60 | a=0 | 90.4 (88.7, 92.0) | 82.9 (80.1, 85.7) | 79.3 (76.3, 81.7) | 69.9 (64.3, 74.9) | 66.6 (60.5, 72.3) | 61.6 (56.1, 67.8)
a=1 | 98.5 (98.1, 99.0) | 99.0 (98.6, 99.3) | 99.2 (98.9, 99.4) | 99.7 (99.5, 99.8) | 99.8 (99.6, 99.9) | 99.9 (99.7, 100.0)
$\Delta\text{xAUC}$ | 8.2 (6.3, 10.5) | 16.1 (13.0, 19.1) | 20.0 (17.4, 23.1) | 30.8 (25.5, 36.8) | 33.3 (27.3, 39.4) | 38.3 (31.9, 43.9)
90 | a=0 | 89.0 (86.9, 90.8) | 81.6 (79.4, 84.7) | 75.4 (71.3, 78.9) | 66.8 (59.3, 71.6) | 65.5 (58.9, 70.8) | 61.3 (54.3, 66.9)
a=1 | 98.6 (98.2, 99.0) | 99.0 (98.6, 99.2) | 99.1 (98.8, 99.4) | 99.7 (99.4, 99.8) | 99.7 (99.5, 99.9) | 99.9 (99.7, 100.0)
$\Delta\text{xAUC}$ | 9.7 (7.7, 11.8) | 17.4 (14.1, 19.6) | 23.8 (20.1, 28.0) | 32.8 (27.8, 40.3) | 34.2 (28.8, 41.0) | 38.6 (32.7, 45.6)
120 | a=0 | 81.8 (79.1, 85.0) | 75.2 (71.3, 79.1) | 71.9 (66.7, 76.1) | 65.1 (59.1, 70.9) | 63.6 (57.3, 70.2) | 60.0 (53.4, 66.2)
a=1 | 98.5 (98.1, 98.9) | 99.0 (98.5, 99.3) | 99.2 (98.8, 99.4) | 99.6 (99.4, 99.8) | 99.8 (99.5, 99.9) | 99.9 (99.7, 100.0)
$\Delta\text{xAUC}$ | 16.8 (13.3, 19.6) | 23.9 (19.7, 27.7) | 27.4 (23.0, 32.6) | 34.6 (28.5, 40.7) | 36.3 (29.4, 42.5) | 39.9 (33.5, 46.5)
150 | a=0 | 78.8 (76.3, 82.8) | 72.5 (67.9, 77.0) | 69.5 (65.1, 74.6) | 63.7 (57.9, 69.4) | 63.7 (57.9, 69.4) | 59.0 (52.2, 65.8)
a=1 | 98.6 (98.1, 99.0) | 99.1 (98.6, 99.4) | 99.3 (98.9, 99.6) | 99.7 (99.5, 99.9) | 99.7 (99.5, 99.9) | 99.9 (99.7, 100.0)
$\Delta\text{xAUC}$ | 19.6 (15.7, 22.3) | 26.6 (21.9, 31.1) | 29.9 (24.6, 34.3) | 36.1 (30.1, 41.9) | 36.1 (30.1, 41.9) | 40.9 (33.8, 47.8)
180 | a=0 | 78.4 (75.4, 81.6) | 71.6 (67.6, 76.7) | 68.8 (64.1, 73.0) | 63.4 (57.7, 68.5) | 61.9 (55.8, 67.4) | 58.7 (52.6, 64.9)
a=1 | 98.5 (98.1, 98.9) | 99.1 (98.6, 99.3) | 99.4 (99.0, 99.6) | 99.8 (99.5, 99.9) | 99.8 (99.6, 100.0) | 99.9 (99.7, 100.0)
$\Delta\text{xAUC}$ | 20.3 (16.7, 23.2) | 27.5 (22.2, 31.6) | 30.6 (26.3, 35.2) | 36.4 (30.9, 42.2) | 38.0 (32.2, 44.1) | 41.2 (35.0, 47.4)
Table 10: Group-wise xAUC and $\Delta\text{xAUC}$ under conditional
distributional difference for all values of $\tau_{1},\phi$, at
$d^{\prime}=6$.

| $\tau_{1}$
---|---
$\phi$ | Group | 5.0 | 5.4 | 5.8 | 6.2 | 6.6 | 7.0
0 | a=0 | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0)
a=1 | 97.6 (97.3, 97.9) | 97.5 (97.1, 97.8) | 97.3 (96.8, 97.6) | 97.7 (97.3, 98.1) | 93.7 (92.1, 94.9) | 83.4 (77.2, 86.9)
$\Delta\text{AUC}$ | 0.2 (0.0, 0.6) | 0.2 (0.0, 0.7) | 0.4 (0.0, 1.0) | 2.4 (1.6, 3.6) | 4.0 (2.9, 5.7) | 14.3 (10.8, 20.5)
30 | a=0 | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0)
a=1 | 96.2 (95.7, 96.9) | 93.9 (92.8, 94.8) | 92.8 (91.1, 94.1) | 90.8 (89.2, 92.0) | 89.8 (87.5, 91.3) | 81.4 (76.4, 85.6)
$\Delta\text{AUC}$ | 1.5 (0.8, 2.0) | 3.8 (2.6, 4.9) | 4.9 (3.3, 6.6) | 7.0 (5.7, 8.6) | 7.1 (5.7, 9.1) | 16.4 (12.1, 21.4)
60 | a=0 | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0)
a=1 | 91.5 (90.2, 92.7) | 87.3 (85.5, 88.8) | 85.2 (83.0, 87.1) | 81.1 (78.4, 83.5) | 80.3 (77.3, 82.6) | 76.5 (72.5, 79.9)
$\Delta\text{AUC}$ | 6.1 (5.0, 7.3) | 10.4 (8.8, 12.4) | 12.4 (10.4, 14.6) | 16.5 (14.2, 19.5) | 17.5 (14.7, 20.3) | 21.2 (17.9, 25.1)
90 | a=0 | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0)
a=1 | 84.9 (83.2, 86.9) | 79.9 (77.5, 81.9) | 77.7 (74.9, 79.9) | 73.2 (70.0, 76.7) | 72.9 (69.6, 75.9) | 71.6 (66.3, 75.1)
$\Delta\text{AUC}$ | 12.8 (10.9, 14.5) | 17.8 (15.8, 20.0) | 20.0 (17.7, 22.9) | 24.5 (21.2, 27.6) | 24.7 (21.8, 28.0) | 26.1 (22.6, 31.4)
120 | a=0 | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0)
a=1 | 78.2 (75.7, 80.9) | 73.1 (69.4, 76.1) | 71.1 (67.7, 73.6) | 69.0 (64.0, 73.0) | 68.5 (63.8, 73.1) | 69.5 (58.7, 73.5)
$\Delta\text{AUC}$ | 19.5 (17.0, 22.0) | 24.6 (21.6, 28.4) | 26.5 (24.0, 30.0) | 28.7 (24.7, 33.7) | 29.1 (24.7, 34.0) | 28.1 (24.4, 38.7)
150 | a=0 | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0)
a=1 | 71.0 (68.3, 74.8) | 73.2 (70.5, 76.2) | 66.2 (61.9, 70.6) | 67.0 (61.0, 72.6) | 67.3 (31.5, 72.4) | 68.1 (32.4, 73.7)
$\Delta\text{AUC}$ | 26.7 (23.0, 29.5) | 30.6 (26.9, 34.7) | 31.4 (26.9, 36.0) | 30.7 (25.0, 36.5) | 30.4 (25.2, 66.4) | 29.5 (24.0, 65.5)
180 | a=0 | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0)
a=1 | 67.7 (63.4, 70.7) | 63.6 (59.9, 68.4) | 63.8 (59.0, 69.0) | 65.8 (35.2, 70.8) | 66.3 (31.7, 72.3) | 67.2 (31.0, 72.7)
$\Delta\text{AUC}$ | 30.0 (27.0, 34.3) | 34.3 (29.3, 38.0) | 33.8 (28.7, 38.6) | 31.7 (26.6, 62.6) | 31.6 (25.4, 66.0) | 30.5 (24.8, 66.6)
Table 11: Group-wise AUC and $\Delta\text{AUC}$ under conditional
distributional difference for all values of $\tau_{1},\phi$, at
$d^{\prime}=8$.

