User:Moremajorthanmajor/15edX: Difference between revisions

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'''15edX''' is the scale which occurs as the dominant minor edX.
'''15edX''' is the scale which occurs as the dominant minor edX (equal division of a tenth interval). It also gives the period of the 15th-decade Alfincavallo branch of the Carrera temperament family.
==Intervals==
 
{| class="wikitable" style="text-align: center"
== Intervals ==
! rowspan="5" |Degrees
{| class="wikitable"
! colspan="2" rowspan="5" |Enneatonic
|+
! colspan="8" |Minor mode Middletown temperament
!Decade
! colspan="2" |Neutral mode Middletown temperament
!Period
! colspan="10" |Major mode Middletown temperament
!Notes
|-
|-
! colspan="2" rowspan="3" |Aeolian-Dorian
|6\5
! colspan="6" |Dorian
|96.000
! colspan="2" rowspan="3" |Dorian-Mixolydian
|Intense Aeolian-Subpental Dorian mode begins
! colspan="8" |Mixolydian
! colspan="2" rowspan="3" |Mixolydian-Ionian
|-
|-
! colspan="2" rowspan="2" |Subpental
|17\14
! colspan="2" rowspan="2" |Pental
|97.143
! colspan="2" rowspan="2" |Superpental
|
! colspan="2" rowspan="2" |Subpental
! colspan="2" rowspan="2" |Pental
! colspan="4" |Superpental
|-
|-
! colspan="2" |Soft
|11\9
! colspan="2" |Intense
|97.778
|Intense Aeolian-Subpental Dorian mode ends, Subpental Dorian mode begins
|-
|-
! width="4%" | Min
|16\13
! width="4%" | Max
|98.462
! width="4%" | Min
|
! width="4%" | Max
! width="4%" | Min
! width="4%" | Max
! width="4%" | Min
! width="4%" | Max
! width="4%" | Min
! width="4%" | Max
! width="4%" | Min
! width="4%" | Max
! width="4%" | Min
! width="4%" | Max
! width="4%" | Min
! width="4%" | Max
! width="4%" | Min
! width="4%" | Max
! width="4%" | Min
! width="4%" | Max
|-
|-
| |1
|21\17
| colspan="2" |F#/Gb
|98.824
| |96
|
| colspan="2" |97.778
| colspan="2" |100
| colspan="2" |101.818
| colspan="2" |102.857
| colspan="2" |104
| colspan="2" |105
| colspan="2" |106.667
| colspan="2" |107.586
| colspan="2" |110.345
| |112
|-
|-
| |2
|26\21
| colspan="2" |G
|99.048
| |192
|
| colspan="2" |195.556
| colspan="2" |200
| colspan="2" |203.636
| colspan="2" |205.714
| colspan="2" |208
| colspan="2" |210
| colspan="2" |213.333
| colspan="2" |215.172
| colspan="2" |220.69
| |224
|-
|-
| |3
|5\4
|G#/Jb
|100.000
|''G#/Ab''
|Subpental Dorian mode ends, Pental Dorian mode begins
| |288
| colspan="2" |293.333
| colspan="2" |300
| colspan="2" |305.4545
| colspan="2" |308.571
| colspan="2" |312
| colspan="2" |315
| colspan="2" |320
| colspan="2" |322.759
| colspan="2" |331.034
| |336
|-
|-
| |4
|19\15
|J
|101.333
|''A''
|
| |384
| colspan="2" |391.111
| colspan="2" |400
| colspan="2" |407.273
| colspan="2" |411.429
| colspan="2" |416
| colspan="2" |420
| colspan="2" |426.667
| colspan="2" |430.345
| colspan="2" |441.379
| |448
|-
|-
| |5
|14\11
|A
|101.818
|''B''
|Pental Dorian mode ends, Superpental Dorian mode begins
| |480
| colspan="2" |488.889
