15edX is the scale which occurs as the dominant minor edX.
Intervals
| Degrees
|
Enneatonic
|
Minor mode Middletown temperament
|
Neutral mode Middletown temperament
|
Major mode Middletown temperament
|
| Aeolian-Dorian
|
Dorian
|
Dorian-Mixolydian
|
Mixolydian
|
Mixolydian-Ionian
|
| Subpental
|
Pental
|
Superpental
|
Subpental
|
Pental
|
Superpental
|
| Soft
|
Intense
|
| Min
|
Max
|
Min
|
Max
|
Min
|
Max
|
Min
|
Max
|
Min
|
Max
|
Min
|
Max
|
Min
|
Max
|
Min
|
Max
|
Min
|
Max
|
Min
|
Max
|
| 1
|
F#/Gb
|
96
|
97.778
|
100
|
101.818
|
102.857
|
104
|
105
|
106.667
|
107.586
|
110.345
|
112
|
| 2
|
G
|
192
|
195.556
|
200
|
203.636
|
205.714
|
208
|
210
|
213.333
|
215.172
|
220.69
|
224
|
| 3
|
G#/Jb
|
G#/Ab
|
288
|
293.333
|
300
|
305.4545
|
308.571
|
312
|
315
|
320
|
322.759
|
331.034
|
336
|
| 4
|
J
|
A
|
384
|
391.111
|
400
|
407.273
|
411.429
|
416
|
420
|
426.667
|
430.345
|
441.379
|
448
|
| 5
|
A
|
B
|
480
|
488.889
|
500
|
509.091
|
514.286
|
520
|
525
|
533.333
|
537.931
|
551.724
|
560
|
| 6
|
A#/Bb
|
B#/Hb
|
576
|
586.667
|
600
|
610.909
|
617.143
|
624
|
630
|
640
|
645.517
|
662.069
|
672
|
| 7
|
B
|
H
|
672
|
684.444
|
700
|
712.727
|
720
|
728
|
735
|
746.667
|
753.103
|
772.414
|
784
|
| 8
|
B#/Hb
|
H#/Cb
|
768
|
782.822
|
800
|
814.5455
|
822.857
|
832
|
845
|
853.333
|
860.69
|
882.759
|
896
|
| 9
|
H
|
C
|
864
|
880
|
900
|
916.364
|
925.714
|
936
|
940
|
960
|
968.276
|
993.103
|
1008
|
| 10
|
C
|
D
|
960
|
977.778
|
1000
|
1018.182
|
1028.571
|
1040
|
1050
|
1066.667
|
1075.862
|
1103.448
|
1120
|
| 11
|
C#/Db
|
D#/Sb
|
1056
|
1075.556
|
1100
|
1120
|
1131.429
|
1144
|
1155
|
1173.333
|
1183.448
|
1213.379
|
1232
|
| 12
|
D
|
S
|
1152
|
1173.333
|
1200
|
1221.818
|
1234.286
|
1248
|
1260
|
1280
|
1291.034
|
1324.138
|
1344
|
| 13
|
D#/Eb
|
S#/Eb
|
1248
|
1271.111
|
1300
|
1323.636
|
1337.143
|
1352
|
1365
|
1386.667
|
1398.621
|
1424.483
|
1456
|
| 14
|
E
|
1344
|
1368.889
|
1400
|
1425.4545
|
1440
|
1456
|
1470
|
1493.333
|
1506.207
|
1544.827
|
1568
|
| 15
|
F
|
1440
|
1466.667
|
1500
|
1527.273
|
1542.857
|
1560
|
1575
|
1600
|
1613.793
|
1655.172
|
1680
|
Most harmonic tunings
| Degrees
|
Enneatonic
|
Minor mode Middletown temperament
|
Neutral mode Middletown temperament
|
Major mode Middletown temperament
|
| 7/3±13¢
|
22/9±13¢
|
5/2±13¢
|
| 1
|
F#/Gb
|
96.925-98.658
|
102.294-104.027
|
