My theory of harmonic functions

[Z1202]

FWIW I decided to share my theory of harmonic functions (of major and minor scales), which I have worked out by myself based on harmony theory textbooks followed by some composition practice. Quite surprisingly, I never encountered these ideas elsewhere, would be happy to hear otherwise. So here we go.

The harmonic function of a chord is mainly a combination of two factors.

1. The strongest and the primary factor is the presence of the 6th or the 7ths degree of the scale in the triad

• any triad containing the 6th degree has subdominant function (II, IV, VI).
• any triad containing the leading tone 7th degree has dominant function (III, V, VII), where the III of the major scale is rather weak as the dominant and rather falls into the third group below.
• any triad containing the natural (Aeolian) 7th degree belongs to the third functional group, the name of which I don't know, if it exists.
• the raised 6th of the melodic minor could be considered as giving rise to a special flavor of subdominant group (with a questionable member of the diminished root position chord on the 6th degree, thus instead one could rather simply apply the second rule below to the remaining II and IV chords with raised 6th degree of the scale)
• note that thereby the tonic triad is the only one not belonging to one of the above mentioned groups.

2. The less strong and the secondary factor is the bass note, which may define the chord function to be the one of the root position triad built on the same bass (where the latter is defined by the first factor above, thus this secondary factor is thereby a derivative of the primary one):

• sometimes this rule makes no difference (e.g. the first inversion of the IV triad is built from the 6th degree of the scale as the bass, which still implies a subdominant-group root position chord)
• sometimes it makes a difference (e.g. the first inversion of the tonic triad is built from the 3rd degree and often can function as the root position III chord)
• in the sense of this rule, all chords built from the same bass have the same function regardless of the notes above the bass (thus, e.g. first inversions of triads should be rather considered as chords with added 6th, and second inversions as chords with added 6ths and 4ths)
• the rule clearly contains the case of adding higher-order tones to the chord (the 7th, the 9th, etc.), but also the 2nd, 4th and 6th (if the latter are not considered as the 9th, 11th and 13th).
• if the tones above the bass include the 6th or (especially raised) 7th degree of the scale, the first rule tends to dominate and the bass note tends not to matter in regards to the harmonic function, especially if this tone belongs to the respective triad (e.g. the first inversion of the leading-tone triad has a stronger tendency to work as a dominant chord rather than as a chord of the 2nd degree of the scale with added 6th)

Note that the above two rules are more or less in agreement with the classical harmonic theory, it's more about putting different focus and a (somewhat) different way of thinking. However sometimes there can be other factors defining the harmonic functions. E.g. once I had the 4th of the scale as a bass note of the dominant chord, and this note didn't function as the chord's 7th at all, more like a weird root. The factors allowing that were the previous history of harmonic changes (kind of harmonic sequence) and the melodic logic of the bass.

In composition practice I find it most convenient to use the two factors in the opposite order, concentrating on the second factor as the primary means to plan the harmonic function and then being aware of exceptions (mostly rising from the cases in which the functions defined by the primary and the secondary factors are different and the primary factor wins).


Disclaimer: these rules are actually my attempt to express my own intuitive understanding rather than a formal list that I'm following. I possibly have missed some points and may refine them later.

Comments, thoughts, feedback?

[Z1202]

A couple of forgotten things.

In the first rule the strength of the function grows (or rather morphs from the tonic function to the respective function) as the chord moves away from tonic, going in thirds. That is the morph from tonic to subdominant function goes I -> VI－＞IV -> II and the morph to the dominant/3rd functional group goes I -> III -> V -> VII. That probably explains why the III in major doesn't really work as a dominant: its dominant function is still too weak (especially since the deciding 7th degree of the scale is the chord's 5th, whereas for VI which has a relatively clear subdominant flavor the deciding degree of the scale is the chord's root). OTOH in harmonic minor the dominant function of the III is reinforced by the presence of the raised 7th of the scale. Note that in Aeolian scale the III is also not a very convincing dominant, and is probably functioning more as a tonic of the relative major.

