Intervals reference

Notation

Every interval is <quality><degree><modifiers>.

Quality

Symbol	Quality	Used for
`R`	Root	shorthand for `P1`
`P`	Perfect	unison, 4th, 5th, 8th, 11th, 12th
`M`	Major	2nd, 3rd, 6th, 7th, 9th, 10th, 13th
`m`	Minor	flattened major intervals
`A`	Augmented	raised by a semitone
`d`	Diminished	lowered by a semitone

Degree

The scale degree, 1 to 13+. Anything above 8 wraps an octave.

Modifiers

Symbol	Meaning
`+`	raise by one semitone (chromatic)
`-`	lower by one semitone (chromatic)
`++`, `--`	stack `+` and `-` for whole-tone shifts
`+Nc`, `-Nc`	raise / lower by N cents (microtonal)

+/- shift by 100 cents; +Nc/-Nc shift by exactly *N* cents. See Microtones.

Standard intervals (12-EDO)

Interval	Semitones	Cents	Name
`R` / `P1`	0	0	Perfect unison
`m2`	1	100	Minor second
`M2`	2	200	Major second
`m3`	3	300	Minor third
`M3`	4	400	Major third
`P4`	5	500	Perfect fourth
`A4` / `d5`	6	600	Tritone
`P5`	7	700	Perfect fifth
`m6`	8	800	Minor sixth
`M6`	9	900	Major sixth
`m7`	10	1000	Minor seventh
`M7`	11	1100	Major seventh
`P8`	12	1200	Perfect octave

Extended intervals

Interval	Semitones	Cents	Name
`m9`	13	1300	Minor ninth
`M9`	14	1400	Major ninth
`m10`	15	1500	Minor tenth
`M10`	16	1600	Major tenth
`P11`	17	1700	Perfect eleventh
`A11`	18	1800	Augmented eleventh
`P12`	19	1900	Perfect twelfth
`m13`	20	2000	Minor thirteenth
`M13`	21	2100	Major thirteenth

Arithmetic

rela

M3 + m3     ; = P5 (4 + 3 = 7 semitones)
P5 + P4     ; = P8 (7 + 5 = 12 semitones)
P8 - P5     ; = P4
M7 - M3     ; = P5
invert(M3)  ; = m6 (12 - 4 = 8 semitones)
invert(P5)  ; = P4

Arithmetic preserves cents, so M3 -13.7c + m3 returns a fifth that is 13.7 cents flat of P5.

Microtonal intervals

Append a cents offset to fine-tune. Chains compose by addition:

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M3 -13.7c       ; just-intonation major third
P5 +1.96c       ; well-temperament perfect fifth (Werckmeister III)
P1++ +50c       ; quarter-tone shy of a major second

See Microtones for the surrounding language features — tuning declarations, set tuning, |> in_tuning.

Common chord shapes

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chord MajorTriad   = [ R, M3, P5 ]            ; 0, 4, 7
chord MinorTriad   = [ R, m3, P5 ]            ; 0, 3, 7
chord Diminished   = [ R, m3, d5 ]            ; 0, 3, 6
chord Augmented    = [ R, M3, A5 ]            ; 0, 4, 8

chord Maj7         = [ R, M3, P5, M7 ]        ; 0, 4, 7, 11
chord Min7         = [ R, m3, P5, m7 ]        ; 0, 3, 7, 10
chord Dom7         = [ R, M3, P5, m7 ]        ; 0, 4, 7, 10
chord Dim7         = [ R, m3, d5, d7 ]        ; 0, 3, 6, 9
chord HalfDim7     = [ R, m3, d5, m7 ]        ; 0, 3, 6, 10
chord MinMaj7      = [ R, m3, P5, M7 ]        ; 0, 3, 7, 11

chord Maj9         = [ R, M3, P5, M7, M9 ]
chord Min9         = [ R, m3, P5, m7, M9 ]
chord Maj11        = [ R, M3, P5, M7, M9, P11 ]
chord Maj13        = [ R, M3, P5, M7, M9, P11, M13 ]

Common scale shapes

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scale Major          = { R, M2, M3, P4, P5, M6, M7 }
scale Minor          = { R, M2, m3, P4, P5, m6, m7 }
scale HarmonicMinor  = { R, M2, m3, P4, P5, m6, M7 }
scale MelodicMinor   = { R, M2, m3, P4, P5, M6, M7 }

scale Pentatonic     = { R, M2, M3, P5, M6 }
scale MinorPentatonic = { R, m3, P4, P5, m7 }
scale Blues          = { R, m3, P4, A4, P5, m7 }

scale Dorian         = { R, M2, m3, P4, P5, M6, m7 }
scale Phrygian       = { R, m2, m3, P4, P5, m6, m7 }
scale Lydian         = { R, M2, M3, A4, P5, M6, M7 }
scale Mixolydian     = { R, M2, M3, P4, P5, M6, m7 }
scale Locrian        = { R, m2, m3, P4, P5, m6, m7 }

scale WholeTone      = { R, M2, M3, A4, A5, A6 }
scale Diminished     = { R, M2, m3, P4, A4, m6, M6, M7 }
scale Chromatic      = { R, m2, M2, m3, M3, P4, A4, P5, m6, M6, m7, M7 }

Enharmonic equivalence

Intervals at the same cents distance compare equal regardless of spelling:

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A4 == d5   ; true (both six semitones)
M3 == d4   ; true (both four semitones)
m6 == A5   ; true (both eight semitones)

Cents offsets are also considered for equality:

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M3 -100c == m3   ; true — 300 cents on both sides