Symmetries in Music / in Time / in 1-D data

Melodies are present with different types of symmetries:

RSR — a mirror, a flip

SRG SRG — a repetition

NS , GM since both pairs are half a note apart and ascending

I would like each type of symmetry to have a vocabulary term.

Is there a categorization for the symmetries?

Similar to space groups and Bravais lattices

In the case of 2-D and 3-D symmetries associated with geometrical objects / crystals — the constraint, “space-filling”, i.e. no empty space between objects limits the number of symmetries possible.

There might be a limit if space-filling constraint is not enforced.

Where does the analogy with space symmetries end?

Why is symmetry interesting?

Symmetry is work half-done. Just take the same thing and apply some transformation, and there is a new thing. A tree branch on the other side.

A pattern.

Entanglement is probably because of symmetry.

Degenerate states are symmetric.

In symmetry, something is the same, something is different. The same offers a portal into what’s different. The familiarity offers a path to the unfamiliar.

Symmetric / skewed datasets. Categorization of datasets leads to insight about processes (ex. growth of cities. Population of cities has a skewed distribution. Ownership of financial assets.)

Is there a technique to find all possible symmetries in time-series data? Some library for that like stumpy for finding repetitions / (FT / autocorrelation) — certain limitations in those.

RSR with symmetric time is different than RSR if the second R is extended.

What are terms that describe these melodic elements.

Symmetry — which makes us see the same entity in different light.

Symmetry — constraint condition satisfied in different ways. Degenerate states.

Symmetry — something familiar and something unfamilar integrated into the symmetric entity.