// thinkbeforecoding

Ukulele Fun for XMas !

2015-12-17T10:44:43 / jeremie chassaing

This post is part of the F# Advent Calendar in English 2015 project. Check out all the other great posts there! And special thanks to Sergey Tihon for organizing this.

Hi something fun and not too technical for end the year !

As everyone knows, the favorite instrument of Santa Claus is Ukulele ! So let's play some music, and especialy some Ukulele !

First thing first, let's create functions for notes. We start with C at octave 0, and have a progression by half tones.

So C is 0, D is 2, E is 4.

Since there is only a half tone between E and F, F is 5.

F is 7, A is 9, B is 11, and we reach next octave at 12, which is C 1 :

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open System

let C n = 12 * n
let D n = C n + 2
let E n = C n + 4
let F n = C n + 5
let G n = C n + 7
let A n = C n + 9
let B n = C n + 11 

For sharps and flat, lets define two functions that had and remove a half tone

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let sharp n = n + 1
let flat n = n - 1

We can now create names for each note :

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let Cd = C >> sharp
let Db = D >> flat
let Dd = D >> sharp
let Eb = E >> flat
let Fd = F >> sharp
let Gb = G >> flat
let Gd = G >> sharp
let Ab = A >> flat
let Ad = A >> sharp
let Bb = B >> flat

There is no E sharp or F flat because it is F and E respectively, same thing for B and C...

Will create a structure with a custome comparison/equality that doesn't take the octave into account by using a 12 modulus, this will prove usefull to work with chords:

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[<Struct>]
[<CustomComparison>]
[<CustomEquality>]
[<StructuredFormatDisplay("{Display}")>]
type Note(note : int) =
    member __.Note = note
    
    override __.GetHashCode() = note % 12 

    override __.Equals other =
        match other with
        | :? Note as other ->
            note % 12 = other.Note % 12
        | _ -> false

    static member names = 
        [| "C"
           "C#"
           "D"
           "D#"
           "E"
           "F"
           "F#"
           "G"
           "G#"
           "A"
           "A#"
           "B" |]
    member __.Display = 
        let name = Note.names.[note % 12]
        let octave = note / 12
        sprintf "%s %d" name octave

    override this.ToString() = this.Display
        
    interface IEquatable<Note> with
        member __.Equals other =
            note % 12 = other.Note % 12
    interface IComparable<Note> with
        member __.CompareTo other =
            compare (note % 12) (other.Note % 12) 
    interface IComparable with
        member __.CompareTo other =
            match other with
            | :? Note as other -> 
                compare (note % 12) (other.Note % 12)
            | _ -> 1 

    static member (+) (string: Note, fret: int) =
        Note (string.Note + fret)

let notes = List.map Note

Ukulele Strings

A Ukulele has 4 strings.

The funy thing is that the 1st one is higher than the second one, where on most string instruments strings are in progressive order.

This is simply due to the limited size of the Ukulele, a low first string would not sound good, so it is adjusted to the next octave.

This gives use the following:

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let strings = notes [G 4;C 4;E 4; A 4]

Chords

Instead of hard-encoding ukulele chords, we will compute them !

So a bit of theory about chords.

Chords are defined by their root note and the chord quality (major, minor).

The chords start on the root note, and the chord quality indicates the distance to other notes to include in the chord.

On string instrument, the order and the height of the actual notes are not really important for the chord to be ok. So we can use a note at any octave.

Now, let's define the chord qualities.

First, Major, uses the root note, 3rd and 5th, for instance for C, it will be C, E, G, which gives intervals of 0, 4 and 7 half tones from root.

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let quality = notes >> Set.ofList

let M n = quality [n ; n + 4; n+7] 

Then, Minor, uses the root note, the lower 3rd and 5th. For C it will be C, E flat, G, so intervals of 0, 3 and 7 half tones for root.

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let m n = quality [n; n + 3; n+7] 

The 7th adds a 4th note on the Major:

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let M7 n = quality [n; n + 4; n+7; n+11 ]

Frets

As on a gitare, a ukulele has frets, places where you press the string with your finger to change the tone of a string.

