Some scales are symmetric when transposed: that is, shifting by some number of steps smaller than 12 leaves the scale unchanged. They're known as modes of limited transposition because the number of distinct transpositions of the scale is smaller than the number of notes it has. Unwieldy and unintuitive, so we'll call them "symmetric scales" on this site.
For the most part I don't find this collection of scales musically interesting (a few classical composers have been drawn to the way their ambiguity makes for smooth modulations). However, they're a complication when counting scales, and enumerating them is an interesting enough problem to discuss.
Here are some examples: the diminished 7th chord shows 4-fold symmetry while the augmented scale shows 3-fold and that other thing has only 2-fold symmetry. When a scale has n-fold symmetry, that means there are n ways to transpose it that leave it unchanged. What about the number of distinct transpositions t? For example, in the augmented scale t=4 with the transpositions shown below.
Each of the t scales has n-fold symmetry for itself, giving n×t total transpositions. Between these, all 12 possible transpositions must be accounted for—it has to land on one of the t unique transpositions! So n×t is 12, meaning than n is a divisor of 12 or 0 = n|12. All right, let's list the possible values of n.
Divisors ← { / 0 = (↕𝕩) | 𝕩 } ⊢ p12 ← Divisors 12 # Proper divisors ⟨ 1 2 3 4 6 ⟩ ⊢ d12 ← p12 ∾ 12 # All divisors ⟨ 1 2 3 4 6 12 ⟩
A scale is symmetric when n>1 or equivalently t<12. One-fold symmetry is no symmetry at all.
To find the total number of symmetric scales, we could add up the number for each t<12, that is, t∊p12. Since a scale with t distinct repetitions repeats every t steps, we might expect that the number of scales is 2⋆t. That is, we choose which notes out of the first t appear, and all other choices are made for us. This doesn't work though: suppose we set t←4 and choose 1‿0‿1‿0. We get the following scale with only 2 unique tranpositions!
Every scale that repeats after t steps also repeats after k×t steps. The number of scales with t=4 is really the number that repeat after 4 steps, minus the number of scales with t=2 and t=1, the divisors of 4. The arithmetic gets complicated quickly but is easily expressed with recursion. No base case needed, because 1 has no divisors!
SymCount ← {(2⋆𝕩) - +´ 𝕊¨ Divisors 𝕩} SymCount 4 12
Here's how many total scales fit into each classification. The number of scales with any degree of symmetry is the total number of scales, minus the non-symmetric ones where t=12.
sym12 ← SymCount¨ d12 d12 ≍ sym12 ┌─ ╵ 1 2 3 4 6 12 2 2 6 12 54 4020 ┘ (2⋆12) - SymCount 12 76
So there are 12 scales with 4 unique transpositions each? Not quite, just 3 of them. Our system counts a new one for each starting point, so we should divide by 4 if we want to treat transpositions as equivalent.
After dividing we get a new breakdown of the number of scale classes, displayed below. We also tack on a cumulative sum: 17 symmetric classes and 352 total classes. The total of 352 is one higher than the one we derived in the tertian scale page because this method counts the scale with no notes.
↗️d12 ∾ (⊢ ≍ +`) sym12 ÷ d12 ┌─ ╵ 1 2 3 4 6 12 2 1 2 3 9 335 2 3 5 8 17 352 ┘
Following the method on that page, we can normalize all scales (including those that don't contain the root) and group them by the number of unique transpositions.
↗️Normalize ← { ⊑ ∨ ⌽⟜𝕩¨ ↕≠𝕩 } ≠ scales ← ⍷ Normalize¨ ⥊↕12⥊2 352 p12 ≍˘ >¨ ¯1 ↓ (+˝¬∨`p12 (⌽≡⊢)⌜ scales) ⊔ scales ┌─ ╵ 1 ┌─ ╵ 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 ┘ 2 ┌─ ╵ 1 0 1 0 1 0 1 0 1 0 1 0 ┘ 3 ┌─ ╵ 1 0 0 1 0 0 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0 1 1 0 ┘ 4 ┌─ ╵ 1 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 1 0 1 1 1 0 1 1 1 0 ┘ 6 ┌─ ╵ 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 1 0 1 0 0 0 1 0 1 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 0 0 1 0 1 1 0 0 1 0 1 1 0 1 0 0 1 1 0 1 0 0 1 1 1 1 0 0 1 1 1 1 0 0 1 1 1 0 1 0 1 1 1 0 1 0 1 1 1 1 1 0 1 1 1 1 1 0 ┘ ┘
All except the 6-fold symmetry are quite boring: just clusters of notes repeated at intervals! However, four of the 6-fold symmetric scales break this pattern.
Out of these, the first three are all subsets of the octatonic scale (although in a twist the French 6th is also a subset of the whole-tone scale; it's kind of its own thing). The last one is a pretty complex beast. Also it can be traversed as a cycle of 4-step and 5-step intervals (4-4-5-5-4-4-5-5) and that's… something?