Tertiaseptal: Difference between revisions

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== Tuning spectra ==
== Tuning spectra ==
=== Tertiaseptal ===
=== Tertiaseptal ===
Gencom: [2 68/65; 243/242 375/374 441/440 625/624 3584/3575]
Gencom mapping: [{{val|1 3 2 3 7 1 1}}, {{val|0 -22 5 -3 -55 42 48}}]


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! [[eigenmonzo|eigenmonzo<br>(unchanged interval]])
! septimal <br>whole tone (¢)
! generator<br>(¢)
! major third<br>(¢)
! perfect fifth<br>(¢)
! comments
! comments
|-
|-
| 8/7
| 8/7
| 231.1741
| 77.0580
| 385.2902
| 704.7233
|  
|  
|-
|-
| 13/10
| 13/10
| 231.4228
| 77.1409
| 385.7046
| 702.8998
|  
|  
|-
|-
| 14/13
| 14/13
| 231.4468
| 77.1489
| 385.7446
| 702.7236
|  
|  
|-
|-
| 16/13
| 16/13
| 231.4663
| 77.1554
| 385.7771
| 702.5807
|  
|  
|-
|-
| 15/13
| 15/13
| 231.4708
| 77.1569
| 385.7847
| 702.5475
|  
|  
|-
|-
| 16/15
| 16/15
| 231.4820
| 77.1607
| 385.8033
| 702.4654
|  
|  
|-
|-
| 13/12
| 13/12
| 231.4956
| 77.1652
| 385.8260
| 702.3656
|  
|  
|-
|-
| 18/13
| 18/13
| 231.5099
| 77.1700
| 385.8499
| 702.2606
|  
|  
|-
|-
| 20/17
| 20/17
| 231.5331
| 77.1777
| 385.8886
| 702.0903
|  
|  
|-
|-
| 17/14
| 17/14
| 231.5370
| 77.1790
| 385.8950
| 702.0618
|  
|  
|-
|-
| 17/15
| 17/15
| 231.5394
| 77.1798
| 385.8990
| 702.0445
|  
|  
|-
|-
| 15/14
| 15/14
| 231.5480
| 77.1827
| 385.9133
| 701.9816
|  
|  
|-
|-
| 4/3
| 4/3
| 231.5516
| 77.1839
| 385.9193
| 701.9550
|  
|  
|-
|-
| 18/17
| 18/17
| 231.5558
| 77.1853
| 385.9264
| 701.9239
|  
|  
|-
|-
| 24/17
| 24/17
| 231.5572
| 77.1857
| 385.9286
| 701.9142
|  
|  
|-
|-
| 7/5
| 7/5
| 231.5579
| 77.1860
| 385.9299
| 7 and 9-odd-limit minimax
| 701.9085
| 7, 9-limit minimax
|-
|-
| 17/16
| 17/16
| 231.5597
| 77.1866
| 385.9329
| 701.8954
|  
|  
|-
|-
| 10/9
| 10/9
| 231.5757
| 77.1919
| 385.9596
| 701.7779
|  
|  
|-
|-
| 9/7
| 9/7
| 231.5792
| 77.1931
| 385.9654
| 701.7524
|  
|  
|-
|-
| 6/5
| 6/5
| 231.5954
| 77.1985
| 385.9924
| 5-odd-limit minimax
| 701.6336
| 5-limit minimax
|-
|-
| 7/6
| 7/6
| 231.6112
| 77.2037
| 386.0187
| 701.5179
|  
|  
|-
|-
| 13/11
| 13/11
| 231.6250
| 77.2083
| 386.0417
| 13, 15 and 17-odd-limit minimax
| 701.4164
| 13, 15, 17-limit minimax
|-
|-
| 22/17
| 22/17
| 231.6593
| 77.2198
| 386.0989
| 701.1648
|  
|  
|-
|-
| 11/8
| 11/8
| 231.7463
| 77.2488
| 386.2438
| 700.5272
|  
|  
|-
|-
| 11/10
| 11/10
| 231.7498
| 77.2499
| 386.2496
| 11-odd-limit minimax
| 700.5016
| 11-limit minimax
|-
|-
| 14/11
| 14/11
| 231.7793
| 77.2598
| 386.2988
| 700.2851
|  
|  
|-
|-
| 5/4
| 5/4
| 231.7882
| 77.2627
| 386.3137
| 700.2197
|  
|  
|-
|-
| 15/11
| 15/11
| 231.8645
| 77.2882
| 386.4409
| 699.6601
|  
|  
|-
|-
| 12/11
| 12/11
| 231.8761
| 77.2920
| 386.4602
| 699.5753
|  
|  
|-
|-
| 17/13
| 17/13
| 232.2139
| 77.4046
| 387.0231
| 697.0983
|  
|  
|-
|-
| 11/9
| 11/9
| 232.5251
| 77.5084
| 387.5418
| 694.8159
|  
|  
|}
|}