| $\tau_{1}$
---|---
$\phi$ | Group | 5.0 | 5.4 | 5.8 | 6.2 | 6.6 | 7.0
0 | a=0 | 97.6 (97.0, 98.1) | 90.9 (89.5, 92.4) | 85.8 (83.6, 88.3) | 72.3 (68.5, 77.0) | 69.0 (64.6, 73.6) | 62.3 (55.8, 67.7)
a=1 | 97.6 (97.2, 98.2) | 99.7 (99.5, 99.8) | 99.9 (99.8, 99.9) | 100.0 (99.9, 100.0) | 100.0 (99.9, 100.0) | 99.9 (99.8, 100.0)
$\Delta\text{xAUC}$ | 0.4 (0.1, 1.1) | 8.8 (7.1, 10.2) | 14.1 (11.6, 16.3) | 27.6 (23.0, 31.4) | 30.9 (26.3, 35.4) | 37.6 (32.0, 44.2)
30 | a=0 | 94.6 (93.6, 95.5) | 87.9 (85.6, 89.8) | 82.8 (80.7, 85.9) | 70.9 (66.2, 75.6) | 68.2 (62.9, 73.9) | 62.4 (56.7, 68.1)
a=1 | 98.5 (98.1, 98.9) | 99.2 (98.9, 99.4) | 99.5 (99.3, 99.7) | 99.9 (99.8, 99.9) | 99.9 (99.8, 100.0) | 99.9 (99.8, 100.0)
$\Delta\text{xAUC}$ | 3.9 (2.9, 5.2) | 11.5 (9.3, 13.6) | 16.6 (13.6, 18.9) | 29.0 (24.2, 33.7) | 31.7 (25.9, 37.0) | 37.5 (31.7, 43.2)
60 | a=0 | 89.1 (86.8, 90.6) | 81.8 (78.7, 84.4) | 77.9 (74.5, 80.3) | 68.4 (61.9, 72.9) | 66.2 (60.5, 71.1) | 62.0 (55.6, 67.4)
a=1 | 98.6 (98.1, 98.9) | 99.0 (98.6, 99.3) | 99.2 (98.8, 99.4) | 99.7 (99.4, 99.8) | 99.8 (99.6, 99.9) | 99.9 (99.7, 100.0)
$\Delta\text{xAUC}$ | 9.5 (7.6, 11.9) | 17.2 (14.4, 20.5) | 21.2 (18.6, 24.9) | 31.3 (26.6, 37.7) | 33.5 (28.5, 39.3) | 38.0 (32.4, 44.3)
90 | a=0 | 83.6 (80.8, 86.0) | 76.5 (72.6, 79.7) | 73.1 (67.8, 76.2) | 66.1 (59.3, 70.5) | 64.6 (58.2, 69.7) | 60.5 (54.2, 66.1)
a=1 | 98.5 (98.1, 98.9) | 98.9 (98.5, 99.2) | 99.1 (98.8, 99.4) | 99.6 (99.4, 99.8) | 99.7 (99.5, 99.9) | 99.9 (99.7, 100.0)
$\Delta\text{xAUC}$ | 15.0 (12.2, 17.6) | 22.4 (19.0, 26.4) | 26.1 (22.2, 31.5) | 33.5 (28.9, 40.3) | 35.2 (29.8, 41.5) | 39.4 (33.6, 45.6)
120 | a=0 | 79.0 (75.1, 81.9) | 72.3 (67.0, 76.5) | 69.4 (63.4, 74.0) | 63.8 (57.9, 69.5) | 62.1 (55.9, 67.5) | 58.9 (52.6, 65.0)
a=1 | 98.6 (98.1, 99.0) | 99.1 (98.7, 99.4) | 99.3 (98.9, 99.6) | 99.8 (99.5, 99.9) | 99.8 (99.6, 100.0) | 99.9 (99.7, 100.0)
$\Delta\text{xAUC}$ | 19.7 (16.3, 23.6) | 26.9 (22.5, 32.0) | 29.8 (25.2, 36.2) | 35.9 (30.1, 42.1) | 36.3 (29.4, 42.5) | 41.1 (34.8, 47.4)
150 | a=0 | 75.1 (71.4, 79.3) | 68.6 (63.4, 73.3) | 66.3 (61.0, 71.5) | 60.7 (54.6, 66.0) | 59.4 (52.3, 65.4) | 58.0 (51.9, 63.8)
a=1 | 98.8 (98.3, 99.1) | 99.4 (99.1, 99.6) | 99.6 (99.3, 99.8) | 99.9 (99.7, 100.0) | 99.9 (99.7, 100.0) | 99.9 (99.8, 100.0)
$\Delta\text{xAUC}$ | 19.6 (15.7, 22.3) | 30.7 (25.9, 35.8) | 33.4 (27.9, 38.8) | 39.2 (33.7, 45.3) | 40.5 (34.4, 47.7) | 41.9 (36.1, 48.1)
180 | a=0 | 74.0 (70.3, 77.8) | 67.5 (62.1, 72.2) | 64.4 (58.9, 69.7) | 59.4 (53.0, 65.2) | 58.3 (51.6, 63.5) | 57.7 (50.7, 63.7)
a=1 | 98.9 (98.4, 99.3) | 99.5 (99.2, 99.7) | 99.7 (99.5, 99.9) | 99.9 (99.7, 100.0) | 99.9 (99.7, 100.0) | 99.9 (99.8, 100.0)
$\Delta\text{xAUC}$ | 24.9 (20.7, 29.0) | 31.9 (27.1, 37.5) | 35.2 (29.8, 40.9) | 40.5 (34.5, 46.9) | 41.7 (36.3, 48.3) | 42.2 (36.1, 49.3)
Table 12: Group-wise xAUC and $\Delta\text{xAUC}$ under conditional
distributional difference for all values of $\tau_{1},\phi$, at
$d^{\prime}=8$.

| $\tau_{1}$
---|---
$\phi$ | Group | 5.0 | 5.4 | 5.8 | 6.2 | 6.6 | 7.0
0 | a=0 | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0)
a=1 | 97.6 (97.3, 97.9) | 97.5 (97.1, 97.8) | 97.3 (96.8, 97.6) | 97.7 (97.3, 98.1) | 93.7 (92.1, 94.9) | 83.4 (77.2, 86.9)
$\Delta\text{AUC}$ | 0.2 (0.0, 0.6) | 0.2 (0.0, 0.7) | 0.4 (0.0, 1.0) | 2.4 (1.6, 3.6) | 4.0 (2.9, 5.7) | 14.3 (10.8, 20.5)
30 | a=0 | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0)
a=1 | 95.9 (95.0, 96.6) | 93.2 (91.9, 94.4) | 92.0 (90.3, 93.2) | 989.4 (87.7, 91.2) | 88.7 (86.1, 90.3) | 81.2 (76.0, 84.5)
$\Delta\text{AUC}$ | 1.8 (1.1, 2.7) | 4.5 (3.2, 5.9) | 5.7 (4.5, 7.4) | 8.2 (6.6, 10.0) | 9.0 (7.3, 11.5) | 16.5 (13.1, 21.7))
60 | a=0 | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0)
a=1 | 89.7 (88.5, 91.4) | 85.3 (83.1, 87.1) | 83.1 (80.5, 85.5) | 78.8 (76.0, 81.0) | 78.1 (75.1, 80.5) | 74.9 (71.0, 78.5)
$\Delta\text{AUC}$ | 8.0 (6.3, 9.2) | 12.4 (10.5, 14.5) | 14.6 (12.3, 17.1) | 18.7 (16.7, 21.5) | 19.6 (17.3, 22.6) | 22.8 (19.3, 26.6)
90 | a=0 | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0)
a=1 | 81.3 (79.2, 83.1) | 76.0 (73.4, 78.3) | 73.8 (70.8, 76.2) | 71.1 (67.1, 73.2) | 70.3 (65.2, 72.9) | 70.0 (65.0, 73.9)
$\Delta\text{AUC}$ | 16.4 (14.6, 18.6) | 21.6 (19.2, 24.2) | 24.0 (21.5, 26.6) | 26.7 (24.2, 30.7) | 27.4 (24.7, 32.5) | 27.7 (24.0, 32.4)
120 | a=0 | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0)
a=1 | 72.4 (69.2, 75.7) | 68.1 (65.2, 71.9) | 67.6 (63.0, 71.8) | 68.4 (61.7, 71.8) | 68.5 (63.8, 73.1) | 68.9 (31.4, 72.8)
$\Delta\text{AUC}$ | 25.4 (22.2, 28.5) | 29.6 (25.7, 32.6) | 30.4 (26.7, 33.9) | 30.0 (26.1, 34.6) | 29.3 (26.0, 35.7) | 28.9 (24.9, 66.1)
150 | a=0 | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0)
a=1 | 64.4 (60.8, 68.5) | 66.2 (57.1, 70.2) | 66.8 (32.9, 71.1) | 67.8 (31.0, 71.6) | 67.8 (31.0, 71.6) | 67.8 (31.0, 71.6)
$\Delta\text{AUC}$ | 33.2 (28.9, 37.0) | 31.4 (27.1, 40.0) | 31.0 (26.4, 64.7) | 30.1 (26.0, 66.7) | 30.1 (26.0, 66.7) | 30.1 (26.0, 66.7)
180 | a=0 | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0) | 97.7 (97.4, 98.0)
a=1 | 66.0 (31.4, 72.1) | 66.0 (31.4, 72.1) | 63.8 (59.0, 69.0) | 65.8 (35.2, 70.8) | 66.3 (31.7, 72.3) | 67.2 (31.0, 72.7)
$\Delta\text{AUC}$ | 31.7 (25.6, 66.2) | 31.7 (25.6, 66.2) | 31.7 (25.6, 66.2) | 31.7 (25.6, 66.2) | 31.7 (25.6, 66.2) | 31.7 (25.6, 66.2)
Table 13: Group-wise AUC and $\Delta\text{AUC}$ under conditional
distributional difference for all values of $\tau_{1},\phi$, at
$d^{\prime}=10$.

