8 | 4 | D6 | 0.968243 | 2.49097 | - | - | - | - | $10^{-4}$
- | - |  D62 | 0.991944 | 2.62267 | - | - | - | - | $10^{-5}$
- | - | D8 | 0.999337 | 2.62288 | 0.805992 | 181.413 | - | - | $10^{-7}$
- | - | D6D8 | 0.99998 | 2.62072 | 0.97182 | 210.724 | - | - | $10^{-7}$
- | - | D10 | 1.0001 | 2.62066 | 1.00395 | 210.728 | 0.764383 | 5681.68 | $10^{-7}$
8 | 5 | D6 | 0.955466 | 1.59391 | - | - | - | - | $10^{-3}$
- | - |  D62 | 0.992516 | 1.72194 | - | - | - | - | $10^{-5}$
- | - | D8 | 0.999046 | 1.72204 | 0.721846 | 116.046 | - | - | $10^{-6}$
- | - | D6D8 | 1.00007 | 1.71997 | 0.981244 | 147.899 | - | - | $10^{-6}$
- | - | D10 | 1.00023 | 1.71989 | 1.0256 | 147.906 | 0.970839 | 3634.1 | $10^{-7}$
8 | 6 | D6 | 0.939784 | 1.10663 | - | - | - | - | $10^{-2}$
- | - |  D62 | 0.993134 | 1.23278 | - | - | - | - | $10^{-4}$
- | - | D8 | 0.998564 | 1.23275 | 0.615123 | 80.5432 | - | - | $10^{-6}$
- | - | D6D8 | 1.00002 | 1.23084 | 0.982674 | 116.017 | - | - | $10^{-6}$
- | - | D10 | 1.00008 | 1.23082 | 0.99788 | 116.019 | 0.300343 | 2522.09 | $10^{-7}$
8 | 7 | D6 | 0.921244 | 0.812837 | - | - | - | - | $10^{-2}$
- | - |  D62 | 0.993822 | 0.937502 | - | - | - | - | $10^{-4}$
- | - | D8 | 0.997995 | 0.937343 | 0.493329 | 59.1427 | - | - | $10^{-5}$
- | - | D6D8 | 0.999975 | 0.93568 | 0.985574 | 99.6911 | - | - | $10^{-5}$
- | - | D10 | 0.999973 | 0.935681 | 0.984953 | 99.691 | -0.0109677 | 1851.87 | $10^{-8}$
8 | 8 | D6 | 0.899895 | 0.622173 | - | - | - | - | $10^{-2}$
- | - |  D62 | 0.994585 | 0.744852 | - | - | - | - | $10^{-4}$
- | - | D8 | 0.997419 | 0.744595 | 0.360089 | 45.2594 | - | - | $10^{-5}$
- | - | D6D8 | 1.00001 | 0.743297 | 0.993138 | 92.4681 | - | - | $10^{-5}$
- | - | D10 | 0.999963 | 0.743309 | 0.980253 | 92.4648 | -0.214335 | 1417.14 | $10^{-7}$
8 | 9 | D6 | 0.875709 | 0.491475 | - | - | - | - | $10^{-1}$
- | - |  D62 | 0.995279 | 0.610833 | - | - | - | - | $10^{-4}$
- | - | D8 | 0.996787 | 0.610569 | 0.216587 | 35.7478 | - | - | $10^{-4}$
- | - | D6D8 | 1.00002 | 0.609804 | 0.993614 | 89.0923 | - | - | $10^{-4}$
- | - | D10 | 0.999969 | 0.609801 | 0.974937 | 89.0889 | -0.42456 | 1119.39 | $10^{-8}$
8 | 10 | D6 | 0.84869 | 0.398008 | - | - | - | - | $10^{-1}$
- | - |  D62 | 0.995711 | 0.511905 | - | - | - | - | $10^{-4}$
- | - | D8 | 0.996071 | 0.511789 | 0.0641191 | 28.9509 | - | - | $10^{-4}$
- | - | D6D8 | 0.999906 | 0.51173 | 0.961513 | 78.9495 | - | - | $10^{-4}$
- | - | D10 | 0.999976 | 0.511688 | 0.970967 | 78.9489 | -0.58538 | 906.709 | $10^{-7}$
9 | 1 | D6 | 0.992295 | 50.4604 | - | - | - | - | $10^{-7}$
