In music theory, the word inversion has several meanings. Music theory is the field of study that deals with the Mechanics of music and how Music works There are inverted chords, inverted melodies, inverted intervals, and (in counterpoint) inverted voices. In Music, counterpoint is the relationship between two or more voices that are independent in contour and Rhythm, and interdependent in Harmony The concept of inversion also plays a role in musical set theory. Musical set theory provides concepts for categorizing musical objects and describing their relationships

Inverted chords

Figure 1: the closing phrase of the hymn-setting Rustington by the English composer Hubert Parry (1848–1918),^[1] showing all three positions of the C major chord. Sir Charles Hubert Hastings Parry 1st Baronet (27 February 1848 &ndash 7 October 1918 was an English Composer, best known for the choral song Jerusalem ^[2] See figured bass below for a description of the numerical symbols.

A chord's inversion describes the relationship of its bass to the other tones in the chord. For instance, a C major triad contains the tones C, E and G; its inversion is determined by which of these tones is used as the bottom note in the chord. In Music and Music theory, a triad is a three- note chord that can be stacked in thirds

The term inversion is often used to categorically refer to the different possibilities, although it may also be restricted to only those chords where the bass note is not also the root of the chord (see root position below). In Music the root ( basse fouhuhuhe) of a chord is the note or pitch upon which that chord is perceived or labelled as being built In texts that make this restriction, the term position may be used instead to refer to all of the possibilities as a category.

Root position

A root-position chord is sometimes known as the parent chord of its inversions. In Music the root ( basse fouhuhuhe) of a chord is the note or pitch upon which that chord is perceived or labelled as being built For example, C is the root of a C major triad and is in the bass when the triad is in root position; the 3rd and the 5th of the triad are sounded above the bass. Thus, a root-position chord is also known as a chord.

Inversions

In an inverted chord, the root is not in the bass (i. This article describes musical chords in traditional Western styles In Music the root ( basse fouhuhuhe) of a chord is the note or pitch upon which that chord is perceived or labelled as being built The bass note of a chord or sonority is the lowest Note played or notated e. , is not the lowest note). The inversions are numbered in the order their bass tones would appear in a closed root position chord (from bottom to top).

In the first inversion of a C major triad, the bass is E—the 3rd of the triad—with the 5th and the root stacked above it (the root now shifted an octave higher), forming the intervals of a 3rd and a 6th above the inverted bass of E, respectively. A first-inversion triad is also known as a chord.

In the second inversion, the bass is G—the 5th of the triad—with the root and the 3rd above it (both again shifted an octave higher), forming a 4th and a 6th above the (inverted) bass of G, respectively. A second-inversion triad is also known as a chord. This inversion can be either consonant or dissonant, and analytical notation sometimes treats it differently depending on the harmonic and voice-leading context in which it occurs (e. g. see The cadential six-four chord below).

Third inversions exist only for chords of four or more tones, such as 7th chords. In a third-inversion chord, the 7th of the chord is in the bass position. For example, a C major 7th chord in third inversion consists of B in the bass position, with C, E and G above it— being intervals of a 2nd, 4th and 6th above the (inverted) bass of B, respectively.

Notating root position and inversions

Figured bass

Figure 2: the common chord positions and their corresponding figured-bass notation in abbreviated form

Main article: Figured bass

In figured bass, Arabic numerals (figures) are written below each bass note. Figured bass, or thoroughbass, is a kind of integer Musical notation used to indicate intervals, chords and Nonchord tones in relation The arabic numerals (often capitalized are the ten Digits (0 1 2 3 4 5 6 7 8 9 which—along with the system These figures refer to intervals above the bass (usually assuming octave equivalence). In Music, an octave ( is the the use of which is "common in most musical systems In a root-position triad, the intervals above the root are a 5th and a 3rd, giving the figures . Normally, however, this is abbreviated by assuming that any bass note given without symbols indicates a chord by default. Similarly, the full figuring of the first inversion ( ) is abbreviated to just ; the full figuring of the second inversion ( ) has no abbreviation.

Figured bass is also applied to 7th chords, which have four tones. A seventh chord is a chord consisting of a triad plus a note forming an interval of a Seventh above the chord's root. A root-position dominant-7th chord contains a 7th, 5th, and 3rd. The full figuring of 7 5 3 is usually abbreviated to just ; the full figuring of the first inversion (6 5 3) is usually rendered as just , the second inversion (6 4 3) as , and the third inversion (6 4 2) as .

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 " chord"; similarly, in harmonic analysis the term refers to a tonic triad in first inversion. A diatonic function, in tonal Music theory, is the specific recognized Roles of Notes or chords in relation to the key.

