Английская Википедия:Augmented sixth chord

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In music theory, an augmented sixth chord contains the interval of an augmented sixth, usually above its bass tone. This chord has its origins in the Renaissance,Шаблон:Sfn was further developed in the Baroque, and became a distinctive part of the musical style of the Classical and Romantic periods.Шаблон:Sfn

Conventionally used with a predominant function (resolving to the dominant), the three most common types of augmented sixth chords are usually called the Italian sixth, the French sixth, and the German sixth.

Augmented sixth interval

The augmented sixth interval is typically between the sixth degree of the minor scale, Шаблон:MusicШаблон:Music, and the raised fourth degree, Шаблон:MusicШаблон:Music. With standard voice leading, the chord is followed directly or indirectly by some form of the dominant chord, in which both Шаблон:MusicШаблон:Music and Шаблон:MusicШаблон:Music have resolved to the fifth scale degree, Шаблон:Music. This tendency to resolve outwards to Шаблон:Music is why the interval is spelled as an augmented sixth, rather than enharmonically as a minor seventh (Шаблон:MusicШаблон:Music and Шаблон:MusicШаблон:Music).

Although augmented sixth chords are more common in the minor mode, they are also used in the major mode by borrowing Шаблон:MusicШаблон:Music of the parallel minor scale.

Types

There are three main types of augmented sixth chords, commonly known as the Italian sixth, the French sixth, and the German sixth.

Шаблон:Block indent 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. According to Kostka and Payne, the other two terms are similar to the Italian sixth, which, "has no historical authenticity-[being] simply a convenient and traditional label."

Italian sixth

Шаблон:Image frame The Italian sixth (It⁺⁶ or It⁶ or Шаблон:Musiciv⁶) is derived from iv⁶ with an altered fourth scale degree, Шаблон:MusicШаблон:Music. This is the only augmented sixth chord comprising just three distinct notes; in four-part writing, the tonic pitch is doubled. Шаблон:Block indent

The Italian sixth is enharmonically equivalent to an incomplete dominant seventh.Шаблон:Sfn Шаблон:MusicVI7=Шаблон:MusicV7: AШаблон:Music, C, (EШаблон:Music,) GШаблон:Music.

French sixth

Файл:French sixth chord in Schubert's Am Feirabend.png

A French sixth chord in Schubert's Die schöne Müllerin, #5: "Am Feierabend" Шаблон:Audio

The French sixth (Fr⁺⁶ or FrШаблон:Su) is similar to the Italian, but with an additional tone, Шаблон:Music. 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), though they also make frequent appearances in Russian music.

This chord has the same notes as a dominant seventh flat five chord and is in fact the second inversion of II^{7Шаблон:Music5}.

German sixth

The German sixth (Ger⁺⁶ or GerШаблон:Su) is also like the Italian, but with an added tone, Шаблон:MusicШаблон:Music.

In Classical music, however, it appears in much the same places as the other variants, though perhaps less often because of the contrapuntal difficulties outlined below. It appears frequently in the works of Beethoven,Шаблон:Efn and in ragtime music. The German sixth chord is enharmonically equivalent to a dominant seventh chord though it functions differently.

Avoiding parallel fifths

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

Other types

Other variants of augmented sixth chords can be found in the repertoire, and are sometimes given whimsical geographical names. For example: 4–Шаблон:Music6–7–Шаблон:Music2; (F–AШаблон:Music–B–DШаблон:Music) is called by one source an Australian sixth, and Шаблон:Music7–1–3–Шаблон:Music5 (BШаблон:Music–C–E–G#), sometimes called the Japanese sixth, Blackadder, or Ikisugi chord. Such anomalies usually have alternative interpretations.

Function

Standard function

From the Baroque to the Romantic periods, augmented sixth chords had the same harmonic function: as a chromatically altered predominant chord (typically, an alteration of [[supertonic|iiШаблон:Su]], [[subdominant|IVШаблон:Su]], 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 to Шаблон:Music from above and below (from Шаблон:MusicШаблон:Music and Шаблон:MusicШаблон:Music); essentially, these two notes act as leading-tones.

