User:Xenwolf/Fifthspan

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Fifthspan examples

Prime → 2 3 5 7 11 13 17 19 23
EDO
↓ 		Unique fifth
(mapping)
↓ 		↓ (octave-reduced corresponding primary intervals) ↓
(1/1) (3/2) (5/4) (7/4) (11/8) (13/8) (17/16) (19/16) (23/16)
1edo N/A
2edo 1\2 0 +1 +1 0 +1 +1 0 0 +1
3edo 2\3 0 +1 -1 +1 -1 +1 0 -1 +1
4edo N/A (2x 2edo)
5edo 3\5 0 +1 -1 -2 -1 -2 0 +2 +1
6edo N/A (2x 3edo)
7edo 4\7 0 +1 -3 -2 -1 +3 +2 -3 +1
8edo 5\8 0 +1 -1 -2 +4 -2 -3 +2 +4
9edo 5\9 0 +1 -3 -4 -1 +3 +2 +4 +1
10edo N/A (2x 5edo)
11edo 6\11 0 +1 -3 -4 -1 +5 +2 -5 +1
12edo 7\12 0 +1 +4 -2 +6 -4 -5 -3 +6
13edo 8\13 0 +1 -6 -2 +4 +6 +5 +2 -4
14edo N/A (2x 7edo)
15edo N/A (3x 5edo)
16edo 9\16 0 +1 -3 +5 -1 +3 -7 +4 +8
17edo 10\17 0 +1 -8 -2 -6 +8 -5 -3 +6
18edo 11\18 0 +1 -6 +3 +4 -7 -8 +2 +9
19edo 11\19 0 +1 +4 -9 +6 -4 -5 -3 -6
20edo N/A (4x 5edo)
21edo N/A (3x 7edo)
22edo 13\22 0 +1 +9 -2 -6 -9 -10 -3 +6
23edo 13\23 0 +1 -3 +5 -8 +3 +9 +4 +8
24edo N/A (2x 12edo)
25edo N/A (5x 5edo)
26edo 15\26 0 +1 +4 -9 +6 -4 -12 -10 -6
27edo 16\27 0 +1 +9 -2 -6 +13 -10 -8 +11
28edo N/A (4x 7edo)
29edo 17\29 0 +1 -8 -14 +11 +8 +7 -3 +6
30edo N/A (6x 5edo)
31edo 18\31 0 +1 +4 +10 -13 +15 -5 -3 -6
32edo 19\32 0 +1 +14 -2 -11 -14 -15 -8 +11
33edo 19\33 0 +1 +11 -9 +6 -4 -12 -10 -13
34edo N/A (2x 17edo)
35edo N/A (5x 7edo)
36edo N/A (3x 12edo)
37edo 22\37 0 +1 +14 -2 -11 +18 -15 -8 +16
38edo N/A (2x 19edo)
39edo 23\39 0 +1 -13 -19 -6 -9 +12 +14 -11
40edo 23\40 0 +1 +11 -16 +6 -4 -19 -10 -13
41edo 24\41 0 +1 -8 -14 -18 +20 +7 -3 +6
42edo 25\42 0 +1 +14 -2 -11 -19 -20 -8 +16
43edo 25\43 0 +1 +4 +10 +18 -16 -5 -3 -18
44edo N/A (2x 22edo)
45edo 26\45 0 +1 +4 -9 +6 +22 +14 +16 -6
46edo 27\46 0 +1 +21 +15 +11 +8 -22 -3 +6
47edo 27\47 0 +1 +11 -16 +13 -4 -19 -10 -13
48edo N/A (4x 12edo)
49edo 29\49 0 +1 +9 -2 +16 +13 -10 +19 -16
50edo 29\50 0 +1 +4 +10 -13 +15 -24 -22 -6
51edo N/A (3x 17edo)
52edo N/A (2x 26edo)
53edo 31\53 0 +1 -8 -14 +23 +20 +7 -3 +18
54edo N/A (2x 27edo)
55edo 32\55 0 +1 +4 +22 -25 +27 -5 -3 -18
56edo 33\56 0 +1 +26 -19 -6 -9 -27 +14 -11
57edo N/A (3x 19edo)
58edo N/A (2x 29edo)
59edo 35\59 0 +1 -18 -2 +21 -14 +17 -8 +11
60edo N/A (5x 12edo)
61edo 36\61 0 +1 -13 +20 -6 +30 +12 -25 +28
62edo N/A (2x 31edo)
63edo 37\63 0 +1 -25 +15 +11 +8 +24 -20 +6
64edo 37\64 0 +1 -15 -28 +25 -23 +14 +16 -6
65edo 38\65 0 +1 -8 -26 -30 +32 +7 -3 +18
66edo N/A (3x 22edo)
67edo 39\67 0 +1 +4 +22 +30 -28 -5 -3 -18
68edo N/A (4x 17edo)
69edo 40\69 0 +1 +4 +29 -13 +15 -24 -22 -6
70edo 41\70 0 +1 +33 +27 -18 -21 -34 -3 -23
71edo 42\71 0 +1 +9 -24 +16 +35 -10 -30 +33
72edo N/A (6x 12edo)
73edo 43\73 0 +1 -30 -19 -6 -9 +29 +14 -11
74edo 43\74 0 +1 +4 +10 +18 -16 -36 -34 +25
75edo 44\75 0 +1 +21 -31 +11 +37 -22 +26 +6
76edo N/A (4x 19edo)
77edo 45\77 0 +1 -8 -26 +35 +32 +7 -3 +18
78edo N/A (2x 39edo)
79edo 46\79 0 +1 +16 +22 -37 -28 -5 -3 -18
80edo 47\80 0 +1 +38 +15 +11 +8 -39 -20 +6
81edo 47\81 0 +1 +4 +10 -13 +15 +26 +28 -6
82edo N/A (2x 41edo)
83edo 49\83 0 +1 -13 +20 -6 -31 +12 +36 -33
84edo N/A (7x 12edo)
85edo N/A (5x 17edo)
86edo N/A (2x 43edo)
87edo N/A (3x 29edo)
88edo 51\88 0 +1 +4 +29 -32 +34 -24 -22 -6
89edo 52\89 0 +1 -8 -26 -42 +32 +7 -3 +30
90edo 53\90 0 +1 +43 -19 -23 -9 -44 +14 -11
91edo 53\91 0 +1 +16 +34 +42 -40 -5 -3 -30
92edo N/A (2x 46edo)
93edo N/A (3x 31edo)
94edo 55\94 0 +1 -8 -14 +23 +20 -46 -3 -35
95edo 56\95 0 +1 +26 +37 -6 +47 -27 +14 +45
96edo N/A (8x 12edo)
97edo 57\97 0 +1 -42 +32 +11 +8 -39 -20 +6
98edo 57\98 0 +1 +4 -33 -25 +27 -5 -46 +37
99edo 58\99 0 +1 -37 -43 -18 -21 +36 -32 -23