| $\tau_{1}$
---|---
$\phi$ | Group | 5.0 | 5.4 | 5.8 | 6.2 | 6.6 | 7.0
0 | a=0 | 97.6 (97.0, 98.1) | 90.9 (89.5, 92.4) | 85.8 (83.6, 88.3) | 72.3 (68.5, 77.0) | 69.0 (64.6, 73.6) | 62.3 (55.8, 67.7)
a=1 | 97.6 (97.2, 98.2) | 99.7 (99.5, 99.8) | 99.9 (99.8, 99.9) | 100.0 (99.9, 100.0) | 100.0 (99.9, 100.0) | 99.9 (99.8, 100.0)
$\Delta\text{xAUC}$ | 0.4 (0.1, 1.1) | 8.8 (7.1, 10.2) | 14.1 (11.6, 16.3) | 27.6 (23.0, 31.4) | 30.9 (26.3, 35.4) | 37.6 (32.0, 44.2)
30 | a=0 | 94.0 (93.0, 95.0) | 87.1 (85.1, 89.2) | 82.2 (79.6, 85.1) | 70.6 (65.8, 75.3) | 67.8 (62.4, 73.5) | 62.2 (56.7, 67.9)
a=1 | 98.5 (98.1, 98.9) | 99.1 (98.8, 99.4) | 99.4 (99.2, 99.6) | 99.9 (99.7, 99.9) | 99.9 (99.8, 100.0) | 99.9 (99.8, 100.0)
$\Delta\text{xAUC}$ | 4.5 (3.4, 5.8) | 12.0 (9.8, 14.0) | 17.3 (14.2, 19.8) | 29.2 (24.4, 34.1) | 32.1 (26.3, 37.5) | 37.6 (31.8, 43.2)
60 | a=0 | 87.4 (85.4, 89.5) | 80.4 (77.2, 83.0) | 77.9 (74.5, 80.3) | 67.7 (61.4, 72.3) | 66.2 (59.3, 70.8) | 61.7 (55.9, 67.9)
a=1 | 98.6 (98.1, 98.9) | 98.9 (98.6, 99.2) | 99.1 (98.8, 99.4) | 99.6 (99.5, 99.8) | 99.8 (99.5, 99.9) | 99.9 (99.7, 100.0)
$\Delta\text{xAUC}$ | 11.1 (9.1, 13.3) | 18.8 (15.5, 22.1) | 22.4 (18.9, 26.1) | 31.9 (27.2, 38.4) | 33.5 (28.8, 40.5) | 38.1 (31.9, 44.0)
90 | a=0 | 81.0 (77.5, 83.8) | 74.1 (70.1, 77.7) | 71.2 (66.4, 75.6) | 64.7 (57.9, 70.0) | 63.2 (57.0, 69.3) | 59.8 (52.5, 65.5)
a=1 | 98.5 (98.1, 98.9) | 99.0 (98.5, 99.2) | 99.2 (98.8, 99.4) | 99.7 (99.5, 99.9) | 99.8 (99.6, 99.9) | 99.9 (99.7, 100.0)
$\Delta\text{xAUC}$ | 17.6 (14.2, 21.1) | 22.4 (19.0, 26.4) | 27.9 (23.5, 32.8) | 35.0 (29.5, 41.8) | 36.5 (30.2, 42.9) | 40.1 (34.3, 47.4)
120 | a=0 | 75.5 (71.5, 79.6) | 69.2 (63.2, 74.1) | 66.4 (58.9, 71.5) | 60.7 (53.8, 66.7) | 59.5 (51.8, 65.5) | 58.2 (49.7, 63.4)
a=1 | 98.8 (98.2, 99.2) | 99.4 (98.9, 99.6) | 99.6 (99.2, 99.8) | 99.9 (99.6, 100.0) | 99.9 (99.7, 100.0) | 99.9 (99.7, 100.0)
$\Delta\text{xAUC}$ | 23.2 (18.8, 27.9) | 30.1 (24.9, 36.5) | 33.1 (28.0, 40.7) | 39.2 (32.8, 46.1) | 40.3 (34.3, 48.1) | 41.7 (36.3, 50.3)
150 | a=0 | 67.9 (62.8, 72.4) | 61.0 (54.3, 67.3) | 58.7 (50.6, 64.8) | 60.7 (54.6, 66.0) | 59.4 (52.3, 65.4) | 57.3 (50.5, 63.9)
a=1 | 99.5 (99.1, 99.7) | 99.9 (99.6, 100.0) | 99.6 (99.3, 99.8) | 99.9 (99.7, 100.0) | 99.9 (99.8, 100.0) | 99.9 (99.8, 100.0)
$\Delta\text{xAUC}$ | 31.6 (26.9, 37.0) | 38.8 (32.4, 45.7) | 33.4 (27.9, 38.8) | 41.2 (35.0, 49.4) | 42.5 (35.9, 49.5) | 42.5 (35.9, 49.5)
180 | a=0 | 57.2 (48.3, 63.9) | 57.2 (48.3, 63.9) | 57.2 (48.3, 63.9) | 57.2 (48.3, 63.9) | 57.2 (48.3, 63.9) | 57.2 (48.3, 63.9)
a=1 | 99.9 (99.7, 100.0) | 99.9 (99.7, 100.0) | 99.9 (99.7, 100.0) | 99.9 (99.7, 100.0) | 99.9 (99.7, 100.0) | 99.9 (99.7, 100.0)
$\Delta\text{xAUC}$ | 42.7 (35.8, 51.7) | 42.7 (35.8, 51.7) | 42.7 (35.8, 51.7) | 42.7 (35.8, 51.7) | 42.7 (35.8, 51.7) | 42.7 (35.8, 51.7)
Table 14: Group-wise xAUC and $\Delta\text{xAUC}$ under conditional
distributional difference for all values of $\tau_{1},\phi$, at
$d^{\prime}=10$.

###  E.4 Additional Results for Varying Marginal Distributional Distance
(Setting 2)

##### Increasing divergence between the marginal risk distributions correlates
with larger performance gaps under disparate censorship.

We set $\tau_{0}=5,\tau_{1}=6.6$, a setting in which we observed performance
gaps under disparate censorship, and vary $\Delta\mu$ (defined as
$\mu_{1}-\mu_{0}$) and $\sigma$. We also fix $\mu_{0}+\mu_{1}=0.35+0.55=0.9$.
We are interested whether varying the KL divergence between the marginals,
given by $\frac{1}{2\sigma}\lVert\Delta\mu\rVert_{2}^{2}$, impacts the
magnitude of performance gaps. To that end, we first explore the impacts of
changing $\sigma$ for both groups on subgroup performance gaps. We then vary
$\Delta\mu$, or the difference of the means between the marginal risk
distributions, on subgroup performance gaps. In summary, we find that
increasing the KL divergence by either decreasing the within-group variance or
increasing the mean difference exacerbates the negative impacts of disparate
censorship.

Figure 17:  $\Delta\text{AUC}$ (left) and $\Delta\text{xAUC}$ (right) with 95%
empirical CIs for the model trained on $y$ (gray) and on $\tilde{y}$ (green).
As $\sigma^{2}$ increases to the right, increasing overlap between the
distributions, the number of missed positives increases, and
$\Delta\text{AUC},\Delta\text{xAUC}$ decrease.