- | - |  D62 | 0.993465 | 50.8251 | - | - | - | - | $10^{-7}$
- | - | D8 | 1.0008 | 50.8254 | 0.999832 | 4653.98 | - | - | $10^{-7}$
- | - | D6D8 | 1.00083 | 50.8236 | 1.00863 | 4686.59 | - | - | $10^{-7}$
- | - | D10 | 1.00137 | 50.8233 | 1.18949 | 4686.61 | 6.22714 | 184513. | $10^{-7}$
9 | 2 | D6 | 0.988606 | 12.6144 | - | - | - | - | $10^{-6}$
- | - |  D62 | 0.993286 | 12.7943 | - | - | - | - | $10^{-6}$
- | - | D8 | 1.00037 | 12.7946 | 0.965824 | 1163.3 | - | - | $10^{-7}$
- | - | D6D8 | 1.00048 | 12.7927 | 1.00082 | 1196.54 | - | - | $10^{-7}$
- | - | D10 | 1.00113 | 12.7924 | 1.21951 | 1196.56 | 7.37889 | 46119. | $10^{-7}$
9 | 3 | D6 | 0.982512 | 5.60584 | - | - | - | - | $10^{-5}$
- | - |  D62 | 0.993028 | 5.75187 | - | - | - | - | $10^{-6}$
- | - | D8 | 0.999467 | 5.75213 | 0.880738 | 516.887 | - | - | $10^{-7}$
- | - | D6D8 | 0.999684 | 5.75039 | 0.953195 | 551.208 | - | - | $10^{-7}$
- | - | D10 | 0.999602 | 5.75043 | 0.925523 | 551.205 | -0.902349 | 20490.7 | $10^{-7}$
9 | 4 | D6 | 0.974543 | 3.15288 | - | - | - | - | $10^{-4}$
- | - |  D62 | 0.993243 | 3.28751 | - | - | - | - | $10^{-5}$
- | - | D8 | 0.999272 | 3.28771 | 0.828786 | 290.646 | - | - | $10^{-7}$
- | - | D6D8 | 0.999663 | 3.286 | 0.957701 | 326.642 | - | - | $10^{-7}$
- | - | D10 | 0.999643 | 3.28601 | 0.951186 | 326.641 | -0.202258 | 11521.1 | $10^{-7}$
9 | 5 | D6 | 0.964463 | 2.01751 | - | - | - | - | $10^{-3}$
- | - |  D62 | 0.993704 | 2.14727 | - | - | - | - | $10^{-5}$
- | - | D8 | 0.999251 | 2.1474 | 0.769484 | 185.935 | - | - | $10^{-7}$
- | - | D6D8 | 0.999876 | 2.14572 | 0.973086 | 224.313 | - | - | $10^{-7}$
- | - | D10 | 0.999922 | 2.1457 | 0.988786 | 224.315 | 0.456807 | 7369.8 | $10^{-7}$
Table 9: As described in Tab. 6. $M_{\Phi}$ | $Y_{\Phi}$ | dim | $c_{6}$ | $\delta c_{6}$ | $c_{8}$ | $\delta c_{8}$ | $c_{10}$ | $\delta c_{10}$ | $\chi^{2}_{\rm min}$
---|---|---|---|---|---|---|---|---|---
9 | 6 | D6 | 0.952123 | 1.40079 | - | - | - | - | $10^{-3}$
- | - |  D62 | 0.99426 | 1.52813 | - | - | - | - | $10^{-5}$
- | - | D8 | 0.999187 | 1.52818 | 0.694104 | 129.061 | - | - | $10^{-6}$
- | - | D6D8 | 1.00011 | 1.52657 | 0.989926 | 170.732 | - | - | $10^{-6}$
- | - | D10 | 1.00019 | 1.52653 | 1.01798 | 170.736 | 0.752688 | 5115.09 | $10^{-8}$
9 | 7 | D6 | 0.93734 | 1.02894 | - | - | - | - | $10^{-2}$
- | - |  D62 | 0.994685 | 1.15479 | - | - | - | - | $10^{-5}$
- | - | D8 | 0.998782 | 1.15474 | 0.592083 | 94.7737 | - | - | $10^{-6}$
- | - | D6D8 | 1.00003 | 1.15327 | 0.988138 | 140.922 | - | - | $10^{-6}$
