Functional Modelling of Musical Harmony

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#Functional Modelling of Musical Harmony
[An Introduction to HarmTrace](http://dreixel.net/research/pdf/fmmh.pdf)
.footnote[By [Soares Chen](http://github.com/soareschen) (Ruo Fei) (U064984M)]

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# Introduction to HarmTrace

- Harmony Analysis and Retrieval of Music with Type-level Representations of Abstract Chords Entities
- Encode musical structure at the type level
- Implemented in Haskell
- Limited to western jazz and pop music

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# Harmony Basics

- Tone
- Note
- Semitone
- Chord
- Root
- Dominant
- Subdominant

## Demo

http://www.multiplayerpiano.com/cs4215

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# Example Chord Progression

- I:Maj, IV:Maj, V:Maj, I:Maj

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# Top Level Types

- A piece has a list of phrases
- A phrase is either tonic or dominant
- Tonic is a chord of I:Maj degree
- Dominant has a main chord of degree V:Maj
  - Preceeded by any number of subdominant
- Subdominant is a chord of IV:Maj degree

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# Degree Type - 1st Attempt

- Naiive approach - allow degree to be encoded in any level and class
- Example
  - V:Maj = Degree 5 Maj
  - IV:Min = Degree 4 Min
  - VI:Dom7 = Degree 6 Dom7

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# Problem - Allow Bad Chords

- I:Maj, III:Maj, V:Maj, I:Maj
- III:Maj is neither tonic, dominant, nor subdominant

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# Degree Type - 2nd Attempt

- Haskell Phantom Type

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# Degree/Deg Conversion

- Make use of Haskell Type Classes
- ToRoot - convert roman numeral data type into integer
- ToClass - identity type?
- Key techique - hide the constructor for bad degrees
- Example - `instance ToRoot II` is hidden
  - Not possible to construct piece containing II:Maj

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# Secondary Dominant

- Use Haskell GADT to encode degree at type level
  - GADT = Generalised Algebraic Data Types
- Degree can either be a base degree or a ConsV
- Base degree - just the dominant itself
- ConsV - The subdominant chord V/δ where δ is the δth scale degree
- Basically calculation of the subdominant chord, but how?

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# Type Family V/

- HarmTrace approach: brute force table using Haskell's type family
- Actual musical approach:
  - Get the note that is fifth from the root of current degree
  - Use that note as root and form a dominant 7th degree.
- Example: Subdominant of V:Dom7 is II:Dom7
  - V + 5 = II
  - V, VI, VII, I, II
- Possible to calculate in Haskell at type level?

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# Example Tree

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# Example Command

## Input

```
C:maj
C:maj;1 F:maj;1 D:7;1 G:7;1 C:maj;1
```

## Command

```bash
$ harmtrace parse -i sample.txt -g jazz 
```

## Input Syntax and Manual

http://www.cs.uu.nl/wiki/GenericProgramming/HarmTrace#Input_file_syntax

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# HarmTrace Result

```
[Piece
  [PT
    [T_1_1[I_1[ C:maj ]]]]
  [PD
    [D_1_1
      [S_3_1[IV_1[ F:maj ]]]
      [D_2_1
        [V/V_1
          [II7_1[ D:7 ]]]
        [V7_1[ G:7 ]]]]]
  [PT
    [T_1_1
      [I_1[ C:maj ]]]]]

```
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# Parser

- Parser implemented using Haskell applicative functor
  - `<|>`, `<$>`, `<*>`
- Use `uu-parsinglib` library to build parser
- Use `instant-generics` library to reduce boilerplates
- Constructor order prioritize ambigious rules
- No left-recursive datatypes in HarmTrace

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# Adhoc Parser

- Parsing is trivial for cases like `Ton` and `Dom`
- Only non-generic part of parser is Degree_Final
  - Use of `ToDegree` and `ToClass`
- Parsing type-level natural number is "undecidable" for GHC
  - Solution: introduce base case instance
  - <img src="images/base-case.png">

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# Chord Similarity

- Expressing music at type level enable use of generic tools
  - Parser
  - Pretty Printer
  - Diff tools

- Generic `diff` for comparing harmonic similarity
- Based on four primitive generic functions
  - `children` - list of all children of term
  - `build` - build term from children
  - `eqCon` - compare equality of terms at constructor
  - `typeOf` - get type of a term

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# Thank you!

Slide available at http://soareschen.github.io/harmtrace-slides

Slideshow created using [remark](http://github.com/gnab/remark).