Illustrative usage of Group, Act and Torsor: manipulation of musical intervals.

The musical distance between two musical notes is a musical interval.

Intervals can be compounded and inverted, so they form a Group.

Notes can be translated by a given interval, which is an Act of intervals on notes.

There's a unique musical interval taking any note to any other given one, so notes are a torsor under intervals.

This functionality is useful in providing enharmonically correct voicings of chords.

We begin by defining note names, which are acted upon by the cyclic group of order 7.

In this case we used DerivingVia to derive the action of C 7, using the CyclicEnum newtype created for this exact purpose.

Note the use of DerivingVia to transfer algebraic operations from Sum Int.

For non-newtypes, one can use generics, for example:

data Klein4 = Klein4 ( C 2 ) ( C 2 )
  deriving stock Generic
  deriving ( Semigroup, Monoid, Group )
    via Generically Klein4

This uses the Generically newtype from the generic-data library.

An interval is represented as a number of scale steps to take (relative to the major scale), together with an additional alteration to apply.

For instance, a major third is two steps up (diatonic steps relative to the root in a major scale):

> Steps ( Sum 2 ) Natural
major 3rd up

A minor sixth is 5 steps up, and then a flat:

> Steps ( Sum 5 ) Flat
minor 6th up

The smart constructor Interval is also provided that is more intuitive to use:

> Interval 3 Natural
major 3rd up

> Interval 7 Flat
minor 7th up

Note that the Semigroup/Group operations on intervals are not the obvious ones, e.g.:

> Steps ( Sum 2 ) Natural
major 3rd up

> Steps ( Sum (-2) ) Natural
minor 3rd down

> inverse ( Steps ( Sum 2 ) Natural )
Steps ( Sum (-2) ) Flat
major 3rd down

To define algebraic operations on intervals, we use an equivariant bijection to the product group ( Sum Int, Sum Int ).

Note that ( Sum Int, Sum Int ) is automatically a Semigroup, Monoid and Group using the product structure.

Example chords:

> majorTriad <&> ( • Note C Natural 4 )
[C4,E4,G4]

> diminished7th <&> ( • Note G Sharp 3 )
[G#3,B3,D4,F4]

> minor11th <&> ( • Note D Natural 3 )
[D3,A3,E4,F4,C5,G5]

Example scales:

> phrygian <&> ( • Note E Natural 3 )
[E3,F3,G3,A3,B3,C4,D4,E4]

> phrygian <&> ( • Note C Sharp 3 )
[C#3,D3,E3,F#3,G#3,A3,B3,C#4]

> lydian <&> ( • Note C Natural 4 )
[C4,D4,E4,F#4,G4,A4,B4,C5]

> wholeTone <&> ( • Note G Natural 5 )
[G5,A5,B5,C#6,Eb6,F6]

End of main example code.

Follows: helper code for reading/showing musical notes and intervals.