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This is the neutral scale of the Vesuvius temperament family.
This is the neutral scale of the Vesuvius temperament family. It also gives every other period of the 16th-decade Roccocavallo branch of the Carrera temperament family.
 
==Intervals==
== Intervals ==
{| class="wikitable"
{| class="wikitable" style="text-align: center"
|+
! rowspan="3" |Degrees
! colspan="5" | Decade
! colspan="3" rowspan="2" |Intense Aeolian-Subpental Dorian
!Period
! colspan="9" |Dorian
!Notes
|-
|6\5
|
|
|
|
|180
|Intense Aeolian-Subpental Dorian mode begins
|-
|-
! colspan="3" |Subpental
|
! colspan="3" |Pental
|17\14
! colspan="3" |Superpental
|
|
|
|182.143
|
|-
|-
!Min-comma
|11\9
!Med
|
! colspan="2" |Max+comma~Min-comma
|
!Med
|
! colspan="2" |Max+comma~Min-comma
|
!Med
|183.{{Overline|3}}
! colspan="2" |Max+comma~Min-comma
|Intense Aeolian-Subpental Dorian mode ends, Subpental Dorian mode begins
!Med
!Max+comma
|-
|-
|1
|
|175-180
|16\13
|182.143
|
| colspan="2" |178.333-188.333
|
|
|184.615
|184.615
| colspan="2" |182.5-192.5
|
|190
|-
| colspan="2" |185.909-195.909
|
|191.667
|
|192.857-197.857
|21\17
|
|
|185.294
|
|-
|
|
|
|26\21
|
|185.714
|
|-
|
|
|
|
|31\25
|186
|
|-
|5\4
|
|
|
|
|187,5
| Subpental Dorian mode ends, Pental Dorian mode begins
|-
|-
|2
|
|350-360
|
|364.286
|
| colspan="2" |356.667-376.667
|
|369.231
|34\27
| colspan="2" |365-385
|188.{{Overline|8}}
|380
|
| colspan="2" |371.818-391.818
|383.333
|385.714-395.714
|-
|-
|3
|
|525-540
|
|546-429
|
| colspan="2" |535-565
|29\23
|553.846
|
| colspan="2" |547.5-577.5
|189.13
|570
|
| colspan="2" |557.727-587.727
|575
|578.571-593.571
|-
|-
|4
|
|700-720
|
|728.571
|24\19
| colspan="2" |713.333-753.333
|
|738.4615
|
| colspan="2" |730-770
|189.474
|760
|
| colspan="2" |743.636-783.636
|766.667
|771.429-791.429
|-
|-
|5
|
|875-900
|19\15
|910.714
|
| colspan="2" |891.667-941.667
|
|923.071
|
| colspan="2" |912.5-962.5
|190
|950
|
| colspan="2" |929.5455-979.5455
|958.333
|964.286-989.286
|-
|-
|6
|14\11
|1050-1080
|
|1092.857
|
| colspan="2" |1070-1130
|
|1107.692
|
| colspan="2" |1095-1155
|190.{{Overline|90}}
|1140
|Pental Dorian mode ends, Superpental Dorian mode begins
| colspan="2" |1115.4545-1175.4545
|1150
|1157.143-1187.143
|-
|-
|7
|
|1225-1260
|23\18
|1275
|
| colspan="2" |1248.333-1318.333
|
|1292.308
|
| colspan="2" |1277.5-1347.5
|191.{{Overline|6}}
|1330
|
| colspan="2" |1301.364-1371.364
|1341.667
|1350-1385
|}
{| class="wikitable" style="text-align: center"
! rowspan="2" |Degrees
! colspan="4" |Mohajira
! colspan="3" |Beatles
|-
|-
!Min-comma
|9\7
!''r¢''
|
!Med
|
! colspan="2" |Max+comma~Min-comma
|
!Med
|
!Max+comma
|192.857
|Superpental Dorian mode ends, Mohajira Dorian-Mixolydian mode begins
|-
|-
|1
|
|187.857-192.857
|31\24
| colspan="2" |''193.75''
|
| colspan="2" |189.118-199.118
|
|194.444
|
|195-200
|193
|
|-
|-
|2
|22\17
|375.714-385.714
|
| colspan="2" |''387.5''
|
| colspan="2" |378.235-398.235
|
|388.889
|
|390-400
|194.118
|Mohajira Dorian-Mixolydian mode ends, Beatles Dorian-Mixolydian mode begins
|-
|-
|3
|
|563.571-578.571
|35\27
| colspan="2" |''581.25''
|
| colspan="2" |567.353-597.353
