20170210

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 27/12 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
steps 322122
Heptatonic
Lydian mode (diatonic)
2221221
steps 2221221Pentatonic
Major Mode 3
32322
steps 32322
Octatonic
Bebop Dominant
22122111
steps 22122111Tetratonic
5232
steps 5232
Nonatonic
221112111
steps 221112111Tritonic
552
steps 552
Decatonic
2111211111
steps 2111211111Duotonic
Alternating Tonic-Dominant
75
steps 65

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 210/17 ≈ 1.5034, comes closest to 3/2 (almost as closely as does 'our' 27/12). 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}').

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}).

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].

steps 331213121

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.

fantasy heptadecaphonic keyboard

20150429

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).

image

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.

image

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.

20150425

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:

image

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.

image

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):

image

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

20150421

Phrygian = subtonic – Dorian

What do we mean by Phrygian = subtonic - Dorian, which you might be tempted to rewrite as Phrygian + Dorian = subtonic? Both look nonsensical. The units are all wrong for a start. It may mean something to ‘add’ a mode to a mode because they’re at least from the same ‘object’ space. But the term ‘subtonic’ is in another realm, that of relationships between the notes of a scale, and independent of scale to boot (e.g. the mediant of the A minor scale is C, a minor third above the tonic, whereas the same term is commonly used for the major third above the tonic, as E is to C in the C major scale). It’s like adding a green apple to a red one and a kangaroo shows up.

Furthermore, just because things are the same type of object, ‘adding’ them doesn’t necessarily make sense. You can add 10 minutes (a duration object) to 14:34 hrs (a time of day object) to produce a new time (14:44 hrs), or you can subtract a time from a time (e.g. 13:10 hrs from 13:58 hrs) to produce a duration (48 minutes) but what would we mean by adding 17:14 hrs to 11:08 hrs? Note that we’re not attempting to add a duration (of 11 hours and 8 minutes) to a time of day – that could make sense (it would take you to 04:02 hrs of the ‘next day’, if we’re pretending we are on Earth and not, say, Mars with its shorter day), we really meant trying to add ‘just after 11AM’ to ‘teatime’. It’s nonsensical. (You might temporarily add them, and divide by two, to get a time of day midway between them, but that’s part of a bigger operation).

Bear in mind that things that look like numbers might not be numbers. It makes no sense to expect any meaning to result from the addition (or subtraction, multiplication or division) of two credit card ‘numbers’. Not even on the way to (say) an average credit card number.

But enough of this. That was all to get you used to the idea that ‘addition’ or ‘subtraction’ may – or may not – legitimately be applied sometimes to like-things and sometimes to unlike-things, i.e. not just numbers.

So, let’s see what we might mean by our article’s title. We know, from our earlier post on reliable inversions, that to phrase-invert a piece of music in the Dorian mode so that it stays Dorian – and thus doesn’t need any further adjustments (such as sharpening a non-Dorian note here, flattening another there), we must subtract every note from the tonic.

For example, here’s a rather familiar bleeding chunk of Dorianistry (we’ll base it in C-Dorian, which looks like the key of B♭ on paper – a two-flats key signature – but its tonic note is C, not B♭):

image

The octave tone pattern of the 7 note Dorian mode scale, recall, is 0, 2, 3, 5, 7, 9, 10 where – in this case – those note values are attached to the ‘real’ notes (in the above) C, D, E♭, F, G, A, B♭ respectively. Subtraction from the tonic means that an inversion will require that every note value in the piece be subtracted from 0 (always the scale representation of the tonic note of any scale whatever). 0-0 is 0 – which means the tonic note will stay put. In the above, that means that every occurrence of the note C goes on being a C. It may shift octave, as the movement of the line dictates, but it’s the only note of the scale which remains as it started.

Now the 2 note is subtracted from 0 to become a –2 note, which, when 12 is added to keep it positive (remember we’re in the modulo 12 number land of what is sometimes called clock arithmetic) becomes a 10. The 3 is subtracted from 0 to become –3, which is 9. The 5 is subtracted from 0 (or 12) to produce –5 (or 7) etc.

So 0, 2, 3, 5, 7, 9, 10 is inverted to the set 0, 10, 9, 7, 5, 3, 2. Thus wherever you have a D in the original you’ll have a B♭ in the inversion – and vice versa. All E♭ and A swap places, as do all F and G. C, as we know, stays put. Recall our dorian knob from our heptagonal representations of the ancient greek (aka church) modes.

