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This scale, a Long Common Practice minor mode, occurs in [[12edo]]; where it is the equal division of the minor ninth. It has the Grandfather 5L 3s MOS with a 2:1 ratio of large against small.
This scale, a Long Common Practice minor mode, occurs in [[12edo]]; where it is the equal division of the minor ninth. It has the Grandfather 5L 3s MOS with a 2:1 ratio of large against small. It also gives the wide values for the period of the Greater [[Orwellismic temperaments #Triskaidekic|Triskaidekic]], [[Octagar temperaments #Tridecatonic|tridecatonic]], [[Ragismic microtemperaments #Trideci|trideci]] and [[aluminium]] temperaments.
==Intervals==
==Intervals==
{| class="wikitable"
{| class="wikitable"
! rowspan="3" |Degrees
|+
! rowspan="3" |Octatonic
!Nonet
! colspan="4" |Phrygian Mode (aka Double) Neapolitan Temperament (natural minor)
!Period
! colspan="9" |Phrygian-Aeolian Mode Neapolitan Temperament (intermediate minor)
!Notes
! colspan="3" |Aeolian Mode Neapolitan Temperament (melodic/harmonic minor)
|-
|1/1
|''92.308''
|Soft Phrygian mode begins
|-
|25\24
|''96.154''
|
|-
|-
! rowspan="2" |Absolute Minimum
|24\23
! rowspan="2" |Minimum
|96.321
! rowspan="2" |Median
|
! rowspan="2" |Maximum
! colspan="3" |Intense Phrygian
! colspan="3" |Intense Phrygian-Soft Aeolian
! colspan="3" |Soft Aeolian
! rowspan="2" |Minimum
! rowspan="2" |Median
! rowspan="2" |Maximum
|-
|-
!Minimum
|23\22
!Median
|96.503
!Maximum
|
!Minimum
!Median
!Maximum
!Minimum
!Median
!Maximum
|-
|-
| |1
|22\21
| |F#/Gb
|96.703
|92.308
|Soft Phrygian mode ends, Thirdtone Phrygian mode begins
|97.582
|98.4615
| colspan="2" |101.5385
|102.024
| colspan="2" |102.564
|103.167
| colspan="2" |103.846
|104.615
| colspan="2" |105.4945
|107.692
|110.769
|-
|-
| |2
|21\20
| |G
|96.923
|184.615
|
|195.165
|196.923
| colspan="2" |203.077
|204.0485
| colspan="2" |205.128
|206.335
| colspan="2" |207.692
|209.231
| colspan="2" |210.989
|215.385
|221.5385
|-
|-
| |3
|20\19
| |G#/Ab
|97.166
|276.923
|
|292.747
|295.385
| colspan="2" |304.615
|306.073
| colspan="2" |307.692
|309.502
| colspan="2" |311.5385
|313.846
| colspan="2" |316.4835
|323.077
|332.308
|-
|-
| |4
|19\18
| |A
|97.436
|369.231
|
|390.33
|393.846
| colspan="2" |406.154
|408.097
| colspan="2" |410.256
|412.67
| colspan="2" |415.385
|418.4615
| colspan="2" |421.978
|430.769
|443.076
|-
|-
| |5
|37\35
| |B
|97.582
|461.5385
|
|487.912
|491.308
| colspan="2" |507.692
|510.1215
| colspan="2" |512.8205
|515.837
| colspan="2" |519.231
|523.077
| colspan="2" |527.4725
|538.4615
|553.846
|-
|-
| |6
|18\17
| |B#/Hb
|97.738
|553.846
