wikipedia
Also, what we hear and what we thing we hear are not always the same thing, for a lot of folks find tones in the higher registers to be flat, so tuners often tweak them a little on the sharp side to account for the difference in perception Piano tuning - Wikipedia
When I trip and stub my toe on my piano, the result is a piercing, pained, and far from perfect 587.330 D. Tuning the piano does not seem to help much with this.
I actually went and fiddled with the numbers this morning and found that:
[ul]
[li]The simplest ratio after the octave (1:2) is actually the fifth (close to 3:2) not the Major 3rd[/li][li]The next three in the series are the fourth (4:3) then Major 3rd (5:4) then Minor 3rd (6:5)[/li][/ul]
So apparently, I was completely talking out of my arse in the first place, and there’s actually not that much difference, mathematically speaking, between a Major or Minor 3rd
OK, I was phrasing that rather misleadingly - yes, there’s something of an outline of a major triad with the 5:4 and 6:5 ratios - but not of a minor triad. That the latter is just as an important element of western tonality suggests that the overtone series really doesn’t serve as an explanation at all.
Additionally, to preempt any suggestion that you could just swap the intervals around, they’d then be derived from two completely different fundamentals.
Yes, and if you have a good ear you can hear the effects of this. Two examples I can think of are a section of the Grand March from Aida and the first trumpet entry in 633 Squadron. In both cases the trumpets, which are non-equally-tempered instruments, are having to play in an awkward key, and some of the notes are slightly out of tune.
Sometimes sad, sometimes comical.
What?
I can do this.
I used to work in a store that sold digital pianos and organs and each had the ability (which we would occasionally demonstrate) to tune the instrument up or down by 0.1 Hz.
I could always tell by sitting down at the instrument and playing a few notes if it was off, even by the tiniest amount.
IIRC the Ride of the Valkyries is another example, with a key which gives a strident rather than triumphal sound to the brass.
Just to kick that back into layman’s terminology again…
A major triad has the harmonic intervals 4:5:6. So that’s a nice, mathematically simple elegant ratio (score 1 for the “mathematical elegance” model of why we like what we like)
A minor triad would be (scribble scribble) - uh - 10:12:15 ? Which is not mathematically elegant at all (score -1 for the “mathematical elegance” model). But I’m not sure that I agree that a minor triad is just as important an element of western music - most music seems to be written in a major key - it seems that people start making music in minor keys simply because they’ve exhausted all they want to do in a major key first?
A diminished chord would be 5:6:8. Which is getting back into nice elegant ratios territory. And it seems to me just from casual listening/playing contemporary music that diminished chords are more common than minor chords … but bearing in mind that I know about two fifths of bugger all actual formal music theory, so I don’t have a handle on what’s considered to be “important” by serious musicians.
Fascinating thread 
5:6:8 is a first inversion major triad. 5:6:7 would be the diminished triad, but that involves the 7th partial, which nobody uses because it’s so flat.
Not at all! Pre-‘tonal’ music (I dislike using the term to mean ‘major/minor’ but it’s commonly employed in this way) has no less in common with familar minor keys than with major ones.
In any case, what I’m emphasising is the equivalence we hear between major and minor chords as well as keys. You can’t use C major as a key (rather than just a scale) without encountering chords of D minor and A minor.
You’re absolutely right that most music is not formed of simple consonant sounds. Controlling how those which ‘sound right’ work alongside those which don’t is the function of keys, and only using major and minor triads in isolation would be very boring indeed!
You’re right, I copied down the wrong column in my notebook. Need more caffeine. 5:6:7 is still a nice mathematical ratio though
It seems to me that people do use diminished chords reasonably frequently in guitar tabs (which was what I was thinking of upthread). But I suspect that the guitar is not exactly an equal tempered instrument, so I don’t know if that makes a difference.
Incidentally, I seem to recall that there was an experimental piano made with alternating white and black keys, something like this:
**C# D# F G A B**
C D E F# G# A#
I can’t recall whether C was a white note or a black note, but you get the idea.
It had a few advantages: it was much easier to transpose from one key to the next (you just move up to the next white key). There would only be two sets of fingering to learn — white-key and black-key fingering. Your hands wouldn’t need to be as big, because there were more keys per inch.
It never caught on. Without labels on the keys it’d be hard to know where you were, especially if you took your hands off of the instrument to go up a few octaves.
That is the greatest advantage of the current piano system. No, it isn’t symmetrical and it isn’t uniform white-black-white-black. Because of this, it’s much easier to see at a glance where you are.