Accelerator physics: Difference between revisions
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[[File:Aug6interval.png|right|frame|The interval of an augmented sixth normally resolves outwards by semitone to an [[octave]]. {{audio|Augmented sixth resolving to octave.ogg|Play}}]] | |||
[[File:Most common augmented sixth chords.png|thumb|250px|The 3 most common augmented sixth chords in the tonalities of C major and C minor: Italian 6th, French 6th, and German 6th.]] | |||
In [[music theory]], an '''augmented sixth chord''' contains the [[interval (music)|interval]] of an [[augmented sixth]], usually above its [[bass note|bass tone]]. This [[chord (music)|chord]] has its origins in the [[Renaissance music|Renaissance]],<ref>{{Cite book | last = Andrews | first = Herbert Kennedy | year = 1950 | title = The Oxford Harmony | edition = 1 | volume = 2 | location = London | publisher = [[Oxford University Press]] | oclc = 223256512 | ref = CITEREFAndrews1950 | pages = 45–46}}</ref> further developed in the [[Baroque music|Baroque]], and became a distinctive part of the musical style of the [[Classical period (music)|Classical]] and [[Romantic period (music)|Romantic]] periods.<ref>{{Harvnb|Andrews|1950|pp=46–52}}</ref> The three more common types of augmented sixth chords as usually called '''Italian 6th''', '''French 6th''', and '''German 6th'''. | |||
==Resolution and chord construction== | |||
The augmented sixth interval is typically between the sixth [[scale degree|degree]] of the minor scale (henceforth '''{{music|flat}}6''') and the raised fourth degree (henceforth '''{{music|sharp}}4'''). With standard [[voice leading]], the chord is followed directly or indirectly by some form of the [[dominant (music)|dominant]] chord, in which both '''{{music|flat}}6''' and '''{{music|sharp}}4''' have [[Resolution (music)|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 [[enharmonic]]ally as a [[minor seventh]] ('''{{music|flat}}6''' and '''{{music|flat}}5'''). Although augmented sixth chords are more common in the [[Major and minor|minor mode]], they are also used in the major mode by [[borrowed chord|borrowing]] '''{{music|flat}}6''' of the [[parallel minor]] scale.<ref name="Aldwell478483">{{Cite book | |||
| last = Aldwell | |||
| first = Edward | |||
| author-link = Edward Aldwell | |||
| coauthors = [[Carl Schachter]] | |||
| year = 1989 | |||
| title = Harmony and Voice Leading | |||
| edition = 2 | |||
| location = San Diego, Toronto | |||
| publisher = [[Harcourt Brace Jovanovich]] | |||
| isbn = 0-15-531519-6 | |||
| oclc = 19029983 | |||
| ref = CITEREFAldwellSchachter1989 | |||
| pages = 478–483 | |||
}}</ref> | |||
===Double-diminished triad=== | |||
In [[music theory]], the '''''double-diminished triad''''' is an archaic concept and term referring to a [[triad (music)|triad]], or three note [[chord (music)|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.<ref>Ernst Friedrich Richter (1912). ''Manual of Harmony'', p.94. Theodore Baker.</ref> | |||
For example, F-A{{music|b}}-C is a minor triad. F{{music|#}}-A{{music|b}}-C is a doubly diminished triad. Note that it is [[enharmonic]]ally equivalent to G{{music|b}}-A{{music|b}}-C (incomplete dominant seventh [missing E{{music|b}}]<!--in D{{music|b}}--> or "backdoor" dominant-seventh substitute {{music|b}}VII<!--in G-->). A{{music|b}}-C-F{{music|#}} is an Italian augmented sixth chord. | |||
==Types== | |||
There are three main types of augmented sixth chords, commonly known as Italian 6th, French 6th, and German 6th. 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.<ref name="Aldwell478483"/><ref>{{Cite book | last = Gauldin | first = Robert | year = 1997 | |||
| title = Harmonic Practice in Tonal Music | edition = 1 | location = New York | publisher = [[W.W. Norton]] | isbn = 0-393-97074-4 | |||
| oclc = 34966355 | ref = CITEREFGauldin1997 | pages = 422–438}}</ref><ref>{{Cite book | last = Christ | first = William | year = 1973 | |||
| title = Materials and Structure of Music | edition = 2 | volume = 2 | location = Englewood Cliffs, NJ | publisher = [[Prentice Hall]] | isbn = 0-13-560342-0 | oclc = 257117 | ref = CITEREFChrist1973 | pages = 141–171}} Offers a detailed explanation of augmented sixth chords as well as [[Neapolitan sixth]] chords.</ref> | |||