|
|585.4945
|589.769
| colspan="2" |609.231
|612.146
| colspan="2" |615.384
|619.0045
| colspan="2" |623.077
|627.692
| colspan="2" |632.967
|646.154
|664.615
|-
|-
| |7
|17\16
| |H
|98.077
|646.154
|
|683.077
|690.231
| colspan="2" |710.769
|714.17
| colspan="2" |717.945
|722.172
| colspan="2" |736.923
|732.308
| colspan="2" |738.4615
|754.846
|775.385
|-
|-
| |8
|16\15
| |C
|98.462
|738.4615
|Thirdtone Phrygian mode ends
|780.66
|788.692
| colspan="2" |812.308
|816.194
| colspan="2" |820.513
|825.339
| colspan="2" |830.769
|836.923
| colspan="2" |843.856
|861.5385
|886.154
|-
|-
| |9
|15\14
| |C#/Db
|98.901
|830.769
|Semitone Phrygian mode begins
|878.242
|896.154
| colspan="2" |913.846
|918.219
| colspan="2" |923.077
|929.507
| colspan="2" |934.615
|941.5385
| colspan="2" |949.4505
|969.231
|996.923
|-
|-
| |10
|14\13
| |D
|99.408
|923.077
|
|975.824
|984.615
| colspan="2" |1015.385
|1020.243
| colspan="2" |1025.641
|1031.674
| colspan="2" |1038.4165
|1046.154
| colspan="2" |1054.945
|1076.923
|1107.692
|-
|-
| |11
|27\25
| |D#/Eb
|99.692
|1015.385
|
|1073.407
|1083.077
| colspan="2" |1116.923
|1123.267
| colspan="2" |1128.205
|1134.842
| colspan="2" |1142.308
|1150.769
| colspan="2" |1160.44
|1184.615
|1218.4615
|-
|-
| |12
|13\12
| |E
|''100.000''
|1107.692
|
|1170.99
|1181.5385
| colspan="2" |1218.4615
|1224.291
| colspan="2" |1230.769
|1238.009
| colspan="2" |1246.154
|1255.385
| colspan="2" |1265.9348
|1292.308
|1329.231
|-
|-
| |13
|12\11
| |F
|100.699
|1200
|
|1268.571
|1280
| colspan="2" |1320
|1326.316
| colspan="2" |1333.333
|1341.1765
| colspan="2" |1350
|1360
| colspan="2" |1371.429
|1400
|1440
|}
Most harmonic tunings
{| class="wikitable"
|-
|-
! rowspan="2" |Degrees
|11\10
! rowspan="2" |Octatonic
|101.538
!Phrygian Mode (natural minor)
|Semitone Phrygian mode ends, Intense Phrygian mode begins
!Phrygian-Aeolian Mode (intermediate minor)
!Aeolian Mode (melodic/harmonic minor)
|-
|-
!15/7±13¢
|21\19
!11/5±13¢
|102.024
!9/4±13¢
|
|-
|-
| |1
|10\9
| |F#/Gb
|102.564
|100.496-102.496
|Intense Phrygian mode ends, Intense Phrygian-Soft Aeolian mode begins
|104-106
|106.993-108.993
|-
|-
| |2
|19\17
| |G
|103.167
|200.991-204.912
|
|208.001-212.001
|213.986-217.986
|-
|-
| |3
|9\8
| |G#/Ab
|''103.846''
|301.487-307.487
|Intense Phrygian-Soft Aeolian mode ends, Soft Aeolian mode begins
|312.001-318.001
|320.979-326.979
|-
|-
| |4
|17\15
| |A
|104.615
|401.982-409.982
|
|416.001-424.001
|427.972-435.972
|-
|-
| |5
|8\7
| |B
|105.495
|502.478-512.478
|Soft Aeolian mode ends, Flattone Aeolian mode begins
|520.002-530.002
|534.965-544.965
|-
|-
| |6
|15\13
| |B#/Hb
|106.509
|602.974-614.974