This characteristic has led many analysts to compare the voice leading of augmented sixth chords to the secondary dominant V of V because of the presence of Шаблон:MusicШаблон:Music, 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, Шаблон:MusicШаблон:Music and Шаблон:MusicШаблон:Music, rather than just Шаблон:MusicШаблон:Music.

In most occasions, the augmented-sixth chords precede either the dominant, or the tonic in second inversion.Шаблон:Sfn The augmented sixths can be treated as chromatically altered passing chords.Шаблон:Sfn

Other functions

In the late Romantic period and other musical traditions, 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 remotely related keys, and possible dissonances with the juxtaposition of remotely related keys.

The French sixth sees a lot of non-functional use in much Russian music of the late-Romantic period. Due to its construction of two tritones separated by a major third, it has transpositional invariance and is often used to create tonal ambiguity in highly chromatic music of the 19th century. This use actually began in Germany with its use by Wagner and Bruckner (eg. the prelude from Tristan und Ysolde and Bruckner's third symphony), but is most notable in Russian works such as Rimsky-Korsakov's Scheherezade and Scriabin's Prometheus: The Poem of Fire. The chord is separated by only a single note from the half-diminished chord, or the "Tristan chord," as well as the German sixth or dominant seventh. Tonal ambiguity is created by the French sixth as it is symmetrical about a tritone, for example, the notes of a French sixth chord built on G are the same as the notes as the chord built on C sharp, up to enharmonic equivalence. Due to this tonal ambiguity, the French sixth is often used in lieu of the triad and carries with it an unresolved and uneasy sound.

Scriabin also begun to add chord extensions to the French sixth, for example, he added a sixth and a ninth to create his 'mystic chord' which is found in his aforementioned Prometheus tone poem. The chord is usually combined with the octatonic, or diminished, scale, as the scale contains two distinct French sixths and thus has similar symmetric properties. This combination can be found ubiquitously in much of Rimsky and Scriabin's music, as well as in some 20th century French works such as Debussy's Nuages and Ravel's Scarbo.

Tchaikovsky considered the augmented sixth chords to be altered dominant chords. He described the augmented sixth chords to be inversions of the diminished triad and of dominant and diminished seventh chords with a lowered second degree (Шаблон:MusicШаблон:Music), 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 2nd degree, but on the altered 6th degree in major and on the natural 6th degree in minor", yet calls this view, "fallacious", insisting that a, "chord of the augmented sixth on the 6th degree is nothing else than a modulatory degression into the key of the dominant".

The example below shows the last nine measures from Schubert's Piano Sonata in A major, D. 959. In m. 352, an Italian sixth chord built on scale degree Шаблон:MusicШаблон:Music functions as a substitute for the dominant.

Inversions

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.Шаблон:Citation needed Sometimes, "inverted" augmented sixth chords occur as a product of voice leading.

Rousseau considered that the chord could not be inverted. Seventeenth century instances of the augmented sixth with the sharp note in the bass are generally limited to German sources.

The excerpt below is from J.S. Bach's Mass in B minor. 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 (CШаблон:Music–EШаблон:Music); these two voices resolve inward to an octave.

Related chords

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. For example, F–AШаблон:Music–C is a minor triad, so FШаблон:Music–AШаблон:Music–C is a doubly diminished triad. This is enharmonically equivalent to GШаблон:Music–AШаблон:Music–C, an incomplete dominant seventh AШаблон:MusicШаблон:Music, missing its fifth), which is a tritone substitute that resolves to G. Its inversion, AШаблон:Music–C–FШаблон:Music, is the Italian sixth chord that resolves to G.

Classical harmonic theory would notate the tritone substitute as an augmented sixth chord on Шаблон:Music2. The augmented sixth chord can either be (i) an It⁺⁶ enharmonically equivalent to a dominant seventh chord (with a missing fifth); (ii) a Ger⁺⁶ equivalent to a dominant seventh chord with (with a fifth); or (iii) a Fr⁺⁶ equivalent to the Lydian dominant (with a missing fifth), all of which serve in a classical context as a substitute for the secondary dominant of V.

