Augmented sixth chord

The interval of an augmented sixth normally resolves outwards by semitone to an octave.  Play (help·info)

In music theory, an augmented sixth chord contains the interval of an augmented sixth above its "root" or bass tone (see below). This chord has its origins in the Renaissance,^[1] further developed in the Baroque, and became a distinctive part of the musical style of the Classical and Romantic periods.^[2]

Resolution and chord construction

The augmented sixth interval is typically between the sixth degree of the minor scale (henceforth ♭6) and the raised fourth degree (henceforth ♯4). With standard voice leading, the chord is followed directly or indirectly by some form of the dominant chord, in which both ♭6 and ♯4 have resolved to the fifth scale degree (henceforth 5). This tendency to resolve outwards to 5 is why the interval is spelled as an augmented sixth, rather than enharmonically as a minor seventh (♭6 and ♭5). Although augmented sixth chords are more common in the minor mode, they are also used in the major mode by borrowing ♭6 of the parallel minor scale.^[3]

Double-diminished triad

In music theory, the double-diminished triad is an archaic concept and term referring to a triad, or three note chord, which, already being minor, has its root raised a semitone, making it doubly diminished. However, this may be used as the derivation of the augmented sixth chord.^[4]

For example, F-A♭-C is a minor triad. F♯-A♭-C is a doubly diminished triad. Note that it is enharmonically equivalent to G♭-A♭-C (incomplete dominant seventh, missing E♭). A♭-C-F♯ is an Italian augmented sixth chord.

Standard harmonic function

From the Baroque to the Romantic period, augmented sixth chords have had the same harmonic function: as a chromatically altered predominant chord (typically, an alteration of ii$_3^4$, IV$_5^6$, vi⁷ or their parallel equivalents in the minor mode) leading to a dominant chord. This movement to the dominant is heightened by the semitonal resolution of both ♭6 to 5 and ♯4 to 5; essentially, these two notes act as leading-tones. This characteristic has led many analysts^{[citation needed]} to compare the voice leading of augmented sixth chords to the secondary dominant V of V because of the presence of ♯4, the leading-tone of V, in both chords. In the major mode, the chromatic voice leading is more pronounced because of the presence of two chromatically altered notes, ♭6, as well as ♯4, rather than just ♯4 in the minor mode.

During the Romantic period, the augmented sixth harmony increased in ambiguity as composers explored other functional possibilities outside of its role as a predominant. See #Extended functions.

Variants

There are several variants of the augmented sixth chord. Though each is named after a European nationality, theorists disagree on their precise origins and have struggled for centuries to define their roots, and fit them into conventional harmonic theory.^[3][5][6]

Italian sixth

The Italian sixth moving to V.  Play (help·info)

The Italian sixth (It ^{+ 6} or It⁶) is derived from iv⁶ with an altered fourth scale degree, ♯4: ♭6—1—♯4; A♭—C—F♯ in C major. This is the only augmented sixth chord comprising just three distinct notes; in four-part writing, the tonic pitch is doubled.

Examples

• The second movement of Beethoven's piano sonata in F-sharp major, op. 78, begins with an Italian sixth chord:

French sixth

The French sixth chord; the distinguishing tone is highlighted in blue.  Play (help·info)

The French sixth (Fr ^{+ 6} or Fr$_3^4$) is similar to the Italian, but with an additional tone, 2: ♭6—1—2—♯4; A♭—C—D—F♯ in C major. The notes of the French sixth chord are all contained within the same whole tone scale, lending a sonority common to French music in the 19th century (especially associated with Impressionist music).^[7]

Examples

• Richard Wagner's famous Tristan chord (indicated below with Tr) from the opening of his opera, Tristan und Isolde, can be interpreted as a French sixth in the key of A minor (F-A-B-D♯) with an upwardly resolving appoggiatura in the upper voice. Note that the D♯ resolves downwards to D♮ instead of E:

German sixth

The German sixth; the distinguishing tone is highlighted in blue.  Play (help·info)

The German sixth (Gr ^{+ 6} or Ger$_5^6$) is also like the Italian, but with an added tone ♭3: ♭6—1—♭3—♯4; A♭—C—E♭—F♯ in C major. In Classical music, however, it appears in much the same places as the other variants, though perhaps less used because of the contrapuntal difficulties outlined below. It appears frequently in the works of Beethoven. The German sixth chord contains the same notes as a dominant seventh chord, though it functions differently.

