Septidiasemi: Difference between revisions

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'''Septidiasemi''', in this article, is the rank-two regular temperament for the 2.3.5.7.13.17 subgroup defined by tempering out 2401/2400 and 2152828125/2147483648 in the 7-limit; 2205/2197 and 4096/4095 in the 13-limit; 833/832, 1275/1274, and 2025/2023 in the 17-limit.
{{todo|cleanup}}
'''Septidiasemi''', in this article, is the [[Rank-2 temperament|rank-two regular temperament]] for the 2.3.5.7.13.17 [[subgroup]] defined by [[tempering out]] 2401/2400 and 2152828125/2147483648 in the 7-limit; 2205/2197 and 4096/4095 in the 13-limit; 833/832, 1275/1274, and 2025/2023 in the 17-limit.


It can be seen as implying a rank-two tuning which is [[generator|generated]] by a large semitone of about 119.3 cents which represents [[15/14]] (septimal diatonic semitone) and from this it derives its name. Equal temperaments that support septidiasemi include [[10edo|10EDO]] (generator 1\10), [[151edo|151EDO]] (generator 15\151), [[161edo|161EDO]] (generator 16\161), and [[171edo|171EDO]] (generator 17\171).
It can be seen as implying a rank-two tuning which is [[generator|generated]] by a large semitone of about 119.3 cents which represents [[15/14]] (septimal diatonic semitone) and from this it derives its name. Equal temperaments that support septidiasemi include [[10edo|10EDO]] (generator 1\10), [[151edo|151EDO]] (generator 15\151), [[161edo|161EDO]] (generator 16\161), and [[171edo|171EDO]] (generator 17\171).
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| 2
| 2
| 238.593
| 238.593
|  
| 39/34
|-
|-
| 3
| 3
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| 4
| 4
| 477.186
| 477.186
|  
| 112/85, 120/91
|-
|-
| 5
| 5
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| 6
| 6
| 715.779
| 715.779
|  
| 68/45
|-
|-
| 7
| 7
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| 10
| 10
| 1192.965
| 1192.965
|  
| 448/225, 255/128, <br>544/273, 576/289
|-
|-
| 11
| 11
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| 13
| 13
| 350.855
| 350.855
|  
| 60/49, 49/40
|-
|-
| 14
| 14
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| 16
| 16
| 708.745
| 708.745
|  
| 128/85
|-
|-
| 17
| 17
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| 19
| 19
| 1066.634
| 1066.634
|  
| (close to 50/27)
|-
|-
| 20
| 20
| 1185.931
| 1185.931
|  
| 119/60, 135/68
|-
|-
| 21
| 21
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| 23
| 23
| 343.820
| 343.820
|  
| 39/32
|-
|-
| 24
| 24
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| 30
| 30
| 1178.896
| 1178.896
|  
| (close to 160/81)
|-
|-
| 31
| 31
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| 36
| 36
| 694.676
| 694.676
|  
| 112/75
|-
|-
| 37
| 37
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| 40
| 40
| 1171.862
| 1171.862
| 63/32~128/65
| 63/32, 128/65
|-
|-
| 41
| 41
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| 60
| 60
| 1157.793
| 1157.793
|  
| 39/20
|-
|-
| 61
| 61
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| 81
| 81
| 63.020
| 63.020
|  
| (close to 28/27)
|-
|-
| 82
| 82
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| 88
| 88
| 898.096
| 898.096
|  
| 42/25
|-
|-
| 89
| 89
| 1017.392
| 1017.392
| 9/5
| 9/5
|-
| 90
| 1136.689
| 27/14
|}
|}


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== See also ==
== See also ==
* [[15/14]] - septimal diatonic semitone
* [[15/14]] - septimal diatonic semitone
* [[15/14ths equal temperament|15/14s equal temperament]]


[[Category:Temperament]]
[[Category:Septidiasemi| ]] <!-- main article -->
[[Category:Breed]]
[[Category:Rank-2 temperaments]]
[[Category:Breedsmic temperaments]]

Latest revision as of 07:46, 2 December 2025

Septidiasemi, in this article, is the rank-two regular temperament for the 2.3.5.7.13.17 subgroup defined by tempering out 2401/2400 and 2152828125/2147483648 in the 7-limit; 2205/2197 and 4096/4095 in the 13-limit; 833/832, 1275/1274, and 2025/2023 in the 17-limit.

It can be seen as implying a rank-two tuning which is generated by a large semitone of about 119.3 cents which represents 15/14 (septimal diatonic semitone) and from this it derives its name. Equal temperaments that support septidiasemi include 10EDO (generator 1\10), 151EDO (generator 15\151), 161EDO (generator 16\161), and 171EDO (generator 17\171).

See Breedsmic temperaments #Septidiasemi for more technical data.

Intervals

Generator
steps 		Cents* Approximate
ratios†
0 0.000 1/1
1 119.297 15/14
2 238.593 39/34
3 357.890 16/13
4 477.186 112/85, 120/91
5 596.483 24/17
6 715.779 68/45
7 835.076 34/21
8 954.372 26/15
9 1073.669 13/7
10 1192.965 448/225, 255/128,
544/273, 576/289
11 112.262 16/15
12 231.559 8/7
13 350.855 60/49, 49/40
14 470.152 21/16
15 589.448 45/32
16 708.745 128/85
17 828.041 21/13
18 947.338
19 1066.634 (close to 50/27)
20 1185.931 119/60, 135/68
21 105.227 17/16
22 224.524
23 343.820 39/32
24 463.117 17/13
25 582.414 7/5
26 701.710 3/2
27 821.007
28 940.303
29 1059.600 24/13
30 1178.896 (close to 160/81)
31 98.193 18/17
32 217.489 17/15
33 336.786 17/14
34 456.082 13/10
35 575.379
36 694.676 112/75
37 813.972 8/5
38 933.269 12/7
39 1052.565
40 1171.862 63/32, 128/65
41 91.158
42 210.455
43 329.751
44 449.048
45 568.344
46 687.641
47 806.937
48 926.234
49 1045.531 64/35
50 1164.827
51 84.124 21/20
52 203.420 9/8
53 322.717
54 442.013
55 561.310 18/13
56 680.606
57 799.903
58 919.199 17/10
59 1038.496
60 1157.793 39/20
61 77.089
62 196.386 28/25
63 315.682 6/5
64 434.979 9/7
65 554.275 (close to 11/8)
66 673.572
67 792.868
68 912.165
69 1031.461
70 1150.758
71 70.054
72 189.351
73 308.648
74 427.944 32/25
75 547.241
76 666.537
77 785.834
78 905.130 27/16
79 1024.427
80 1143.723
81 63.020 (close to 28/27)
82 182.316 (close to 10/9)
83 301.613
84 420.910
85 540.206
86 659.503
87 778.799
88 898.096 42/25
89 1017.392 9/5
90 1136.689 27/14
* in 2.3.5.7.13.17 POTE tuning
† 2.3.5.7.13.17 ratio interpretations

See also

  • 15/14 - septimal diatonic semitone
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