Popular-music notation

Main article: Slash chord

A notation for chord inversion often used in popular music is to write the name of a chord followed by a forward slash and then the name of the bass note. In Popular music a slash chord or slashed chord is a chord whose Bass note or inversion is indicated by the addition of a slash Popular music is Music belonging to any of a number of musical styles that are accessible to the general public and are disseminated by one or more For example, the C chord above, in first inversion (i. e. , with E in the bass) may be notated as C/E. This notation works even when a note not present in a triad is the bass; for example, F/G is a way of notating a particular approach to voicing a G11th chord (G–F–A–C). (This is quite different from analytical notations of function of function; e. A diatonic function, in tonal Music theory, is the specific recognized Roles of Notes or chords in relation to the key. g. , the use of IV/V or S/D to represent the subdominant of the dominant).

Lower-case letters

Lower-case letters may be placed after a chord symbol to indicate root position or inversion. ^[3] Hence, in the key of C major, the C major chord below in first inversion may be notated as Ib, indicating chord I, first inversion. (Less commonly, the root of the chord is named, followed by a lower-case letter: Cb). If no letter is added, the chord is assumed to be in root inversion, as though a had been inserted.

Arabic numerals

A less common notation is to place the number 1, 2 or 3 etc. after a chord to indicate that it is in first, second, or third inversion respectively. The C chord above in root position is notated as C, and in first inversion as C1. (This notation is quite different from the Arabic numerals placed after note names to indicate the octave of a tone, typically used in acoustical contexts; for example, C4 is often used to mean the single tone middle C, and C3 the tone an octave below it. C or Do is the first Note of the fixed-Do Solfege. In Western Music, the expression " Middle C " refers to the note )

The cadential six-four chord

Figure 3: a cadential progression^[4]

The cadential (Figure 3) is a common harmonic phenomenon that is analyzed in two different ways: the first labels it as a second-inversion chord; the second treats it instead as part of a horizontal progression involving voice leading above a stationary bass. In Music, voice leading is the relationship between the successive pitches of simultaneous moving parts or voices.

1. In the first option, the cadential chord is considered a second inversion tonic triad because of the tones it contains. Under this designation, the progression is labeled: . Unlike the alternative analysis (see below), this label does not indicate any difference between a cadential and other uses of chords. Most older harmony textbooks use this label, and it can be traced back to the early 19th century. ^[5]
2. In the second option, this chord is not considered an inversion of a tonic triad^[6] but as a dissonance resolving to a consonant dominant harmony. ^[7] This is notated as , in which the is not the inversion of the chord, but a figured dissonance which resolves to . Figured bass, or thoroughbass, is a kind of integer Musical notation used to indicate intervals, chords and Nonchord tones in relation This function is very similar to the resolution of a 4–3 suspension. A nonchord tone, nonharmonic tone, or non-harmony note is a note in a piece of Music which is not a part of the chord that is formed Several modern textbooks prefer this conception of the cadential , which can also be traced back to the early 19th century. ^[8]

Inverted intervals

An interval is inverted by raising or lowering either of the notes the necessary number of octaves, so that both retain their names (pitch class) and the one which was higher is now lower and vice versa, changing the perspective or relation between the pitch classes. In Music theory, the term interval describes the relationship between the pitches of two Notes Intervals may be described as vertical In Music, an octave ( is the the use of which is "common in most musical systems In Music, a pitch class is a set of all pitches that are a whole number of Octaves apart e For example, the inversion of an interval consisting of a C with an E above it is an E with a C above it - to work this out, the C may be moved up, the E may be lowered, or both may be moved.

Under inversion, perfect intervals remain perfect, major intervals become minor and the reverse, augmented intervals become diminished and the reverse. (Double diminished intervals become double augmented intervals, and the reverse. ) Traditional interval names sum to nine: seconds become sevenths and the reverse, thirds become sixes and the reverse, and fourths become fifths and the reverse. Thus a perfect fourth becomes a perfect fifth, an augmented fourth becomes a diminished fifth, and a simple interval (that is, one that is narrower than an octave) and its inversion, when added together, will equal an octave. See also complement (music). In traditional Music theory a complement is the interval which when added to the original interval spans an Octave in total

Inversion in counterpoint

Contrapuntal inversion requires that two melodies, having accompanied each other once, do it again with the melody that had been in the high voice now in the low, and vice versa. In Music, a melody (from Greek μελῳδία - melōidía, "singing chanting" also tune, voice, or Also called "double counterpoint" (if two voices are involved) or "triple counterpoint" (if three), themes that can be developed in this way are said to involve themselves in "invertible counterpoint. " The action of changing the voices is called "textural inversion".

