Log(music) with Undertone

Whether you’re a physicist, biologist, economist, data scientist, whatever, the chances are you’ll meet the Markov chains sooner or later. And today we’ll try them on… music!

Christmas

If you’re not up to the math part, feel free to skip it.

[Originally posted here].

Math

The system states changes are represented with a help of transition matrix, describing the probabilities of possible transitions. This matrix is a key to determine the state of the model after n steps.

The math behind that is very straightforward - we need just a matrix power. Oh, wait - what if it is fractional, for example, when representing time? There’re many ways to compute it, let’s take a look at some of them.

1. Eigen vectors

This method is based on properties of eigen values decomposition:

That’s actually how Matlab mpower function looks like:

 1: [V,D] = eig(M)
 2: An = V * (D .^ n) * inv(V) 

We can take advantage of Math.NET matrix decompositions to implement this method:

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open namespaces, define constants

/// Matrix power: eigen values decomposition method
/// M^p = V * (D .^ p) / V
let mpowEig (m: SparseMatrix) p =
    let evd = m.Evd()
    let v, d = evd.EigenVectors(), evd.D()
    mapInPlace (fun x -> x ** p) d
    v * d * v.Inverse()

What are the pros/cons?

+ the method is rather efficient
- inverse matrix might not exist 1

- some matrices cannot be diagonalised:

2. Generator matrices

If we can find a matrix , such as , then (remember series?)

We are interested in finding a generator with zero row sums and nonnegative off-diagonal entries. So we’ll use Mercator’s series:

In matrix case it’s

The common concern is that the series methods are not very efficient and may be not accurate enough. It’s easy to compare the methods by computing the squared error between the actual and expected values (say, compute ).

Another problem is that a generator matrix (and power too) might have negative elements - not really what you want when dealing with probabilities. We can make them non-negative and keep the rows sums equal to zero… but the power properties won’t be preserved. For example, you can find out that .

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/// generator matrix
let genm (a: SparseMatrix) =
    let size = a.RowCount
    let mid = constDiag size 1.0
    let a_i = a - mid

    let rec logm (i, qpow: SparseMatrix) res =
        if i = MAX_ITER then res
        else
           let qdiv = qpow.Divide (float i)
           if sumSq qdiv < PREC then res + qdiv
           else 
               logm (i+1, -qpow * a_i) (res + qdiv)
    // find log(A - I)
    // update negative off-diagonal entries, the row-sum = 0
    logm (1, a_i) (zeroCreate size size) |> updateneg

/// exp series approximation
let expSeries (m: SparseMatrix)  =
    let size = m.RowCount
    
    let rec sum (i, k, mpow: SparseMatrix) res =
        if i = MAX_ITER then res
        else
            let mpowi, ki = mpow * m, k * float i
            let mdiv = mpowi.Divide ki
            if sumSq mdiv < PREC then res + mdiv
            else
                sum (i+1, ki, mpowi) (res + mdiv) 

    let mid = constDiag size 1.0
    sum (1, 1.0, mid) mid

/// Matrix power: series method
let mpowSeries m (p: float) = SparseMatrix.OfMatrix (genm m * p) |> expSeries

+ Works for non-diagonizable matrices
- Not very efficient
- May be not accurate enough or even fail to converge

These are just the most simple algorithms - would be interesting to check the others, but let’s move on.

References

  1. 19 Dubious Ways to Compute the Exponential of a Matrix
  2. Finding Generators for Markov Chains via empirical transition matrices, with applications to credit ratings

Music

It’s not a secret, that Markov models are often used for generative music, here’s one of the examples: Programming Instrumental Music From Scratch (there’re also some interesting references in the comments). We’ll do a different thing, though: given notes, we can calculate the transition probabilities and… generate new melodies.

But everything in its time.

We need to somehow define the sequences of notes and play them. And thanks to Undertone that’s pretty easy! First of all, let’s download the piano samples - University of Iowa has a collection of recordings here. You can get them manually or using this script. It’s also ok to generate the notes programmatically - but somewhat a bit more realistic is more fun ^_^

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open namespaces

// Size: 2/4

/// 1/16 (semiquaver)
[<Literal>] 
let Sq = 0.0625
/// 1/8 (quaver)
[<Literal>] 
let Q1 = 0.125
/// 3/8
[<Literal>] 
let Q3 = 0.375

...

