I am going to be talking about a specific symmetry found in music, that being the symmetry of a piano or keyboard.
First, let me explain what I mean by symmetry of a piano. If we start with the C key, the order is C, C#, D, D#, E, F, F#, G, G#, A, A#, B, C. So, if you played the set of those keys from left to right, you would hear those tones. But if we used a reflection symmetry about the center (F#), and then played left to right, you would hear C, B, A#, A, G#, G, F#, F, E, D#, D, C#, C. This is simply a permutation of the set of keys. Using the set of the entire piano (there are 88 keys), if we numbered each of them from left to right, the permutation would be (88 1)(87 2)(86 3)…(45 44). Looking at a sheet of music then, any note with a high pitch would now have a low pitch because, say we read an E. We would play the key that corresponds to an E normally, but with the symmetry, it would correspond to an F. The switching of these keys is represented by (44 45), or any other transposition that corresponds to the switching of an E and an F.
So, what is the significance of this symmetry? First, the key of a musical piece is changed. If the piece is in the key of D with two sharps, playing that piece with the reflection will be in the key of Bb which has two flats. If you are familiar with the Circle of Fifths, playing the reflected song just reflects the key across the vertical axis of the Circle of Fifths. This is the Circle of Fifths: (http://files.meetup.com/227762/Circle%20of%20Fifths%20wheel.gif). In addition to the key, the mood of the piece is changed. If playing the first five notes of a major scale, the mirror will produce the first five notes of the corresponding minor scale in the reverse order. So if the original is ascending, the mirror will be descending. Therefore, playing C, D, E, F, G will produce A, G, F, E, D. Chords also switch from the major to the corresponding minor. A C-major switches to an A-minor. So, in general, a happy song will sound sad and a sad song will sound happy.
I really thought this was a cool application of symmetry and permutations that we have been learning about. While I was browsing the internet, and I found something called a Keyboard Mirror. For newer keyboards, this device is plugged into the keyboard and will basically perform the transposition for each note that is played. But even without this device, you can perform the transposition by hand on each note that you are playing and then play the new song that is formed. Either way, you are able to hear what a permutation does to music, which is pretty cool.