3L 2s (3/2-equivalent): Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
Inthar (talk | contribs)
autopatrolled, emailconfirmed
15,668 edits
No edit summary
Line 12: Line 12:
Because uranian is a fifth-repeating scale, each tone has a 3/2 perfect fifth above it. The scale has three major chords and two minor chords, all voiced so that the third of the triad is an octave higher, a tenth. Uranian also has two harmonic 7th chords.
Because uranian is a fifth-repeating scale, each tone has a 3/2 perfect fifth above it. The scale has three major chords and two minor chords, all voiced so that the third of the triad is an octave higher, a tenth. Uranian also has two harmonic 7th chords.


[[Step ratio|Basic]] uranian is in [[8edf]], which is a very good fifth-based equal tuning similar to [[88cET]].
[[Step ratio|Basic]] uranian is in [[8edf]], which is a very good fifth-based equal temperament similar to [[88cET]].


== Temperaments ==
==Temperaments==
The most basic rank-2 temperament interpretation of uranian is semiwolf, which has 4:7:10 chords spelled <code>root-(p+1g)-(3p-2g)</code> (p = 3/2, g = the approximate 7/6). The name "semiwolf" comes from two [[7/6]] generators approximating a [[27/20]] wolf fourth.
The most basic rank-2 temperament interpretation of uranian is semiwolf, which has 4:7:10 chords spelled <code>root-(p+1g)-(3p-2g)</code> (p = 3/2, g = the approximate 7/6). The name "semiwolf" comes from two [[7/6]] generators approximating a [[27/20]] wolf fourth.
=== Semiwolf ===
===Semiwolf===
[[Subgroup]]: 3/2.7/4.5/2
[[Subgroup]]: 3/2.7/4.5/2


[[Comma]] list: [[245/243]]
[[Comma]] list: [[245/243]]


Optimal "[[inharmonic TE]]" pure-3/2 generator: ~7/6 = 262.8529
[[POL2]] generator: ~7/6 = 262.1728
 
Optimal "[[subgroup TE]]" pure-3/2 generator: ~7/6 = 262.1728


[[Mapping]]: [{{val|1 1 3}}, {{val|0 1 -2}}]
[[Mapping]]: [{{val|1 1 3}}, {{val|0 1 -2}}]


[[Vals]]: {{val list|8edf, 11edf, 13edf}}
[[Vals]]: {{val list|8edf, 11edf, 13edf}}
==== Semilupine ====
====Semilupine====
[[Subgroup]]: 3/2.7/4.5/2.11/4
[[Subgroup]]: 3/2.7/4.5/2.11/4


[[Comma]] list: [[245/243]], [[100/99]]
[[Comma]] list: [[245/243]], [[100/99]]


Optimal "[[inharmonic TE]]" pure-3/2 generator: ~7/6 = 264.3198
[[POL2]] generator: ~7/6 = 264.3771
 
Optimal "[[subgroup TE]]" pure-3/2 generator: ~7/6 = 264.3771


[[Mapping]]: [{{val|1 1 3 4}}, {{val|0 1 -2 -4}}]
[[Mapping]]: [{{val|1 1 3 4}}, {{val|0 1 -2 -4}}]


[[Vals]]: {{val list|8edf, 13edf}}
[[Vals]]: {{val list|8edf, 13edf}}
 
====Hemilycan====
==== Hemilycan ====
[[Subgroup]]: 3/2.7/4.5/2.11/4
[[Subgroup]]: 3/2.7/4.5/2.11/4


[[Comma]] list: [[245/243]], [[441/440]]
[[Comma]] list: [[245/243]], [[441/440]]


Optimal "[[inharmonic TE]]" pure-3/2 generator: ~7/6 = 261.8554
[[POL2]] generator: ~7/6 = 261.5939
 
Optimal "[[subgroup TE]]" pure-3/2 generator: ~7/6 = 261.5939


[[Mapping]]: [{{val|1 1 3 1}}, {{val|0 1 -2 4}}]
[[Mapping]]: [{{val|1 1 3 1}}, {{val|0 1 -2 4}}]