| colspan="2" |500
| colspan="2" |509.091
| colspan="2" |514.286
| colspan="2" |520
| colspan="2" |525
| colspan="2" |533.333
| colspan="2" |537.931
| colspan="2" |551.724
| |560
|-
|-
| |6
|23\18
|A#/Bb
|102.222
|''B#/Hb''
|
| |576
| colspan="2" |586.667
| colspan="2" |600
| colspan="2" |610.909
| colspan="2" |617.143
| colspan="2" |624
| colspan="2" |630
| colspan="2" |640
| colspan="2" |645.517
| colspan="2" |662.069
| |672
|-
|-
| |7
|9\7
|B
|102.857
|''H''
|Superpental Dorian mode ends, Mohajira Dorian-Mixolydian mode begins
| |672
| colspan="2" |684.444
| colspan="2" |700
| colspan="2" |712.727
| colspan="2" |720
| colspan="2" |728
| colspan="2" |735
| colspan="2" |746.667
| colspan="2" |753.103
| colspan="2" |772.414
| |784
|-
|-
|8
|31\24
|B#/Hb
|103.333
|''H#/Cb''
|
|768
| colspan="2" |782.822
| colspan="2" |800
| colspan="2" |814.5455
| colspan="2" |822.857
| colspan="2" |832
| colspan="2" |840
| colspan="2" |853.333
| colspan="2" |860.69
| colspan="2" |882.759
|896
|-
|-
|9
|22\17
|H
|103.529
|''C''
|Mohajira Dorian-Mixolydian mode ends, Beatles Dorian-Mixolydian mode begins
|864
| colspan="2" |880
| colspan="2" |900
| colspan="2" |916.364
| colspan="2" |925.714
| colspan="2" |936
| colspan="2" |945
| colspan="2" |960
| colspan="2" |968.276
| colspan="2" |993.103
|1008
|-
|-
|10
|35\27
|C
|103.704
|''D''
|
|960
| colspan="2" |977.778
| colspan="2" |1000
| colspan="2" |1018.182
| colspan="2" |1028.571
| colspan="2" |1040
| colspan="2" |1050
| colspan="2" |1066.667
| colspan="2" |1075.862
| colspan="2" |1103.448
|1120
|-
|-
|11
|13\10
|C#/Db
|104.000
|''D#/Sb''
|Beatles Dorian-Mixolydian mode ends, Subpental Mixolydian mode begins
|1056
| colspan="2" |1075.556
| colspan="2" |1100
| colspan="2" |1120
| colspan="2" |1131.429
| colspan="2" |1144
| colspan="2" |1155
| colspan="2" |1173.333
| colspan="2" |1183.448
| colspan="2" |1213.379
|1232
|-
|-
|12
|17\13
|D
|104.615
|''S''
|
|1152
| colspan="2" |1173.333
| colspan="2" |1200
| colspan="2" |1221.818
| colspan="2" |1234.286
| colspan="2" |1248
| colspan="2" |1260
| colspan="2" |1280
| colspan="2" |1291.034
| colspan="2" |1324.138
|1344
|-
|-
|13
|21\16
|D#/Eb
|105.000
|''S#/Eb''
|Subpental Mixolydian mode ends, Pental Mixolydian mode begins
|1248
| colspan="2" |1271.111
| colspan="2" |1300
| colspan="2" |1323.636
| colspan="2" |1337.143
| colspan="2" |1352
| colspan="2" |1365
| colspan="2" |1386.667
| colspan="2" |1398.621
| colspan="2" |1424.483
|1456
|-
|14
| colspan="2" |E
|1344
| colspan="2" |1368.889
| colspan="2" |1400
| colspan="2" |1425.4545
| colspan="2" |1440
| colspan="2" |1456
| colspan="2" |1470
| colspan="2" |1493.333
| colspan="2" |1506.207
| colspan="2" |1544.827
|1568
|-
|15
| colspan="2" |F
|1440
| colspan="2" |1466.667
| colspan="2" |1500
| colspan="2" |1527.273
| colspan="2" |1542.857
| colspan="2" |1560
| colspan="2" |1575
| colspan="2" |1600
| colspan="2" |1613.793
| colspan="2" |1655.172
|1680
|}
 
