Chain-of-fifths notation: Difference between revisions

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The '''circle-of-fifths notation''' is suitable to open up the variety of tones of a selection of EDOs and regular temperaments of fifth generator. The principle is based on one of the intervals taking over the role of the fifth of the traditional classical notation system (in [[12-EDO]] or the [[meantone]] tuning). The classical notation system uses seven root notes and accidentals (<span style="font-size:larger">♯, ♭</span> and their multiples) to sharpen and flatten these root notes by the same amount (which is an octave-reduced stack of 7 fifths).  
The '''circle-of-fifths notation''' is suitable to open up the variety of tones of a selection of EDOs and regular temperaments of fifth generator. The principle is based on one of the intervals taking over the role of the fifth of the traditional classical notation system (in [[12-EDO]] or the [[meantone]] tuning). The classical notation system uses seven root notes and accidentals (<span style="font-size:larger">♯, ♭</span> and their multiples) to sharpen and flatten these root notes by the same amount (which is an octave-reduced stack of 7 fifths).  


EDOs that are best supported by this system are those whose fifth does not deviate too much from the pure fifth [[3/2]] (702 cent) and that can be represented by only one ring of fifths (24edo, as a counter-example, contains two rings). These include {{EDOs| 12, 17, 19, 22, 26, 29, and 31edo }}.
EDOs that are best supported by this system are those whose fifth does not deviate too much from the pure fifth [[3/2]] (702 cent) 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), semitones (3*P8 - 5*P5), and accidentals (7*P5 - 4*P8) use a positive amount of steps, we lose all EDOs below 12 EDO and also {{EDOs| 13, 16, 18, and 23 }}. The remaining EDOs up to 100 are:


{| class="wikitable center-all"
{| class="wikitable center-all"
! EDO
! Octave
! Fifth (cents)
! Fifth
! Delta !!
! Detuning %
! Wholetone
! Whole tone
! Semitone
! Accidental
! Accidental
|-
|-
| 12
| 12 ||  7 ||  -2.0 || 2 ||  1 || 1
| 7\12 (700.0)
| -2.0 ||
| 2\12
| 1\12
|-
|-
| 17
| 17 || 10 ||  +5.6 || 3 || 1 || 2
| 10\17 (705.9)
| +3.9 ||
| 3\17
| 2\17
|-
|-
| 19
| 19 || 11 || -11.4 || 3 || 2 || 1
| 11\19 (694.7)
| -7.2 ||
| 3\19
| 1\19
|-
|-
| 22
| 22 || 13 || +13.1 || 4 ||  1 || 3
| 13\22 (709.1)
| +7.1 ||
| 4\22
| 3\22
|-
|-
| 26
| 26 || 15 || -20.9 || 4 ||  3 || 1
| 15\26 (692.3)
| -9.6 ||
| 4\26
| 1\26
|-
|-
| 29
| 27 || 16 || +20.6 ||  5 || 1 || 4
| 17\29 (703.4)
| +1.5 ||
| 5\29
| 3\29
|-
|-
| 31
| 29 || 17 ||  +3.6 ||  5 ||  2 ||  3
| 18\31 (696.8)
|-
| -5.2 ||
| 31 || 18 || -13.4 ||  5 ||  3 ||  2
| 5\31
|-
| 2\31
| 32 || 19 || +28.1 ||  6 ||  1 ||  5
|-
| 33 || 19 || -30.4 ||  5 ||  4 ||  1
|-
| 37 || 22 || +35.6 ||  7 ||  1 ||  6
|-
| 39 || 23 || +18.6 ||  7 ||  2 ||  5
|-
| 40 || 23 || -39.9 ||  6 ||  5 ||  1
|-
| 41 || 24 ||  +1.7 ||  7 ||  3 ||  4
|-
| 42 || 25 || +43.2 ||  8 ||  1 ||  7
|-
| 43 || 25 || -15.3 ||  7 ||  4 ||  3
|-
| 45 || 26 || -32.3 ||  7 ||  5 ||  2
|-
| 46 || 27 ||  +9.2 ||  8 ||  3 ||  5
|-
| 47 || 27 || -49.3 ||  7 ||  6 ||  1
|-
| 49 || 29 || +33.7 ||  9 ||  2 || 7
|-
| 50 || 29 || -24.8 ||  8 ||  5 ||  3
|-
| 53 || 31 ||  -0.3 ||  9 ||  4 ||  5
|-
| 55 || 32 || -17.3 ||  9 ||  5 ||  4
|-
| 56 || 33 || +24.2 || 10 ||  3 ||  7
|-
| 59 || 35 || +48.7 || 11 ||  2 ||  9
|-
| 61 || 36 || +31.7 || 11 ||  3 ||  8
|-
| 63 || 37 || +14.7 || 11 ||  4 ||  7
|-
| 64 || 37 || -43.8 || 10 ||  7 ||  3
|-
| 65 || 38 ||  -2.3 || 11 ||  5 ||  6
|-
| 67 || 39 || -19.2 || 11 ||  6 ||  5
|-
| 69 || 40 || -36.2 || 11 ||  7 ||  4
|-
| 70 || 41 ||  +5.3 || 12 ||  5 ||  7
|-
| 71 || 42 || +46.8 || 13 ||  3 || 10
|-
| 73 || 43 || +29.8 || 13 ||  4 ||  9
|-
| 74 || 43 || -28.7 || 12 ||  7 ||  5
|-
| 75 || 44 || +12.8 || 13 ||  5 ||  8
|-
| 77 || 45 ||  -4.2 || 13 ||  6 ||  7
|-
| 79 || 46 || -21.2 || 13 ||  7 ||  6
|-
| 80 || 47 || +20.3 || 14 ||  5 ||  9
|-
| 81 || 47 || -38.2 || 13 ||  8 ||  5
|-
| 83 || 49 || +44.8 || 15 ||  4 || 11
|-
| 88 || 51 || -47.7 || 14 ||  9 ||  5
|-
| 89 || 52 ||  -6.2 || 15 ||  7 ||  8
|-
| 90 || 53 || +35.3 || 16 ||  5 || 11
|-
| 91 || 53 || -23.2 || 15 ||  8 ||  7
|-
| 94 || 55 ||  +1.4 || 16 ||  7 ||  9
|-
| 95 || 56 || +42.9 || 17 ||  5 || 12
|-
| 97 || 57 || +25.9 || 17 ||  6 || 11
|-
| 98 || 57 || -32.6 || 16 ||  9 ||  7
|-
| 99 || 58 ||  +8.9 || 17 ||  7 || 10
|}
|}


