Chain-of-fifths notation
The circle-of-fifths notation (aka extended Pythagorean notation) is suitable for a variety of tuning systems which are octave repeating and generated by the fifth. A good number of edos and regular temperaments can be notated this way, as it generalizes the traditional classical notation system for the Pythagorean tuning, the meantone tunings, and 12edo. It uses the seven natural notes of the diatonic scale and accidentals (♯, ♭ and their multiples) to sharpen and flatten these seven notes by the augmented unison aka the chromatic semitone.
Circle-of-fifths notation only works for single-ring edos. A counter-example is 24edo, which is double-ring. This notation works best for 7-edo and edos of sharpness 1 or -1. For all other edos, this notation causes the notes to run out of order. For example, 17-edo would run C Db C# D Eb D# E... One can avoid this by using ups and downs notation, or for certain edos by using half-sharps (see below).
Any regular rank-2 temperament generated by the 8ve and the 5th (i.e. one with the unsplit pergen) can be notated this way. Because it's rank-2, the circle of fifths is actually a (theoretically infinite) chain of fifths.
The circle-of-half-fifths notation (aka circle-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 are useless). Not all even-sharpness edos allow this notation. For example, 34-edo (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, 41-edo (sharp-4) has C Ddb Ct Db C# Dd C#t D.
Circle-of-third-fifths notation, circle-of-quarter-fifths notation, etc., are theoretical possibilities. In practice, ups and downs are usually used for third-sharps or quarter-sharps.
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 | Fifth-detuning abs(¢), rel(%) |
Major
2nd |
Minor
2nd |
Augmented
unison |
|---|---|---|---|---|---|
| 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 | Fifth-detuning abs(¢), rel(%) |
Major
2nd |
Minor
2nd |
Augmented
unison |
|---|---|---|---|---|---|
| 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
- Syntonic-rastmic subchroma notation – built on neutral circle-of-fifths notation
- Ups and downs notation – built on circle-of-fifths notation
- Neutral ups and downs notation (→ Alternative symbols for ups and downs notation)
- Sagittal notation (evo flavor) – built on circle-of-fifths notation or neutral circle-of-fifths notation
See also
- Nominal-accidental chain
- Circle of fifths
- Fifthspan
- User:Xenwolf/cofn – sortable table with more intervals (all fifths within the interval [4\7, 3\5], the "diatonic range")