===Italian sixth=== | |||
[[File:ItalianSixth.png|thumb|right|The '''Italian sixth''' moving to V. {{audio|Italian sixth moving to V.ogg|Play}}]] | |||
The '''Italian sixth''' ('''It'''<math>^{+6}</math> or '''It'''<math>^6</math>) is derived from '''iv'''<math>^6_{}</math> with an altered fourth scale degree, {{music|sharp}}4: '''{{music|flat}}6'''—'''1'''—'''{{music|sharp}}4'''; A{{music|flat}}—C—F{{music|sharp}} in C major and C minor. This is the only augmented sixth chord comprising just three distinct notes; in four-part writing, the tonic pitch is doubled<!--wouldn't it be because it's the best voice leading strategy?-->. | |||
[[File:BeethovenOp78.png|center|thumb|300px|The second movement of [[Beethoven's]] [[Piano Sonata No. 24 (Beethoven)|piano sonata in F-sharp major]], op. 78, begins with an Italian sixth chord. {{audio|BeethovenOp78.mid|Play}}]] | |||
{{clear}} | |||
The Italian 6th is [[enharmonic]]ally equivalent to an incomplete [[dominant seventh]].{{sfn|Rimsky|p=121}} | |||
===French sixth=== | |||
[[File:FrenchSixth.png|thumb|right|The '''French sixth''' chord; the distinguishing tone is highlighted in blue. {{audio|French sixth moving to V.ogg|Play}}]] | |||
The '''French sixth''' ('''Fr'''<math>^{+6}_{}</math> or '''Fr'''<math>_3^4</math>) is similar to the Italian, but with an additional tone, '''2''': '''{{music|flat}}6'''—'''1'''—'''2'''—'''{{music|sharp}}4'''; A{{music|flat}}—C—D—F{{music|sharp}} in C major and C minor. 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]]).<ref>Blatter, Alfred (2007). ''Revisiting Music Theory: a Guide to the Practice'', p.144. ISBN 978-0-415-97440-0. "One may note that the French sixth contains the elements of a whole tone scale commonly associated with French impressionistic composers."</ref> | |||
{{clear}} | |||
===German sixth=== | |||
The '''German sixth''' ('''Gr'''<math>^{+6}_{}</math> or '''Ger'''<math>_5^6</math>) is also like the Italian, but with an added tone '''{{music|flat}}3''': {{music|flat}}'''6'''—'''1'''—'''{{music|flat}}3'''—'''{{music|sharp}}4'''; A{{music|flat}}—C—E{{music|flat}}—F{{music|sharp}} in C major and C minor. In Classical music, however, it appears in much the same places as the other variants, though perhaps less used because of the [[counterpoint|contrapuntal]] difficulties outlined below. It appears frequently in the works of [[Ludwig van Beethoven|Beethoven]].<ref>Notable examples include the themes of the slow movements (both in variation form) of the opp. 57 ("Appassionata") and 109 piano sonatas.</ref> 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 ''[[Consecutive fifths#Mozart fifths|Mozart fifths]]'', were occasionally accepted by [[Common practice period|common practice]] composers. There are two ways they can be avoided: | |||
#The '''{{music|flat}}3''' can move to either '''1''' or '''2''', thereby generating an Italian or French sixth, respectively, and eliminating the perfect fifth between '''{{music|flat}}6''' and '''{{music|flat}}3'''.<ref>{{Cite book | |||
| last = Benjamin | |||
| first = Thomas | |||
| coauthors = Michael Horvit & Robert Nelson | |||
| year = 2008 | |||
| title = Techniques and Materials of Music: From the Common Practice Period Through the Twentieth Century | |||
| edition = 7 | |||
| location = Belmont, CA | |||
| publisher = Thomson & Schirmer | |||
| isbn = 978-0-495-18977-0 | |||
| oclc = 145143714 | |||
| ref = CITEREFBenjaminHorvitNelson2008 | |||
| page = 165 | |||
}} Beethoven frequently moves from one form of the chord to another in such a way, sometimes passing through all three.</ref> | |||