[[Vals]]: {{val list|8edf, 11edf}}
[[Vals]]: {{val list|8edf, 11edf}}
== Notation==
Since 1-7/4-5/2 is fifth-equivalent to a tone cluster of 1-10/9-7/6, it is more convenient to notate uranian scales as repeating at multiple fifths. This way, 7/4 is its own pitch class, distinct from 7/6. Notating this way produces a major ninth which is the Aeolian mode of Annapolis[6L 4s]:
{| class="wikitable"
|+
!Note
!18edf
!13edf
!21edf
!8edf
!19edf
!11edf
!14edf
|-
|1#
|1\18
38.9975
|1\13
53.9965
|2\21
66.8529
| rowspan="2" |1\8
87.7444
|3\19
110.835
|2\11
127.6282
|3\14
150.4189
|-
|2b
|3\18
116.9925
|2\13
107.9931
|3\21
100.2793
|2\19
73.89
|1\11
63.814
|1\14
50.1396
|-
|2
|4\18
155.99
|3\13
161.9896
|5\21
167.1321
|2\8
175.48875
|5\19
184.725
|3\11
191.4423
|4\14
200.5586
|-
|2#
|5\18
194.9875
|4\13
215.9862
|7\21
233.985
! rowspan="2" |'''3\8'''
'''263.2331'''
|8\19
295.56
|5\11
319.07045
|7\14
350.9775
|-
!3b
!7\18
272.9825
!5\13
269.9829
!8\21
267.4114
!7\19
258.615
!4\11
255.2564
!5\14
250.6982
|-
|3
|8\18
311.98
|6\13
323.9792
|10\21
334.2643
|4\8
350.9775
|10\19
369.45
|6\11
382.88455
|8\14
401.1171
|-
|3#
|9\18
350.9775
| rowspan="2" |7\13
377.9758
|12\21
401.1171
|5\8
438.7219
|13\19
470.285
|8\11
510.5128
|11\14
551.536
|-
|4b
|10\18
389.975
|11\21
367.9607
|4\8
350.9775
|9\19
332.505
|5\11
319.07045
|6\14
300.8379
|-
|4
|11\18
428.9725
|8\13
431.9723
|13\21
434.5436
|5\8
438.7219
|12\19
443.34
|7\11
446.6986
|9\14
451.2568
|-
|4#
|12\18
467.97
|9\13
485.9688
|15\21
501.3964
| rowspan="2" |6\8
526.46625
|15\19
554.175
|9\11
574.3268
|12\14
601.6757
|-
|5b
|13\18
506.9675
|10\13
539.9653
|16\21
534.8229
|14\19
516.23
|8\11
510.5128
|10\14
501.3964
|-
|5
|15\18
584.9625
|11\13
593.9619
|18\21
601.6757
|7\8
614.2106
|17\19
628.065
|10\11
638.1409
|13\14
651.8154
|-
|5#
|16\18
622.96
| rowspan="2" |12\13
646.9585
|20\21
668.5286
|8\8
701.955
|20\19
738.9
|12\11
765.769
|16\14
802.2343
|-
|6b
|17\18
662.9575
|19\21
635.1021
|7\8
614.2106
|16\19
591.12
|9\11
574.3268
|11\14
551.636
|-
!6
! colspan="7" |701.955
|-
|6#
|19\18
740.9525
|14\13
754.9515
|23\21
768.8021
| rowspan="2" |9\8
789.6994
|22\19
812.79
|13\11
829.5832
|17\14
852.3739
|-
|7b
|21\18
818.9475
|15\13
809.9481
|24\21
802.2343
|21\19
775.845
|12\11
765.769
|15\14
752.0946
|-
|7
|22\18
857.945
|16\13
862.9446
|26\21
868.0871
|10\8
877.44375
|24\19
886.68
|14\11
893.3973
|18\14
902.5136
|-
|7#
|23\18
896.9425
|17\13
917.9412
|28\21
935.94
! rowspan="2" |11\8
965.1881
|27\19
997.515
|16\11
1021.02545
|21\14
1052.9235
|-
!8b
!25\18
974.9375
!18\13
971.9379
!29\21
969.3664
!26\19
960.57
!15\11
957.2114
!19\14
952.6532
|-
|8
|26\18
1012.935
|19\13
1025.9342
|31\21
1036.2193
|12\8
1052.9235
|29\19
1071.405
|17\11
1084.83955
|22\14
1103.0721
|-
|8#
|27\18
1052.9325
| rowspan="2" |20\13
1079.9308
|33\21
1103.0721
|13\8
1140.7769
|32\19
1172.24
|19\11
1212.5678
|25\14
1253.4911
|-
|9b
|28\18
1091.93
|32\21
1069.9157
|12\8
1052.9235
|28\19
1034.46
|16\11
1021.02545
|20\14
1002.7929
|-
|9
|29\18
1130.9275
|21\13
1133.9273
|34\21
1136.4986
|13\8
1140.7769
|31\19
1145.295
|18\11
1148.6536
|23\14
1153.2118
|-
|9#
|30\18
1169.925
|22\13
1187.9238
|36\21
1203.3514
| rowspan="2" |14\8
1228.42125
|34\19
1256.13
|20\11
1276.2818
|26\14
1303.6307
|-
|0b
|31\18
1208.9225
|23\13
1241.9203
|37\21
1236.7779
|33\19
1218.285
|19\11
1212.5678
|24\14
1203.3514
|-
|0
|33\18
1286.9175
|24\13
1295.9169
|39\21
1303.6307
|15\8
1316.1656
|36\19
1330.02
|21\11
1340.0959
|27\14
1353.8704
|-
|0#
|34\18
1323.915
| rowspan="2" |25\13
1348.9135
|41\21
1370.4836
|16\8
1403.91
|39\19
1440.855
|23\11
1468.724
|30\14
1504.1892
|-
|1b’
|35\18
1364.9125
|40\21
1337.0571
|15\8
1316.1656
|35\19
1293.075
|20\11
1276.2818
|25\14
1253.591
|-
!1’
! colspan="7" |1403.91
|}
[[Category:Scales]]
[[Category:Scales]]
[[Category:Abstract MOS patterns]]
[[Category:Abstract MOS patterns]]
[[Category:Nonoctave]]
[[Category:Nonoctave]]

