Inversion (music)#Musical set theory

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Figured bass is a notation in which chord inversions are indicated by Arabic numerals (the figures) either above or below the bass notes, indicating a harmonic progression. Each numeral expresses the interval that results from the voices above it (usually assuming octave equivalence). For example, in root-position triad C–E–G, the intervals above bass note C are a third and a fifth, giving the figures {{su|p=5|b=3}}. If this triad were in first inversion (e.g., E–G–C), the figure {{su|p=6|b=3}} would apply, due to the intervals of a third and a sixth appearing above the bass note E.

Certain conventional abbreviations exist in the use of figured bass. For instance, root-position triads appear without symbols (the {{su|p=5|b=3}} is understood), and first-inversion triads are customarily abbreviated as just {{music|6 chord}}, rather than {{su|p=6|b=3}}. The table to the right displays these conventions.

Figured-bass numerals express distinct intervals in a chord only as they relate to the bass note. They make no reference to the key of the progression (unlike Roman-numeral harmonic analysis), they do not express intervals {{em|between}} pairs of upper voices themselves – for example, in a C–E–G triad, the figured bass does not signify the interval relationship between E–G, and they do not express notes in upper voices that double, or are unison with, the bass note.

However, the figures are often used on their own (without the bass) in music theory simply to specify a chord's inversion. This is the basis for the terms given above such as "64 chord" for a second inversion triad. Similarly, in harmonic analysis the term I⁶ refers to a tonic triad in first inversion.

{{Clear}}

{{anchor|Invertible counterpoint}}

{{Image frame|content=

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|width=400|align=right|caption=An example of contrapuntal inversion in one measure of J.S. Bach's Invention No. 13 in A minor, BWV 784.}}

In contrapuntal inversion, two melodies, having previously accompanied each other once, accompany each other again but with the melody that had been in the high voice now in the low, and vice versa. The action of changing the voices is called textural inversion. This is called double counterpoint when two voices are involved and triple counterpoint when three are involved. The inversion in two-part invertible counterpoint is also known as rivolgimento.{{Cite encyclopedia|title=Rivolgimento (It.)|encyclopedia=Grove Music Online|date=2001 |doi=10.1093/gmo/9781561592630.article.23544|quote=The inversion of the parts in two-part Invertible counterpoint.}}

In J.S. Bach's The Art of Fugue, the first canon is at the octave, the second canon at the tenth, the third canon at the twelfth, and the fourth canon in augmentation and contrary motion. Other exemplars can be found in the fugues in [http://www2.nau.edu/tas3/wtc/ii16.html#movie G minor] {{Webarchive|url=https://web.archive.org/web/20100327103142/http://www2.nau.edu/tas3/wtc/ii16.html#movie |date=2010-03-27 }} and [https://web.archive.org/web/20100220110325/http://www2.nau.edu/tas3/wtc/ii21.html#movie B{{music|flat}} major] [external Shockwave movies] from J.S. Bach's The Well-Tempered Clavier, Book 2, both of which contain invertible counterpoint at the octave, tenth, and twelfth.

When this passage returns in bars 25–35 these lines are exchanged:File:Bach Prelude in A flat from WTC1 bars 25-36.wav

File:Bach Prelude in A flat from WTC1 bars 25-35.png

J.S. Bach's Three-Part Invention in F minor, BWV 795 involves exploring the combination of three themes. Two of these are announced in the opening two bars. A third idea joins them in bars 3–4. When this passage is repeated a few bars later in bars 7–9, the three parts are interchanged:File:Bach Prelude in A flat from WTC1 bars 25-38.wavFile:Bach 3-part Invention BWV 795, bars 1-9.png

The piece goes on to explore four of the six possible permutations of how these three lines can be combined in counterpoint.

One of the most spectacular examples of invertible counterpoint occurs in the finale of Mozart's Jupiter Symphony. Here, no less than five themes are heard together: File:Mozart Jupiter Finale final section, bars 389-397 all five themes together.wavFile:Mozart Jupiter Finale final section, bars 389-396 all five themes together.png

The whole passage brings the symphony to a conclusion in a blaze of brilliant orchestral writing. According to Tom Service:

{{Quote|Mozart's composition of the finale of the Jupiter Symphony is a palimpsest on music history as well as his own. As a musical achievement, its most obvious predecessor is really the fugal finale of his G major String Quartet K. 387, but this symphonic finale trumps even that piece in its scale and ambition. If the story of that operatic tune first movement is to turn instinctive emotion into contrapuntal experience, the finale does exactly the reverse, transmuting the most complex arts of compositional craft into pure, exhilarating feeling. Its models in Michael and Joseph Haydn are unquestionable, but Mozart simultaneously pays homage to them – and transcends them. Now that's what I call real originality.Service, Tom. (2014) "Symphony Guide: Mozart's 41st (Jupiter)", The Guardian, 27 May.}}

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d8 c16 d e d c8 b16 c d c

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|width=450|align=right|caption=Two lines from the fugue in G major from J. S. Bach's The Well-Tempered Clavier, Book 1. The lowest voice in mm. 28–30 is an inversion of the opening melody in mm. 1–3.}}

A melody is inverted by flipping it "upside-down", reversing the melody's contour. For instance, if the original melody has a rising major third, then the inverted melody has a falling major third (or, especially in tonal music, perhaps a falling minor third).