=== Tertia ===
=== Tertia ===
Gencom: [2 22/21; 352/351 385/384 561/560 625/624 715/714]
Gencom mapping: [{{val|1 3 2 3 5 1 1}}, {{val|0 -22 5 -3 -24 42 48}}]


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! eigenmonzo<br>(unchanged interval)
! eigenmonzo<br>(unchanged interval)
! septimal <br>whole tone (¢)
! generator<br>(¢)
! major third<br>(¢)
! perfect fifth<br>(¢)
! comments
! comments
|-
|-
| 12/11
| 12/11
| 225.9556
| 75.3185
| 376.5926
| 742.9924
|  
|  
|-
|-
| 15/11
| 15/11
| 230.1218
| 76.7073
| 383.5363
| 712.4404
|  
|  
|-
|-
| 14/11
| 14/11
| 231.0726
| 77.0242
| 385.1209
| 705.4678
|  
|  
|-
|-
| 11/8
| 11/8
| 231.0853
| 77.0284
| 385.1421
| 705.3748
|  
|  
|-
|-
| 8/7
| 8/7
| 231.1741
| 77.0580
| 385.2902
| 704.7233
|  
|  
|-
|-
| 11/10
| 11/10
| 231.2065
| 77.0688
| 385.3441
| 704.4860
|  
|  
|-
|-
| 13/11
| 13/11
| 231.3277
| 77.1092
| 385.5462
| 703.5968
|  
|  
|-
|-
| 22/17
| 22/17
| 231.4016
| 77.1339
| 385.6693
| 703.0552
|  
|  
|-
|-
| 13/10
| 13/10
| 231.4228
| 77.1409
| 385.7046
| 702.8998
|  
|  
|-
|-
| 14/13
| 14/13
| 231.4468
| 77.1489
| 385.7446
| 702.7236
|  
|  
|-
|-
| 16/13
| 16/13
| 231.4663
| 77.1554
| 385.7771
| 702.5807
|  
|  
|-
|-
| 15/13
| 15/13
| 231.4708
| 77.1569
| 385.7847
| 702.5475
|  
|  
|-
|-
| 16/15
| 16/15
| 231.4820
| 77.1607
| 385.8033
| 702.4654
|  
|  
|-
|-
| 13/12
| 13/12
| 231.4956
| 77.1652
| 385.8260
| 702.3656
|  
|  
|-
|-
| 18/13
| 18/13
| 231.5099
| 77.1700
| 385.8499
| 13, 15 and 17-odd-limit minimax
| 702.2606
| 13, 15, 17-limit minimax
|-
|-
| 20/17
| 20/17
| 231.5331
| 77.1777
| 385.8886
| 702.0903
|  
|  
|-
|-
| 17/14
| 17/14
| 231.5370
| 77.1790
| 385.8950
| 702.0618
|  
|  
|-
|-
| 17/15
| 17/15
| 231.5394
| 77.1798
| 385.8990
| 702.0445
|  
|  
|-
|-
| 15/14
| 15/14
| 231.5480
| 77.1827
| 385.9133
| 701.9816
|  
|  
|-
|-
| 4/3
| 4/3
| 231.5516
| 77.1839
| 385.9193
| 11-odd-limit minimax
| 701.9550
| 11-limit minimax
|-
|-
| 18/17
| 18/17
| 231.5558
| 77.1853
| 385.9264
| 701.9239
|  
|  
|-
|-
| 24/17
| 24/17
| 231.5572
| 77.1857
| 385.9286
| 701.9142
|  
|  
|-
|-
| 7/5
| 7/5
| 231.5579
| 77.1860
| 385.9299
| 7 and 9-odd-limit minimax
| 701.9085
| 7, 9-limit minimax
|-
|-
| 17/16
| 17/16
| 231.5597
| 77.1866
| 385.9329
| 701.8954
|  
|  
|-
|-
| 10/9
| 10/9
| 231.5757
| 77.1919
| 385.9596
| 701.7779
|  
|  
|-
|-
| 9/7
| 9/7
| 231.5792
| 77.1931
| 385.9654
| 701.7524
|  
|  
|-
|-
| 6/5
| 6/5
| 231.5954
| 77.1985
| 385.9924
| 5-odd-limit minimax
| 701.6336
| 5-limit minimax
|-
|-
| 7/6
| 7/6
| 231.6112
| 77.2037
| 386.0187
| 701.5179
|  
|  
|-
|-
| 5/4
| 5/4
| 231.7882
| 77.2627
| 386.3137
| 700.2197
|  
|  
|-
|-
| 11/9
| 11/9
| 232.1112
| 77.3704
| 386.8520
| 697.8513
|  
|  
|-
|-
| 17/13
| 17/13
| 232.2139
| 77.4046
| 387.0231
| 697.0983
|  
|  
|}
|}


