Type triads as C, F♯, Bb major, or c, g#, e♭ minor (also Cm, C-, Cmaj). Uppercase means major and lowercase means minor unless you add a quality. P, L and R keep two common tones and move one note by a half step (P, L) or a whole step (R). S keeps one common tone (the third), N keeps one (the major chord's root), and H keeps none, moving every voice by a half step. Each counts as a single move. Piano: Salamander Grand Piano by Alexander Holm (CC BY 3.0).
Each transformation turns a major triad into a minor one (or back) while holding as many notes as it can and moving the rest by step. Pick a start triad and every example, diagram and recording below follows it.
| Move | As P, L, R | Result | Holds | Moves | Semitones |
|---|
P, L and R are Riemann's basic relations as Hyer and Cohn recast them: flip the triangle across one of its edges. The two notes on that edge stay put and the third note moves by step.
Each of these is a chain of three primary moves treated as one step. Formulas read left to right: LPR means L, then P, then R. On the diagrams the thin line traces that chain, and outlined notes are the ones held.
Two primary moves in a row keep the mode and move the root. Four pairs give the chromatic mediants (roots a third apart, one common tone) and two give the fifth relations.
Alternate two moves and you eventually come back to where you started. On the flat lattice each cycle runs along a strip and ends on another copy of the start triad; on the torus the two ends are the same face.
Piano samples: Salamander Grand Piano by Alexander Holm, CC BY 3.0.
Major and minor divide the octave unevenly, which is what gives them a home note. The other scales here divide it into equal, repeating segments, so they sound the same from several starting notes and have no single center. Pick a tonic, then hear each scale up and down.
| Scale | Notes | Steps (semitones) | Transpositions | Major / minor triads |
|---|
All four share the tonic, second, fourth and fifth. They differ in the third, sixth and seventh degrees. Major and natural minor on the same tonic are a P move apart at the tonic triad (C ↔ c). Natural minor uses the same notes as the major scale a minor third higher, its relative major, an R move away (c ↔ E♭).
Eight notes alternating half and whole steps. The pattern repeats every minor third, so there are only three different octatonic collections. Starting on a half step or a whole step gives the two forms, which are modes of each other.
Six notes, all a whole step apart. It repeats every whole step, so there are only two whole-tone collections, and every note is equally the “tonic.”
Six notes alternating minor thirds and half steps. It is two augmented triads a half step apart, it repeats every major third, and there are four hexatonic collections.
Olivier Messiaen catalogued seven scales in Technique de mon langage musical (1944) that divide the octave into identical segments. Because of that symmetry each can be transposed only a few times before it lands back on the same notes, and it has only as many modes as there are notes in one segment. He heard this built-in limit as a “charm of impossibilities.” Modes 1 and 2 are the whole-tone and octatonic scales above. On each clock, dashed spokes mark where one segment ends and the next begins.
Spellings: major and minor scales use one letter per degree. The symmetric scales have no standard spelling, so they use common note names. Piano samples: Salamander Grand Piano by Alexander Holm, CC BY 3.0.