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It only dawned on me recently, but it's quite a fundamental and important feature of scales that they are not mirrored under their own inversion.

The intervalic formula for major is:

W W H W W W H

If we invert this, we end up with phrygian:

H W W W H W W 

But this actually matters a lot, because if melodies in a given key are descending, they have different intervals than if they're ascending. I noticed this when writing a song in pentatonic major, where I kept using a descending minor sixth (my favourite interval). Any time the melody was descending, I was using really fun, spicy intervals that were distinctly non-major. So for example, consider the ascending melody in C major pentatonic: C E A C. This has intervals above the tonic of major third, major sixth, octave.

But if we had some descending melody like: C E D C , the intervals below the tonic are: minor sixth, minor seventh, octave.

But it's not just in melody. Whenever chords are voiced like: 1 3 8, you can hear the very distinct minor sixth interval. In other words, when a chord is voiced in such a way that the root is played in a higher octave, the other notes in the chord have these intriguing "descending interval" relationships with it.

With all that said, I never read about this when reading up on music theory. I Googled it and the term "scale inversion" just takes you to investment websites. The closest concept in music theory I could find is "melodic inversion", but that discusses the opposite case - keeping intervals unchanged under inversion.

Is there a reason why this seemingly important property of scales is not considered in music theory?

My guess at an answer, after discussing with someone: the psychoacoustics of harmony is that we consider the root of the chord to, well, root the chord. The chord's character is largely determined by the root. And in the cases when we do things like invert the chord in funny ways, we create ambiguity about what the chord is. For example, if we take the second inversion of C major: G E C, we might hear it as C major, but we'll also kind of hear it like a Gsus4.

So to summarise my question: wow, scale inversions are really important, why is nobody talking about them?

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    In the major chord with the root at the top there is neither a descending minor sixth nor an ascending minor sixth, because the notes sound simultaneously. There is simply a minor sixth. Furthermore, the phenomenon you're discussing seems to be effectively covered by the concept of interval inversion, not scale inversion. What do scales have to do with it? To put it another way, the discussion in this question does not leave me with the impression "wow, scale inversions are really important." – phoog Nov 12 '20 at 12:48
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    I've seen stuff about what you're asking about, but not with the phrase "scale inversion" used, which might be why you haven't found it. Seems like scale symmetry is a related concept - in the sense that you are talking about scales that aren't symmetrical. (see en.wikipedia.org/wiki/Symmetric_scale) Also, as you noted, "inverting" a scale gives you a different mode, and of course modes are discussed. So I suggest that for asymmetric scales, we talk about the different "modes" which are essentially the "inversions" you are asking about under a different name. – Todd Wilcox Nov 12 '20 at 13:02
  • Is the notion of an "axis of symmetry" more in line with what you're looking for? – Richard Nov 12 '20 at 15:52
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    Disagree with your penultimate paragraph. CEG, in any order, is C major. Gsus4 contains GCD.Chord character is more determined by 3rd and 7th than by root, which only gives it an initial name. – Tim Nov 12 '20 at 16:35
  • It could be just me, but I'm a bit confused by this question... I would say that "mirroring" in the sense that Phrygian is the "mirror" of major is one concept; inversion in the sense of chord inversion, which is essentially "same root but different starting position", is another concept, and playing a given set of intervals descending rather than ascending is a different idea again. As far as I can see, you've mentioned all three of those, and if you're saying that one or more of those ideas are related in an important way, I'm not quite seeing why... – topo Reinstate Monica Nov 14 '20 at 7:36
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In the common practice period, at least, melodic inversion was a common concept. But note that this was often done with adjustments to the inverted melody to make it fit a recognized scale and/or a predefined harmonic scheme. So the prevalence of melodic inversion needn't necessarily lead to the discovery of scale inversion.

And indeed, as far as I know your observation about scale inversion is correct -- hardly anybody seems to talk about it. I don't know for sure, but the reason might be that the scales that were (and remain) actually used in tonal music invert to modes of themselves, so you don't get anything new from the process. Well, almost:

  • Major scale inverts to a mode of itself (TTSTTTS -> STTTSTT = Phrygian)
  • Melodic minor inverts to a mode of itself
  • Octatonic and whole-tone scales invert to themselves / modes thereof
  • The usual pentatonic scale inverts to a mode of itself
  • Harmonic minor inverts to... Harmonic major

So we do get something out of the ordinary -- Harmonic Major appears to be a rarely-used scale.

Personally I think studying inversions of scales as a way to find new material for music-making is extremely interesting, but only if you're working with more exotic scales than the ones mentioned above and looking for relationships between them. It's prima facie likely that scales that are inversions of each other will sound similar because of having the same interval content in reverse order.

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Scale inversions as you describe them are often discussed re the music of Debussy, Ravel and (especially) Stravinsky. Particularly in regards to how modes such as the dorian and phrygian interact with symmetrical constructs such as the whole-tone and octatonic collections. Van den Toorn’s 1983 Stravinsky book is exhaustive but exhausting, if you have a spare week or two.

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  • Olivier Messiaen was also interested in symmetrical scale; he referred to the whole-tone, octatonic, and similar scales as "modes of limited transposition," and his theories are somewhat related to this discussion. – Peter Jan 4 at 16:39
  • indeed, although Messiaen’s modes don’t interact with conventional diatonic modes in the same way. – user71850 Jan 4 at 19:02
  • +1 It drives me crazy when better answer like this one have few upvotes and the selected answer is inferior! – Michael Curtis Jan 5 at 18:06

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