Overlapping Transpositions

Every scale has twelve transpositions, some of which are modes and some hypermodes of the original one. Unless the scale has only one note in it (!) there will be some overlap between the original scale and at least some of its transposed copies. Indeed, for seven-note scales, one of the most common categories we work with, every transposition will share at least two notes in common with the original. Let's see if we can use this to our advantage.

(This is one of those hand-waving "speculation" posts, but I think the results are strong enough that I'll have something you can play based on this soon.)

Before we start, here's my motivation for this. When I work with harmonic resources I generally want to partition the 12 available notes into two (or more -- but usually two) groups to bring out musically distinctive sounds. Say I've got hold of Superaugmented, which is 1 #2 #3 #4 #5 #6 7. That's seven notes of the total chromatic; the other five are what I call the "coscale" of superaugmented, i.e. b2, 2, 3, 5, 6. So there are really two things to learn: the scale and its corresponding coscale, which together cover the twelve available notes.

But what if I'm lazy and don't want to learn the coscale, or if I just want a bit more flexibility? One idea is to take the original scale and transpose it until it contains the coscale as well as some extra notes. So instead of "switching to the coscale" I simply transpose the scale to a new root note. This is a very cheap way to discover new sounds, and it's easy on the listener's ear because we're still using the same intervallic structure.

Anyway, let's start with the C major scale, because it's nice and familiar. Here it is transposed to all 12 keys -- I've chosen note-names that emphasise overlaps with the original:

     C   D   E   F   G   A   B   
     C#  D#  F   F#  G#  A#  C            2 overlaps   
     D   E   F#  G   A   B   C#           5 overlaps
     D#  F   G   G#  A#  C   D            4 overlaps
     E   F#  G#  A   B   C#  D#           3 overlaps
     F   G   A   A#  C   D   E            6 overlaps
     F#  G#  A#  B   C#  D#  F            2 overlaps
     G   A   B   C   D   E   F#           6 overlaps
     G#  A#  C   C#  D#  F   G            3 overlaps
     A   B   C#  D   E   F#  G#           4 overlaps
     A#  C   D   D#  F   G   A            4 overlaps
     B   C#  D#  E   F#  G#  A#           2 overlaps

Two things interest me here.

First, those transpositions that have a large overlap with the original, which ought to sound similar to it. In this case the 6-overlap examples are C Lydian and C Mixolydian -- no big surprises there. Then the next most similar is the one with 5 overlaps, which is a hypermode, so that's potentially interesting. These all offer variations on the original scale without straying too far away from it.

Second, those transpositions that are far away from the original. The 2-overlap examples are C Locrian and a pair of hypermodes; with 3 overlaps we have Phrygian and another hypermode. Again, no big surprises here. In fact, "side-slipping" up or down a semitone -- which obtains two out of the three 2-overlap cases -- is a common trick used by jazz players to temporarily "go outside". It works because usually you get something close to the coscale of whatever you were originally playing.

Crucially, note that whenever a heptatonic scale is transposed so that it only has 2 overlaps with the original, it contains the entire coscale of the original; this is inevitable since the original and the transposition have 2x7 = 14 notes; if they share two incommon they must cover 12, which is all the notes available to us. By definition the notes in the coscale are not in the original scale, so they must all be in the transposed version.

I usually work with seven-note scales, which have five-note coscales. But these pentatonic scales don't always harmonize out in a satisfactory way; heptatonics have a couple of extra notes that make forming chords a bit easier. So I'm inclined to use minimal-overlap transpositions as a way to "thicken" or "fill out" the coscale in a musically meaningful way.

C Superaugmented will illustrate what often seems to happen. Here are the twelve transpositions. Again, enharmonic names have been chosen to make spotting overlaps easy:

     C  D# E# F# G# A# B            
     C# E  F# G  A  B  C            3 overlaps
     D  E# G  G# A# C  C#           4 overlaps
     D# F# G# A  B  C# D            4 overlaps
     E  G  A  A# C  D  D#           3 overlaps
     E# G# A# B  C# D# E            5 overlaps
     F# A  B  C  D  E  E#           4 overlaps
     G  A# C  C# D# E# F#           5 overlaps
     G# B  C# D  E  F# G            3 overlaps
     A  C  D  D# E# G  G#           4 overlaps
     A# C# D# E  F# G# A            4 overlaps
     B  D  E  E# G  A  A#           3 overlaps

None of these contains the coscale -- they all have three or more overlaps. So we only have "almost-coscale-containing transpositons" at our disposal. These are the ones built on the b2, 3, #5 and 7, which are most easily visualized as a minor seventh arpeggio a semitone above the original root. This is rather a distinctive and interesting structure that belongs uniquely (I suspect) to Superaugmented. But none of these is quite satisfactory: combining it with the original scale covers 11 notes, not 12.

I tried a bunch of heptatonic scales that I happen to have in a Python-friendly form and here are the ones I found that do have at least one 2-overlap transposition:

Major 1 2 3 4 5 6 7
Melodic Minor 1 2 b3 4 5 6 7
Neapolitan 1 b2 b3 4 5 6 7
Rasikapriya 1 b3 3 b5 5 b7 7
Melodic Minor b4 1 2 b3 3 5 6 7
Neetimati 1 2 b3 b5 5 b7 7
1maj+b2maj 1 b2 b3 3 4 5 b6
Yagapriya 1 b3 3 4 5 b6 6
Varunapriya 1 2 b3 4 5 b7 7
Gayakapriya 1 b2 3 4 5 b6 6
Salagam 1 b2 2 b5 5 b6 6

(Note this is absolutely not a complete list, just a selection I could lay my hands on easily.)

Lots of interesting things here.

To start with the familiar, Melodic Minor's 2-overlap transpositions are Super Locrian and the hypermode you get from playing the scale up a semitone; no big surprises there. These are the two you would guess would have minimal overlap if you didn't have any more information about the scale, although very often it's not that way with other scales.

Neetimati and Gayakapriya both have three 2-overlap transpositions, and they're the same: the ones built on the 2, the b5 and the b7. The b5 seems to be quite a common one, which isn't too surprising if you think of it as the "most distant key" on the circle of fourths.

Incidentally, 2, b5, b7 is a symmetrical set, i.e. it's unchanged by inverting it about the 1. To put it another way, the interval map including 1 is tone, major third, major third, tone. This symmetry is always present because we're superimposing two identical things. Putting Copy B a tone up from Copy A (transposing Copy B to 2) is the same as putting Copy A a tone down from Copy B (transposing Copy B to b7). This means that if a scale has an odd number of 2-overlap transpositions, one of them must be the b5.

Here's a lovely thing, too: all six modes of Neapolitan are 2-overlap transpositions of the original scale. It's not so surprising when you think of how Neapolitan is structured, but still, it's certainly given me a new perspective on that scale and made me think about some ways this post from a few days ago could perhaps be extended.