Posts tagged: music

QuasiMusic is a Java applet that generates music, functioning a lot like a player-piano roll, by exploiting the patterns in quasiperiodic tilings of the 2 dimensional plane. The animations show two different instances of the applet in action.

These are tilings that fill the entire infinite plane using different shapes called tiles that are arranged in a certain way with out overlapping each other and without leaving any empty space.

If its possible to take a copy of a tiling and shift it over in some direction by a certain amount relative to the original tiling and have it line up exactly with the original, then that tiling is periodic. This minimum shifting distance needed to make the tiling match itself is called the period.

Whats special about quasiperiodic tilings is that there is no such period. So no matter which direction you shifted the tiling, it would never exactly line up with itself.

Using these quasiperiodic tilings has interesting consequences for the music they generate. Since the pattern never repeats, this implies that the music will never repeat. Although there may be moments which sound similar they can never be exactly the same when considered in a longer time interval.

To understand how the applet creates the music from a given tiling first assign to each type of tile a sound (instrument and pitch). Next, imagine a series of vertical lines passing over the tiling. Then as the whole tiling moves upwards passing a horizontal axis, play a sound each time one type of tiling changes to another type on one of these vertical lines. In the animations, the tiles light up when they trigger a sound, and the series of vertically arranged white dots specify when the tiles change. These white dots are reminiscent of the holes in a player-piano roll.

The applet allows the user to control many different parameters, and their are over 350 different instruments and sounds to choose from literally allowing for endless possibilities. Check it out here.

Also check out a recently released dark ambient album which makes use of QuasiMusic on some tracks by master mathematics expositor John Baez.

Making use of music theory, group theory, and category theory

From Musical Actions of Dihedral Groups

Abstract:
The sequence of pitches which form a musical melody can be transposed or inverted. Since the 1970s, music theorists have modeled musical transposition and inversion in terms of an action of the dihedral group of order 24. More recently music theorists have found an intriguing second way that the dihedral group of order 24 acts on the set of major and minor chords. We illustrate both geometrically and algebraically how these two actions are {\it dual}. Both actions and their duality have been used to analyze works of music as diverse as Hindemith and the Beatles.

Summary:
This paper connects the twelve musical tones to elements in the dihedral group of order 24 (the symmetries of a regular dodecagon). The translation from pitch classes to integers modulo 12 allows for the modeling of musical works using abstract algebra. The first action on major and minor chords described in the paper is based on the musical techniques of transposition and inversion. A transposition moves a sequence of pitches up or down and an inversion reflects a melody about a fixed axis. The other action arises from the P, L, and R operations of the 19th-century music theorist Hugo Riemann. It is through these operations that the dihedral group of order 24 acts on the set of major and minor triads. The paper also describes how the P, L, and R operations have beautiful geometric presentations in terms of graphs. In particular the authors describe a connection between the PLR-group and chord progressions in Beethoven’s 9th Symphony, which leads to a proof that the PLR-group is dihedral. Another musical example is Pachelbel’s Canon in D. In summary, the paper gives a very pretty explanation of what we commonly hear in tonal music in terms of elementary group theory.

matthen:

Pick a whole number, if it is even divide it by 2, if it is odd then multiply it by 3 and add 1. Repeat. For example, 12 is even, so goes to 6, then 3 which is odd, so goes to 10, then 5, then 16, then 8, 4, 2, 1.  Your number will have also gone down to 1 (I imagine).  Will the numbers always go down to 1? This seemingly simple question hasn’t been answered yet, and is called the Collatz conjecture.  The image shows my visualisation of how the first few hundred numbers whittle down to 1.  It is such an easy question to ask, but we can’t prove it yet; a good example to show there is still plenty of work to be done in maths! [more]

I made a post about this a while back, and also posted a song that supposedly represents the “oneness of the first 300 integers”. However, I have not been able to figure out the mapping which takes the integers to the sequence of notes in the song.I was thinking the oneness of some integer (mod 8), would get assigned to some pitch in some scale (perhaps A minor).Can anyone make sense of this?Here is the sheet music created from a midi of the song.

matthen:

Pick a whole number, if it is even divide it by 2, if it is odd then multiply it by 3 and add 1. Repeat. For example, 12 is even, so goes to 6, then 3 which is odd, so goes to 10, then 5, then 16, then 8, 4, 2, 1.  Your number will have also gone down to 1 (I imagine).  Will the numbers always go down to 1? This seemingly simple question hasn’t been answered yet, and is called the Collatz conjecture.  The image shows my visualisation of how the first few hundred numbers whittle down to 1.  It is such an easy question to ask, but we can’t prove it yet; a good example to show there is still plenty of work to be done in maths! [more]


I made a post about this a while back, and also posted a song that supposedly represents the “oneness of the first 300 integers”. However, I have not been able to figure out the mapping which takes the integers to the sequence of notes in the song.
I was thinking the oneness of some integer (mod 8), would get assigned to some pitch in some scale (perhaps A minor).
Can anyone make sense of this?
Here is the sheet music created from a midi of the song.


Listen to the oneness of the first 300 integers