- Dominant chords may feature an augmented or diminished fifth, indicated in analysis by “+” beside the Roman numeral. Diminished fifths are indicated by a “o” beside the Roman numeral.
- Augmented dominant chords typically have the augmented fifth resolving up to the third of the chord. These most commonly are found resolving to major triads
- Diminished fifths typically resolve down by semitone to the root of the following chord
- A 9th replaces a doubled root, and resolves downward by step
- An 11th replaces a third, and resolves via common-tone
- A 13th replaces a fifth, and resolves downward by third
- Extended chords are almost exclusively found in root position
To this point, many of the chromatic harmonies we have explored have been explained in terms of either secondary function or modal borrowing. None of these harmonies include alterations to the perfect fifth of the chord; a practice we will now discuss in some detail. When the fifth of a triad is raised or lowered, it becomes either augmented or diminished. While we have seen diminished triads and seventh chords that feature diminished fifths, we have yet to discuss the possibility of a chord with a major third and either an augmented or diminished fifth. These chords are commonly referred to as “altered chords.” Although there is no standard analytical notation, augmented fifths are often indicated in analysis by “+” beside the Roman numeral. Diminished fifths are indicated by a “o” beside the Roman numeral. We will adopt this practice here.
The Dominant with an augmented fifth
If you raise the fifth of a major triad, it will become an augmented triad. Typically, raised fifths resolve upward by step to the third of its target chord. In Example 1, Franck abruptly modulates from f-sharp minor to D major by way of a dominant seventh chord with an augmented fifth. Note the resolution moves upward to the third of the target chord (in this case, tonic in D major). This is the most common resolution of this chromatic pitch.
The dominant chord with augmented fifth can be used as an applied chord, as well, as you can hear in the excerpt from Strauss’s “Till Eulenspiegel’s Merry Pranks” in Example 2.
Importantly, these chords do not resolve easily to minor triads, since the augmented fifth would not be able to resolve upward by step. In Example 3, the dominant chord with an augmented fifth is used in first inversion, with the chromaticized fifth resolving upward again.
Note that the augmented triad is a symmetrical chord than can be interpreted in multiple ways, making it difficult to identify its root without proper surrounding context. In Example 3, Brahms makes it clear that chord is functioning as the dominant of Bb, but one could easily re-interpret the bass note as the root of an A augmented triad. Like the diminished-seventh chord, the augmented triad can be a pathway to distant, chromatic modulations. Example 4 shows the three possible enharmonic interpretations and resolutions of the C augmented triad.
The Dominant with Diminished Fifth
Flattening the fifth of a major triad results in a strange chord quality that we have not encountered thus far. This alteration makes a diminished fifth above the root, and this tone tends to resolve downward by step to the root of its target chord. Although it is possible to use this triad, it is far more common to use the diminished fifth in a dominant-seventh chord. In Example 5, an excerpt from the opera Salome by Richard Strauss, the cadential dominant includes a diminished fifth, which provides a delightful dissonance leading to the tonic. In this example, Strauss is more liberal with the resolution of the flattened fifth, and doesn’t bother resolving the pitch to the root of the tonic chord.
Like the augmented triads and diminished-seventh chords, the dominant seventh with a diminished fifth is symmetrical. Moreover, the chord is equivalent to a French augmented-sixth chord. This equivalence becomes even clearer when you use the Vo7 chord in second inversion, leaving the lowered fifth in the bass voice to resolve downward by step. For example, in C major, the Vo4/3 chord is identical to the French augmented-sixth chord in F major. Similarly, the Vo4/3/V is equivalent to the French augmented-sixth chord in the home key of C major. Technically, the root-position dominant seventh chord with a diminished fifth is also equivalent to a French augmented-sixth chord, though this requires some enharmonic respelling to become clear. Example 6 shows all of these equivalences.
To finish off our discussion of altered dominants, consider Example 7, from the second movement of Tchaikovsky’s Symphony No. 5. In this excerpt, emerging from a chromatic quagmire of half-diminished and fully-diminished seventh chords is a relatively surprising diatonic ii7 chord. This chord resolves via stepwise motion, through another half-diminished seventh chord, to the cadential dominant of the passage: a Vo4/3 chord. Typically, cadential dominants are only effective in root position, however this chord is emblematic of nineteenth-century chromaticism and ambiguity. The lowered fifth resolves downward by step, much like an augmented-sixth chord, to the D major tonic.
To this point, we have only used harmonies that form triads or seventh chords. It is possible, and indeed, many Romantic-era composers and beyond did this, to continue to superimpose thirds beyond the seventh above the root of a chord. You can do this only so far, though, before you reach the root of the chord again, as you can see in Example 8a). In jazz theory, chord members beyond the seventh are referred to as extensions, and we will adopt this practice here. Importantly, extensions rely on their voicing to determine their meaning. In Example 8b) and 8c), two different voicings of the same four pitches are given. In Example 8b), the chord can clearly be contextualized as V13/7 in the key of C major. Example 8c) is much less clear; it could be some kind of inversion of the chord in 8b), or we could consider it to be an extended iii chord. In either case, the context is made unclear by the chord’s voicing. Because of this, we tend to only use extended chords in root position (as the extensions are truly defined by the chord root).