0 usually represent when you don't press a string at all, and pinching the string will play the string note.

When pressing fret 1, the note is one half tone higher, fret 2, two half tone (or one tone) higher.

So pressing the second fret on the C 4 string give a D 4.

Our first function will try pressing on frets to find frets for notes that belong to the chord

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let findFrets chord (string: Note) =
    [0..10]
    |> List.filter (fun fret -> 
        Set.contains (string + fret) chord)
    |> List.map (fun fret -> fret, string + fret)

The result is list of pair, (fret, note) that can be used on the strnig

The second function will explore the combinaison of frets/note and keep only those that contains all notes of the chords.

Ex: for a C Major chord, we need at least a C, a E and a G.

using frets 0 on string G, 0 on string C, 3 on string E, and 3 on string A, we get G, C, G, C.

All notes are part of the chord, but there is no E... not enough. 0,0,0,3 is a better solution.

The function explore all possible solution by checking notes on string that belong to the chord, and each time remove a note from the chord. At the end, there should be no missing note.

At each level sub solutions are sorted by a cost. Standard Ukulele chords try to place fingers as close to the top as possible. So lewer frets are better.

The cost function for a chords is to sum square of frets. If there is any solution, we keep the one with the lowest cost.

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let rec filterChord chord missingNotes solution stringFrets  =
    match stringFrets with
    | [] -> 
        if Set.isEmpty missingNotes then 
            Some (List.rev solution)
        else
            None
    | string :: tail -> 
        string
        |> List.filter (fun (_,note) -> 
            chord |> Set.contains note)
        |> List.choose (fun (fret,note) -> 
            filterChord chord (Set.remove note missingNotes) ((fret,note) :: solution) tail)
        |> List.sortBy(fun s -> 
            List.sumBy (fun (fret,_) -> fret*fret) s)
        |> List.tryHead
       

making a cord is now simple.

Compute the note in the chord using quality and root.

For each string, map possible frets the belong to the chord, then filter it.

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let chord root quality =    
    let chord = quality (root 4)
    strings
    |> List.map (findFrets chord)
    |> filterChord chord chord []
    |> Option.get
    

We can now try with classic chords:

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let CM = chord C M

and the result is:

[(0, G 4); (0, C 4); (0, E 4); (3, C 5)]

Now C minor:

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let Cm = chord C m

which is exactly what you can find on a tab sheet:

[(0, G 4); (3, D# 4); (3, G 4); (3, C 5)]
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chord D m
    
chord A M
chord A m

chord G m
chord E M

Printing chords

To print chords, we will simply use pretty unicode chars, and place a small 'o' on the fret where we should place fingers:

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let print chord  =
    let fret n frt = 
        if n = frt then 
            "o" 
        else 
            "│"
    let line chord n  =
            chord 
            |> List.map (fst >> fret n)
            |> String.concat ""       
    printfn "┬┬┬┬"
    [1..4] 
    |> List.map (line chord)
    |> String.concat "\n┼┼┼┼\n" 
    |> printfn "%s"

Let's try it

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chord C M |> print

It prints

┬┬┬┬
││││
┼┼┼┼
││││
┼┼┼┼
│││o
┼┼┼┼
││││

Another one

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chord G M |> print

and we get

┬┬┬┬
││││
┼┼┼┼
│o│o
┼┼┼┼
││o│
┼┼┼┼
││││

Playing chords

We can also play chords using NAudio.

You can find NAudio on nuget.org

For simplicity I will use the midi synthetizer:

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// #r @"../packages\NAudio\lib\net35\NAudio.dll"

// open NAudio.Midi
// let device = new MidiOut(0)
// MidiOut.DeviceInfo 0
// let midi (m:MidiMessage) =  device.Send m.RawData
module MidiMessage =
    let StartNote(_,_,_) = ()
    let StopNote(_,_,_) = ()

let midi = ignore

let startNote note volume = 
    MidiMessage.StartNote(note, volume, 2) |> midi

let stopNote note volume = 
    MidiMessage.StopNote(note, volume, 2) |> midi

// let sleep n = System.Threading.Thread.Sleep(n: int)

Now we can define a function that will play a chord.

