Tertiaseptal: Difference between revisions

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Tuning spectrum by Eigenmonzos: improve and standardize tuning spectra tables
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<sup>a</sup> in 11-limit POTE tuning
<sup>a</sup> in 11-limit POTE tuning


== Tuning spectrum by Eigenmonzos ==
== Tuning spectra ==
=== Tertiaseptal ===
=== Tertiaseptal ===


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! Eigenmonzo
! [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! Septimal <br>whole tone
! septimal <br>whole tone (¢)
! Major third
! major third<br>(¢)
! Perfect fifth
! perfect fifth<br>(¢)
! comments
|-
|-
| 8/7
| 8/7
Line 1,002: Line 1,003:
| 385.2902
| 385.2902
| 704.7233
| 704.7233
|
|-
|-
| 13/10
| 13/10
Line 1,007: Line 1,009:
| 385.7046
| 385.7046
| 702.8998
| 702.8998
|
|-
|-
| 14/13
| 14/13
Line 1,012: Line 1,015:
| 385.7446
| 385.7446
| 702.7236
| 702.7236
|
|-
|-
| 16/13
| 16/13
Line 1,017: Line 1,021:
| 385.7771
| 385.7771
| 702.5807
| 702.5807
|
|-
|-
| 15/13
| 15/13
Line 1,022: Line 1,027:
| 385.7847
| 385.7847
| 702.5475
| 702.5475
|
|-
|-
| 16/15
| 16/15
Line 1,027: Line 1,033:
| 385.8033
| 385.8033
| 702.4654
| 702.4654
|
|-
|-
| 13/12
| 13/12
Line 1,032: Line 1,039:
| 385.8260
| 385.8260
| 702.3656
| 702.3656
|
|-
|-
| 18/13
| 18/13
Line 1,037: Line 1,045:
| 385.8499
| 385.8499
| 702.2606
| 702.2606
|
|-
|-
| 20/17
| 20/17
Line 1,042: Line 1,051:
| 385.8886
| 385.8886
| 702.0903
| 702.0903
|
|-
|-
| 17/14
| 17/14
Line 1,047: Line 1,057:
| 385.8950
| 385.8950
| 702.0618
| 702.0618
|
|-
|-
| 17/15
| 17/15
Line 1,052: Line 1,063:
| 385.8990
| 385.8990
| 702.0445
| 702.0445
|
|-
|-
| 15/14
| 15/14
Line 1,057: Line 1,069:
| 385.9133
| 385.9133
| 701.9816
| 701.9816
|
|-
|-
| 4/3
| 4/3
Line 1,062: Line 1,075:
| 385.9193
| 385.9193
| 701.9550
| 701.9550
|
|-
|-
| 18/17
| 18/17
Line 1,067: Line 1,081:
| 385.9264
| 385.9264
| 701.9239
| 701.9239
|
|-
|-
| 24/17
| 24/17
Line 1,072: Line 1,087:
| 385.9286
| 385.9286
| 701.9142
| 701.9142
|
|-
|-
| 7/5 <br>(7, 9-limit minimax)
| 7/5
| 231.5579
| 231.5579
| 385.9299
| 385.9299
| 701.9085
| 701.9085
| 7, 9-limit minimax
|-
|-
| 17/16
| 17/16
Line 1,082: Line 1,099:
| 385.9329
| 385.9329
| 701.8954
| 701.8954
|
|-
|-
| 10/9
| 10/9
Line 1,087: Line 1,105:
| 385.9596
| 385.9596
| 701.7779
| 701.7779
|
|-
|-
| 9/7
| 9/7
Line 1,092: Line 1,111:
| 385.9654
| 385.9654
| 701.7524
| 701.7524
|
|-
|-
| 6/5 <br>(5-limit minimax)
| 6/5
| 231.5954
| 231.5954
| 385.9924
| 385.9924
| 701.6336
| 701.6336
| 5-limit minimax
|-
|-
| 7/6
| 7/6
Line 1,102: Line 1,123:
| 386.0187
| 386.0187
| 701.5179
| 701.5179
|
|-
|-
| 13/11 <br>(13, 15, 17-limit<br>minimax)
| 13/11
| 231.6250
| 231.6250
| 386.0417
| 386.0417
| 701.4164
| 701.4164
| 13, 15, 17-limit minimax
|-
|-
| 22/17
| 22/17
Line 1,112: Line 1,135:
| 386.0989
| 386.0989
| 701.1648
| 701.1648
|
|-
|-
| 11/8
| 11/8
Line 1,117: Line 1,141:
| 386.2438
| 386.2438
| 700.5272
| 700.5272
|
|-
|-
| 11/10 <br>(11-limit minimax)
| 11/10
| 231.7498
| 231.7498
| 386.2496
| 386.2496
| 700.5016
| 700.5016
| 11-limit minimax
|-
|-
| 14/11
| 14/11
Line 1,127: Line 1,153:
| 386.2988
| 386.2988
| 700.2851
| 700.2851
|
|-
|-
| 5/4
| 5/4
Line 1,132: Line 1,159:
| 386.3137
| 386.3137
| 700.2197
| 700.2197
|
|-
|-
| 15/11
| 15/11
Line 1,137: Line 1,165:
| 386.4409
| 386.4409
| 699.6601
| 699.6601
|
|-
|-
| 12/11
| 12/11
Line 1,142: Line 1,171:
| 386.4602
| 386.4602
| 699.5753
| 699.5753
|
|-
|-
| 17/13
| 17/13
Line 1,147: Line 1,177:
| 387.0231
| 387.0231
| 697.0983
| 697.0983
|
|-
|-
| 11/9
| 11/9
Line 1,152: Line 1,183:
| 387.5418
| 387.5418
| 694.8159
| 694.8159
|
|}
|}


