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Canopus

Revision as of 03:14, 28 October 2019 by Moremajorthanmajor (talk | contribs)

Canopus is the rank two 3.5.7 temperament tempering out 16875/16807. Having a generator of ~7:5, it possesses non-trivial MOS of the families 1L 2s (triad), 3L 1s (tetrad), 3L 4s ("neutral" diatonic) and 3L 7s (augmented neutral decatonic). On either side the greater region where it appears, there appear the most important, though as yet unnamed, tritave-equivalent temperaments which retain twos, they being important for using a (smeary) ~4:3 or 3:2 as a generator.

The Sigma and Anti-Sigma (Mu) MOS families of 8L+3s and 3L+8s (unfair) or 4L+7s and 7L+4s (fair), but especially the unfair families which by definition include an interval for the function of an "ordinary" ~2:1, are good scales to know for the conceptualizations they provide of how an "ordinary" diatonic or anti-diatonic scale extends into a tritave equivalence (8L+3s being in fact the Obikhod pitch set used in Russian Orthodox chants). These scales are neighbors of the 7&3 region where the 3L+7s Canopus decatonic scale appears. Below is a list of equal temperaments which contain these scales using generators between or 475.5 and 713.2 cents:

L=1 s=0 8 edt L=1 s=0 7 edt L=1 s=0 3 edt
L=7 s=1 59 L=7 s=1 53 L=7 s=1 28
L=6 s=1 51 L=6 s=1 46 L=6 s=1 25
L=5 s=1 43 L=5 s=1 39 L=5 s=1 22
L=4 s=1 35 L=4 s=1 32 L=4 s=1 19
L=7 s=2 62 L=7 s=2 57 L=7 s=2 35
L=3 s=1 27 L=3 s=1 25 L=3 s=1 16
L=5 s=2 46 L=5 s=2 43 L=5 s=2 29
L=7 s=3 65 L=7 s=3 61 L=7 s=3 42
L=2 s=1 19 L=2 s=1 18 L=2 s=1 13
L=7 s=4 68 L=7 s=4 65 L=7 s=4 49
L=5 s=3 49 L=5 s=3 47 L=5 s=3 36
L=3 s=2 30 L=3 s=2 29 L=3 s=2 23
L=7 s=5 71 L=7 s=5 69 L=7 s=5 56
L=4 s=3 41 L=4 s=3 40 L=4 s=3 33
L=5 s=4 52 L=5 s=4 51 L=5 s=4 43
L=6 s=5 63 L=6 s=5 62 L=6 s=5 53
L=7 s=6 74 L=7 s=6 73 L=7 s=6 63
L=1 s=1 11 edt L=1 s=1 10 edt
L=7 s=6 69 L=7 s=6 70 L=7 s=6 67
L=6 s=5 58 L=6 s=5 59 L=6 s=5 57
L=5 s=4 47 L=5 s=4 48 L=5 s=4 47
L=4 s=3 36 L=4 s=3 37 L=4 s=3 37
L=7 s=5 61 L=7 s=5 63 L=7 s=5 64
L=3 s=2 25 L=3 s=2 26 L=3 s=2 27
L=5 s=3 39 L=5 s=3 41 L=5 s=3 44
L=7 s=4 53 L=7 s=4 56 L=7 s=4 61
L=2 s=1 14 L=2 s=1 15 L=2 s=1 17
L=7 s=3 45 L=7 s=3 49 L=7 s=3 58
L=5 s=2 31 L=5 s=2 30 L=5 s=2 41
L=3 s=1 17 L=3 s=1 19 L=3 s=1 24
L=7 s=2 37 L=7 s=2 42 L=7 s=2 55
L=4 s=1 20 L=4 s=1 23 L=4 s=1 31
L=5 s=1 23 L=5 s=1 27 L=5 s=1 38
L=6 s=1 26 L=6 s=1 31 L=6 s=1 45
L=7 s=1 29 L=7 s=1 35 L=7 s=1 52
L=1 s=0 3 edt L=1 s=0 4 edt L=1 s=0 7 edt

