Lesson 5: Diatonic Substitute functions, beginning prolongation

Remember last week the chord P6?
In context of C major and a given piece, this sonority is Predominant, but has a sixth above the fa in the bass, instead of a fifth. In a different context, these notes (re fa la) might have a different function. That is one of the most important things to remember about Functional Analysis: the same notes might function differently in different context, meaning one chord might have multiple labels even if it looks or sounds the same out of context.

We already know that there are more than three chords in a given key, even if there are only three primary functions. Today we will discuss how these other, substitute functions relate to the primary functions.


Now, you can do a lot with only three chords. Some famous 20th Century examples include Johny Cash “Ring of Fire” https://www.youtube.com/watch?v=It7107ELQvY and Lynryd Skynyrd “Sweet Home Alabama” https://www.youtube.com/watch?v=IwWUOmk7wO0

Both of those songs are also good examples of the plagal cadences; most of the phrases end S T. [*]

In fact, the catchiest songs that get stuck in your head the most probably have no more than four chords. Think of the four chord pop song as demonstrated by Axis of Awesome (some swearing): https://www.youtube.com/watch?v=5pidokakU4I Another version of four chord songs is the Pachelbel[†] type of progression (also some swearing): https://www.youtube.com/watch?v=JdxkVQy7QLM


Clearly both of these are exaggerations, with most full performances of any of these songs having elaborations and the occasional embellishing chords. But the Pachelbel outline leads us to the first substitute chord we want to talk about, the Relative. In D major, the Pachelbel type progression is D Bm G A. The solfège for each of those chords: do mi so, la do mi, fa la do, so ti re. That means that with what we already know we can label the chords’ functions as T ? P D. Notice that D and Bm have two notes in common, do mi. This means that the Bm chord is functioning as a prolongation of the tonic chord.

Since we generally want to describe harmony with the three primary functions, we will spend a good bit of time talking about how the sense of these three functions is carried even when the literal tonic (or predominant or dominant) chord might not be present. Prolongation is that sense of still being in the primary function area, even if other embellishing chords are sounding. The most common functional area to prolong is the tonic (with dominant as a close second). Generally, a phrase will start on the tonic chord, explore the tonic area thru various prolongations, and then close with a P moving to D to cadence.

There are many ways in which to prolong any primary function, and if you want to delve into detail about what each type of prolongation is called and how it works, you certainly can.[‡] For the purposes of this introductory Functional Analysis series, we will say that prolongation exists in many forms, and we explore some of them without necessarily covering all the details right away.


This progression, D Bm, uses a type of chord called the tonic relative. Since D and Bm triads share two notes in common, our ears can hear Bm as a place holder and perceive the tonic area to continue. We abbreviate this Tr, using a lower case “r” because the chord is minor, but the upper case “T” because the tonic it is based off of is major. We use the term relative because the keys D major and B minor have the same key signature (2 sharps) and are know as relative keys. In general, the relative chord of any major function is going to be a minor triad formed on the note a third below the root, and share two notes in common.

The re fa la example from the top of the post is the Predominant relative in C major. Predominant is F (fa la do), the relative is a triad with the root down a third (F-E-D), and they share two notes, fa la. Now context determines whether we hear the fa functioning more strongly that the re to differentiate between Pr and the P6 we had last week. If the re is in the bass, chances are that slides the scales towards Pr. If the fa is strongly doubled or accented, it might shift towards P6. Each case must be taken individually.

Dominant relatives (mi so ti in major keys) are possible, but fairly uncommon. (This triad is more often used as a Tv, see below.)

What if we’re in a minor key? Since we base this relation off the relative key identity, relative relations are reversed in minor. To find the relative minor of D major, we went down a third. But, to find the relative major of B minor, we would be going up a third instead. Therefore, for a piece with the Bm tonic (t, solfège do me so), the relative would start a third up and be abbreviated tR (minor tonic, major chord sounding, solfège me so te). This stays the same for predominants (p – fa le do, pR – le do me) and dominants (d – so te re, dR – te re fa). The pR chord will be important when we get to chromatic alterations. The dR gets used as a cadential pre-tonic in some current vernacular genres (good examples are Bon Jovi “Livin’ on a Prayer” chorus https://www.youtube.com/watch?v=lDK9QqIzhwk and Simon and Garfunkel “Scarborough Fair” https://www.youtube.com/watch?v=-BakWVXHSug )


