Extended meantone notation

Most musicians are familiar with the circle of fifths. This is a way of organizing and showing relationships between pitches as a sequence of fifths, and applies to any tuning system that can be generated by fifths and octaves. The generalized chain of fifths involves the 7 base note letters of the C major scale, along with sharps, double-sharps, flats, and double-flats (and beyond):

... 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𝄪 ...

In a general meantone tuning, a sharp is split into 2 different parts, the diesis and the kleisma.

Generalizing accidentals

Most musicians are familiar with single and double sharps and flats—these denote raising and lowering by one or two chromatic semitones, respectively. In a general meantone tuning, there are two additional intervals: the diesis, which is the difference between adjacent accidentals (e.g. C♯–D♭ and D♯–E♭), and the kleisma, which is the amount by which B♯ exceeds C♭ and E♯ exceeds F♭ (that is, C♭ – B♯ and F♭ – E♯).

Symbol		Interval	Number of fifths
Raise	Lower
♯	♭	Chromatic semitone	7
↑	↓	Diesis	12
+	−	Kleisma	19

A meantone chromatic semitone consists of one diesis and one kleisma. The diesis represents the just intervals 128/125 and 648/625, while the meantone kleisma represents 15625/15552 or 3125/3072. 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.

An octave is made of 19 dieses and 12 kleismas.

Unlike semisharps and semiflats, the diesis and kleisma can be generalized to other tunings:

Notes per octave	Approximate syntonic comma fraction	Steps				Explanation
		Chromatic semitone (e.g. C–C♯)	Diatonic semitone (e.g. C–D♭)	Diesis	Kleisma
7edo		0	1	1	−1	Chromatic semitone is tempered out, diesis is positive, and kleisma is negative
12edo (standard tuning)	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
33edo (c mapping)	1⁄2 comma	1	4	3	−2
31edo	1⁄4 comma	2	3	1	1	Diesis is equal to kleisma
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

There are of course 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

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 steps, such as 24-tone equal temperament (quarter tones) and 31-tone equal temperament (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