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I recently read the great and free infographics pamphlet over here: http://tobyrush.com/theorypages/
And I was wondering why the dominant and the leading-tone harmonies are having a "dominant function" when they resolve in the tonic? Wouldn't you call it a subtonic function? Especially considering there seem to be no other functions with such name, and the principle of subdominant function would mean to resolve in a dominant.
Note that "subXXX" doesn't mean "resolving to XXX". It just means that the respective chord is as far (in scale steps) below the tonic as the XXX-chord is above the tonic.
The term subdominant function comes from the fact that it happens to be the subdominant chord that is one of the most used predominant chords (resolving to the dominant); but it does not imply that "subXXX" means a chord resolving to XXX.
The subdominant chord is only one example of a predomiant chord. Also the supertonic, the submediant, or the V7/V (among others) can have that function.
The dominant function by definition is the creation of tension resolving to the tonic, so there's nothing to argue about. The leading-tone triad is just the upper structure of the dominant seventh chord, e.g. in the key of C:
dominant-seventh chord: G7 = G-B-D-F leading-tone triad: Bdim = B-D-F
So they naturally have the same (dominant) function.
Finally, note that the term subtonic is used for the scale degree a whole step below the tonic, i.e., not for the leading tone (which is half step below the tonic).
Subdominant means lower dominant, ie the chord or tone a fifth below the tonic. The name has nothing to do with how this chord, or chords with a subdominant function, "tend to resolve". The tonic is the chord around which the naming convention revolves. Compare with mediant (a third above the tonic) and submediant (a third below).
And I was wondering why the dominant and the leading-tone harmonies are having a "dominant function" when they resolve in the tonic?
Because that is what it means to have a dominant function.
[...] and the principle of subdominant function would mean to resolve in a dominant.
Yes, but the name does not reflect that.
Wikipedia has a rather good article about functional harmony: https://en.wikipedia.org/wiki/Diatonic_function, which may help clarify things.
"Sub" doesn't mean "dominant of". It isn't a function like "dominant". It just means "lower".
The dominant is five up from the tonic, the subdominant is five down. The mediant is half-way up to the dominant, the sub-mediant is half-way down to the sub dominant. The tonic is zero steps up from the tonic, the subtonic would be zero steps down from it. Meaningless.
The "dominant of the dominant" isn't IV (the subdominant) it's ii (the supertonic).
Others have already answered the question adequately but I would like to address the issue that seems to be causing a problem: that of the apparent inconsistency in naming between those degrees of the scale beginning with 'sub'.
Tonic: the 1st degree
Dominant: the 5th degree
Subdominant: the 4th degree (5th degree 'down' the scale, hence 'sub')
Mediant: the 3rd degree
Submediant: the 6th degree (3rd degree 'down' the scale)
Leading note: the 7th degree of the scale
Note that we don't call the 7th degree the 'subtonic' when we consider scale degree for a major key, when there are no accidentals.
Tonic: the 'home key' chord
Dominant: the one that resolves to the tonic
Subdominant: usually resolves to the dominant
Supertonic: Chord ii, resolves to dominant, like the subdominant, so we can say it has 'subdominant function'
Leading note: Chord vii, resolves to the tonic, like the dominant, so we can say it has 'dominant function'
The term 'subtonic' is used solely to define the note one whole tone below the tonic.
Consider: the term 'subdominant' is telling you to count the same number of steps as the dominant (5) below the tonic. It is something of coincidence that the subdominant is immediately below the dominant, and quite reasonably you might think that the 'sub' just means 'the note below'. When in fact, the prefix 'sub' instructs you which direction to count in from the tonic!
Things become clearer when we consider 'mediant' and 'submediant': the former means count up three degrees, the latter means count down three degrees. (Bonus: the name mediant is derived from being between, or in the 'median position' between tonic and dominant. The same goes for submediant, which is in between the tonic and the subdominant, counting down, of course!).
Now coming to the 'subtonic' and 'supertonic', the same logic applies - count down one step from the tonic in the former case, and count up one in the latter case.
In the Common Practice Era, when considering chord function in a major key, the 7th degree is (almost) always sharpened and therefore always appears as the 'leading note' at cadences. This is not to say that the subtonic will not appear - it will, if there are modulations to the dominant, or if the natural minor scale is used, but in basic music theory exercises in a major key, you probably won't see the subtonic.
I think the key point is that 'sub' in this context does not mean 'a note immediately below', or even more generally 'a note below', rather that it tells you in which direction a note's degree is counted relative to the tonic.
The (excellent) internet source, you mentioned is here, for those who are interested, specifically this page.
See also the Wiki page on Diatonic Function for more information.