Most harmonic tunings
{| class="wikitable"
|-
|-
! rowspan="2" |Degrees
|25\19
! colspan="2" rowspan="2" |Enneatonic
|105.263
!Minor mode Middletown temperament
|
!Neutral mode Middletown temperament
!Major mode Middletown temperament
|-
|-
!7/3±13¢
|4\3
!22/9±13¢
|106.667
!5/2±13¢
|Pental Mixolydian mode ends, Soft Superpental Mixolydian mode begins
|-
|-
|1
|31\23
| colspan="2" |F#/Gb
|107.826
|96.925-98.658
|
|102.294-104.027
|104.888-106.621
|-
|-
| |2
|27\20
| colspan="2" |G
|108.000
|193.8495-197.316
|
|204.588-208.054
|209.77-213.242
|-
|-
|3
|23\17
|G#/Jb
|108.235
|''G#/Ab''
|
|290.774-295.974
|306.8815-312.0815
|314.663-319.863
|-
|-
| |4
|19\14
|J
|108.571
|''A''
|
|387.699-394.632
|409.1755-416.109
|419.559-426.484
|-
|-
| |5
|15\11
|A
|109.091
|''B''
|Soft Superpental Mixolydian mode ends, Intense Superpental Mixolydian mode begins
|484.624-493.29
|511.469-520.136
|524.438-533.105
|-
|-
|6
|26\19
|A#/Bb
|109.474
|''B#/Hb''
|
|581.548-591.948
|613.763-624.163
|629.3255-639.7255
|-
|-
| |7
|11\8
|B
|110.000
|''H''
|Intense Superpental Mixolydian mode ends, Mixolydian-Ionian mode begins
|678.473-690.606
|716.057-728.19
|734.213-746.346
|-
|-
|8
|18\13
|B#/Hb
|110.769
|''H#/Cb''
|
|775.398-789.2645
|818.351-832.218
|839.101-852.967
|-
|-
|9
|25\18
|H
|111.111
|''C''
|
|872.3225-887.9225
|920.645-936.245
|943.988-959.588
|-
|-
|10
|7\5
|C
|112.500
|''D''
|Mixolydian-Ionian mode ends
|969.247-986.561
|}
|1022.939-1040.272
Relative cents
|1048.876-1066.209
|-
| |11
|C#/Db
|''D#/Sb''
|1066.172-1085.239
|1125.2325-1144.299
|1153.763-1172.83
|-
|12
|D
|''S''
|1163.097-1183.897
|1227.526-1248.326
|1258.651-1279.451
|-
| |13
|D#/Eb
|''S#/Eb''
|1260.0215-1282.555
|1329.82-1352.3535
|1363.539-1386.072
|-
|14
| colspan="2" |E
|1356.946-1381.213
|1432.114-1456.381
|1468.426-1492.693
|-
|15
| colspan="2" |F
|1453.871-1479.871
|1534.408-1560.408
|1573.314-1599.314
|}Golden tunings±13¢
{| class="wikitable" style="text-align: center"
{| class="wikitable" style="text-align: center"
! rowspan="4" |Degrees
!Degrees
! colspan="2" rowspan="4" |Enneatonic
! colspan="2" |Enneatonic
! colspan="4" |Minor mode Middletown temperament
!Intense Aeolian-Subpental Dorian
!Neutral mode Middletown temperament
!Dorian
! colspan="6" |Major mode Middletown temperament
!Dorian-Mixolydian
|-
!Subpental-Soft Superpental Mixolydian
! rowspan="3" |Aeolian-Dorian
!Intense Superpental Mixolydian ~ Mixolydian-Ionian
! colspan="3" |Dorian
! colspan="2" rowspan="3" |Dorian-Mixolydian
! colspan="4" |Mixolydian
! rowspan="3" |Mixolydian-Ionian
|-
! rowspan="2" |Subpental
! rowspan="2" |Pental
! rowspan="2" |Superpental
! rowspan="2" |Subpental
! rowspan="2" |Pental
! colspan="2" |Superpental
|-
!Soft
!Intense
|-
|-
| |1
| |1
| colspan="2" |F#/Gb
|G#/Jb
| |96.457-98.19
|''G#/Ab''
|97.39-99.123
|96.{{Overline|6}}
|100.618-102.351
|100
|101.245-102.978
|103.{{Overline|3}}
| colspan="2" |102.788-104.5215
|106.{{Overline|6}}
|103.855-105.788
|110
|104.306-106.04
|106.664-108.398
|108.4245-110.158
|109.6375-111.371
|-
|-
| |2
| |2
| colspan="2" |G
|J
| |192.913-196.38
|''A''
|194.78-198.247
|193.{{Overline|3}}
|201.236-204.702
|200
|202.49-205.956
|206.{{Overline|6}}
| colspan="2" |205.576-209.043
|213.{{Overline|3}}
|207.709-211.576
|220
|208.613-212.08
|213.329-216.795
|216.849-220.316
|219.275-222.742
|-
|-
| |3
| |3
|G#/Jb
|J#/Ab
|''G#/Ab''
|''A#/Bb''
| |289.37-294.57
|290
|292.17-297.37
|300
|301.854-204.702
|310
|303.734-308.934
|320
| colspan="2" |308.365-313.565
|330
|311.564-317.364
|312.919-318.119
|319.993-325.193
|325.273-330.473