=== Tertia ===
=== Tertia ===


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! Eigenmonzo
! eigenmonzo<br>(unchanged interval)
! Septimal <br>whole tone
! septimal <br>whole tone (¢)
! Major third
! major third<br>(¢)
! Perfect fifth
! perfect fifth<br>(¢)
! comments
|-
|-
| 12/11
| 12/11
Line 1,167: Line 1,200:
| 376.5926
| 376.5926
| 742.9924
| 742.9924
|
|-
|-
| 15/11
| 15/11
Line 1,172: Line 1,206:
| 383.5363
| 383.5363
| 712.4404
| 712.4404
|
|-
|-
| 14/11
| 14/11
Line 1,177: Line 1,212:
| 385.1209
| 385.1209
| 705.4678
| 705.4678
|
|-
|-
| 11/8
| 11/8
Line 1,182: Line 1,218:
| 385.1421
| 385.1421
| 705.3748
| 705.3748
|
|-
|-
| 8/7
| 8/7
Line 1,187: Line 1,224:
| 385.2902
| 385.2902
| 704.7233
| 704.7233
|
|-
|-
| 11/10
| 11/10
Line 1,192: Line 1,230:
| 385.3441
| 385.3441
| 704.4860
| 704.4860
|
|-
|-
| 13/11
| 13/11
Line 1,197: Line 1,236:
| 385.5462
| 385.5462
| 703.5968
| 703.5968
|
|-
|-
| 22/17
| 22/17
Line 1,202: Line 1,242:
| 385.6693
| 385.6693
| 703.0552
| 703.0552
|
|-
|-
| 13/10
| 13/10
Line 1,207: Line 1,248:
| 385.7046
| 385.7046
| 702.8998
| 702.8998
|
|-
|-
| 14/13
| 14/13
Line 1,212: Line 1,254:
| 385.7446
| 385.7446
| 702.7236
| 702.7236
|
|-
|-
| 16/13
| 16/13
Line 1,217: Line 1,260:
| 385.7771
| 385.7771
| 702.5807
| 702.5807
|
|-
|-
| 15/13
| 15/13
Line 1,222: Line 1,266:
| 385.7847
| 385.7847
| 702.5475
| 702.5475
|
|-
|-
| 16/15
| 16/15
Line 1,227: Line 1,272:
| 385.8033
| 385.8033
| 702.4654
| 702.4654
|
|-
|-
| 13/12
| 13/12
Line 1,232: Line 1,278:
| 385.8260
| 385.8260
| 702.3656
| 702.3656
|
|-
|-
| 18/13 <br>(13, 15, 17-limit<br>minimax)
| 18/13
| 231.5099
| 231.5099
| 385.8499
| 385.8499
| 702.2606
| 702.2606
| 13, 15, 17-limit minimax
|-
|-
| 20/17
| 20/17
Line 1,242: Line 1,290:
| 385.8886
| 385.8886
| 702.0903
| 702.0903
|
|-
|-
| 17/14
| 17/14
Line 1,247: Line 1,296:
| 385.8950
| 385.8950
| 702.0618
| 702.0618
|
|-
|-
| 17/15
| 17/15
Line 1,252: Line 1,302:
| 385.8990
| 385.8990
| 702.0445
| 702.0445
|
|-
|-
| 15/14
| 15/14
Line 1,257: Line 1,308:
| 385.9133
| 385.9133
| 701.9816
| 701.9816
|
|-
|-
| 4/3 <br>(11-limit minimax)
| 4/3
| 231.5516
| 231.5516
| 385.9193
| 385.9193
| 701.9550
| 701.9550
| 11-limit minimax
|-
|-
| 18/17
| 18/17
Line 1,267: Line 1,320:
| 385.9264
| 385.9264
| 701.9239
| 701.9239
|
|-
|-
| 24/17
| 24/17
Line 1,272: Line 1,326:
| 385.9286
| 385.9286
| 701.9142
| 701.9142
|
|-
|-
| 7/5 <br>(7, 9-limit minimax)
| 7/5
| 231.5579
| 231.5579
| 385.9299
| 385.9299
| 701.9085
| 701.9085
| 7, 9-limit minimax
|-
|-
| 17/16
| 17/16
Line 1,282: Line 1,338:
| 385.9329
| 385.9329
| 701.8954
| 701.8954
|
|-
|-
| 10/9
| 10/9
Line 1,287: Line 1,344:
| 385.9596
| 385.9596
| 701.7779
| 701.7779
|
|-
|-
| 9/7
| 9/7
Line 1,292: Line 1,350:
| 385.9654
| 385.9654
| 701.7524
| 701.7524
|
|-
|-
| 6/5 <br>(5-limit minimax)
| 6/5
| 231.5954
| 231.5954
| 385.9924
| 385.9924
| 701.6336
| 701.6336
| 5-limit minimax
|-
|-
| 7/6
| 7/6
Line 1,302: Line 1,362:
| 386.0187
| 386.0187
| 701.5179
| 701.5179
|
|-
|-
| 5/4
| 5/4
Line 1,307: Line 1,368:
| 386.3137
| 386.3137
| 700.2197
| 700.2197
|
|-
|-
| 11/9
| 11/9
Line 1,312: Line 1,374:
| 386.8520
| 386.8520
| 697.8513
| 697.8513
|
|-
|-
| 17/13
| 17/13
Line 1,317: Line 1,380:
| 387.0231
| 387.0231
| 697.0983
| 697.0983
|
|}
|}


