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Inventionstrompete
(Ger.).
Generic term used from the mid-17th to the mid-19th centuries to refer to a trumpet provided with some kind of mechanism designed to increase its performing compass (see Trumpet (ii)). Similarly, the Inventionshorn was a horn provided with a system of sliding crooks to alter the pitch of the instrument; it was conceived by A.J. Hempel and first built in about 1753 (see Horn, §2(iii)).
Invernici [Invernizzi], Ottavio.
See Vernizzi, Ottavio.
Inversion
(Ger. Umkehrung).
(1) The rearrangement of the notes of a chord built in 3rds so that the lowest note is not the root of the chord. Ex.1 shows the three positions of the C major triad, each in both close and open spacings. If the lowest note of the chord is the 3rd of the triad it is said to be ‘in first inversion’ (see Sixth chord). If its lowest note is the 5th, then it is ‘in second inversion’ (see Six-four chord). The inversion of triads can be extended to 7th chords, where an extra position of the chord – ‘third inversion’, with the 7th as lowest note – is possible (ex.2). Much the same can be applied to 9th chords. In Roman numeral chord notation (see Harmony, §2(ii)), inversions are indicated by arabic numerals or letters: in a C major context, for example, the three chords in ex.1 are either I, I6 and I6-4 or I, Ib and Ic, while ex.2 shows I7, I7b, I7c and I7d.
(2) The complement of an interval with respect to some fixed interval often assumed to be an octave. Within an octave, a 2nd inverts to a 7th, a 3rd to a 6th, a 4th to a 5th and vice versa; within a 10th, a 2nd inverts to a 9th, a 3rd to a an octave, etc. This type of inversion is the basis of Invertible counterpoint, where the functioning of different polyphonic parts as the bass part is dependent on the consonant intervals between them inverting to other consonant intervals.
(3) The mirroring of a succession of notes about a fixed note, usually the first note or interval in the succession of the most easily identified form of inversion because of its association with melodic contour. It is a common feature of contrapuntal music, since it offers new thematic combinations, for whose workability a structurally sound original is usually a necessary and sufficient condition. For some writers the discipline of invertible counterpoint also includes counterpoint based on thematic inversion. This type of inversion has been extended by the theory of Twelve-note composition to include the inversion of the 12 pitch classes making up a set about a fixed pitch class. Thus the set used at the beginning of each movement of Schoenberg’s Suite op.25, shown in ex.4a, yields the set given in ex.4b when inverted about its first note.
WILLIAM DRABKIN
Inverted cadence.
Medial cadence.
Invertible counterpoint.
The contrapuntal design of two or more parts in a polyphonic texture so that any of these parts may serve as the highest part or as the bass. Invertible counterpoint involving two (three, four) parts is called double (triple, quadruple) counterpoint.
The underlying principle of invertible counterpoint is the Inversion of intervals with respect to some fixed interval. For instance, ex.1a shows a cantus firmus (given in semibreves) with a note-against-note solution beneath, and in ex.1b the same two parts are inverted ‘at the octave’. The lower part is moved up an octave, and the intervals between the two parts become the complements of those in the original setting: unisons become octaves, 3rds become 6ths and 6ths become 3rds.
The inversion of a consonant interval is not always consonant. For instance the 5th, inverted at the octave, becomes a 4th (which is treated as a dissonance in note-against-note counterpoint); the 6th, inverted at the 12th, becomes a 7th. This means that the choice of intervals between the parts of a passage in invertible counterpoint is restricted, dependent on the interval of inversion.
Moreover, not only must the concept of consonance and dissonance be respected, but the rules of Part-writing must also be observed. For example: at the interval of a 10th, consecutive 3rds and 6ths invert to consecutive octaves and fifths, and must therefore be avoided. Ex.1c illustrates the unsuitability of the ‘solution’ in ex.1a for inversion at the 10th; an alternative solution to the same cantus firmus – one which can be inverted at this interval – is shown in Ex.2.
An exhaustive discussion of the applications of invertible counterpoint is found in S.I. Taneyev’s Podvizhnoy kontrapunkt strogogo pis'ma (Leipzig and Moscow, 1909; Eng. trans., 1962, as Convertible Counterpoint in the Strict Style). The standard English textbook has been Ebenezer Prout’s Double Counterpoint and Canon (London, 1891).
As a musical discipline, invertible counterpoint has remained within the domain of strict counterpoint since it was introduced by Vicentino in L’antica musica ridotta alla moderna prattica (1555). As a compositional technique, though, it was used infrequently until the Baroque era, when the bass was established as a fully developed line with a recognizable contour and a melodic identity of its own. In fugue the relationship of counter-subject to subject makes their invertibility almost an a priori condition; invertible counterpoint flourished in fugues and other compositions in a strict polyphonic style. In such works as Bach’s Three-Part Invention in F minor, invertibility may be said to be used as a structural principle; and in works like Die Kunst der Fuge and the Canonic Variations on Vom Himmel hoch he lifted the technique to a level far above that of an ‘academic’ exercise.
For bibliography see Counterpoint.
WILLIAM DRABKIN
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