|328.913-334.113
|-
|-
| |4
| |4
|J
|A
|''A''
|''B''
| |385.829-392.76
|386.{{Overline|6}}
|389.56-396.493
|400
|402.4715-307.054
|413.{{Overline|3}}
|404.979-411.9215
|426.{{Overline|6}}
| colspan="2" |411.153-418.086
|440
|415.419-423.152
|417.226-424.159
|426.657-433.5905
|433.698-440.631
|438.55-445.4835
|-
|-
| |5
| |5
|A
|B
|''B''
|''C''
| |482.284-490.95
|483.{{Overline|3}}
|486.95-495.617
|500
|503.089-511.756
|516.{{Overline|6}}
|509.502-514.891
|533.{{Overline|3}}
| colspan="2" |513.941-526.308
|550
|519.2735-528.94
|521.532-530.199
|533.321-541.988
|542.122-550.789
|548.188-556.854
|-
|-
| |6
| |6
|A#/Bb
|B#/Cb
|''B#/Hb''
|''C#/Qb''
| |578.74-589.14
|580
|584.334-594.74
|600
|603.707-614.107
|620
|607.469-617.869
|640
| colspan="2" |616.729-627.129
|660
|623.128-634.728
|625.8385-636.238
|639.987-650.386
|650.547-660.947
|657.825-668.225
|-
|-
| |7
| |7
|B
|C
|''H''
|''Q''
| |675.197-687.33
|676.{{Overline|6}}
|681.73-693.863
|700
|704.325-716.4585
|723.{{Overline|3}}
|708.7135-720.847
|746.{{Overline|6}}
| colspan="2" |719.5175-731.651
|770
|726.983-740.516
|730.145-742.278
|746.65-758.783
|758.971-771.105
|767.463-779.596
|-
|-
|8
|8
|B#/Hb
|C#/Qb
|''H#/Cb''
|''Q#/Db''
|771.654-785.52
|773.{{Overline|3}}
|779.12-792.9865
|800
|804.943-818.81
|826.{{Overline|6}}
|809.958-823.825
|853.{{Overline|3}}
| colspan="2" |822.306-836.172
|880
|830.8375-844.704
|834.451-848.318
|853.314-867.181
|867.396-881.262
|877.1-890.967
|-
|-
|9
|9
|H
|Q
|''C''
|''D''
|868.1105-883.7105
|870
|876.51-892.11
|900
|905.561-921.161
|930
|911.203-926.823
|960
| colspan="2" |925.094-940.694
|990
|934.693-952.092
|938.758-954.358
|959.9785-975.5785
|975.82-991.42
|986.738-1002.338
|-
|-
|10
|10
|C
|D
|''D''
|''S''
|964.567-981.901
|966.{{Overline|6}}
|973.9-991.233
|1000
|1006.179-1023.512
|1033.{{Overline|3}}
|1012.448-1029.781
|1066.{{Overline|6}}
| colspan="2" |1027.882-1045.2155
|1100
|1038.547-1057.88
|1043.064-1060.3975
|1066.643-1083.976
|1084.245-1101.578
|1096.3755-1113.709
|-
|-
|11
|11
|C#/Db
|D#/Eb
|''D#/Sb''
|''S#/Eb''
|1061.024-1080.091
|1063.{{Overline|3}}
|1071.39-1090.356
|1100
|1106.797-1125.863
|1136.{{Overline|6}}
|1113.693-1132.759
|1173.{{Overline|3}}
| colspan="2" |1130.67-1149.737
|1210
|1142.402-1163.668
|1147.3705-1166.437
|1173.307-1192.374
|1192.669-1211.736
|1206.013-1225.08
|-
|-
|12
|12
|D
| colspan="2" |E
|''S''
|1160
|1157.481-1178.281
|1200
|1168.68-1189.48
|1240
|1207.4145-1228.2145
|1280
|1214.9375-1235.7375
|1320
| colspan="2" |1233.459-1254.259
|1246.256-1269.456
|1251.677-1272.477
|1279.971-1300.771
|1301.0395-1321.8935
|1315.651-1336.451
|-
|-
|13
|13
|D#/Eb
| colspan="2" |E#/Fb
|''S#/Eb''
|1256.{{Overline|6}}
|1253.937-1276.471
|1300
|1266.07-1288.603
|1343.{{Overline|3}}
|1308.032-1330.566
|1386.{{Overline|6}}
|1316.182-1338.716
|1430
| colspan="2" |1336.247-1358.78
|1353.111-1275.244
|1355.983-1378.517
|1386.636-1409.169
|1409.518-1425.288
|1438.993-1447.8215
|-
|-
|14
|14
| colspan="2" |E
| colspan="2" |F
|1350.394-1374.661
|1353.{{Overline|3}}
|1363.46-1387.726
|1400
|1408.65-1432.917
|1446.{{Overline|6}}
|1417.427-1441.694
|1493.{{Overline|3}}
| colspan="2" |1439.035-1463.302
|1540
|1453.966-1481.032
|1460.29-1484.556
|1493.3-1517.567
|1517.9425-1534.926
|1541.778-1559.192
|-
|-
|15
|15
| colspan="2" |F
| colspan="2" |G
|1446.851-1472.851
|1450
|1460.85-1486.85
|1500
|1509.268-1535.268
|1550
|1518.672-1544.672
|1600
| colspan="2" |1541.823-1567.823
|1650
|1557.82-1583.82
|1564.596-1590.596
|1599.964-1625.964
|1626.366-1652.366
|1644.563-1670.563
|}
|}


