Inversion

Musical chords, multiple melodic lines, as well as the intervals between tones -- all may all be "inverted."
Thus the terms "inversion" or "invertible" in music might refer to three (3) different things: 1) the way the notes of a chord are arranged in vertical sequence (i.e. which note is the lowest, or the bass note), 2) the rearrangement of 2 or more moving melodic lines above or below one another, or 3) how the consecutive tones in a melodic line may be reversed and turned upside down by making upward intervals move downward and downward intervals move upward, which creates a different but related melody.
In the first case, called "harmonic inversion," when the root of a triad (3-note chord) is in the bass (the lowest note), the chord is said to be in root position, for example. a C Major chord built upon the notes C-E-G.
When the root of a triad is NOT in the bass, it is said to be inverted.
When the 3rd of the chord is in the bass (or is the lowest note), we have a 1st inversion of a triad (E-C-G).
When the 5th of the chord is in the bass (or is the lowest note), we have a 2nd inversion of a triad (G-C-E).
In the case of 4 note chords, when the 7th of the chord is in the bass (or is the lowest note), we have a 3rd inversion (in the case of a G7 chord, on the notes F-G-B-D).
A so-called "pedal" or "organ point" is a sustained note entering into the construction of harmony of which it is not always properly a part; it's most frequently found in the bass, although it cannot be considered as the bass and must not be figured [See blog, Figured Bass].
When the sustained tone is in an upper voice it's called an "inverted pedal point."
When the pedal point is in the bass the part next above it is to be considered the real bass temporarily and is written in accordance with the principles of chord progression independently of the sustained tone.
The part immediately above a pedal point in the bass serves as the bass of the harmony and is figured to indicate the chord.
When the pedal point is in the bass, it's most commonly placed on the 1st (tonic) or 5th (dominant) scale degrees -- less commonly, although sometimes, on the 3rd (mediant) degree; a pedal point could also be placed in the tenor, alto, or soprano voices.
The Op. 11 Prelude is an example of a composition where a dominant pedal point appears several times in the bass and as an inverted pedal point in the tenor [See blog, Free Stuff].
In the second case, we also find that 2 separate lines (parts, or voices) of music can be "inverted" or switched so that the original lower one is made to sound above the original high one.
This is called "inversion of counterpoint, double counterpoint, or invertible counterpoint" and can be applied to any number of parts.
When developing a melody in the third case however, the term "inversion" takes on an entirely different meaning.
In this third meaning we can have the original interval and its inversion of equal size but extending in different directions from a reference point, usually the starting note; in a fugue it would mean the vertical reflection of the subject.
The term "mirror writing" (using inverse movement) is the systematic application of this; some authors refer to this as "strict" or "real" inversion because the reversal of half tones is exactly the same in terms of number of half tones.
NOTE: Inverse movement in, let's say, a fugue represents statement of the subject or countersubjects exactly upside down, upward intervals being downward ones, and vice versa; this is sometimes called "inversion," but "inverse movement" is a preferable term since it avoids any possible confusion with the true melodic inversion of a subject or countersubject.
With inverse movement all of the intervals are exactly the same in terms of half tones, only in reverse (upside down); with true melodic inversion the intervals change; 4ths become 5ths when inverted, 5ths become 4ths, unisons become octaves, and octaves become unisons, etc., according to mathematical formula where 9 minus the interval = the inverted interval.
Thus 2nds become 7ths when so inverted, and 7ths become 2nds; 3rds become 6ths, and 6ths become 3rds.
When so inverted, perfect intervals remain perfect; major intervals become minor, minor intervals become major, diminished intervals become augmented, and augmented intervals become diminished.
A moment's thought will make it clear that slightly different results will be obtained depending upon whether we're making a true inversion of a melodic line or we're working it in inverse movement; while the term "inversion" is often used to apply to both, technically there's a difference (and in any given situation one may manage to work better harmonically than the other).
There are also situations where, in turning a melody upside-down by either inversion or inverse movement, a tonal statement may be made in that the intervals are adjusted to accommodate the harmonic structure of the passage.