=== Hemitert ===
=== Hemitert ===
Gencom: [2 45/44; 2401/2400 3025/3024 65625/65536]
Gencom mapping: [{{val|1 3 2 3 6}}, {{val|0 -44 10 -6 -79}}]


{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
! eigenmonzo<br>(unchanged interval)
! eigenmonzo<br>(unchanged interval)
! septimal <br>whole tone (¢)
! generator<br>(¢)
! major third<br>(¢)
! perfect fifth<br>(¢)
! comments
! comments
|-
|-
| 8/7
| 8/7
| 231.1741
| 38.5290
| 385.2902
| 704.7233
|  
|  
|-
|-
| 16/15
| 16/15
| 231.4820
| 38.5803
| 385.8033
| 702.4654
|  
|  
|-
|-
| 12/11
| 12/11
| 231.5378
| 38.5896
| 385.8963
| 702.0563
|  
|  
|-
|-
| 11/8
| 11/8
| 231.5455
| 38.5909
| 385.9091
| 701.9999
|  
|  
|-
|-
| 15/14
| 15/14
| 231.5480
| 38.5913
| 385.9133
| 701.9816
|  
|  
|-
|-
| 4/3
| 4/3
| 231.5516
| 38.5919
| 385.9193
| 701.9550
|  
|  
|-
|-
| 7/5
| 7/5
| 231.5579
| 38.5930
| 385.9299
| 7, 9 and 11-odd-limit minimax
| 701.9085
| 7, 9, 11-limit minimax
|-
|-
| 11/10
| 11/10
| 231.5727
| 38.5955
| 385.9546
| 701.7998
|  
|  
|-
|-
| 10/9
| 10/9
| 231.5757
| 38.59596
| 385.9596
| 701.7779
|  
|  
|-
|-
| 14/11
| 14/11
| 231.5760
| 38.59600
| 385.9600
| 701.7760
|  
|  
|-
|-
| 9/7
| 9/7
| 231.5792
| 38.5965
| 385.9654
| 701.7524
|  
|  
|-
|-
| 15/11
| 15/11
| 231.5934
| 38.5989
| 385.9891
| 701.6481
|  
|  
|-
|-
| 6/5
| 6/5
| 231.5954
| 38.5992
| 385.9924
| 5-odd-limit minimax
| 701.6336
| 5-limit minimax
|-
|-
| 11/9
| 11/9
| 231.6053
| 38.6009
| 386.0088
| 701.5612
|  
|  
|-
|-
| 7/6
| 7/6
| 231.6112
| 38.6019
| 386.0187
| 701.5179
|  
|  
|-
|-
| 5/4
| 5/4
| 231.7882
| 38.6314
| 386.3137
| 700.2197
|  
|  
|}
|}

