5ed5/4: Difference between revisions

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'''5ED5/4''' is the [[Equal-step tuning|equal division]] of the [[5/4|just major third]] into five parts of 77.2627 [[cent|cents]] each, corresponding to every second step of [[31edo]]. It is related to the [[Breedsmic temperaments|tertiaseptal temperament]] (which tempers out 2401/2400 and 65625/65536 in the 7-limit) and the [[Starling temperaments|valentine temperament]] (which tempers out 126/125 and 1029/1024 in the 7-limit).
{{Infobox ET}}
'''5ED5/4''' is the [[Equal-step tuning|equal division]] of the [[5/4|just major third]] into five parts of 77.2627 [[cent|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 [[Breedsmic temperaments|tertiaseptal temperament]] and the [[Starling temperaments|valentine temperament]]).


==Intervals==
{| class="wikitable"
{| class="wikitable"
|-
|-
! | degree
! | degree
! | cents value
! | cents value
! | corresponding <br>JI intervals
! | ratio
! | comments
|-
|-
| | 0
| | 0
| | 0.0000
| | 0.0000
| | '''exact [[1/1]]'''
| | '''[[1/1]]'''
| |
|-
|-
| | 1
| | 1
| | 77.2627
| | 77.2627
| | [[25/24]] (valentine)<br>[[22/21]] (valentine)<br>[[21/20]] (valentine)
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">1/5</font>
| |
|-
|-
| | 2
| | 2
| | 154.5255
| | 154.5255
| | [[12/11]] (valentine)<br>35/32<br>[[11/10]] (valentine)
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">2/5</font>
| |
|-
|-
| | 3
| | 3
| | 231.7882
| | 231.7882
| | [[8/7]]
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">3/5</font>
| |
|-
|-
| | 4
| | 4
| | 309.0510
| | 309.0510
| | [[6/5]] (valentine)
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">4/5</font>
| |
|-
|-
| | 5
| | 5
| | 386.3137
| | 386.3137
| | '''exact [[5/4]]'''
| | '''[[5/4]]'''
| | just major third
|-
|-
| | 6
| | 6
| | 463.5765
| | 463.5765
| | [[21/16]] (valentine)
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">6/5</font>
| |
|-
|-
| | 7
| | 7
| | 540.8392
| | 540.8392
| | [[11/8]] (valentine)
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">7/5</font>
| |
|-
|-
| | 8
| | 8
| | 618.1019
| | 618.1019
| | [[10/7]]
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">8/5</font>
| |
|-
|-
| | 9
| | 9
| | 695.3647
| | 695.3647
| | [[3/2]] (valentine)
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">9/5</font>
| |
|-
|-
| | 10
| | 10
| | 772.6274
| | 772.6274
| | '''exact [[25/16]]'''
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">2</font> = [[25/16]]
| | classic augmented fifth
|-
|-
| | 11
| | 11
| | 849.8902
| | 849.8902
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">11/5</font>
| |
|-
|-
| | 12
| | 12
| | 927.1529
| | 927.1529
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">12/5</font>
| |
|-
|-
| | 13
| | 13
| | 1004.4157
| | 1004.4157
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">13/5</font>
| |
|-
|-
| | 14
| | 14
| | 1081.6784
| | 1081.6784
| | [[28/15]] (tertiaseptal)
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">14/5</font>
| |
|-
|-
| | 15
| | 15
| | 1158.9411
| | 1158.9411
| | '''exact 125/64'''
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">3</font> = 125/64
| | octave minus diesis
|-
|-
| | 16
| | 16
| | 1236.2039
| | 1236.2039
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">16/5</font>
| |
|-
|-
| | 17
| | 17
| | 1313.4666
| | 1313.4666
| | [[16/15|32/15]] (tertiaseptal)
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">17/5</font>
| |
|-
|-
| | 18
| | 18
| | 1390.7294
| | 1390.7294
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">18/5</font>
| |
|-
|-
| | 19
| | 19
| | 1467.9921
| | 1467.9921
| | [[7/3]] (tertiaseptal)
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">19/5</font>
| |
|-
|-
| | 20
| | 20
| | 1545.2549
| | 1545.2549
| | '''exact 625/256'''
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">4</font> = 625/256
| | double augmented ninth
|-
|-
| | 21
| | 21
| | 1622.5176
| | 1622.5176
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">21/5</font>
| |
|-
|-
| | 22
| | 22
| | 1699.7803
| | 1699.7803
| | [[8/3]] (tertiaseptal)
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">22/5</font>
| |
|-
|-
| | 23
| | 23
| | 1777.0431
| | 1777.0431
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">23/5</font>
| |
|-
|-
| | 24
| | 24
| | 1854.3058
| | 1854.3058
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">24/5</font>
| |
|-
|-
| | 25
| | 25
| | 1931.5686
| | 1931.5686
| | '''exact 3125/1024'''
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">5</font> = 3125/1024
| | triple augmented eleventh<br>([[3/1]] plus [[magic comma]])
|-
|-
| | 26
| | 26
| | 2008.8313
| | 2008.8313