[[Category:Notation]]
[[Category:Notation]]
[[Category:Method]]
[[Category:Method]]

Revision as of 21:30, 15 November 2020

The circle-of-fifths notation is suitable to open up the variety of tones of a selection of EDOs and regular temperaments of fifth generator. The principle is based on one of the intervals taking over the role of the fifth of the traditional classical notation system (in 12-EDO or the meantone tuning). The classical notation system uses seven root notes and accidentals (♯, ♭ and their multiples) to sharpen and flatten these root notes by the same amount (which is an octave-reduced stack of 7 fifths).

EDOs that are best supported by this system are those whose fifth does not deviate too much from the pure fifth 3/2 (702 cent) 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), semitones (3*P8 - 5*P5), and accidentals (7*P5 - 4*P8) use a positive amount of steps, we lose all EDOs below 12 EDO and also 13, 16, 18, and 23. The remaining EDOs up to 100 are:

Octave Fifth Detuning % Whole tone Semitone Accidental
12 7 -2.0 2 1 1
17 10 +5.6 3 1 2
19 11 -11.4 3 2 1
22 13 +13.1 4 1 3
26 15 -20.9 4 3 1
27 16 +20.6 5 1 4
29 17 +3.6 5 2 3
31 18 -13.4 5 3 2
32 19 +28.1 6 1 5
33 19 -30.4 5 4 1
37 22 +35.6 7 1 6
39 23 +18.6 7 2 5
40 23 -39.9 6 5 1
41 24 +1.7 7 3 4
42 25 +43.2 8 1 7
43 25 -15.3 7 4 3
45 26 -32.3 7 5 2
46 27 +9.2 8 3 5
47 27 -49.3 7 6 1
49 29 +33.7 9 2 7
50 29 -24.8 8 5 3
53 31 -0.3 9 4 5
55 32 -17.3 9 5 4
56 33 +24.2 10 3 7
59 35 +48.7 11 2 9
61 36 +31.7 11 3 8
63 37 +14.7 11 4 7
64 37 -43.8 10 7 3
65 38 -2.3 11 5 6
67 39 -19.2 11 6 5
69 40 -36.2 11 7 4
70 41 +5.3 12 5 7
71 42 +46.8 13 3 10
73 43 +29.8 13 4 9
74 43 -28.7 12 7 5
75 44 +12.8 13 5 8
77 45 -4.2 13 6 7
79 46 -21.2 13 7 6
80 47 +20.3 14 5 9
81 47 -38.2 13 8 5
83 49 +44.8 15 4 11
88 51 -47.7 14 9 5
89 52 -6.2 15 7 8
90 53 +35.3 16 5 11
91 53 -23.2 15 8 7
94 55 +1.4 16 7 9
95 56 +42.9 17 5 12
97 57 +25.9 17 6 11
98 57 -32.6 16 9 7
99 58 +8.9 17 7 10
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