A B♭ in my bonnet

There’s no shortage of material about George Russell’s “Lydian Chromatic Concept” on the ‘tubes, but much of it is geometrically justified by some interpreters and – consequently - may come across as something from the Green Ink Brigade, i.e. a little cranky.
But the geometry seems sound, being as it is just a representational consequence of a harmonic ‘reality’ – at least no less so than the “cycle of fifths” is based on the fifth’s frequency’s being (classically, anyway) 3/2 times the root note’s frequency (or 2^7/12 times, if you’re well-tempered). As this is aurally (arguably, I suppose, since everything’s arguable) the next most obvious interval after the octave’s 2/1, its importance in music is well established.

If you allow the standard 12 hour clock (or dodecagon) as both a useful and reasonable model for talking about dodecaphonically partitioned octaves then you’re already happy about (by which I mean that you are mathematically and unavoidably led to) using twelve 7 semitone jumps having as much legitimacy as a ‘generator’ of all 12 notes as is the more direct single 1 semitone stepping up the sequence. That's because 7 (and 5) is relatively prime to 12, just as is 11 (and, trivially, 1).

If you’re comfortable (and many are not) with letting the maths repurpose its role from being merely a usefully descriptive modeller to its being a prescriptive constructor of musics, with a constructor's often concomitant value judgements, then the Lydian ends up as ‘tops’. It cannot help it!

So if you decide that the (diatonic, seven note) scale you’re generating is to begin on the tonic note of that scale (bearing in mind that - modally – you're quite free not to) then starting on (for example) C takes you to G then D then A then E then B then F#, at which point you stop (you've got your 7 notes) and reorder those notes into the (tada – Lydian, not Ionian) scale/mode with that telltale sharpened fourth. And its relative ‘minor’ is of course three semitones back to starting on the A, with its F# making it a Dorian and not an Aeolian (which would have the F).

Another way of seeing the ‘distinguishedness’ of the Lydian is to order all 7 diatonic modes alphabetically (which, as it happens, turns out to be numerically) with their halfstep/wholestep descriptions (not their names - that would be silly).

• 1221222 Locrian
• 1222122 Phrygian
• 2122122 Aeolian
• 2122212 Dorian
• 2212212 Mixolydian
• 2212221 Ionian
• 2221221 Lydian

Which is – effectively – the modes ordered by their ‘majorness’ starting from the most minorish. And there’s the Lydian right at the end of the list, with the Ionian coming in only as the runner-up. Naturally the 'ugly duckling' Locrian brings up the rear (but personally I'm quite fond of that next 'loser', the Phrygian).

Note that these (key independent) semitone-step-determinatives of the diatonic modes are the exact same consequence of the ‘generative fifthiness’ – there’s no new information there - but it’s still interesting.

This kind of modelling will work with any sized scale built up from stacked fifths – perhaps the next most familiarly the pentatonic (with its five modes) embedded within 12-note systems.

There’ll be a ‘most major’ ordering (the ordering with all the biggest skips at the beginning of the scale) of an octatonic scale too. It’s 22122111, the dominant bebop scale (=Ionian plus an extra – functionally dominant - seventh), since you ask. As to why you'd select that particular octatonic (and its eight - permutationally cycling - modes) pattern of steps (as opposed to - say - 22221111, or 23112111) it's because we're (here) considering only scales constructed with stacked fifths:

The 'Majorest' modes built from stacked fifths, for scales of varying degree
		Hexatonic 322122
Heptatonic Lydian mode (diatonic) 2221221		Pentatonic Major Mode 3 32322
Octatonic Bebop Dominant 22122111		Tetratonic 5232
Nonatonic 221112111		Tritonic 552
Decatonic 2111211111		Duotonic Alternating Tonic-Dominant 75

Note that - as is typical with paired n-note and 12-n note scales - the 'majorest' 12-n note scale is one of the modes of the scale constructed from the notes missing from the n-note scale. (Hexatonic scales are, naturally, their own 'anti-scales').