We first evaluate the effect of within-group variance. As we decrease
$\sigma^{2}$ from 0.2 to 0.05, we observe increasing AUC and xAUC gaps. As
seen in Figure 17, at a variance of 0.05 (left side, all graphs), the median
AUC and xAUC gaps are 0.09 and 0.22. However, as we increase the within-group
variance to 0.2 (right side, all graphs), the median AUC and xAUC gaps narrow
to 0.01 and 0.06. Decreasing $\sigma^{2}$ also decreases the overlap between
the two distributions, leading to greater performance disparities.

Figure 18:  $\Delta\text{AUC}$ (left) and $\Delta\text{xAUC}$ (right) with 95%
empirical CIs for the model trained on $y$ (gray) and on $\tilde{y}$ (green).
As $\Delta\mu$ increases to the right, the AUC and xAUC gaps widen. Note that
$P_{0}(\tilde{y}=0\mid y=1)=0$—there are no missed positives in group $a=0$.
The number of missed positives first increases, then decreases, since
sufficiently high $\Delta\mu$ means that untested patients’ risk largely lies
in one tail of the risk distribution.

We then evaluate the effect of mean difference on performance disparities. As
between-group mean difference increases, disparities between groups generally
worsen with respect to oracle performance. We increase the per-element mean
difference between the distributions from 0 to 0.4, keeping the decision
boundary constant. Figure 18 shows that as mean difference $\Delta\mu$
increases, disparities in AUC, xAUC emerge. At $\Delta\mu=0$, the performance
is identical to the oracle, as expected. At $\Delta\mu=0$, we are in Setting
1: there are no marginal or conditional differences between groups. However,
as $\Delta\mu$ grows to 0.3, the AUC and xAUC gaps increase to 0.03, 0.14,
respectively.

Our results suggest that distributional distance between patient subgroups as
measured using mean difference and variance is correlated with performance
disparities. Characterizing distributional differences in patient subgroup
risk could provide expected levels of performance disparities under disparate
censorship. Full results are provided in Table 15 (differences in $\Delta\mu$)
and Table 16 (differences in $\sigma^{2}$).

Intergroup mean diff. ($\Delta\mu$) | Group | AUC | xAUC
---|---|---|---
0 | $a=0$ | 95.8 (95.2, 96.2) | 95.7 (95.0, 96.2)
$a=1$ | 95.8 (95.2, 96.3) | 95.9 (95.2, 96.3)
$\Delta$ | 0.1 (0.0, 0.4) | 0.2 (0.0, 0.5)
0.1 | $a=0$ | 94.7 (93.8, 95.4) | 97.1 (96.5, 97.5)
$a=1$ | 95.8 (95.0, 96.3) | 92.4 (91.1, 93.2)
$\Delta$ | 1.0 (0.7, 1.5) | 4.8 (4.2, 5.6)
0.2 | $a=0$ | 95.1 (94.0, 95.7) | 98.6 (98.3, 98.8)
$a=1$ | 96.8 (96.2, 97.3) | 89.5 (87.8, 90.7)
$\Delta$ | 1.8 (1.3, 2.3) | 9.2 (8.0, 10.6)
0.3 | $a=0$ | 96.5 (95.8, 97.1) | 99.6 (99.5, 99.7)
$a=1$ | 98.5 (98.1, 98.7) | 88.2 (86.0, 89.9)
$\Delta$ | 1.9 (1.5, 2.5) | 11.4 (9.8, 13.4)
Table 15: AUC, xAUC with empirical 95% CIs at varying settings of $\Delta\mu$, $\tau_{0}=5,\tau_{1}=6.6$. Within-group var. ($\sigma^{2}$) | Group | AUC | xAUC
---|---|---|---
0.05 | $a=0$ | 93.7 (91.6, 95.5) | 98.2 (97.5, 98.7)
$a=1$ | 94.1 (92.7, 95.3) | 81.0 (78.3, 83.5)
$\Delta$ | 0.5 (0.0, 1.3) | 17.2 (15.2, 19.2)
0.1 | $a=0$ | 95.9 (95.1, 96.6) | 98.4 (98.1, 98.7)
$a=1$ | 95.7 (95.0, 96.4) | 89.0 (87.3, 90.4)
$\Delta$ | 0.2 (0.0, 0.5) | 9.4 (8.2, 10.7)
0.15 | $a=0$ | 96.4 (95.9, 96.9) | 98.3 (98.0, 98.6)
$a=1$ | 96.1 (95.5, 96.6) | 91.8 (90.5, 92.9)
$\Delta$ | 0.3 (0.0, 0.7) | 6.5 (5.6, 7.5)
0.2 | $a=0$ | 96.7 (96.3, 97.2) | 98.2 (97.9, 98.6)
$a=1$ | 96.4 (95.9, 97.0) | 93.4 (92.5, 94.4)
$\Delta$ | 0.2 (0.0, 0.6) | 4.8 (4.1, 5.4)
Table 16: AUC, xAUC with empirical 95% CIs at varying settings of
$\sigma^{2}$, $\tau_{0}=5,\tau_{1}=6.6$.

##  Appendix F Code

For reproducibility, code used to generate all figures and experimental
results after review is provided at the MLD3 Github.


11institutetext: IDLab, Department of Information Technology at Ghent
University – imec
Technologiepark-Zwijnaarde 126, B-9052 Ghent, Belgium
11email<EMAIL_ADDRESS>
# Rhythm, Chord and Melody Generation for Lead Sheets using Recurrent Neural
Networks

Cedric De Boom  Stephanie Van Laere  Tim Verbelen  Bart Dhoedt

###### Abstract

Music that is generated by recurrent neural networks often lacks a sense of
direction and coherence. We therefore propose a two-stage LSTM-based model for
lead sheet generation, in which the harmonic and rhythmic templates of the
song are produced first, after which, in a second stage, a sequence of melody
notes is generated conditioned on these templates. A subjective listening test
shows that our approach outperforms the baselines and increases perceived
musical coherence.

###### Keywords:

Music generation lead sheets neural networks

##  1 Lead Sheets

Lead sheets are widely used to represent the fundamental musical information
about almost any contemporary song: they contain a chord scheme, a melody
line, some navigation and repetition markers, and sometimes lyrics. They
seldom contain information about the instrumentation or accompaniment, so any
band can take a lead sheet as a guideline and make the song their own,
sometimes even by improvising over the chord schemes. In this paper we focus
on generating chords and melody lines for lead sheets from scratch.

A major difficulty in music generation is that harmony, melody and rhythm all
influence each other. For example, a melody note can change whenever the
underlying harmony changes, and vice versa. Rhythmic patterns can influence
which notes are played, and rhythm and harmony together define the overall
groove of the piece. To tackle this issue, we split the generation process
into two stages. First, we generate a harmonic progression using chord
sequences, while simultaneously picking the most appropriate rhythmic
patterns. And in a second step the melody is generated on top of this harmonic
and rhythmic template.

We are, however, not the first to tackle the problem of lead sheet generation
and, in general, music generation. Briot et al. provide a recent and extensive
overview of all deep learning based techniques in this field [1]. Regarding
lead sheets specifically, Liu et al. use GAN-based models on piano roll
representations, but the melody and chords are still predicted independently
by different generators [9]. Roy et al. devise a lead sheet generator with
user constraints defined by Markov models, and harmonic synchronization
between melody and chords through a probabilistic model that encodes which
melody notes fit on which chords [11]. There have also been many efforts in
the past that learn to generate chords for a given melody [8, 10, 3] or the
other way round [12]. In this paper we want to show that, on the one hand, a
two-stage generation process greatly improves the perceived quality of the
music. And, on the other hand, we show that melodic coherence improves when
the melody generator gets to look ahead at the entire harmonic template of the
song.

We formally define a lead sheet $x_{1:n}$ of length $n$, characterized by a
sequence of chords $c_{1:n}$, rhythms $r_{1:n}$ and melody pitches $m_{1:n}$.
At each time step $i$ in the piece, each of these quantities take some value:

$\displaystyle x_{1:n}$
$\displaystyle=\left\\{c_{1:n},r_{1:n},m_{1:n}\right\\},\quad
x_{i}=\left\\{c_{i},r_{i},m_{i}\right\\}.$ (1)

In this equation, the compact $x_{j:k}$ notation denotes the sequence
$\left(x_{j},x_{j+1}\dots x_{k}\right)$. Figure 1 shows an example of this
decomposition. Notice that the chords are repeated until there is a change in
harmony, thereby allowing us to model the entire lead sheet using a single
shared time scale. We also point out that there is only one melody note per
time step, which is not a severe restriction, since most lead sheets only
contain monophonic melodies. Finally, we choose to treat the barlines as
separate elements in the sequence, which is indicated by the vertical bars in
Figure 1.

