Chainoffifths notation
The chainoffifths notation, also known as extended Pythagorean notation, is a musical notation system that supports a variety of tuning systems which are octaverepeating 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 Pythagorean tuning and meantone tunings (including 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 chromatic semitone. Any regular rank2 temperament generated by the octave and fifth (i.e. one with the unsplit pergen) can be notated this way.
Chainoffifths notation only works for singlering edos; some tunings have multiple circles of fifths, such as 24edo which has two, and 36edo which has three. This notation works best for edos of sharpness 1, and for 7edo, where accidentals have no effects. In tunings where sharps raise by multiple steps, notes in the chromatic scale will run out of order. For example, 17edo's chromatic scale would be C – D♭ – C♯ – D – E♭ – D♯ – E – F – G♭ – F♯ – G – A♭ – G♯ – A – B♭ – A♯ – B – C. If the fifth is flatter than 685.714 ¢, the order of the sharps and flats will be inverted. One can avoid these by using ups and downs notation, or for certain edos by using halfsharps (see below). Edos whose fifth has a high relative error makes more sense considered as dualfifth, and notated using subset notation, such as in the case of 13edo, which 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 chainoffifths notation to a dualfifth tuning is generally called the native fifth notation.
The neutral chainoffifths notation (aka chainofhalffifths notation, chainofneutralthirds notation, or less accurately, quartertone notation) uses an extended accidental set including halfsharps and halfflats. It works for any rank2 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 sharp0 edos, in which sharps and flats have no effects). Not all evensharpness edos allow this notation. For example, 34edo (sharp4) does not, because its halffifth is 10\34, and 10 and 34 are not coprime. The GCD is 2, thus there are two rings of halffifths. In other words, the edo must be singlering with respect to the halffifth. 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, in 41edo, which is sharp4, the notes within a (major) whole tone are C – D – C – D♭ – C♯ – D – C – D.
Finer divisions (chainofthirdfifths, chainofquarterfifths, and beyond) are also theoretical possibilities. In practice, ups and downs are usually used when sharps raise by three or more steps.
Accidentals
The Standard Music Font Layout (SMuFL) specification provides Unicode codepoints for the standard accidentals of chainoffifths notation and for the SteinZimmermann accidentals of neutral circleoffifths 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.
Style \ Offset  −2  −1½  −1  −½  ±0  +½  +1  +1½  +2 

Name  Double flat  Sesquiflat  Flat  Halfflat Demiflat Semiflat 
Natural  Halfsharp Demisharp Semisharp 
Sharp  Sesquisharp  Double sharp 
Standard accidentals^{[2]}  𝄫 (U+1D12B) 
♭ (U+266D) 
♮ (U+266E) 
♯ (U+266F) 
𝄪 (U+1D12A)  
Standard accidentals + SteinZimmermann 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}} {{sesquiflat2}} 
{{flat}}  {{demiflat}} {{demiflat2}} 
{{natural}}  {{demisharp}} {{demisharp2}} 
{{sharp}}  {{sesquisharp}} {{sesquisharp2}} 
{{sharp2}} 
Alternative accidentals
While the SteinZimmermann accidentals appear to be the most widespread for neutral circleoffifths 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 SteinZimmermann 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.
Style \ Offset  −2  −1½  −1  −½  ±0  +½  +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) 

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.
Edo  Fifth  Fifthdetuning 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 
Edo  Fifth  Fifthdetuning 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
 Syntonicrastmic subchroma notation – built on neutral chainoffifths notation
 Ups and downs notation – built on chainoffifths notation
 Neutral ups and downs notation (→ Alternative symbols for ups and downs notation)
 Diamondmos notation – built on neutral chainoffifths notation
 Sagittal notation (evo flavor) – built on chainoffifths notation or neutral chainoffifths notation
 Extended meantone notation – built on chainoffifths notation or neutral chainoffifths notation
See also
 Nominalaccidental chain
 Chain of fifths
 Fifthspan
 Pythagorean tuning
 User:Xenwolf/cofn – sortable table with more intervals (all fifths within the interval [4\7, 3\5], the "diatonic range")
Notes
 ↑ Symbols for fivequartertones accidentals are also available.
 ↑ 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.
 ↑ Wyschnegradsky accidentals also include twelfthtone (72edo) accidentals.
References
 ↑ SMuFL  Introducing SMuFL
 ↑ Standard Music Font Layout  Standard accidentals (12EDO)
 ↑ Standard Music Font Layout  SteinZimmermann accidentals (24EDO)
 ↑ Standard Music Font Layout  Gould arrow quartertone accidentals (24EDO)
 ↑ Standard Music Font Layout  Persian accidentals
 ↑ Standard Music Font Layout  Spartan Sagittal singleshaft accidentals
 ↑ Standard Music Font Layout  Spartan Sagittal multishaft accidentals
 ↑ Standard Music Font Layout  Wyschnegradsky accidentals (72EDO)