How do I use a markov chain matrix to generate music?

Question

I have a matrix (square - nxn) <--Depends on how many chords there are.

------------------------------------------
            Transition Matrix: (FROM)
------------------------------------------
 A     Bm   C#m    D     E    F#m    G#   
------------------------------------------
0.07  0.00  0.00  0.00  0.66  0.14  0.86 | A   | 
0.15  0.17  0.00  0.05  0.07  0.30  0.00 | Bm  | 
0.00  0.00  0.00  0.00  0.00  0.37  0.00 | C#m | 
0.19  0.00  0.39  0.16  0.14  0.00  0.00 | D   | (TO)
0.52  0.83  0.29  0.43  0.00  0.19  0.00 | E   | 
0.07  0.00  0.32  0.36  0.14  0.00  0.14 | F#m | 
0.00  0.00  0.00  0.00  0.00  0.00  0.00 | G#  |

This matrix gets multiplied by another matrix which is:

This means the starting note is an A (1st row corresponds to A).

The next note is determined by the column of the Markov Transition Matrix which says there is a:

7% chance to become A

15% chance to become Bm

0% chance to become C#m, etc. etc.

(Done by matrix multiplication, using numPy).

So the resulting matrix is a 1col x nrows matrix.

This matrix will look like this:

0.07
0.15
0.00
0.19
0.52
0.07
0.00

^ This means that there is a 0.07% that the next note is an A, etc.

^ Using these probabilities can I play back those notes? Or generate a midi file based on it?

I should only get back one note (plus the first one), and based on the probability it should be an E.

Which midi library are you using to generate the music from the notes? If all you want to do is generate the markov chain from the transition matrix, [this question](https://stackoverflow.com/questions/59858123/) may help. — CDJB, Feb 05 '20 at 11:28
@CDJB I don't know, I'm new to python, anything simple is fine. Thanks for the link for the random generator. — LightninBolt74, Feb 05 '20 at 12:18

eraoul · Accepted Answer · 2020-02-23T04:31:53.697

I wrote a small program to demonstrate Markov generation, including output to a MIDI file using the music21 library.

Note that your transition matrix has two problems:

The probabilities in the 5th column do not sum to 1.00. I replaced 0.66 with 0.65 to make it work.
This is a matrix of chord transition probabilities, not note transitions. So my example involves chords (triads), not individual notes. Your question uses the word "note" many times where I think you meant "chord" instead, because all your examples involve chords such as B minor.

I wrote a program to repeat each chord 4 times, so that each chord lasts an entire measure of 4 beats.

I picked C major as the default instead of your A major. If you want to generate chords in a different key you can change the scale root in the calls to chord_to_names to chord_to_midi, using the "root_midi" parameter. For instance, call it with 57 instead of the default 60 to get A major.

Once you run this program it will generate chords.mid, which you can open in a program such a MuseScore or a midi editor.

Sample output:

Generated chords: [0, 0, 1, 4, 0, 3, 4, 5]
Note names:
['C', 'E', 'G']
['C', 'E', 'G']
['D', 'F', 'A']
['G', 'B', 'D']
['C', 'E', 'G']
['F', 'A', 'C']
['G', 'B', 'D']
['A', 'C', 'E']
MIDI Notes:
[60, 64, 67]
[60, 64, 67]
[62, 65, 69]
[55, 59, 62]
[60, 64, 67]
[53, 57, 60]
[55, 59, 62]
[57, 60, 64]

Which looks like this rendered in MuseScore:

import numpy as np

from music21.chord import Chord
from music21.stream import Part
from music21.midi.translate import streamToMidiFile


CHORD_TRANSITIONS = np.matrix([
  #              Transition Matrix: (FROM)
  # ---------------------------------------------
  #  I     ii     iii    IV      V     vi    viio   
  # ---------------------------------------------
    [0.07,  0.00,  0.00,  0.00,  0.65,  0.14,  0.86],  # | I    | 
    [0.15,  0.17,  0.00,  0.05,  0.07,  0.30,  0.00],  # | ii   | 
    [0.00,  0.00,  0.00,  0.00,  0.00,  0.37,  0.00],  # | iii  | 
    [0.19,  0.00,  0.39,  0.16,  0.14,  0.00,  0.00],  # | IV   | (TO)
    [0.52,  0.83,  0.29,  0.43,  0.00,  0.19,  0.00],  # | V    | 
    [0.07,  0.00,  0.32,  0.36,  0.14,  0.00,  0.14],  # | vi   |  
    [0.00,  0.00,  0.00,  0.00,  0.00,  0.00,  0.00]]) # | viio |  

CHORD_TRANSITIONS = CHORD_TRANSITIONS.T  # swap axes for simpler lookup based on current chord

TRIADS = [
  (0, 4, 7),  # I
  (2, 5, 9),  # ii
  (4, 7, 11),  # iii
  (-7, -3, 0),  # IV
  (-5, -1, 2),  # V
  (-3, 0, 4),  # vi
  (-1, 2, 5),  # viio
] 

PC_TO_NAME = ('C', 'C#', 'D', 'Eb', 'E', 'F', 'F#', 'G', 'Ab', 'A', 'Bb', 'B')

def midi_to_pc(midi_note):
  return midi_note % 12

def midi_to_name(midi_note):
  return PC_TO_NAME[midi_to_pc(midi_note)]

def chord_to_midi(chord_index, root_midi=60):  # chord_index is from 0 to 6
  return [root_midi + x for x in TRIADS[chord_index]]

def chord_to_names(chord_index, root_midi=60):  # chord_index is from 0 to 6
  return [midi_to_name(note) for note in chord_to_midi(chord_index, root_midi)]

num_chords = len(CHORD_TRANSITIONS)

def generate(start_chord=0, length=8):
  cur = start_chord
  chords = [cur]
  for _ in range(length - 1):
    probs = CHORD_TRANSITIONS[cur,:].tolist()[0]
    chord = np.random.choice(num_chords, p=probs)
    chords.append(chord)
    cur = chord
  return chords

chords = generate()

print(f'Generated chords: {chords}')

print(f'Note names:')
for chord in chords:
  print(chord_to_names(chord))

print(f'MIDI notes:')
for chord in chords:
  print(chord_to_midi(chord))

print('Converting to MIDI')
part = Part()
for chord in chords:
  midis = chord_to_midi(chord)

  # Repeat each chord 4 times.
  for i in range(4):  
    part.append(Chord(midis, quarterLength=1))

mf = streamToMidiFile(part)

mf.open('chords.mid', 'wb')
mf.write()
mf.close()

Hello, I hope my answer/code was useful to you. If your question has been answered, could you please make sure to mark the answer as accepted? — eraoul, Mar 05 '20 at 02:04

How do I use a markov chain matrix to generate music?

1 Answers1