|
|624.002-636.002
|641.9585-653.9585
|-
|-
| |7
|22\19
| |H
|106.883
|703.469-717.469
|Flattone Aeolian mode ends, Meantone Aeolian mode begins
|728.002-742.0022
|748.9515-762.9515
|-
|-
| |8
|29\25
| |C
|107.077
|803.965-819.965
|
|832.003-848.002
|855.945-871.945
|-
|-
| |9
|7\6
| |C#/Db
|''107.692''
|904.46-922.46
|Meantone Aeolian mode ends, Superpyth Intense Aeolian mode begins
|936.003-924.003
|962.938-980.938
|-
|-
| |10
|20\17
| |D
|108.597
|1004.956-1024.956
|
|1040.003-1060.003
|1069.931-1089.931
|-
|-
| |11
|13\11
| |D#/Eb
|109.091
|1105.452-1127.452
|Superpyth Intense Aeolian mode ends, Ultrapyth Intense Aeolian mode begins
|1144.004-1166.004
|1177-924-1199.924
|-
|-
| |12
|19\16
| |E
|109.615
|1205.947-1299.947
|
|1248.004-1272.004
|1283.917-1307.917
|-
|-
| |13
|6\5
| |F
|110.769
|1306.443-1332.443
|Ultrapyth Intense Aeolian mode ends
|1352.004-1378.004
|}Golden/Pyrite Tunings±13¢
|1390.91-1416.91
|}
Golden Tunings
 
Phrygian Mode (natural minor)
{| class="wikitable"
{| class="wikitable"
|-
|-
!Degrees
! rowspan="4" |Degrees
!Octatonic
! colspan="2" rowspan="3" |Octatonic
!Min-Med±13¢
! colspan="2" |Soft Phrygian
!Min-Max±13¢
! colspan="4" |Phrygian
!Med-Max±13¢
|-
! rowspan="3" |Min-Med
! rowspan="3" |Min-Max/Med-Max
! colspan="2" |Thirdtone
! colspan="2" |Semitone
|-
! rowspan="2" |Min-Med/Min-Max
! rowspan="2" |Med-Max
! rowspan="2" |Min-Med/Min-Max
! rowspan="2" |Med-Max
|-
!Natural
!Subset of 26edn
|-
|-
| |1
| |1
| |F#/Gb
| |G#/Ab
|96.7665-98.7665
|G#v/Ab^
|97.176-99.176
|95.389-97.389
|98.359-100.359
|95.578-97.578
|95.957-98.345
|96.785-98.844
|98.359-100.709
|99.058-101.523
|-
|-
| |2
| |2
| |G
| colspan="2" |A
|193.533-197.533
|190.778-194.778
|194.352-198.352
|191.155-195.155
|196.719-200.719
|191.914-196.689
|193.569-197.688
|196.719-201.417
|198.1165-203.046
|-
|-
| |3
| |3
| |G#/Ab
| |A#/Bb
|290.2995-296.2995
|A#v/Bb^
|291.528-297.528
|286.1665-292.1665
|295.078-301.078
|286.733-292.733
|287.871-295.034
|290.354-296.531
|295.078-302.126
|297.175-304.569
|-
|-
| |4
| |4
| |A
| colspan="2" |B
|387.063-395.063
|381.555-389.555
|388.705-396.705
|382.3105-390.3105
|393.4375-401.4375
|383.828-393.378
|387.138-395.375
|393.4375-402.835
|396.233-406.092
|-
|-
| |5
| |5
| |B
| |C
|483.833-493.833
|B#v/Cb^
|485.881-495.881
|476.944-486.944
|491.797-501.797
|477.888-487.888
|479.785-491.723
|483.923-494.219
|491.797-503.544
|495.291-507.615
|-
|-
| |6
| |6
| |B#/Hb
| |C#/Qb
|580.299-592.299
|C
|583.057-595.057
|572.333-584.333
|590.156-602.156
|573.466-585.466
|575.742-590.068
|580.707-593.063
|590.156-604.253
|594.349-609.138
|-
|-
| |7
| |7
| |H
| |Q
|677.366-691.366
