Chain-of-fifths notation

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The chain-of-fifths notation, also known as extended Pythagorean notation, is a musical notation system that supports a variety of tuning systems which are octave-repeating and generated by the fifth (just or tempered). A good number of edos and regular temperaments can be notated this way, as it generalizes the classical notation system for the Pythagorean tuning, the meantone tunings, and 12edo. It uses the seven natural notes of the diatonic scale (A to G) and accidentals (♯, ♭ and their multiples) to sharpen and flatten these seven notes by the augmented unison aka the chromatic semitone. Any regular rank-2 temperament generated by the 8ve and the 5th (i.e. one with the unsplit pergen) can be notated this way.

Chain-of-fifths notation only works for single-ring edos. A counter-example is 24edo, which is double-ring. This notation works best for edos of sharpness 1, and for 7edo, where accidentals have no effects. For any multi-sharpness edos, this notation causes the notes to run out of order. For example, 17edo would run C D C D E D E… For negative sharpness edos the accidentals will be inverse. One can avoid these by using ups and downs notation, or for certain edos by using half-sharps (see below). Edos whose fifth has a high relative error makes more sense considered as dual-fifth, and notated using subset notation. For example, 13edo can be notated as a subset of 26edo. Nonetheless, such tunings may also be notated without resorting to subset notation, and the direct application of the chain-of-fifths notation to a dual-fifth tuning is generally called the native fifth notation.

The neutral chain-of-fifths notation (aka chain-of-half-fifths notation, chain-of-neutral-thirds notation, or less accurately, quartertone notation) uses an extended accidental set including half-sharps and half-flats. It works for any rank-2 temperament generated by an octave and a neutral third, i.e. those with a pergen of (P8, P5/2), such as the mohaha temperament. It also works for certain edos of even sharpness (except sharp-0 edos, in which sharps and flats have no effects). Not all even-sharpness edos allow this notation. For example, 34edo (sharp-4) does not, because its half-fifth is 10\34, and 10 and 34 are not coprime. The GCD is 2, thus there are two rings of half-fifths. In other words, the edo must be single-ring with respect to the half-fifth. All edos with sharpness 2 or -2 qualify. If a qualifying edo's sharpness is not ±2, the notes will run out of order. For example, 41edo (sharp-4) has C D C D C D C D.

Chain-of-third-fifths notation, chain-of-quarter-fifths notation, etc., are theoretical possibilities. In practice, ups and downs are usually used for third-sharps or quarter-sharps.

Accidentals

The Standard Music Font Layout (SMuFL) specification provides Unicode codepoints for the standard accidentals of chain-of-fifths notation and for the Stein-Zimmermann accidentals of neutral circle-of-fifths notation. Some fonts may not include all symbols, so fonts designed for musical notation, such as Bravura or Leland[1], are recommended.

In circumstances where the fonts or codepoints are not quickly accessible, ASCII substitute symbols are used instead of the regular symbols. In addition, the Xenharmonic Wiki provides character templates to enter these symbols easily in wiki pages. The following table includes these equivalences.

Accidentals in (neutral) circle-of-fifths notation
Style \ Offset -2 -1½ -1 ±0 +1 +1½ +2
Name Double flat Sesquiflat Flat Half-flat
Demiflat
Semiflat
Natural Half-sharp
Demisharp
Semisharp
Sharp Sesquisharp Double sharp
Standard accidentals[2] 𝄫
(U+1D12B)

(U+266D)

(U+266E)

(U+266F)
𝄪
(U+1D12A)
Standard accidentals
+ Stein-Zimmermann accidentals[3]

(U+E264)

(U+E281)

(U+E260)

(U+E280)

(U+E261)

(U+E282)

(U+E262)

(U+E283)

(U+E263)
Substitute symbols bb db b d h t # t# x
Xen Wiki character templates {{flat2}} {{sesquiflat}} {{flat}} {{demiflat}} {{natural}} {{demisharp}} {{sharp}} {{sesquisharp}} {{sharp2}}

Alternative accidentals

While the Stein-Zimmermann accidentals appear to be the most widespread for neutral circle-of-fifths notation nowadays, and are most likely to be understood by professional musicians, other accidental sets have been developed and used by various musicians.

Note that certain symbols may be very similar or identical to standard or Stein-Zimmermann accidentals despite having different Unicode codepoints.

A particular case is ups and downs notation, which uses arrows placed to the left of accidentals (e.g. ^#) or note names (e.g. ^C#). Since different tuning systems associate a different number arrows to different offsets, they are not included below, but the most basic notation can be found at 24edo #Notation.

Alternative accidentals in (neutral) circle-of-fifths notation
Style \ Offset -2 -1½ -1 ±0 +1 +1½ +2
Gould arrow quartertone accidentals[4][note 1]
(U+E271)
(U+1D12D)

(U+E278)

(U+E270)
(U+1D12C)

(U+E273)
(U+1D12F)

(U+E275)
(U+1D131)

(U+E272)
(U+1D12E)

(U+E274)
(U+1D130)

(U+E277)
Persian accidentals[5]
Koron
(U+E460)

Sori
(U+E461)
Sagittal accidentals[6][7][note 2]
(U+E335)

(U+E327)

(U+E319)

(U+E30B)

(U+E30A)

(U+E318)

(U+E326)

(U+E334)
Wyschnegradsky accidentals[8][note 3]
(U+E433)

(U+E430)

(U+E42D)

(U+E422)

(U+E425)

(U+E428)
  1. Symbols for five-quarter-tones accidentals are also available.
  2. In mixed Sagittal notation, standard sharps and flats may be used instead of sagittal sharps and flats, and sagittal accidentals may be used to the left of those to alter them. Also, Sagittal notation includes many more accidentals besides those included in the table.
  3. Wyschnegradsky accidentals also include twelfth-tone (72edo) accidentals.