- | - | D10 | 1.00007 | 1.15325 | 1.00163 | 140.924 | 0.329028 | 3755.93 | $10^{-8}$
9 | 8 | D6 | 0.920304 | 0.78761 | - | - | - | - | $10^{-2}$
- | - |  D62 | 0.995174 | 0.912112 | - | - | - | - | $10^{-4}$
- | - | D8 | 0.998365 | 0.911974 | 0.48056 | 72.5267 | - | - | $10^{-5}$
- | - | D6D8 | 0.999986 | 0.910686 | 0.989533 | 124.691 | - | - | $10^{-5}$
- | - | D10 | 0.999983 | 0.910687 | 0.988484 | 124.691 | -0.0231992 | 2874.14 | $10^{-8}$
9 | 9 | D6 | 0.901044 | 0.622172 | - | - | - | - | $10^{-2}$
- | - |  D62 | 0.995732 | 0.74486 | - | - | - | - | $10^{-4}$
- | - | D8 | 0.997972 | 0.744657 | 0.360517 | 57.2809 | - | - | $10^{-5}$
- | - | D6D8 | 1.00003 | 0.743632 | 0.996549 | 117.031 | - | - | $10^{-5}$
- | - | D10 | 1. | 0.743639 | 0.987559 | 117.029 | -0.189147 | 2269.97 | $10^{-7}$
9 | 10 | D6 | 0.879495 | 0.503851 | - | - | - | - | $10^{-1}$
- | - |  D62 | 0.996208 | 0.623678 | - | - | - | - | $10^{-4}$
- | - | D8 | 0.997503 | 0.623466 | 0.231618 | 46.3826 | - | - | $10^{-5}$
- | - | D6D8 | 1.00002 | 0.622811 | 0.996653 | 113.328 | - | - | $10^{-5}$
- | - | D10 | 0.999982 | 0.622811 | 0.98118 | 113.325 | -0.407428 | 1838.18 | $10^{-8}$
10 | 1 | D6 | 0.993549 | 62.297 | - | - | - | - | $10^{-7}$
- | - |  D62 | 0.994497 | 62.7196 | - | - | - | - | $10^{-8}$
- | - | D8 | 1.0008 | 62.7199 | 1.06003 | 7093.48 | - | - | $10^{-8}$
- | - | D6D8 | 1.00082 | 62.7183 | 1.06755 | 7133.69 | - | - | $10^{-8}$
- | - | D10 | 1.0012 | 62.7181 | 1.22298 | 7133.7 | 6.61533 | 347198. | $10^{-8}$
10 | 2 | D6 | 0.990706 | 15.5735 | - | - | - | - | $10^{-6}$
- | - |  D62 | 0.994495 | 15.7679 | - | - | - | - | $10^{-7}$
- | - | D8 | 1.00035 | 15.7681 | 0.985429 | 1773.13 | - | - | $10^{-8}$
- | - | D6D8 | 1.00042 | 15.7666 | 1.01394 | 1813.96 | - | - | $10^{-8}$
- | - | D10 | 1.00079 | 15.7664 | 1.1659 | 1813.97 | 6.36299 | 86785.3 | $10^{-8}$
10 | 3 | D6 | 0.985607 | 6.92102 | - | - | - | - | $10^{-5}$
- | - |  D62 | 0.994117 | 7.07335 | - | - | - | - | $10^{-6}$
- | - | D8 | 0.999337 | 7.07357 | 0.880505 | 787.887 | - | - | $10^{-8}$
- | - | D6D8 | 0.999477 | 7.07218 | 0.938416 | 829.77 | - | - | $10^{-8}$
- | - | D10 | 0.999427 | 7.0722 | 0.917419 | 829.768 | -0.855342 | 38561. | $10^{-8}$
10 | 4 | D6 | 0.979654 | 3.89265 | - | - | - | - | $10^{-4}$
- | - |  D62 | 0.994815 | 4.03078 | - | - | - | - | $10^{-6}$
- | - | D8 | 1.0001 | 4.03097 | 0.893974 | 443.056 | - | - | $10^{-7}$
- | - | D6D8 | 1.00037 | 4.0295 | 1.00552 | 486.609 | - | - | $10^{-7}$
- | - | D10 | 1.00062 | 4.02938 | 1.10805 | 486.621 | 4.01508 | 21682.9 | $10^{-8}$
10 | 5 | D6 | 0.971129 | 2.49097 | - | - | - | - | $10^{-4}$