|
|583.333
|
|585-600
|194.{{Overline|4}}
|
|-
|-
|4
|13\10
|751.429-771.429
|
| colspan="2" |''775''
|
| colspan="2" |756.471-796.471
|
|777.778
|
|780-800
|195
|Beatles Dorian-Mixolydian mode ends, Subpental Mixolydian mode begins
|-
|-
|5
|17\13
|939.286-964.286
|
| colspan="2" |''968.75''
|
| colspan="2" |945.588-995.588
|
|972.222
|
|975-1000
|196.154
|
|-
|-
|6
|21\16
|1127.143-1157.143
|
| colspan="2" |''1162.5''
|
| colspan="2" |1134.706-1194.706
|
|1166.667
|
|1170-1200
|196.875
|Subpental Mixolydian mode ends, Pental Mixolydian mode begins
|-
|-
|7
|
|1315-1350
|25\19
| colspan="2" |''1356.25''
|
| colspan="2" |1323.8235-1393.8235
|
|1361.111
|
|1365-1400
|197.37
|}
|
{| class="wikitable" style="text-align: center"
|-
|-
! rowspan="4" |Degrees
|
! colspan="12" |Mixolydian
|
! colspan="3" rowspan="3" |Mixolydian-Ionian
|29\22
|
|
|197.{{Overline|72}}
|
|-
|-
! colspan="3" rowspan="2" |Subpental
|
! colspan="3" rowspan="2" |Pental
|
! colspan="6" |Superpental
|
|33\25
|
|198
|
|-
|-
! colspan="3" |Soft
|
! colspan="3" |Intense
|
|
|
|37\28
|
|
|-
|-
! width="4%" |Min-comma
|4\3
!Med
|
! colspan="2" width="4%" |Max+comma~Min-comma
|
!Med
|
! colspan="2" width="4%" |Max+comma~Min-comma
|
!Med
|100
! colspan="2" width="4%" |Max+comma~Min-comma
| Pental Mixolydian mode ends, Soft Superpental Mixolydian mode begins
!Med
! colspan="2" width="4%" |Max+comma~Min-comma
!Med
!Max+comma
|-
|-
| |1
|
|190-195
|19\14
|196.154
|
| colspan="2" |191.875-201.875
|
|197.368
|
| colspan="2" |195-205
|101.786
|201.5625
|
| colspan="2" |196.724-206.724
|204.31
| colspan="2" |201.897-211.897
|207.353
|210-215
|-
|-
| |2
|15\11
|380-390
|
|392.308
|
| colspan="2" |383.75-403.75
|
|394.737
|
| colspan="2" |390-410
|102.{{Overline|27}}
|403.125
|Soft Superpental Mixolydian mode ends, Intense Superpental Mixolydian mode begins
| colspan="2" |393.448-413.448
|408.621
| colspan="2" |403.793-423.793
|414.706
|420-430
|-
|-
| |3
|
|570-585
|26\19
|588.4615
|
| colspan="2" |575.625-605.625
|
|592.105
|
| colspan="2" |585-615
|102.632
|604.6875
|
| colspan="2" |590.172-620.172
|612.931
| colspan="2" |605.69-635.69
|622.059
|630-645
|-
|-
|4
|11\8
|760-780
|
|784.615
|
| colspan="2" |767.5-807.5
|
|789.474
|
| colspan="2" |780-820
|103.125
|806.25
|Intense Superpental Mixolydian mode ends, Mixolydian-Ionian mode begins
| colspan="2" |786.897-826.897
|817.241
| colspan="2" |807.586-847.586
|829.412
|840-860
|-
|-
|5
|
|950-975
|18\13
|980.769
|
| colspan="2" |959.375-1008.375
|
|986.842
|
| colspan="2" |975-1025
|103.846
|1007.8125
|
| colspan="2" |983.621-1033.621
|1021.552
| colspan="2" |1009.483-1059.483
|1036.765
|1050-1075
|-
|-
|6
|
|1140-1170
|
|1176.923
| 25\18
| colspan="2" |1151.25-1211.25
|
|1184.2105
|
| colspan="2" |1170-1230
|104.1{{Overline|6}}
|1209.375
|
| colspan="2" |1180.345-1240.345
|1225.862
| colspan="2" |1211.379-1271.379
|1244.118
|1260-1290
|-
|-
|7
|7\5
|1330-1365
|
|1373.077
|
| colspan="2" |1343.125-1413.125
|
|1381.579
|
| colspan="2" |1365-1435
|105
|1410.9375
|Mixolydian-Ionian mode ends
| colspan="2" |1377.069-1447.069
|}Relative cents
|1430.172
| colspan="2" |1413.276-1483.276
|1451.471
|1470-1505
|}
Relative cents
 
