Microtonal Notation

Akkado supports microtonal pitch notation through micro-step operators (^, v, +) and the tune() function, which selects how those operators map to frequency.

Micro-step operators

Micro-step operators go between the note name/accidentals and the octave number:

Operator	Meaning	Example
`^`	Step up	`c^4`: C4 raised by one EDO step
`v`	Step down	`cv4`: C4 lowered by one EDO step
`+`	Step up (alias for `^`)	`c+4`: same as `c^4`

Operators stack: c^^4 raises by two steps, cvvv4 lowers by three. Mixed operators cancel: c^v4 = c4.

Standard accidentals (#, b, x) remain unchanged and always shift by semitones. They can combine with micro-steps:

// Sharp + micro step up
n"c#^4"   // C#4 + one EDO step

// Flat + micro step down
n"Bbv4"   // Bb4 - one EDO step

The tune() function

tune() wraps a pattern and sets the EDO tuning for micro-step resolution:

// 31-EDO: each ^ step = 38.7 cents
tune("31edo", n"c4 c^4 c^^4")
    |> ((f) -> sine(f) * ar(trigger(3)))
    |> out(@)

// 24-EDO: each ^ step = 50 cents (quarter tone)
tune("24edo", n"a4 a^4 a^^4")
    |> ((f) -> sine(f) * ar(trigger(3)))
    |> out(@)

Without tune(), the default is 12-EDO where each ^ step equals one semitone (100 cents), so c^4 is the same as c#4.

Accepted formats: "31edo", "31-edo", "31-EDO".

How EDO steps work

An EDO system divides the octave (1200 cents) into N equal steps. The ^/v operators move by one step in the active tuning:

EDO	Step size	`^` =	Notes
12	100.0 cents	semitone	Standard Western tuning
17	70.6 cents	~3/4 semitone	Narrower intervals, wider major thirds
19	63.2 cents	~2/3 semitone	Near-just minor thirds
24	50.0 cents	quarter tone	Arabic/Turkish maqam music
31	38.7 cents	~1/3 semitone	Near-just thirds and sevenths
53	22.6 cents	Holdrian comma	Near-just for all intervals

Important: Notes without micro-steps sound identical in all tunings. n"c4 e4 g4" produces the same frequencies whether wrapped in tune("31edo", ...) or not. tune() only changes how ^ and v are resolved.

Micro-step arithmetic

Sharps/flats and micro-steps are independent axes:

// # shifts the MIDI note by a semitone (always 100 cents)
// ^ shifts by one EDO step (depends on tuning)
tune("31edo", n"c4 c#4 c^4 c#^4")
// c4   = 261.6 Hz (Middle C)
// c#4  = 277.2 Hz (+ 100 cents, semitone)
// c^4  = 267.5 Hz (+ 38.7 cents, one 31-EDO step)
// c#^4 = 283.3 Hz (+ 100 + 38.7 = 138.7 cents)

Example: Fur Elise retuned

The opening motif of Beethoven’s Fur Elise, the E/D# trill, is a useful testbed for microtonal retuning because the character of a trill depends on interval size.

Standard (12-EDO)

The E-D# trill spans exactly 100 cents:

n"e5 d#5 e5 d#5 e5 b4 d5 c5 a4@2 ~ c4 e4 a4 b4@2 ~ e4 g#4 b4 c5@2"
    |> ((f) -> tri(f) * ar(trigger(20), 0.005, 0.3))
    |> out(@)

17-EDO: the narrow trill

In 17-EDO the semitone shrinks to 70.6 cents. D#-to-E becomes tighter and more restless. The trill wants to resolve but can’t quite settle:

m = n"e5 d#5 e5 d#5 e5 b4 d5 c5 a4@2 ~ c4 e4 a4 b4@2 ~ e4 g#4 b4 c5@2"
tune("17edo", m)
    |> ((f) -> tri(f) * ar(trigger(20), 0.005, 0.3))
    |> out(@)

Add micro-step ornaments for a xenharmonic flavor. e^5 and ev5 nudge notes by 70.6 cents:

tune("17edo", n"e5 d#5 e^5 d#5 ev5 d#5 e5 b4 d5 c5 a4@2")
    |> ((f) -> tri(f) * ar(trigger(12), 0.005, 0.3))
    |> out(@)

31-EDO: microtonal cascade

In 31-EDO each step is 38.7 cents. Instead of a binary E/D# trill, replace it with a micro-step descent. Each v lowers by a third of a semitone, creating a portamento-like effect:

tune("31edo", n"e5 ev5 evv5 ev5 e5 b4 d5 c5 a4@2 ~ c4 e4 a4 b4@2")
    |> ((f) -> tri(f) * ar(trigger(14), 0.005, 0.3))
    |> out(@)

Deeper cascade with three micro-steps, a glide spanning ~116 cents:

tune("31edo", n"e5 ev5 evv5 evvv5 evv5 ev5 e5 b4 d5 c5 a4@2")
    |> ((f) -> tri(f) * ar(trigger(11), 0.005, 0.3))
    |> out(@)

Side-by-side: the trill

Minimal examples to hear the difference in trill character:

// 12-EDO: 100-cent trill (standard semitone)
n"e5 d#5 e5 d#5 e5 d#5 e5 d#5"
    |> ((f) -> sine(f) * ar(trigger(8), 0.005, 0.1))
    |> out(@)

// 17-EDO: 70.6-cent trill (narrow, restless)
tune("17edo", n"e5 d#5 e5 d#5 e5 d#5 e5 d#5")
    |> ((f) -> sine(f) * ar(trigger(8), 0.005, 0.1))
    |> out(@)

// 31-EDO: 38.7-cent micro-step shimmer
tune("31edo", n"e5 ev5 e5 ev5 e5 ev5 e5 ev5")
    |> ((f) -> sine(f) * ar(trigger(8), 0.005, 0.1))
    |> out(@)

Tips

Start with 24-EDO: quarter tones are the most intuitive entry point to microtonal music.
^^ in 31-EDO (77.4 cents) is close to a quarter tone, useful as a mental anchor.
tune() only affects ^/v: standard notes, sharps, and flats are unchanged across all tunings.
Samples are unaffected: bd, sn, and other sample tokens ignore micro-step operators.
Velocity still works: c^4:0.5 is a micro-stepped note at half velocity.