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♭.
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
View • Talk • EditMusical notation | |
---|---|
Universal | Sagittal notation |
Just intonation | Functional Just System • Ben Johnston's notation (Johnston–Copper notation) • Helmholtz–Ellis notation • Color notation |
MOS scales | Diamond-mos notation • KISS notation (Quasi-diatonic MOS notation) |
Temperaments | Circle-of-fifths notation • Ups and downs notation (alternative symbols) • Syntonic–rastmic subchroma notation • Extended meantone notation • Fractional sharp notation |
See musical notation for a longer list of systems by category. See Category:Notation for the most complete, comprehensive list, but not sorted by category. |