|Db
|680.233-694.233
|667.722-681.722
|688.516-702.516
|669.043-683.043
|671.699-688.412
|677.492-691.907
|688.516-704.962
|693.408-710.661
|-
|-
| |8
| |8
| |C
| |D
|774.132-790.132
|Q#
|777.409-793.409
|763.111-781.111
|786.875-802.875
|764.621-780.621
|767.656-786.757
|774.2765-790.75
|786.875-805.67
|792.466-812.184
|-
|-
| |9
| |9
| |C#/Db
| |D#/Eb
|870.899-888.899
|Eb
|874.585-892.585
|858.5-876.5
|885.234-903.234
|860.199-878.199
|863.613-885.1015
|871.061-889.594
|885.234-906.379
|891.524-913.707
|-
|-
| |10
| |10
| |D
| |E
|967.665-987.665
|D#
|971.761-991.761
|953.8885-973.8885
|983.594-1003.594
|955.776-975.776
|959.57-983.446
|967.846-988.438
|983.594-1007.088
|990.582-1015.23
|-
|-
| |11
| |11
| |D#/Eb
| |E#/Fb
|1064.432-1086.432
|Fb
|1068.9375-1090.9375
|1049.277-1071.277
|1081.953-1103.953
|1051.354-1073.354
|1055.527-1081.791
|1064.63-1087.282
|1081.953-1107.797
|1089.641-1116.75
|-
|-
| |12
| |12
| |E
| |F
|1161.198-1185.198
|E#
|1166.114-1190.114
|1144.666-1168.666
|1180.312-1204.312
|1146.932-1170.932
|1151.484-1180.135
|1161.415-1186.126
|1180.312-1208.506
|1188.699-1218.276
|-
|-
| |13
| |13
| |F
| colspan="2" |G
|1257.965-1283.965
|1240.055-1266.055
|1263.29-1289.29
|1242.506-1268.509
|1278.672-1304.672
|1247.441-1278.48
|1258.199-1284.969
|1278.672-1309.2145
|1287.757-1319.799
|}
|}
Phrygian-Aeolian Mode (intermediate minor)
 
{| class="wikitable"
{| class="wikitable"
|-
|-
! rowspan="2" |Degrees
! rowspan="3" |Degrees
! rowspan="2" |Octatonic
! colspan="4" rowspan="2" |Octatonic
! colspan="2" |Intense Phrygian
! colspan="2" |Intense Phrygian
! colspan="2" |Intense Phrygian-Soft Aeolian
! colspan="2" |Intense Phrygian-Soft Aeolian
! colspan="2" |Soft Aeolian
! colspan="2" |Soft Aeolian
|-
|-
!Min-Med/Min-Max±13¢
! rowspan="2" |Min-Med/Min-Max
!Med-Max±13¢
! rowspan="2" |Med-Max
!Min-Med/Min-Max±13¢
! rowspan="2" |Min-Med/Min-Max
!Med-Max±13¢
! rowspan="2" |Med-Max
!Min-Med/Min-Max±13¢
! rowspan="2" |Min-Med/Min-Max
!Med-Max±13¢
! rowspan="2" |Med-Max
|-
! colspan="2" |Natural
! colspan="2" |Subset of 26edn
|-
|-
| |1
| |1
| |F#/Gb
| |F#/Gb
|G#/Ab
|F#v/Gb^
|G#v/Ab^
|100.905-102.905
|100.905-102.905
|101.1465-103.1465
|101.1465-103.1465
Line 431: Line 315:
| |2
| |2
| |G
| |G
|A
|G
|A
|201.81-205.81
|201.81-205.81
|202.293-206.293
|202.293-206.293
Line 440: Line 327:
| |3
| |3
| |G#/Ab
| |G#/Ab
|A#/Bb
|G#v/Ab^
|A#v/Bb^
|302.715-308.715
|302.715-308.715
|303.44-309.44
|303.44-309.44
Line 449: Line 339:
| |4
| |4
| |A
| |A
|B
|A
|B
|403.62-411.62
|403.62-411.62
|404.586-412.586
|404.586-412.586
Line 458: Line 351:
| |5
| |5
| |B
| |B
|C
|A#v/Bb^
|B#v/Cb^
|504.525-514.525