Revision as of 01:55, 2 May 2021

↖ 2L 1s⟨3/2⟩ ↑ 3L 1s⟨3/2⟩ 4L 1s⟨3/2⟩ ↗
← 2L 2s⟨3/2⟩ 3L 2s (3/2-equivalent) 4L 2s⟨3/2⟩ →
↙ 2L 3s⟨3/2⟩ ↓ 3L 3s⟨3/2⟩ 4L 3s⟨3/2⟩ ↘
Scale structure
Step pattern LLsLs
sLsLL
Equave 3/2 (702.0 ¢)
Period 3/2 (702.0 ¢)
Generator size(edf)
Bright 3\5 to 2\3 (421.2 ¢ to 468.0 ¢)
Dark 1\3 to 2\5 (234.0 ¢ to 280.8 ¢)
Related MOS scales
Parent 2L 1s⟨3/2⟩
Sister 2L 3s⟨3/2⟩
Daughters 5L 3s⟨3/2⟩, 3L 5s⟨3/2⟩
Neutralized 1L 4s⟨3/2⟩
2-Flought 8L 2s⟨3/2⟩, 3L 7s⟨3/2⟩
Equal tunings(edf)
Equalized (L:s = 1:1) 3\5 (421.2 ¢)
Supersoft (L:s = 4:3) 11\18 (429.0 ¢)
Soft (L:s = 3:2) 8\13 (432.0 ¢)
Semisoft (L:s = 5:3) 13\21 (434.5 ¢)
Basic (L:s = 2:1) 5\8 (438.7 ¢)
Semihard (L:s = 5:2) 12\19 (443.3 ¢)
Hard (L:s = 3:1) 7\11 (446.7 ¢)
Superhard (L:s = 4:1) 9\14 (451.3 ¢)
Collapsed (L:s = 1:0) 2\3 (468.0 ¢)
View • Talk • Edit

3L 2s<3/2> (sometimes called uranian), is a fifth-repeating MOS scale. The notation "<3/2>" means the period of the MOS is 3/2, disambiguating it from octave-repeating 3L 2s.

Because uranian is a fifth-repeating scale, each tone has a 3/2 perfect fifth above it. The scale has three major chords and two minor chords, all voiced so that the third of the triad is an octave higher, a tenth. Uranian also has two harmonic 7th chords.

Basic uranian is in 8edf, which is a very good fifth-based equal temperament similar to 88cET.

Temperaments

The most basic rank-2 temperament interpretation of uranian is semiwolf, which has 4:7:10 chords spelled root-(p+1g)-(3p-2g) (p = 3/2, g = the approximate 7/6). The name "semiwolf" comes from two 7/6 generators approximating a 27/20 wolf fourth.

Semiwolf

Subgroup: 3/2.7/4.5/2

Comma list: 245/243

POL2 generator: ~7/6 = 262.1728

Mapping: [⟨1 1 3], ⟨0 1 -2]]

Vals: Template:Val list

Semilupine

Subgroup: 3/2.7/4.5/2.11/4

Comma list: 245/243, 100/99

POL2 generator: ~7/6 = 264.3771

Mapping: [⟨1 1 3 4], ⟨0 1 -2 -4]]

Vals: Template:Val list

Hemilycan

Subgroup: 3/2.7/4.5/2.11/4

Comma list: 245/243, 441/440

POL2 generator: ~7/6 = 261.5939

Mapping: [⟨1 1 3 1], ⟨0 1 -2 4]]

Vals: Template:Val list

Notation

Since 1-7/4-5/2 is fifth-equivalent to a tone cluster of 1-10/9-7/6, it is more convenient to notate uranian scales as repeating at multiple fifths. This way, 7/4 is its own pitch class, distinct from 7/6. Notating this way produces a major ninth which is the Aeolian mode of Annapolis[6L 4s]:

