Chain-of-fifths notation: Difference between revisions

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autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
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I think chroma is the correct term for the interval caused by one accidental
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
added absolute de-tuning and named units
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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 to this, contains two rings. If we as well demand that whole tones (2*P5 - P8), semitones (3*P8 - 5*P5), and Chromas, shifts caused by one accidental, (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:
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 Chromas, shifts caused by one accidental, (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 }}.  
 
EDOs up to 100 are listed in the following table. The unit (if not stated otherwise) is ''steps'' of the corresponding EDO which is given in the first column of each row.


{| class="wikitable center-all"
{| class="wikitable center-all"
! Octave
! Octave/ <br> Edo
! Fifth
! Fifth
! Detuning %
! Fifth-detuning <br> abs(¢), rel(%)
! Whole tone
! Whole tone
! Semitone
! Semitone
! Chroma
! Chroma
|-
|-
| 12 ||  7 ||  -2.0 ||  2 ||  1 ||  1
| 12 ||  7 ||  -2.0 ( -2.0%) ||  2 ||  1 ||  1
|-
|-
| 17 || 10 ||  +5.6 ||  3 ||  1 ||  2
| 17 || 10 ||  +3.9 ( +5.6%) ||  3 ||  1 ||  2
|-
|-
| 19 || 11 || -11.4 ||  3 ||  2 ||  1
| 19 || 11 || -7.2 (-11.4%) ||  3 ||  2 ||  1
|-
|-
| 22 || 13 || +13.1 ||  4 ||  1 ||  3
| 22 || 13 || +7.1 (+13.1%) ||  4 ||  1 ||  3
|-
|-
| 26 || 15 || -20.9 ||  4 ||  3 ||  1
| 26 || 15 || -9.6 (-20.9%) ||  4 ||  3 ||  1
|-
|-
| 27 || 16 || +20.6 ||  5 ||  1 ||  4
| 27 || 16 || +9.2 (+20.6%) ||  5 ||  1 ||  4
|-
|-
| 29 || 17 ||  +3.6 ||  5 ||  2 ||  3
| 29 || 17 ||  +1.5 ( +3.6%) ||  5 ||  2 ||  3
|-
|-
| 31 || 18 || -13.4 ||  5 ||  3 ||  2
| 31 || 18 || -5.2 (-13.4%) ||  5 ||  3 ||  2
|-
|-
| 32 || 19 || +28.1 ||  6 ||  1 ||  5
| 32 || 19 || +10.5 (+28.1%) ||  6 ||  1 ||  5
|-
|-
| 33 || 19 || -30.4 ||  5 ||  4 ||  1
| 33 || 19 || -11.0 (-30.4%) ||  5 ||  4 ||  1
|-
|-
| 37 || 22 || +35.6 ||  7 ||  1 ||  6
| 37 || 22 || +11.6 (+35.6%) ||  7 ||  1 ||  6
|-
|-
| 39 || 23 || +18.6 ||  7 ||  2 ||  5
| 39 || 23 || +5.7 (+18.6%) ||  7 ||  2 ||  5
|-
|-
| 40 || 23 || -39.9 ||  6 ||  5 ||  1
| 40 || 23 || -12.0 (-39.9%) ||  6 ||  5 ||  1
|-
|-
| 41 || 24 ||  +1.7 ||  7 ||  3 ||  4
| 41 || 24 ||  +0.5 ( +1.7%) ||  7 ||  3 ||  4
|-
|-
| 42 || 25 || +43.2 ||  8 ||  1 ||  7
| 42 || 25 || +12.3 (+43.2%) ||  8 ||  1 ||  7
|-
|-
| 43 || 25 || -15.3 ||  7 ||  4 ||  3
| 43 || 25 || -4.3 (-15.3%) ||  7 ||  4 ||  3
|-
|-
| 45 || 26 || -32.3 ||  7 ||  5 ||  2
| 45 || 26 || -8.6 (-32.3%) ||  7 ||  5 ||  2
|-
|-
| 46 || 27 ||  +9.2 ||  8 ||  3 ||  5
| 46 || 27 ||  +2.4 ( +9.2%) ||  8 ||  3 ||  5
|-
|-
| 47 || 27 || -49.3 ||  7 ||  6 ||  1
| 47 || 27 || -12.6 (-49.3%) ||  7 ||  6 ||  1
|-
|-
| 49 || 29 || +33.7 ||  9 ||  2 ||  7
| 49 || 29 || +8.2 (+33.7%) ||  9 ||  2 ||  7
|-
|-
| 50 || 29 || -24.8 ||  8 ||  5 ||  3
| 50 || 29 || -6.0 (-24.8%) ||  8 ||  5 ||  3
|-
|-
| 53 || 31 ||  -0.3 ||  9 ||  4 ||  5