The tempo is used as a multiplicator for a the chord length.

Longer tempo means slower.

For better result we introduce an arpegio, a small delay between each note. Don't forget to remove this time from the waiting length...

The direction indicate if the cords are strumed Up, or Down. In the Up case we reverse the chord.

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type Direction = Dn of int | Up of int

let play tempo arpegio (chord, strum)  =
    let strings, length = 
        match strum with 
        | Dn length -> chord, length
        | Up length -> List.rev chord, length 

    strings
    |> List.iter (fun (_,(n: Note)) -> 
        startNote n.Note 100 ; (*sleep arpegio*) )

    let arpegioLength = 
        List.length chord * arpegio

    // sleep (length * tempo - arpegioLength)

    strings
    |> List.iter (fun (_,(n: Note)) -> 
        stopNote n.Note 100 )

To strum a chord, we give a list of length, and a chord, and it will apply the cord to each length:

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let strum strm chord =
    let repeatedChord = 
        strm 
        |> List.map (fun _ -> chord)
    
    List.zip repeatedChord strm

Now here is Santa Clause favorite song, Get Lucky by Daft Punk.

First the chords :

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let luckyChords = 
    [ //Like the legend of the Phoenix,
      chord B m
      // All ends with beginnings.
      chord D M
      // What keeps the planets spinning,
      chord (Fd) m
      // The force from the beginning.
      chord E M ]

Then strum, this is the rythm used to play the same chord, it goes like, Dam, Dam, Dam Dala Dam Dam:

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let luckyStrum = 
    [ Dn 4; Dn 3; Dn 2; Dn 1; Up 2; Dn 2; Up 2]

and the full song :

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let getLucky =
    luckyChords
    |> List.collect (strum luckyStrum)

And now, let's play it :

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getLucky
|> List.replicate 2
|> List.concat
|> List.iter (play 130 25)

And the tab notations for the song !

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luckyChords
|> List.iter print
┬┬┬┬
││││
┼┼┼┼
│ooo
┼┼┼┼
││││
┼┼┼┼
o│││
┬┬┬┬
││││
┼┼┼┼
ooo│
┼┼┼┼
││││
┼┼┼┼
││││
┬┬┬┬
│o││
┼┼┼┼
o│o│
┼┼┼┼
││││
┼┼┼┼
││││
┬┬┬┬
o│││
┼┼┼┼
│││o
┼┼┼┼
││││
┼┼┼┼
│o││

Conclusion

I hope this small thing was entertaining and that it'll get you into ukulele !

For excercise you can:

  • implements more chords
  • Better printing
  • add more liveliness and groove by adding some jitter to the strum...
  • add the lyrics for Karaoke !
  • try with other songs !
  • try the same for a 6 string gitar !

Now it's your turn to rock !

namespace System
val C : n:int -> int
val n : int
val D : n:int -> int
val E : n:int -> int
val F : n:int -> int
val G : n:int -> int
val A : n:int -> int
val B : n:int -> int
val sharp : n:int -> int
val flat : n:int -> int
val Cd : (int -> int)
val Db : (int -> int)
val Dd : (int -> int)
val Eb : (int -> int)
val Fd : (int -> int)
val Gb : (int -> int)
val Gd : (int -> int)
val Ab : (int -> int)
val Ad : (int -> int)
val Bb : (int -> int)
Multiple items
type StructAttribute =
  inherit Attribute
  new : unit -> StructAttribute

--------------------
new : unit -> StructAttribute
Multiple items
type CustomComparisonAttribute =
  inherit Attribute
  new : unit -> CustomComparisonAttribute

