Extended meantone notation

This page is currently reworked at User:PiotrGrochowski/Extended meantone notation, see also Talk:Extended meantone notation #under construction.

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 some tunings, such as 24-tone equal temperament (quarter tones) and 31-tone equal temperament (extended quarter-comma meantone), sharps can be split in half. Thus, some notes can be notated using semisharps and semiflats, or with ups and downs.

For example, in 31edo, 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

The generalized chain of fifths, however, does not have a single semisharp or semiflat. In a general meantone tuning, a sharp is split into 2 different parts, the diesis and the kleisma.

Generalizing accidentals

Most people are familiar with sharps and flats—these denote raising and lowering, respectively, by a chromatic semitone. In a general meantone tuning, there are two additional intervals: the diesis, 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. Note that in most meantone tunings, the diesis and kleisma are roughly a quarter tone. 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	Syntonic comma fraction	Steps				Explanation
		Chromatic semitone	Diatonic semitone	Diesis	Kleisma
7		0	1	1	−1	Chromatic semitone is tempered out, diesis is positive, and kleisma is negative
12 (standard tuning)	1⁄11 comma	1	1	0	1	Chromatic semitone is equal to kleisma, diesis is tempered out
19	1⁄3 comma	1	2	1	0	Chromatic semitone is equal to diesis, kleisma is tempered out
26		1	3	2	−1	Diesis is larger than chromatic semitone, kleisma is negative
33 (c mapping)	1⁄2 comma	1	4	3	−2
31	1⁄4 comma	2	3	1	1	Diesis is equal to kleisma
43	1⁄5 comma	3	4	1	2	Diesis is smaller than kleisma
55	1⁄6 comma	4	5	1	3
50	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↑	B♭	D	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	⁠9/5⁠	⁠9/7⁠	⁠16/9⁠	⁠10/9⁠	⁠14/9⁠

Two dieses or two kleismas cannot be stacked to produce a chromatic semitone, and notation for 11-limit and 13-limit intervals (intervals involving the 11th and 13th harmonics) can vary (see meantone vs meanpop).

True half-sharps and half-flats

If true half-sharps and true half-flats are desired, which exactly bisect the chromatic semitone, the meantone fifth is split in half. This creates a 2D new tuning system which is generated by a chain of neutral thirds, with meantone existing as every other note in the generator chain. This adds true half-sharps and half-flats, and creates a "neutral" version of each interval class.

Real-world Arabic and Persian music often involve many fine microtonal details (such as the use of multiple unequal neutral intervals) and exhibit significant regional variations, and as a result they are very difficult to notate exactly. However, they are commonly notated using half-sharps and-half flats. If we take these to be exactly equal to half of a chromatic semitone, then mathematically, this notation system results in a two-dimensional lattice that is generated by a neutral third and an octave. If adjacent sharps and flats, such as C♯ and D♭, are made enharmonically equivalent, this lattice degenerates further into 24edo, which is often suggested as a simplified framework and tuning system for notating and playing Arabic and Persian music. But the usual written notation typically lets musicians and composers treat adjacent sharps and flats as two distinct entities if it is decided that they should be different.

The chain-of-neutral thirds tuning system is not a true "temperament," because it is contorted: the neutral third does not have any JI interval mapping to it in the 7-limit. But, if we go to the 11-limit, and decide that there should only be a single neutral second (i.e. 121/120 should be tempered out, resulting in the small neutral second of 12/11 and large neutral second of 11/10 both being mapped to a single equally-tempered interval), we obtain mohajira, an exceptionally good 11-limit temperament. The neutral third approximates 11/9, and two of them make a perfect fifth. Furthermore, flattening a minor third by a half-flat results in an approximation of 7/6, while sharpening a major third by a half-sharp gives an approximation of 9/7. Mohajira is supported very well by 24edo and 31edo.