User:Xenwolf/Fifthspan

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

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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