|504.525-514.525
|505.733-515.733
|505.733-515.733
Line 467: Line 363:
| |6
| |6
| |B#/Hb
| |B#/Hb
|C#/Qb
|B
|C
|605.43-617.43
|605.43-617.43
|606.879-618.879
|606.879-618.879
Line 476: Line 375:
| |7
| |7
| |H
| |H
|Q
|Cb
|Db
|706.335-720.335
|706.335-720.335
|708.026-722.026
|708.026-722.026
Line 485: Line 387:
| |8
| |8
| |C
| |C
|D
|H#
|Q#
|807.24-823.24
|807.24-823.24
|809.172-825.172
|809.172-825.172
Line 494: Line 399:
| |9
| |9
| |C#/Db
| |C#/Db
|D#/Eb
|Db
|Eb
|908.145-926.145
|908.145-926.145
|910.319-928.319
|910.319-928.319
Line 503: Line 411:
| |10
| |10
| |D
| |D
|E
|C#
|D#
|1009.0505-1029.0505
|1009.0505-1029.0505
|1011.465-1031.465
|1011.465-1031.465
Line 512: Line 423:
| |11
| |11
| |D#/Eb
| |D#/Eb
|E#/Fb
|Eb
|Fb
|1109.9555-1131.9555
|1109.9555-1131.9555
|1112.61-1134.612
|1112.61-1134.612
Line 521: Line 435:
| |12
| |12
| |E
| |E
|F
|D#
|E#
|1210.861-1234.861
|1210.861-1234.861
|1213.758-1227.758
|1213.758-1227.758
Line 530: Line 447:
| |13
| |13
| |F
| |F
|G
|F
|G
|1311.767-1337.767
|1311.767-1337.767
|1314.905-1340.905
|1314.905-1340.905
Line 537: Line 457:
|1349.558-1375.558
|1349.558-1375.558
|}
|}
Aeolian Mode (melodic/harmonic minor)
 
{| class="wikitable"
{| class="wikitable"
! rowspan="4" |Degrees
! colspan="2" rowspan="3" |Octatonic
! colspan="4" |Aeolian
! colspan="4" |Intense Aeolian
|-
! colspan="2" |Flattone
! colspan="2" |Meantone
! colspan="2" |Superpyth
! colspan="2" |Ultrapyth
|-
|-
!Degrees
! rowspan="2" |Min-Med
!Octatonic
! rowspan="2" |Med-Max~Min-Max
!Min-Med±13¢
! rowspan="2" |Min-Med/Min-Max
!Min-Max±13¢
! rowspan="2" |Med-Max
!Med-Max-13c
! rowspan="2" |Min-Med
! rowspan="2" |Min-Max/Med-Max
! rowspan="2" |Min-Med/Min-Max
! rowspan="2" |Med-Max
|-
!Natural
!Subset of 26edn
|-
|-
| |1
| |1
| |F#/Gb
| |F#/Gb
|F#v/Gb^
|105.256-107.256
|105.256-107.256
|106.11-108.11
| colspan="2" |105.625-108.015
|106.156-108.156
|107.435-109.435
|107.738-109.738
|107.738-109.738
|108.459-110.459
|108.802-110.802
|-
|-
| |2
| |2
| |G
| colspan="2" |G
|210.511-214.511
|210.511-214.511
|212.22-216.22
| colspan="2" |211.25-216.03
|215.477-219.477
|212.313-216.313
|214.8705-218.8705
|215.4765-219.4765
|216.918-220.918
|217.604-221.604
|-
|-
| |3
| |3
| |G#/Ab
| |G#/Ab
|G#v/Ab^
|315.767-321.767
|315.767-321.767
|318.33-324.33
| colspan="2" |316.875-324.044
|323.215-329.215
|318.469-324.469
|321.306-329.306
|322.215-330.215
|325.377-331.377
|326.4065-332.4065
|-
|-
| |4
| |4
| |A
| colspan="2" |A
|421.022-429.022
|421.022-429.022
|424.44-432.44
| colspan="2" |422.501-432.059
|424.626-432.626
|429.741-437.741
|430.953-438.953
|430.953-438.953
|433.836-437.836
|435.209-443.209