According to this theory the double dominant is nothing more than a special flavor of subdominant. Remarkably, IIRC, I have heard somewhere that in Russian Saint Peterburgh's school of harmony they talk of an altered supertonic rather than double dominant. In the same sense the dominant of the subdominant is just a flavor of the tonic. Other secondary dominants in the tonality may be interpreted as dominants of the tonic/subdominant/dominant of the respective relative minor or major. E.g. in minor that includes V of III, V of VI and V of VII (natural) and in major that is V of II, V of III and V of VI. This interpretation is practical, because the dominant function reinforces the relative minor or major tonality.

[Z1202]

Actually upon thinking a bit more, there are two factors contributing to the strength of the function: the distance away from tonic (the strength grows away from tonic) and the position of the deciding scale degree among the triad notes (which grows from 5th of the triad to its root). For dominant function both factors are aligned and the strength grows rapidly III->V->VII, where for III it's so weak that the function is barely present and for VII it's very strong. For subdominant both factors are counteraligned, so the chord with the most clear subdominant function is IV, although it still feels stronger in some sense in II (thus the distance from tonic may be a stronger factor than the position within the triad)

Edit: some further notes.

As we move away from tonic there is a sudden jump from dominant to subdominant function: I -> III -> V -> VII --> II -> IV -> VI -> I. The jump occurs where the chord loses the scale's 7th and obtains the scale's 6th.

The seventh chord on the VII scale degree still has a clear dominant function, because the 7th scale degree is in the root while the 6th degree is only in the chord's 7th. In harmonic minor the dominant function is even more reinforced by the raised 7th of the scale.

[Pythagorean]

The determining factor of tonic, dominant and subdominant function is in the end the root of the chord.
Under your rule 1 you've simply shifted this to the third above the root (the 6th of the scale for the subdominant and the 7th of the scale for the dominant).
Functions like you've stated them are not set in stone. For instance in major the minor triad on ii does not necessarily have a subdominant function (in which case it serves as a IV with added sixth) but can also be a true ii chord with the 2nd scale as its true root often described as the superdominant.
Also, chords do not have to be triads of course, and chords do not have to have their root present nor their fifth or third. And chords get their function based on context.
There are many theories of harmony which work with function. To see everything as a tonic, subdominant or dominant chord / function is too limited a view in my opinion.
My personal view is that all the natural scale steps can have a true / unique harmonic function and be the true root of a chord except the 7th scale step in major and the 2nd scale step in natural minor.

[Forgotten]

Sorry, but what you have written is a little confusing in places. Maybe you can go back and make some edits for clarity?

You’re not always telling us which scale you are talking about and if it is all major (except where noted) the diatonic chord symbols don’t match with major. For example where you appear to be referencing major you used “III”.

Also your use of the word “dominant” related to chords is a little confusing, as you use it both to refer to the 5th note in a scale and then to other notes.

Maybe read back what you wrote and see if you can see these instances?

[jancivil]

I have no sense at all of what a “true” ii chord is supposed to be, as opposed to subdominant; the three terms regard function. ii to say ‘I’ is functionally always pretty much Plagalesque at least. ii to V is same subdominant as IV, and first inversion pretty prevalent for CPP music, ie., same bass.

I think we’re on the brink of being unnecessarily verbose there. EG: Is I ii iii really functional? Where’s the real interest?

I left the OP alone all this time. Bit of a clusterfuck, but whatever.
It’s quite idiosyncratic stuff, I mean these rules, no, not to anyone but you. The “degree of the scale” you look at for function is to identify strong vs weak root (or eg., ii6, bass) movement. Function, all rather simple, really but we get a lot of lingo that doesn’t really travel, hate to say.

[Ploki]

[*]in the sense of this rule, all chords built from the same bass have the same function regardless of the notes above the bass (thus, e.g. first inversions of triads should be rather considered as chords with added 6th, and second inversions as chords with added 6ths and 4ths)

They don't function like that in chord progressions tho.
(they can, but not always)
this rule is not really a rule and its wrong mostly.

Remarkably, IIRC, I have heard somewhere that in Russian Saint Peterburgh's school of harmony they talk of an altered supertonic rather than double dominant. In the same sense the dominant of the subdominant is just a flavor of the tonic.

