Having completed the exposition of most concepts common to common-practice tonality, we move on to chords that don’t quite fit Functional Analysis as I originally conceived it. None of the following chords can really be said to be complicated, uncommon, or exotic. However, in the context of Functional Analysis’s common-practice tonality basic metric, they are considered unusual, and some were initially difficult to label. These more colorful harmonies show up more often in later genres, such as late-Romantic era music (mediants) or 20th/21st century pop music (dominants other than V).
Chromaticism became more and more elaborate through the 19th century. Composers like Reger, Liszt, and Wagner among others began to experiment with chord successions that did not necessarily follow functional logic. Some of this chromatic music can be explained with tonal relationships; others cannot. Functional Analysis may be only partially useful for musics in which functional tonality is not the main driving factor – but still may be useful on some level. On the other hand, much chromatic music can still be explained with some references to a tonic, and then Functional Analysis can be extended to include these chromaticisms. For example, a chord might include altered tones that resolve in a functional manner. In fact, the increasing prevalence of chords like the augmented sixth or Neapolitan is a good example of altered tones that resolve functionally.
Another type of chromatic color is chromatic mediants, which begin to occur both as key areas and as chords. They are fairly easily described as alterations of relative and variant third relationships discussed in Diatonic Harmony. The following examples show a few of these more chromatic relationships, starting with the chromatic mediants available in a major key in Example 3.43. Some mediants may be functional, alterations of a function, or not functional at all.
Because the capital T refers to the home tonic, we know the variant would normally be built on the major third mi (second chord), and that substitute functions are the opposite quality of the primary – Tv. However, if we have TV (third chord), that would be a major chord built on mi, not a closely related key at all. Likewise with TR (last chord), which is the parallel major of the relative minor. Other chromatic mediants can be seen as more familiar chords borrowed from the parallel minor, such as tR, but Example 3.44 shows a few more possibilities.
These chords are now the minor version (tr, tv) of those that would normally be major triads in a minor key (tR, tV). Since much music of the 19th century already uses modal borrowing (chords from the parallel major or minor are available in any key, see Example 3.28), any of these chromatic mediants from major or minor could hypothetically be found in either major or minor keys.
These chords expand the realm of harmonic possibilities when considering modal borrowing and other chromatic techniques. Students may enjoy exploring the relationships and startling color shifts between distantly related sonorities by writing chromatic progressions or modulations of their own, or analyzing the opening of the second movement of Dvořák’s Ninth Symphony, as in Example 3.45 below. While the chords sound very striking, and look very unrelated – sharps in a flat key – the Functional Analysis shows that there are still prolongational relationships to be found even in this chromatic music; specifically, this passage consists of upper and lower chromatic mediants surrounding the emergent D♭ tonic, ending with a plagal p–T progression.
3.3.2 Non-V Dominants
Functional Analysis is designed for common-practice tonal music. While common- practice music does sometimes use non-standard chords for any given function, the primary reason for exploring non-V dominants is modern popular music. This section does not comprise a complete adaptation of Functional Analysis for non-common-practice genres; however, these sections may give insight on how to begin adapting my methods for different musics.
Having chords other than so–ti–re leading to tonic was initially a large stumbling block for me using Functional Analysis with pop music, because to call something dominant implies tonic is coming, but in Functional Analysis it also implies the specific notes so–ti–re. And what to do when so–ti–re doesn’t imply tonic? I struggled for some time with to find a way to describe chords I heard as moving to tonic, but yet were not the pitches so–ti–re that was easy to read and understand and no more complicated than the rest of the system.
Then I discovered Nobile and Doll. These two recent pop scholars are talking about harmony and function in pop or rock music, and have some very helpful insights as to how to describe function. Up to this point we have not spoken in detail of what/how to define function theoretically – only practically, aurally, by cadence. Drew Nobile offers three different theoretical versions of function in his forthcoming article in Journal of Music Theory. These are function-as-category – in which function is defined by chord identity, or a chord’s intrinsic notes; function-as-progression – function as defined by what follows what, such as predominant is what it is because it is followed by dominant; and finally function-as-syntax – function as defined by context, usually a combination of the context of a key or of a form.
Using syntax to define function relies more on formal inputs than the individual notes. A chord is described as tonic because it is the end of the phrase and feels stable. Dominant is not only the chord that implies tonic, but also any chord that gives a feeling of half cadence or motion – whether or not a tonic comes next. Syntax function also emphasizes prolongation, looking for only one T P D T circuit per formal unit/phrase.