It is more difficult to avoid parallel fifths when resolving a German sixth chord to the dominant, V. These parallel fifths, referred to as Mozart fifths, were occasionally accepted by common practice composers. There are two ways they can be avoided:

1. The ♭3 can move to either 1 or 2, thereby generating an Italian or French sixth, respectively, and eliminating the perfect fifth between ♭6 and ♭3.^[8]
2. The chord can resolve to a "six-four" chord, functionally either as a cadential six-four intensification of V, or as the second inversion of I; the cadential six-four, in turn, resolves to a root-position V. This progression ensures that, in its voice leading, each pair of voices moves either by oblique motion or contrary motion and avoids parallel motion altogether. In minor modes, both 1 and ♭3 do not move during the resolution of the German sixth to the cadential six-four. In major modes, ♭3 can be enharmonically respelled as ♯2 if it resolves upwards to ♮3, similar in voice leading to the resolution of French sixth to the cadential six-four. This respelled chord is sometimes referred to as the English, Swiss or Alsatian sixth chord. (The respelled chord is sometimes ambiguously labeled a "doubly augmented sixth chord"; however, the interval of a fourth between ♭6 and ♯2 (i.e., A♭ to D♯ in C major) is doubly augmented, not the sixth.)

The German sixth is typically followed by a I$_4^6$ chord to avoid parallel fifths.

German sixth chord respelled with doubly augmented fourth (highlighted in blue) for voice-leading purposes. Also referred to as English, Swiss or Alsatian.

Examples

• A German sixth can be found in the high passage heard twice in the "Passepied" from Debussy's Suite Bergamasque.
• A German sixth chord from Michael Haydn's Requiem in C minor, first movement:

( Listen)

Other variants

Other variants of augmented sixth chords are sometimes found in the repertoire, and are sometimes given whimsical geographical names. For example: 4—♭6—7—♯2; (F—A♭—B—D♯) is called by one source an Australian sixth.^[9] Such anomalies usually have alternative interpretations.

"Inverted" augmented sixth chords

Augmented sixth chords are occasionally used with a different chord member in the bass. Since there is no consensus among theorists that they are in root position in their normal form, the word "inversion" isn't necessarily accurate, but is found in some textbooks, nonetheless. Sometimes, "inverted" augmented sixth chords occur as a product of voice leading.

The French philosopher composer, Jean-Jacques Rousseau considered that the chord could not be inverted (Dictionnaire de Musique). Seventeenth-century instances of the augmented sixth with the sharp note in the bass are generally limited to German sources.^[10]

Examples

• Tchaikovsky's, Symphony no. 5 (op. 64, I), Allegro con anima (bars 3–4).
• The following excerpt shows an augmented sixth chord in inversion used by Bach. At the end of the second measure, the augmented sixth is inverted to create a diminished third or tenth between the bass and the soprano (C♯—E♭); these two voices resolve inward to an octave:

Excerpt from Bach's Mass in B minor

"Roots" of augmented sixth chords

Simon Sechter, in 'Die Grundsätze der musicalischen Komposition', explains the chord of the French Sixth A♭—C—D—F♯ in the key of C as being a chromatically altered version of a seventh chord on the second degree of the scale, and therefore gives the root as 'D'. The German Sixth A♭—C—E♭—F♯ is explained as a chromatically altered ninth chord on the same root, but with the root omitted. (In Sechter's theory, the diminished seventh chord F♯—A—C—E♭ is invariably described in the same way, i.e. a ninth chord on D with the root omitted, hence its equivalence to the augmented sixth.)

The tendency of the interval of the augmented sixth to resolve outwards is therefore explained by the fact that the A♭, being a dissonant note, a diminished fifth above the root (D), and flatted, must fall, whilst the F♯ - being chromatically raised - must rise.