As the table shows, the two families overlap at several equal temperaments within the first sixteen proper members of each tree due to the fact that the chain of ~4:3s forms an index-2 subtemperament of a chain of ~3:2s under tritave equivalence. Beyond that, the unfair Sigma and Mu scales match the EDO-EDT correspondences due to their definition including an interval with the function of an "ordinary" ~2:1 which can nevertheless be off by up to +68.0 cents and the fair scales compare to 5a+2b edos in a completely backwards way, with 7L+4s actually comparing to the anti-diatonic scale but being contained in the larger edts. This backward way that the fair scales compare to edos creates an interesting coincidence between 27edt and 27edo both as generated by an ~4:3.

Generator cents

hekts

L s notes
3\8 713.23

487.5

237.74

162.5

0
22\59 709.2

484.75

225.66

154.24

32.24

22.03

19\51 708.57

484.31

223.76

152.94

37.29

25.49

35\94 708.175

484.04

222.57

152.13

40.47

27.66

16\43 707.74

483.72

221.16

151.16

44.23

29.23

45\121 707.34

483.47

220.06

150.41

47.16

32.23

29\78 707.14

483.33

219.46

150

48.77

33.33

42\113 706.92

483.19

218.81

149.56

50.49

34.51

13\35 706.44

482.86

217.37

148.57

54.34

35.71

49\132 706.03

482.58

216.13

147.73

57.635

39.39

36\97 705.88

482.47

215.69

147.42

58.82

40.21

59\159 705.76

482.39

216.32

147.17

59.81

40.88

23\62 705.56

482.26

214.74

146.77

61.35

41.935

56\151 705.36

482.12

214.13

146.36

62.98

43.05

33\89 705.22

482.02

213.7

146.07

64.11

43.82

43\116 705.035

481.9

213.15

145.69

65.585

44.83

10\27 704.43

481.48

211.33

144.44

70.44

48.15

47\127 703.87

481.1

209.66

143.31

74.88

51.18

37\100 703.72

481

209.215

143

76.08

52

64\173 703.61

480.925

208.885

142.775

76,96

52.6

27\73 703.46

480.82

208.43

142.47

78.16

53.425

71\192 703.33

480.73

208.03

142.19

79.25

54.17

44\119 703.24

480.67

207.78

142.02

79.91

54.62

61\165 703.15

480.61

207.49

141.82

80.69

55.15

17\46 702.9

480.435

206.73

141.3

82.69

56.52

58\157 702.63

480.255

205.94

140.76

84.8

57.96

41\111 702.52

480.18

205.62

140.54

85.67

58.56

65\176 702.43

480.11

205.325

140.34

86.45

59.09

24\65 702.26

480

204.83

140

87.78

60

55\149 702.06

479.87

204.24

139.6

89.35

61.07

31\84 701.91

479.76

203.78

139.29

90.57

61.905

38\103 701.69

479.61

203.12

138.835

92.34

63.11

7\19 700.72

478.95

200.21

136.84

100.1

68.42

Boundary of propriety for unfair Sigma scale
39\106 699.78

478.3

197.37

134.91

107.66

73.585

32\87 699.57

478.16

196.75

134.48

109.31

74.71

57\155 699.43

478.065

196.33

134.19

110.44

75.48

25\68 699.25

477.94

195.71

133.82

111.88

76.47

68\185 699.1

477.84

195.34

133.51

113.09

77.3

43\117 699.01

477.78

195.07

133.33

113.79

77.78

61\166 698.91

477.71

194.78

133.13

114.58

78.31

18\49 698.68

477.55

194.08

132.65

116.45

79.59