Supertonic just means "above tonic", or II.
on its own it doesn't mean anything. In practice, it was usually the chord that preceded the dominant, hence the "double dominant" term.

That's the bitch about chords, same three tones can be a bunch of different chords if the context is right.

[anomandaris1]

Your didn't define anything, wow, why do you think your "theory" is worth publishing on public forums?; I would stick to Riemann-Oettingen or Schenker-Schoenberg for now (Some of these should be out of copyright these days). (Btw, their theories also fail in several ways, but I don't want to go in details - this topic is worthy of a book on its own).

[Ploki]

it's a forum, not a peer-reviewed journal, and anyway, that's one of the way to learn - to think and develop ideas, despite some being wrong. It's a process.

[jancivil]

Yeah, this is light years from ‘publishing’ in any critical sense. I’ll have just let it fade into oblivion but it got necro’d. One does learn from mistakes, ideally. The OP just barks up a wrong tree and keeps going, tho

[jancivil]

I’m not into theory at the level of caring a lot about eg., Riemann, or Schenker, really at all. Schenker is interesting and deep, but who has time for one individual’s takes. Except maybe Schoenberg, Structural Functions of Harmony. IE., what did Riemann write we care a lot about?
I struggled with Perle and all the Webern sets and series (the interest for me was how it involved timbre) and so on, for some time. I don’t know I was up to retaining all that.

Just my take, don’t mind me.

[jancivil]

Remarkably, IIRC, I have heard somewhere that in Russian Saint Peterburgh's school of harmony they talk of an altered supertonic rather than double dominant. In the same sense the dominant of the subdominant is just a flavor of the tonic.

bII7b5 is equivalent to V7b5.
Tristan chord is iv in first inversion, a “6 chord” but augmented 6th (& a #4, “French” alteration replacing what for the “German 6th” would be the 5 in a construction exactly like bVI dom. 7, just like our 7b5 seen above); so the putative subdominant is made into a *secondary dominant*. It turned heads because its sonority is that of a half-diminished 7th harmony (due to a further alteration, G#).
There is nothing unique to any School there, I got all this at community college.

”Double dominant” is probably a term from some School, but you have one dominant at a time (unless this is supposed to describe bitonality).

“just a flavor of the tonic”? Tonic function, dominant function, two things. I may become V/IV but then it’s the new function.

[jancivil]

IE: key of A minor:
F A D is now F A D#; German 6th looks like F A C Eb, bVI7, and we’re headed to V7. B instead of C, French 6th. Appoggiatura G#, to A to a big ol’ A# on E7. People thought he was mad, no one saw that one coming, sounding like Fm7b5* (owing to the G#) in the key of A minor but it morphs into dominant of E. (*: diminished/minor 7, ‘half diminished’: now F B D# G#, cf., F Cb Eb Ab.).

Flat five substitution principle also, too.

[jancivil]

any tone whatsoever can be the true root of any harmony. That is axiomatic and perfectly clear. The two exceptions supposedly are diminished triads. No. I see why one could think that, because the all m3 and symmetry produces ambiguity.

You could not proceed in a part-writing course not believing diminished chords ever have a true root in a tonality. This may be demonstrated by the fact of eg., ii7 in minor: key = A minor, ii7 is B D F A. Root, 3rd, 5th, 7th. The supposed problem owing to symmetrical construction is obviated, poof. 2 there is its true root, there is no denying it. One may want to make a Major 7th on the diminished triad, as well as a diminished seventh. In analysis you name it or your grade lost some points.

The 7th is typically described apart from the triadic basis, eg., diminished/minor 7. Major/minor 7; minor/major 7... diminished/diminished seventh is what happens in a four note chord on 7 in minor.
B D F Ab. You may spin it into another direction and you might need a new name...

[Ploki]

”Double dominant” is probably a term from some School, but you have one dominant at a time (unless this is supposed to describe bitonality).

“just a flavor of the tonic”? Tonic function, dominant function, two things. I may become V/IV but then it’s the new function.

English isn't my native nor the language i studied in, but i think double dominant is referring to "dominant of the dominant" or V/V, which is technically on the II degree (supertonic).
I found the literal translation in one of our papers and it supports this assumption, but it's not a widely used term at all.