$i$ | $c_{i}$ | $r_{i}$ | $m_{i}$
---|---|---|---
1 | C | quarter | G4
2 | C | eighth | E4
3 | C | eighth | G4
4 | C | half | C5
5 | $\mid$ | $\mid$ | $\mid$
6 | G | quarter | D5
7 | G | quarter | B4
8 | G | half | G4
9 | $\mid$ | $\mid$ | $\mid$

Figure 1: Example of a lead sheet decomposition into chords, rhythms and
melodies.

##  2 The Wikifonia Dataset

In this paper we will make use of the Wikifonia dataset, a former public lead
sheet repository hosted by wikifonia.org. It contains more than 6,500 lead
sheets in MusicXML format, and in all sorts of modern genres. This section
goes over the different preprocessing and encoding steps that are executed on
the dataset in order to obtain a clean collection of lead sheets.

###  2.1 Preprocessing

#### Eliminate polyphony

Whenever multiple notes sound at the same time, we only retain the note with
the highest pitch, as it is often the note that characterizes the melody.

#### Ignore ties

Connections between two notes with the same pitch that extend the first note’s
duration are ignored. The two notes are therefore treated as two separate
notes with their original duration.

#### Delete anacruses

Incomplete bars that often appear at the start of a piece, are removed from
all lead sheets.

#### Unfold repetitions

Lead sheets can contain repetition and other navigation markers. If a section
should be repeated, we duplicate that particular section, thereby unfolding
the piece into a single linear sequence.

#### Remove ornaments

Since such ornaments do not contribute much to the overall melody, we leave
them out.

###  2.2 Data encoding and features

After preprocessing, we encode the melody, rhythm and chord symbols into
feature vectors such that they can be used as input to our generators.

#### Encoding rhythms

We retain the 12 most common rhythm types in the dataset, which are given in
Appendix 0.B. We remove 184 lead sheets from the dataset that contain other
than these 12 types. Together with the representation for a barline, we encode
rhythm into a 13-dimensional one-hot vector $\vec{r}_{i}$.

#### Encoding chords

A chord is described by both its root and its mode. There are 12 possible
roots (C, C$\sharp$, D, D$\sharp$, …, B) and we choose to convert all
accidentals to either no alteration or one sharp.We count 47 different modes
in the dataset, which we map to one of the following four: major, minor,
diminished or augmented. This mapping only very slightly reduces musical
expressivity and interestingness. The mapping table can be found in Appendix
0.A. The 12 roots and 4 modes give 48 chord options in total, resulting in a
49-dimensional one-hot vector $\vec{c}_{i}$ if we include the barline.

#### Encoding melody

The MIDI standard defines 128 possible pitches. We assign two additional
dimensions for rests and barlines, resulting in a 130-dimensional one-hot
encoded melody vector $\vec{m}_{i}$.

##  3 Recurrent Neural Network Design

As mentioned in Section 1, the lead sheet generation process happens in two
stages: in stage one the rhythm and chord template of the song is learned, and
in stage two the melody notes are learned on top of that template. We will use
separate LSTM-based models for both stages [2]; the models are trained
independently of each other, but they are combined at inference time to
generate an entire lead sheet from scratch. Figure 2 shows the complete
architecture.

#### Stage one

In this stage, the rhythm and chord vectors are first concatenated and are
subsequently given as inputs to two LSTM layers followed by a dense layer. All
LSTM layers have a output dimensionality of 512 states, as indicated in the
figure. The output of the dense layer is cut in two vectors, both on which we
apply a softmax nonlinearity with temperature $\tau$, controlling the
concentration of the output distribution. This way we are effectively modeling
a distribution over the chord and rhythm symbols that come next in the
sequence.

#### Stage two

The second model will process the generated sequence of predicted chords and
rhythms. To this end, each chord and rhythm vector is again concatenated
before being processed by two BiLSTM layers. These BiLSTM states allow the
pitch generator to look back and also ahead at the harmonic sequence, as
inspired by [8]. The dimensionality of the BiLSTM layers is 512 in both
directions, adding up to a total of 1024 states. After concatenation with the
previous melody vector, the BiLSTM states are fed through another stack of two
LSTM layers. The output of the last dense layer is used to predict the next
melody note, again using a softmax nonlinearity controlled by a temperature
parameter.

$\vec{c}_{i}$ $\vec{r}_{i}$ LSTM 512 LSTM 512 Dense 61 $\hat{\vec{c}}_{i+1}$
$\hat{\vec{r}}_{i+1}$ $\hat{\vec{c}}_{i+1}$ $\hat{\vec{r}}_{i+1}$ BiLSTM 1024
BiLSTM 1024 $\vec{m}_{i}$ LSTM 512 LSTM 512 Dense 130 $\hat{\vec{m}}_{i+1}$

Figure 2: The RNN architecture for both stage one (left) and stage two
(right). The output dimensionality for every layer is written in each of the
blocks. Whenever two blocks appear next to each other, the (output) vectors
are concatenated.

###  3.1 Optimization details

In this paper we train both stages separately; it is possible to jointly train
both models through reparameterization tricks [4], but we leave this as future
work. While training is done separately, inference of lead sheets from scratch
is easily done by feeding the output of the first model to the input of the
second model. We use a standard cross-entropy loss function on all outputs. In
stage one we sum the losses on both the chord and rhythm outputs with
hyperparameter $\alpha$:

$\displaystyle\mathcal{L}_{\mathrm{stage\,1}}\\!\left(\hat{\vec{c}},\hat{\vec{r}}\right)=\alpha\cdot\mathcal{L}_{\mathrm{CE}}\\!\left(\hat{\vec{c}}\right)+(1-\alpha)\cdot\mathcal{L}_{\mathrm{CE}}\\!\left(\hat{\vec{r}}\right).$
(2)

In this equation, $\mathcal{L}_{\mathrm{CE}}\\!\left(\cdot\right)$ indicates
the cross-entropy loss. In stage two the loss function is equal to the cross-
entropy loss on the melody output.

$\displaystyle\mathcal{L}_{\mathrm{stage\,2}}\\!\left(\hat{\vec{m}}\right)=\mathcal{L}_{\mathrm{CE}}\\!\left(\hat{\vec{m}}\right).$
(3)

We use Adam with learning rate $\lambda$ to optimize both models [6]. During
optimization, the training data is augmented by shifting the pitches and
chords with a random number of semitones between -12 and 12, that is, between
plus or minus one octave. This virtually increases the amount of training data
by a factor of 25, thereby aiding model generalization towards less frequently
appearing keys.

##  4 Experiments

#### Hyperparameters

In all experiments we use a batch size of 128 sequences, each of length 100.
The learning rate $\lambda$ is set fixed to 0.001. We empirically found that a
value $\alpha$ of 0.5 leads to good results, so we set it fixed to that value.
We also set the temperature $\tau$ slightly lower than 1 during inference of
the melody, which helps to improve the perceived quality of the generated
music; we varied it between 0.75 and 1.0 during the experiments. For the
rhythm and chord patterns a temperature of 1.0 gives the most pleasing
results.

#### Baselines

We will compare our model against two baselines:

  1. 1.

An unconditioned LSTM-based model similar to the stage one model in Figure 2,
but now the melody is also concatenated to the input and output. The melody is
no longer conditioned on the entire chord and rhythm sequence. We also add an
extra LSTM layer, adding up to a total of three.

  2. 2.

A two-stage model where the BiLSTM layers are replaced by regular LSTM layers,
so that the melody cannot look ahead at the harmonic sequence. We keep all
other parameters identical to the original model.

#### Subjective listening test

We conducted an online listening test in which we asked 40 participants to
score 12 short audio clips, each of approximately one minute long. The
following songs were included in the test111Listen to the audio clips at
https://users.ugent.be/~cdboom/music:

  * •

3 pieces generated by the two-stage model from scratch,

  * •

3 pieces generated by the two-stage model, but conditioned on the chord and
rhythm scheme of existing songs: I Have a Dream (Abba), Autumn Leaves (jazz
standard) and Colors of the Wind (Alan Menken),

  * •

2 pieces generated by the one-stage baseline model,

  * •

2 pieces generated by the two-stage baseline model,

  * •

2 (relatively unknown) human-composed songs: You Belong to my Heart (Bing
Crosby) and One Small Photograph (Kevin Shegog).

As stated in Section 1, a lead sheet only encodes the basic template of a
song, and it ideally needs to be played by a real musician. We therefore gave
all lead sheets to a semi-professional pianist; the pianist stayed true to the
sheet music, but was free to create an accompaniment that suited the piece. In
our regards, this evaluation method reflects best how a lead sheet, produced
by an AI model, would in practice be used and experienced by musicians and
listeners.