65\177 698.46

477.4

193.42

132.2

118.2

80.79

47\128 698.37

477.34

193.17

132.03

118.87

81.25

76\207 698.3

477.295

192.95

131.88

119.45

81.64

Golden unfair Sigma scale is near here
29\79 698.19

477.215

192.6

131.65

120.38

82.28

69\188 698.05

477.13

192.22

131.38

121.4

82.98

40\109 697.965

477.06

191.94

131.19

122.14

83.49

51\139 697.84

476.98

191.56

130.935

123.15

84.17

11\30 697.38

476.67

190.2

130

126.8

86.67

48\131 696.9

476.34

188.74

129.01

130.67

89.31

37\101 696.76

176.24

188.31

128.71

131.82

90.1

63\172 696.65

176.16

187.98

128.49

132.695

90.7

26\71 696.49

476.06

187.52

128.17

133.94

91.55

67\183 696.34

475.96

187.08

127.87

135.11

92.35

41\112 696.25

475.89

186.8

127.68

135.85

92.86

56\153 696.14

475.82

186.47

127.45

136.74

93.46

15\41 695.84

475.61

185.56

126.83

139.17

95.12

49\134 695.49

475.37

184.52

126.12

141.94

97.01

34\93 695.34

475.27

184.06

125.81

143.16

97.85

53\145 695.2

475.17

183.64

125.51

144.29

98.62

19\52 694.945

475

182.88

125

146.3

100

42\115 694.63

474.78

181.93

124.35

148.85

131.74

23\63 694.365

474.6

181.14

123.81

150.95

103.175

27\74 693.96

474.32

179.915

122.97

154.21

105.405

4\11 691.62

472.73

172.905

118.18

Separatrix of unfair Sigma and Mu scales
25\69 689.11

471.01

192.95

131.88

165.39

113.04

21\58 688.64

470.69

196.75

134.48

163.96

112.07

38\105 688.33

470.48

199.25

136.19

163.025

111.43

17\47 687.94

470.21

202.34

138.3

161.87

110.64

47\130 687.63

470

204.83

140

160.935

110

30\83 687.45

469.88

206.24

140.96

160.41

109.64

43\119 687.26

469.75

207.78

142.02

159.83

109.24

13\36 686.82

469.44

211.33

144.44

158.5

108.33

48\133 686.42

469.17

214.51

146.62

157.305

107.52

35\97 686.27

469.07

215.69

147.42

156.86

107.22

57\158 686.15

468.99

216.68

148.1

156.49

106.96

22\61 685.95

468.85

218.26

149.18

155.9

106.56

53\147 685.74

468.71

219.95

150.34

155.26

106.12

31\86 685.59

468.605

221.16

151.16

154.81

105.81

40\111 685.39

468.47

222.75

152.25

154.21

105.405

9\25 684.7

468

228.235

156

152.16

104

41\114 684.04

467.54

233.57

159.65

150.15

102.63

32\89 683.85

467.42

235.07

160.67

149.59

102.25

55\153 683.71

467.37

236.19

161.44

149.17

101.96

23\64 683.515

467.19

237.74

162.5

148.5

101.56

60\167 683.34

467.07

239.17

163.47

148.06

101.2

Golden unfair Mu scale is near here
37\103 683.23

466.99

240.05

164.08

147.725

100.97

51\142 683.1

466.9

241.09

164.79

147.335

100.7

14\39 682.75

466.67

243.84

166.67

146.3

100

47\131 682.38

466.41

246.815

168.7

145.19

99.24

33\92 682.22

466.3

248.08

169.565

144.71

98.91

52\145 682.08

466.21

249.22

170.345

144.29

98.62

19\53 681.83

466.04

251.2

171.7

143.54

98.11

43\120 681.53

465.83

253.59

173.33

142.65

97.5

24\67 681.3

465.67

255.49

174.63

141.94

97.015