The horizontal white lines on the blue heptagon show, for each one of the seven notes in the scale, what its inverted note number is (mentally add outward-pointing arrowheads on the lines, if it helps you see this as a transformation showing how 7 maps to 5 and 5 maps to 7, for example). Each pair connected to the white lines sums to 12 (even the 0 to 0 at the top, because 0 is 12, right?).

The fully inverted piece is shown below. You should maybe check everything about it (the ups and downs corresponding to the inversion’s downs and ups, the note mappings as described above, etc). When played, it will probably remind you of “What shall we do with the drunken sailor”, but – being an inversion of a piece originally ending on the tonic chord of C-Dorian (which sounds rather like Cm – C E♭ G), it’ll now end on C A F. Which is F-major, a slightly weird major-sounding subdominant of a minor-sounding tonic key. But that’s the Dorian mode for you:

image

Note the complete absence of accidentals. And we didn’t need to tweak anything to get this pure, flawless inversion. It happened automatically because we subtracted from the tonic, we didn’t reflect in any silly old pitch axis.

Now let’s do it wrong

As the title of this article suggests, let’s subtract the original Dorian piece instead from its subtonic. The subtonic note of a scale is the one before (below) the tonic, often used in minor keys as an alternative lead in, instead of the dominant, to the final tonic. It’s the minor seventh (Roman vii) in such keys (and in the minor-sounding church/greek modes). It’s note numbered 10 in our Dorian clock (but it would be VII, sometimes notated as ♮VII, or note number 11 in other, more major sounding modes).

Subtracting each of our Dorian notes 0, 2, 3, 5, 7, 9, 10 from 10 is a lot easier (no pesky 12s to add to keep things positive!) and we get 10, 8, 7, 5, 3, 1, 0.

As expected, the inversion is (technical term) buggered up. The 0, 3, 5, 7 and 10 are OK – they’re Dorian notes – but we’ve got 8 and 1 where we had 9 and 2. Our (Roman) II  (or supertonic) has been flattened, as has our (Roman) VI (or submediant).

Let’s look at the top line of our original four bars of Dorian. We can usefully get away with considering only this one line because it just happens to contain every one of the Dorian scale notes. Which means that every scale note gets to be ‘exercised’ by the inversion. Compare it with its (faulty) inversion (top – original ‘Drunken Sailor’ melody line, bottom – not-quite-completely inverted ‘Drunken Sailor’):

image

The three points where it has ‘gone wrong’ are pretty obvious – they’re where the accidentals turn up in bar 2, bar 3 and bar 4.

The D♭ in bar 3 came from (in the ‘traditional’ way of performing pitch inversions) the original step up of a whole tone (from G to A) which required us to move correspondingly a whole tone down from E♭. Thus we’ve been taken out of the scale.

Similarly in bar 2 of the original we moved up a major third (two whole tones) from B♭ to D, where – in the inversion – we’ve reached a C (corresponding to the B♭) so requiring a major third step down from that C to the (out-of-scale) A♭.

You may work out for yourself why a traditional reflection gifts us with the accidental A♭ in bar 4.

In ‘pitch axis’ terms it looks as if the inversion is being done along (i.e. reflected in) the F between the two bottom E and G lines of the treble staff. Indeed F inverts to F (because, of course, 10-5=5).

At this point we may care to examine what actually happened when we subtracted every one of the ‘Dorian note values’ from the ‘Dorian subtonic value’ of 10. We ended up with notes which, when reassembled into ascending order, looked like this:

0, 1, 3, 5, 7, 8, 10

Which is itself a seven note scale. Is it one we know? Well, yes it is. We’ve seen it before as one of the seven possible orientations of our single heptagon, specifically the one which ‘points’ at the 10 o’clock position – the Phrygian mode:

And that is what we mean by Phrygian = subtonic – Dorian.

