Transformation Groups

As pointed out by Leon Harkleroad in his fine introduction to mathematics and music, The Math Behind the Music (p. 54):  "Any time you analyze how things, particularly operations, combine with each other, groups probably lurk in the background.  So, it's really not to extraordinary that groups crop up in music."  Some examples that we have discussed here, are the dilations and translations (group transformation operations in the time- frequency plane) that occur in Stravinsky's Firebird Suite, and in a Chinese folk melody.  There is also the "clock time" group Z_12 playing a role in some musical illusions related to Shepard's tones (including the famous melodic introduction to a Bach piece).