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Came across an article at openMsicTheory.com about harmonic function. Below excerpt was about finding function of a chord
Each of the three harmonic functions — tonic (T), subdominant (S), and dominant (D) — have characteristic scale degrees. Tonic’s characteristic scale degrees are 1, 3, 5, 6, and 7. Subdominant’s characteristic scale degrees are 1, 2, 3, 4, and 6. Dominant’s characteristic scale degrees are 2, 4, 5, 6, and 7.
Can anyone advise what the writer is attempting to say?
It looks like the author is drawing this from Riemann's functional theory. In it, each of the triads built with scale degrees has a function: T, D, or S. I, III and VI are all T, II and IV are S, and V and VII are D.
T: I, III, and VI have (1, 3, 5), (3, 5, 7), and (6, 1, 3), which gives the set (1, 3, 5, 6, 7).
S: II and IV have (2, 4, 6) and (4, 6, 1), for (1, 2, 4, 6). The 3 comes from IV7 (4, 6, 1, 3) for the set (1, 2, 3, 4, 6).
D: V and VII7 have (5, 7, 2) and (7, 2, 4, 6), which gives the set (2, 4, 5, 6, 7).
Lyd is right in that 7s are involved in the determination and NickQuant is right in that the intersections are important, but they are not the whole story as that would include the V in T, which cannot be true in functional harmony. VI is included instead, as it is the relative minor to I and can have a tonic function.
There is one oddity about the 7s. For almost all of the triads, you can take them as 7s and it doesn't change anything; they just share more of the same notes. The oddity here is III7, which would include the 2. The discrepancy is due to the fact that the III has a dual nature in the theory, where it can be dominant (as it is dominant to the VI) or tonic. In fact, III7 is very dominant in that it contains the V, so while III can be T, III7 is D.
It's saying that each harmonic function is built up using different notes of the reference key, or tonic scale. Those are the notes that are used to build harmony that satisfies those functions. So, if you want to imply an harmonic function, or find the function of a set of notes, you'll be using those notes in some form and extent.
In other words, it's describing those three functions using pitches, notes. Each function is different, so it is described by different notes.
In the key of C we have:
1 2 3 4 5 6 7 C D E F G A B
That makes our tonic, or tonal center, or "home", to be C major. The subdominant is in the 4th scale degree, so F major, and the dominant is at the 5th scale degree, so G major. (For the chord qualities, we are just stacking thirds)
The tonic chord is C major, Cmaj7 chord is C E G B, which are the scale degrees 1, 3, 5, 7. Article mentions 1, 3, 5, 6, and 7.
The subdominant is F major, Fmaj7 chord is F A C E, which are the scale degrees 1, 3, 4, 6. Article mentions 1, 2, 3, 4, and 6.
The dominant is G major, G7 chord is G B D F, which are the scale degrees 2, 4, 5, 7. Article mentions 2, 4, 5, 6, and 7.
So, each harmonic function is characterized, described, achieved, by using those notes. Which makes sense, since they are built using mainly the diatonic chords of that scale degree (1 for tonic, 4 for subdominant, 5 for dominant).
On each function we are still missing one note, though: 6 for tonic, 2 for subdominant, 6 for dominant. According to the article, the 6 in tonic is an "associate", 2 in subdominant is an "associate", and 6 in dominant is a "dissonance". It also classifies all notes, but doesn't seem to explain exactly what it means by this categorization.
I guess, he means the following: (I will do it on the example of Tonic characteristic scales)
1, 3 and 5 are obviously characteristic degrees of a tonic, right? The other two triads, namely on the 6th degree (6, 1, 3) and on the 3rd degree (3, 5, 7) have an intersection of two notes each with the tonic triad (1, 3, 5), and apart from that each one has an additional degree 6 and 7. So each of these notes - 1, 3, 5, 6, 7 may point at at chords with a "big" intersection with tonic (triads on 6, 3 and 1 - the tonic itself).
Each of the three harmonic functions — tonic (T), subdominant (S), and dominant (D) — have characteristic scale degrees. Tonic’s characteristic scale degrees are 1, 3, 5, 6, and 7. Subdominant’s characteristic scale degrees are 1, 2, 3, 4, and 6. Dominant’s characteristic scale degrees are 2, 4, 5, 6, and 7.
If those scale degrees (indicated by ^) are re-arranged as...
subdominant : ^2 ^4 ^6 ^1 ^3
dominant: ^5 ^7 ^2 ^4 ^6
tonic : ^1 ^3 ^5 ^6 ^7
...or as letters in C major...
subdominant : D F A C E
dominant: G B D F A
tonic : C E G A B
...you get...
subdominant : ii9
dominant : V9
tonic : I6 or IM7
That is very much a tonal view from the world of jazz.
In classical harmony ^3 isn't part of the subdominant, and the tonic is only ^1 ^3 ^5.
The basic idea is the primary chords - the tonic, dominant, and subdominant - are built of thirds above the roots ^1 ^5 ^4. In jazz those thirds are stacked up to make ninth chord. In the classical world the primary chord are triads with seventh chord being used under somewhat strict voice leading procedures.
The description on the linked page may not be clear, because it doesn't make a clear connection between the 'characteristic scale degrees' and their connection to the chord tones.
A final point seems worth mentioning: the subdominant or pre-dominant function can be fulfilled by either the subdominant chord IV (scale degrees ^4 ^6 ^1) or the supertonic chord ii7 - often inverted as ii6/5 - (scale degrees ^2 ^4 ^6 ^1.)
