enlambdment / scale-degrees

Representing Euterpea pitches as scale degrees with respect to a tonic key & scale mode

scale-degrees

Goal

Extend the PitchClass / Pitch / etc. data types available via the Haskell library Euterpea with additional functionality for calculating, given a pitch and choice of tonic / scale mode, which scale degree of the scale the pitch is (including any accidentals, if applicable.)

Examples

Intermediate newtype PitchClassEquiv and type alias OctaveEquiv

> c0 = (C, 0) :: Pitch
> b_1 = (Bs, -1) :: Pitch
> absPitch c0
12
> absPitch b_1
12

Observe how both c0 and b_1, values of the Euterpea newtype Pitch, have the same absolute pitch (measured in semitone distance from the lowest nonnegative pitch, i.e. for (C, -1).) This is in spite of the fact that the second pair element for each, the Octave (Int) part, differs. This is by design and has to do in particular with the Euterpea datatype PitchClass, and how its pairings with Octave get mapped to absolute pitch values.

I sought an alternative representation in which

• the PitchClasses would have a canonical representation as Int modulo 12, in line with the typical chromatic circle picture whereby e.g. C / B-sharp / D-double-flat all lie at "zero", then C-sharp / B-double-sharp / D-flat at "one", etc. etc. until the pitch classes B / A-double-sharp / C-flat lie at "eleven"
• pairings of this new version of PitchClass with some newtype of Octave bear the following relationship to their absolute pitch: (chromatic-circle representation of PitchClass) + (12 * (Octave - 1)) == absolute semitone pitch where the term (Octave - 1) is in order to respect the convention that the lowest-sounding instance having non-negative semitone count, for most Euterpea pitch classes, lies in octave #(-1).

Certain tasks for calculating semitone distances are simplified by using this representation, where the second pair element o of (p, o) :: (PitchClassEquiv, OctaveEquiv) indicates (regardless of the pitch class part) that the pitch in question has semitone value lying within the range 12 * (o + 1) <= semitone value < 12 * (o + 2).

toScaleDegreeRepr

Given the following data:

• an input pitch, in Euterpea representation (i.e. as pair (PitchClass, Octave))
• a tonic pitch class, as PitchClassEquiv (i.e. using the alternative pitch-class representation for which there is a mapping into the chromatic circle of mod-12 integers defined)
• a scale mode (presently supporting the two principal musical modes of major, MajorMode, and natural minor, MinorMode, as ScaleMode values at this time)

the function toScaleDegreeRepr will convert the input Pitch into a ScalePitch. The algebraic data type (ADT) ScalePitch is defined so as to convey the following information:

• which 'scale octave' the pitch in question lies in. This is calculated with respect to the tone (tonic :: PitchClassEquiv, -1 :: OctaveEquiv) as the 1st scale degree of the (-1)'st scale (which is followed by the 0th scale, the 1st scale, etc.)
• what 'scale degree' the pitch in question relates to, either as an exact match or a modification up / down by one or more accidentals. + / - marks following the scale degree indicate semitones up (sharps) or down (flats), respectively
• what tonic and mode this scale-degree position is being reported with respect to.

Using this representation on the two Pitch values defined above, we can see that e.g.:

> toScaleDegreeRepr c0 (MkPCE A) MajorMode
-1th octave S3- (A MajorMode)
> toScaleDegreeRepr b_1 (MkPCE A) MajorMode
-1th octave S2+ (A MajorMode)

which informs us that both of these pitches lie within the (-1)'st octave of the A major scale (using the convention that this octave of the scale starts from (MkPCE A, -1).) Furthermore, while (C,0) is a lowering of the 3rd scale degree in A major (namely, C-sharp), the alternative spelling for this pitch using B-sharp is in fact a raising of the 2nd scale degree in this scale (namely, B-natural.)

Validation

To validate that this representation in terms of scale degrees is reasonable, i.e. expresses pitches with respect to the most typically encountered key signatures - those with tonic spelled using up to one sharp or flat - and in a manner that is consistent with the semitone-count calculation for pitches, toAbsPitchRepr is defined on ScalePitch values, and this computation of semitone count for ScalePitch form of arbitrary pitches is compared with what Euterpea computes using its own absPitch function on the original pitches, before being re-cast into this alternative form: this property test is called chkScaleDegreeRepr and currently is confirmed to work for all the typical key signatures mentioned above.

(The implementation as well as validation for functionality e.g. deriving the spellings for all diatonic scale-degree pitch classes of a given scale, casting into scale-degree representation, etc. had to be limited to these typical key signatures because the PitchClass datatype which this work is based upon includes only pitch-class spellings containing up to 2 sharps or flats, whereas a datatype supporting pitch-class spellings for sharpened / flattened pitch classes with respect to totally arbitrary key signatures would in fact have to be infinite in principle.)