This bit of analysis was prompted by an interesting question asked on the Music Theory forum on Reddit. It ended up as a question about which scales you can transpose and get a completely different set of notes from the set you started with, with no overlap.

As an example, the major triad C E G can be transposed up a semitone to C# F G#, which is a completely different set of notes. On the other hand this can't be done with a major scale because it has 7 notes, and there are only 12 in the chromatic scale, so any pair of major scales must overlap by at least two notes.

Initially I took a stab-in-the-dark guess at how this would pan out across all scales. Not long afterwards I'd proved this guess completely wrong. I'm delighted about this because the picture is far weirder and more lumpy than I would have guessed. I ought to have learned by now: in scale theory, never bet against things being weird and lumpy.

Anyway, in this post I don't want to post a full set of findings, because I don't have that yet, but instead just point to a possible application of this idea for improvisors. It bears a close relationship to what I call coscales, though I won't use that idea here.

Roomy Pentatonic Pairs

Let's call a scale "roomy" if it "leaves enough room" for another copy of itself to coexist alongside it in the chromatic scale with no overlap. I'm going to focus only on pentatonics today, for reasons I'll get to. The musical idea here is that if we can learn a roomy pentatonic, we can transpose it to give a total of 10 out of the 12 possible pitches -- almost, but not quite, the total chromatic.

Improvising in a way that moves between these two sets of notes like two different palettes of colour is likely, it seems to me, to produce musically interesting results while being a practical thing to do "on the fly".

My analysis turned up 23 roomy pentatonics. Some of them contain runs of semitones; I discarded these as I don't think they're very interesting. I kept hold of the following examples, first the ones containing no semitones at all:

Binary representation	Interval map	Spelling	Comments
100010101010	MT, t, t, t, t	1, 3, #4, #5, #6	Whole-Tone Scale minus a note
100100101010	mT, mT, t, t, t	1, b3, b5, b6, b7	m7b5b13 arpeggio
100101001010	mT, t, mT, t, t	1, b3, 4, #5, b7	Common pentatonic (mode)

...then the ones containing one or more non-consecutive semitones:

Binary representation	Interval map	Spelling	Comment
100001010110	Fo, t, t, s, t	1, 4, 5, 6, b7	Mixolydian subset
100001011010	Fo, t, s, t, t	1, 4, 5, b6, b7	Aeolian subset
100010001101	MT, MT, s, t, s	1, 3, #5, 6, 7
100010010101	MT, mT, t, t, s	1, 3, 5, 6, b7	Mixolydian subset, 7 add 13
100010011001	MT, mT, s, mT, s	1, 3, 5, b6, 7	Harmonic Major subset
100011001001	MT, s, mT, mT, s	1, 3, 4, #5, 7
100011001100	MT, s, mT, s, mT	1, 3, 4, #5, 6
100011010100	MT, s, t, t, mT	1, 3, 4, 5, 6	Major scale subset

If you want to find diagrams for these in my free scales ebook, the best way is to search the PDF for the interval map; they're all in there, of course.

Application 1

A general observation is that roomy scales tend to have more bunched-up notes than average, which tends to make them less musically interesting. Hence the first set of pentatonics with no semitones is especially interesting. Of them the most promising seems to me to be this one:

Binary representation	Interval map	Spelling	Comments
100100101010	mT, mT, t, t, t	1, b3, b5, b6, b7	m7b5b13 arpeggio

Here it is in C, along with the transposition up a semitone that shares none of the same pitch classes; on the far right are the two pitch classes not included in either of them:

To me the obvious thing to look at here is situations where I want to avoid those two notes, D and F; and the obvious time I want to do that is on Bbm7b5, where they're the major third and perfect fifth respectively. Both of these are tough notes to sell in this context. Here's how the chord tones for Bbm7b5 map out onto this pair of pentatonics:

What's nice about this is that neither really contains the whole chord; both contain good chord tones and interesting tension tones. So this really does seem like it would be a setup that gives two different approaches to the chord, with no overlap, that together cover all ten notes you might want to play on it. This strikes me as an extremely promising way to approach building new vocabulary for this kind of chord.

Application 2

Let's now look at a couple of those more exotic examples from the second table:

Binary representation	Interval map	Spelling	Comment
100011001001	MT, s, mT, mT, s	1, 3, 4, #5, 7	+Maj7 add 11

I picked this just because it's one of the less familiar-looking ones. Here's the pitch classes you get if you start on C:

The missing pitches this time are separated by a tritone. Let's say again we want to avoid the 3, this time perhaps when playing over a minor chord. Then we can achieve this in two ways:

• On Cm, play Db+Maj7 add 11 and Eb+Maj7 add 11
• On Cm, play G+Maj7 add 11 and A+Maj7 add 11

Either option will give us exactly the same notes but organised differently; this is a kind of symmetry I haven't seen before. So this suggests that I can combine the two approaches to give me the following resource: on Cm, play any of those four pentatonics at will.

We'd better learn our +Maj7 add 11 arpeggios:

As its name implies, it turns out that this scale is covered by a major and a minor triad arpeggio a semitone apart, so these shapes shouldn't be hard to learn. The fact that they give us access to this very interesting material over a minor chord is pretty good inducement to do so.