{| class="wikitable" style="text-align: center"
{| class="wikitable" style="text-align: center"
!Degrees
!Degrees
! colspan="3" |Enneatonic
!Intense Aeolian-Subpental Dorian
!Intense Aeolian-Subpental Dorian
!Dorian
!Dorian
!Subpental-Soft Superpental Mixolydian  
!Dorian-Mixolydian
!Intense Superpental Mixolydian ~ Mixolydian-Ionian
|-
|-
|-
|1
|1
|181.25
|1#/2b
|187.5
|G#/Jb
|200
|''G#/Ab''
|206.25
|''90.625''
|''93.75''
|''96.875''
|-
|-
|2
|2
|362.5
|2
|375
|J
|400
|''A''
|412.5
|''181.25''
|''187.5''
|''193.75''
|-
|-
|3
|3
|543.75
|2#/3b
|562.5
|J#/Ab
|600
|''A#/Bb''
|618.75
|''271.875''
|''281.25''
|''290.625''
|-
|-
|4
|4
|543.75
|3
|750
|J
|800
|''B''
|825
|''362.5''
|''375''
|''387.5''
|-
|-
|5
|5
|906.25
|3#/4b
|937.5
|J#/Ab
|1000
|B#/Cb
|1031.25
|''453.125''
|''468.75''
|''484.375''
|-
|-
|6
|6
|1087.5
|4
|1125
|A
|1200
|''C''
|1237.5
|''543.75''
|''562.5''
|''581.25''
|-
|-
|7
|7
|1268.75
|5
|1312.5
|A#/Bb
|1400
|''C#/Qb''
|1443.75
|''633.375''
|}
|''656.25''
Golden/Pyrite Tunings±13¢
|''678.125''
{| class="wikitable" style="text-align: center"
|-
! rowspan="2" |Degrees
|8
! rowspan="2" |Intense Aeolian-Subpental Dorian
|5#/6b
! colspan="3" |Dorian
|B
|''Q''
|''725''
|''750''
|''775''
|-
|-
!Subpental
|9
!Pental
|6
!Superpental
|C
|''D''
|''815.625''
|''843.75''
|''871.875''
|-
|-
|1
|10
|180.856-184.106
|6#/7b
|182.606-185.856
|C#/Db
|188.2585-191.9085
|D#/Sb
|189.834-193.084
|''906.25''
|''937.5''
|''968.75''
|-
|-
|2
|11
|361.712-368.213
|7
|365.212-371.712
|D
|377.317-383.817
|''S''
|379.668-386.168
|''996.875''
|''1031.25''
|''1065.625''
|-
|-
|3
|12
|542.569-552.319
|7#/8b
|547.819-557.569
|D#/Eb
|565.974-575.726
|''S#/Eb''
|569.502-579.252
|''1087.5''
|''1125''
|''1162.5''
|-
|-
|4
| 13
|723.425-736.425
|8
|730.425-743.425
| colspan="2" |E
|754.634-767.634
|''1178.125''
|759.336-771.336
|''1218.75''
|''1259.375''
|-
|-
|5
|14
|904.282-920.532
|8#/9b
|913.031-920.281
| colspan="2" |E#/Fb
|943.293-959.543
|''1268.75''
|949.17-965.42
|''1312.5''
|''1356.25''
|-
|-
|6
|15
|1085.138-1104.638
|9
|1095.637-1115.137
| colspan="2" |F
|1131.951-1151.451
|''1359.375''
|1139.004-1158.503
|''1406.25''
|''1453.125''
|-
|-
|7
|16
|1265.9945-1288.7445
|1
|1278.2435-1300.9935
| colspan="2" |G
|1320.61-1343.36
|''1450''
|1328.838-1351.588
|''1500''
|''1550''
|}
|}
{| class="wikitable" style="text-align: center"
{| class="wikitable" style="text-align: center"
!Degrees
!Degrees
!Mohajira
! colspan="3" |Enneatonic
!Beatles
!Subpental-Soft Superpental Mixolydian
!Intense Superpental Mixolydian ~ Mixolydian-Ionian
|-
|-
|1
| | 1
|192.237-195.487
|1#/2b
