Until now we’ve been working in musical scales, all of which allow melodic inversion. We’ve looked at the standard western major and minor seven-note (heptatonic) scales, the old Greek or Church modes (Aeolian, Dorian, Mixolydian etc), a couple of pentatonic scales (major and minor) and an octatonic scale or two for jazz, and slightly more unusual heptatonic scales such as the Hungarian Minor and its relatives.
All of those scales will permit you to invert a musical phrase, i.e. a phrase using only the notes of the scale it is written in, where the inversion may be forced to stay in the same scale without having to fudge things. Such inversions are performed by a subtraction from a fixed note (or, as in the case of the two octatonic scales presented, from any of a choice of four) within that scale – and no others.
Furthermore, we’ve seen how if you choose the ‘wrong’ (mathematically, not musically – there’s nothing ‘wrong’ in music!) note of the scale to subtract each note of the phrase from, you will fail to get a true inversion (because some notes in the inverted phrase will stray out of the original scale). But, notwithstanding such failures, you will end up in a different scale – one of a family of related scales analogous to the modes of the standard western scale.
For example, we know it’s possible to properly, flawlessly, invert a phrase in the Hungarian Minor. Just subtract each note of the phrase from the quasi-supertonic of that scale, which happens to be the business-as-usual major 2nd. Or if you’re still unwilling to dispense with the idea of pitch-axis then reflect your phrase along a horizontal line set on the minor 2nd of the scale (a pitch which isn’t even in the scale for heaven’s sake – that’s how silly ‘pitch axis’ is) – and trudge along, following the melody line and moving your new line in the opposite direction.
We also know that if, instead, we rebelliously subtract a Hungarian Minor melody from its augmented subdominant (a Devil’s Interval) our new phrase leaps out of the Hungarian Minor scale and ends up in its sibling scale, the Double Harmonic (aka Gypsy, aka, Byzantine, etc).
So, what about the Hungarian Major scale? Yet another of one of the many heptatonic scales, its ‘pitch class signature’ is 0, 3, 4, 6, 7, 9, 10. If we insist on using scale degree terminology, its sequence starts (as always) with the tonic, its supertonic is an augmented 2nd, its mediant a major 3rd, its subdominant an augmented 4th, its dominant and submediant are perfectly cromulent 5th and 6th, and its subtonic a minor 7th. Here it is in the key of C, and also represented more abstractly with its (key-independent) polygonal representation. (It’s pink, not blue, for a reason).
As usual, to see which inversions work, we will take each of the scale pitch classes in turn and subtract the entire scale sequence from that selected pitch. Thus:
|0, 3, 4, 6, 7, 9, 10|
The ‘reordered’ column is there only for information, to show the ‘scale’ of the transformed notes. It’s the ‘modulo 12’ column which embodies the actual ‘note to note transformation’.
By looking down the column headed 'reordered' it's easy to see that none of the scales are Hungarian Major. The one in the second row, beginning "0,3" is indeed the only one which even begins with a match on the first two pitches,
This means that, however hard you try, no musical phrase in the Hungarian Major scale can be inverted along any axis so that every note in the resultant phrase remains in the Hungarian Major scale.
Musically speaking of course, this is no tragedy. A musician need not commit suicide on learning of such an impossibility, If a musical – as opposed to a scale-preserving mathematical - inversion is required, the musician will either allow the notes to go out of scale as they will, or else they will relax the requirement that every transition between successive notes in the ‘inversion’ exactly match (but in reverse direction) the interval transitions taken by the original phrase, A minor third up for a major third down here, a minor sixth leap for a major sixth drop there, and the like.
Nonetheless, let’s just have a quick look at the plesio-inversion represented by the first working row of the above table – the row having us subtracting the scale from 0. I.e. the one which moves 0 to 0, 3 to 9, 4 to 8, 6 to 6, 7 to 5, 9 to 3 and 10 to 2 (the modulo 12 column).
If in the key of C, then C and F# keep their pitches, D# (aka an enharmonic E♭) swaps with A, E is replaced by G#, G by F and B♭ by D. Altogether, only four of the original pitch classes remain and three are swapped out of the scale entirely.
None of the other attempted inversions fare any better. But the sharp-eyed amongst you will have noticed that this second heptagon is just a lateral inversion (in the optical sense, reflected in the vertical axis) of the first one. And it should come as no surprise, therefore, that the scales turning up in the second through seventh rows are simply the other six (we always require a vertex at 0) rotations of this second polygon. So here are those seven polygons, row by row:
It would be tempting to say that these seven scales were all ‘musical modes’ of essentially the same scale and that they just started on a different (tonic) note. But as this author is unaware that any of these seven scales is used anywhere in the world. that might be stretching a point. Rotations of the same polygons they definitely are, Maybe you can find one that sounds pleasing enough to noodle around in? If so, then you might plesio-invert out of it and find yourself in the Hungarian Major (as it were).
It will not have escaped your attention that – corresponding to the Hungarian Major – there will be another six scales, rotations of the Hungarian Major’s polygon, and that each one of these is a one for one (vertical axis) reflection of the ones above. Here they all are (the Hungarian Major is the first of them):
Together, these fourteen scales form a nicely complicated, closed, family in which any musical phrase written entirely within one of them can be plesio-inverted into any of the others, As far as I know, the only actual scale in use out of these fourteen is that Hungarian Major. But of course I would be delighted to hear of any others.
The especially eagle-eyed – and non-colour-blind, will note that these polygons are pink. We’ve been using blue ones in all the previous articles. They’re pink because they’re representations of uninvertible scales. All our blue polygons represent invertible scales – and they’re all symmetric. All of our pink polygons are non-symmetric and occur in families with corresponding mirror image families. There are many more pink polygons than blue ones.