One of the most challenging aspects of composition is thematic development; it's usually not too difficult to come up with some kind of convincing theme, melodic, or textual ideas, but we can struggle with writing an entire piece.
When we don't know how to develop our musical ideas our progress as a composer or improvisor can come to an abrupt halt.
How to develop their piece can leave students in composition classes wondering what to do, they can "get stuck," and eventually some of them get so frustrated that they decide to give up.
This is sad because this doesn't have to be.
To be able to develop a musical theme requires certain tools, one of which is inversion [See blog, Retrograde, Augmentation / Diminution, Sequencing, Segmentation] and is arguably the mostly widely known and beloved compositional tool.
The 2 men in the above sketch (photo) are, line for line and shadow for shadow, identical and one and the same individual; in one case, he's right side up; in the other, he's upside down.
When he's upside down he takes on a different character, a lingering sense of familiar unfamiliarity; this comes about via the illusion of inversion.
Inverse movement of a musical line, in the same way, simply means that all of the notes in a particular melody are the same distance apart but upside down, which spawns a brand new, but related, melody that bears a distant family resemblance to the original, as in the photo.
Inverting the intervals changes them according to formula, which creates a little different line than using inverse movement but has essentially the same effect.
Many composers from history, particularly fugue writers like J.S. Bach and his school, made frequent use of inversion and inverse movement to help draw from a subject or its countersubject(s) all of their potential.
The next time you feel yourself "stuck" about how to work a free theme or develop an episode for a fugue, why not apply this compositional tool to your melody and see if it generates new, usable material.
If you like your original theme or subject, you'll probably like its inversion or hearing it in inverse movement because the intervallic and rhythmic content in each case is the same.
It generally follows that an interesting theme or subject will make an interesting inversion.
The score for the A Major "jig" Fugue Op. 13 [See menu bar, Free Stuff] is an example of a fugue in which there is an exposition immediately following the opening exposition where the subject and both countersubjects are treated with inverse movement; this adds color and interest to the harmony.
As we work with true inversion and inverse movement (such passages in a fugue usually involve a blending of both) it is not uncommon to discover that at times the resulting inverted lines may cross, move outside normal voice ranges, or make hidden or parallel fifths or octaves (which are mistakes in composition) when they're inverted combine [See blog, Getting Started With Writing, Part XX].
These are red flags that beg to be tweaked -- and we should expect that not every moving line can be inverted like this without doing some tweaking to the outline of the melodic line, such as by moving a note or two to a different octave or making a tonal adjustment in a note or two, to get it to sound in consonant harmony with what's going on around it.
This does not make that line any less of a line than one which can be worked without tweaking, it's just an indicator, a sign post telling us how we need to handle it.
In this situation, no harm, no foul; we either proceed by making the inverted line fit with the prevailing harmonic conditions or simply keep inversions out of the music altogether.
This is NOT a dead end if and when we notice this, and it should never dismay or disappoint any new fugue writer; on the contrary, it should be counted all blessing because it frees the inverted forms for use as connecting material in the 2 voice modulating episodes which join the entries to each other.
The inverted forms in fact are perfect for this purpose [See blog, Getting Started With Writing, Part XVIII] and sometimes save the day when we're stuck on what to write for an episode.
If we construct an episode or two this way, it's surprising how well they blend with the entries; the music seems to move naturally and gradually from one entry to the other, like the changing colors in a kaleidoscope.
This is well illustrated in the score of the F Major Fugue from Op. 19 [See menu bar, Free Stuff).
Melodic inversions are often the musical antidote for those pesky little blank episodes in the score which seem at times to "gang up" on the fugue writer, devour much time, and slow down progress as (s)he tries to figure out where to go next for the right material.
An entire Prelude can be developed from nothing more than a single line which outlines, in inverse movement, one of the free voices in the fugue which follows it; this is illustrated in the score of the a minor Prelude from Op. 25 [See menu bar, Free Stuff].
Inverting the intervals in a melodic line like this is just one of the time-honored compositional tools that can be applied to virtually any style of music.