29\81 680.95

465.43

258.29

176.54

140.89

96.3

5\14 679.27

464.29

271.71

185.71

135.85

92.86

Boundary of propriety for unfair Mu scale
26\73 677.48

463.01

286.6

195.89

130.27

89.04

21\59 676.97

462.71

290.13

198.305

128.95

88.14

37\104 676.66

462.5

292.61

200

128.02

87.5

16\45 676.25

462.22

295.86

202.22

126.78

86.67

43\121 675.9

461.98

298.65

204.13

125.75

85.95

27\76 675.695

461.84

300.31

205.26

125.13

85.53

38\107 675.46

461.68

302.18

206.54

124.43

85.05

11\31 674.89

461.29

306.77

209.68

122.71

83.87

39\110 674.33

460.91

311.23

212.73

121.03

82.73

28\79 674.12

460.76

312.98

213.92

120.38

82.28

45\127 673.92

460.63

314.5

214.96

119.81

81.89

17\48 673.61

460.42

316.99

216.67

118.87

81.25

40\113 673.26

460.18

319.8

218.58

117.82

80.53

23\65 673

460

321.89

220

117.04

80

29\82 672.64

459.76

324.72

221.95

115.97

79.27

6\17 671.28

458.82

335.64

229.41

111.88

76.47

25\71 669.7

457.75

348.245

238.03

107.15

73.24

19\54 669.21

457.41

352.21

240.74

105.66

72.22

32\91 668.82

457.14

355.31

242.86

104.5

71.43

13\37 668.25

456.76

359.83

245.95

102.81

70.27

33\94 667.71

456.38

364.2

248.94

101.17

69.15

20\57 667.35

456.14

367.04

250.88

100.1

68.42

27\77 666.92

455.84

370.51

253.25

98.8

67.53

7\20 665.68

455

380.39

260

95.1

65

22\63 664.175

453.97

392.37

268.25

90.57

61.905

15\43 663.47

453.49

398.08

272.09

88.46

60.465

23\66 662.8

453.03

403.445

275.76

86.45

59.09

8\23 661.55

452.17

413.47

282.61

82.69

56.52

17\49 659.86

451.02

426.97

291.84

73.63

53.06

9\26 658.37

450

439.81

300

73.15

50

10\29 655.85

448.28

459.09

313.79

65.585

44.83

1\3 633.985

433.33

0
9\28 611.34 475.49 67.92
8\25 608.63 456.47 76.08
15\47 607.01 445.39 80.93
7\22 605.18 432.26 86.45
20\63 603.795 422.66 90.57
13\41 603.06 417.50 92.78
19\60 602.29 412.09 95.10
6\19 600.62 400.41 100.11
23\73 599.25 390.81 104.22
17\54 598.76 387.425 105.66
28\89 598.37 384.665 106.85
11\35 597.76 380.39 108.68
27\86 597.125 375.97 110.58
16\51 596.69 372.93 111.88
21\67 596.135 369.04 113.55
5\16 594.36 356.62 118.87
24\77 592.82 345.81 123.50
19\61 592.41 342.975 124.72
33\106 592.12 340.92 125.60
14\45 591.72 338.125 126.80
37\119 591.36 335.64 127.86
23\74 591.15 334.13 128.51
32\103 590.90 332.38 129.26
9\29 590.26 327.92 131.17
31\100 589.61 323.33 133.14
22\71 589.34 321.46 133.94
35\113 589.10 319.80 134.65
13\42 588.70 316.99 135.85
30\97 588.23 313.725 137.25
17\55 587.88 311.23 138.32
21\68 587.37 307.67 139.85
4\13 585.22 292.61 146.30
23\75 583.27 278.95 152.16
19\62 582.86 276.09 153.38
34\111 582.58 274.16 154.21
15\49 582.23 271.71 155.26
41\134 581.94 269.68 156.13
26\85 581.77 268.51 156.63
37\121 581.59 267.22 157.19
11\36 581.15 264.16 158.50
40\131 580.75 261.34 150.71
29\95 580.60 260.27 160.165
47\154 580.47 259.36 160.555
18\59 580.26 257.89 161.18
43\141 580.03 259.29 161.87