=== Hemitert ===
=== Hemitert ===


{| class="wikitable"
{| class="wikitable center-all"
|-
|-
! Eigenmonzo
! eigenmonzo<br>(unchanged interval)
! Septimal <br>whole tone
! septimal <br>whole tone (¢)
! Major third
! major third<br>(¢)
! Perfect fifth
! perfect fifth<br>(¢)
! comments
|-
|-
| 8/7
| 8/7
Line 1,332: Line 1,397:
| 385.2902
| 385.2902
| 704.7233
| 704.7233
|
|-
|-
| 16/15
| 16/15
Line 1,337: Line 1,403:
| 385.8033
| 385.8033
| 702.4654
| 702.4654
|
|-
|-
| 12/11
| 12/11
Line 1,342: Line 1,409:
| 385.8963
| 385.8963
| 702.0563
| 702.0563
|
|-
|-
| 11/8
| 11/8
Line 1,347: Line 1,415:
| 385.9091
| 385.9091
| 701.9999
| 701.9999
|
|-
|-
| 15/14
| 15/14
Line 1,352: Line 1,421:
| 385.9133
| 385.9133
| 701.9816
| 701.9816
|
|-
|-
| 4/3
| 4/3
Line 1,357: Line 1,427:
| 385.9193
| 385.9193
| 701.9550
| 701.9550
|
|-
|-
| 7/5 <br>(7, 9, 11-limit<br>minimax)
| 7/5
| 231.5579
| 231.5579
| 385.9299
| 385.9299
| 701.9085
| 701.9085
| 7, 9, 11-limit minimax
|-
|-
| 11/10
| 11/10
Line 1,367: Line 1,439:
| 385.9546
| 385.9546
| 701.7998
| 701.7998
|
|-
|-
| 10/9
| 10/9
Line 1,372: Line 1,445:
| 385.9596
| 385.9596
| 701.7779
| 701.7779
|
|-
|-
| 14/11
| 14/11
Line 1,377: Line 1,451:
| 385.9600
| 385.9600
| 701.7760
| 701.7760
|
|-
|-
| 9/7
| 9/7
Line 1,382: Line 1,457:
| 385.9654
| 385.9654
| 701.7524
| 701.7524
|
|-
|-
| 15/11
| 15/11
Line 1,387: Line 1,463:
| 385.9891
| 385.9891
| 701.6481
| 701.6481
|
|-
|-
| 6/5 <br>(5-limit minimax)
| 6/5
| 231.5954
| 231.5954
| 385.9924
| 385.9924
| 701.6336
| 701.6336
| 5-limit minimax
|-
|-
| 11/9
| 11/9
Line 1,397: Line 1,475:
| 386.0088
| 386.0088
| 701.5612
| 701.5612
|
|-
|-
| 7/6
| 7/6
Line 1,402: Line 1,481:
| 386.0187
| 386.0187
| 701.5179
| 701.5179
|
|-
|-
| 5/4
| 5/4
Line 1,407: Line 1,487:
| 386.3137
| 386.3137
| 700.2197
| 700.2197
|
|}
|}



Revision as of 00:21, 30 September 2021

Tertiaseptal is a temperament for the 7, 11, 13, and 17 limit. EDOs that support tertiaseptal include 31edo, 140edo, and 171edo.

See Breedsmic temperaments #Tertiaseptal for more information.

Interval chain

Tertiaseptal and tertia

generator cents value a
(octave-reduced)
17-limit ratio
(octave-reduced)
tertiaseptal
(31&171)
tertia
(31&140)
1 77.2 117/112, 256/245, 68/65 117/112, 256/245, 68/65, 22/21
2 154.4 130/119, 35/32 12/11, 130/119, 35/32
3 231.6 8/7
4 308.8 117/98, 140/117
5 386.0 5/4
6 463.1 17/13