On the contrary, Tchaikovsky considers the augmented sixth chords, rather than being built on the minor sixth degree (A♭ in C), as being altered dominants.^[11] In his Guide to the practical study of harmony considers the augmented sixth chords to be inversions of the diminished triad and of dominant and diminished seventh chords with the second degree chromatically lowered, and accordingly resolving into the tonic. He notes that, "some theorists insist upon [augmented sixth chord's] resolution not into the tonic but into the dominant triad, and regard them as being erected not on the altered 2-nd degree, but on the altered 6-th degree in major and on the natural 6-th degree in minor", yet calls this view, "fallacious", insisting that a, "chord of the augmented sixth on the 6-th degree is nothing else than a modulatory degression into the key of the dominant".^[12] This would make the chord of the augmented sixth a member of a large group of chords with an altered second degree (which includes the Neapolitan chord). For an exhaustive discussion of the possibility of augmented sixth chords resolving to tonic (or other scale degrees), see Daniel Harrison's article, "A Supplement to the Theory of Augmented-Sixth Chords."

Extended functions

In the late Romantic period and other musical genres, especially jazz, other harmonic possibilities of augmented sixth variants and sonorities outside its function as a predominant were explored, exploiting their particular properties. An example of this is through the "reinterpretation" of the harmonic function of a chord: Since a chord could simultaneously have more than one enharmonic spelling with different functions (i.e., both predominant as a German sixth and dominant as a dominant seventh), its function could be reinterpreted mid-phrase. This heightens both chromaticism by making possible the tonicization of remote keys, and possible dissonances with the juxtaposition of remotely related keys.

Enharmonic equivalency of the French sixth

The French sixth has two characteristics in common with the diminished seventh chord:

1. Both chords are constructed of two superimposed tritones; in the French sixth, between ♭6—2 (A♭—D) and 1—♯4 (C—F♯). Thus, both have inversional symmetry;
2. Both are enharmonically equivalent at the tritone; i.e., both chords transposed up or down a tritone will result in the same pitches as the original.

As with the diminished seventh chord, the latter property allows the chord to be used in modulating to very remote keys. For instance: ♭6—1—2—♯4; (A♭—C—D—F♯ in C), could be interpreted identically in F♯ if reordered and respelled as D—F♯—G♯—B♯, i.e., the French sixth of the ♯IV key area, displaced an interval of a tritone relative to the tonic key, I.

Dominant functions

German sixth, G♭-B♭-C♯-E, and equivalent dominant seventh, F♯-A♯-C♯-E,^[13]  Play (help·info).

Irregular resolution through augmented sixth equivalence  Play (help·info).^[14]

All variants of augmented sixth chords are closely related to the applied dominant V⁷ of ♭II; both Italian and German variants are enharmonically identical to dominant seventh chords. For example, in the key of C (I), the German sixth chord, A♭—C—E♭—F♯, could be reinterpreted as A♭—C—E♭—G♭, the applied dominant of D♭ (V/D♭).

Classical harmonic theory would notate the "tritone substitute" as an augmented sixth chord on ♭2. The Augmented sixth chord can either be the It⁺⁶ enharmonic to a dominant 7th chord without the 5th, Gr⁺⁶, enharmonically equivalent to a dominant 7th chord with the 5th, or Fr⁺⁶ enharmonically equivalent to the Lydian dominant without the 5th, all of which serve in a classical context as a substitute for the secondary dominant of V.^[15][16]

Below is the original dominant-tonic progression, that progression with the tritone substitution, and the same progression with the substitution notated as an Italian augmented sixth chord:

Original V⁷-I  Play (help·info), tritone substitution (♭V⁷/V-I)  Play (help·info), augmented sixth chord (It⁶-I)  Play (help·info).

French sixth sonority as dominant

In jazz, the French sixth sonority functions as a dominant instead of a predominant chord; exploiting the enharmonically equivalent property of the French sixth is a common technique is referred to as tritone substitution.

The French sixth sonority, for example A♭—C—D—F♯ in the key of C, is interpreted as a specific variation of a dominant seventh chord in the following keys:

• V⁷ of D♭ with ♯4; A♭ as the root: A♭—C—D—G♭; or
• V⁷ of G with ♯4; D as the root: D—F♯—G♯—C.

This chord is called the Lydian dominant (A♭^7♯11—D^7♯11), see jazz chord.