--------------------
new : unit -> CustomComparisonAttribute
Multiple items
type CustomEqualityAttribute =
  inherit Attribute
  new : unit -> CustomEqualityAttribute

--------------------
new : unit -> CustomEqualityAttribute
Multiple items
type StructuredFormatDisplayAttribute =
  inherit Attribute
  new : value:string -> StructuredFormatDisplayAttribute
  member Value : string

--------------------
new : value:string -> StructuredFormatDisplayAttribute
Multiple items
type Note =
  struct
    interface IComparable
    interface IComparable<Note>
    interface IEquatable<Note>
    new : note:int -> Note
    override Equals : other:obj -> bool
    override GetHashCode : unit -> int
    override ToString : unit -> string
    member Display : string
    member Note : int
    static member names : string []
    ...
  end

--------------------
Note ()
new : note:int -> Note
val note : int
Multiple items
val int : value:'T -> int (requires member op_Explicit)

--------------------
type int = int32

--------------------
type int<'Measure> = int
val __ : inref<Note>
val other : obj
val other : Note
property Note.Note: int
val name : string
property Note.names: string []
val octave : int
val sprintf : format:Printf.StringFormat<'T> -> 'T
val this : inref<Note>
type IEquatable<'T> =
  member Equals : other:'T -> bool
Multiple items
type IComparable =
  member CompareTo : obj:obj -> int

--------------------
type IComparable<'T> =
  member CompareTo : other:'T -> int
val compare : e1:'T -> e2:'T -> int (requires comparison)
Multiple items
val string : Note

--------------------
type string = String
val fret : int
val notes : (int list -> Note list)
Multiple items
module List

from Microsoft.FSharp.Collections

--------------------
type List<'T> =
  | ( [] )
  | ( :: ) of Head: 'T * Tail: 'T list
    interface IReadOnlyList<'T>
    interface IReadOnlyCollection<'T>
    interface IEnumerable
    interface IEnumerable<'T>
    member GetSlice : startIndex:int option * endIndex:int option -> 'T list
    member Head : 'T
    member IsEmpty : bool
    member Item : index:int -> 'T with get
    member Length : int
    member Tail : 'T list
    ...
val map : mapping:('T -> 'U) -> list:'T list -> 'U list
val strings : Note list
val quality : (int list -> Set<Note>)
Multiple items
module Set

from Microsoft.FSharp.Collections

--------------------
type Set<'T (requires comparison)> =
  interface IReadOnlyCollection<'T>
  interface IComparable
  interface IEnumerable
  interface IEnumerable<'T>
  interface ICollection<'T>
  new : elements:seq<'T> -> Set<'T>
  member Add : value:'T -> Set<'T>
  member Contains : value:'T -> bool
  override Equals : obj -> bool
  member IsProperSubsetOf : otherSet:Set<'T> -> bool
  ...

--------------------
new : elements:seq<'T> -> Set<'T>
val ofList : elements:'T list -> Set<'T> (requires comparison)
val M : n:int -> Set<Note>
val m : n:int -> Set<Note>
val M7 : n:int -> Set<Note>
val findFrets : chord:Set<Note> -> string:Note -> (int * Note) list
val chord : Set<Note>
val filter : predicate:('T -> bool) -> list:'T list -> 'T list
val contains : element:'T -> set:Set<'T> -> bool (requires comparison)
val filterChord : chord:Set<'a> -> missingNotes:Set<'a> -> solution:(int * 'a) list -> stringFrets:(int * 'a) list list -> (int * 'a) list option (requires comparison)
val chord : Set<'a> (requires comparison)
val missingNotes : Set<'a> (requires comparison)
val solution : (int * 'a) list (requires comparison)
val stringFrets : (int * 'a) list list (requires comparison)
val isEmpty : set:Set<'T> -> bool (requires comparison)
union case Option.Some: Value: 'T -> Option<'T>
val rev : list:'T list -> 'T list
union case Option.None: Option<'T>
Multiple items
val string : (int * 'a) list (requires comparison)

