We’ve seen how to invert a piece of music so that it’s guaranteed to stay in the same scale (or mode) as the original melody (again assumed to consist exclusively of scale notes). The subject here is melodic, or phrase, or even *horizontal* inversion. It concerns neither chord nor *vertical* inversion.

We know how to do this for any of the traditional western classical 7 note diatonic scales in all their modes (Ionian, Aeolian, etc.). We can do it in major and minor pentatonic scales (and their further three but seldom mentioned sibling modes). We can even invert *four* different ways when the original melody is either of the two common octatonic (aka diminished) scales used in jazz.

But what about other scales, those off the beaten track (to use a phrase very much *on* the beaten track)?

Let’s pretend to pick one at random. Say, the Hungarian Minor. As you may (or may not) know, this is another flavour of 7 note scale which – when rooted on the A above middle C (just so we can stick to as many ‘white notes on the piano’ as possible) – looks like this:

The top A is of course present just to keep you from becoming too tense with an unresolved sequence. It really is a 7 note scale.

There are a couple of sesquitonic (one and a half tone) leaps larger than whole tones in this scale. In terms of (key independent) pitch numbers, it can be represented as 0, 2, 3, 6, 7, 8, 11. It’s pretty easy to see that it’s quite similar in shape to the standard Western Minor except that its IV and vii have been sharpened. Compare their heptagonal clock diagrams:

2131131 | 2122122 |
---|---|

The one on the left is the Hungarian one, the one on the right is the standard (“all the white notes” if you start on A) minor (or the Aeolian Mode). It may not be easy to spot, but both are symmetric heptagons, The Hungarian’s prow points at seven o’clock (an E for an A at twelve o’clock) and the Aeolian’s points at five o’clock (a D for its A). The headings are the intervals (number of semitones) between the scale notes clockwise from the top.

Now we already know that the Aeolian mode is invertible (in our strict scale-preserving sense) and that to invert an Aeolian tune, you subtract each note from the Aeolian’s subtonic (i.e. the modulo 12 number 10 corresponding to vii, the minor seventh degree, in that scale). So if you’re in A minor then the quickest way to an inversion of a phrase is to subtract its notes from G. Which boils down to leaving all the Ds alone, swapping all C with E, all B with F and all G with A and adjust the contour of the melodic line, as you commit it to the staff, by appropriate octave shifts where necessary. This is far quicker than trying to follow the original phrase and move correspondingly down and up the exact same interval for its every up and down, as if you were reflecting with a silly old mirror placed on an imaginary pitch-axis located on the G# line (a note which isn't even *in* the scale we're working in).

*Aeolian Inversion, showing the pitch swaps (including the ‘do-nothing’ swapat five o’clock) resulting from subtraction from ten o’clock*

And like the Aeolian, the Hungarian Minor’s heptagon is symmetric. So we might hope that we can invert a tune in this scale too. However, it should be immediately apparent that there’s no 10 in the scale to subtract from – it got sharpened. As before, we can try subtracting each note in the Hungarian Minor (the 0, 2, 3, 6, 7, 8, 11) from each of its scale notes (modulo 12) in turn. If one (or more) of those subtractions results in the same numbers then we’re done.

Let’s try one. Subtract each of the seven in turn from 3 and we get 3, 1, 0, –3, –4, –5, –8, which in modulo 12 is 3, 1, 0, 9, 8, 7, 4, which rearranged into ascending order is 0, **1**, 3. **4**, 7, 8, **9**. Clearly this is not at all the same scale (the out of scale notes are in **bold red**). But fear not – there is one answer (and only one – can you see why it’s only one?) – and it is by subtraction from 2. Here we go:

- 2 – 0 = 2 → 2
- 2 – 2 = 0 → 0
- 2 – 3 = -1 → 11
- 2 – 6 = -4 → 8
- 2 – 7 = -5 → 7
- 2 – 8 = -6 → 6
- 2 – 11 = -9 → 3

Here’s the heptagon again with the inversion mappings:

*Hungarian Minor Inversion, showing the pitch swaps (including the ‘do-nothing’ swapat seven o’clock) resulting from subtraction from two o’clock*

To demonstrate, let’s be slightly more adventurous and pen a quick phrase in D Hungarian Minor. The root note is of course D, the supertonic is E (which is the note we’ll eventually be subtracting *from*), the mediant F, the (augmented) subdominant G#, the dominant A, the submediant B♭ and the (augmented) subtonic C#. As you know, the D minor scale is usually written with the single (i.e. B) flat associated with the major key of F, so we’ll go along with that, and use the accidentals at C and G to bring out the phrase’s Hungarianism.