- | - |  D62 | 0.994808 | 2.62277 | - | - | - | - | $10^{-5}$
- | - | D8 | 0.999612 | 2.62291 | 0.818614 | 283.455 | - | - | $10^{-7}$
- | - | D6D8 | 1.00003 | 2.62151 | 0.98696 | 329.256 | - | - | $10^{-7}$
- | - | D10 | 1.0001 | 2.62147 | 1.01644 | 329.26 | 1.09605 | 13871.1 | $10^{-8}$
10 | 6 | D6 | 0.961023 | 1.72957 | - | - | - | - | $10^{-3}$
- | - |  D62 | 0.995134 | 1.85822 | - | - | - | - | $10^{-5}$
- | - | D8 | 0.999501 | 1.8583 | 0.751808 | 196.764 | - | - | $10^{-7}$
- | - | D6D8 | 1.00011 | 1.85694 | 0.994318 | 245.626 | - | - | $10^{-7}$
- | - | D10 | 1.0002 | 1.8569 | 1.03128 | 245.631 | 1.28776 | 9628.09 | $10^{-7}$
10 | 7 | D6 | 0.948945 | 1.27048 | - | - | - | - | $10^{-3}$
- | - |  D62 | 0.995368 | 1.39733 | - | - | - | - | $10^{-5}$
- | - | D8 | 0.999135 | 1.39735 | 0.659486 | 144.499 | - | - | $10^{-6}$
- | - | D6D8 | 0.999944 | 1.39608 | 0.980288 | 197.418 | - | - | $10^{-6}$
- | - | D10 | 0.999964 | 1.39607 | 0.98884 | 197.419 | 0.275652 | 7070.13 | $10^{-8}$
10 | 8 | D6 | 0.935095 | 0.972527 | - | - | - | - | $10^{-2}$
- | - |  D62 | 0.995734 | 1.09813 | - | - | - | - | $10^{-5}$
- | - | D8 | 0.99891 | 1.09808 | 0.570479 | 110.583 | - | - | $10^{-6}$
- | - | D6D8 | 0.999973 | 1.09691 | 0.987468 | 168.874 | - | - | $10^{-6}$
- | - | D10 | 0.999988 | 1.0969 | 0.993603 | 168.875 | 0.181087 | 5410.37 | $10^{-8}$
10 | 9 | D6 | 0.919405 | 0.768265 | - | - | - | - | $10^{-2}$
- | - |  D62 | 0.996141 | 0.892628 | - | - | - | - | $10^{-5}$
- | - | D8 | 0.998658 | 0.892509 | 0.470451 | 87.3373 | - | - | $10^{-6}$
- | - | D6D8 | 1.00001 | 0.891482 | 0.993735 | 152.599 | - | - | $10^{-6}$
- | - | D10 | 1.00001 | 0.891482 | 0.993769 | 152.599 | 0.000898094 | 4272.91 | $10^{-7}$
10 | 10 | D6 | 0.901816 | 0.62217 | - | - | - | - | $10^{-2}$
- | - |  D62 | 0.996482 | 0.744854 | - | - | - | - | $10^{-5}$
- | - | D8 | 0.998283 | 0.744691 | 0.357783 | 70.7167 | - | - | $10^{-5}$
- | - | D6D8 | 0.999924 | 0.743874 | 0.985524 | 144.47 | - | - | $10^{-5}$
- | - | D10 | 0.999892 | 0.743882 | 0.970747 | 144.466 | -0.383684 | 3459.76 | $10^{-7}$
Table 10: As described in Tab. 6. $M_{X}$ | $\beta$ | dim | $c_{6}$ | $\delta c_{6}$ | $c_{8}$ | $\delta c_{8}$ | $c_{10}$ | $\delta c_{10}$ | $\chi^{2}_{\rm min}$
---|---|---|---|---|---|---|---|---|---
3 | 3 | D6 | 1.12788 | 0.353318 | - | - | - | - | $10^{-1}$
- | - |  D62 | 1.13203 | 0.356668 | - | - | - | - | $10^{-1}$
- | - | D8 | 0.980971 | 0.356442 | 1.45476 | 2.38857 | - | - | $10^{-2}$
- | - | D6D8 | 0.979411 | 0.357435 | 1.49589 | 2.44542 | - | - | $10^{-2}$