Revision as of 12:18, 14 April 2022

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

Gencom: [2 68/65; 243/242 375/374 441/440 625/624 3584/3575]

Gencom mapping: [⟨1 3 2 3 7 1 1], ⟨0 -22 5 -3 -55 42 48]]

eigenmonzo
(unchanged interval) 		generator
(¢) 		comments
8/7 77.0580
13/10 77.1409
14/13 77.1489
16/13 77.1554
15/13 77.1569
16/15 77.1607
13/12 77.1652
18/13 77.1700
20/17 77.1777
17/14 77.1790
17/15 77.1798
15/14 77.1827
4/3 77.1839
18/17 77.1853
24/17 77.1857
7/5 77.1860 7 and 9-odd-limit minimax
17/16 77.1866
10/9 77.1919
9/7 77.1931
6/5 77.1985 5-odd-limit minimax
7/6 77.2037
13/11 77.2083 13, 15 and 17-odd-limit minimax
22/17 77.2198
11/8 77.2488
11/10 77.2499 11-odd-limit minimax
14/11 77.2598
5/4 77.2627
15/11 77.2882
12/11 77.2920
17/13 77.4046
11/9 77.5084

Tertia

Gencom: [2 22/21; 352/351 385/384 561/560 625/624 715/714]

Gencom mapping: [⟨1 3 2 3 5 1 1], ⟨0 -22 5 -3 -24 42 48]]

eigenmonzo
(unchanged interval) 		generator
(¢) 		comments
12/11 75.3185
15/11 76.7073
14/11 77.0242
11/8 77.0284
8/7 77.0580
11/10 77.0688
13/11 77.1092
22/17 77.1339
13/10 77.1409
14/13 77.1489
16/13 77.1554
15/13 77.1569
16/15 77.1607
13/12 77.1652
18/13 77.1700 13, 15 and 17-odd-limit minimax
20/17 77.1777
17/14 77.1790
17/15 77.1798
15/14 77.1827
4/3 77.1839 11-odd-limit minimax
18/17 77.1853
24/17 77.1857
7/5 77.1860 7 and 9-odd-limit minimax
17/16 77.1866
10/9 77.1919
9/7 77.1931
6/5 77.1985 5-odd-limit minimax
7/6 77.2037
5/4 77.2627
11/9 77.3704
17/13 77.4046

Hemitert

Gencom: [2 45/44; 2401/2400 3025/3024 65625/65536]

Gencom mapping: [⟨1 3 2 3 6], ⟨0 -44 10 -6 -79]]

eigenmonzo
(unchanged interval) 		generator
(¢) 		comments
8/7 38.5290
16/15 38.5803
12/11 38.5896
11/8 38.5909
15/14 38.5913
4/3 38.5919
7/5 38.5930 7, 9 and 11-odd-limit minimax
11/10 38.5955
10/9 38.59596
14/11 38.59600
9/7 38.5965
15/11 38.5989
6/5 38.5992 5-odd-limit minimax
11/9 38.6009
7/6 38.6019
5/4 38.6314
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