Formal definition of an inverted chord

Question

Question:

Is there a formal definition that would apply to all chords / genres of what an inverted chord is? Given a chord, I want to formally prove on paper (so not using my ears) that the chord is inverted or in root position.

What I tried:

I tried finding a proof using intervals.

We know that the lowest note played in a chord denotes its inversion, regardless of the order of the higher notes

We also know that all triades and 7th chords are built on thirds.

In this context, a chord is inverted if the interval between two consecutive notes is less than 3 or more than 4 semi-tones. Using this approach we can prove E G B C is inverted (it is actually the first inversion of Cmaj7). This approach works with a lot of common chords such as C, Cmaj7, F7...

Problem:

This rule does not apply to other chords like sixth chords (e.g. Cm6), ninth chords (e.g. C9b5), suspended chords (e.g. Csus2, Csus4) and other chords like C13, Cmaj13, etc...

So is there a formal definition which would apply to all the aforementioned chords to prove if they are inverted or in root position?

Related question:

What implications do the formal rules of inversions have for suspended chords?

Some answers to this question seem to suggest that there is no universal definition which would apply to all types of genres / musical context / types of chords.

But surely we are able to tell if a chord is inverted or not, so there must be some kind of formal proof/definition (or a set of definitions) somewhere.

Answer 1

This is not possible.

PROOF: Given the notes A-C-E-G and no other information, it's impossible to know whether this is a C6 or A-7 chord. Therefore, it cannot be determined whether the chord is in root position or not.

See also How would you identify the root of a non-standard chord / cluster?, which is fundamentally the same question. Since one cannot clearly identify the root of an arbitrary chord, it follows that one cannot determine whether or not a chord is inverted.

Answer 2

The concept of "inversion" is usable in some harmonic styles and situations, and sometimes it is even useful. In some conditions or styles, it is possible to unambiguously answer the inversion or not question. Outside those conditions the question can be ambiguous. In the following, I formulate a style or discipline of using chords, where it IS possible to say if a chord is an inversion, and where it's also possible to say if a chord falls outside this area of applicability. In other styles the concept may still be useful, for example for finding a roughly equivalently behaving chord, even if a canonical non-inverted chord cannot be found.

I'm not saying that this is the ONLY possible harmonic system where the inversion question is unambiguous, but it is ONE such system.

Definition

• By this Definition, the "inversion or not" question only applies to chords that are Pure Stacks of Thirds.
• A chord is (within this definition) a Pure Stack of Thirds, if and only if its note names (ignoring octaves) can be ordered so that the notes form a continuous chain of thirds, and thirds only. If you want maths and more formal-looking things, you can imagine something about modulo 7 arithmetic, where accidentals are thrown away from note names, and the remaining letters are assigned numbers C=0, D=1, E=2, F=3, G=4, A=5 and B=6, and you have to find a permutation for the notes where the sequence goes like note (x) = note (x-1) + 2 (modulo 7), for every 1 < x <= N, where N is the number of notes in the chord. If you can find no such permutation, it is not a stack of thirds. If you can, it is.
• By this Definition, note combinations of less than 3 or more than 6 notes are not considered.
• By this Definition, if a note combination CAN be expressed as a stack of thirds, then the stack-of-thirds permutation is its canonical form, and other names are non-canonical aliases.
• By this Definition, if a note combination CANNOT be expressed as a stack of thirds, then it is outside the scope of this "inversion or root position" logic. For example C-F-G.

So, our isInversion(noteArray) algorithm gives the following possible outputs:

• 0 : noteArray can be expressed as a stack of thirds, and the root note is the lowest note: it is in root position
• 1 : noteArray can be expressed as a stack of thirds, and the root note is not the lowest note: it is an inversion
• 2 : noteArray cannot be expressed as a stack of thirds.

With these requirements and logic, we can say that "C6", C-E-G-A, is a non-canonical alias name for an inversion of Am7: A-C-E-G. Its canonical systematic tertian-harmony name would be Am7/C.

I set an extra limitation to exclude 7-note chords, which allow a full circle around the 7-note modulo arithmetic. If all seven note names are included, then the stack of thirds is ambiguous. For example, C-D-E-F-G-A-B could be said to be Cmaj13 in root position, or Fmaj13(b5) with C in the bass.

If we compare the theoretical model defined above with example chords you can find in the Real World (tm), we'll find lots of chords that either aren't stacks of thirds according to the Definition (and thus it's not possible to give an unambiguous answer to the inversion problem), or which are "incorrectly" named.

• sus chords (no stack of thirds permutation)
• 6 chords (misnomer of m7 inversion)
• "2" chords (no stack of thirds permutation)
• "add" chords (no stack of thirds permutation)
• "5" chords (no stack of thirds permutation)
• anything with the a number "8" or "10" etc.