According to The Harvard Dictionary of Music, "The intervals between successive pitches may remain exact or, more often in tonal music, they may be the equivalents in the diatonic scale. Hence c'–d–e' may become c'–b–a (where the first descent is by a semitone rather than by a whole tone) instead of c'–b{{music|flat}}–a{{music|flat}}."{{Cite book|url=https://archive.org/details/harvarddictionar0004unse/page/418|title=The Harvard Dictionary of Music|date=2003|publisher=Belknap Press of Harvard University Press|isbn=0674011635|editor-last=Randel|editor-first=Don Michael|editor-link=Don Michael Randel|edition=fourth|location=Cambridge, Massachusetts|pages=[https://archive.org/details/harvarddictionar0004unse/page/418 418]|oclc=52623743}} Moreover, the inversion {{em|may}} start on the same pitch as the original melody, but it does not have to, as illustrated by the example to the right.

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In set theory, the inverse operation is sometimes designated as $T\_nI$, where $I$ means "invert" and $T\_n$ means "transpose by some interval $n$" measured in number of semitones. Thus, inversion is a combination of an inversion followed by a transposition. To apply the inversion operation $I$, you subtract the pitch class, in integer notation, from 12 (by convention, inversion is around pitch class 0). Then we apply the transposition operation $T\_n$ by adding $n$. For example, to calculate $T\_5I(3)$, first subtract 3 from 12 (giving 9) and then add 5 (giving 14, which is equivalent to 2). Thus, $T\_5I(3)=2$.{{Cite book|title=Introduction to Post-Tonal Theory|last=Straus|first=Joseph N.|publisher=Prentice Hall|year=1990|isbn=0136866921|location=Englewood Cliffs, New Jersey|pages=34–35|oclc=20012239}} To invert a set of pitches, simply invert each pitch in the set in turn.{{sfn|Straus|1990|p=36}}

Sets are said to be inversionally symmetrical if they map onto themselves under inversion. The pitch that the sets must be inverted around is said to be the axis of symmetry (or center). An axis may either be at a specific pitch or halfway between two pitches (assuming that microtones are not used). For example, the set C–E{{music|flat}}–E–F{{music|sharp}}–G–B{{music|flat}} has an axis at F, and an axis, a tritone away, at B if the set is listed as F{{music|sharp}}–G–B{{music|flat}}–C–E{{music|flat}}–E. As another example, the set C–E–F–F{{music|sharp}}–G–B has an axis at the dyad F/F{{music|sharp}} and an axis at B/C if it is listed as F{{music|sharp}}–G–B–C–E–F.{{Citation|last=Wilson|first=Paul|title=The Music of Béla Bartók|pages=10–11|year=1992|publisher=Yale University Press |isbn=0-300-05111-5|author-link=Paul Wilson (music theorist)}}

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\once \override NoteHead.color = #red c4^\markup { Melody } \once \override NoteHead.color = #red c g' g \once \override NoteHead.color = #red a \once \override NoteHead.color = #red a g2 f4 f e e d d c2 \bar "||"

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\once \override NoteHead.color = #red c4^\markup { Diatonic inversion about pitch axis C } \once \override NoteHead.color = #red c f, f e e f2 g4 g a a b b c2 \bar "||"

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\once \override NoteHead.color = #red c4^\markup { Chromatic inversion about pitch axis C } \once \override NoteHead.color = #red c f, f es es f2 g4 g aes aes bes bes c2 \bar "||"

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f4^\markup { Diatonic inversion about pitch axis A } f b, b \once \override NoteHead.color = #red a \once \override NoteHead.color = #red a b2 c4 c d d e e f2 \bar "||"

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|width=350|align=right|caption=Pitch axis inversions of "Twinkle, Twinkle, Little Star" about C and A}}

In jazz theory, a pitch axis is the center around which a melody is inverted.Pease, Ted (2003). Jazz Composition: Theory and Practice, p.152. {{ISBN|0-87639-001-7}}.

The "pitch axis" works in the context of the compound operation transpositional inversion, where transposition is carried out after inversion. However, unlike in set theory, the transposition may be a chromatic or diatonic transposition. Thus, if D-A-G (P5 up, M2 down) is inverted to D-G-A (P5 down, M2 up) the "pitch axis" is D. However, if it is inverted to C-F-G the pitch axis is G while if the pitch axis is A, the melody inverts to E-A-B.

The notation of octave position may determine how many lines and spaces appear to share the axis. The pitch axis of D-A-G and its inversion A-D-E either appear to be between C/B{{music|natural}} or the single pitch F.

{{Twelve-tone technique}}

{{Voicing (music)}}

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Category:Melody

Category:Musical symmetry

Category:Harmony

Category:Voicing (music)