5ed5/4: Difference between revisions

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==5ED5/4 as tertiaseptal generator==
==5ED5/4 as a generator==
===Tertiaseptal===
Aside from 2100875/2097152, tertiaseptal temperament tempers out 2401/2400, 65625/65536, and 703125/702464 in the 7-limit. It can be described as the 31&171 temperament, and the step interval of 5ED5/4 (tuned between 256/245 and 68/65) can serve as its generator. In the 17-limit, it tempers out 243/242, 375/374, 441/440, 625/624, and 3584/3575.
Aside from 2100875/2097152, tertiaseptal temperament tempers out 2401/2400, 65625/65536, and 703125/702464 in the 7-limit. It can be described as the 31&171 temperament, and the step interval of 5ED5/4 (tuned between 256/245 and 68/65) can serve as its generator. In the 17-limit, it tempers out 243/242, 375/374, 441/440, 625/624, and 3584/3575.


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|-
|-
! | generator
! | generator
! | cents value <sup>a</sup>
! | cents value <sup>a</sup><br>(octave-reduced)
! | ratio<br>(octave-reduced)
! | 17-limit ratio<br>(octave-reduced)
|-
|-
| | 1
| | 1
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| | 154.4
| |  
| | [[35/32]]
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| | 849.2
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| | [[18/11]]
| | 44/27, [[18/11]]
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| | 1158.0
| | 1158.0
| |  
| | 39/20
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|-
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| | 16
| | 35.2
| | 35.2
| |  
| | 55/54, 52/51, 51/50, [[50/49]], [[49/48]], [[45/44]]
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| | 17
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| |  
| | 39/35
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| | 344.0
| | 344.0
| |  
| | 39/32
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| | 421.2
| | 421.2
| |  
| | [[51/40]]
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| | 575.6
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| |  
| | 39/28
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|-
| | 24
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| | 652.8
| | 652.8
| |  
| | [[35/24]]
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|-
| | 25
| | 25
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| | 26
| | 26
| | 807.2
| | 807.2
| |  
| | 51/32
|-
|-
| | 27
| | 27
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| | 28
| | 28
| | 961.6
| | 961.6
| |  
| | 68/39
|-
|-
| | 29
| | 29
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| | 39
| | 39
| | 610.9
| | 610.9
| |  
| | [[64/45]]
|-
|-
| | 40
| | 40
| | 688.1
| | 688.1
| |  
| | 52/35
|-
|-
| | 41
| | 41
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| | 46
| | 46
| | 1151.3
| | 1151.3
| |  
| | 68/35, [[35/18]]
|-
|-
| | 47
| | 47
| | 28.5
| | 28.5
| |  
| | [[65/64]], [[64/63]], [[56/55]]
|-
|-
| | 48
| | 48
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| | 50
| | 50
| | 260.1
| | 260.1
| |  
| | [[64/55]]
|-
|-
| | 51
| | 51
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| | 57
| | 57
| | 800.5
| | 800.5
| |  
| | 35/22
|-
|-
| | 58
| | 58
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| | 63
| | 63.7
| | 63.7
| |  
| | [[28/27]]
|-
|-
| | 64
| | 64
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| | 66
| | 295.3
| | 295.3
| |  
| | [[32/27]]
|-
|-
| | 67
| | 67
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| | 68
| | 68
| | 449.7
| | 449.7
| |  
| | [[35/27]]
|-
|-
| | 69
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| | 71
| | 681.3
| | 681.3
| |  
| | [[40/27]]
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|-
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| | 74
| | 912.9
| | 912.9
| |  
| | 56/33
|-
|-
| | 75
| | 75
| | 990.1
| | 990.1
| |  
| | 39/22
|-
|-
| | 76
| | 76
| | 1067.3
| | 1067.3
| |  
| | 50/27
|-
|-
| | 77
| | 77
| | 1144.5
| | 1144.5
| |  
| | 64/33
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|-
| | 78
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| | 98.9
| | 98.9
| |  
| | 35/33
|-
|-
| | 80
| | 80
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| | 81
| | 81
| | 253.3
| | 253.3
| |  
| | 52/45
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|-
| | 82
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| | 330.5
| | 330.5
| |  
| | 40/33
|-
|-
| | 83
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| | 716.5
| | 716.5
| |  
| | 68/45, 50/33
|-
|-
| | 88
| | 88
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| | 751.7
| | [[17/11]]
| | [[17/11]]
|-
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| | 828.9
| |
|-
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| | 906.1
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|-
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| | 983.3
| |
|-
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| | 1060.5
| |
|-
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| | 1137.8
| | 52/27
|-
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| | 15.0
| |
|-
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| | 92.2
| |
|-
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| | 169.4
| |
|-
| | 112
| | 246.6
| |
|-
| | 113
| | 323.8
| |
|-
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| | 401.0
| | 34/27
|-
| | 115
| | 478.2
| |
|-
| | 116
| | 555.4
| |
|-
| | 117
| | 632.6
| |
|-
| | 118
| | 709.8
| |
|-
| | 119
| | 787.0
| | 52/33
|-
| | 120
| | 864.2
| |
|-
| | 121
| | 941.4
| |
|-
| | 122
| | 1018.6
| |
|-
| | 123
| | 1095.8
| |
|-
| | 124
| | 1173.0
| | 65/33
|-
| | 125
| | 50.2
| | 34/33
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<sup>a</sup> in 17-limit POTE tuning
<sup>a</sup> in 17-limit POTE tuning

