Tonnetz (p,r,L) transformations from C Maj to Cm, to Em, to Am, etc .
This is “Group” Transformation in disguise in 1800 AD, 20 Years before Galois invented Group Theory ?
Gemini 3 confirms my suspicion above: Tonnetz by Euler is a Group Action in disguise.
——-Gemini 3 Reply —-
It is a brilliant observation to connect Euler’s Speculum Musicum (the basis for the Tonnetz) with the formal birth of Group Theory. While Euler wasn’t intentionally writing down group axioms in the late 1700s, he was absolutely working with the algebraic structure that Galois would later formalize.
Technically, you are right: Euler’s work on musical intervals and transformations describes a group in everything but name.
The Mathematical “Disguise”
Leonhard Euler’s Tentamen novae theoriae musicae (1739) and his later work on the Tonnetz (the “tone web”) utilized what we now call a Group Action.
Here is how Euler’s work pre-dated Galois by using group-like properties:
* *Closure* : Euler recognized that moving by a perfect fifth or a major third always landed you on another note within the pitch-class space.
* *Inverses* : He understood that moving “up” a fifth could be undone by moving “down” a fifth.
* *Commutativity* : On a 2D Tonnetz, moving a Major Third then a Perfect Fifth gets you to the same place as a Fifth then a Major Third. This describes an *Abelian Group* .
*Why it wasn’t “Group Theory” yet* ?
While Euler was using the structure, he lacked the Group abstraction.
*The Neo-Riemannian Connection*
The specific transformations you mentioned—P (Parallel), R (Relative), and L (Leittonwechsel)—were actually formalized much later by Hugo Riemann and modern theorists like David Lewin.
However, these transformations form the *PRL Group* . If we look at them through Euler’s lens:
* They are *involutions* 内卷: Applying P twice (P²) returns you to the original chord, meaning P = P-¹.
* They operate on a set of 24 major and minor triads.
Euler was essentially navigating the Cayley Graph of a group before Arthur Cayley or Évariste Galois were even born. He had the “map” (the Tonnetz), but Galois provided the “laws of motion” that explained why the map worked.
1584 AD 朱载堉 (明朝 太子) invented 12 tones (copied by French Jesuit 白晋 in China 康熙 Court -> informed Germany Lebniz-> taught Sebastian Bach who implemented it in first modern piano today).
Any closed system of musical transformations has implicit group structure (just like the circle of fifths is ℤ/12ℤ).
with 4 operations : {+ – × ÷}
is none other than Identity Automorphism (mirror image of itself).
and its conjugate. (a+ib) conjugate with (a-ib)
is right coset,
Math Online Tom Circle 