Appendix IV: The Dual Function of the Tritone
Obviously, the tritone divides the octave exactly in half, and so it is a symmetrical musical structure (The inversions are identical in sound, if not in notated interval: Diminished fifth, and augmented fourth). Since there are two notes involved, there are only six tritones in the dodecophonic (twelve tone) system, and yet there are twelve - twice as many - overtone chords. This is because either note of the tritone can function as the leading tone, and either note in the tritone can function as the leaning tone: It is the passive tones - the root and perfect fifth - that put the tritone into context.
For example, the overtone chord for the key of C major is a G(m7) (Or G7, to be less precise), and its notes are G, B, D, and F. The B and F are the tritone, and the G to D perfect fifth - the passive tones - give the tritone its context. Here, the B is the active leading tone, and the F is the active leaning tone. However, this same tritone, if enharmonically notated as F and C-flat, can belong to a D-flat(m7) (D-flat7) when spelled D-flat, F, A-flat, C-flat. Here, the passive perfect fifth from D-flat to A-flat reverses the role of the active tones of the tritone from their functions in the previous G(m7) (Or G7): The F is now the leading tone and the C-flat (Formerly B-natural) is the leaning tone.
Jazz theorists use this shared tritone to justify so-called Substitute Secondary Dominants. For example, to us jazzers, a G7 to C progression can be replaced by a D-flat7 to C. This leads to jazzy cliches such as e(m7), A7, d(m7), G7, to C(M7) being reinterpreted as e(m7), E-flat7, d(m7), D-flat7, C(M7), where the E-flat7 and D-flat7 are substituting for the A7 and G7 respectively. In jazz harmonic analysis, the first progression would be iii(m7), V7/ii, ii(m7), V7/I, I(M7), while the second one would be ii(m7), subV7/ii, ii(m7), subV7/I, I(M7).
In classical theory, however, the jazz subV7/V (A-flat7: A-flat, C, E-flat, and G-flat) appears as the idiotically so-called German Augmented Sixth sonority, and is notated A-flat, C, E-flat, and F-sharp. The augmented sixth interval from A-flat to F-sharp is enharmonically exactly the same as the minor seventh from A-flat to G-flat in the jazz subV7/V.
Though the classical nomenclature is foolish - Ger.+6 in an analysis tells one less than nothing about what is going on or how the chord functions - the notation is, in fact, right in line with what the harmonic overtone series implies. Fact is, a German Augmented Sixth chord is an altered secondary dominant (!), which is, in this case, and altered D(m7) (Or D7): V7/V.
The way this chord is created - as well as what the overtone series implies is the logical basis for it - goes back to the passive tone/active tone dichotomy. The D(m7) is spelled D, F-sharp, A, and C, where the D to A perfect fifth represents the passive tones, and the F-sharp to C represent the active tritone. In order to increase the resolutional impetus of this D(m7) chord, we can either change the A to A-flat, thereby making it a leaning tone (Which creates a D(d5m7) chord (Or D7(flat-5), in jazz terminology)), or we can replace the root with E-flat, thereby making it a leaning tone (Which creates an F-sharp fully diminished seventh chord in a 4/2 orientation). If we perform BOTH of these alterations simultaneously, we get the so-called German Augmented Sixth chord, with the traditional notation.
So, though the jazz terminology is more useful than the classical terminology from a certain descriptive viewpoint, it is the traditional notation that represents what the overtone series implies as the logical origin of the sonority. Again, by looking at these chords a simple cases of altered dominants and secondary dominants, they become avilable to target any chord in the integrated modal system. This is, literally, light years beyond jazz or traditional theory.
Another few points reised in comments - one by a traditional theorist who thought it necessary to mention that he had a PhD in music theory, and was a "lecturer" in music theory at a university *eyes roll* - and also by a student can be answered with an email I sent to said student:
"The tritone is the only dissonant interval within the harmonic interval range of the overtone series (that does not invert to a melodic interval: The minor seventh from G to F is also a dissonance, but it inverts to a second. - ed), which is partials 1 - 8. If we start at G1 the series goes, G1, G2, D2, G3, B3, D3, F3, and G4. Every harmonic interval within those eight partials is consonant (The major second from F3 to G4 is a melodic interval) except for the tritone. We only need the G4 in there to get the 8:5 minor sixth from B3 to G4 (Which is the inversion of the major third from G3 to B3).
The tritone "wants" to resolve because it is dissonant: It wants to become a consonance, specifically a major third if it is within a root position overtone chord (Dominant seventh). A distinction must be made between "wants" to resolve and "has to" resolve: The overtone chord does not "have to" do anything, as its function as a tonic in Blues tonality proves. The "wants to resolve" is best viewed as a desire, and not a mandate."
So, sure, in theory the overtone series continues infinitely, but music is a harmonic system, and so music doesn't "care" about the overtone series once the harmonic intervals are exceeded.
As a basketball fan and a Pamela Anderson fan, I always suspected this was the case.