Music.Pitch.Common.Interval

Provides standard intervals.

Quality

data Quality

Interval quality is either perfect, major, minor, augmented, and diminished. This representation allows for an arbitrary number of augmentation or diminishions, so augmented is represented by Augmented 1, doubly augmented by Augmented 2 and so on.

The quality of a compound interval is the quality of the simple interval on which it is based.

Constructors

Major

Minor

Perfect

Augmented Integer

Diminished Integer

Instances

Eq Quality

Ord Quality

Show Quality

HasQuality Quality

class HasQuality a where

Methods

quality :: a -> Quality

Instances

HasQuality Interval

HasQuality Quality

invertQuality :: Quality -> Quality

Invert a quality.

Perfect is unaffected, major becomes minor and vice versa, augmented becomes diminished and vice versa.

isPerfect :: HasQuality a => a -> Bool

Returns whether the given quality is perfect.

isMajor :: HasQuality a => a -> Bool

Returns whether the given quality is major.

isMinor :: HasQuality a => a -> Bool

Returns whether the given quality is minor.

isAugmented :: HasQuality a => a -> Bool

Returns whether the given quality is augmented (including double augmented etc).

isDiminished :: HasQuality a => a -> Bool

Returns whether the given quality is diminished (including double diminished etc).

Number

data Number

The number portion of an interval (i.e. second, third, etc).

Note that the interval number is always one step larger than number of steps spanned by the interval (i.e. a third spans two diatonic steps). Thus number does not distribute over addition:

 number (a + b) = number a + number b - 1

Instances

Enum Number

Eq Number

Integral Number

Num Number

Ord Number

Real Number

Show Number

HasNumber Number

class HasNumber a where

Methods

number :: a -> Number

Returns the number portion of an interval.

The interval number is negative if and only if the interval is negative.

See also quality, octaves and semitones.

Instances

HasNumber Interval

HasNumber Number

unison :: Number

A synonym for 1.

prime :: Number

A synonym for 2.

second :: Number

A synonym for 3.

third :: Number

A synonym for 4.

fourth :: Number

A synonym for 5.

fifth :: Number

A synonym for 6.

sixth :: Number

A synonym for 7.

seventh :: Number

A synonym for 8.

octave :: Number

A synonym for 9.

ninth :: Number

A synonym for 10.

tenth :: Number

A synonym for 11.

twelfth :: Number

A synonym for 13.

duodecim :: Number

A synonym for 14.

thirteenth :: Number

A synonym for 15.

fourteenth :: Number

A synonym for 16.

fifteenth :: Number

A synonym for 17.

Intervals

data Interval

An interval is the difference between two pitches, incuding negative intervals.

Intervals and pitches can be added using .+^. To get the interval between two pitches, use .-..

 c .+^ minor third == eb
 f .-. c           == perfect fourth

Adding intervals preserves spelling. For example:

 m3 ^+^ _M3 = _P5
 d5 ^+^ _M6 = m10

The scalar type of Interval is Integer, using ^* to stack intervals of a certain type on top of each other. For example _P5 ^* 2 is a stack of 2 perfect fifths, or a major ninth. The Num instance works as expected for +, negate and abs, and (arbitrarily) uses minor seconds for multiplication. If you find yourself *, or signum on intervals, consider switching to *^ or normalized.

Intervals are generally described in terms of Quality and Number. To construct an interval, use the interval constructor, the utility constructors major, minor, augmented and diminished, or the interval literals:

 m5  == minor   fifth    == interval Minor   5
 _P4 == perfect fourth   == interval Perfect 5
 d5  == diminished fifth == diminish (perfect fifth)

Instances

Eq Interval

Num Interval

Ord Interval

Show Interval

Typeable Interval

Monoid Interval

IsInterval Interval

Augmentable Interval

Semigroup Interval

VectorSpace Interval

AdditiveGroup Interval

HasSemitones Interval

HasNumber Interval

HasQuality Interval

Creating intervals

mkInterval :: Quality -> Number -> Interval

Creates an interval from a quality and number.

Given Perfect with an number not indicating a perfect consonant, interval returns a major interval instead. Given Major or Minor with a number indicating a perfect consonance, interval returns a perfect or diminished interval respectively.

perfect :: Number -> Interval

Creates a perfect interval. If given an inperfect number, constructs a major interval.

major :: Number -> Interval

Creates a major interval. If given a perfect number, constructs a perfect interval.

minor :: Number -> Interval

Creates a minor interval. If given a perfect number, constructs a diminished interval.

augmented :: Number -> Interval

Creates an augmented interval.

diminished :: Number -> Interval

Creates a diminished interval.

doublyAugmented :: Number -> Interval

Creates a doubly augmented interval.

doublyDiminished :: Number -> Interval

Creates a doubly diminished interval.

Inspecting intervals

octaves :: Interval -> Octaves

Returns the non-simple part of an interval.

 (perfect octave)^*x + y = z  iff  y = simple z

isNegative :: Interval -> Bool

Returns whether the given interval is negative.

isPositive :: Interval -> Bool

Returns whether the given interval is positive.

isNonNegative :: Interval -> Bool

Returns whether the given interval is non-negative. This implies that it is either positive or a perfect unison.

isPerfectUnison :: Interval -> Bool

Returns whether the given interval a perfect unison.

isStep :: Interval -> Bool

Returns whether the given interval is a step (a second or smaller).

Only diatonic number is taken into account, so _A2 is considered a step and m3 a leap, even though they have the same number of semitones.

isLeap :: Interval -> Bool

Returns whether the given interval is a leap (larger than a second).

Only the diatonic number is taken into account, so _A2 is considered a step and m3 a leap, even though they have the same number of semitones.

Simple and compound intervals

isSimple :: Interval -> Bool

Returns whether the given interval is simple.

A simple interval is a non-negative interval spanning less than one octave.

isCompound :: Interval -> Bool

Returns whether the given interval is compound.

A compound interval is either a negative interval, or a positive interval spanning more than octave.

separate :: Interval -> (Octaves, Interval)

Separate a compound interval into octaves and a simple interval.

 (perfect octave)^*x + y = z  iff  (x, y) = separate z

simple :: Interval -> Interval

Returns the simple part of an interval.

 (perfect octave)^*x + y = z  iff  y = simple z

Inversion

invert :: Interval -> Interval

Intervallic inversion.

The inversion an interval is determined as follows:

• The number of a simple interval the difference of nine and the number of its inversion.
• The quality of a simple interval is the inversion of the quality of its inversion.
• The inversion of a compound interval is the inversion of its simple component.

Utility

asInterval :: Interval -> Interval

This is just the identity function, but is useful to fix the type of Interval.

intervalDiff :: Interval -> Int

Deprecated: This should be hidden

mkInterval' :: Int -> Int -> Interval

Deprecated: This should be hidden