The audio clips were presented to each user in randomized order. For each clip
we asked the user to rate on a scale of 1 to 5 how much he likes the piece, if
the melody is musically coherent, and whether the piece is composed by a
computer (1) or a human (5). We also asked the user to indicate if he
recognizes the piece. Since each user has his own rating bias and spread [7,
5], we converted the ratings for each user to a standardized $Z$ score between
$-0.5$ and $0.5$:

$\displaystyle
Z_{c,u}=\frac{R_{c,u}-\mu_{u}}{\max_{c^{\prime}}R_{c^{\prime},u}-\min_{c^{\prime}}R_{c^{\prime},u}}.$
(4)

In this formula, $R_{c,u}$ is the rating of user $u$ for clip $c$, $\mu_{u}$
is the average rating of user $u$, and $Z_{c,u}$ is the associated
standardized score. Table 1 reports the average $\bar{Z}$ score across the
audio clips for each of the three questions in the survey, along with the
standard deviation. A negative score means that the ratings are below average
overall, and a positive score indicates an overall above-average rating.

We observe that the scores are, by far, better for the two-stage models
compared to the unconditioned one-stage model. This shows that first sampling
a harmonic and rhythmic sequence, and conditioning the melody on top of this
sequence, is more beneficial than sampling all quantities simultaneously. Next
to this, we also notice that adding the BiLSTM layers improves the score for
all three questions. And although by a small margin, we can conclude that the
musical quality improves when the melody generator can look ahead in the
harmonic sequence. When we condition the melody generator on an existing chord
and rhythm scheme, it is remarkable that the human-composed and AI-composed
songs perform almost on par. The AI-composed songs are even considered most
pleasing. Related to this observation, 4 participants indicated having
recognized a piece from the two-stage model, 5 recognized a piece that was
generated based on existing chords, and 3 participants recognized a human-
composed song.

Finally, we also want to point out that the standard deviations are very
substantial, which shows that there is a high level of disagreement between
the reviewers. It is however interesting to point out that the standard
deviation is slightly higher for better performing models. This might indicate
that there is more consensus on what it means for music to sound ‘badly’, but
that the definition of ‘good’ music is more subjective and person-dependent.

Model | Pleasing $\bar{Z}$ | Coherence $\bar{Z}$ | Turing $\bar{Z}$
---|---|---|---
One-stage | $-0.23\pm 0.34$ | $-0.24\pm 0.31$ | $-0.22\pm 0.29$
Two-stage, without BiLSTM | $-0.04\pm 0.35$ | $-0.07\pm 0.33$ | $-0.09\pm 0.33$
Two-stage, with BiLSTM | $-0.01\pm 0.36$ | $0.01\pm 0.34$ | $-0.02\pm 0.34$
Two-stage, with existing chords | $\textbf{0.15}\pm 0.37$ | $0.09\pm 0.36$ | $\textbf{0.13}\pm 0.36$
Human-composed songs | $0.03\pm 0.34$ | $\textbf{0.13}\pm 0.36$ | $\textbf{0.13}\pm 0.34$
Table 1: Results of the subjective listening experiments. We report averaged
$\bar{Z}$-scores for each of the questions, along with the standard
deviations.

##  5 Conclusion

We have proposed a two-stage LSTM-based model to generate lead sheets from
scratch. In the first stage, a sequence of chords and rhythm patterns is
generated, and in the second stage the sequence of melody notes is generated
conditioned on the output of the first stage. We conducted a subjective
listening test of which the results showed that our approach outperformed the
baselines. We can therefore conclude that conditioning helps the quality of
the generated music, and that this approach can be explored further in the
future.

## References

  * [1] Briot, J.P., et al.: Deep Learning Techniques for Music Generation - A Survey (2017)
  * [2] Hochreiter, S., Schmidhuber, J.: Long short-term memory. Neural Computation (1997)
  * [3] Huang, C.Z.A., et al.: Music Transformer: Generating Music with Long-Term Structure. In: ICLR (2019)
  * [4] Jang, E., et al.: Categorical Reparameterization with Gumbel-Softmax. In: ICLR (2017)
  * [5] Jin, R., Si, L.: A study of methods for normalizing user ratings in collaborative filtering. In: SIGIR (2004)
  * [6] Kingma, D., Ba, J.: Adam: A Method for Stochastic Optimization. In: ICLR (2015)
  * [7] Koren, Y., et al.: Matrix Factorization Techniques for Recommender Systems. Computer (2009)
  * [8] Lim, H., et al.: Chord Generation from Symbolic Melody Using BLSTM Networks. In: ISMIR (2017)
  * [9] Liu, H.M., Yang, Y.H.: Lead Sheet Generation and Arrangement by Conditional Generative Adversarial Network. To appear (2019)
  * [10] Pachet, F., Roy, P.: Non-Conformant Harmonization - the Real Book in the Style of Take 6. (2014)
  * [11] Roy, P., et al.: Sampling Variations of Lead Sheets. arXiv.org (2017)
  * [12] Yang, L.C., et al.: MidiNet - A Convolutional Generative Adversarial Network for Symbolic-Domain Music Generation. In: ISMIR (2017)

##  Appendix 0.A Mode Mapping for Chords

In Table 2 we show how different chord modes are mapped to one of the
following four options: major, minor, diminished or augmented.

Original mode | Mapped mode | Original mode | Mapped mode
---|---|---|---
6 | major | major-6-9 | major
7 | major | major-7 | major
9 | major | major-9 | major
augmented | augmented | major-minor | major
augmented-7 | augmented | minor | minor
augmented-9 | augmented | minor-11 | minor
diminished | diminished | minor-13 | minor
diminished-7 | diminished | minor-6 | minor
dominant | major | minor-7 | minor
dominant-11 | major | minor-7-b5 | diminished
dominant-13 | major | minor-9 | minor
dominant-7 | major | minor-major | minor
dominant-9 | major | minor-major-7 | minor
half-diminished | diminished | power | major
major | major | sus2 | major
major-13 | major | sus4 | major
major-6 | major | sus4-7 | major
Table 2: Chord modes are mapped to one of four options.

##  Appendix 0.B Rhythm types

Table 3 provides an overview of the twelve rhythmic figures that are used.

Textual description | Musical symbol
---|---
32nd note |
32nd dotted note |
16th note |
8th triplet note | $\overset{\;\;\;\;3}{\AchtBR}$
8th note |
quarter triplet note | $\overset{\quad\;\;3}{\Vier}$
8th dotted note |
quarter note |
quarter dotted note |
half note |
half dotted note |
whole note |
Table 3: The rhythm types that are considered in this paper.


[orcid=0000-0003-0145-8358] [1]

# A DFT Study of the Electronic, Optical, and Mechanical Properties of a
Recently Synthesized Monolayer Fullerene Network

Raphael M. Tromer  Luiz A. Ribeiro Junior  Douglas S. Galvão
<EMAIL_ADDRESS>Applied Physics Department, ’Gleb Wataghin’ Institute of
Physics, State University of Campinas, Campinas,SP, 13083-970, Brazil  Center
for Computing in Engineering & Sciences, State University of Campinas,
Campinas, SP, 13083-970, Brazil  Institute of Physics, University of Brasília,
Brasília, 70910-900, Brazil

###### Abstract

Closely packed quasi-hexagonal and quasi-tetragonal crystalline phase of C60
molecules (named qHPC60) was recently synthesized. Here, we used DFT
simulations to investigate the electronic, optical, and mechanical properties
of qHPC60 monolayers. qHPC60 has a moderate direct electronic bandgap, with
anisotropic mechanical properties. Their elastic modulus ranges between 50 and
62 GPa. The results for optical properties suggest that qHPC60 can act as UV
collectors for photon energies until 5.5 eV since they present low
reflectivity and refractive index greater than one. The estimated optical
bandgap (1.5-1.6 eV) is in very good agreement with the experimental one (1.6
eV).

###### keywords:

Density Functional Theory Electronic Properties Optical Properties Mechanical
Properties 2D C60 Crystals

##  1 Introduction

The advent of graphene in 2004 [1] has stimulated the search for new carbon-
based 2D materials that can share some of its remarkable properties [2], such
as high mechanical strength, flexibility, thermal, and electrical conductivity
[3]. These unique properties made graphene one of the most promising materials
for developing a wide range of applications [4, 5]. However, in spite of these
remarkable properties, graphene is zero semi-metal bandgap, which limits its
use in some electronics applications [6]. For this reason, much effort has
been dedicated in the search of new 2D carbon allotropes [7, 8, 9, 10, 11, 12]
that can present a non-zero bandgap while preserving some of the other
graphene properties.

