# Circle-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 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 Db C# D Eb 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 as subsets. For example, 13edo can be notated as a subset of 26edo.

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 **neutral circle-of-fifths notation** (aka **circle-of-half-fifths notation**, **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 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 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 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 | 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

- 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")