/// Mapping between note enum and file name convention 
let noteToPianoName note =
    match note with
    | Note.C         -> "C" 
    | Note.Csharp    -> "Db"
    | Note.D         -> "D"
    | Note.Dsharp    -> "Eb"
    | Note.E         -> "E"
    | Note.F         -> "F"
    | Note.Fsharp    -> "Gb"
    | Note.G         -> "G"
    | Note.Gsharp    -> "Ab"
    | Note.A         -> "A"
    | Note.Asharp    -> "Bb"
    | Note.B         -> "B"
    | _ -> failwith "invalid note"

/// Read piano note from .aiff file
let readPianoNote() =
    let cache = Dictionary<_,_> HashIdentity.Structural
    let inline path (noteName, octave) = 
        Path.Combine(dir, "Piano.ff." + noteName + string octave + ".aiff")

    fun (note, octave) ->
        let noteKey = noteToPianoName note, octave
        match cache.TryGetValue noteKey with
        | true, wave -> wave
        | _ -> 
            let wave = IO.read (path noteKey) |> Seq.toArray
            cache.Add(noteKey, wave)
            wave

let makeNote = readPianoNote() 
let getMelody = Seq.collect (fun (noct, time) -> makeNote noct |> Seq.take (noteValue time))

The note is defined with its symbol (C = Do, F# = Fis = Fa-diesis = F sharp), octave and time (♪ - 1/8, ♬ - 2 notes * 1/16). The sequence of notes gives a melody, which can be played and even saved later. Can you guess the following one?

Original

Sounds recognizable, but awkwardly stumbling - makes me feel much better about my piano skills. I’d recommend to watch this performace, it’s amazing! (and that pianist-feeling suddenly goes down again)). I believe there’s a way to make the generated sound smoother and so on, but that’s not the goal now.

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Define commonly used notes

/// Sample melody
let ent = [
    D3,Sq; Ds3,Sq
    E3,Sq; C4,Q1; E3,Sq; C4,Q1; E3,Sq; C4,Q3
    C4,Sq; D4,Sq; Ds4,Sq
    E4,Sq; C4,Sq; D4,Sq; E4,Q1; C4,Sq; D4,Q1
    C4,Sq; D3,Sq; Ds3,Sq ...
    C4,Q3
]

Player.Play (getMelody ent)

(Note, the example above has 3 times faster tempo that the one you’ll get with downloaded notes)

The next thing to do is to calculate the transition probabilities. Consider the following sequence: E3 - 1/16, C4 - 1/8, E3 - 1/16, C4 - 3/8, C4 - 1/16.

The first option is to simply count transitions between notes and then normalize the rows, so the sum are equal to 1:


But wait, what about time? There’re different notes, we can take that into account too: if the ‘smallest’ length is 1/16, then we can count, how many such intervals each note takes, multiplying by 162:

E3 - 1/16, C4 - 1/8, E3 - 1/16, C4 - 3/8, C4 - 1/16

So, we know the transitions and times when the next note should be played. A uniform random number generator will choose this next note: e.g. if transitions are 0.4 0.2 0.1 0.3 and generated number is 0.65, that’s the 3rd one (this operation is called choose below).
The note at time t, given transition matrix P is

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/// Compute the times and probabilities of transitions between notes
let transitions xs =
    let noteToId = Seq.distinctBy fst xs |> Seq.mapi (fun i (note, _) -> note, i) |> dict

    // # of unique notes and # of notes in melody
    let n, m = noteToId.Count, List.length xs

    let matrix = Array.init n (fun _ -> Array.zeroCreate n)
    let times = Array.create m 0.