These functions could also be spelled as a dominant seventh with a flatted fifth:

• V⁷ of D♭, with ♭5; A♭ as the root: A♭—C—E—G♭; or
• V⁷ of G, with ♭5; D as the root: D—F♯—A♭—C.

and could be notated A♭^7♭5 and D^7♭5. This spelling does not suggest the Lydian-dominant mode, but it does suggest a dominant function. Also, C—E and F♯—A♭ are inverted augmented sixths (diminished thirds), where C—D and F♯—G♯ are inverted minor sevenths (major seconds).

Augmented sixths built on scale degrees other than ♭6

The augmented sixth chord may be built on notes other than ♭6. Often, this is the result of a temporary tonicization, and the resulting augmented sixth chord is borrowed from the key of the secondary dominant which follows it. However, there are examples in the literature of these chords appearing without such a context. Schubert used it in some of his last compositions in dramatic final cadences.

Examples

• An Italian sixth chord built on scale degree ♭2 in Schubert's piano sonata D. 959, preceded by a ♭II chord in root position (B♭—D—F♮). Instead of functioning as a predominant, here, both the ♭II and the Italian sixth chord serve in a sort of "neighbour motion,"^{[citation needed]} or "plagal cadence"^{[citation needed]} (usually I—IV—I) in the coda. I—♭II—(It⁺⁶)—I:

Schubert's A major sonata, D. 959.

See also

• Acoustic scale

References

1. ^ Andrews, Herbert Kennedy (1950). The Oxford Harmony. 2 (1 ed.). London: Oxford University Press. pp. 45–46. OCLC 223256512. 
2. ^ Andrews 1950, pp. 46–52
3. ^ ^{a b} Aldwell, Edward; Carl Schachter (1989). Harmony and Voice Leading (2 ed.). San Diego, Toronto: Harcourt Brace Jovanovich. pp. 478–483. ISBN 0155315196. OCLC 19029983. 
4. ^ Ernst Friedrich Richter (1912). Manual of Harmony, p.94. Theodore Baker.
5. ^ Gauldin, Robert (1997). Harmonic Practice in Tonal Music (1 ed.). New York: W.W. Norton. pp. 422–438. ISBN 0393970744. OCLC 34966355. 
6. ^ Christ, William (1973). Materials and Structure of Music. 2 (2 ed.). Englewood Cliffs, NJ: Prentice Hall. pp. 141–171. ISBN 0135603420. OCLC 257117.  Offers a useful, detailed explanation of augmented sixth chords as well as Neapolitan sixth chords.
7. ^ Blatter, Alfred (2007). Revisiting Music Theory: a Guide to the Practice, p.144. ISBN 9780415974400. "One may note that the French sixth contains the elements of a whole tone scale commonly associated with French impressionistic composers."
8. ^ Benjamin, Thomas; Michael Horvit & Robert Nelson (2008). Techniques and Materials of Music: From the Common Practice Period Through the Twentieth Century (7 ed.). Belmont, CA: Thomson & Schirmer. p. 165. ISBN 9780495189770. OCLC 145143714.  Beethoven frequently moves from one form of the chord to another in such a way, sometimes passing through all three.
9. ^ Burnard, Alex (1950). Harmony and Composition: For the Student and the Potential Composer. Melbourne: Allans Music (Australia). pp. 94–95. OCLC 220305086. 
10. ^ Ellis, Mark (2010). A Chord in Time: The Evolution of the Augmented Sixth from Monteverdi to Mahler, pp. 92-94. Farnham: Ashgate. ISBN 978-0-7546-6385-0.
11. ^ Roberts, Peter Deane (1993). Modernism in Russian Piano Music: Skriabin, Prokofiev, and Their Russian Contemporaries, p.136. ISBN 0253349923.
12. ^ Tchaikovsky, Peter Ilyich [1871] trans Emil Krall and James Liebling, Guide to the Practical Study of Harmony,^{[page needed]}. [ISBN 0486442721.]
13. ^ Benward & Saker (2008). Music in Theory and Practice, Vol. II, p.222. ISBN 978-0-07-310188-0.
14. ^ Owen, Harold (2000). Music Theory Resource Book, p.132. ISBN 0195115392.
15. ^ Satyendra, Ramon. "Analyzing the Unity within Contrast: Chick Corea's Starlight", p.55. Cited in Stein.
16. ^ Stein, Deborah (2005). Engaging Music: Essays in Music Analysis. New York: Oxford University Press. ISBN 0-19-517010-5.