Now, relatives aren’t the only type of substitute function. The same solfège that make a tonic relative (la do mi) also share two notes in common with the predominant (fa la do). So if that chord shows up in a predominant prolongation, or leading to some sort of dominant chord, it could be heard as a predominant substitute. We’ll call it the Variant, Pv. Again, since the predominant is a major chord, upper case “P” and the literal chord is minor, lower case “v”. The variants are similar to relatives in that they share two notes in common with their primary function, and have a root a third a way. It’s just that the third is the opposite direction of the relative.

In major, Pv is the most common, but Tv (mi so ti) is also possible, if somewhat uncommon. To build a minor triad up from ti to have a Dv would require a chromatic note, so while theoretically possible, Dv is almost unheard of.[§]

In minor, the tonic variant relation is fairly common. Again reversing directions from major, the tonic variant in minor (tV) is le do me (same notes as pR). However, dV (me so te) is more likely to be a tR in context, and pV is chromatic.[**]


To summarize:

A relative of a primary function/chord shares a similar third relationship as relative major and minor key signatures. A variant relation is a third in the opposite direction as the relative.

Common substitute functions and their solfège:

Tr,            Pr,         Pv,         (Tv)

la do mi,     re fa la,    la do mi,   mi so ti


tV,            pR,         dR,         tR

le do me,    le do me,   te re fa,   me so te


One use of Tr (or tV in minor) is as an unexpected cadential resolution known as the deceptive cadence. This is when, at a cadence, the Dominant chord has sharpened our expectations for the coming tonic, but then instead the chord based on la (or le) appears instead, deceiving the resolution. In this case the Tr or tV substitutes entirely for the tonic (in a less stable fashion) instead of prolonging the tonic part of the phrase. Compare 0:14 to 0:25 of this recording of the second movement of Schumann’s piano quintet: https://www.youtube.com/watch?v=zh__fSU3ABY

The first is the deceptive resolution D tV, the second is the expected, D t. This alternation between deceptive and authentic cadences underlies most of the movement. It provides an unsettled feeling a propels the next phrase to answer the metaphorical question and return to the tonic.


Next week: diminished triads


Main Ideas

The same notes will function differently in different context!

Prolongation – the idea that a primary function can continue on some level even while other notes that might not belong to the triad are present. We will mainly deal with identifying the pillars (primary functions) and some of the other chords or intervals (numbers from last week) that help prolong the function. However, ideas like non-chord tones play off of this idea, especially when tracking melodic prolongation (vs. harmonic prolongation).

Substitute function – a chord that serves the function of tonic, predominant, or dominant without having the notes of the T P or D triad. Specific types are relative and variant (below). Substitute functions might serve a prolongational purpose or completely replace the primary function for variety, novelty, or subtly. If the primary function is major, the substitute will be minor, and if the primary function is minor, the substitute will be major.

Relative – similar to how relative major and minor keys have the same key signature, a primary function and it’s relative triad share a third root relation, and two notes in common. In major, it’s a diatonic third down from the primary root, in minor it’s a diatonic third up.

Variant – the opposite of the relative substitute. A minor triad with a root a third above the primary function in major, and a major triad with the root a third below in minor.

Deceptive cadence – generally D Tr or D tV. Technically any unexpected resolution of the dominant could be labeled deceptive, but generally the two listed are what is referred to, unless otherwise noted. Often feels like a question.

[*] https://tabs.ultimate-guitar.com/j/johnny_cash/ring_of_fire_crd.htm and https://tabs.ultimate-guitar.com/l/lynyrd_skynyrd/sweet_home_alabama_crd.htm for those interested in chords/lyrics

[†] A baroque version of Pachelbel includes more than four chords, but we’ll talk about that more later when we get to sequences: https://www.youtube.com/watch?v=JvNQLJ1_HQ0

[‡] Try Schenker!

[§] The diatonic ti re fa triad is diminished, and we’ll get to labeling that soon. Dv would be ti re fi (raised 4).

[**] ra fa le

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