#The chord can resolve to a "six-four" chord, functionally either as a [[Inversion (music)#Cadential six-four chord|cadential six-four]] intensification of '''V''', or as the second [[Inversion (music)#Inverted chords|inversion of '''I''']]; the cadential six-four, in turn, resolves to a root-position '''V'''. This [[chord progression|progression]] ensures that, in its voice leading, each pair of voices moves either by [[contrapuntal motion|oblique motion or contrary motion]] and avoids parallel motion altogether. In minor modes, both '''1''' and '''{{music|flat}}3''' do not move during the resolution of the German sixth to the cadential six-four. In major modes, '''{{music|flat}}3''' can be [[enharmonic]]ally respelled as '''{{music|sharp}}2''' if it resolves upwards to '''{{music|natural}}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,{{Citation needed|date=March 2012}}<!--cite: DeVoto's editions of Piston; David Cohen; respectively; will find exact cites asap.--> or as a "'doubly augmented sixth chord"', as it contains two augmented intervals. However, other sources describe it as a German sixth.<ref name=grove>Drabkin, William. [http://www.oxfordmusiconline.com/subscriber/article/grove/music/01517? Augmented sixth chord]. Grove Music Online (subscription needed). Accessed March 2012.</ref> | |||
{|align="center" | |||
|[[File:GermanSixth.png|thumb|The '''German sixth'''; the distinguishing tone is highlighted in blue. {{audio|German sixth moving to V.ogg|Play}}]] | |||
|[[File:GermanSixth64.png|thumb|The '''German sixth''' is typically followed by a '''I'''<math>_4^6</math> chord to avoid parallel fifths. {{audio|GermanSixth64.mid|Play}}]] | |||
|[[File:DoublyAugmentedFourth.png|thumb|'''German sixth''' chord respelled with doubly augmented fourth (highlighted in blue) for voice-leading purposes. Also referred to as '''English''', '''Swiss''' or '''Alsatian'''.]] | |||
|} | |||
{{Clear}} | |||
<!--image displayed wider than 400px, WP:IMGSIZE, for clarity-->[[File:M Haydn - German sixth chord.jpg|thumb|center|575px|A German sixth chord from [[Michael Haydn]]'s ''[[Requiem (Michael Haydn)|Requiem]]'' in C minor, first movement. ([[File:Loudspeaker.svg|11px]] [[Media:M Haydn - German sixth chord.mid|Listen]])]] | |||
{{Clear}} | |||
===Other variants=== | |||
Other variants of augmented sixth chords are sometimes found in the repertoire, and are sometimes given whimsical geographical names. For example: '''4'''—'''{{music|flat}}6'''—'''7'''—'''{{music|sharp}}2'''; (F—A{{music|flat}}—B—D{{music|sharp}}) is called by one source an ''Australian sixth''.<ref>{{Cite book | |||
| last = Burnard | |||
| first = Alex | |||
| year = 1950 | |||
| title = Harmony and Composition: For the Student and the Potential Composer | |||
| location = Melbourne | |||
| publisher = [[Allans Music]] (Australia) | |||
| oclc = 220305086 | |||
| ref = CITEREFBurnard1950 | |||
| pages = 94–95 | |||
}}</ref> Such anomalies usually have alternative interpretations. | |||
==Standard harmonic function== | |||
From the Baroque to the Romantic period, augmented sixth chords have had the same [[Diatonic_function#Diatonic_functions_of_notes_and_chords|harmonic function]]: as a [[chromatic]]ally altered '''[[predominant]]''' chord (typically, an alteration of '''[[supertonic|ii<math>_3^4</math>]]''', '''[[subdominant|IV<math>_5^6</math>]]''', '''[[submediant|vi<sup>7</sup>]]''' or their [[Parallel key|parallel]] equivalents in the minor mode) leading to a [[dominant (music)|dominant]] chord. This movement to the dominant is heightened by the [[semitone|semitonal]] resolution of both '''{{music|flat}}6''' to '''5''' and '''{{music|sharp}}4''' to '''5'''; essentially, these two notes act as [[leading-tone]]s. This characteristic has led many analysts<ref name="Piston419">{{Cite book | last = Piston | first = Walter | author-link = Walter Piston | |||
| coauthors = Mark DeVoto | year = 1987 | title = Harmony | edition = 5 | location = New York | publisher = [[W. W. Norton & Company, Inc.]] | isbn = 0-393-95480-3 | ref = CITEREFPiston419 | page = 419 | |||
}}</ref> to compare the voice leading of augmented sixth chords to the [[secondary dominant]] '''V''' of '''V''' because of the presence of '''{{music|sharp}}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, '''{{music|flat}}6''', as well as '''{{music|sharp}}4''', rather than just '''{{music|sharp}}4''' in the [[minor mode]]. | |||
==Root position and inversion of 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. | |||