- | - | D10 | 1.00498 | 0.357087 | 0.850995 | 2.44694 | 1.8517 | 8.11232 | $10^{-4}$
3 | 6 | D6 | 1.10815 | 0.0870852 | - | - | - | - | $10^{0}$
- | - |  D62 | 1.12538 | 0.0905217 | - | - | - | - | $10^{0}$
- | - | D8 | 0.987427 | 0.0903332 | 1.29106 | 0.572319 | - | - | $10^{-2}$
- | - | D6D8 | 0.981081 | 0.0913225 | 1.45242 | 0.631675 | - | - | $10^{-2}$
- | - | D10 | 1.00166 | 0.0910396 | 0.943074 | 0.633042 | 1.34751 | 1.92171 | $10^{-4}$
3 | 9 | D6 | 1.07091 | 0.0378047 | - | - | - | - | $10^{1}$
- | - |  D62 | 1.11237 | 0.0413785 | - | - | - | - | $10^{1}$
- | - | D8 | 1.00085 | 0.0412695 | 1.00748 | 0.238168 | - | - | $10^{-2}$
- | - | D6D8 | 0.986527 | 0.0422007 | 1.35066 | 0.303184 | - | - | $10^{-2}$
- | - | D10 | 0.997849 | 0.0420497 | 1.07877 | 0.304116 | 0.622311 | 0.787609 | $10^{-3}$
3 | 12 | D6 | 1.00923 | 0.0206546 | - | - | - | - | $10^{0}$
- | - |  D62 | 1.08999 | 0.0243725 | - | - | - | - | $10^{1}$
- | - | D8 | 1.02154 | 0.0243805 | 0.616649 | 0.12527 | - | - | $10^{0}$
- | - | D6D8 | 0.997353 | 0.0251004 | 1.14917 | 0.202589 | - | - | $10^{0}$
- | - | D10 | 0.992205 | 0.0251541 | 1.26864 | 0.201985 | -0.222281 | 0.411097 | $10^{-1}$
3 | 15 | D6 | 0.926368 | 0.0129133 | - | - | - | - | $10^{2}$
- | - |  D62 | 1.06471 | 0.0167086 | - | - | - | - | $10^{1}$
- | - | D8 | 1.04683 | 0.0167598 | 0.180373 | 0.0785712 | - | - | $10^{1}$
- | - | D6D8 | 1.01233 | 0.0171405 | 0.866336 | 0.179652 | - | - | $10^{1}$
- | - | D10 | 0.984882 | 0.0171841 | 1.49054 | 0.174775 | -0.983607 | 0.26331 | $10^{0}$
3 | 18 | D6 | 0.842816 | 0.00888742 | - | - | - | - | $10^{3}$
- | - |  D62 | 1.05498 | 0.0126621 | - | - | - | - | $10^{2}$
- | - | D8 | 1.06967 | 0.0125303 | -0.20599 | 0.0574116 | - | - | $10^{2}$
- | - | D6D8 | 1.01002 | 0.0127117 | 1.18099 | 0.169176 | - | - | $10^{2}$
- | - | D10 | 0.977171 | 0.0122382 | 1.69327 | 0.166058 | -1.47187 | 0.203686 | $10^{1}$
3 | 21 | D6 | 0.771161 | 0.00654558 | - | - | - | - | $10^{3}$
- | - |  D62 | 1.06933 | 0.0101629 | - | - | - | - | $10^{3}$
- | - | D8 | 1.08616 | 0.00970972 | -0.508954 | 0.0464134 | - | - | $10^{2}$
- | - | D6D8 | 0.993852 | 0.00885815 | 2.06728 | 0.119261 | - | - | $10^{2}$
- | - | D10 | 0.96982 | 0.00879942 | 1.84537 | 0.122417 | -1.65923 | 0.177694 | $10^{1}$
3 | 24 | D6 | 0.710911 | 0.00505784 | - | - | - | - | $10^{4}$
- | - |  D62 | 1.10199 | 0.0082592 | - | - | - | - | $10^{3}$
- | - | D8 | 1.09643 | 0.0076162 | -0.744376 | 0.0398252 | - | - | $10^{3}$
- | - | D6D8 | 0.972171 | 0.00620991 | 2.37603 | 0.0755521 | - | - | $10^{3}$