Note 18edf 13edf 21edf 8edf 19edf 11edf 14edf
1# 1\18

38.9975

1\13

53.9965

2\21

66.8529

1\8

87.7444

3\19

110.835

2\11

127.6282

3\14

150.4189

2b 3\18

116.9925

2\13

107.9931

3\21

100.2793

2\19

73.89

1\11

63.814

1\14

50.1396

2 4\18

155.99

3\13

161.9896

5\21

167.1321

2\8

175.48875

5\19

184.725

3\11

191.4423

4\14

200.5586

2# 5\18

194.9875

4\13

215.9862

7\21

233.985

3\8

263.2331

8\19

295.56

5\11

319.07045

7\14

350.9775

3b 7\18

272.9825

5\13

269.9829

8\21

267.4114

7\19

258.615

4\11

255.2564

5\14

250.6982

3 8\18

311.98

6\13

323.9792

10\21

334.2643

4\8

350.9775

10\19

369.45

6\11

382.88455

8\14

401.1171

3# 9\18

350.9775

7\13

377.9758

12\21

401.1171

5\8

438.7219

13\19

470.285

8\11

510.5128

11\14

551.536

4b 10\18

389.975

11\21

367.9607

4\8

350.9775

9\19

332.505

5\11

319.07045

6\14

300.8379

4 11\18

428.9725

8\13

431.9723

13\21

434.5436

5\8

438.7219

12\19

443.34

7\11

446.6986

9\14

451.2568

4# 12\18

467.97

9\13

485.9688

15\21

501.3964

6\8

526.46625

15\19

554.175

9\11

574.3268

12\14

601.6757

5b 13\18

506.9675

10\13

539.9653

16\21

534.8229

14\19

516.23

8\11

510.5128

10\14

501.3964

5 15\18

584.9625

11\13

593.9619

18\21

601.6757

7\8

614.2106

17\19

628.065

10\11

638.1409

13\14

651.8154

5# 16\18

622.96

12\13

646.9585

20\21

668.5286

8\8

701.955

20\19

738.9

12\11

765.769

16\14

802.2343

6b 17\18

662.9575

19\21

635.1021

7\8

614.2106

16\19

591.12

9\11

574.3268

11\14

551.636

6 701.955
6# 19\18

740.9525

14\13

754.9515

23\21

768.8021

9\8

789.6994

22\19

812.79

13\11

829.5832

17\14

852.3739

7b 21\18

818.9475

15\13

809.9481

24\21

802.2343

21\19

775.845

12\11

765.769

15\14

752.0946

7 22\18

857.945

16\13

862.9446

26\21

868.0871

10\8

877.44375

24\19

886.68

14\11

893.3973

18\14

902.5136

7# 23\18

896.9425

17\13

917.9412

28\21

935.94

11\8

965.1881

27\19

997.515

16\11

1021.02545

21\14

1052.9235

8b 25\18

974.9375

18\13

971.9379

29\21

969.3664

26\19

960.57

15\11

957.2114

19\14

952.6532

8 26\18

1012.935

19\13

1025.9342

31\21

1036.2193

12\8

1052.9235

29\19

1071.405

17\11

1084.83955

22\14

1103.0721

8# 27\18

1052.9325

20\13

1079.9308

33\21

1103.0721

13\8

1140.7769

32\19

1172.24

19\11

1212.5678

25\14

1253.4911

9b 28\18

1091.93

32\21

1069.9157

12\8

1052.9235

28\19

1034.46

16\11

1021.02545

20\14

1002.7929

9 29\18

1130.9275

21\13

1133.9273

34\21

1136.4986

13\8

1140.7769

31\19

1145.295

18\11

1148.6536

23\14

1153.2118

9# 30\18

1169.925

22\13

1187.9238

36\21

1203.3514

14\8

1228.42125

34\19

1256.13

20\11

1276.2818

26\14

1303.6307

0b 31\18

1208.9225

23\13

1241.9203

37\21

1236.7779

33\19

1218.285

19\11

1212.5678

24\14

1203.3514

0 33\18

1286.9175

24\13

1295.9169

39\21

1303.6307

15\8

1316.1656

36\19

1330.02

21\11

1340.0959

27\14

1353.8704

0# 34\18

1323.915

25\13

1348.9135

41\21

1370.4836

16\8

1403.91

39\19

1440.855

23\11

1468.724

30\14

1504.1892

1b’ 35\18

1364.9125

40\21

1337.0571

15\8

1316.1656

35\19

1293.075

20\11

1276.2818

25\14

1253.591

1’ 1403.91
Retrieved from "https://en.xen.wiki/index.php?title=3L_2s_(3/2-equivalent)&oldid=69124"