[[Jean-Jacques Rousseau|Rousseau]] considered that the chord could not be inverted.<ref>{{cite book|last=Rousseau|first=Jean Jaques|title=Dictionnaire de Musique}}</ref> 17th century instances of the augmented sixth with the sharp note in the bass are generally limited to German sources.<ref>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.</ref> | |||
<!--image displayed wider than 400px, WP:IMGSIZE, for clarity--> | |||
[[File:Excerpt from Bach's Mass in B Minor.jpg|thumb|900px|center|Excerpt from 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 (C{{music|sharp}}—E{{music|flat}}); these two voices resolve inward to an octave{{audio|Excerpt from Bach's Mass in B Minor.mid|Play}}]] | |||
[[File:Augmented 6th cord roots according to Simon Sechter.png|thumb|left|150px|Root positions of the augmented 6th chords in the tonality of C, according to Simon Sechter.<ref name="Sechter"/>]] | |||
[[Simon Sechter]] explains the chord of the French Sixth as being a chromatically altered version of a [[seventh chord]] on the second degree of the scale. The German Sixth is explained as a chromatically altered [[ninth chord]] on the same root, but with the root omitted.<ref name="Sechter">{{cite book|last=Sechter|first=Simon|title=Die Grundsätze der musicalischen Komposition|year=1853|publisher=Breitkopf und Härtel|location=Leipzig|language=German}}</ref> | |||
The tendency of the interval of the augmented sixth to resolve outwards is therefore explained by the fact that the A{{music|flat}}, being a dissonant note, a diminished fifth above the root (D), and flatted, must fall, whilst the F{{music|sharp}} - being chromatically raised - must rise. | |||
{{clear}} | |||
==Extended 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 remote keys, and possible [[Consonance and dissonance|dissonances]] with the juxtaposition of remotely related keys. | |||
===Augmented sixth chords as altered dominant chords with flattened 2nd degree=== | |||
{{see also|tritone substitution}} | |||
The augmented sixth chord may be built on notes other than '''{{music|flat}}6'''. | |||
[[Tchaikovsky]] considered the augmented sixth chords as being [[altered dominant]]s.<ref>Roberts, Peter Deane (1993). ''Modernism in Russian Piano Music: Skriabin, Prokofiev, and Their Russian Contemporaries'', p.136. ISBN 0-253-34992-3.</ref> He described the augmented sixth chords as 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".<ref name="Tchaik">{{cite book|last=Tschaikovsky|first=Peter|title=Guide to the Practica Study of Harmony|year=1900|publisher=P. Jurgenson|location=Leipzig|pages=106, 108|url=http://imslp.org/wiki/Guide_to_the_Practical_Study_of_Harmony_%28Tchaikovsky,_Pyotr%29|edition=English translation|editor=Translated from the German version by Emil Krall and James Liebling|chapter=XXVII}}</ref> | |||
[[File:Aug 6ths Tchaikovsky.png|thumb|350px|center|Augmented sixths as dominants in the tonality of C, according to Tchaikovsky. Notice the preluminary resolution to avoid [[parallel fifths]] in the last example.<ref name="Tchaik"/>]] | |||
[[File:SchubertAug6.png|center|thumb|350px|An Italian sixth chord built on scale degree '''{{music|flat}}2''' preceded by a '''{{music|flat}}II''' chord in root position. Here the Italian sixth chord functions as a substitute for the dominant, with which it shares the third and seventh. Schubert's A major sonata, D. 959. {{audio|SchubertAug6.mid|Play}}]] | |||
Classical harmonic theory would notate the "[[tritone substitute]]" as an augmented sixth chord on {{music|flat}}2. The Augmented sixth chord can either be the [[Augmented sixth chord#Italian sixth|It<sup>+6</sup>]] enharmonic to a dominant 7th chord without the 5th, [[Augmented sixth chord#German sixth|Gr<sup>+6</sup>]], enharmonically equivalent to a dominant 7th chord with the 5th, or [[Augmented sixth chord#French sixth|Fr<sup>+6</sup>]] 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.<ref>Satyendra, Ramon. "Analyzing the Unity within Contrast: Chick Corea's ''Starlight''", p.55. Cited in Stein.</ref><ref>Stein, Deborah (2005). ''Engaging Music: Essays in Music Analysis''. New York: Oxford University Press. ISBN 0-19-517010-5.</ref> | |||
