## 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♭):

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:

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’):

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!