|-
|-
| |5
| |5
| |B
| |B
|A#v/Bb^
|526.278-536.278
|526.278-536.278
|530.55-540.55
| colspan="2" |528.126-540.074
|530.782-540.782
|547.176-547.176
|538.6915-548.6915
|538.6915-548.6915
|542.295-552.295
|544.011-554.011
|-
|-
| |6
| |6
| |B#/Hb
| |B#/Hb
|B
|631.534-643.534
|631.534-643.534
|636.6595-648.6595
| colspan="2" |633.751-648.089
|636.9385-648.9385
|644.611-656.611
|646.43-658.43
|646.43-658.43
|650.754-666.754
|652.813-664.813
|-
|-
| |7
| |7
| |H
| |H
|Cb
|736.789-750.789
|736.789-750.789
|742.769-756.769
| colspan="2" |739.376-756.104
|743.095-757.095
|752.047-766.047
|754.168-768.168
|754.168-768.168
|759.213-773.213
|761.615-775.615
|-
|-
| |8
| |8
| |C
| |C
|H#
|842.045-858.045
|842.045-858.045
|848.879-864.879
| colspan="2" |845.001-864.1185
|849.251-865.521
|859.482-875.482
|861.906-877.906
|861.906-877.906
|867.672-883.672
|870.417-886.417
|-
|-
| |9
| |9
| |C#/Db
| |C#/Db
|Db
|947.3-965.3
|947.3-965.3
|954.989-972.989
| colspan="2" |950.626-972.133
|958.408-976.408
|966.917-984.917
|969.645-986.645
|969.645-986.645
|976.131-994.131
|979.219-997.219
|-
|-
| |10
| |10
| |D
| |D
|C#
|1052.556-1072.556
|1052.556-1072.556
|1061.099-1081.099
| colspan="2" |1056.2515-1080.148
|1061.564-1081.564
|1074.352-1094.352
|1077.383-1077.383
|1077.383-1077.383
|1084.59-1104.59
|1088.022-1108.022
|-
|-
| |11
| |11
| |D#/Eb
| |D#/Eb
|Eb
|1157.8115-1179.8115
|1157.8115-1179.8115
|1167.209-1189.209
| colspan="2" |1161.877-1188.163
|1167.7205-1189.7205
|1181.7875-1203.7875
|1185.121-1207.121
|1185.121-1207.121
|1193.049-1215.049
|1196.824-1218.824
|-
|-
| |12
| |12
| |E
| |E
|D#
|1264.067-1287.067
|1264.067-1287.067
|1273.319-1297.319
| colspan="2" |1267.502-1296.178
|1273.877-1298.877
|1288.223-1312.223
|1292.86-1316.86
|1292.86-1316.86
|1301.508-1325.508
|1304.626-1328-626
|-
|-
| |13
| |13
| |F
| colspan="2" |F
|1368.322-1394.322
|1368.323-1394.323
|1379.429-1405.429
| colspan="2" |1373.127-1404.193
|1380.033-1406.033
|1396.658-1422.658
|1400.598-1426.598
|1400.598-1426.598
|1409.967-1435.967
|1414.428-1440.428
|}
|}
[[Category:13-tone]]
 
[[Category:ed9/4]]
== See also ==
[[Category:edIX]]
[[13ed33/16]] - equal division of the harmonic subminor ninth
[[Category:todo:add_definition]]
 
[[Category:todo:unify_precision]]
[[13ed44/21]] - equal division of the Neogothic minor ninth
 
[[13ed21/10]] - equal division of the septimal chromatic minor ninth
 
[[13ed15/7]] - equal division of the septimal diatonic minor ninth
 
[[13ed11/5]] - equal division of the undecimal neutral ninth
 
[[13ed9/4]] - equal division of the classic major ninth
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