104.888-106.621
|
| 2
|
G
|
193.8495-197.316
|
204.588-208.054
|
209.77-213.242
|
| 3
|
G#/Jb
|
G#/Ab
|
290.774-295.974
|
306.8815-312.0815
|
314.663-319.863
|
| 4
|
J
|
A
|
387.699-394.632
|
409.1755-416.109
|
419.559-426.484
|
| 5
|
A
|
B
|
484.624-493.29
|
511.469-520.136
|
524.438-533.105
|
| 6
|
A#/Bb
|
B#/Hb
|
581.548-591.948
|
613.763-624.163
|
629.3255-639.7255
|
| 7
|
B
|
H
|
678.473-690.606
|
716.057-728.19
|
734.213-746.346
|
| 8
|
B#/Hb
|
H#/Cb
|
775.398-789.2645
|
818.351-832.218
|
839.101-852.967
|
| 9
|
H
|
C
|
872.3225-887.9225
|
920.645-936.245
|
943.988-959.588
|
| 10
|
C
|
D
|
969.247-986.561
|
1022.939-1040.272
|
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
|
E
|
1356.946-1381.213
|
1432.114-1456.381
|
1468.426-1492.693
|
| 15
|
F
|
1453.871-1479.871
|
1534.408-1560.408
|
1573.314-1599.314
|
Golden tunings
Minor mode Middletown temperament
| Degrees
|
Enneatonic
|
Aeolian-Dorian
|
Dorian
|
| -13¢
|
Mean
|
-13¢ (s5D)
|
Subpental
|
Pental
|
Superpental
|
| +13¢ (A-D)
|
Mean
|
+13¢
|
-13¢
|
-13¢ (S5)
|
Mean
|
Mean
|
+13¢ (5)
|
-13¢ (D-M)
|
| 1
|
F#/Gb
|
96.457
|
97.323
|
97.39
|
98.19
|
98.257
|
99.123
|
100.618
|
101.245
|
101.4845
|
102.1115
|
102.351
|
102.788
|
| 2
|
G
|
192.913
|
194.647
|
194.78
|
196.38
|
196.513
|
198.247
|
201.236
|
202.49
|
202.969
|
204.223
|
204.702
|
205.576
|
| 3
|
G#/Jb
|
G#/Ab
|
289.37
|
291.97
|
292.17
|
294.57
|
294.77
|
297.37
|
301.854
|
303.734
|
304.454
|
306.334
|
307.054
|
308.365
|
| 4
|
J
|
A
|
385.829
|
389.294
|
389.56
|
392.76
|
394.027
|
396.493
|
402.4715
|
404.979
|
405.938
|
408.446
|
409.405
|
411.153
|
| 5
|
A
|
B
|
482.284
|
486.617
|
486.95
|
490.95
|
491.283
|
495.617
|
503.089
|
569.502
|
507.423
|
510.557
|
511.756
|
513.941
|
| 6
|
A#/Bb
|
B#/Hb
|
578.74
|
583.94
|
584.334
|
589.14
|
589.54
|
594.74
|
603.707
|
607.469
|
608.907
|
612.669
|
614.107
|
616.729
|
| 7
|
B
|
H
|
675.197
|
681.364
|
681.73
|
687.33
|
687.7965
|
693.863
|
704.325
|
708.7135
|
710.392
|
714.78
|
716.4585
|
719.5175
|
| 8
|
B#/Hb
|
H#/Cb
|
771.654
|
778.587
|
779.12
|
785.52
|
786.053
|
792.9865
|
804.943
|
809.958
|
811.877
|
816.892
|
818.81
|
822.306
|
| 9
|
H
|
C
|
868.1105
|
875.9105
|
876.51
|
883.7105
|
884.31
|
892.11
|
905.561
|
911.203
|
913.361
|
919.003
|
921.161
|
925.094
|
| 10
|
C
|
D
|
964.567
|
973.234
|
973.9
|
981.901
|
982.5665
|
991.233
|