    // fill the matrix
    Seq.pairwise xs
    |> Seq.iteri (fun i ((note1, time1), (note2, _)) ->
        let n1, n2 = noteToId.[note1], noteToId.[note2]
        // option #1 - ignore times
        //   matrix.[n1].[n2] <- matrix.[n1].[n2] + 1.
        // option #2 - add extra probability for transition to the same note
        //    matrix.[n1].[n1] <- matrix.[n1].[n1] + time1 * 16.0
        //    matrix.[n1].[n2] <- matrix.[n1].[n2] + 1.0
        // option #3 - take time into account (scale by 16)
        let p = time1*16.-1. in if p > 0. then matrix.[n1].[n1] <- matrix.[n1].[n1] + p
        matrix.[n1].[n2] <- matrix.[n1].[n2] + 1.0
        
        times.[i+1] <- times.[i] + time1)
    
    normalizeRows matrix, times, Seq.toArray noteToId.Keys

let matrix, times, idToNote = transitions ent

/// Find index of the next note given probability
let transTo (ps: _[]) = ...

/// Generate music from transition matrices in a file
let genMusic matricesFileName (fstNote, idToNote: _[]) =
    // transition matrices and times
    let ps, ts = readMatrices matricesFileName
    let n = ps.Count
    let rand = System.Random()
    
    let rec gen i prevInd res =
        if i >= n-1 then List.rev res
        else
            let j = transTo ps.[i].[prevInd] (rand.NextDouble())
            let next = idToNote.[j], ts.[i]
            gen (i+1) j (next::res)

    gen 0 0 [fstNote] 

I tried to compute the transition matrix - and both method failed! Turns out this is the real-life ‘unlucky’ matrix. To be sure that doesn’t work, checked R’s expm package:

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logm(x, method = "Eigen")
Error in logm(x, method = "Eigen") : non diagonalisable matrix
logm(x, method = "Higham08")
Error in solve.default(X[ii, ii] + X[ij, ij], S[ii, ij] - sumU) :
system is computationally singular: reciprocal condition number = 0
In addition: Warning messages:
1: In sqrt(S[ij, ij]) : NaNs produced
2: In sqrt(S[ij, ij]) : NaNs produced

Fortunately, we don’t need fraction power here: if we count time in 1/16th, it becomes integer! 1/16 is 1, 3/8 is 6 and so on. Good old multiplication saves the situation.

Here’s several examples of generated melodies - there’re only 10 notes are in the game, but the results are already pretty different from the original:

Generated-1
Generated-2
Generated-3

What if we add an ‘extra-probability’ for note to stay in the same state?

E3 - 1/16, C4 - 1/8, E3 - 1/16, C4 - 3/8, C4 - 1/16

For this matrix, the generator does exist and we can programmatically compose something like that:

Generated-series-1
Generated-series-2

Why stop here? We could define the similar transitions for note length; or add chords - say, generated according to the current tonality; or switch tonalities/modes… Well, there’s a whole ‘musical math’, solfeggio, full of different functions - plenty of space to experiment!

Happy Holidays!

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let jingleBells = [
    E4,Sq; E4,Sq; E4,Q1
    E4,Sq; E4,Sq; E4,Q1
    E4,Sq; G4,Sq; C4,Sq; D4,Sq
    E4,Q1*2.0
    F4,Sq; F4,Sq; F4,Sq; F4,Sq
    F4,Sq; E4,Sq; E4,Sq; E4,Sq
    E4,Sq; D4,Sq; D4,Sq; E4,Sq
    D4,Q1; G4,Q1
]

Player.Play (getMelody jingleBells)

     

open MathNet.Numerics.LinearAlgebra.Double
open MathNet.Numerics.LinearAlgebra.Generic
open Matrix
open SparseMatrix

[<Literal>]
let MAX_ITER = 10000
[<Literal>]
let PREC = 1e-60

val mpowEig : m:SparseMatrix -> p:float -> Matrix<float>

Full name: LogMusic.mpowEig

 Matrix power: eigen values decomposition method
 M^p = V * (D .^ p) / V

val m : SparseMatrix
Multiple items
type SparseMatrix =
  inherit Matrix
  new : storage:SparseCompressedRowMatrixStorage<float> -> SparseMatrix + 6 overloads
  member CreateMatrix : numberOfRows:int * numberOfColumns:int * ?fullyMutable:bool -> Matrix<float>
  member CreateVector : size:int * ?fullyMutable:bool -> Vector<float>
  member FrobeniusNorm : unit -> float
  member IndexedEnumerator : unit -> IEnumerable<Tuple<int, int, float>>
  member InfinityNorm : unit -> float
  member IsSymmetric : bool
  member KroneckerProduct : other:Matrix<float> * result:Matrix<float> -> unit
  member LowerTriangle : unit -> Matrix<float> + 1 overload
  member NonZerosCount : int
  ...