By a surprising coincidence, the 15ed of the Pyrite tenth (7φ+6)\(5φ^2)edo is almost exactly every third degree of [[34edo]]. Additionally, those of the modal Golden and Pyrite tenths are almost exactly +1/28-syntonic comma 4ed(5/4) (Aeolian-Dorian), 9ed(5/3)/equal multiples of 18/17 (Subpental Dorian), 13ed(15/7) (Pental Dorian), 2ed(9/8) (Superpental Dorian), -1/12-syntonic comma 3ed(6/5) (Dorian-Mixolydian), 14ed(7/3)/equal multiples of 17/16/100π\3 cents (Subpental Mixolydian), 3ed(6/5) (Pental Mixolydian), -1/20-septimal comma 4ed(9/7)/-1/28-syntonic comma 14ed(12/5)/9ed(7/4) (Soft Superpental Mixolydian), 12ed(32/15) (Intense Superpental Mixolydian) and 8ed(5/3)/-1/9 schismic 9ed(16/9)/14ed(22/9) (Mixolydian-Ionian) respectively.
By a surprising coincidence, the 15ed of the Pyrite tenth (7φ+6)\(5φ^2)edo is almost exactly every third degree of [[34edo]]. Additionally, those of the modal Golden and Pyrite tenths are almost exactly +1/28-syntonic comma 4ed(5/4) (Intense Aeolian-Subpental Dorian), [[9ed5/3|9ed(5/3)]]/equal multiples of 18/17 (Subpental Dorian), 13ed(15/7) (Pental Dorian), 2ed(9/8) (Superpental Dorian), -1/12-syntonic comma 3ed(6/5) (Dorian-Mixolydian), 14ed(7/3)/equal multiples of 17/16/100π\3 cents (Subpental Mixolydian), 3ed(6/5) (Pental Mixolydian), 4ed(9/7)/+1/36-syntonic comma 14ed(12/5) (Soft Superpental Mixolydian), 12ed(32/15) (Intense Superpental Mixolydian) and 8ed(5/3)/-1/9 schismic 9ed(16/9)/14ed(22/9) (Mixolydian-Ionian) respectively.
[[Category:15-tone]]
== See also ==
[[Category:ed7/3]]
* [[15ed5/2]] - equal division of the classic major tenth
[[Category:edX]]
* [[15ed7/3]] - equal division of the septimal minor tenth
[[Category:todo:add_definition]]
 