Invertible counterpoint can occur at various intervals, usually the octave (8va), less often at the 10th or 12th. To calculate the interval of inversion, add the intervals by which each voice has moved and subtract one. For example: If motive A in the high voice moves down a 6th, and motive B in the low voice moves up a 5th, in such a way as to result in A and B having exchanged registers, then the two are in double counterpoint at the 10th (6+5)-1 = 10.

Invertible counterpoint achieves its highest expression in the four canons of J. S. Bach's Art of Fugue, with the first canon at the 8va, the second canon at the 10th, the third canon at the 12th, and the fourth canon in augmentation and contrary motion. The Art of Fugue or The Art of the Fugue (original German Die Kunst der Fuge) BWV 1080 is an incomplete masterpiece Other exemplars can be found in the fugues in G minor and B-flat major [external Shockwave movies] from Book II of Bach's Well-Tempered Clavier, both of which contain invertible counterpoint at the 8va, 10th, and 12th. The Well-Tempered Clavier ( Das Wohltemperirte Clavier in the original old German spelling BWV 846–893 is a collection of solo keyboard music composed by

Inverted melodies

When applied to melodies, the inversion of a given melody is the melody turned upside-down. In Music, a melody (from Greek μελῳδία - melōidía, "singing chanting" also tune, voice, or The notes are reflected about the center line of the staff. For instance, if the original melody has a rising major third (see interval), the inverted melody has a falling major third (or perhaps more likely, in tonal music, a falling minor third, or even some other falling interval). In Music theory, the term interval describes the relationship between the pitches of two Notes Intervals may be described as vertical Similarly, in twelve-tone technique, the inversion of the tone row is the so-called prime series turned upside-down. Twelve-tone technique (also dodecaphony, especially in British usage twelve-note composition) is a method of musical composition devised by Arnold In Music, a tone row or note row ( German: Reihe or Tonreihe) also series and set, refers to a non-repetitive

Inversional equivalency

Inversional equivalency or inversional symmetry is the concept that intervals, chords, and other sets of pitches are the same when inverted. In Music theory, the term interval describes the relationship between the pitches of two Notes Intervals may be described as vertical This article describes musical chords in traditional Western styles It is similar to enharmonic equivalency and octave equivalency and even transpositional equivalency. In modern Music and notation, an enharmonic equivalent is a Note ( enharmonic tone) interval ( enharmonic interval) or In Music, an octave ( is the the use of which is "common in most musical systems In Music transposition refers to the process of moving a collection of notes ( pitches) up or down in pitch by a constant interval. Inversional equivalency is used little in tonal theory, though it is assumed a set which may be inverted onto another are remotely in common. However, taking them to be identical or near-identical is only assumed in musical set theory.

All sets of pitches with inversional symmetry have a center or axis of inversion. For example, the set C-E-F-F♯-G-B has one center at the dyad F and F♯ and another at the tritone, B/C, if listed F♯-G-B-C-E-F. For C-E♭-E-F♯-G-B♭ the center is F and B if listed F♯-G-B♭-C-E♭-E. (Wilson 1992, p. 10-11)

Inversion in musical set theory

Main article: set (music)

In musical set theory inversion may be usefully thought of as the compound operation transpositional inversion, which is the same sense of inversion as in the Inverted melodies section above, with transposition carried out after inversion. In musical set theory, a set is a collection of discrete entities for example pitch sets duration sets and Timbre sets (DeLone et al Musical set theory provides concepts for categorizing musical objects and describing their relationships In Music theory, the word inversion has several meanings There are inverted chords, inverted melodies, inverted intervals, and Pitch inversion by an ordered pitch interval may be defined as:

which equals

First invert the pitch or pitches, x=-x, then transpose, -x+n.

Pitch class inversion by a pitch class interval may be defined as:

History

In the theories of Rameau (1722), chords in different positions were considered functionally equivalent. In Music, a pitch class is a set of all pitches that are a whole number of Octaves apart e Jean-Philippe Rameau (ʒɑ̃filip ʀaˈmo in French (September 25 1683 – September 12 1764 was one of the most important French Composers and music theorists However, theories of counterpoint before Rameau spoke of different intervals in different ways, such as the regola delle terze e seste ("rule of sixths and thirds") which required the resolution of imperfect consonances to perfect ones, and would not propose a similarity between and sonorities, for instance.

See also

• Voicing (music)

References

• Wilson, Paul (1992). The Music of Béla Bartók. ISBN 0-300-05111-5.