1006.179
|
1012.448
|
1014.8455
|
1021.115
|
1023.512
|
1027.882
|
| 11
|
C#/Db
|
D#/Sb
|
1061.024
|
1070.557
|
1071.39
|
1080.091
|
1080.823
|
1090.356
|
1106.797
|
1113.693
|
1116.33
|
1123.226
|
1125.863
|
1130.67
|
| 12
|
D
|
S
|
1157.481
|
1167.881
|
1168.68
|
1178.281
|
1179.08
|
1189.48
|
1207.4145
|
1214.9375
|
1217.8145
|
1225.3375
|
1228.2145
|
1233.459
|
| 13
|
D#/Eb
|
S#/Eb
|
1253.937
|
1265.204
|
1266.07
|
1176.471
|
1277.336
|
1288.603
|
1308.032
|
1316.182
|
1319.299
|
1327.449
|
1330.566
|
1336.247
|
| 14
|
E
|
1350.394
|
1362.527
|
1363.46
|
1374.661
|
1375.593
|
1387.726
|
1408.65
|
1417.427
|
1420.783
|
1429.56
|
1432.917
|
1439.035
|
| 15
|
F
|
1446.851
|
1459.851
|
1460.85
|
1472.851
|
1473.85
|
1486.85
|
1509.268
|
1518.672
|
1522.268
|
1531.672
|
1535.268
|
1541.823
|
Neutral (Dorian-Mixolydian) mode Middletown temperament
| Degrees
|
Enneatonic
|
+13¢ (S5D)
|
Mean
|
-13¢ (s5M)
|
| 1
|
F#/Gb
|
102.978
|
103.655
|
103.855
|
| 2
|
G
|
205.956
|
207.31
|
207.709
|
| 3
|
G#/Jb
|
G#/Ab
|
308.934
|
310.965
|
311.564
|
| 4
|
J
|
A
|
411.9215
|
414.6195
|
415.419
|
| 5
|
A
|
B
|
514.891
|
518.274
|
519.2735
|
| 6
|
A#/Bb
|
B#/Hb
|
617.869
|
621.929
|
623.128
|
| 7
|
B
|
H
|
720.847
|
725.584
|
726.983
|
| 8
|
B#/Hb
|
H#/Cb
|
823.825
|
829.239
|
830.8375
|
| 9
|
H
|
C
|
926.823
|
932.893
|
934.693
|
| 10
|
C
|
D
|
1029.781
|
1036.549
|
1038.547
|
| 11
|
C#/Db
|
D#/Sb
|
1132.759
|
1140.204
|
1142.402
|
| 12
|
D
|
S
|
1235.7375
|
1243.859
|
1246.256
|
| 13
|
D#/Eb
|
S#/Eb
|
1338.716
|
1347.513
|
1353.111
|
| 14
|
E
|
1441.694
|
1451.168
|
1453.966
|
| 15
|
F
|
1544.672
|
1554.823
|
1557.82
|
Major mode Middletown temperament
| Degrees
|
Enneatonic
|
Mixolydian
|
Mixolydian-Ionian
|
| Subpental
|
Pental
|
Superpental
|
Mean
|
+13¢
|
| -13¢ (5)
|
+13¢ (D-M)
|
Mean
|
Mean
|
+13¢ (s5)
|
+13¢
|
-13¢ (Soft s5)
|
Soft
|
Intense
|
| +13¢ (Soft s5)
|
-13¢
|
Mean
|
-13¢ (M-I)
|
+13¢
|
| 1
|
F#/Gb
|
104.306
|
104.5215
|
104.721
|
105.173
|
105.788
|
106.04
|
106.664
|
107.531
|
108.398
|
108.4245
|
109.291
|
109.6375
|
110.158
|
110.504
|
111.371
|
| 2
|
G
|
208.613
|
209.043
|
209.443
|
210.346
|
211.576
|
212.08
|
213.329
|
215.032
|
216.795
|
216.849
|
218.582
|
219.275
|
220.316
|
221.008
|
222.742
|
| 3
|
G#/Jb
|
G#/Ab
|
312.919
|
313.565
|
314.164
|
315.519
|
317.364
|
318.119
|
319.993
|
322.593
|
325.193
|
325.273