7 540.3 175/128 15/11, 175/128
8 617.5 10/7
9 694.7 112/75
10 771.9 25/16
11 849.1 44/27, 80/49, 49/30, 85/52, 18/11 80/49, 49/30, 85/52
12 926.3 128/75
13 1003.5 25/14
14 1080.7 28/15
15 1157.9 39/20 39/20, 88/45
16 35.1 55/54, 52/51, 51/50, 50/49, 49/48, 45/44 56/55, 52/51, 51/50, 50/49, 49/48
17 112.3 16/15
18 189.4 39/35
19 266.6 7/6
20 343.8 39/32 39/32, 11/9
21 421.0 51/40 14/11, 51/40
22 498.2 4/3
23 575.4 39/28
24 652.6 35/24 16/11, 35/24
25 729.8 32/21
26 807.0 51/32 35/22, 51/32
27 884.2 5/3
28 961.4 68/39
29 1038.6 51/28 20/11, 51/28
30 1115.7 40/21, 21/11 40/21
31 1192.9
32 70.1 26/25, 25/24
33 147.3 49/45, 12/11 49/45
34 224.5 91/80 25/22, 91/80
35 301.7 25/21
36 378.9 56/45, 96/77 56/45
37 456.1 13/10
38 533.3 34/25, 15/11 34/25
39 610.5 64/45
40 687.7 52/35
41 764.8 14/9
42 842.0 13/8
43 919.2 17/10
44 996.4 16/9
45 1073.6 13/7
46 1150.8 68/35, 35/18 64/33, 68/35, 35/18
47 28.0 65/64, 64/63, 56/55 78/77, 65/64, 64/63, 55/54
48 105.2 17/16
49 182.4 10/9
50 259.6 65/56, 64/55 65/56
51 336.8 17/14
52 414.0 80/63, 14/11 80/63
53 491.1 65/49
54 568.3 25/18
55 645.5 16/11
56 722.7 85/56 50/33, 85/56
57 799.9 100/63, 35/22 100/63
58 877.1 128/77
59 954.3 26/15
60 1031.5 136/75, 20/11 136/75
61 1108.7 91/48, 256/135
62 1185.9 208/105 196/99, 208/105
63 63.1 28/27
64 140.3 13/12
65 217.4 17/15, 25/22 17/15
66 294.6 32/27 13/11, 32/27
67 371.8 26/21
68 449.0 35/27
69 526.2 65/48
70 603.4 17/12
71 680.6 40/27
72 757.8 65/42 17/11, 65/42
73 835.0 34/21
74 912.2 56/33
75 989.4 39/22 136/77, 85/48
76 1066.6 50/27
77 1143.7 64/33 85/44
78 20.9 91/90, 85/84, 78/77 100/99, 91/90, 85/84
79 98.1 35/33
80 175.3 195/176
81 252.5 52/45
82 329.7 40/33
83 406.9 91/72
84 484.1 119/90
85 561.3 112/81
86 638.5 13/9
87 715.7 68/45, 50/33 68/45
88 792.8 128/81 52/33, 128/81
89 870.0 119/72
90 947.2 140/81
91 1024.4 65/36
92 1101.6 17/9
93 1178.8 160/81, 196/99, 240/121 65/33, 160/81
94 56.0 91/88 34/33
95 133.2 68/63
96 210.4 112/99
97 287.6 13/11
98 364.8 68/55
99 442.0 128/99
100 519.1 104/77
101 596.3
102 673.5
103 750.7 17/11
104 827.9 160/99
105 905.1
106 982.3 136/77
107 1059.5
108 1136.7 52/27, 85/44 52/27
109 13.9 100/99
110 91.1 128/121, 256/243 104/99, 256/243
111 168.3
112 245.4
113 322.6
114 399.8 34/27
115 477.0
116 554.2
117 631.4
118 708.6
119 785.8 52/33
120 863.0
121 940.2
122 1017.4
123 1094.5
124 1171.7 65/33
125 48.9 34/33