Edos up to 100

Edos up to 100 are listed in the following tables. The unit (if not stated otherwise) is edosteps of the corresponding edo which is given in the first column of each row. The tables contain only diatonic edos (i.e. A1 and m2 have edostepspans > 0). The last two columns are the edo's pentasharpness and sharpness respectively.

Diatonic edos fit for chain-of-fifths notation
Edo Fifth Fifth-detuning
abs (¢), rel (%)
Major
2nd
Minor
2nd
Augmented
1sn
12 7 -2.0 ( -2.0%) 2 1 1
17 10 +3.9 ( +5.6%) 3 1 2
19 11 -7.2 (-11.4%) 3 2 1
22 13 +7.1 (+13.1%) 4 1 3
26 15 -9.6 (-20.9%) 4 3 1
27 16 +9.2 (+20.6%) 5 1 4
29 17 +1.5 ( +3.6%) 5 2 3
31 18 -5.2 (-13.4%) 5 3 2
32 19 +10.5 (+28.1%) 6 1 5
33 19 -11.0 (-30.4%) 5 4 1
37 22 +11.6 (+35.6%) 7 1 6
39 23 +5.7 (+18.6%) 7 2 5
40 23 -12.0 (-39.9%) 6 5 1
41 24 +0.5 ( +1.7%) 7 3 4
42 25 +12.3 (+43.2%) 8 1 7
43 25 -4.3 (-15.3%) 7 4 3
45 26 -8.6 (-32.3%) 7 5 2
46 27 +2.4 ( +9.2%) 8 3 5
47 27 -12.6 (-49.3%) 7 6 1
49 29 +8.2 (+33.7%) 9 2 7
50 29 -6.0 (-24.8%) 8 5 3
53 31 -0.1 ( -0.3%) 9 4 5
55 32 -3.8 (-17.3%) 9 5 4
56 33 +5.2 (+24.2%) 10 3 7
59 35 +9.9 (+48.7%) 11 2 9
61 36 +6.2 (+31.7%) 11 3 8
63 37 +2.8 (+14.7%) 11 4 7
64 37 -8.2 (-43.8%) 10 7 3
65 38 -0.4 ( -2.3%) 11 5 6
67 39 -3.4 (-19.2%) 11 6 5
69 40 -6.3 (-36.2%) 11 7 4
70 41 +0.9 ( +5.3%) 12 5 7
71 42 +7.9 (+46.8%) 13 3 10
73 43 +4.9 (+29.8%) 13 4 9
74 43 -4.7 (-28.7%) 12 7 5
75 44 +2.0 (+12.8%) 13 5 8
77 45 -0.7 ( -4.2%) 13 6 7
79 46 -3.2 (-21.2%) 13 7 6
80 47 +3.0 (+20.3%) 14 5 9
81 47 -5.7 (-38.2%) 13 8 5
83 49 +6.5 (+44.8%) 15 4 11
88 51 -6.5 (-47.7%) 14 9 5
89 52 -0.8 ( -6.2%) 15 7 8
90 53 +4.7 (+35.3%) 16 5 11
91 53 -3.1 (-23.2%) 15 8 7
94 55 +0.2 ( +1.4%) 16 7 9
95 56 +5.4 (+42.9%) 17 5 12
97 57 +3.2 (+25.9%) 17 6 11
98 57 -4.0 (-32.6%) 16 9 7
99 58 +1.1 ( +8.9%) 17 7 10
Diatonic edos fit for neutral chain-of-fifths notation
Edo Fifth Fifth-detuning
abs (¢), rel (%)
Major
2nd
Minor
2nd
Augmented
1sn
17 10 +3.9 ( +5.6%) 3 1 2
24 14 -4.0 (-4.0%) 4 2 2
27 16 +9.2 (+20.6%) 5 1 4
31 18 -5.2 (-13.4%) 5 3 2
37 22 +11.6 (+35.6%) 7 1 6
38 22 -7.2 (-22.9%) 6 4 2
41 24 +0.5 ( +1.7%) 7 3 4
44 26 +7.1 (+26.2%) 8 2 6
45 26 -8.6 (-32.3%) 7 5 2
52 30 -9.6 (-41.8%) 8 6 2
55 32 -3.8 (-17.3%) 9 5 4
58 34 +1.5 ( +3.6%) 10 4 6
61 36 +6.2 (+31.7%) 11 3 8
65 38 -0.4 ( -2.3%) 11 5 6
69 40 -6.3 (-36.2%) 11 7 4
71 42 +7.9 (+46.8%) 13 3 10
75 44 +2.0 (+12.8%) 13 5 8
78 46 +5.7 (+37.3%) 14 4 10
79 46 -3.2 (-21.2%) 13 7 6
86 50 -4.3 (-30.7%) 14 8 6
89 52 -0.8 ( -6.2%) 15 7 8
92 54 +2.4 ( +18.3%) 16 6 10
95 56 +5.4 (+42.9%) 17 5 12
99 58 +1.1 ( +8.9%) 17 7 10

Expansions

See also

References