Full name: MathNet.Numerics.LinearAlgebra.Double.SparseMatrix

--------------------
SparseMatrix(storage: MathNet.Numerics.LinearAlgebra.Storage.SparseCompressedRowMatrixStorage<float>) : unit
SparseMatrix(order: int) : unit
SparseMatrix(rows: int, columns: int) : unit

val p : float
val evd : Factorization.Evd
Matrix.Evd() : Factorization.Evd
val v : Matrix<float>
val d : Matrix<float>
Factorization.Evd.EigenVectors() : Matrix<float>
Factorization.Evd.D() : Matrix<float>
val mapInPlace : f:(float -> float) -> A:#Matrix<float> -> unit

Full name: MathNet.Numerics.LinearAlgebra.Double.Matrix.mapInPlace

val x : float
Matrix.Inverse() : Matrix<float>
val genm : a:SparseMatrix -> Matrix<float>

Full name: LogMusic.genm

 generator matrix

val a : SparseMatrix
val size : int
property Matrix.RowCount: int
val mid : SparseMatrix
val constDiag : n:int -> f:float -> SparseMatrix

Full name: MathNet.Numerics.LinearAlgebra.Double.SparseMatrix.constDiag

val a_i : SparseMatrix
val logm : (int * SparseMatrix -> Matrix<float> -> Matrix<float>)
val i : int
val qpow : SparseMatrix
val res : Matrix<float>
val MAX_ITER : int

Full name: LogMusic.MAX_ITER

val qdiv : Matrix<float>
Matrix.Divide(scalar: float) : Matrix<float>
Matrix.Divide(scalar: float, result: Matrix<float>) : unit
Multiple items
val float : value:'T -> float (requires member op_Explicit)

Full name: Microsoft.FSharp.Core.Operators.float

--------------------
type float = System.Double

Full name: Microsoft.FSharp.Core.float

--------------------
type float<'Measure> = float

Full name: Microsoft.FSharp.Core.float<_>

val sumSq : m:Matrix<float> -> float

Full name: LogMusic.sumSq

 sum of squares

val PREC : float

Full name: LogMusic.PREC

val zeroCreate : rows:int -> cols:int -> SparseMatrix

Full name: MathNet.Numerics.LinearAlgebra.Double.SparseMatrix.zeroCreate

val updateneg : m:Matrix<float> -> Matrix<float>

Full name: LogMusic.updateneg

 update negative

val expSeries : m:SparseMatrix -> Matrix<float>

Full name: LogMusic.expSeries

 exp series approximation

val sum : (int * float * SparseMatrix -> Matrix<float> -> Matrix<float>)
val k : float
val mpow : SparseMatrix
val mpowi : SparseMatrix
val ki : float
val mdiv : Matrix<float>
val mpowSeries : m:SparseMatrix -> p:float -> Matrix<float>

Full name: LogMusic.mpowSeries

 Matrix power: series method

SparseMatrix.OfMatrix(matrix: Matrix<float>) : SparseMatrix
open System.IO
open System.Collections.Generic
open Undertone
open Undertone.Waves
Multiple items
type LiteralAttribute =
  inherit Attribute
  new : unit -> LiteralAttribute