Additionally, syntax-based function works well in a hierarchy (like Schenkerian analysis, or Functional Analysis levels), because in different context the same chord may function different ways. Even in diatonic, common-practice, historical tonality, a so–ti–re chord can also be an embellishing chord when not at a cadence. Thus, with syntax function, there are two types of function: predictive (i.e., a chord having a pre-tonic function – predicting tonic), and non-predictive (a chord that gives the impression of serving a different hierarchical level).
My earlier definition using the cadence most ideally follows the syntax definition (as you will see in a moment), but my initial aural identification of function depends somewhat on all three types. When we hear the leading tone resolve, or we identify specific notes as being likely markers of a given function, then it is function-as-category. When we hear V follow IV in progression, or I follow V, then it is function-as-progression. When we understand dominant and tonic in context of cadence, phrase, form, and key, it is function- as-syntax. (This is interesting, because we define function in context, but we also define the context via the function.)
In functional tonality, these different definitions of function tend to reinforce each other. In other genres, that is not always the case, and the definitions may be in conflict. For cases where we wish to stretch the limits of Functional Analysis, the syntax definition helps us the most, and I find it to be most aurally salient. In pop music, even when traditional tonal harmony is not in play, we can still identify a feeling of function, of stability versus instability, formal closure, and the desire to resolve.
For syntax purposes, Nobile advocates divorcing the pitch labeling from the function label. If any given analysis or labeling only showed syntax function, a different system may be needed to show what pitches are present – which in pop music is often already obvious based on the chord symbols. However, since Functional Analysis is designed to show both pitch identity and syntactical function simultaneously – an advantage in common-practice music where these identities reinforce the syntax, we must stretch Functional Analysis somewhat to use it when analyzing music where pitch identity and syntax are not equivalent.
Since students are, in general, more familiar with pop music than tonal music when they begin music theory core programs, being able to demonstrate function with music with which they are familiar can be important to help them learn basic concepts. Using music they like can also draw students into the idea of analysis in general, and present them with opportunities for inquiry relevant to their interests.
The first idea of syntax-based function is that “dominant” function (historically the so–ti–re pitches) is defined more by context and emotional drive to tonic than intrinsic notes or progressions – leading to a multiplicity of chords that can come before tonic, which Doll calls pre-tonic chords. Pre-tonic is technically a function-as-progression type of term (this chord sounds dominant because tonic comes next), but I find it to be a better term than dominant (which is so strongly linked to so–ti–re) to encompass the multivariate chords that we will talk about in this section because it shows what it is about these chords that make us feel like they have that function – they imply that tonic and stability are coming, whether they are strictly acting as pre-tonics (tonic does come next) or syntactical dominants (tonic does not necessarily come next).
No more is it only ti that can pull to do, but depending on the tonal or formal context, other pitches or chords may be more successful in implying the oncoming tonic. When I first teach function to the most beginning students, I ask if they can hear open versus closed cadences, or the sense of desire to resolve. This desire is present with more than just the pitches so, ti, re in many genres. After all, “… functions are ultimately determined by specific musical context, not by any unalterable fate of their pitch-class content or intervallic relationship to tonal center.”
According to Doll, any chord that gives the aural impression of leading to or predicting tonic can function as a pre-tonic chord. A beginning, basic pop-adapted Functional Analysis might include first identifying tonic, pre-tonic, and pre-pre-tonic syntactical structural areas, before zooming in to identify specific chords and pitches using some of the following labels.
Since Functional Analysis places emphasis on hearing the functions – the desires – of chords, while also showing the pitch content (function-as-category), I had to come up with additional labels for pre-tonics that were not so–ti–re in order to interact with music where this was common. My labeling decisions for a few of the more common non-V dominants follow, with explanations of why I am using certain abbreviations. Some of the examples will have melody and chords, and some will use only the chord symbols. Ultimately, any primary function, whether T, P, D, or S, can have relatives and variants, and this increases the labeling possibilities to describe chords serving non-common-practice functions. The following examples come from a wide variety of genres – pop, jazz, video games – because these chords can be used in multiple styles (including “classical” ones) and are quite common in many styles currently.
One common pre-tonic chord in many styles of music is the subdominant, which is a pre-tonic chord that pulls to tonic built on scale degree 4, fa. It has the same pitches as common-practice predominant but a different function. Sometimes le–so is substituted as an opposite direction leading tone for dualistic purposes, which leads to more plagal progressions and resolutions. When this chord is not merely a cadential extension of a more traditional PAC, or the analyst wishes to highlight the plagal/dualist cadential potential, we can label fa–la–do or fa–le–do as subdominant (using S or s), a different type of dominant on the opposite side of tonic – the original meaning of Rameau’s term. The Dvořák in Example 3.45 above also includes this type of cadence.
