As music is composed by people with great talent, it doesn't seem reasonable to expect to use mathematical tools and techniques to explore music, and to potentially even create music. However, there is in fact a field called Algorithmic Music Composition, that tries to do exactly that.
For this purpose, a useful music file format is the midi format, which contains information on which notes should be played at what times. Of course, it handles chords and allows multiple tracks to be overlaid. GarageBand, or similar software, takes a midi file and gives a clear visual depiction of this, as well as playing the music. Note that midi files denote each note by an integer from 0 to 127, known as the midi key.
At its simplest, and we shall be making lots of simplifying assumptions, a piece of music consists of a sequence of notes (12 semi-tones per octave) and the length of time that each note should be played. We are ignoring chords and assuming there is just a melody, as we shall throughout this project. We shall also ignore rhythm completely and consider only the sequence of notes plsyed, not the timing.
Therefore, if we consider the set of possible notes as the state space, then a piece of music can be thought of as the realisation of a stochastic process; in this case a discrete-time process. Given how a midi file denotes a note, our state space would naturally be the set of midi keys, S = {0, 1, 2, ... , 127}. So each new note can be considered to be randomly chosen from S. How the new note is chosen depends on the type of stochastic process. In this project, we shall explore a few different stochastic processes, with increasing complexity.
In your stochastic process model, let X,, E S,n = 1, 2, ... be the midi key of then'" note to be played. For your particular piece of music answer the following questions, giving appropriate explanations (give any numerical answers to 4 decimal places).
2) Assume that each new midi key is chosen randomly, depending only on the current midi key, and independently of all previous mid keys.
(a) Argue that this stochastic process is Markovian.
(b) Use the piece of music you were given to determine the proportion of times that the midi key j immediately follows the midi key i, for all i,j E S. What did you do with the last midi-key in your piece of music? Think of two ways you could treat this key, and discuss the fundamental consequences of both choices.
(c) Use this to construct a P-matrix representation for the evolution of your piece of music, including all states i E S.
(i) Are there any zero columns, and if so, explain why and say how many? What do the associated rows look like? Do you have any choice in this?
(ii) Ignoring any states associated with zero columns, classify the character of all remain¬ing states.
(d) What are the probabilities P(X, = 60IX0 = i) for i e {49, 50, , 72}?
(e) What are the probabilities P(X,o = 601X5 = i) for i E {49, 50,..., 72}?
(f) Write down the equations that determine the expected time until the first Middle-C (midi key 60) is played, given that you start your music with mis key i E S .
(g) Solve these equations for all i e S.