| 53 || 31 ||  -0.1 ( -0.3%) ||  9 ||  4 ||  5
|-
|-
| 55 || 32 || -17.3 ||  9 ||  5 ||  4
| 55 || 32 || -3.8 (-17.3%) ||  9 ||  5 ||  4
|-
|-
| 56 || 33 || +24.2 || 10 ||  3 ||  7
| 56 || 33 || +5.2 (+24.2%) || 10 ||  3 ||  7
|-
|-
| 59 || 35 || +48.7 || 11 ||  2 ||  9
| 59 || 35 || +9.9 (+48.7%) || 11 ||  2 ||  9
|-
|-
| 61 || 36 || +31.7 || 11 ||  3 ||  8
| 61 || 36 || +6.2 (+31.7%) || 11 ||  3 ||  8
|-
|-
| 63 || 37 || +14.7 || 11 ||  4 ||  7
| 63 || 37 || +2.8 (+14.7%) || 11 ||  4 ||  7
|-
|-
| 64 || 37 || -43.8 || 10 ||  7 ||  3
| 64 || 37 || -8.2 (-43.8%) || 10 ||  7 ||  3
|-
|-
| 65 || 38 ||  -2.3 || 11 ||  5 ||  6
| 65 || 38 ||  -0.4 ( -2.3%) || 11 ||  5 ||  6
|-
|-
| 67 || 39 || -19.2 || 11 ||  6 ||  5
| 67 || 39 || -3.4 (-19.2%) || 11 ||  6 ||  5
|-
|-
| 69 || 40 || -36.2 || 11 ||  7 ||  4
| 69 || 40 || -6.3 (-36.2%) || 11 ||  7 ||  4
|-
|-
| 70 || 41 ||  +5.3 || 12 ||  5 ||  7
| 70 || 41 ||  +0.9 ( +5.3%) || 12 ||  5 ||  7
|-
|-
| 71 || 42 || +46.8 || 13 ||  3 || 10
| 71 || 42 || +7.9 (+46.8%) || 13 ||  3 || 10
|-
|-
| 73 || 43 || +29.8 || 13 ||  4 ||  9
| 73 || 43 || +4.9 (+29.8%) || 13 ||  4 ||  9
|-
|-
| 74 || 43 || -28.7 || 12 ||  7 ||  5
| 74 || 43 || -4.7 (-28.7%) || 12 ||  7 ||  5
|-
|-
| 75 || 44 || +12.8 || 13 ||  5 ||  8
| 75 || 44 || +2.0 (+12.8%) || 13 ||  5 ||  8
|-
|-
| 77 || 45 ||  -4.2 || 13 ||  6 ||  7
| 77 || 45 ||  -0.7 ( -4.2%) || 13 ||  6 ||  7
|-
|-
| 79 || 46 || -21.2 || 13 ||  7 ||  6
| 79 || 46 || -3.2 (-21.2%) || 13 ||  7 ||  6
|-
|-
| 80 || 47 || +20.3 || 14 ||  5 ||  9
| 80 || 47 || +3.0 (+20.3%) || 14 ||  5 ||  9
|-
|-
| 81 || 47 || -38.2 || 13 ||  8 ||  5
| 81 || 47 || -5.7 (-38.2%) || 13 ||  8 ||  5
|-
|-
| 83 || 49 || +44.8 || 15 ||  4 || 11
| 83 || 49 || +6.5 (+44.8%) || 15 ||  4 || 11
|-
|-
| 88 || 51 || -47.7 || 14 ||  9 ||  5
| 88 || 51 || -6.5 (-47.7%) || 14 ||  9 ||  5
|-
|-
| 89 || 52 ||  -6.2 || 15 ||  7 ||  8
| 89 || 52 ||  -0.8 ( -6.2%) || 15 ||  7 ||  8
|-
|-
| 90 || 53 || +35.3 || 16 ||  5 || 11
| 90 || 53 || +4.7 (+35.3%) || 16 ||  5 || 11
|-
|-
| 91 || 53 || -23.2 || 15 ||  8 ||  7
| 91 || 53 || -3.1 (-23.2%) || 15 ||  8 ||  7
|-
|-
| 94 || 55 ||  +1.4 || 16 ||  7 ||  9
| 94 || 55 ||  +0.2 ( +1.4%) || 16 ||  7 ||  9
|-
|-
| 95 || 56 || +42.9 || 17 ||  5 || 12
| 95 || 56 || +5.4 (+42.9%) || 17 ||  5 || 12
|-
|-
| 97 || 57 || +25.9 || 17 ||  6 || 11
| 97 || 57 || +3.2 (+25.9%) || 17 ||  6 || 11
|-
|-
| 98 || 57 || -32.6 || 16 ||  9 ||  7
| 98 || 57 || -4.0 (-32.6%) || 16 ||  9 ||  7
|-
|-
| 99 || 58 ||  +8.9 || 17 ||  7 || 10
| 99 || 58 ||  +1.1 ( +8.9%) || 17 ||  7 || 10
|}
|}


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

Revision as of 17:34, 16 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 Chromas, shifts caused by one accidental, (7*P5 - 4*P8) use a positive amount of steps, we lose all EDOs below 12 EDO and also 13, 16, 18, and 23.

EDOs up to 100 are listed in the following table. The unit (if not stated otherwise) is steps of the corresponding EDO which is given in the first column of each row.

Octave/
Edo 		Fifth Fifth-detuning
abs(¢), rel(%) 		Whole tone Semitone Chroma
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
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