You'll find lots of that stuff in practice. So the next time you see someone claiming that a chord "actually is" something, for example that C6 is "really" or "actually" an inversion Am7/C, then maybe that person is coming from a theory fantasy land. Take it with a grain of salt. Use theory if it helps you make music.

Actual Implementation

A computer program is a formal definition. Here is a Python program which answers the "inversion or not" question.

import music21
import itertools

def pitchNumber(pitch):
    return (7 + ord(pitch.name[0]) - ord('C')) % 7

def isCanonicalStackOfThirds(pitches):
    if (2 < len(pitches) < 7):
        for i in range(1,len(pitches)):
            n_i = pitchNumber(pitches[i])
            n_p = pitchNumber(pitches[i-1])
            if ((n_i + 5) % 7 != n_p):
                return False
    else:
        return False
    return True

def findCanonicalStackOfThirds(chord):
    for p in itertools.permutations(chord.pitches, len(chord.pitches)):
        if isCanonicalStackOfThirds(p):
            return p
    return None

def inversionStatus(chord):
    canonical = findCanonicalStackOfThirds(chord)
    if canonical is None:
        return False, canonical
    if chord.pitches[0] != canonical[0]:
        return True, [pitch.unicodeName for pitch in canonical]
    else:
        return False, [pitch.unicodeName for pitch in canonical]

for noteSet in ([ \
        ['F#', 'D', 'A'],
        ['D', 'A', 'F#'],
        ['C', 'F', 'G'],
        ['C', 'E', 'G', 'A'],
        ['Db', 'Ab', 'F', 'C'],
        ['C', 'Ab', 'F', 'Db'],
        ['G', 'F', 'B', 'Eb'],
        ['G', 'F', 'B', 'D#'],
        ['C', 'D', 'G'],
        ['F', 'C', 'E', 'G', 'Bb', 'D'],
    ]):
    chord = music21.chord.Chord(noteSet)
    isInversion, canonicalChord = inversionStatus(chord)
    if canonicalChord is None:
        print (str(noteSet), ' cannot be said to be an inversion of any canonical stack of thirds.')
    elif isInversion:
        print (str(noteSet), ' is an inversion of ', str(canonicalChord))
    else:
        print (str(noteSet), ' is in root position. Canonical form: ', str(canonicalChord))

Here is the printout from a run:

['F#', 'D', 'A']  is an inversion of  ['D', 'F♯', 'A']
['D', 'A', 'F#']  is in root position. Canonical form:  ['D', 'F♯', 'A']
['C', 'F', 'G']  cannot be said to be an inversion of any canonical stack of thirds.
['C', 'E', 'G', 'A']  is an inversion of  ['A', 'C', 'E', 'G']
['Db', 'Ab', 'F', 'C']  is in root position. Canonical form:  ['D♭', 'F', 'A♭', 'C']
['C', 'Ab', 'F', 'Db']  is an inversion of  ['D♭', 'F', 'A♭', 'C']
['G', 'F', 'B', 'Eb']  cannot be said to be an inversion of any canonical stack of thirds.
['G', 'F', 'B', 'D#']  is in root position. Canonical form:  ['G', 'B', 'D♯', 'F']
['C', 'D', 'G']  cannot be said to be an inversion of any canonical stack of thirds.
['F', 'C', 'E', 'G', 'Bb', 'D']  is an inversion of  ['C', 'E', 'G', 'B♭', 'D', 'F']

Answer 3

In addition, No, the interval analysis approach will not be accurate for another reason: chord voicings may not come in the tightest possible form (close position), so there are issues with even the simplest chord qualities.

For example, [Db Ab F] is a root position Db major triad:

a chord is inverted if the interval between two consecutive notes is > less than 3 or more than 4 semi-tones

Applying the logic from the question itself, the interval between Db and Ab is wider than 4 semitones, and thus the algorithm incorrectly detects an inversion.

This could be remedied by checking each note against the entire set of input notes for the pitch class 1 semitone higher (or any enharmonics, depending on desired behaviour), then 2 semitones, 3, and finally 4.

But for a true useful algorithm, you'd probably have to compare the notes in the set against every possible chord root and then check for every single valid chord type to really search for inversions, and then your output is more like a collection of possibilities rather than a binary "inversion/root position" identifier.

Answer 4

The inversion concerns itself with what note is at the bottom or in the bass. Which note of the chord is the lowest.

If you have a C Major triad, if the C note is the lowest then you would say the triad or chord is in root position.

If it is the E note, then the chord is in first inversion, and if it is the G note is in the bass, then the chord is in second inversion.

There is also a third inversion for four note chords. If you, for example have the Dominant seventh chord of C major (GBDF) and you have F note in the bass then your chord is in third inversion.

There may also be a plethora of ways the chord is harmonized in the other voices, but this is not relevant to inversion as that only concerns itself with what is happening in the bass.