Extended meantone notation

Meantone can be notated with a chain of fifths consisting of the 7 natural notes along with sharps and flats:

... F𝄫 — C𝄫 — G𝄫 — D𝄫 — A𝄫 — E𝄫 — B𝄫 — F♭ — C♭ — G♭ — D♭ — A♭ — E♭ — B♭ — F — C — G — D — A — E — B — F♯ — C♯ — G♯ — D♯ — A♯ — E♯ — B♯ — F𝄪 — C𝄪 — G𝄪 — D𝄪 — A𝄪 — E𝄪 — B𝄪 ...

The chain is theoretically infinite, and C♯ and D♭ are not (in general) equivalent. When meantone is extended beyond 12 notes, it may require double-sharps, double-flats, and beyond. To avoid this, two new accidental pairs are introduced that raise/lower by the diesis and the kleisma.

Symbol		Interval		Examples	Fifthspan
Raise	Lower
♯	♭	Chromatic semitone	Augmented unison (A1)	C–C♯ E♭–E	+7
↑	↓	Diesis	Diminished 2nd (d2)	C♯–D♭ D♯–E	−12
+	−	Kleisma	Negative double- diminished 2nd (-dd2)	C♭ – B♯ F♭ – E♯	+19

Because 19 − 12 = 7, d2 + −dd2 = A1, and a diesis plus a kleisma equals a chromatic semitone.

An octave is made up of:

• 7 diatonic semitones and 5 chromatic semitones = 7 m2 + 5 A1 = 12 steps
• 12 chromatic semitones and 7 dieses = 12 A1 + 7 d2 = 19 steps
• 19 dieses and 12 kleismas = 19 d2 + 12 −dd2 = 31 steps

The diesis represents the just intervals 128/125 and 648/625 among others, while the meantone kleisma represents 15625/15552 = [-6 -5 6⟩ and 3125/3072 = [-10 -1 5⟩ among others. In septimal meantone, where 7/4 is an augmented sixth, the diesis also represents 36/35, 50/49, and 64/63, while the kleisma also represents 49/48 and 245/243.

The enharmonic unisons ↓d2 and −↓A1 create various notational equivalences:

• B♯↑ and B𝄪− are equal to C
• C+↑ is equal to C♯ (because the two semisharps add up)
• D𝄫↓ and D♭♭♭− are equal to C

If the fifth is wider than 7\12 = 700 ¢, C♯ is higher in pitch than D♭ and the diesis becomes a descending pythagorean comma. In 12edo, the tempering out of the diesis means that C♯ = D♭.

If the fifth is narrower than 11\19 = ~695¢, B♯ is lower in pitch than C♭ and the kleisma becomes a descending double-diminished 2nd. In 19edo, the tempering out of the kleisma means that B♯ = C♭.

Various EDOs that support meantone

EDO	Approximate syntonic comma fraction	Steps				Relative sizes of the chromatic semitone, diesis, and kleisma
		Chromatic semitone	Diatonic semitone	Diesis	Kleisma
		A1	m2	d2	−dd2
12edo	1⁄11 comma	1	1	0	1	Chromatic semitone is equal to kleisma, diesis is tempered out
19edo	1⁄3 comma	1	2	1	0	Chromatic semitone is equal to diesis, kleisma is tempered out
26edo		1	3	2	−1	Chromatic semitone is smaller than diesis, kleisma is negative
31edo	1⁄4 comma	2	3	1	1	Diesis is equal to kleisma
33c-edo	1⁄2 comma	1	4	3	−2	Chromatic semitone is smaller than diesis, kleisma is negative
43edo	1⁄5 comma	3	4	1	2	Diesis is smaller than kleisma
55edo	1⁄6 comma	4	5	1	3
50edo	2⁄7 comma	3	5	2	1	Diesis is larger than kleisma

In 33c-edo, 5/4 is mapped to 10\33 = 364 ¢ instead of 11\33 = 400 ¢.

9-odd-limit intervals and their notation relative to C:

Note	C	G	F	E	A	E♭	A♭	A♯ B♭↓	D♯ E♭↓	F♯ G♭↓	E D↓	B A↓	G♭ F♯↓	D		B♭		F♭ E↑	G♯ A♭↓
Just interval	⁠1/1⁠	⁠3/2⁠	⁠4/3⁠	⁠5/4⁠	⁠5/3⁠	⁠6/5⁠	⁠8/5⁠	⁠7/4⁠	⁠7/6⁠	⁠7/5⁠	⁠8/7⁠	⁠12/7⁠	⁠10/7⁠	⁠9/8⁠	⁠10/9⁠	⁠9/5⁠	⁠16/9⁠	⁠9/7⁠	⁠14/9⁠

Two dieses or two kleismas cannot be stacked to produce a chromatic semitone except in 31edo, and notation for 11-limit and 13-limit intervals (intervals involving the 11th harmonic and 13th harmonic) can vary.

True half-sharps and half-flats

If sharps raise by an even number of edosteps, such as 24-tone equal temperament (quarter tones) and 31-tone equal temperament (approximately extended quarter-comma meantone), they (along with flats) can be split in half. Thus, some notes can be notated using semisharps and semiflats, or with ups and downs.

For example, in 31 equal, the chromatic scale becomes:

C — D𝄫 — C♯ — D♭ — C𝄪 — D — E𝄫 — D♯ — E♭ — D𝄪 — E — F♭ — E♯ — F — G𝄫 — F♯ — G♭ — F𝄪 — G — A𝄫 — G♯ — A♭ — G𝄪 — A — B𝄫 — A♯ — B♭ — A𝄪 — B — C♭ — B♯ — C

Note that the base note letters alternate.

Using semisharps and semiflats, this can be re-written as:

C — C⁠ ⁠ — C♯ — D♭ — D⁠ ⁠ — D — D⁠ ⁠ — D♯ — E♭ — E⁠ ⁠ — E — E⁠ ⁠ — F⁠ ⁠ — F — F⁠ ⁠ — F♯ — G♭ — G⁠ ⁠ — G — G⁠ ⁠ — G♯ — A♭ — A⁠ ⁠ — A — A⁠ ⁠ — A♯ — B♭ — B⁠ ⁠ — B — B⁠ ⁠ — C⁠ ⁠ — C