|
327.873
|
328.913
|
330.473
|
331.613
|
334.113
|
| 4
|
J
|
A
|
417.226
|
418.086
|
418.885
|
420.639
|
423.152
|
424.159
|
426.657
|
430.124
|
433.5905
|
433.698
|
437.1345
|
438.55
|
440.631
|
442.017
|
445.4835
|
| 5
|
A
|
B
|
521.532
|
526.308
|
523.607
|
525.865
|
528.94
|
530.199
|
533.321
|
537.655
|
541.988
|
542.122
|
546.456
|
548.188
|
550.789
|
552.521
|
556.854
|
| 6
|
A#/Bb
|
B#/Hb
|
625.8385
|
627.129
|
628.328
|
631.0385
|
634.728
|
636.238
|
639.987
|
645.186
|
650.386
|
650.547
|
655.747
|
657.825
|
660.947
|
663.025
|
668.225
|
| 7
|
B
|
H
|
730.145
|
731.651
|
733.0495
|
736.212
|
740.516
|
742.278
|
746.65
|
752.717
|
758.783
|
758.971
|
765.038
|
767.463
|
771.105
|
773.5235
|
779.596
|
| 8
|
B#/Hb
|
H#/Cb
|
834.451
|
836.172
|
837.771
|
841.385
|
846.304
|
853.314
|
899.98
|
867.181
|
914.605
|
867.396
|
874.329
|
877.1
|
881.262
|
884.033
|
890.967
|
| 9
|
H
|
C
|
938.758
|
940.694
|
942.492
|
946.558
|
952.092
|
954.358
|
999.978
|
967.7785
|
1016.228
|
975.82
|
983.62
|
986.738
|
991.42
|
994.538
|
1002.338
|
| 10
|
C
|
D
|
1043.064
|
1045.2155
|
1047.214
|
1051.731
|
1057.88
|
1060.3975
|
1066.643
|
1075.3095
|
1083.976
|
1084.245
|
1092.911
|
1096.3755
|
1101.578
|
1105.042
|
1113.709
|
| 11
|
C#/Db
|
D#/Sb
|
1147.3705
|
1149.737
|
1151.935
|
1156.904
|
1163.668
|
1166.437
|
1173.307
|
1182.84
|
1192.374
|
1192.669
|
1202.202
|
1206.013
|
1211.736
|
1215.546
|
1225.08
|
| 12
|
D
|
S
|
1251.677
|
1254.259
|
1256.656
|
1262.077
|
1269.456
|
1272.477
|
1279.971
|
1290.371
|
1300.771
|
1301.0395
|
1311.4935
|
1315.651
|
1321.8935
|
1326.051
|
1336.451
|
| 13
|
D#/Eb
|
S#/Eb
|
1355.983
|
1358.78
|
1361.378
|
1367.25
|
1275.244
|
1378.517
|
1386.636
|
1397.902
|
1409.169
|
1409.518
|
1420.785
|
1438.993
|
1425.288
|
1436.555
|
1447.8215
|
| 14
|
E
|
1460.29
|
1463.302
|
1466.099
|
1472.423
|
1481.032
|
1484.556
|
1493.3
|
1505.433
|
1517.567
|
1517.9425
|
1530.076
|
1541.778
|
1534.926
|
1547.059
|
1559.192
|
| 15
|
F
|
1564.596
|
1567.96
|
1570.82
|
1577.596
|
1583.82
|
1590.596
|
1599.964
|
1612.964
|
1625.964
|
1626.366
|
1639.366
|
1644.563
|
1652.366
|
1657.563
|
1670.563
|
By a surprising coincidence, the 15ed of the Golden tenth (7φ+6)\(5φ^2)edo is almost exactly every third degree of 34edo. Additionally, those of the modal Golden 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.