Full name: Microsoft.FSharp.Core.LiteralAttribute

--------------------
new : unit -> LiteralAttribute

val Sq : float

Full name: LogMusic.Sq

 1/16 (semiquaver)

val Q1 : float

Full name: LogMusic.Q1

 1/8 (quaver)

val Q3 : float

Full name: LogMusic.Q3

 3/8

let inline noteValue time = Time.noteValue 15. time

/// Note samples directory
[<Literal>]
let dir = @"C:\samples"

val noteToPianoName : note:Note -> string

Full name: LogMusic.noteToPianoName

 Mapping between note enum and file name convention

val note : Note
type Note =
  | Cflat = -1
  | C = 0
  | Csharp = 1
  | Dflat = 1
  | D = 2
  | Dsharp = 3
  | Eflat = 3
  | E = 4
  | Fflat = 4
  | Esharp = 5
  | F = 5
  | Fsharp = 6
  | Gflat = 6
  | G = 7
  | Gsharp = 8
  | Aflat = 8
  | A = 9
  | Asharp = 10
  | Bflat = 10
  | B = 11
  | Bsharp = 12

Full name: Undertone.Note

Note.C: Note = 0
Note.Csharp: Note = 1
Note.D: Note = 2
Note.Dsharp: Note = 3
Note.E: Note = 4
Note.F: Note = 5
Note.Fsharp: Note = 6
Note.G: Note = 7
Note.Gsharp: Note = 8
Note.A: Note = 9
Note.Asharp: Note = 10
Note.B: Note = 11
val failwith : message:string -> 'T

Full name: Microsoft.FSharp.Core.Operators.failwith

val readPianoNote : unit -> (Note * int -> float [])

Full name: LogMusic.readPianoNote

 Read piano note from .aiff file

val cache : Dictionary<(string * int),float []>
Multiple items
type Dictionary<'TKey,'TValue> =
  new : unit -> Dictionary<'TKey, 'TValue> + 5 overloads
  member Add : key:'TKey * value:'TValue -> unit
  member Clear : unit -> unit
  member Comparer : IEqualityComparer<'TKey>
  member ContainsKey : key:'TKey -> bool
  member ContainsValue : value:'TValue -> bool
  member Count : int
  member GetEnumerator : unit -> Enumerator<'TKey, 'TValue>
  member GetObjectData : info:SerializationInfo * context:StreamingContext -> unit
  member Item : 'TKey -> 'TValue with get, set
  ...
  nested type Enumerator
  nested type KeyCollection
  nested type ValueCollection

Full name: System.Collections.Generic.Dictionary<_,_>

--------------------
Dictionary() : unit
Dictionary(capacity: int) : unit
Dictionary(comparer: IEqualityComparer<'TKey>) : unit
Dictionary(dictionary: IDictionary<'TKey,'TValue>) : unit
Dictionary(capacity: int, comparer: IEqualityComparer<'TKey>) : unit
Dictionary(dictionary: IDictionary<'TKey,'TValue>, comparer: IEqualityComparer<'TKey>) : unit

module HashIdentity

from Microsoft.FSharp.Collections

val Structural<'T (requires equality)> : IEqualityComparer<'T> (requires equality)

Full name: Microsoft.FSharp.Collections.HashIdentity.Structural

val path : (string * 'a -> string)
val noteName : string
val octave : 'a
type Path =
  static val DirectorySeparatorChar : char
  static val AltDirectorySeparatorChar : char
  static val VolumeSeparatorChar : char
  static val InvalidPathChars : char[]
  static val PathSeparator : char
  static member ChangeExtension : path:string * extension:string -> string
  static member Combine : params paths:string[] -> string + 3 overloads
  static member GetDirectoryName : path:string -> string
  static member GetExtension : path:string -> string
  static member GetFileName : path:string -> string
  ...

Full name: System.IO.Path

Path.Combine(params paths: string []) : string
Path.Combine(path1: string, path2: string) : string
Path.Combine(path1: string, path2: string, path3: string) : string
Path.Combine(path1: string, path2: string, path3: string, path4: string) : string
val dir : string

Full name: LogMusic.dir

 Note samples directory

Multiple items
val string : value:'T -> string

Full name: Microsoft.FSharp.Core.Operators.string

--------------------
type string = System.String

Full name: Microsoft.FSharp.Core.string

val octave : int
val noteKey : string * int
Dictionary.TryGetValue(key: string * int, value: byref<float []>) : bool
val wave : float []
module IO

from Undertone

val read : file:string -> seq<float>

Full name: Undertone.IO.read

module Seq

from Microsoft.FSharp.Collections

val toArray : source:seq<'T> -> 'T []