a in 7-limit POTE tuning

Hemitert

generator cents value a
(octave-reduced)
11-limit ratio
(octave-reduced)
1 38.6 45/44
2 77.2 256/245
3 115.8
4 154.4 35/32
5 193.0
6 231.6 8/7
7 270.2
8 308.8
9 347.4 11/9
10 386.0 5/4
11 424.6
12 463.1 64/49
13 501.7
14 540.3
15 578.9
16 617.5 10/7
17 656.1
18 694.7
19 733.3
20 771.9 25/16
21 810.5
22 849.1
23 887.7
24 926.3
25 964.9
26 1003.5 25/14
27 1042.1
28 1080.7 28/15
29 1119.3 21/11
30 1157.9
31 1196.5
32 35.1 50/49, 49/48
33 73.7
34 112.2 16/15
35 150.8 12/11
36 189.4
37 228.0
38 266.6 7/6
39 305.2
40 343.8
41 382.4
42 421.0
43 459.6
44 498.2 4/3
45 536.8 15/11
46 575.4
47 614.0
48 652.6
49 691.2
50 729.8 32/21
51 768.4
52 807.0
53 845.6
54 884.2 5/3
55 922.8
56 961.4
57 999.9
58 1038.5
59 1077.1
60 1115.7 40/21
61 1154.3
62 1192.9
63 31.5
64 70.1 25/24
65 108.7
66 147.3
67 185.9
68 224.5
69 263.1
70 301.7
71 340.3
72 378.9
73 417.5 14/11
74 456.1
75 494.7
76 533.3
77 571.9
78 610.5
79 649.1 16/11
80 687.6
81 726.2
82 764.8 14/9
83 803.4
84 842.0
85 880.6
86 919.2
87 957.8
88 996.4 16/9
89 1035.0 20/11
90 1073.6
91 1112.2
92 1150.8
93 1189.4
94 28.0 64/63
95 66.6
96 105.2
97 143.8
98 182.4 10/9