[[Category:todo:unify_precision]]
[[Category:15-tone scales]]
[[Category:EdX]]

Latest revision as of 02:25, 27 November 2024

15edX is the scale which occurs as the dominant minor edX (equal division of a tenth interval). It also gives the period of the 15th-decade Alfincavallo branch of the Carrera temperament family.

Intervals

Decade Period Notes
6\5 96.000 Intense Aeolian-Subpental Dorian mode begins
17\14 97.143
11\9 97.778 Intense Aeolian-Subpental Dorian mode ends, Subpental Dorian mode begins
16\13 98.462
21\17 98.824
26\21 99.048
5\4 100.000 Subpental Dorian mode ends, Pental Dorian mode begins
19\15 101.333
14\11 101.818 Pental Dorian mode ends, Superpental Dorian mode begins
23\18 102.222
9\7 102.857 Superpental Dorian mode ends, Mohajira Dorian-Mixolydian mode begins
31\24 103.333
22\17 103.529 Mohajira Dorian-Mixolydian mode ends, Beatles Dorian-Mixolydian mode begins
35\27 103.704
13\10 104.000 Beatles Dorian-Mixolydian mode ends, Subpental Mixolydian mode begins
17\13 104.615
21\16 105.000 Subpental Mixolydian mode ends, Pental Mixolydian mode begins
25\19 105.263
4\3 106.667 Pental Mixolydian mode ends, Soft Superpental Mixolydian mode begins
31\23 107.826
27\20 108.000
23\17 108.235
19\14 108.571
15\11 109.091 Soft Superpental Mixolydian mode ends, Intense Superpental Mixolydian mode begins
26\19 109.474
11\8 110.000 Intense Superpental Mixolydian mode ends, Mixolydian-Ionian mode begins
18\13 110.769
25\18 111.111
7\5 112.500 Mixolydian-Ionian mode ends

Relative cents

Degrees Enneatonic Intense Aeolian-Subpental Dorian Dorian Dorian-Mixolydian Subpental-Soft Superpental Mixolydian Intense Superpental Mixolydian ~ Mixolydian-Ionian
1 G#/Jb G#/Ab 96.6 100 103.3 106.6 110
2 J A 193.3 200 206.6 213.3 220
3 J#/Ab A#/Bb 290 300 310 320 330
4 A B 386.6 400 413.3 426.6 440
5 B C 483.3 500 516.6 533.3 550
6 B#/Cb C#/Qb 580 600 620 640 660
7 C Q 676.6 700 723.3 746.6 770
8 C#/Qb Q#/Db 773.3 800 826.6 853.3 880
9 Q D 870 900 930 960 990
10 D S 966.6 1000 1033.3 1066.6 1100
11 D#/Eb S#/Eb 1063.3 1100 1136.6 1173.3 1210
12 E 1160 1200 1240 1280 1320
13 E#/Fb 1256.6 1300 1343.3 1386.6 1430
14 F 1353.3 1400 1446.6 1493.3 1540
15 G 1450 1500 1550 1600 1650

By a surprising coincidence, the 15ed of the Pyrite tenth (7φ+6)\(5φ^2)edo is almost exactly every third degree of 34edo. Additionally, those of the modal Golden and Pyrite tenths are almost exactly +1/28-syntonic comma 4ed(5/4) (Intense Aeolian-Subpental Dorian), 9ed(5/3)/equal multiples of 18/17 (Subpental Dorian), 13ed(15/7) (Pental Dorian), 2ed(9/8) (Superpental Dorian), -1/12-syntonic comma 3ed(6/5) (Dorian-Mixolydian), 14ed(7/3)/equal multiples of 17/16/100π\3 cents (Subpental Mixolydian), 3ed(6/5) (Pental Mixolydian), 4ed(9/7)/+1/36-syntonic comma 14ed(12/5) (Soft Superpental Mixolydian), 12ed(32/15) (Intense Superpental Mixolydian) and 8ed(5/3)/-1/9 schismic 9ed(16/9)/14ed(22/9) (Mixolydian-Ionian) respectively.

See also

  • 15ed5/2 - equal division of the classic major tenth
  • 15ed7/3 - equal division of the septimal minor tenth
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