While you could include all chord members and extensions in textures with 5, 6, or 7 voices, it is often undesirable to use such dense chords. When composing these chords in a four-voice texture, you will need to decide which notes to leave out. It’s useful to consider these chords as being linearly derived, with the extension displacing a regular chord member. Importantly, when we write these chords, we will always include the root and the chordal seventh, as these are essential for identifying the chord’s quality.
Example 9 shows all of the derivations of these extensions as applied to a dominant-seventh chord.
In example 9a), the 9th displaces the root, doubled an octave above the bass. The 9th may appear together with the root (found in the bass), but make sure that it sounds in the octave above the root; this is a 9th, and not a 2nd. Here, the 5th is omitted, but could well be present in a five-voice texture. The ninth should resolve down by step.
Example 9b) shows the 11th displacing the 10th (3rd). Since the 3rd (the leading tone) is not normally doubled, the 11th and the 3rd should not sound together. Paradoxically, the 11th “resolves by common-tone. Since the leading tone would typically resolve to the tonic, so should the 11th. This chord sometimes includes both the ninth and the eleventh, and resembles a IV chord with scale-degree 5 in the bass.
Example 9c) shows the 13th displacing the 12th (5th). It is uncommon to double the 5th in a dominant-seventh chord, so the 13th and the 5th don’t normally sound together. Since the 5th (scale-degree 2) would often resolve down by step to the tonic, the 13th sometimes “resolves” by leaping down by third to scale-degree 1. It is also possible for the 13th to be held as a common tone, becoming the 3rd of the tonic chord.
Lastly, Example 9d) shows that you can combine extensions and alterations to make some truly interesting dominant sonorities.
“Fake” Extensions vs. “Real” Extensions
It is important to note that in the first part of each of the above examples, the extension resolves to a regular chord member before the chord changes. This makes it much more explainable as an instance of accented dissonance, rather than an extension. Only when the extension lasts for the duration of the chord, resolving with the chord change, do we consider it a “real” extension.
In Example 10, we can see that Schumann uses a dominant ninth chord in the opening of this song. The 9th is treated like a chord member, rather than as an embellishment; notice that it is approached by leap, within the chord, and does not resolve until the chord changes. Compare this with Example 11, in which Schubert uses an Fb to create a 9th above the dominant in measure 5. This 9th resolves down by step before the chord changes. This is an example of a “fake” 9th that is better explained as an accented dissonance, rather than a “real” chord extension.
Similarly, listen to Schubert’s Valse noble in Example 12, below. Notice that the 13th, while prominent in the melody, always resolves down by step before the chord changes. This 13th is best explained as appoggiatura rather than a “real” member of the chord. Compare this with Schubert’s Valse sentimentale in Example 13. Along with several “real” 9ths used throughout the passage, The cadential dominant in m. 7 includes a 13th above the bass that is approached and left by leap–a true chord extension.
The Case for Elevenths
Unlike 9ths and 13ths, 11ths are considerably more rare in the Western European Art Music repertoire. This is likely because the 11th displaces the 3rd of a chord, which is the chord member that indicates a chord’s quality (and in the case of a dominant chord, its function as well). Consider Example 14, below, by Fauré. Throughout the passage, the dominant chords lack 3rds, and in most cases, use a 4th (or 11th) in its place. The overall sense of function from dominant to tonic is particularly weak, a sense which is further weakened by the tonic pedal heard throughout the passage.
Example 14: Fauré, “Les Roses d’Ispahan,” Op. 39 No. 4, mm. 1-14
Sometimes, chord extensions can be applied to non-dominant chords, as is the case in Example 15. Here, the ii chord has a 7th, along with a 9th and an 11th. The 11th is presented as a “true” 11th, voiced above the 7th and 9th, unlike in the Fauré example above, where the 11th is written as a 4th (and sounds more like an unresolved suspension). Voicings like this are far more common in popular music, such as in Example 16 by The Emotions. Here, the dominant is presented as a “real” 11th chord, or, as many practicing pop musicians would put it, a “IV over V” chord in which F major is sounded above a G bass.
Example 16: The Emotions, “Best of My Love,” mm. 1-2
Examples like the last two reinforce the fact that extensions are defined by voicing and linear context, and indeed, begin to call into question the entire notion of function in tertian harmony. Is it the bass that determines function, or is it the chord’s content? For our last example, we will consider the beautiful opening to César Franck’s Violin Sonata in A major, found in Example 17. This movement begins in tonal obscurity, alternating between two chords whose function seems completely suppressed. The opening V9 chord loses its sense of dominant function when it progresses “backwards” to what appears to be a ii13 chord that also includes an 11th. Alternatively, one could interpret this chord as an “inversion” of the same V9 chord heard in the first measure. When the violin enters, more evidence is given for hearing the chord in m.2 as ii, this time supported by an arpeggiated b minor triad in the violin. Eventually, the tonality settles in A major with a tonic arrival. Of course, the tonic is weakened somewhat by the presence of an F sharp, which resolves downward by step, but is left dangling in the violin’s melody.
- Coming soon!