- | - | D10 | 0.962486 | 0.00644627 | 1.94384 | 0.0768416 | -1.62891 | 0.165102 | $10^{1}$
3 | 27 | D6 | 0.658785 | 0.00405035 | - | - | - | - | $10^{4}$
- | - |  D62 | 1.13855 | 0.00654542 | - | - | - | - | $10^{3}$
- | - | D8 | 1.10128 | 0.00600319 | -0.934216 | 0.0354559 | - | - | $10^{3}$
- | - | D6D8 | 0.957567 | 0.00465887 | 2.35815 | 0.0502491 | - | - | $10^{3}$
- | - | D10 | 0.954571 | 0.00486197 | 1.98925 | 0.050523 | -1.51136 | 0.158513 | $10^{2}$
3 | 30 | D6 | 0.612118 | 0.00333483 | - | - | - | - | $10^{4}$
- | - |  D62 | 1.16495 | 0.00506135 | - | - | - | - | $10^{3}$
- | - | D8 | 1.10119 | 0.00476017 | -1.09737 | 0.0323498 | - | - | $10^{3}$
- | - | D6D8 | 0.946954 | 0.00364767 | 2.27092 | 0.0356695 | - | - | $10^{3}$
- | - | D10 | 0.945524 | 0.00379038 | 1.97817 | 0.0357588 | -1.49372 | 0.154855 | $10^{2}$
4 | 3 | D6 | 1.06432 | 0.629608 | - | - | - | - | $10^{-2}$
- | - |  D62 | 1.06679 | 0.632873 | - | - | - | - | $10^{-2}$
- | - | D8 | 0.997294 | 0.632768 | 1.19634 | 7.6072 | - | - | $10^{-4}$
- | - | D6D8 | 0.996878 | 0.633219 | 1.21572 | 7.7096 | - | - | $10^{-4}$
- | - | D10 | 1.00371 | 0.633126 | 0.908069 | 7.71031 | 1.5892 | 46.0403 | $10^{-5}$
4 | 6 | D6 | 1.05434 | 0.15631 | - | - | - | - | $10^{-1}$
- | - |  D62 | 1.06434 | 0.159615 | - | - | - | - | $10^{-1}$
- | - | D8 | 0.997126 | 0.159518 | 1.14089 | 1.86438 | - | - | $10^{-3}$
- | - | D6D8 | 0.995423 | 0.159976 | 1.21881 | 1.96918 | - | - | $10^{-3}$
- | - | D10 | 1.00099 | 0.159899 | 0.969922 | 1.96979 | 1.23001 | 11.2285 | $10^{-5}$
4 | 9 | D6 | 1.03827 | 0.0686636 | - | - | - | - | $10^{0}$
- | - |  D62 | 1.06135 | 0.072045 | - | - | - | - | $10^{0}$
- | - | D8 | 0.999435 | 0.0719638 | 1.02919 | 0.802011 | - | - | $10^{-4}$
- | - | D6D8 | 0.995538 | 0.0724228 | 1.20253 | 0.911658 | - | - | $10^{-4}$
- | - | D10 | 1.00004 | 0.0723604 | 1.00338 | 0.912217 | 0.911276 | 4.7932 | $10^{-5}$
4 | 12 | D6 | 1.01404 | 0.0379978 | - | - | - | - | $10^{-1}$
- | - |  D62 | 1.05654 | 0.0414815 | - | - | - | - | $10^{0}$
- | - | D8 | 1.00303 | 0.0414276 | 0.870016 | 0.431865 | - | - | $10^{-2}$
- | - | D6D8 | 0.996004 | 0.0418767 | 1.17039 | 0.549872 | - | - | $10^{-2}$
- | - | D10 | 0.998972 | 0.0418359 | 1.04102 | 0.55031 | 0.528222 | 2.55695 | $10^{-4}$
4 | 15 | D6 | 0.979859 | 0.023833 | - | - | - | - | $10^{0}$
- | - |  D62 | 1.04913 | 0.0274298 | - | - | - | - | $10^{1}$
- | - | D8 | 1.00774 | 0.0274152 | 0.666482 | 0.26301 | - | - | $10^{-1}$
- | - | D6D8 | 0.996731 | 0.0278271 | 1.11312 | 0.395232 | - | - | $10^{-1}$
- | - | D10 | 0.997378 | 0.0278188 | 1.08536 | 0.395355 | 0.0971082 | 1.544 | $10^{-3}$