|192.728-195.978
| colspan="2" |F#/Gb
|''100''
|''103.125''
|-
|-
| |2
|2
|2
|384.474-390.974
| colspan="2" |G
|385.456-391.956
|''200''
|''206.25''
|-
|-
| |3
|2#/3b
|G#/Jb
|''G#/Ab''
|''300''
|''309.375''
|-
| |4
|3
|3
|576.711-585.961
|J
|578.184-587.934
|''A''
|''400''
|''412.5''
|-
|-
| |5
|3#/4b
|J#/Ab
|''A#/Bb''
|''500''
|''515.625''
|-
| |6
|4
|4
|768.948-781.948
|A
|770.912-783.912
|''B''
|''600''
|''618.75''
|-
|-
| | 7
|5
|5
|961.185-977.435
|B
|963.6395-979.8895
|''H''
|''700''
|''721.875''
|-
|-
|8
| 5#/6b
|B#/Hb
|''H#/Cb''
|''800''
|''825''
|-
|9
|6
|6
|1153.422-1172.921
|H
|1156.367-1175.867
|''C''
|''900''
|''928.125''
|-
|-
|10
|6#/7b
|H#/Cb
|''C#/Db''
|''1000''
|''1031.25''
|-
|11
|7
|7
|1345.659-1368.409
|C
|1349.095-1372.345
|''D''
|}
|''1100''
{| class="wikitable" style="text-align: center"
|''1134.375''
|-
|-
! rowspan="3" |Degrees
|12
! colspan="4" |Mixolydian
|7#/8b
! rowspan="3" |Mixolydian-Ionian
|C#/Db
|''D#/Sb''
|''1200''
|''1237.5''
|-
|-
! rowspan="2" |Subpental
|13
! rowspan="2" |Pental
|8
! colspan="2" |Superpental
|D
|''S''
|''1300''
|''1340.625''
|-
|-
!Soft
|14
!Intense
|8#/9b
|D#/Eb
|''S#/Eb''
|''1400''
|''1443.75''
|-
|-
| |1
|15
|194.7275-197.9775
|9
|195.5745-198.8245
| colspan="2" |E
|199.9955-203.2455
|''1500''
|203.296-206.546
|''1546.875''
|205.57-208.82
|-
|-
| |2
|16
|389.455-395.955
|1
|391.149-397.649
| colspan="2" |F
|399.991-406.491
|''1600''
|406.592-413.092
|''1650''
|411.141-417.641
|}By a surprising coincidence, neutral 16edX turns out to be a false Pelogic temperament with a very pure 5:4 (mistuned by no more than 3.686 cents), and tuning 5:3 pure creates a tenth almost exactly equal to 38/29edo or almost exactly the Golden Subpental Mixolydian step or every ninth step of [[110edo]], and tuning 4:3 pure creates an almost exact 43/74edo fifth. Also, the Golden Soft Superpental Mixolydian step is almost exactly (25/24)^(10/7) and tuning 12:7 pure creates almost exactly the Golden Mixolydian-Ionian step.
|-
==See also==
| |3
*[[16ed5/2]] - equal division of the classic major tenth
|584.183-593.933
*[[16ed7/3]] - equal division of the septimal minor tenth
|586.724-596.474
|599.987-609.737
|609.888-619.638
|616.711-626.461
|-
|4
|778.91-791.91
|782.298-795.298
|799.982-812.982
|813.1835-826.1835
|822.282-835.282
|-
|5
|973.638-989.888
|977.873-994.123
|999.978-1016.238
|1016.479-1032.729
|1027.852-1044.102
|-
|6
|1168.365-1187.865
|1173.447-1192.947
|1199.973-1219.473
|1219.775-1239.275
|1233.422-1252.922
|-
|7
|1363.093-1385.843
|1369.022-1391.772
|1399.968-1422.719
|1423.071-1445.821
|1438.993-1461.743
|}
 