Novel 2D carbon-based materials with atomic thickness have emerged in the last
two years through experimental investigations: the monolayer amorphous carbon
(MAC) [13, 14, 15] and the 2D biphenylene network (BPN) [16]. Laser-assisted
chemical vapor deposition was the synthesis route used to fabricate a free-
standing, continuous, and stable MAC [13, 14, 15]. This material possesses a
pure carbon structure composed of randomly distributed five, six, seven, and
eight atom rings. Its lattice arrangement differs from disordered graphene.
BPN consists of the periodic lattice composed of a set of fused rings
containing four, six, and eight carbon atoms. In the synthetic route used to
produce BPN, an adsorbed halogenated terphenyl molecule undergoes a two-step
interpolymer dehydrofluorination polymerization that creates the four- and
eight-membered rings through carbon-carbon bond formation, employing a gold
surface as substrate [16]. Similar to graphene, MAC and BPN present a zero
semi-metal bandgap.

Very recently, a single-crystal 2D carbon material with a semiconducting
bandgap of about 1.6 eV, named monolayer quasi-hexagonal-phase fullerene
(C60), was experimentally realized overcoming the problem of a null bandgap
shown by other 2D carbon-based materials [17]. Through an organic cation
slicing strategy, the exfoliation of quasi-hexagonal bulk single crystals
yields the monolayer polymeric C60. The C60 2D crystals showed a large size
via an interlayer bonding cleavage strategy. In these structures, C60 polymers
form covalently bonded cluster cages of C60 in a plane. This clustering
mechanism yielded two stable crystals of polymeric C60 in closely packed
quasi-hexagonal (qHPC60) and quasi-tetragonal (qTPC60) phases [17]. qHPC60 and
qTPC60 exhibit high crystallinity and good thermodynamic stability. Their
moderate bandgap and unique topological structure are desirable for developing
new applications in nanoelectronics. Despite the insightful discussion on the
physical properties of these 2D C60 crystals presented in the original work
[17], a detailed description of their electronic, optical, and mechanical
properties is still lacking.

In this study, we numerically investigate the electronic, optical, and
mechanical properties of qHPC60 using density functional theory (DFT) methods.
We restricted our study to the qHPC60 phase since no monolayer phase was
experimentally observed for the qTPC60 one. The calculated electronic and
estimated optical bandgaps agree with the experimental findings. qHPC60 has a
direct bandgap with anisotropic mechanical properties. Consistent with the
experimental findings the DFT results show that qHPC60 is a 2D monolayer
formed by covalent bonds of C60 molecules along both directions of the
crystal.

##  2 Methodology

The lattice structure of qHPC60 is available from reference [17]. Its square
unit cell consists of 120 carbon atoms, (see Figure 1). We fully optimized
these structures and calculated their electronic, optical, and mechanical
properties through DFT calculations, as implemented in the SIESTA code [18,
19, 20]. The van der Waals corrections were adopted to describe the exchange-
correlation term [21, 22]. The calculations were carried out within the
framework of the generalized gradient approximation (GGA) with the Perdew-
Burke-Ernzerhof (PBE) functional [23, 24] for the exchange-correlation term.
We chose the norm-conserving Troullier-Martins pseudopotential to describe the
core electrons [25]. The wave functions are based on localized atomic orbitals
and a double-zeta plus polarization (DZP) basis set.

Figure 1: Schematic representation of the optimized atomic configuration for
the qHPC${}_{6}0$ monolayers.

We used 250 Ry for kinetic energy cut-off, sampling the reciprocal space with
20 $\times$ 20 $\times$ 2 k-point grid. The vacuum region along the
z-direction is set to 30 Å to prevent spurious interactions among layers. The
basis vector along the z-direction was fixed during the optimization process.
The other vectors and atoms were fully relaxed. The convergence criterion for
maximum forces on each atom was 0.001 eV/Å. Once the 2D C60 crystals were
fully optimized, we obtained the formation energy using the following
expression: $E_{formation}=(E_{2D}-N_{C}E_{C})/N_{C}$, where E2D is the total
energy of isolated 2D qHPC60, EC is the energy of each isolated carbon atom,
and NC is the total number of the carbon atoms.

For the electronic band structure calculation, we considered the following
path along the square lattice: $\Gamma=(0.0,0.0)$ to $X=(0.5,0.0)$ to
$U=(0.5,0.5)$ to $Y=(0.0,0.5)$ to $\Gamma=(0.0,0.0)$. To obtain the bulk
modulus from stress-strain relationship, we assumed the linear regime
corresponding to 1% percent of the total applied strain.

To obtain the optical coefficients for each direction, we considered an
external electric field of value 1.0 V/Å [26] applied along three possible
polarization planes. The optical coefficients were extracted from complex
dielectric function $\epsilon=\epsilon_{1}+i\epsilon_{2}$, in which the
imaginary part, $\epsilon_{2}$, is derived from the direct interband
transitions through Fermi’s golden rule [27],

$\epsilon_{2}(\omega)=\frac{4\pi^{2}}{\Omega\omega^{2}}\displaystyle\sum_{i\in\mathrm{VB},\,j\in\mathrm{CB}}\displaystyle\sum_{k}W_{k}\,|\rho_{ij}|^{2}\,\delta(\varepsilon_{kj}-\varepsilon_{ki}-\hbar\omega).$
(1)

In Equation 1, $\omega$ is the photon frequency, $VB$ and $CB$ are the valence
and conduction bands, respectively, $\rho_{ij}$ is the dipole transition
matrix element, $W_{k}$ is the weight of the respective k-point in the
reciprocal space, and $\Omega$ is the unit cell volume.

The real part of the dielectric constant is obtained from the Kramers-Kronig
relation [28]:

$\epsilon_{1}(\omega)=1+\frac{1}{\pi}P\displaystyle\int_{0}^{\infty}d\omega^{\prime}\frac{\omega^{\prime}\epsilon_{2}(\omega^{\prime})}{\omega^{\prime
2}-\omega^{2}},$ (2)

where $P$ denotes the principal value of the logarithm function. Once the
imaginary and real parts of the dielectric functions are obtained, one can
derive, from simple relations, the other relevant properties, such as the
absorption coefficient $\alpha$, the refractive index $\eta$, and the
reflectivity $R$:

$\alpha(\omega)=\sqrt{2}\omega\bigg{[}(\epsilon_{1}^{2}(\omega)+\epsilon_{2}^{2}(\omega))^{1/2}-\epsilon_{1}(\omega)\bigg{]}^{1/2},$
(3)
$\eta(\omega)=\frac{1}{\sqrt{2}}\bigg{[}(\epsilon_{1}^{2}(\omega)+\epsilon_{2}^{2}(\omega))^{1/2}+\epsilon_{1}(\omega)\bigg{]}^{2},$
(4)

and,

$R(\omega)=\bigg{[}\frac{(\epsilon_{1}(\omega)+i\epsilon_{2}(\omega))^{1/2}-1}{(\epsilon_{1}(\omega)+i\epsilon_{2}(\omega))^{1/2}+1}\bigg{]}^{2}.$
(5)

It is well-known that in DFT methodologies, there is intrinsic imprecision in
determining the bandgap values when contrasted to experimental results [29,
30]. This imprecision propagates in the calculation of optical coefficients.
In the SIESTA code, this problem can be avoided from the optical properties by
applying an ad-hoc shift of the unoccupied bands, known as the scissor
operator. Here, we used a scissor equal to 0.6 eV.

##  3 Results

We begin our discussion by analyzing the optimized structure shown in Figure
1. After optimization, the for qHPC60 lattice parameters are: a $=15.8794$ Å,
b $=9.1908$ Å, c $=30.0000$ Å, and $\alpha=\beta=\gamma=90^{\circ}$. The
formation energy (Eformation) is -9.14 eV/atom, which is similar to those for
other 2D carbon allotropes, like phagraphene [31] and graphene [32]. It is
worthwhile to stress that qHPC60 is composed of sp3 and sp2 hybridized carbon
atoms (see Supplementary Material).

Figure 2: qHPC60 electronic band structure and related projected density of
states (PDOS), considering the valence atomic orbitals 2$s$ and 2$p$.