Full name: Microsoft.FSharp.Collections.Seq.toArray

Dictionary.Add(key: string * int, value: float []) : unit
val makeNote : (Note * int -> float [])

Full name: LogMusic.makeNote

val getMelody : (((Note * int) * float) list -> seq<float>)

Full name: LogMusic.getMelody

val collect : mapping:('T -> #seq<'U>) -> source:seq<'T> -> seq<'U>

Full name: Microsoft.FSharp.Collections.Seq.collect

val noct : Note * int
val time : float
val take : count:int -> source:seq<'T> -> seq<'T>

Full name: Microsoft.FSharp.Collections.Seq.take

val noteValue : time:float -> int

Full name: LogMusic.noteValue

let D3 = Note.D, 3
let Ds3 = Note.Dsharp, 3
let E3 = Note.E, 3
let Fs3 = Note.Fsharp, 3
let G3 = Note.G, 3
let A3 = Note.A, 3
let C4 = Note.C, 4
let D4 = Note.D, 4
let Ds4 = Note.Dsharp, 4
let E4 = Note.E, 4

let G4 = Note.G, 4
let F4 = Note.F, 4

val ent : ((Note * int) * float) list

Full name: LogMusic.ent

 Sample melody

val D3 : Note * int

Full name: LogMusic.D3

val Ds3 : Note * int

Full name: LogMusic.Ds3

val E3 : Note * int

Full name: LogMusic.E3

val C4 : Note * int

Full name: LogMusic.C4

val D4 : Note * int

Full name: LogMusic.D4

val Ds4 : Note * int

Full name: LogMusic.Ds4

val E4 : Note * int

Full name: LogMusic.E4

E3,Sq; C4,Q1; E3,Sq; C4,Q1; E3,Sq; C4,Q3
    A3,Sq; G3,Sq
    Fs3,Sq; A3,Sq; C4,Sq; E4,Q1; D4,Sq; C4,Sq; A3,Sq
    D4,Q3; D3,Sq; Ds3,Sq
    E3,Sq; C4,Q1; E3,Sq; C4,Q1; E3,Sq; C4,Q3
    C4,Sq; D4,Sq; Ds4,Sq
    E4,Sq; C4,Sq; D4,Sq; E4,Q1; C4,Sq; D4,Q1
    C4,Q3; D4,Sq; Ds4,Sq
    E4,Sq; C4,Sq; D4,Sq; E4,Q1; C4,Sq; D4,Sq; C4,Sq
    E4,Sq; C4,Sq; D4,Sq; E4,Q1; C4,Sq; D4,Sq; C4,Sq
    E4,Sq; C4,Sq; D4,Sq; E4,Q1; C4,Sq; D4,Q1
type Player =
  private new : unit -> Player
  static member Play : sampleSource:seq<float> -> IPlayer

Full name: Undertone.Player

static member Player.Play : sampleSource:seq<float> -> IPlayer
val transitions : xs:('a * float) list -> float [] [] * float [] * 'a [] (requires equality)

Full name: LogMusic.transitions

 Compute the times and probabilities of transitions between notes

val xs : ('a * float) list (requires equality)
val noteToId : IDictionary<'a,int> (requires equality)
val distinctBy : projection:('T -> 'Key) -> source:seq<'T> -> seq<'T> (requires equality)

Full name: Microsoft.FSharp.Collections.Seq.distinctBy

val fst : tuple:('T1 * 'T2) -> 'T1

Full name: Microsoft.FSharp.Core.Operators.fst

val mapi : mapping:(int -> 'T -> 'U) -> source:seq<'T> -> seq<'U>

Full name: Microsoft.FSharp.Collections.Seq.mapi

val note : 'a (requires equality)
val dict : keyValuePairs:seq<'Key * 'Value> -> IDictionary<'Key,'Value> (requires equality)

