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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?

Alan
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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 https://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... – Нет войне Nov 14 '20 at 07:36
  • @phoog, I agree with there "simply being a minor sixth" when the chord is played simultaneously. I wanted to highlight that when the chord is voiced with the root on the top, the minor sixth is far more obvious than when voiced with the root at the bottom. In the Cmajor voice C4 E4 G4, the minor sixth appears between the E4 and the first harmonic of C4, which is C5. So it's way less evident. I'm interested in the increase in intensity of the minor sixth by changing the voicing so that the non-root notes are a lower pitch than the root. Does that clarify what I meant? – Alan Nov 14 '20 at 12:49
  • Oh and @phoog, to your second point, yes interval inversions are exactly what I am talking about. But isn't there value to consider "scale inversions" in parallel to how it's useful to discuss both intervals, and scales? Just from my experience writing music, I certainly see a lot of value in it. – Alan Nov 14 '20 at 12:50
  • @ToddWilcox, thanks for that source. I checked it out and it's certainly a related, but still different topic. Symmetry in scales seems to be about creating subunits of scales that can be repeated. In the example given, the octave can be divided into two groups of 1->1->4. i.e C -> Db -> D -> F#; F# -> G -> Ab -> C. Interesting stuff though. I followed the reference listed, and it took to a page in a book which brought up the interesting point that scales in all cultures commonly DON'T have this property; which means when we hear intervals in a scale, we know where we are in it. – Alan Nov 14 '20 at 12:57
  • @Tim, yeah you're 100% right, looking back not only did I wrongly say what a sus chord is, I wasn't even making a good point. What I was trying to get across was the thing I mentioned in my comment to phoog, about how differently voicing a chord lets you hear different intervals better. – Alan Nov 14 '20 at 12:59
  • Tangential, but I find that even the humble major triad , being made up of M3 and m3, is, and sounds, major, while the minor triad, made up of m3 and M3, is, and sounds, minor. Of course, it's M3 and P5 (or m3 and P5) really, but those 3rd intervals still exist within the triad. Strange and stranger... – Tim Nov 14 '20 at 13:21
  • OP, you're not inverting, but retrograding interval content within a scale.They're not the same motion (inverting would be transform W in H and vice versa). Said that, inversion is one of the founding thoughts from neo riemannian theory: axis of chord/collection inversion as originally Riemann proposed. Check this out. – Rodrigo B. Furman Nov 14 '20 at 18:04
  • Closely related question:[Do all modes make another mode when mirrored (i.e., intervals reversed)?](https://music.stackexchange.com/questions/51131/do-all-modes-make-another-mode-when-mirrored-i-e-intervals-reversed) – Aaron Jan 01 '21 at 23:53

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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.

helveticat
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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.

  • 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 04 '21 at 16:39
  • indeed, although Messiaen’s modes don’t interact with conventional diatonic modes in the same way. –  Jan 04 '21 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 05 '21 at 18:06