[[File:Tritone substitutions.png|thumb|center|400px|Original dominant 7th-tonic progression {{audio|Tritone substitutions V7-I.mid|Play}}, tritone substitution ({{music|b}}V<sup>7</sup>/V-I) {{audio|Tritone substitutions bV7ofV-I.mid|Play}}, augmented sixth chord (It<sup>6</sup>-I) {{audio|Tritone substitutions It6-I.mid|Play}}.]] | |||
{{Clear}} | |||
===Enharmonicity to other chords=== | |||
====Enharmonic equivalency of the French sixth==== | |||
The French sixth has two characteristics in common with the [[diminished seventh chord]]: | |||
# Both chords are constructed of two superimposed tritones; in the French sixth, between '''{{music|flat}}6'''—'''2''' (A{{music|flat}}—D) and '''1'''—'''{{music|sharp}}4''' (C—F{{music|sharp}}). Thus, both have [[Inversion (music)#Inversional equivalency|inversional symmetry]]; | |||
# Both are [[enharmonic]]ally 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 [[modulation (music)|modulating]] to very remote keys. For instance: '''{{music|flat}}6'''—'''1'''—'''2'''—'''{{music|sharp}}4'''; (A{{music|flat}}—C—D—F{{music|sharp}} in C), could be interpreted identically in F{{music|sharp}} if reordered and respelled as D—F{{music|sharp}}—G{{music|sharp}}—B{{music|sharp}}, i.e., the French sixth of the '''{{music|sharp}}IV''' key area, displaced an interval of a tritone relative to the tonic key, '''I'''. | |||
===Dominant functions=== | |||
{|align="center" | |||
|[[File:German sixth equals dominant seventh.png|thumb|right|German sixth, A{{music|b}}-C-D{{music|#}}-F{{music|#}}, and equivalent dominant seventh, A{{music|b}}-C-E{{music|b}}-G{{music|b}}<ref>Benward & Saker (2008). ''Music in Theory and Practice'', Vol. II, p.222. ISBN 978-0-07-310188-0.</ref> {{audio|German sixth equals dominant seventh.mid|Play}}]] | |||
|[[File:Irregular resolution through augmented sixth equivalence.png|thumb|right|[[Irregular resolution]] through augmented sixth equivalence {{audio|Irregular resolution through augmented sixth equivalence.mid|Play}}.<ref>Owen, Harold (2000). ''Music Theory Resource Book'', p.132. ISBN 0-19-511539-2.</ref>]] | |||
|} | |||
All variants of augmented sixth chords are closely related to the [[applied dominant]] '''V<sup>7</sup>''' of '''{{music|flat}}II'''; both [[#Italian sixth|Italian]] and [[#German sixth|German]] variants are enharmonically identical to [[dominant seventh]] chords. For example, in the key of C (I), the German sixth chord, A{{music|flat}}—C—E{{music|flat}}—F{{music|sharp}}, could be reinterpreted as A{{music|flat}}—C—E{{music|flat}}—G{{music|flat}}, the applied dominant of D{{music|flat}} (V/D{{music|flat}}). | |||
{{clear}} | |||
==Function== | |||
In most occasions, the augmented 6th chords precede either the [[dominant (music)|dominant]], or the [[tonic (music)|tonic]] in second inversion.{{sfn|Rimsky|p=121}} | |||
The augmented sixths can be treated as chromatically altered passing chords.{{sfn|Rimsky|p=121}} | |||
===Tristan chord=== | |||
{{main|Tristan chord}} | |||
[[Richard Wagner]]'s [[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 (music)|key]] of A minor (F-A-B-D{{music|sharp}}) with an upwardly resolving long [[appoggiatura]] in the upper voice (G{{music|sharp}} to A). Note that the D{{music|sharp}} resolves downwards to D{{music|natural}} instead of E: | |||
[[File:Tristanchord.png|center|thumb|350px]] | |||
==See also== | |||
*[[Acoustic scale]] | |||
==References== | |||
{{Reflist|30em}} | |||
==Books== | |||
*{{cite book|last=Rimsky-Korsakov|first=Nikolai|title=Practical Treatise on Harmony|year=1886|publisher=A. Büttner.|location=St. Petersburg|edition=13th - 1924}} | |||
{{Chords}} | |||
{{Chromaticism}} | |||
{{DEFAULTSORT:Augmented Sixth Chord}} | |||
[[Category:Chords]] | |||
[[Category:Chromaticism]] | |||
Revision as of 04:58, 9 January 2014
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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,[1] further developed in the Baroque, and became a distinctive part of the musical style of the Classical and Romantic periods.[2] The three more common types of augmented sixth chords as usually called Italian 6th, French 6th, and German 6th.
