# Music And Geometry In 19C, there was a project called Erlangen program. It basically means that geometry is studying invariants under given transformations. For Euclidean geometry, the transformations consist of translations, rotations, and reflections. They constitute an Euclidean group acting on an Euclidean space. Euclidean geometry is studying the properties which do not change by those transformations. Distances, angles, areas, &c. do not change regardless of how you translate things, rotate things, and reflect things. Therefore Euclidean geometry is all about distances, angles, areas, &c. Also you can add additional transformations to the group. Then it gives you an expansion of the geometry. You can add transformations like scaling, sheer mapping, &c. to an Euclidean group. These transformations are called affine transformations. They make rectangles into parallograms and circles into ellipses. Affine transformations give you affine geometry. Although rectangles became parallograms, the parallel lines are still parallel. But the group no more preserves things like distances. Hence affine geometry is all about lines, parallelism, &c. In general the larger the group becomes, the fewer invariants remain. And vice versa. This kind of relations is prevalent all over mathematics. They are usually called Galois connections. There are things that are exactly like this in music. One of it appears in the twelve-tone technique. The twelve-tone technique is an atonal method to make all 12 tones (C C# D ... A# B) appear equally often. To avoid any "center" in music. Decentralisation. A permutation of the 12 tones is called a twelve-tone row. Serial music consists of tone rows. Here is a tone row. It is a famous one from "Concerto for Nine Instruments, Op.24" by Anton Webern. Since there is no "center", it is simply written with numbers. B B- D E- G F# A- E F C C# A 0 e 3 4 8 7 9 5 6 1 2 t t=10, e=11. To avoid double digits. Also again there is no "center". So the first one just becomes 0. People transform a row into another row with these transformations. This is analogous to the group of transformations in geometry. ## Transformations * Prime: The original tone row 0 e 3 4 8 7 9 5 6 1 2 t * Transposition: Add the same number. 3 2 6 7 e t 0 8 9 4 5 1 (+3) t 9 1 2 6 5 7 3 4 e 0 8 (-2) * Retrograde: The reverse order t 2 1 6 5 9 7 8 4 3 e 0 * Inversion: (+) <-> (-). For example -3 (= +9) instead of +3. 0 1 9 8 4 5 3 7 6 e t 2 * Retrograde Inversion: Inversion then Retrograde. 2 t e 6 7 3 5 4 8 9 1 0 These are permutations. So these preserve the "equally often"-ness. And since consonance is a thing about two adjacent notes, these preserve consonance and dissonance. Symmetry and conservation. And this is exactly what Bach did in his masterpieces of tonal music. His works are full of these transformations in addition to other transformations of tonal music. Like Bach, these are relevant in tonal music too. An example is that (Inversion) = (Complement) in the classical sense. So (Prime) + (Inversion) = 0 (mod 12). 0+0 = 1+e = 3+9 = ... = t+2 = 12 = 0 (mod 12) In tonal music, Perfect <-> Perfect Major <-> Minor Augmented <-> Diminished Accute <-> Grave by inversion. For example the complement of C-F is F-C'. So the complement of the perfect fourth is the perfect fifth. Perfect <-> Perfect. By the way why this tone row is famous is that Webern made the rest of it just by applying the transformations to the first 3 notes. 0 e 3 | 4 8 7 | 9 5 6 | 1 2 t 0 e 3 (-1 +4) Prime 4 8 7 (+4 -1) Retrograde Inversion with (+7) transposition 9 5 6 (-4 +1) Retrograde with (+6) transposition 1 2 t (+1 -4) Inversion with (+1) transposition The transpositions are to make it a twelve-tone row. The things inside the parentheses are called interval vectors. Those mean the dispositions between the pitches. It is a very geometric concept. There are other transformations that do not necessarily preserve the twelve-tone. * Multiplication: Multiply the same number. 0 e 3 4 8 7 9 5 6 1 2 t (*1) 0 9 9 0 0 9 3 3 6 3 6 6 (*3) 0 7 3 8 4 e 9 1 6 5 t 2 (*5) 0 1 9 8 4 5 3 7 6 e t 2 (* -1) Note that Prime = (*1), Inversion = (* -1). Considering multiplication, numbers in Z12 that are relatively prime with 12 are 1,5,7,11. So only (*1), (*5), (*7), (*11) map a twelve-tone row to another twelve-tone row. Only they are permutations. Also 5,7 mean the perfect fourth and the perfect fifth. They obviously do not preserve dissonance. For example (0 7) from the row of (*5) is a perfect fifth. But the original (0 e) is dissonant. Accepting transformations like these might be analogous to expanding to affine geometry from Euclidean geometry. I don't know. * Rotation: The final note comes the first. t 0 e 3 4 8 7 9 5 6 1 2 This is analogous to the rotations of an Euclidean group. Rotation was tried by composers like Stravinsky. But it does not preserve dissonance. The first note and the final note are not necessarily dissonant. Because of this, there is retrograde instead. Then why do things like these exist? As I wrote on the post "The Best Introduction To Representation Theory", every finite group is isomorphic to a subgroup of a matrix group consisting of permutation matrices. Because of Cayley's theorem. Which is a special case of the Yoneda lemma. And you can regard a matrix as a coordinate of a linear map. Hence you can regard it as a transformation acting on a space. That is geometric.