Revision as of 12:45, 22 January 2019

5ED5/4 is the equal division of the just major third into five parts of 77.2627 cents each, corresponding to every second step of 31edo. It is related to Carlos Alpha and the 7-limit temperaments which temper out 2100875/2097152 (including the tertiaseptal temperament and the valentine temperament).

degree cents value ratio
0 0.0000 1/1
1 77.2627 (5/4)1/5
2 154.5255 (5/4)2/5
3 231.7882 (5/4)3/5
4 309.0510 (5/4)4/5
5 386.3137 5/4
6 463.5765 (5/4)6/5
7 540.8392 (5/4)7/5
8 618.1019 (5/4)8/5
9 695.3647 (5/4)9/5
10 772.6274 (5/4)2 = 25/16
11 849.8902 (5/4)11/5
12 927.1529 (5/4)12/5
13 1004.4157 (5/4)13/5
14 1081.6784 (5/4)14/5
15 1158.9411 (5/4)3 = 125/64
16 1236.2039 (5/4)16/5
17 1313.4666 (5/4)17/5
18 1390.7294 (5/4)18/5
19 1467.9921 (5/4)19/5
20 1545.2549 (5/4)4 = 625/256
21 1622.5176 (5/4)21/5
22 1699.7803 (5/4)22/5
23 1777.0431 (5/4)23/5
24 1854.3058 (5/4)24/5
25 1931.5686 (5/4)5 = 3125/1024
26 2008.8313 (5/4)26/5
27 2086.0941 (5/4)27/5
28 2163.3568 (5/4)28/5
29 2240.6195 (5/4)29/5
30 2317.8823 (5/4)6 = 15625/4096
31 2395.1450 (5/4)31/5
32 2472.4078 (5/4)32/5
33 2549.6705 (5/4)33/5
34 2626.9333 (5/4)34/5
35 2704.1960 (5/4)7 = 78125/16384
36 2781.4587 (5/4)36/5
37 2858.7215 (5/4)37/5
38 2935.9842 (5/4)38/5
39 3013.2470 (5/4)39/5
40 3090.5097 (5/4)8 = 390625/65536
41 3167.7725 (5/4)41/5
42 3245.0352 (5/4)42/5
43 3322.2979 (5/4)43/5
44 3399.5607 (5/4)44/5
45 3476.8234 (5/4)9 = 1953125/262144
46 3554.0862 (5/4)46/5
47 3631.3489 (5/4)47/5
48 3708.6117 (5/4)48/5
49 3785.8744 (5/4)49/5
50 3863.1371 (5/4)10 = 9765625/1048576
51 3940.3999 (5/4)51/5
52 4017.6626 (5/4)52/5
53 4094.9254 (5/4)53/5
54 4172.1881 (5/4)54/5
55 4249.4509 (5/4)11 = 48828125/4194304
56 4326.7136 (5/4)56/5
57 4403.9763 (5/4)57/5
58 4481.2391 (5/4)58/5
59 4558.5018 (5/4)59/5
60 4635.7646 (5/4)12 = 244140625/16777216
61 4713.0273 (5/4)61/5
62 4790.2901 (5/4)62/5
63 4867.5528 (5/4)63/5
64 4944.8155 (5/4)64/5
65 5022.0783 (5/4)13 = 1220703125/67108864