Microtonality

This construction principle will also work with scales embedded within the more exotic world of microtonality. Consider, for example, a scale divided into 17 'equal' (or as near as dammit) divisions. This, by the way, is a real thing. To generate the 'best' (value judgment!) scale/mode from some root note of this scale, you'd ascertain which of the 16 remaining notes was nearest in frequency to 3/2 times the root note. If it's an even-tempered microtonality (it need not be - that's a human choice, not some law of the cosmos) then 2^10/17 ≈ 1.5034, comes closest to 3/2 (almost as closely as does 'our' 2^7/12). In other words, the most consonant sounding scales within a 17 note microtonality would be generated from its 'cycle of fifths' based on 10 (as opposed to 7) semitone jumps.

Regardless of 'key', you may generate the whole set of 17 from the sequence 0, 10, 3, 13, 6, 16, 9, 2, 12, 5, 15, 8, 1, 11, 4, 14, 7 (successive remainders of successive multiples of 10 when divided by the 17 - see 'star polygon {17,10}').
Compare that with the cycle of fifths 0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5, familiar to our dodecaphonic culture (a star polygon {12, 7}).

Again, you might choose to elect some equivalent to a 'diatonic' scale comprising just over half of the available tones from the beginning of that 17 note sequence, i.e. a nine note 'octave'. Then you'd arrange those 9 numbers in ascending order to generate your 'diatonic' scale. It would be 0, 2, 3, 6, 9, 10, 12, 13, 16. Analogously there'd be nine modes, and the most major of those modes would be the one with the largest internal steppings up front.
As the generated stepping is 213312131 (a scale built from three types of skip - 4 semitones, 2 tones and 3 sesquitones!), the alphabetically highest one would be 331213121 (i.e. the scale sequence 0, 3, 6, 7, 9, 10, 13, 14, 16) [see right].
Build up some analogous triads (0, 7, 10) as a 'major chord' in this scale. A 'minor' would correspondingly be (0, 6, 10). Note that both contain the fifth (the 10) and that the minor has a 'flattened third' (a 6 instead of a 7).
This 17 note scale still has room for a separate 4th, close to the fifth for that super major 'Lydian' feel. Furthermore you have two sevenths (like the two thirds) at 13 and 16 - a dominant one and a major one for a leading tone, in the same scale.

Below's a picture of an imaginary heptadecaphonic piano with 17 note scale support. You could play with both thirds (minor and major) and with both kinds of sevenths (minor/dominant and major) without ever leaving the white keys, in its "C-Major" mode. Although we - like Miles Davis - would think of the piano layout as being based on a white-noted F-Lydian, with a middle F.

The Hungarian Major won’t be Inverted

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
from	subtracted	modulo 12	reordered
0	0,-3,-4,-6,-7,-9,-10	0,9,8,6,5,3,2	0,2,3,5,6,8,9
3	3,0,-1,-3,-4,-6,-7	3,0,11,9,8,6,5	0,3,5,6,8,9,11
4	4,1,0,-2,-3,-5,-6	4,1,0,10,9,7,6	0,1,4,6,7,9,10
6	6,3,2,0,-1,-3,-4	6,3,2,0,11,9,8	0,2,3,6,8,9,11
7	7,4,3,1,0,-2,-3	7,4,3,1,0,10,9	0,1,3,4,7,9,10
9	9,6,5,3,2,0,-1	9,6,5,3,2,0,11	0,2,3,5,6,9,11
10	10,7,6,4,3,1,0	10,7,6,4,3,1,0	0,1,3,4,6,7,10

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:

2121213	3212121	1321212	2132121	1213212	2121321	1212132
0,2,3,5,6,8,9	0,3,5,6,8,9,11	0,1,4,6,7,9,10	0,2,3,6,8,9,11	0,1,3,4,7,9,10	0,2,3,5,6,9,11	0,1,3,4,6,7,10

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):

3121212	1212123	2121231	1212312	2123121	1231212	2312121
0,3,4,6,7,9,10	0.1.3.4.6.7.9	2,3,5,6,8,11	0,1,3,4,6,9,10	0,2,3,5,8,9,11	0,1,3,6,7,9,10	0,2,5,6,8,9,11

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.

The Invertible Hungarian Minor

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’ swap
at 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’ swap
at 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 Gypsy, the Byzantine, the Chahargah, and the Arabian. 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.

Minor Hungarian Minor Waltz by LemoUtan