Xenharmonic Wiki

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

← Older editNewer edit →
VisualWikitext
Inthar (talk | contribs)
autopatrolled, emailconfirmed
15,004 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]]
Retrieved from "https://en.xen.wiki/w/3L_2s_(3/2-equivalent)"