When any chord can be a pre-tonic, this also opens up possibilities for pre- dominants (or pre-pre-tonics, if you like). When S is being used as the pre-tonic chord, the chord built on so sometimes provides a pre-subdominant function, which I have abbreviated PS. What follows are some examples of S as a pre-tonic, as well as pre-subdominants. Example 3.46 shows the chorus of the Beatles tune “Let it Be.” This chorus uses a fa-based subdominant as the pre-tonic chord, as well as the so based triad as the pre-subdominant chord. This example also shows that chords are can still be used in a prolongational manner in non-common-practice tonality idioms; at the final cadence, the C/E and the Dm7 provide a prolongational fill and smooth bass line from the structural subdominant (F) to the tonic (C).
Example 3.47 shows the refrain of theme song from the video game Portal, “Still Alive,” by Jonathan Coulton. The verses are in D major, but the cadence of the verse provides an unusual deceptive cadence, D–tR (A major to F major), that provides a tonic substitute while still taking our ears for a very unexpected turn. The chorus below is in F, and whether described as one phrase or three, the first four measures use PS and S to return to T. Measure 5 of the example also shows a non-cadential pre-tonic as the standard so-based dominant (C major), so that within five measures there have been two different pre-tonics on approximately the same prolongational level. The primary cadence of the chorus (B♭–A– D) returns to D major is very tonal with pR–D–T.
Just as tonic and predominant can be replaced with substitute chords, such as the relative relation, some other common pre-tonic chords can be described as substitutions for either the standard so dominant or the fa plagal subdominant. One is dR: the major relative of the minor dominant, te–re–fa. This chord often uses te instead of ti to pull to do. This is shown below in Example 3.48. (In this expanded harmonic vocabulary, minor v dominant is also a cadential possibility.)
Pop songs that use dR as the pre-tonic chord include the chorus of “Living on a Prayer” by Bon Jovi: (Last phrase of chorus, with tonic resolution on verse)
tR pR dR [t]
G C D7sus4 [Emin]
Wooo, livin’ on a prayer
as well as Simon and Garfunkel’s “Scarborough Fair:”
t dR t
Emin D Emin
Are you going to Scarborough Fair?
Continuing with replacements, just as dR can be used, so can sR (le–do–me). The refrain from “Carry On My Wayward Son” by Kansas provides a good example of this. This refrain also uses te–re–fa (VII) as a pre-tonic and a pre-pre-tonic (dR, psR).
t tR psR sR
Em G D C
Carry on my wayward son
t tR dR
Em G D
There’ll be peace when you are done
t tR psR sR
Em G D C
Lay your weary head to rest
Don’t you cry no more
Many modern pop songs are rotational or looping, repeating four chords over and over. This repetition can lead to multiple interpretations. Depending on the context, you may hear no chord as most important (tonic), or more than one chord as the home base. Invite students to explore multiple interpretations! My favorite song that is an example of this is “Radioactive” by Imagine Dragons, which repeats the progression A minor – C major – G major – D major. This could be read as:
[Amin C G D]
a: t – tR – dR – S
C: Tr – T – D – DD
a: t – tR – (D)[tR] – S
which might be heard as having common-practice-type dominants or more modal subdominants as the pre-tonic chord. My ears usually hear the first row, with A minor as tonic, but yours might be different!
The last pre-tonic chord I want to mention is the tritone substitution, using an excerpt from an arrangement of Duke Ellington’s “Satin Doll”. Shown below in Example 3.49, the chord in question is the D♭13, in m. 6. (The rest of the analysis is more or less common-practice functional plus tertian extensions, as is common in many styles of jazz.)
This chord appears at a cadential point, as a pre-tonic, and is preceded by a dominant of the traditional dominant. It has all the proper chord qualities of a dominant function – major-minor seventh, extra 9 and 13 for emphasis. But we would be expecting G as the root, not D♭, if this were traditional tonality. Jazz practitioners know this chord as the “tritone substitution,” a chord that takes a tritone relation away from the standard dominant and uses it in a dominant functioning place. For Functional Analysis purposes, some of the reason this works is because it uses a le–so tendency tone to replace the ti–do. Additionally, the ♭2 scale degree, or ra–do, reinforces the plagal le–so resolution. While some may argue that this chord includes the leading tone ti, ti does not resolve up as a leading tone in this instance – it is the seventh of the chord and is better described as ♭do, which resolves down, conceptually, to ti in the next chord, even if they are the same key on the piano.
As a le–so pre-tonic chord, this chord is a variation of the s–T plagal resolution discussed earlier. With s as f–a♭–c in this key, D♭ can be seen as a third relation – sV (similar to the Neapolitan pV). This tritone sub is foreshadowed somewhat by the use of altered dominants that include lowered fifths earlier in the progression – the one time a G chord does appear as a dominant, it uses a voice-leading including D♭.