Table 11: Table of fit $\chi^{2}$ fit values for the $X$ model as described in Sec. 4 and Tab. 2. Table is continued below. $M_{X}$ | $\beta$ | dim | $c_{6}$ | $\delta c_{6}$ | $c_{8}$ | $\delta c_{8}$ | $c_{10}$ | $\delta c_{10}$ | $\chi^{2}_{\rm min}$
---|---|---|---|---|---|---|---|---|---
4 | 18 | D6 | 0.934817 | 0.016197 | - | - | - | - | $10^{1}$
- | - |  D62 | 1.03896 | 0.0198842 | - | - | - | - | $10^{1}$
- | - | D8 | 1.01327 | 0.0199112 | 0.426714 | 0.174712 | - | - | $10^{0}$
- | - | D6D8 | 0.997761 | 0.0202352 | 1.01528 | 0.330998 | - | - | $10^{0}$
- | - | D10 | 0.994955 | 0.0202625 | 1.1344 | 0.330258 | -0.347859 | 1.02276 | $10^{-2}$
4 | 21 | D6 | 0.881011 | 0.0116809 | - | - | - | - | $10^{2}$
- | - |  D62 | 1.0278 | 0.0153866 | - | - | - | - | $10^{1}$
- | - | D8 | 1.01883 | 0.0154218 | 0.167914 | 0.125567 | - | - | $10^{1}$
- | - | D6D8 | 0.99813 | 0.0156184 | 0.898149 | 0.316049 | - | - | $10^{1}$
- | - | D10 | 0.991452 | 0.0156319 | 1.18145 | 0.313794 | -0.771298 | 0.742141 | $10^{-1}$
4 | 24 | D6 | 0.824176 | 0.0088429 | - | - | - | - | $10^{3}$
- | - |  D62 | 1.01956 | 0.0124572 | - | - | - | - | $10^{1}$
- | - | D8 | 1.02311 | 0.0124232 | -0.0876536 | 0.0974413 | - | - | $10^{1}$
- | - | D6D8 | 0.991305 | 0.0125123 | 1.05743 | 0.29586 | - | - | $10^{1}$
- | - | D10 | 0.986642 | 0.0124081 | 1.21565 | 0.295296 | -1.16688 | 0.591249 | $10^{-1}$
4 | 27 | D6 | 0.769791 | 0.00696577 | - | - | - | - | $10^{3}$
- | - |  D62 | 1.01711 | 0.0103706 | - | - | - | - | $10^{2}$
- | - | D8 | 1.02478 | 0.0102001 | -0.326512 | 0.080718 | - | - | $10^{1}$
- | - | D6D8 | 0.977765 | 0.0100053 | 1.52128 | 0.221022 | - | - | $10^{1}$
- | - | D10 | 0.980274 | 0.00996182 | 1.22494 | 0.220395 | -1.59332 | 0.510147 | $10^{-1}$
4 | 30 | D6 | 0.719886 | 0.00566208 | - | - | - | - | $10^{3}$
- | - |  D62 | 1.01993 | 0.00874775 | - | - | - | - | $10^{2}$
- | - | D8 | 1.02303 | 0.00844388 | -0.550255 | 0.0701249 | - | - | $10^{2}$
- | - | D6D8 | 0.965893 | 0.0079549 | 1.7356 | 0.147362 | - | - | $10^{2}$
- | - | D10 | 0.972029 | 0.00807064 | 1.1986 | 0.145733 | -2.19289 | 0.465412 | $10^{0}$
5 | 3 | D6 | 1.04054 | 0.984653 | - | - | - | - | $10^{-3}$
- | - |  D62 | 1.04214 | 0.987883 | - | - | - | - | $10^{-3}$
- | - | D8 | 1.00196 | 0.987822 | 1.08267 | 18.6226 | - | - | $10^{-5}$
- | - | D6D8 | 1.00181 | 0.988081 | 1.09388 | 18.7829 | - | - | $10^{-5}$
- | - | D10 | 1.00562 | 0.98803 | 0.825618 | 18.7835 | 2.17667 | 176.233 | $10^{-5}$
5 | 6 | D6 | 1.03348 | 0.245126 | - | - | - | - | $10^{-2}$