EDO information with fifthspans

5edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0 +18 +94 −9 −71 +119 −105 −58 +92
relative (%) +0 +8 +39 −4 −30 +50 −44 −24 +38
Mapping patent val v 5 8 12 14 17 19 20 21 23
v (mod 5) 0 3 2 4 2 4 0 1 3
Fifthspan 0 +1 −1 −2 −1 −2 0 +2 +1

7edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 −16.2 −43.5 +59.7 −37.0 +16.6 +66.5 +45.3 +57.4
relative (%) +0 −9 −25 +35 −22 +10 +39 +26 +34
Mapping patent val v 7 11 16 20 24 26 29 30 32
v (mod 7) 0 4 2 6 3 5 1 2 4
Fifthspan 0 +1 −3 −2 −1 +3 +2 −3 +1

8edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 +48.0 +63.7 −68.8 +48.7 +59.5 +45.0 +2.5 −28.3
relative (%) +0 +32 +42 −46 +32 +40 +30 +2 −19
Mapping patent val v 8 13 19 22 28 30 33 34 36
v (mod 8) 0 5 3 6 4 6 1 2 4
Fifthspan 0 +1 −1 −2 +4 −2 −3 +2 +4

9edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 −35.3 +13.7 −35.5 −18.0 −40.5 +28.4 −30.8 +38.4
relative (%) +0 −26 +10 −27 −13 −30 +21 −23 +29
Mapping patent val v 9 14 21 25 31 33 37 38 41
v (mod 9) 0 5 3 7 4 6 1 2 5
Fifthspan 0 +1 −3 −4 −1 +3 +2 +4 +1

11edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 −47.4 +50.0 +13.0 −5.9 +32.2 +4.1 +29.8 +26.3
relative (%) +0 −43 +46 +12 −5 +30 +4 +27 +24
Mapping patent val v 11 17 26 31 38 41 45 47 50
v (mod 11) 0 6 4 9 5 8 1 3 6
Fifthspan 0 +1 −3 −4 −1 +5 +2 −5 +1

12edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 −2.0 +13.7 +31.2 +48.7 −40.5 −5.0 +2.5 −28.3
relative (%) +0 −2 +14 +31 +49 −41 −5 +2 −28
Mapping patent val v 12 19 28 34 42 44 49 51 54
v (mod 12) 0 7 4 10 6 8 1 3 6
Fifthspan 0 +1 +4 −2 +6 −4 −5 −3 +6

13edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 +36.5 −17.1 −45.7 +2.5 −9.8 −12.6 −20.6 +17.9
relative (%) +0 +40 −19 −50 +3 −11 −14 −22 +19
Mapping patent val v 13 21 30 36 45 48 53 55 59
v (mod 13) 0 8 4 10 6 9 1 3 7
Fifthspan 0 +1 −6 −2 +4 +6 +5 +2 −4

16edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 −27.0 −11.3 +6.2 −26.3 −15.5 −30.0 +2.5 −28.3
relative (%) +0 −36 −15 +8 −35 −21 −40 +3 −38
Mapping patent val v 16 25 37 45 55 59 65 68 72
v (mod 16) 0 9 5 13 7 11 1 4 8
Fifthspan 0 +1 −3 +5 −1 +3 −7 +4 +8

17edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 +3.9 −33.4 +19.4 +13.4 +6.5 −34.4 −15.2 +7.0
relative (%) +0 +6 −47 +27 +19 +9 −49 −21 +10
Mapping patent val v 17 27 39 48 59 63 69 72 77
v (mod 17) 0 10 5 14 8 12 1 4 9
Fifthspan 0 +1 −8 −2 −6 +8 −5 −3 +6

18edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 +31.4 +13.7 +31.2 −18.0 +26.1 +28.4 −30.8 −28.3
relative (%) +0 +47 +21 +47 −27 +39 +43 −46 −42
Mapping patent val v 18 29 42 51 62 67 74 76 81
v (mod 18) 0 11 6 15 8 13 2 4 9
Fifthspan 0 +1 −6 +3 +4 −7 −8 +2 +9

19edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 −7.2 −7.4 −21.5 +17.1 −19.5 +21.4 +18.3 +3.3
relative (%) +0 −11 −12 −34 +27 −31 +34 +29 +5
Mapping patent val v 19 30 44 53 66 70 78 81 86
v (mod 19) 0 11 6 15 9 13 2 5 10
Fifthspan 0 +1 +4 −9 +6 −4 −5 −3 −6

22edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 +7.1 −4.5 +13.0 −5.9 −22.3 +4.1 −24.8 +26.3
relative (%) +0 +13 −8 +24 −11 −41 +8 −45 +48
Mapping patent val v 22 35 51 62 76 81 90 93 100
v (mod 22) 0 13 7 18 10 15 2 5 12
Fifthspan 0 +1 +9 −2 −6 −9 −10 −3 +6

23edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 −23.7 −21.1 +22.5 +22.6 −5.7 −0.6 +15.5 −2.2
relative (%) +0 −45 −40 +43 +43 −11 −1 +30 −4
Mapping patent val v 23 36 53 65 80 85 94 98 104
v (mod 23) 0 13 7 19 11 16 2 6 12
Fifthspan 0 +1 −3 +5 −8 +3 +9 +4 +8

26edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 −9.6 −17.1 +0.4 +2.5 −9.8 −12.6 −20.6 +17.9
relative (%) +0 −21 −37 +1 +5 −21 −27 −45 +39
Mapping patent val v 26 41 60 73 90 96 106 110 118
v (mod 26) 0 15 8 21 12 18 2 6 14
Fifthspan 0 +1 +4 −9 +6 −4 −12 −10 −6

27edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 +9.2 +13.7 +9.0 −18.0 +3.9 −16.1 +13.6 −6.1
relative (%) +0 +21 +31 +20 −40 +9 −36 +31 −14
Mapping patent val v 27 43 63 76 93 100 110 115 122
v (mod 27) 0 16 9 22 12 19 2 7 14
Fifthspan 0 +1 +9 −2 −6 +13 −10 −8 +11

29edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 +1.5 −13.9 −17.1 −13.4 −12.9 +19.2 −7.9 −7.6
relative (%) +0 +4 −34 −41 −32 −31 +46 −19 −18
Mapping patent val v 29 46 67 81 100 107 119 123 131
v (mod 29) 0 17 9 23 13 20 3 7 15
Fifthspan 0 +1 −8 −14 +11 +8 +7 −3 +6

31edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 −5.2 +0.8 −1.1 −9.4 +11.1 +11.2 +12.2 −8.9
relative (%) +0 −13 +2 −3 −24 +29 +29 +31 −23
Mapping patent val v 31 49 72 87 107 115 127 132 140
v (mod 31) 0 18 10 25 14 22 3 8 16
Fifthspan 0 +1 +4 +10 −13 +15 −5 −3 −6

39edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 +5.7 +13.7 −15.0 +2.5 −9.8 −12.6 +10.2 −12.9
relative (%) +0 +19 +44 −49 +8 −32 −41 +33 −42
Mapping patent val v 39 62 91 109 135 144 159 166 176
v (mod 39) 0 23 13 31 18 27 3 10 20
Fifthspan 0 +1 −13 −19 −6 −9 +12 +14 −11

41edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 +0.5 −5.8 −3.0 +4.8 +8.3 +12.1 −4.8 −13.6
relative (%) +0 +2 −20 −10 +16 +28 +41 −17 −47
Mapping patent val v 41 65 95 115 142 152 168 174 185
v (mod 41) 0 24 13 33 19 29 4 10 21
Fifthspan 0 +1 −8 −14 −18 +20 +7 −3 +6

46edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.0 +2.4 +5.0 −3.6 −3.5 −5.7 −0.6 −10.6 −2.2
relative (%) +0 +9 +19 −14 −13 −22 −2 −40 −8
Mapping patent val v 46 73 107 129 159 170 188 195 208
v (mod 46) 0 27 15 37 21 32 4 11 24
Fifthspan 0 +1 +21 +15 +11 +8 −22 −3 +6

53edo

Prime number 2 3 5 7 11 13 17 19 23
Error absolute (¢) +0.00 −0.07 −1.41 +4.76 −7.92 −2.79 +8.25 −3.17 +5.69
relative (%) +0 −0 −6 +21 −35 −12 +36 −14 +25
Mapping patent val v 53 84 123 149 183 196 217 225 240
v (mod 53) 0 31 17 43 24 37 5 13 28
Fifthspan 0 +1 −8 −14 +23 +20 +7 −3 +18

Separate mod steps

Shown on the example of 17edo.

All fifthspans are given as positive values. (This can be transformed into mixed mode by subtracting 17 from all values > 17/2 == 8.5)

prime 2 prime 3 prime 5 prime 7 prime 11 prime 13 prime 17 prime 19 prime 23
Error absolute (¢) 0.0 +3.9 -33.4 +19.4 +13.4 +6.5 -34.3 -15.2 +7.0
relative (%) 0 +6 -47 +27 +19 +9 -49 -21 +10
Mapping value 17 27 39 48 59 63 69 72 77
~ (mod 17) 0 10 5 14 8 12 1 4 9
Fifthspan (in steps) 0 1 9 15 11 8 12 14 6
Approximation of prime intervals in 12 EDO
Prime interval 2 3 5 7 11 13 17 19
Error absolute (¢) 0.0 -2.0 +13.7 +31.2 +48.7 -40.5 -5.0 +2.5
relative (%) 0 -2 +14 +31 +49 -41 -5 +2
Mapping patent val v 12 19 28 34 42 44 49 51
v mod 12 0 7 4 10 6 8 1 3
Approximation of prime intervals in 17 EDO
Prime number 2 3 5 7 11 13 17 19
Error absolute (¢) +0.0 +3.9 −33.4 +19.4 +13.4 +6.5 −34.4 −15.2
relative (%) +0 +6 −47 +27 +19 +9 −49 −21
Mapping patent val v 17 27 39 48 59 63 69 72
v (mod 17) 0 10 5 14 8 12 1 4
Approximation of prime intervals in 17 EDO
Prime number 2 3 5 7 11 13 17 19
Error absolute (¢) +0.0 +3.9 −33.4 +19.4 +13.4 +6.5 −34.4 −15.2
relative (%) +0 +6 −47 +27 +19 +9 −49 −21
Mapping patent val v 17 27 39 48 59 63 69 72
v (mod 17) 0 10 5 14 8 12 1 4
Fifthspan 0 1 −8 −2 −6 8 −5 −3

Fifthspans of 17edo steps

step span
0 0
1 12
2 7
3 2
4 14
5 9
6 4
7 16
8 11
9 6
10 1
11 13
12 8
13 3
14 15
15 10
16 5
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