5ED5/4 as a generator

Tertiaseptal

Aside from 2100875/2097152, tertiaseptal temperament tempers out 2401/2400, 65625/65536, and 703125/702464 in the 7-limit. It can be described as the 31&171 temperament, and the step interval of 5ED5/4 (tuned between 256/245 and 68/65) can serve as its generator. In the 17-limit, it tempers out 243/242, 375/374, 441/440, 625/624, and 3584/3575.

generator cents value a
(octave-reduced) 		17-limit ratio
(octave-reduced)
1 77.2
2 154.4 35/32
3 231.6 8/7
4 308.8
5 386.0 5/4
6 463.2 17/13
7 540.4
8 617.6 10/7
9 694.8
10 772.0 25/16
11 849.2 44/27, 18/11
12 926.4
13 1003.6 25/14
14 1080.8 28/15
15 1158.0 39/20
16 35.2 55/54, 52/51, 51/50, 50/49, 49/48, 45/44
17 112.4 16/15
18 189.6 39/35
19 266.8 7/6
20 344.0 39/32
21 421.2 51/40
22 498.4 4/3
23 575.6 39/28
24 652.8 35/24
25 730.0 32/21
26 807.2 51/32
27 884.4 5/3
28 961.6 68/39
29 1038.8
30 1116.0 40/21, 21/11
31 1193.2
32 70.4 26/25, 25/24
33 147.6 12/11
34 224.8
35 302.0 25/21
36 379.3
37 456.5 13/10
38 533.7 34/25, 15/11
39 610.9 64/45
40 688.1 52/35
41 765.3 14/9
42 842.5 13/8
43 919.7 17/10
44 996.9 16/9
45 1074.1 13/7
46 1151.3 68/35, 35/18
47 28.5 65/64, 64/63, 56/55
48 105.7 17/16
49 182.9 10/9
50 260.1 64/55
51 337.3 17/14
52 414.5 14/11
53 491.7
54 568.9 25/18
55 646.1 16/11
56 723.3
57 800.5 35/22
58 877.7
59 954.9 26/15
60 1032.1 20/11
61 1109.3
62 1186.5
63 63.7 28/27
64 140.9 13/12
65 218.1 17/15, 25/22
66 295.3 32/27
67 372.5 26/21
68 449.7 35/27
69 526.9
70 604.1 17/12
71 681.3 40/27
72 758.5
73 835.7 34/21
74 912.9 56/33
75 990.1 39/22
76 1067.3 50/27
77 1144.5 64/33
78 21.7
79 98.9 35/33
80 176.1
81 253.3 52/45
82 330.5 40/33
83 407.7
84 484.9
85 562.1
86 639.3 13/9
87 716.5 68/45, 50/33
88 793.7
89 870.9
90 948.1
91 1025.3
92 1102.5 17/9
93 1179.7
94 56.9
95 134.1
96 211.3
97 288.5 13/11
98 365.7
99 442.9
100 520.1
101 597.3
102 674.5
103 751.7 17/11
104 828.9
105 906.1
106 983.3
107 1060.5
108 1137.8 52/27
109 15.0
110 92.2
111 169.4
112 246.6
113 323.8
114 401.0 34/27
115 478.2
116 555.4
117 632.6
118 709.8
119 787.0 52/33
120 864.2
121 941.4
122 1018.6
123 1095.8
124 1173.0 65/33
125 50.2 34/33

a in 17-limit POTE tuning

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