# Circle-of-fifths notation

The **circle-of-fifths notation** (aka **extended Pythagorean notation**) is suitable to open up the 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 later 12edo. It uses seven root notes of the diatonic scale and accidentals (♯, ♭ and their multiples) to sharpen and flatten these root notes by the chromatic semitone (which is an octave-reduced stack of 7 fifths).

To notate edos, one of the intervals must be selected as the fifth. Edos that are best supported by this system are those whose fifth does not deviate too much from the pure fifth 3/2 (702 ¢) and that can be represented by only one ring of fifths. 24edo, as a counter-example to this, contains two rings. If we as well demand that whole tones (2 × P5 - P8), diatonic semitones (3 × P8 - 5 × P5), and chromatic semitones (shifts caused by one accidental, 7 × P5 - 4 × P8), use a positive number of steps, we exclude all edos below 12 and also 13, 16, 18, and 23. They make more sense 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 **quartertone notation**) uses an extended accidental set including **demisharps** and **demiflats**. 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). It also works for any edo of sharpness 2. Examples are the mohaha temperament and its typical edo tunings (17edo, 24edo, 31edo, 38edo, 45edo).

## Edos up to 100

Edos up to 100 are listed in the following tables. The unit (if not stated otherwise) is *steps* of the corresponding edo which is given in the first column of each row. The list contains only those edos whose all degrees can be reached by stacking the direct approximation of the fifth in the respective edo. The last two columns are the edo's pentasharpness and sharpness respectively.

Edo | Fifth | Fifth-detuning abs(¢), rel(%) |
Whole tone |
Diatonic semitone |
Chromatic semitone |
---|---|---|---|---|---|

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(%) |
Whole tone |
Diatonic semitone |
Chromatic semitone |
---|---|---|---|---|---|

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")