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When figures are used to describe chords, they represent the intervals above the lowest pitch, and these seem to be treated as unique. For example, basic triads come in 5-3, 6-3, and 6-4 forms; and seventh chords come in 7-5-3, 6-5-3, 6-4-3, and 6-4-2 forms.

But could a "misfit" chord be constructed so that two of its inversions share exactly the same figures (without resorting to enharmonically equivalent spellings)?

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    Is it a question about Am7 and C6 being inversions of each other? Aug 24, 2022 at 4:47
  • @user1079505 No.
    – Aaron
    Aug 24, 2022 at 5:03
  • Then is it a question about C7b5 with the second inversion being identical to Gb7b5? Aug 24, 2022 at 5:28
  • @user1079505 They aren't identical. The figure for the root position would be 7-5-3, and the figure for the second inversion would be 6-4-3 (leaving aside accidentals). I think you should look at the answer (and, more specifically, who left it).
    – Aaron
    Aug 24, 2022 at 5:32
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    The boundaries are now clear, but I don't understand the motivation. Is it a maths/combinatorics question, or request to solve a specific issue that may occur when performing or composing some type of music? Aug 24, 2022 at 8:16

3 Answers 3

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  1. One can represent a chord by writing a sequence of all 7 consecutive diatonic notes, marking with 1 the ones that are played, and 0 the ones not played. E.g. taking a major scale, a major triad can be represented as 1010100. We obtain an inversion by rotating (see) the whole sequence, e.g. the first inversion would be 1010010, and the second inversion 1001010. One can then easily see that no inversions yield the same pattern, except for a single note 1000000 (which becomes the same when rotated by 7 steps), and a chord including all 7 notes 1111111 (which is the same rotated by any number of steps). The reason is that 7 is a prime number.

  2. I think there is a general agreement that a single note, or even an octave is not really a chord. A chord including all 7 notes of a diatonic (let's pick major for example) scale is a bit problematic as well. The identity of such chord is a bit vague, but if one considers a particular voicing and harmonic context it may work.

  3. I'm quite sure that this was not a chord typically used in the baroque and classical period, when the figured bass notation was widely used, which makes the discussion a bit academic.

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  • I think you're hedging your answer more than needed. Nothing in the question refers to musical period or the definition of a chord, and it specifically mentions "constructing" a "misfit" — so, a cluster that includes all seven diatonic pitches is a perfect answer. Pointing out in chat the figure 7-6-5-4-3-2 to me was perfect, with the binary representation being nice, but secondary.
    – Aaron
    Aug 24, 2022 at 23:44
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Your question gives figured bass figures, so we are working with that system, not jazz chord symbols or some other system.

The figures in figured bass are just interval numbers. When a chord is inverted the intervals of those figures just get inverted.

Although in a confusing way the inversion of the intervals results in a new bass and the figured bass intervals relative to that new bass are not the same as the inverted intervals. Ex. C E G would be 3/5 above a C, a third and fifth invert to a sixth and fourth and the pitches would be E G C, so while the inverted intervals are a sixth and fourth, the figured bass intervals are not 4/6, because E G C in figured bass would be 6/3 above an E. Sorry for the digression. Combining "inversion" as an interval process with "inversion" as a chord voicing is a muddle.

Anyway, I think the question is just a matter of "is there any interval whose inversion results in the same interval?" Because when a chord is "inverted" - as in changing the voicing - you are also "inverting" intervals.

Inverting an interval as a process means one of the intervals is moved by one octave. If one of the voices moves, there will be a change in distance, a change of interval size, so any interval inversion results in a different interval. Those changed intervals would require different figured bass intervals. So, no, you can't have two chord inversions that would have the same figured bass intervals.

But, figured bass does reduce compound intervals to simple, which is a transposition by octave. So, if your "chord" were an incomplete root position chord, like C3 C4 C5, and you "inverted" it to C4 C4 C5, or something like that, figure bass would just reduce them all to a C with either no figures or I suppose an 8, and if the key were C major, and you did it as Roman numeral analysis, all the "inversions" would simply get labelled I, so that "chord" would be an example of all inversions having the same figure. But this example is really silly and contrived.

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A silly little point: one of the reasons "tritone substitution" (at least in jazz and pop) works, is that an inversion of a tritone interval is again a tritone interval. So, for example, the 7th+3rd F-B of a G7 chord, inverted once, give B-F, which is the 7th+3rd of a Dflat7.

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  • Right, but this includes enharmonic change from B to Cb, which Aaron wants to avoid. Aug 25, 2022 at 13:42
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    B-F tritone is a diminished fifth, and its inverse, F-B is an augmented fourth, so a different "figure". Aug 25, 2022 at 13:47

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