Theory Lab › Scales from symmetry

Scales from symmetry

The octave is 12 semitones, and 12 divides evenly by 2, 3, 4 and 6. Put a note at each cut, add the same one or two notes above every cut, and you have a scale that repeats itself every few frets. Whole-tone, diminished and augmented scales are all this one idea.

Root Cut the octave into Start from Sound

The cell: notes above each cut (the cut itself is always in)

Dominant Diminished
H-W-H-W-H-W-H-W · repeats every 3 frets

CD♭E♭EF♯GAB♭C 1♭2♯23♯456♭78HWHWHWHW

Boxed: the cuts. ▶ plays the scale up to the octave and back down over a low drone on the root; "Hear the cuts only" plays just the boxed notes, so you can hear the frame the cell is hung on.

Half-whole diminished. Fits dominant seventh chords with altered ninths.

Open in Scales & Modes

The lesson

Start with the whole-tone scale. Cut the octave into 6 parts: a note every 2 frets. That's already a scale, and it has no cell at all; it's nothing but cuts. Hear it: it floats, because every note is as far from its neighbors as every other note. Nothing pulls home.

Cut into 4, add one note. Four cuts, a minor third apart (0, 3, 6, 9 frets). Add +1 above each: eight notes, half-step whole-step all the way round. Jazz players call it the dominant diminished scale and play it over 7♭9 chords. Switch the cell to +2 and you get its sibling, the diminished (whole–half) scale.

Cut into 3. Three cuts, a major third apart. Add +3: the augmented scale, the sound of Coltrane's "Giant Steps" harmony laid flat. Try +1 instead: the same six notes started one step later, which is why it has no name of its own.

Why guitarists should care. Whatever shape you learn for a symmetric scale is the whole scale: move it up by the span between cuts (2, 3, 4 or 6 frets) and you're still inside it. One fingering, the entire neck. The diminished scale has only three different ones in existence; the whole-tone scale, two.

Cut into 2. Two cuts a tritone apart leave five slots to fill, so this is where you can make scales nobody has named. The "two-tonic" preset is one of them. Take any one you like to Scales & Modes and it will tell you if it already has a name.

Schillinger worked this out as the "third group" of pitch-scales (Theory of Pitch-Scales, chapters 6 and 7), listing every symmetric scale inside one octave. Divisions into 5, 7 and the rest aren't possible: they don't divide 12.

Bonus: how many seven-note symmetrical scales are there?

None. A scale that repeats every few frets is made of equal copies: one cell copied 2, 3, 4 or 6 times around the octave. So its number of notes has to divide by 2, 3, 4 or 6, and seven doesn't divide by any of them. Out of all 66 seven-note scales, not one repeats. That's why every seven-note scale has one home note, and why the scales we lean on most have seven notes.

There is a second kind of symmetry: a scale whose steps read the same forwards and backwards, like Dorian (W H W W W H W). Ten of the 66 seven-note scales are mirror-symmetric that way, but none of them repeats.

Practice task

Find one fingering for the dominant diminished scale in A. Play it at the 5th fret, then the 8th, the 11th, the 14th. Same notes.
Put A7 on the backing-track page and play that one shape, sliding it every 3 frets.
Journal it as "dim scale, one shape × 4 positions".

Thoughts? How can we make this better?

Erasmus cares.