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">26/5</font>
| |
|-
|-
| | 27
| | 27
| | 2086.0941
| | 2086.0941
| | [[10/3]] (tertiaseptal)
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">27/5</font>
| |
|-
|-
| | 28
| | 28
| | 2163.3568
| | 2163.3568
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">28/5</font>
| |
|-
|-
| | 29
| | 29
| | 2240.6195
| | 2240.6195
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">29/5</font>
| |
|-
|-
| | 30
| | 30
| | 2317.8823
| | 2317.8823
| | '''exact 15625/4096'''
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">6</font> = 15625/4096
| | triple augmented thireenth<br>(two octaves minus double diesis)
|-
|-
| | 31
| | 31
| | 2395.1450
| | 2395.1450
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">31/5</font>
| |
|-
|-
| | 32
| | 32
| | 2472.4078
| | 2472.4078
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">32/5</font>
| |
|-
|-
| | 33
| | 33
| | 2549.6705
| | 2549.6705
| | [[12/11|48/11]]
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">33/5</font>
| |
|-
|-
| | 34
| | 34
| | 2626.9333
| | 2626.9333
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">34/5</font>
| |
|-
|-
| | 35
| | 35
| | 2704.1960
| | 2704.1960
| | '''exact 78125/16384'''
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">7</font> = 78125/16384
| |
|-
|-
| | 36
| | 36
| | 2781.4587
| | 2781.4587
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">36/5</font>
| |
|-
|-
| | 37
| | 37
| |  
| | 2858.7215
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">37/5</font>
| |
|-
|-
| | 38
| | 38
| |  
| | 2935.9842
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">38/5</font>
| |
|-
|-
| | 39
| | 39
| |  
| | 3013.2470
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">39/5</font>
| |
|-
|-
| | 40
| | 40
| |  
| | 3090.5097
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">8</font> = 390625/65536
| |
|-
|-
| | 41
| | 41
| |  
| | 3167.7725
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">41/5</font>
| |
|-
|-
| | 42
| | 42
| |  
| | 3245.0352
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">42/5</font>
| |
|-
|-
| | 43
| | 43
| |  
| | 3322.2979
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">43/5</font>
| |
|-
|-
| | 44
| | 44
| |  
| | 3399.5607
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">44/5</font>
| |
|-
|-
| | 45
| | 45
| |  
| | 3476.8234
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">9</font> = 1953125/262144
| |
|-
|-
| | 46
| | 46
| |  
| | 3554.0862
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">46/5</font>
| |
|-
|-
| | 47
| | 47
| |  
| | 3631.3489
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">47/5</font>
| |
|-
|-
| | 48
| | 48
| |  
| | 3708.6117
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">48/5</font>
| |
|-
|-
| | 49
| | 49
| |  
| | 3785.8744
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">49/5</font>
| |
|-
|-
| | 50
| | 50
| |  
| | 3863.1371
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">10</font> = 9765625/1048576
| |
|-
|-
| | 51
| | 51
| |  
| | 3940.3999
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">51/5</font>
| |
|-
|-
| | 52
| | 52
| |  
| | 4017.6626
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">52/5</font>
| |
|-
|-
| | 53
| | 53
| |  
| | 4094.9254
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">53/5</font>
| |
|-
|-
| | 54
| | 54
| |  
| | 4172.1881
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">54/5</font>
| |
|-
|-
| | 55
| | 55
| |  
| | 4249.4509
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">11</font> = 48828125/4194304
| |
|-
|-
| | 56
| | 56
| |  
| | 4326.7136
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">56/5</font>
| |
|-
|-
| | 57
| | 57
| |  
| | 4403.9763
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">57/5</font>
| |
|-
|-
| | 58
| | 58
| |  
| | 4481.2391
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">58/5</font>
| |
|-
|-
| | 59
| | 59
| |  
| | 4558.5018
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">59/5</font>
| |
|-
|-
| | 60
| | 60
| |  
| | 4635.7646
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">12</font> = 244140625/16777216
| |
|-
|-
| | 61
| | 61
| |  
| | 4713.0273
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">61/5</font>
| |
|-
|-
| | 62
| | 62
| |  
| | 4790.2901
| |  
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">62/5</font>
| |  
|-
| | 63
| | 4867.5528
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">63/5</font>
|-
| | 64
| | 4944.8155
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">64/5</font>
|-
| | 65
| | 5022.0783
| | (5/4)<font style="vertical-align:super;font-size:0.8em;">13</font> = 1220703125/67108864
|}
|}