== See also ==
* [[8ed5/2]] - equal division of the classic major tenth
* [[8ed7/3]] - equal division of the septimal minor tenth


[[Category:16-tone scales]]
[[Category:EdX]]
[[Category:EdX]]

Revision as of 21:43, 7 June 2023

This is the neutral scale of the Vesuvius temperament family. It also gives every other period of the 16th-decade Roccocavallo branch of the Carrera temperament family.

Intervals

Decade Period Notes
6\5 180 Intense Aeolian-Subpental Dorian mode begins
17\14 182.143
11\9 183.3 Intense Aeolian-Subpental Dorian mode ends, Subpental Dorian mode begins
16\13 184.615
21\17 185.294
26\21 185.714
31\25 186
5\4 187,5 Subpental Dorian mode ends, Pental Dorian mode begins
34\27 188.8
29\23 189.13
24\19 189.474
19\15 190
14\11 190.90 Pental Dorian mode ends, Superpental Dorian mode begins
23\18 191.6
9\7 192.857 Superpental Dorian mode ends, Mohajira Dorian-Mixolydian mode begins
31\24 193
22\17 194.118 Mohajira Dorian-Mixolydian mode ends, Beatles Dorian-Mixolydian mode begins
35\27 194.4
13\10 195 Beatles Dorian-Mixolydian mode ends, Subpental Mixolydian mode begins
17\13 196.154
21\16 196.875 Subpental Mixolydian mode ends, Pental Mixolydian mode begins
25\19 197.37
29\22 197.72
33\25 198
37\28
4\3 100 Pental Mixolydian mode ends, Soft Superpental Mixolydian mode begins
19\14 101.786
15\11 102.27 Soft Superpental Mixolydian mode ends, Intense Superpental Mixolydian mode begins
26\19 102.632
11\8 103.125 Intense Superpental Mixolydian mode ends, Mixolydian-Ionian mode begins
18\13 103.846
25\18 104.16
7\5 105 Mixolydian-Ionian mode ends

Relative cents

Degrees Enneatonic Intense Aeolian-Subpental Dorian Dorian Dorian-Mixolydian
1 1#/2b G#/Jb G#/Ab 90.625 93.75 96.875
2 2 J A 181.25 187.5 193.75
3 2#/3b J#/Ab A#/Bb 271.875 281.25 290.625
4 3 J B 362.5 375 387.5
5 3#/4b J#/Ab B#/Cb 453.125 468.75 484.375
6 4 A C 543.75 562.5 581.25
7 5 A#/Bb C#/Qb 633.375 656.25 678.125
8 5#/6b B Q 725 750 775
9 6 C D 815.625 843.75 871.875
10 6#/7b C#/Db D#/Sb 906.25 937.5 968.75
11 7 D S 996.875 1031.25 1065.625
12 7#/8b D#/Eb S#/Eb 1087.5 1125 1162.5
13 8 E 1178.125 1218.75 1259.375
14 8#/9b E#/Fb 1268.75 1312.5 1356.25
15 9 F 1359.375 1406.25 1453.125
16 1 G 1450 1500 1550
Degrees Enneatonic Subpental-Soft Superpental Mixolydian Intense Superpental Mixolydian ~ Mixolydian-Ionian
1 1#/2b F#/Gb 100 103.125
2 2 G 200 206.25
3 2#/3b G#/Jb G#/Ab 300 309.375
4 3 J A 400 412.5
5 3#/4b J#/Ab A#/Bb 500 515.625
6 4 A B 600 618.75
7 5 B H 700 721.875
8 5#/6b B#/Hb H#/Cb 800 825
9 6 H C 900 928.125
10 6#/7b H#/Cb C#/Db 1000 1031.25
11 7 C D 1100 1134.375
12 7#/8b C#/Db D#/Sb 1200 1237.5
13 8 D S 1300 1340.625
14 8#/9b D#/Eb S#/Eb 1400 1443.75
15 9 E 1500 1546.875
16 1 F 1600 1650

By a surprising coincidence, neutral 16edX turns out to be a false Pelogic temperament with a very pure 5:4 (mistuned by no more than 3.686 cents), and tuning 5:3 pure creates a tenth almost exactly equal to 38/29edo or almost exactly the Golden Subpental Mixolydian step or every ninth step of 110edo, and tuning 4:3 pure creates an almost exact 43/74edo fifth. Also, the Golden Soft Superpental Mixolydian step is almost exactly (25/24)^(10/7) and tuning 12:7 pure creates almost exactly the Golden Mixolydian-Ionian step.

See also

  • 16ed5/2 - equal division of the classic major tenth
  • 16ed7/3 - equal division of the septimal minor tenth
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