Circle-of-fifths notation

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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.

The neutral circle-of-fifths notation (aka quartertone notation) uses an extended accidental set including demisharps and demiflats. It works for any tuning system generated by a neutral third. The mohaha temperament and its typical edo tunings (17edo, 24edo, 31edo, 38edo, 45edo) are well represented by this system.

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.

Edos fit for circle-of-fifths notation
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
Edos fit for neutral circle-of-fifths notation
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

See also