[[Category:5/4]]
== Harmonics ==
{{Harmonics in equal
| steps = 5
| num = 5
| denom = 4
}}
{{Harmonics in equal
| steps = 5
| num = 5
| denom = 4
| start = 12
| collapsed = 1
}}
 
==5ED5/4 as a generator==
===Valentine===
Aside from 2100875/2097152, [[valentine]] temperament tempers out 126/125, 1029/1024, 6144/6125, and 64827/64000 in the 7-limit. It can be described as the 31&amp;46 temperament, and the step interval of 5ED5/4 (tuned between 256/245 and 22/21) can serve as its generator. In the 11-limit, it tempers out 121/120, 176/175, and 441/440.
 
===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&amp;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.
 
===Tertia===
Aside from 2100875/2097152, [[tertiaseptal|tertia]] temperament tempers out 385/384, 1331/1323, and 1375/1372 in the 11-limit. It can be described as the 31&amp;140 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 352/351, 385/384, 561/560, 625/624, and 715/714.
 
[[Category:Major third]]
[[Category:Equal-step tuning]]
[[Category:Equal-step tuning]]
[[Category:Edonoi]]

Latest revision as of 19:23, 1 August 2025

← 4ed5/4 5ed5/4 6ed5/4 →
Prime factorization 5 (prime)
Step size 77.2627 ¢ 
Octave 16\5ed5/4 (1236.2 ¢)
Twelfth 25\5ed5/4 (1931.57 ¢) (→ 5\1ed5/4)
Consistency limit 3
Distinct consistency limit 3

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).

Intervals

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

Harmonics

Approximation of harmonics in 5ed5/4
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +36.2 +29.6 -4.9 -4.9 -11.4 +30.7 +31.3 -18.0 +31.3 +20.9 +24.8
Relative (%) +46.9 +38.3 -6.3 -6.3 -14.8 +39.8 +40.6 -23.3 +40.6 +27.0 +32.0
Steps
(reduced) 		16
(1) 		25
(0) 		31
(1) 		36
(1) 		40
(0) 		44
(4) 		47
(2) 		49
(4) 		52
(2) 		54
(4) 		56
(1)
Approximation of harmonics in 5ed5/4
Harmonic 13 14 15 16 17 18 19 20 21 22 23
Error Absolute (¢) -36.6 -10.3 +24.8 -9.7 -37.4 +18.2 +1.8 -9.7 -16.9 -20.2 -19.9
Relative (%) -47.3 -13.4 +32.0 -12.6 -48.4 +23.5 +2.4 -12.6 -21.9 -26.1 -25.7
Steps
(reduced) 		57
(2) 		59
(4) 		61
(1) 		62
(2) 		63
(3) 		65
(0) 		66
(1) 		67
(2) 		68
(3) 		69
(4) 		70
(0)

5ED5/4 as a generator

Valentine

Aside from 2100875/2097152, valentine temperament tempers out 126/125, 1029/1024, 6144/6125, and 64827/64000 in the 7-limit. It can be described as the 31&46 temperament, and the step interval of 5ED5/4 (tuned between 256/245 and 22/21) can serve as its generator. In the 11-limit, it tempers out 121/120, 176/175, and 441/440.

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.

Tertia

Aside from 2100875/2097152, tertia temperament tempers out 385/384, 1331/1323, and 1375/1372 in the 11-limit. It can be described as the 31&140 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 352/351, 385/384, 561/560, 625/624, and 715/714.

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