25\82 579.86 255.14 162.36
32\105 579.64 253.59 163.025
7\23 578.86 248.08 165.39
31\102 578.045 242.41 167.82
24\79 577.81 240.75 168.53
41\135 577.63 239.505 169.06
17\56 577.38 237.74 169.82
44\145 577.145 236.105 170.52
27\89 577.00 235.07 170.96
37\122 576.82 233.85 171.49
10\33 576.35 230.54 172.905
33\109 575.82 226.84 174.49
23\76 575.59 225.23 175.18
36\119 575.38 223.76 175.81
13\43 575.01 221.16 176.93
29\96 574,55 217.93 178.31
16\53 574.175 215.32 179.43
19\63 573.605 211.33 181.14
3\10 570.59 190.20
20\67 567.75 198.72 170.32
17\57 567.25 200.21 166.84
31\104 566.93 201.17 164.50
14\47 566.54 202.34 161.87
39\131 566.23 203.26 159.71
25\84 566.06 203.78 158.50
36\121 565.87 204.34 157.19
11\37 565.45 205.62 154.21
41\138 565.07 206.73 151.605
30\101 564.94 207.14 150.65
49\165 564.82 207.49 149.85
19\64 564.64 208.03 148.59
46\155 564.45 208.60 147.25
27\91 564.32 209.00 146.30
35\118 564.14 209.54 145.06
8\27 563.54 211.33 140.89
37\125 562.98 213.02 136.94
29\98 562.82 213.485 135.85
50\169 562.71 213.83 135.05
21\71 562.55 214.30 133.94
55\186 562.41 214.74 132.93
34\115 562.32 215.00 132.31
47\159 562.21 215.32 131.58
13\44 561.94 216.13 129.68
44\149 561.65 217.00 127.65
31\105 561.53 217.37 126.80
49\166 561.42 217.69 126.03
18\61 561.23 218.26 124.72
41\139 561.01 218.93 123.15
23\78 560.83 219.46 121.92
28\95 560.58 220.23 120.12
5\17 559.40 223.76 111.88
27\92 558.18 227.41 103.37
22\75 557.91 228.235 101.44
39\133 557.72 228.81 100.10
17\58 557.47 229.55 98.38
46\157 557.26 230.17 96.915
29\99 557.14 230.54 96.06
41\140 557.00 230.95 95.10
12\41 556.67 231.95 92.78
43\147 556.35 232.89 90.57
31\106 556.23 233.26 89.715
50\171 556.13 233.57 88.98
19\65 555.96 234.09 87.78
45\154 555.77 234.66 86.45
26\89 555.63 235.07 85.48
33\113 555.44 235.64 84.16
7\24 554.74 237.74 79.25
30\103 553.97 240.05 73.86
23\79 553.73 240.75 72.23
39\134 553.55 241.29 70.97
16\55 553.30 242.07 69.16
41\141 553.05 242.80 67.445
25\86 552.89 243.27 66.35
34\117 552.805 243.84 65.02
9\31 552.18 245.41 61.35
29\100 551.57 247.25 57.06
20\69 551.29 248.08 55.13
31\107 551.03 248.85 53.33
11\38 550.57 250.26 50.05
24\83 549.96 252.07 45.83
13\45 549.45 253.59 42.27
15\52 548.64 256.03 36.58
2\7 543.42 271.71 0
15\53 538.29 251.20 35.89
13\46 537.51 248.08 41.35
24\85 537.02 246.135 44.75
11\39 536.45 243.84 48.77
31\110 536.00 242.07 51.87
20\71 535.76 241.09 53.58
29\103 535.50 240.05 55.40
9\32 534.925 237.74 59.44
34\121 534.43 235.78 62.875
25\89 534.26 235.07 64.11
41\146 534.11 234.49 65.135
16\57 533.88 233.57 66.735
39\139 533.6 232.61 68.42
23\82 533.475 231.95 69.58
30\107 533.26 231.09 71.10
7\25 532.55 228.235 76.08
33\118 531.90 225.66 80.59
26\93 531.73 224.96 81.805
45\161 531.60 224.45 82,69
19\68 531.43 223.76 83.91
50\179 531.27 223.13 85.00
31\111 531.18 222.75 85.67
43\154 531.065 222.31 86.45
12\43 530.78 221.16 88.46
41\147 530.48 218.95 90.57
29\104 530.35 219.46 91.44
46\165 530.24 219.01 92.22