Resolution and chord construction
The augmented sixth interval is typically between the sixth degree of the minor scale (henceforth Template:Music6) and the raised fourth degree (henceforth Template:Music4). With standard voice leading, the chord is followed directly or indirectly by some form of the dominant chord, in which both Template:Music6 and Template:Music4 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 (Template:Music6 and Template:Music5). Although augmented sixth chords are more common in the minor mode, they are also used in the major mode by borrowing Template:Music6 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-ATemplate:Music-C is a minor triad. FTemplate:Music-ATemplate:Music-C is a doubly diminished triad. Note that it is enharmonically equivalent to GTemplate:Music-ATemplate:Music-C (incomplete dominant seventh [missing ETemplate:Music] or "backdoor" dominant-seventh substitute Template:MusicVII). ATemplate:Music-C-FTemplate:Music is an Italian augmented sixth chord.
Types
There are three main types of augmented sixth chords, commonly known as Italian 6th, French 6th, and German 6th. 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
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The Italian sixth (It or It) is derived from iv with an altered fourth scale degree, Template:Music4: Template:Music6—1—Template:Music4; ATemplate:Music—C—FTemplate:Music in C major and C minor. This is the only augmented sixth chord comprising just three distinct notes; in four-part writing, the tonic pitch is doubled.
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50 year old Petroleum Engineer Kull from Dawson Creek, spends time with interests such as house brewing, property developers in singapore condo launch and camping. Discovers the beauty in planing a trip to places around the entire world, recently only coming back from . The Italian 6th is enharmonically equivalent to an incomplete dominant seventh.Template:Sfn
French sixth
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The French sixth (Fr or Fr) is similar to the Italian, but with an additional tone, 2: Template:Music6—1—2—Template:Music4; ATemplate:Music—C—D—FTemplate:Music in C major and C minor. 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] 50 year old Petroleum Engineer Kull from Dawson Creek, spends time with interests such as house brewing, property developers in singapore condo launch and camping. Discovers the beauty in planing a trip to places around the entire world, recently only coming back from .
German sixth
The German sixth (Gr or Ger) is also like the Italian, but with an added tone Template:Music3: Template:Music6—1—Template:Music3—Template:Music4; ATemplate:Music—C—ETemplate:Music—FTemplate:Music in C major and C minor. 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.[8] 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:
- The Template:Music3 can move to either 1 or 2, thereby generating an Italian or French sixth, respectively, and eliminating the perfect fifth between Template:Music6 and Template:Music3.[9]
- 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 Template:Music3 do not move during the resolution of the German sixth to the cadential six-four. In major modes, Template:Music3 can be enharmonically respelled as Template:Music2 if it resolves upwards to Template:Music3, 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,Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. or as a "'doubly augmented sixth chord"', as it contains two augmented intervals. However, other sources describe it as a German sixth.[10]
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Other variants
Other variants of augmented sixth chords are sometimes found in the repertoire, and are sometimes given whimsical geographical names. For example: 4—Template:Music6—7—Template:Music2; (F—ATemplate:Music—B—DTemplate:Music) is called by one source an Australian sixth.[11] Such anomalies usually have alternative interpretations.
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, IV, 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 Template:Music6 to 5 and Template:Music4 to 5; essentially, these two notes act as leading-tones. This characteristic has led many analysts[12] to compare the voice leading of augmented sixth chords to the secondary dominant V of V because of the presence of Template:Music4, 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, Template:Music6, as well as Template:Music4, rather than just Template:Music4 in the minor mode.
Root position and inversion of 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.
