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I'm looking for clarity over how to best describe two different meanings of the word "inversion" I have come across. The two different aspects I see are:

  1. I have three notes C-E-G. The intervals between them are a major third followed by a minor third. By using octave equivalence this can be inverted to E-G-C (minor third followed by perfect fourth) and G-C-E (perfect fourth followed by major third). Other voicings of these inversions C-G-E, E-C-G and G-E-C also follow on from octave equivalence.

  2. I have three notes C-E-G. The intervals between them are +4 semitones C-E and +3 semitones E-G. I can "invert" this by starting at C, going down 4 semitones to Ab, then going down 3 semitones to F. This gives me C-A♭-F.

The first meaning appears to be common usage. The second definition of inversion is the one used in pitch class set theory. They appear distinctly different to me and I think its confusing to use the same word to describe them.

I would like to know if there is another word or term I can use to differentiate between these two aspects?

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Yeah, this can be very confusing. First of all, I would clarify that the standard understanding of the first type of inversion is too restricted. I can invert a root-position C-major chord by either putting the third in the bass, or the 5th in the bass. It doesn't have to be literally a m3-P4 to be a 1st inversion CM chord, it just has to be the notes of a CM chord, with E as the lowest pitch. Type 1 inversion is purely concerned with which member of the triad is lowest regardless of voicing, doubling, etc; the voicing and number of notes could be identical except for the bass note, and that would still be an inversion. The only intervals necessarily inverted, are the possible intervals above the bass.

I think the primary semantic confusion is removed as long as you refer to "1st inversion", "second inversion", "inversions of a CM chord", etc. That terminology doesn't exist in set theory, where we would call your example a T0I inversion of [0,4,7] (apologies if you learned a different notation for this, I'm following Straus' use in his Post-Tonal textbook). Since set class terminology is entirely octave- and inversionally-equivalent, there is no distinctions about what note is on bottom.

I've heard people refer to your type 1 as "bass inversion" and your type 2 as "pitch inversion", "mirror inversion" or "axis inversion," but I'm not sure how widespread that terminology is. Straus' solution is to refer to type 1 without the word inversion at all, he just calls the three positions of a triad "bass positions".

But remember, your type 2 is the most obvious example of pitch inversion—around a literal PC0 axis—but the mapping shows that it is less dissimilar from chord inversion than it might seem. If I move a CM chord from root position to second inversion, one way to visualize it is that the G moves from being above the C and E to being below (that's just one way to do it, but bear with me...). It was a P5 (pi7) above C and becomes a P4 (pi5) below C. It was a m3 (pi3) above E and becomes a M6 (pi9) below E. If you do T0I on [0,4,7], than any Gs that are pi7 above C become pi5 above C (F) and any Gs that are pi3 above E become pi9 above C (Ab). In both cases, you can see pi7s becoming pi5s and pi3s becoming pi9s, it's just a slightly more abstract relationship in the case of (037).

Type 1 inversion must always have the same PC content, Type 2 inversion almost always (except for a handful of inversionally-symmetric sets at just the right TnI values) have a least some different PCs, and often are an entirely new collection.

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It does make sense. I was aiming to be concise so I avoided using the phrase "1st inversion", "2nd inversion" etc.I like the term "mirror inversion"; it seems to capture the notion that a pi7 above becomes a pi7 below etc. In some sense type 1 is more like 'rotation' or 'permutation' ... Both are of course equivalent when you only have two pitches. –  Richard Padley Feb 8 at 18:19
    
"Rotation" and "permutation" are great terms for it, I'm going to start using that in post-tonal classes! But I don't think they're equivalent even with two pitches (except for the tritone). Rotating changes E–C to C–E while inverting changes E–C to C–Ab. –  Pat Muchmore Feb 8 at 18:34
    
Good point. I was adding in a transposition on top. I was getting at the fact that E-C and C-A♭ are still both a pi8. –  Richard Padley Feb 8 at 18:39
    
BTW I added extra cases to type 1, so I think the your comment on my definition being restricted probably does not hold any more? Maybe needs a light correction. –  Richard Padley Feb 8 at 18:42
    
Ah, I hadn't seen that. But still, those other cases aren't additional inversions, just different voicings of the same inversions already mentioned. It's only a semantic distinction, but I think it actually gets right to the heart of the difference between the two "inversions". Type 1 inversion is purely concerned with which member of the triad is lowest regardless of voicing, doubling, etc. –  Pat Muchmore Feb 8 at 19:36

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