17\61 530.05 218.26 93.54
39\140 529.83 217.37 95.10
22\79 529.66 216.68 96.30
27\97 529.41 215.69 98.04
5\18 528.32 211.33 105.66 Boundary of propriety for fair Mu scale
28\101 527.275 207.14 112.99
23\83 527.05 206.23 114.58
41\148 526.89 205.62 115.66
18\65 526.695 204.83 117.04
49\177 526.53 204.165 118.20
31\112 526.43 203.78 118.87
44\159 526.53 203.35 119.62
13\47 526.07 202.34 121.40
47\170 525.835 201.38 123.07
34\123 525.74 201.02 123.70
55\199 525.67 200.71 124.25 Golden fair Mu scale is near here
21\76 525.54 200.21 125.13
50\181 525.40 199.65 126.10
29\105 525.30 199.25 126.80
37\134 525.17 198.71 127.74
8\29 524.68 196.75 131.17
35\127 524.16 194.69 134.78
27\98 524.01 194.08 135.85
46\167 523.89 193.61 136.67
19\69 523.73 192.95 137.82
49\178 523.57 192.33 138.91
30\109 523.47 191.94 139.59
41\149 523.37 191.47 140.41
11\40 523.04 190.20 142.65
36\131 522.675 188.74 145.19
25\91 522.515 188.105 146.30
39\142 522.37 187.52 147.335
14\51 522.105 186.466 149.17
31\113 521.78 185.15 151.48
17\62 521.50 184.06 153.38
20\73 521.08 182.38 156.325
3\11 518.715 172.905 Separatrix of fair Sigma and Mu scales
19\70 516.24 190.20 163.025
16\59 512.78 193.42 161.18
29\107 515.48 195.53 159.98
13\48 515.11 198.12 158.50
36\133 514.815 200.21 157.305
23\85 514.65 201.38 156.63
33\122 514.46 202.67 155.90
10\37 514.04 205.62 154.21
37\137 513.67 208.24 152.71
27\100 513.53 209.215 152.16
44\163 513.41 210.03 151.69
17\63 513.23 211.33 150.95
41\152 513.03 212.72 150.15
24\89 512.89 213.70 149.59
31\115 512.70 215.00 148.85
7\26 512.59 219.68 146.30
32\119 511.45 223.76 143.845
25\93 511.28 224.96 143.16
43\160 511.10 225.86 142.65
18\67 510.97 227.10 141.94
47\175 510.81 228.23 141.29 Golden fair Sigma scale is near here
29\108 510.71 228.94 140.89
40\149 510.59 229.77 140.41
11\41 510.28 231.95 139.17
37\138 509.94 234.30 137.82
26\97 509.80 235.29 137.25
41\153 509.67 236.19 136.71
15\56 509.45 237.74 135.85
34\127 509.185 239.62 134.78
19\71 507.97 241.09 133.94
23\86 506.66 243.27 132.695
4\15 507.19 253.59 126.80 Boundary of propriety for fair Sigma scale
21\79 505.58 264.83 120.38
17\64 505.21 267.42 118.87
30\113 504.94 269.30 117.82
13\49 504.60 271.71 116.45
35\132 504.30 273.77 115.27
22\83 504.13 274.98 114.58
31\117 503.94 265.35 113.79
9\34 503.46 279.70 111.88
32\121 503.00 282.935 110.03
23\87 502.82 284.20 109.31
37\140 502.66 285.29 108.68
14\53 502.40 287.09 107.66
33\125 502.12 289.10 106.51
19\72 501.90 290.58 105.66
24\91 501.615 292.61 104.50
5\19 500.51 300.31 100.10
21\80 499.26 309.07 95.10
16\61 498.87 311.80 93.54
27\103 498.57 313.915 92.33
11\42 498.13 316.99 90.57
28\107 497.71 319.955 88.88
17\65 497.43 321.87 87.78
23\88 497.10 324.20 86.42
6\23 496.16 330.775 82.69
19\73 495.03 338.70 78.16
13\50 494.51 342.35 76.08
20\77 494.01 345.81 74.10
7\27 493.10 352.21 70.44
15\58 491.885 360.72 65.585
8\31 490.83 368.12 61.35
9\35 489.07 380.39 54.34
1\4 475.49 0
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