All variants of augmented sixth chords are closely related to the applied dominant V⁷ of Шаблон:MusicII. Both Italian and German variants are enharmonically identical to dominant seventh chords. For example, in the key of C, the German sixth chord could be reinterpreted as the applied dominant of DШаблон:Music.

Simon Sechter explains the chord of the French sixth chord as being a chromatically altered version of a seventh chord on the second degree of the scale, Шаблон:Music. The German sixth is explained as a chromatically altered ninth chord on the same root but with the root omitted.

The tendency of the interval of the augmented sixth to resolve outwards is therefore explained by the fact that the AШаблон:Music, being a dissonant note, a diminished fifth above the root (D), and flatted, must fall, whilst the FШаблон:Music – being chromatically raised – must rise.

Relationship between the different types

The following "curious chromatic sequence", graphed by Dmitri Tymoczko as a four-dimensional tesseract, outlines the relationships between the augmented sixth chords in 12TET tuning:

Файл:Schlegel wireframe 8-cell.png

A tesseract. The diminished seventh chords occupy points on two diagonally opposite corners.

• Starting with a diminished seventh chord, lower any factor by a semitone. The result is equivalently to a German sixth chord.
• From the German sixth chord, lower any factor by a semitone so that the result is ancohemitonic (i.e.: possesses no half steps). The result is a French sixth chord or minor seventh chord possibly posing as a virtual augmented sixth.
• From the French sixth chord (or minor seventh chord posing as augmented sixth), there exists a factor which, when lowered by semitone, gives a result equivalent to a half-diminished seventh chord possibly posing as a virtual augmented sixth.
• From the half-diminished seventh chord as augmented sixth, there exists a factor which, when lowered by a semitone, is equivalent to a diminished seventh chord at the interval one semitone lower than the diminished seventh chord which started the sequence.
• Three repetitions of the above complete the cycle in modulo-12 note space, forming a necklace of three tesseracts joined at opposite corners by diminished seventh chords and subsuming all 12 notes of the chromatic scale.

Minor seventh as virtual augmented sixth chord

The minor seventh chord may also have its interval of minor seventh (between the root and seventh degree (i.e.: C–BШаблон:Music in C–EШаблон:Music–G–BШаблон:Music) rewritten as an augmented sixth (C–EШаблон:Music–G–AШаблон:Music). Rearranging and transposing, this gives AШаблон:Music–CШаблон:Music–EШаблон:Music–FШаблон:Music, a virtual minor version of the German sixth chord. Again like the typical +6, this enharmonic interpretation gives a resolution irregular for the minor seventh but normal for the augmented sixth, where the two voices at the enharmonic major second converge to a unison or diverge to an octave.

Half-diminished seventh as virtual augmented sixth chord

The half-diminished seventh chord is the inversion of the German sixth chord (it is its inversion as a set, rather than as a chord). Its interval of minor seventh (between root and seventh degree (i.e.: C–BШаблон:Music in C–EШаблон:Music–GШаблон:Music–BШаблон:Music) can be written as an augmented sixth (C–EШаблон:Music–GШаблон:Music–AШаблон:Music). Rearranging and transposing, this gives AШаблон:Music–CШаблон:Music–D–FШаблон:Music, a virtual minor version of the French sixth chord.Шаблон:Quote needed Like the typical +6, this enharmonic interpretation gives a resolution irregular for the half-diminished seventh but normal for the augmented sixth, where the two voices at the enharmonic major second converge to a unison or diverge to an octave.

Tristan chord

Шаблон:Main Richard Wagner's Tristan chord, the first vertical sonority in his opera, Tristan und Isolde, can be interpreted as a half-diminished seventh that transitions to a French sixth in the key of A minor (F–A–B–DШаблон:Music, in red below). The upper voice continues upward with a long appoggiatura (GШаблон:Music to A). Note that the DШаблон:Music resolves down to DШаблон:Music instead of up to E:

See also

• Neapolitan chord
• Acoustic scale

Further reading

• Шаблон:Cite book
• Шаблон:Cite book
• Шаблон:Cite book

External links

• Augmented sixths in Commons.