Rousseau considered that the chord could not be inverted.[13] 17th century instances of the augmented sixth with the sharp note in the bass are generally limited to German sources.[14]
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Simon Sechter explains the chord of the French Sixth as being a chromatically altered version of a seventh chord on the second degree of the scale. The German Sixth is explained as a chromatically altered ninth chord on the same root, but with the root omitted.[15]
The tendency of the interval of the augmented sixth to resolve outwards is therefore explained by the fact that the ATemplate:Music, being a dissonant note, a diminished fifth above the root (D), and flatted, must fall, whilst the FTemplate:Music - being chromatically raised - must rise. 50 year old Petroleum Engineer Kull from Dawson Creek, spends time with interests such as house brewing, property developers in singapore condo launch and camping. Discovers the beauty in planing a trip to places around the entire world, recently only coming back from .
Extended 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 remote keys, and possible dissonances with the juxtaposition of remotely related keys.
Augmented sixth chords as altered dominant chords with flattened 2nd degree
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The augmented sixth chord may be built on notes other than Template:Music6.
Tchaikovsky considered the augmented sixth chords as being altered dominants.[16] He described the augmented sixth chords as 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".[17]
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Classical harmonic theory would notate the "tritone substitute" as an augmented sixth chord on Template:Music2. 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.[18][19]
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Enharmonicity to other chords
Enharmonic equivalency of the French sixth
The French sixth has two characteristics in common with the diminished seventh chord:
- Both chords are constructed of two superimposed tritones; in the French sixth, between Template:Music6—2 (ATemplate:Music—D) and 1—Template:Music4 (C—FTemplate:Music). Thus, both have inversional symmetry;
- 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: Template:Music6—1—2—Template:Music4; (ATemplate:Music—C—D—FTemplate:Music in C), could be interpreted identically in FTemplate:Music if reordered and respelled as D—FTemplate:Music—GTemplate:Music—BTemplate:Music, i.e., the French sixth of the Template:MusicIV key area, displaced an interval of a tritone relative to the tonic key, I.
Dominant functions
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All variants of augmented sixth chords are closely related to the applied dominant V7 of Template:MusicII; both Italian and German variants are enharmonically identical to dominant seventh chords. For example, in the key of C (I), the German sixth chord, ATemplate:Music—C—ETemplate:Music—FTemplate:Music, could be reinterpreted as ATemplate:Music—C—ETemplate:Music—GTemplate:Music, the applied dominant of DTemplate:Music (V/DTemplate:Music). 50 year old Petroleum Engineer Kull from Dawson Creek, spends time with interests such as house brewing, property developers in singapore condo launch and camping. Discovers the beauty in planing a trip to places around the entire world, recently only coming back from .
Function
In most occasions, the augmented 6th chords precede either the dominant, or the tonic in second inversion.Template:Sfn
The augmented sixths can be treated as chromatically altered passing chords.Template:Sfn
Tristan chord
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. Richard Wagner's 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-DTemplate:Music) with an upwardly resolving long appoggiatura in the upper voice (GTemplate:Music to A). Note that the DTemplate:Music resolves downwards to DTemplate:Music instead of E:
See also
References
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Books
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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Template:Chords Template:Chromaticism
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Template:Harvnb
- ↑ 3.0 3.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Ernst Friedrich Richter (1912). Manual of Harmony, p.94. Theodore Baker.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Offers a detailed explanation of augmented sixth chords as well as Neapolitan sixth chords. - ↑ Blatter, Alfred (2007). Revisiting Music Theory: a Guide to the Practice, p.144. ISBN 978-0-415-97440-0. "One may note that the French sixth contains the elements of a whole tone scale commonly associated with French impressionistic composers."
- ↑ Notable examples include the themes of the slow movements (both in variation form) of the opp. 57 ("Appassionata") and 109 piano sonatas.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 Beethoven frequently moves from one form of the chord to another in such a way, sometimes passing through all three. - ↑ Drabkin, William. Augmented sixth chord. Grove Music Online (subscription needed). Accessed March 2012.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ 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.
- ↑ 15.0 15.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Roberts, Peter Deane (1993). Modernism in Russian Piano Music: Skriabin, Prokofiev, and Their Russian Contemporaries, p.136. ISBN 0-253-34992-3.
- ↑ 17.0 17.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Satyendra, Ramon. "Analyzing the Unity within Contrast: Chick Corea's Starlight", p.55. Cited in Stein.
- ↑ Stein, Deborah (2005). Engaging Music: Essays in Music Analysis. New York: Oxford University Press. ISBN 0-19-517010-5.
- ↑ Benward & Saker (2008). Music in Theory and Practice, Vol. II, p.222. ISBN 978-0-07-310188-0.
- ↑ Owen, Harold (2000). Music Theory Resource Book, p.132. ISBN 0-19-511539-2.