--------------------
type string = String
val tail : (int * 'a) list list (requires comparison)
Multiple items
val string : value:'T -> string

--------------------
type string = String
val note : 'a (requires comparison)
val choose : chooser:('T -> 'U option) -> list:'T list -> 'U list
val remove : value:'T -> set:Set<'T> -> Set<'T> (requires comparison)
val sortBy : projection:('T -> 'Key) -> list:'T list -> 'T list (requires comparison)
val s : (int * 'a) list (requires comparison)
val sumBy : projection:('T -> 'U) -> list:'T list -> 'U (requires member ( + ) and member get_Zero)
val tryHead : list:'T list -> 'T option
val chord : root:(int -> 'a) -> quality:('a -> Set<Note>) -> (int * Note) list
val root : (int -> 'a)
val quality : ('a -> Set<Note>)
module Option

from Microsoft.FSharp.Core
val get : option:'T option -> 'T
val CM : (int * Note) list
val Cm : (int * Note) list
val print : chord:(int * 'a) list -> unit
val chord : (int * 'a) list
val fret : ('b -> 'b -> string) (requires equality)
val n : 'b (requires equality)
val frt : 'b (requires equality)
val line : (('b * 'c) list -> 'b -> string) (requires equality)
val chord : ('b * 'c) list (requires equality)
val fst : tuple:('T1 * 'T2) -> 'T1
Multiple items
type String =
  new : value:char[] -> string + 8 overloads
  member Chars : int -> char
  member Clone : unit -> obj
  member CompareTo : value:obj -> int + 1 overload
  member Contains : value:string -> bool + 3 overloads
  member CopyTo : sourceIndex:int * destination:char[] * destinationIndex:int * count:int -> unit
  member EndsWith : value:string -> bool + 3 overloads
  member Equals : obj:obj -> bool + 2 overloads
  member GetEnumerator : unit -> CharEnumerator
  member GetHashCode : unit -> int + 1 overload
  ...

--------------------
String(value: char []) : String
String(value: nativeptr<char>) : String
String(value: nativeptr<sbyte>) : String
String(value: ReadOnlySpan<char>) : String
String(c: char, count: int) : String
String(value: char [], startIndex: int, length: int) : String
String(value: nativeptr<char>, startIndex: int, length: int) : String
String(value: nativeptr<sbyte>, startIndex: int, length: int) : String
String(value: nativeptr<sbyte>, startIndex: int, length: int, enc: Text.Encoding) : String
val concat : sep:string -> strings:seq<string> -> string
val printfn : format:Printf.TextWriterFormat<'T> -> 'T
val StartNote : 'a * 'b * 'c -> unit
val StopNote : 'a * 'b * 'c -> unit
val midi : ('a -> unit)
val ignore : value:'T -> unit
val startNote : note:'a -> volume:'b -> unit
val note : 'a
val volume : 'b
module MidiMessage

from Ukulele
val stopNote : note:'a -> volume:'b -> unit
type Direction =
  | Dn of int
  | Up of int
union case Direction.Dn: int -> Direction
union case Direction.Up: int -> Direction
val play : tempo:'a -> arpegio:int -> chord:('b * Note) list * strum:Direction -> unit
val tempo : 'a
val arpegio : int
val chord : ('b * Note) list
val strum : Direction
val strings : ('b * Note) list
val length : int
val iter : action:('T -> unit) -> list:'T list -> unit
val n : Note
val arpegioLength : int
val length : list:'T list -> int
val strum : strm:'a list -> chord:'b -> ('b * 'a) list
val strm : 'a list
val chord : 'b
val repeatedChord : 'b list
val zip : list1:'T1 list -> list2:'T2 list -> ('T1 * 'T2) list
val luckyChords : (int * Note) list list
val luckyStrum : Direction list
val getLucky : ((int * Note) list * Direction) list
val collect : mapping:('T -> 'U list) -> list:'T list -> 'U list
val replicate : count:int -> initial:'T -> 'T list
val concat : lists:seq<'T list> -> 'T list