Full name: Microsoft.FSharp.Core.ExtraTopLevelOperators.dict

val n : int
val m : int
property ICollection.Count: int
Multiple items
type List<'T> =
  new : unit -> List<'T> + 2 overloads
  member Add : item:'T -> unit
  member AddRange : collection:IEnumerable<'T> -> unit
  member AsReadOnly : unit -> ReadOnlyCollection<'T>
  member BinarySearch : item:'T -> int + 2 overloads
  member Capacity : int with get, set
  member Clear : unit -> unit
  member Contains : item:'T -> bool
  member ConvertAll<'TOutput> : converter:Converter<'T, 'TOutput> -> List<'TOutput>
  member CopyTo : array:'T[] -> unit + 2 overloads
  ...
  nested type Enumerator

Full name: System.Collections.Generic.List<_>

--------------------
List() : unit
List(capacity: int) : unit
List(collection: IEnumerable<'T>) : unit

val length : list:'T list -> int

Full name: Microsoft.FSharp.Collections.List.length

val matrix : float [] []
module Array

from Microsoft.FSharp.Collections

val init : count:int -> initializer:(int -> 'T) -> 'T []

Full name: Microsoft.FSharp.Collections.Array.init

val zeroCreate : count:int -> 'T []

Full name: Microsoft.FSharp.Collections.Array.zeroCreate

val times : float []
val create : count:int -> value:'T -> 'T []

Full name: Microsoft.FSharp.Collections.Array.create

val pairwise : source:seq<'T> -> seq<'T * 'T>

Full name: Microsoft.FSharp.Collections.Seq.pairwise

val iteri : action:(int -> 'T -> unit) -> source:seq<'T> -> unit

Full name: Microsoft.FSharp.Collections.Seq.iteri

val note1 : 'a (requires equality)
val time1 : float
val note2 : 'a (requires equality)
val n1 : int
val n2 : int
val normalizeRows : m:float [] [] -> float [] []

Full name: LogMusic.normalizeRows

 Normalize rows, so the row-sums are equal to 1

property IDictionary.Keys: ICollection<'a>
val matrix : float [] []

Full name: LogMusic.matrix

val times : float []

Full name: LogMusic.times

val idToNote : (Note * int) []

Full name: LogMusic.idToNote

val transTo : ps:float [] -> (float -> int)

Full name: LogMusic.transTo

 Find index of the next note given probability

val ps : float []
let last = ps.Length-1
    let rec findj j v =
        if ps.[j] >= v || j = last then j
        else findj (j+1) (v - ps.[j])
    findj 0
val genMusic : matricesFileName:string -> fstNote:('a * float) * idToNote:'a [] -> ('a * float) list

Full name: LogMusic.genMusic

 Generate music from transition matrices in a file

val matricesFileName : string
val fstNote : 'a * float
val idToNote : 'a []
val ps : List<ResizeArray<float []>>
val ts : float []
val readMatrices : path:string -> List<ResizeArray<float []>> * float []

Full name: LogMusic.readMatrices

 Read transition matrices and times

property List.Count: int
val rand : System.Random
namespace System
Multiple items
type Random =
  new : unit -> Random + 1 overload
  member Next : unit -> int + 2 overloads
  member NextBytes : buffer:byte[] -> unit
  member NextDouble : unit -> float

Full name: System.Random

--------------------
System.Random() : unit
System.Random(Seed: int) : unit

val gen : (int -> int -> ('a * float) list -> ('a * float) list)
val prevInd : int
val res : ('a * float) list
val rev : list:'T list -> 'T list

Full name: Microsoft.FSharp.Collections.List.rev

val j : int
System.Random.NextDouble() : float
val next : 'a * float
val jingleBells : ((Note * int) * float) list

Full name: LogMusic.jingleBells

val G4 : Note * int

Full name: LogMusic.G4

val F4 : Note * int

Full name: LogMusic.F4

  1. the example matrix isn’t actually a transition matrix - the row sums should be equal to 1 - it’s here just to demonstrate a possible problem.

  2. the sequence C4 - 1/8, E3 - 1/16 is almost the same as C4 - 1/16, C4 - 1/16, E3 - 1/16. That gives us 50/50 transitions: after each period of time C4 can either stay as C4 or transform to E3.

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