 Daniel Harrison, Chromatic Harmony, 1.
 The Chopin analysis of Section 4.3.2 is an example of functionality that works consistently only at the largest level, but does not use tonal function to move from one note to the next. For ideas on how to approach the non-functional parts of late Romantic music, see Daniel Harrison’s “A Renewed Dualist Theory of Harmonic Function,” in Harmonic Function in Chromatic Music. Additionally, David Kopp has developed another approach to chromatic third relations positing common-tone tonality – a version of chromatic pitch space that privileges chords that keep common-tones. This realm is neither diatonic in the 18th-century sense, but it is also not entirely atonal/chromatic in the 20th-century sense. Kopp, Chromatic Transformations in Nineteenth-century Music, Cambridge University Press, 2002.
 Kopp, Chromatic Transformations, 8.
 Christopher Doll, Listening to Rock Harmony; Drew Nobile, A Structural Approach to the Analysis of Rock Music. Doll is also working on a forthcoming book, Hearing Harmony: Towards a Tonal Theory for the Rock Era (University of Michigan Press) from which he kindly shared the first two chapters with me, “Tonic and Pre-tonic,” and “Chains, Numerals, and Levels.”
 Nobile, “Harmonic Function,” 4.
 Nobile, “Harmonic Function,” 13.
 Nobile, “Harmonic Function,” 12.
 Nobile, “Harmonic Function,” 10, 13.
 Nobile, “Harmonic Function,” 11.
 Milo Fultz, classroom brainstorming, 10 May 2014.
 Nobile, “Harmonic Function,” 2.
 This is not unlike the current versions used in say, the Laitz textbook, that show pitch with RNs and function with T P D in a different layer.
 Doll, Listening to Rock Harmony, 16; Nobile, A Structural Approach, 32.
 Nobile, “Harmonic Function,” 12.
 Doll, “Tonics and Pre-Tonics,” 9.
 Nobile, “Harmonic Function,” 14, Doll, Listening to Rock Harmony, 16, Doll, “Tonics and Pre-Tonics,” 12. Doll focuses quite a bit on pitch membership when distinguishing different types of pre-tonics. In “Tonics and Pre-Tonics,” he discusses designations of upper and lower subdominant based on le or la presence as well as lead/rogue dominants with ti or te (25–27). He also has classification for upper and lower mediant pre-tonics (30–31). All these can be described with Functional Analysis relationships: S and s (and relatives and variants), D and d, and mediants could be dV or Dr.
 If, in the case such as where Nobile shows numerous examples which include almost every chord leading to tonic, even one Talking Heads example where i7 leads to i, (Harmonic Function,” 15–17) we needed to label a do–me–so–te chord as a dominant, we could extend relations further than two steps such as: dVr – the minor relative of minor dominant’s variant; or in some instances it may make sense to treat it similar to the p6 : dV56 dominant variant with both sixth and fifth, but sixth in the bass. This issue certainly bears more exploration, but there are multiple possibilities, and each of those possibilities opens up other avenues of labeling and discussion.
 The plagal cadence is the “Amen” cadence that in CPP typically follows and reinforces the standard D–T cadence.
 Joel Lester, “Rameau and eighteenth-century harmonic theory,” The Cambridge History of Western Music Theory, 768.
 Doll, Listening to Rock Harmony, 16.
 John Lennon and Paul McCartney, “Let it Be,” 218.
 This is not necessarily the best example structural subdominants and presubdominants, but as a familiar musical example it helps show the expected pitch relationships.
 Jonathan Coulton, “Still Alive.”
 Examples of this are available in Biamonte, “Triadic Modal and Pentatonic Patterns in Rock Music,” 97, 101–102; Doll, “Tonics and Pre-tonics,” 27; and Allan Moore, “The So-Called ‘Flattened Seventh’ in Rock,” 185–201.
 Biamonte, “Triadic Modal and Pentatonic Patterns in Rock Music,” 103; Bon Jovi. “Living on a Prayer Chords.”
 Paul Simon, “Scarborough Fair,” 24.
 Kansas; Kansas.
 Nobile, “Counterpoint in Rock Music,” 193–194. For a detailed look at the mechanics of looping see Chapters 11 and 12 of Phillip Tagg’s Everyday Tonality, “Chord loops 1” and “Modal loops and bimodality,” 199–240.
 Imagine Dragons, “Radioactive.”
 Green Day’s “Boulevard of Broken Dreams” amongst others uses a similar progression.
 Duke Ellington, “Satin Doll,” 90.