We’ve already had another such statement right under our nose all this time. It is Dorian = tonic – Dorian. In fact this last is but one of the seven from our original list in our attack on the concept of pitch axis, viz

  • Ionian = mediant – Ionian
  • Dorian = tonic – Dorian
  • Phrygian = submediant – Phrygian
  • Lydian = subdominant – Lydian
  • Mixolydian = supertonic – Mixolydian
  • Aeolian = subtonic – Aeolian
  • Locrian = dominant – Locrian

Of course this kind of ‘arithmetic’ must be carefully interpreted. The above statements should be more accurately expressed as

  • Ionian.pitch = Ionian.mediant.pitch – Ionian.pitch

Etc. And our ‘contentious arithmetic’ example is intended to be read as

  • Phrygian.pitch = Dorian.subtonic.pitch – Dorian.pitch

On the right hand side of the equals sign we have ‘subtonic’. But, scalewise (or modewise), subtonic is a relative term. Although note-like, or pitch-class-like, it does depend upon which key you’re in, or more generally within which mode. Sometimes its note value is 10, sometimes 11. Sometimes the mediant note is 4 (e.g. in Ionian) and sometimes it’s 3 (as in the Dorian). But once you’ve chosen your mode, the scale degree is effectively a constant.

You might want to make the case that a scale degree is not a pitch, but an interval – specifically understood to be implicitly from the tonic. But things get a tad recursive if you regard the tonic degree as an interval from the tonic, umm, degree. This may be why musicological terminology is not systematic enough to be scientific.

Whenever one of these context dependent ‘scale degrees’ (such as mediant, tonic etc) occurs, the context is supplied by the mode you’re in. So when we say Ionian = mediant – Ionian, we mean the Ionian mode’s mediant note, which is always 4 semitones up from its (always) 0 tonic note. No matter what actual key you’re in.

Right at the beginning we gave an example where subtracting a pair of ‘time-of-day’ objects yielded a ‘duration’ object. What we are doing here is something similar in that we are subtracting a note from another note within the context of a specific musical mode, i.e. Dorian.subtonic.note – Dorian.note.

Now every musician knows that note subtraction (i.e. the difference between a pair of notes) does not give you a note but an ‘interval’. So this arithmetic should seem a bit suspicious, or at least peculiar.

It’s only because we’re working with numbers, specifically those integers found in modulo 12 arithmetic which are applicable to both pitch classes and interval classes, that we’re able to get away with such ‘arithmetic’. It’s as if we’re coercing an object type of note out of an object type of interval. We can get away with this because musical intervals themselves are commonly regarded as interval classes (like pitch classes) insofar as intervals spanning distances larger than an octave are in some way equivalent to intervals within an octave (e.g. a major 9th is ‘the same’ as a major 2nd, a 13th is the same as a 6th, etc – just keep subtracting those 12s – they’re always executed as the same note; they’re just in different octaves).

But - regardless of whether or not you’re happy with the equivalence of intervals, or the legitimacy of type-overloading from interval to note, it remains an undeniable fact that the bunch of seven numbers you get out of those subtractions of each of your original mode’s pitches from your original mode’s single distinguished degree note is another bunch of seven numbers which, modulo 12, is indistinguishable from a bunch of seven numbers characterising a mode - which may be the original mode, but more often is a sister mode.

The point of the inversion list above is to invert into the exact same mode where the degree you’re subtracting from remains in exactly the same place. But when you use one of these scale degrees in non-inverting transformations the mode you end up in is different, as is perforce its degree.

All traditional modes can in fact be transformed into each of the seven modes (including themselves) by their notes’ subtraction from each of their seven degrees. We’ll use our (already established) three letter abbreviations for the modes, and the following two letter labels for the degrees

  • tn tonic
  • ut supertonic
  • md mediant
  • bd subdominant
  • dm dominant
  • bm submediant
  • bt subtonic

The following table can be used to transform pieces from mode to mode. To use it, select your starting mode from the seven in the top row, then look at the left hand column to pick out your desired target mode. Where the row and column intersect you will find the abbreviation for the degree (always relative to your original mode, here your column heading) from which you must subtract each note value of your original piece to produce the corresponding note value in your target mode.

to\from ion dor phr lyd mix aeo loc
ion md ut tn bt bm dm bd
dor ut tn bt bm dm bd md
phr tn bt bm dm bd md ut
lyd bt bm dm bd md ut tn
mix bm dm bd md ut tn bt
aeo dm bd md ut tn bt bm
loc bd md ut tn bt bm dm

more usefully, perhaps, we can use the note numbers of the two letter degree names corresponding to their column heading mode:

to\from ion dor phr lyd mix aeo loc
ion 4 2 0 11 9 7 5
dor 2 0 10 9 7 5 3
phr 0 10 8 7 5 3 1
lyd 11 9 7 6 4 2 0
mix 9 7 5 4 2 0 10
aeo 7 5 3 2 0 10 8
loc 5 3 1 0 10 8 6

The bold red diagonals of these tables represent, of course, the mode preserving pure inversion transformations. It may look a bit peculiar (considering we have all numbers 0 to 11 available to us) that only even numbers appear down that diagonal – and that, furthermore, 6 turns up twice. The 6 (the tritone note, yer actual Devil’s Interval, in any scale) turns up twice because it’s the dominant of the Locrian and the subdominant of the Lydian. So all seven degrees do turn up for inversions (as the first table shows) despite the second table’s (misleading) suggestion that only six of them are used.