In Figure 2, we present the qHPC60 electronic band structure and related
projected density of states (PDOS) considering the valence atomic orbitals
2$s$ and 2$p$. From the band structure, we can see that qHPC60 is a small
bandgap semiconductor. It has a direct bandgap at the symmetry point $\Gamma$
with a value of 0.9 eV, estimated from the energy difference between the
lowest unoccupied molecular orbital (LUCO) and the highest occupied molecular
orbital (HOCO). The PDOS show that HOCO and LUCO are predominantly composed of
2$p$ carbon atoms. We will discuss later the spatial distribution of these
frontier orbitals.

Figure 3: (a) Stress-strain curves for qHPC60 monolayer for the x and
y-directions. (b) Energy bandgap values as a function of the applied strain.
Red, black, and blue refer to stress-strain interplay along the x, y, and x
and y directions, respectively.

As mentioned above, it is well-known that standard DFT methods usually
underestimate the bandgap values. For small bandgap semiconductors, the
difference between computed and experimental bandgaps can reach 0.5 eV [30,
29]. It was reported in reference [17] that 2D C${}_{6}0$ crystal has a
moderate bandgap of approximately 1.6 eV. Based on these facts, we can
consider that the bandgap calculated here for qHPC60 (0.9 eV) is in relative
good agreement with the experimental value.

To investigate the qHPC60 mechanical properties, we applied a tensile strain
(uniaxially and biaxially) and measured the tensile stress for the range of 0
to 8% of strain. In Figures 3(a) and 3(b) we present the stress-strain curves
under uniaxial strain loading along the $x$ and $y$ directions, respectively.
To calculate Young’s modulus values, ($\varepsilon_{XX}$ and
$\varepsilon_{YY}$ along the x- and y-directions), we considered the linear
regime until 1% of strain. We can see from Figures 3(a) and 3(b) that there
are two well-defined regimes for the stress-strain relationship. In the first
regime, between 0 and 6% of strain, the stress rises linearly. Above this
critical threshold (second regime defined by 6 to 8 % of strain) occurs a
change in the slope of the line, and the stress increases slower when
contrasted with the first regime. This trend for the stress-strain interplay
suggests a structural phase transition.

The estimated elastic modulus were $\varepsilon_{XX}=49.6$ GPa and
$\varepsilon_{XX}=61.1$ GPa. These results revealed that 2D C60 crystals have
clear in-plane anisotropic properties, as reported in the experiment [17] and
expected from the structure topology. The square ring formed among the next-
neighboring C${}_{6}0$ units (see Figure 1) results in a higher resilience to
structural deformations along the y-direction.

Figure 4: Schematic representation of the charge spatial distribution for the
(left) HOCO and (right) LUCO energy bands. In the color scheme, black and
yellow represent localized charges and carbon atoms/carbon-carbon bonds,
respectively.

We further explored the effects of strain engineering on the bandgaps values
by applying uniaxial and biaxial strain in the range of 0 to 8%. In Figure
3(b) we present the results of the calculated bandgaps as a function of the
applied strain. We can see from the Figure that the bandgap values decrease
linearly with increasing strain. As a general trend, the bond stretching
process imposed by the applied tension increases the delocalization of the
HOCO and LUCO orbitals decreasing the bandgap value. The exception is the
bandgap increasing from 6% to 8% of strain in the biaxial case, in which the
bandgap values increased from about 0.4 to 0.6 eV.

As mentioned above, we calculated the HOCO and LUCO frontier crystalline
orbitals. A schematic representation of their localization is illustrated in
Figure 4. We can see that HOCO and LUCO are well-spread over the carbon-carbon
bonds, forming an electron transfer channel that can lead to high electron
mobility. As mentioned above, the electronic states of HOCO and LUCO originate
mainly from 2$p$ electrons. This HOCO and LUCO localization are consistent
with the understanding that the higher the charge delocalization, the smaller
the bandgap value.

Figure 5: qHPC60 absorption coefficient ($\alpha$), refractive ($\eta$), and
reflectivity ($R$) indexes as a function of the photon energy value. Black,
red, and blue lines refer to X, Y, and Z polarizations.

Finally, we discuss the optical properties of the 2D C60 crystal studied here.
As mentioned in the Methodology section, the optical calculations considered
the scissor correction to obtain a more accurate description of the optical
transitions. Figures 6(a-c) show the absorption coefficient ($\alpha$),
refractive ($\eta$), and reflectivity ($R$) indexes as a function of the
photon energy value. From Figure 6(a), we can see that photon absorption
begins approximately at 1.5-1.6 eV, which is in very good agreement with the
experimental result, in which the optical bandgap value was also estimated to
be at 1.55 eV [17].

As a general feature, we observe a clear anisotropy among the three directions
of the light propagation. For the $\alpha$ profile shown in Figure 6(a), we
can see that the first optical transition peak occurs for nearly the same
photon energy in all directions, about 2.2 eV (i.e., within the visible
spectrum). This optical transition refers to the HOCO-LUCO transition. The
other optical transitions occur within the violet and ultra-violet (UV)
regions. The maximum absorption intensities are approximately $3.6\times
10^{5}$ cm-1. In the context of carbon-based materials, intense absorption
within the UV region denotes the presence of sp3-like carbons (see
Supplementary Material).

Figures 6(b) and 6(e) show the refractive index as a function of the photon
energy for X, Y, and Z polarizations. The maximum peak occurs at 1.8 eV
(E$||$X). Light attenuation occurs for photon energies higher than 3.2 eV.
$\eta$ values converge to 1.0, indicating that the incident UV light tends to
be refracted similarly in all directions. The refractive index exhibits the
same trend for photon energies between 3.2 and 6.0 eV. In Figures 6(c) we can
see that the reflectivity has the maximum peak at 3.3 eV (E$||$X, violet
region) and 3.5 eV (E$||$Y, UV region), respectively. In both cases, almost
30% of the incident light is reflected, and the other reflectivity peaks occur
in the UV region. These results suggest that qHPC60 can act as UV collectors
for photon energies until 5.5 eV since they present low reflectivity and
refractive index greater than one.

##  4 Conclusions

In summary, we carried out DFT calculations to investigate the electronic,
optical, and mechanical properties of the recently synthesized 2D C60
crystals. In these materials, C60 polymers form covalently bonded cluster
cages of C60 in a plane. This clustering mechanism yielded two stable
structures named qHPC60 and qTPC60, but only qHPC60 formed monolayers. From
the electronic band structure, one can conclude that qHPC60 is a small bandgap
semiconductor with a direct bandgap with a value of 0.9 eV, consistent with
the experimental data [17]. The estimated optical absorption bandgap is .5-1.6
eV, which is in very good agreement with the experimental result (1.55 ev).

Regarding the mechanical properties analysis, the calculated elastic modulus
ranged from 50 to 62 GPa. The results also revealed that 2D C60 crystals have
clear in-plane anisotropic properties, as reported in the experiment [17] and
expected from its topology with a higher mechanical resilience along the
y-direction. The patterns of the calculated HOCO and LUCO frontier crystalline
orbitals suggest a structure with high electron mobility.

The results obtained for the optical properties suggest that qHPC60 can act as
UV collectors for photon energies until 5.5 eV since it presents low
reflectivity and refractive index greater than one. Moreover, our analysis
points to the fact that these materials can serve as active layers in
optoelectronic applications with photon energies varying from 1.5 to 5.5 eV,
including the infrared and ultraviolet regions.

## Acknowledgements

This work was financed in part by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, CNPq, and
FAPESP. We thank the Center for Computing in Engineering and Sciences at
Unicamp for financial support through the FAPESP/CEPID Grants #2013/08293-7
and #2018/11352-7. We also thank Conselho Nacional de Desenvolvimento
Cientifico e Tecnológico (CNPq) for their financial support. L.A.R.J
acknowledges the financial support from a Brazilian Research Council FAP-DF
grants $00193-00000857/2021-14$, $00193-00000853/2021-28$, and
$00193-00000811/2021-97$ and CNPq grant $302236/2018-0$, respectively.
L.A.R.J. gratefully acknowledges the support from ABIN grant 08/2019. L.A.R.J.
acknowledges Núcleo de Computação de Alto Desempenho (NACAD) for providing the
computational facilities through the Lobo Carneiro supercomputer. L.A.R.J.
thanks Fundação de Apoio à Pesquisa (FUNAPE), Edital 02/2022 - Formulário de
Inscrição N.4, for the financial support. L.A.R.J acknowledges CENAPAD-SP for
providing the computational facilities.

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## Supplementary Material

Figure 6: Bond distances within the unit cell for qHPC60 and $qTPC_{60}$. We
consider sp2 hybridized carbon atoms the ones involved in carbon-carbon bonds
with size until 1.48 Å. sp3 hybridized carbon atoms form bonds with size
higher than this critical value. This bond configuration is in accordance with
XPS measures reported in reference [17].