a in 11-limit POTE tuning

Tuning spectra

Tertiaseptal

eigenmonzo
(unchanged interval
)
septimal
whole tone (¢)
major third
(¢)
perfect fifth
(¢)
comments
8/7 231.1741 385.2902 704.7233
13/10 231.4228 385.7046 702.8998
14/13 231.4468 385.7446 702.7236
16/13 231.4663 385.7771 702.5807
15/13 231.4708 385.7847 702.5475
16/15 231.4820 385.8033 702.4654
13/12 231.4956 385.8260 702.3656
18/13 231.5099 385.8499 702.2606
20/17 231.5331 385.8886 702.0903
17/14 231.5370 385.8950 702.0618
17/15 231.5394 385.8990 702.0445
15/14 231.5480 385.9133 701.9816
4/3 231.5516 385.9193 701.9550
18/17 231.5558 385.9264 701.9239
24/17 231.5572 385.9286 701.9142
7/5 231.5579 385.9299 701.9085 7, 9-limit minimax
17/16 231.5597 385.9329 701.8954
10/9 231.5757 385.9596 701.7779
9/7 231.5792 385.9654 701.7524
6/5 231.5954 385.9924 701.6336 5-limit minimax
7/6 231.6112 386.0187 701.5179
13/11 231.6250 386.0417 701.4164 13, 15, 17-limit minimax
22/17 231.6593 386.0989 701.1648
11/8 231.7463 386.2438 700.5272
11/10 231.7498 386.2496 700.5016 11-limit minimax
14/11 231.7793 386.2988 700.2851
5/4 231.7882 386.3137 700.2197
15/11 231.8645 386.4409 699.6601
12/11 231.8761 386.4602 699.5753
17/13 232.2139 387.0231 697.0983
11/9 232.5251 387.5418 694.8159

Tertia

eigenmonzo
(unchanged interval)
septimal
whole tone (¢)
major third
(¢)
perfect fifth
(¢)
comments
12/11 225.9556 376.5926 742.9924
15/11 230.1218 383.5363 712.4404
14/11 231.0726 385.1209 705.4678
11/8 231.0853 385.1421 705.3748
8/7 231.1741 385.2902 704.7233
11/10 231.2065 385.3441 704.4860
13/11 231.3277 385.5462 703.5968
22/17 231.4016 385.6693 703.0552
13/10 231.4228 385.7046 702.8998
14/13 231.4468 385.7446 702.7236
16/13 231.4663 385.7771 702.5807
15/13 231.4708 385.7847 702.5475
16/15 231.4820 385.8033 702.4654
13/12 231.4956 385.8260 702.3656
18/13 231.5099 385.8499 702.2606 13, 15, 17-limit minimax
20/17 231.5331 385.8886 702.0903
17/14 231.5370 385.8950 702.0618
17/15 231.5394 385.8990 702.0445
15/14 231.5480 385.9133 701.9816
4/3 231.5516 385.9193 701.9550 11-limit minimax
18/17 231.5558 385.9264 701.9239
24/17 231.5572 385.9286 701.9142
7/5 231.5579 385.9299 701.9085 7, 9-limit minimax
17/16 231.5597 385.9329 701.8954
10/9 231.5757 385.9596 701.7779
9/7 231.5792 385.9654 701.7524
6/5 231.5954 385.9924 701.6336 5-limit minimax
7/6 231.6112 386.0187 701.5179
5/4 231.7882 386.3137 700.2197
11/9 232.1112 386.8520 697.8513
17/13 232.2139 387.0231 697.0983

Hemitert

eigenmonzo
(unchanged interval)
septimal
whole tone (¢)
major third
(¢)
perfect fifth
(¢)
comments
8/7 231.1741 385.2902 704.7233
16/15 231.4820 385.8033 702.4654
12/11 231.5378 385.8963 702.0563
11/8 231.5455 385.9091 701.9999
15/14 231.5480 385.9133 701.9816
4/3 231.5516 385.9193 701.9550
7/5 231.5579 385.9299 701.9085 7, 9, 11-limit minimax
11/10 231.5727 385.9546 701.7998
10/9 231.5757 385.9596 701.7779
14/11 231.5760 385.9600 701.7760
9/7 231.5792 385.9654 701.7524
15/11 231.5934 385.9891 701.6481
6/5 231.5954 385.9924 701.6336 5-limit minimax
11/9 231.6053 386.0088 701.5612
7/6 231.6112 386.0187 701.5179
5/4 231.7882 386.3137 700.2197

Template:IoT