It should go without saying that these so-called ‘failures’ of inversion (the ones that are not in bold red, i.e. 42 of them, i.e. 72%, i.e. most) by no means result in bad music. Or that ‘real’ musicians must justify themselves (if they happen to want to stay in scale – nobody’s forcing them to!) when they find they must tweak notes as they invert. Inversion as a musical process has been around much longer than any formalisations introduced by maths.

You might want to investigate these transformations as useful (or at least novel) ways of modulating from one mode to another. By useful, I mean ‘no thought required’ – you don’t have to tweak, you just do the appropriate subtractions.

For example, to move (the threepenny word for modulate) from (say) Lydian to Phrygian, just subtract all of your Lydian notes (turning up in various octaves, as they do, as 0, 2, 4, 6, 7, 9, 11) from 7 (to get 7, 5, 3, 1, 0, –2, –4 which yield 7, 5, 3, 1, 0, 10, 8) which, reordered, are notes in your desired Lydian scale of 0, 1, 3, 5, 7, 8, 10.

Inversions, even if they’re not quite ‘right’ are still close enough to a theme (a sixpenny word for a musical motif) you may have developed as a useful transitioning mechanism to get you into another mode on your way to another key.

Try switching from an Ionian D mode (the traditional ‘D major’) into some intermediate non-major mode such as Mixolydian – but in another key, say D♭-with a recognisable ‘pseudo-inverted’ phrase still reminiscent of your motif. From there, do a second ‘pseudo inversion’ (thus recovering your original motif, or something very close to it) by switching from Mixolydian back to Ionian (use the table to show you this is a ‘subtract from 9’ this time), but this time in C. Thus performing a double semitone dropping modulation from D major to C major. Try it and see!

Major minor teatime diner

We’ve seen in a previous post that a musical phrase inversion in the major scale is guaranteed to remain within the major scale if and only if each note in the phrase is numerically subtracted from the mediant note of the scale. And we’ve seen that to achieve the same guarantee in the minor scale, the subtraction must be done from its subtonic.

Thus, for example, to invert the following phrase in the key of F#m:

image

The (blue) numbers underneath the notes are not figured-bass – they’re too big and too colourful for that – but are the numerical values of the notes within the scale they are embedded. Thus F# is note number zero (aka the tonic note) in the F#m key, and you will find the subtonic E of that scale (marked in red) ten semitones up.

Since the subtonic of the minor scale acts as the ‘inversion rail’ (we’re deprecating both the phrase and idea of ‘pitch axis’ and are replacing it with something of utility) from which all notes of the phrase are to be subtracted, we can see that – from left to right – the inverted phrase will begin on the subtonic (10 – 0 = 10), will drop three semitones down to the dominant (10 – 3 = 7), will then rise a semitone to the submediant (10 – 2 = 8) … etc and will end on the mediant (10 – 7 = 3):

image

Subtraction is simpler than reflection because you need no longer traverse the whole phrase from beginning to end, tracking the changes in direction in your first phrase and making, correspondingly, the exact opposite moves in your new phrase. Instead, you take each note on its own (start anywhere you like, jump around) and simply subtract its value from 10. There’s no need to remember where you were, so that you add or subtract some number dependent on that - you simply need to know where you are, and always subtract from 10. If your notes were in a minor key, they’ll stay there. Guaranteed.

If you use a clock face to represent all twelve chromatic notes, within which the seven notes of the minor scale are embedded, it’s relatively straightforward to see why subtractions from 10 yield inversions which remain in the same minor key.

minor inversions: { 0 ↔ 10, 2 ↔ 8, 3 ↔ 7, 5 ↔ 5 }

The thick white lines represent the inversions. The tonic note 0 is taken to the subtonic 10, and vice versa (10–0=10 and 10–10=0). The supertonic 2 is taken to the submediant 8 (10-2=8, 10-8=2), the mediant 3 to the dominant 7 (and vice versa) and the subdominant 5 (necessarily, shown with a very short white line) remains unaltered.