To invert the phrase, certain that we’re going to stay in the exact same scale, with the exact same tonic note of D, we subtract each of its notes from E (the scale’s 2 pitch). This amounts to keeping any A where it is, swapping all B♭ with G#, all F with C#, and all D with E. After adjusting octaves appropriately (we can use this whilst moving from note to note left to right without needing to worry if the interval movements are exact mirrors – they *will* be. it’s guaranteed):

So there you go. Easy, isn’t it?

Recall that in the last post we deliberately did a ‘wrong’ inversion (by subtracting Dorian scale notes from its subtonic instead of the tonic. thus landing in the Phrygian) and showed that for each of the seven traditional western 7 note modes, subtraction from – in turn – each of each scale’s scale notes (yes, that’s hard to parse) will always result in seven numbers characterising another one of the modes. In other words the *collection* is self preserving even if the true inversion is only one of them. Is this also true of the Hungarian Minor?

Of course it is. It *must* be. All subtractions, modulo 12, from each of the seven notes in the scale will result in the same heptagon. It’s just that it will be rotated. This is because the heptagon is symmetric. We have in fact already seen one of them by subtraction from the 3, which gave us the scale (0, 1, 3, 4, 7, 8, 9) shown here.

It’s just the Hungarian Minor scale with the ‘cooker knob’ turned one click forward (clockwise) – where the rules of our cooker knob turning require that there’s always a vertex (of whatever polygon we have) at twelve o’clock (because obviously we need the root note of the scale to be in the scale). This is a legitimate seven note scale (for who is to say it is not?) which – rooted-on-C-wise – would comprise the notes C, D♭, E♭, E, G, A♭, A. As the little white blob at the four o’clock position indicates, *this* scale is invertible (in our strict sense) by subtractions from its (quasi)-subdominant.

Note the use of the *quasi*- there by the way. That four o’clock position is usually occupied by the mediant, but we’ve already ‘used up’ the mediant’s (albeit minor) slot at three o’clock (because the supertonic looks like it turned up early at one o’clock). These scale degree terms aren’t going to scale well (no pun intended) in more general cases. The Pentatonic Major may appear to be simply deficient in the subdominant and subtonic departments, but what are we to do with only seven scale degree terms in an *octatonic* (or nonatonic) scale? But that’s by the way.

As it happens, subtracting Hungarian Minor notes from its (augmented) subdominant (i.e subtracting each of 0, 2, 3, 6, 7, 8, 11 from 6) yields the following rotation - not one, but three big clicks - clockwise:

This scale (0, 1, 4, 5, 7, 8, 11) which, rooted on C, would comprise C, D♭, E, F, G, A♭, B is invertible by subtraction from its tonic. All strictly invertible scales which subtract from their tonics (such as the Dorian mode) have the rather visibly obvious axis of symmetry vertically down the middle of the (perforce) symmetric polygon they live on. They are self-negative (which is subtractions of their integer members from 0) scales. You might think of this scale and the one above as, respectively, the quasi-Dorian and the quasi-Mixolydian (in addition to the quasi-Aeolian orientation of the Hungarian Minor) modes of an alternative system of seven modes.

As it happens, this scale has a name, the * Double Harmonic*. In fact it has other names. It’s also called

*, the*

**Gypsy****, the**

*Byzantine***, and the**

*Chahargah***. That’s quite a few genres you can now accurately invert in! And use it to try out those double backflip modulations by first inverting (say) out of the Hungarian Minor into (say) the Double Harmonic and once more by further inversion from Double Harmonic back into the Hungarian Minor.**

*Arabian*
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