Now, grasp the big blue heptagonal knob (consider the white lines as being gouged into the polygon) and turn it 3 clicks anticlockwise. After that, you’re here:

major inversions: { 0 ↔ 4, 2 ↔ 2, 5 ↔ 11, 7 ↔ 9 }

Congratulations. We have just switched into the major scale.

The tonic note of the scale, 0, is always - necessarily - at the top. The knob is engineered so that one of the heptagonal vertices is pointing at zero, so there are only seven distinct positions and not the twelve you might think.

The subdominant (5) and dominant (7) happen to be in the same place, but the mediant (3), submediant (8) and subtonic (10) have each moved up a notch, to 4, 9 and 11, respectively. Which is of course exactly what you’d expect when switching from minor to major. What were the minor third, minor sixth and minor seventh become the (major) third, (major) sixth and (major) seventh (except you don’t usually bother to say ‘major’). The three ‘flattenings’ required to take us from any major key to the minor of the same name are evident here.

It should also be clear (from whichever gouged white line connects the tonic to its inversion rail) why inverting a phrase written in a major scale requires that you subtract each note of the original phrase from 4.

Just to show that the ‘knob’ view works, we can re-present our original phrase (at the beginning of this post) as one in the key of A major. The only difference is our blue annotations numbers under each note, which will now be the note numbers found in a major key.

image

phrase now interpreted as being in A

This time we’ve reddened the ‘inversion rail’ required for the major scale, which is the scale’s mediant (which in the key of A major is the C#). As we know, the mediant in a major key is the III (in roman numeral notation), the major third, and is a 4 semitone interval above the tonic.

These are exactly the same notes as before. Nothing has changed. The phrase is simply being re-viewed as one in a major key (necessarily the one corresponding to its earlier, relative minor, re-shown here for comparison)

image

phrase originally interpreted as being in F#m

Inverting it, by subtracting every note from the value 4 (and adding 12, wherever necessary to re-notate it as a positive note number) gives us the new phrase:

image

Compare it with the earlier inversion, where the original was considered as being in a minor key:

image

Again, exactly the same notes as the inversion before.

At this point, we remember our old Greek modes, and that the modern major and minor scales used to be known as, respectively, the Ionian and Aeolian modes. The following diagram shows all seven (where each vertex of the heptagon gets a turn at pointing to the zero) knobby orientations, each of which corresponds to one of the modes.

dor ion loc aeo mix lyd phr

Abbreviations should be obvious, but here they are anyway – ion=Ionian, dor=Dorian, phr=Phrygian, lyd=Lydian, mix=Mixolydian, aeo=Aeolian, loc=Locrian.

If you traverse the edges of each of these polygonal orientations clockwise from 0, and write down a 2 for a whole tone skip and a 1 for a semitone skip (almost, but not quite, the lengths of the polygonal edges clockwise from the top), you may represent the seven orientations/modes as dor=2122212, ion=2212221, loc=1221222, aeo=2122122, mix=2212212, lyd=2221221 and phr=1222122.

These are the seven possible permutations of a permutation group, typically notated with parentheses as, for example, (2212221) – where the parentheses contain a string of digits allowed to ‘rotate’ inside the parentheses by moving the last digit from the end to the beginning, or vice versa. E.g. take the final 1 of the 2212221 inside the parentheses and move it to the first position and you get (1221222). Which represents a rotation of the knob (clockwise) from an ionian to a locrian orientation.

It may be slightly surprising that it’s the Dorian mode which most clearly shows the vertical symmetry of the modern seven note diatonic scale, but it must be that way because it’s only the inversions of dorian phrases (intended to remain dorian) which subtract from the tonic (or 0). The short white line has to be there.

It’s an accident of symmetry, first in that the heptagon is symmetric at all (it need not have been) and second in that there’s no law of the universe which says that the 0 direction is ‘up’.

So it can surely be no accident that the Ionian (the modern major) and the Aeolian (the modern minor) appear to be set at two o’clock and five o’clock. Representing, as they do, the twin pinnacles of civilised society of maximal satisfaction and self-assuredness just after lunch and the more sober contemplation of the approaching teatime.