- | - |  D62 | 1.03991 | 0.248375 | - | - | - | - | $10^{-2}$
- | - | D8 | 0.999576 | 0.248316 | 1.07771 | 4.60098 | - | - | $10^{-4}$
- | - | D6D8 | 0.998928 | 0.248584 | 1.12409 | 4.76346 | - | - | $10^{-4}$
- | - | D10 | 1.00132 | 0.248551 | 0.956224 | 4.76386 | 1.32458 | 43.4194 | $10^{-5}$
5 | 9 | D6 | 1.02388 | 0.108177 | - | - | - | - | $10^{-1}$
- | - |  D62 | 1.03854 | 0.111473 | - | - | - | - | $10^{-1}$
- | - | D8 | 0.999888 | 0.111418 | 1.01896 | 2.00528 | - | - | $10^{-5}$
- | - | D6D8 | 0.998403 | 0.111689 | 1.12332 | 2.17231 | - | - | $10^{-5}$
- | - | D10 | 1.00038 | 0.111662 | 0.985568 | 2.17267 | 1.03623 | 18.8382 | $10^{-5}$
5 | 12 | D6 | 1.00998 | 0.0602473 | - | - | - | - | $10^{-2}$
- | - |  D62 | 1.03656 | 0.0636086 | - | - | - | - | $10^{0}$
- | - | D8 | 1.0007 | 0.063563 | 0.930206 | 1.09799 | - | - | $10^{-4}$
- | - | D6D8 | 0.998018 | 0.0638348 | 1.11413 | 1.2723 | - | - | $10^{-4}$
- | - | D10 | 0.999613 | 0.0638126 | 1.00372 | 1.27262 | 0.774643 | 10.2533 | $10^{-5}$
5 | 15 | D6 | 0.990933 | 0.0380704 | - | - | - | - | $10^{-1}$
- | - |  D62 | 1.03351 | 0.0415125 | - | - | - | - | $10^{0}$
- | - | D8 | 1.00163 | 0.04148 | 0.813574 | 0.679671 | - | - | $10^{-2}$
- | - | D6D8 | 0.997397 | 0.041748 | 1.09566 | 0.864998 | - | - | $10^{-2}$
- | - | D10 | 0.998511 | 0.0417325 | 1.01919 | 0.865256 | 0.488463 | 6.30339 | $10^{-4}$
5 | 18 | D6 | 0.965893 | 0.0260397 | - | - | - | - | $10^{0}$
- | - |  D62 | 1.02891 | 0.0295697 | - | - | - | - | $10^{0}$
- | - | D8 | 1.00242 | 0.029555 | 0.669774 | 0.454578 | - | - | $10^{-1}$
- | - | D6D8 | 0.996306 | 0.0298097 | 1.06217 | 0.656493 | - | - | $10^{-1}$
- | - | D10 | 0.996742 | 0.0298038 | 1.03251 | 0.656615 | 0.168076 | 4.1875 | $10^{-4}$
5 | 21 | D6 | 0.934215 | 0.0188138 | - | - | - | - | $10^{1}$
- | - |  D62 | 1.02235 | 0.0224234 | - | - | - | - | $10^{0}$
- | - | D8 | 1.00275 | 0.0224291 | 0.500178 | 0.321628 | - | - | $10^{-1}$
- | - | D6D8 | 0.994565 | 0.0226532 | 1.00184 | 0.548546 | - | - | $10^{-1}$
- | - | D10 | 0.993985 | 0.0226603 | 1.04119 | 0.548338 | -0.193425 | 2.94854 | $10^{-3}$
5 | 24 | D6 | 0.895982 | 0.0141668 | - | - | - | - | $10^{2}$
- | - |  D62 | 1.0137 | 0.0178232 | - | - | - | - | $10^{0}$
- | - | D8 | 1.00221 | 0.017844 | 0.307827 | 0.238731 | - | - | $10^{0}$
- | - | D6D8 | 0.992008 | 0.0180137 | 0.896514 | 0.501956 | - | - | $10^{0}$
- | - | D10 | 0.989903 | 0.0180328 | 1.0405 | 0.500969 | -0.619413 | 2.188 | $10^{-3}$
5 | 27 | D6 | 0.852562 | 0.0110361 | - | - | - | - | $10^{2}$
- | - |  D62 | 1.00348 | 0.014677 | - | - | - | - | $10^{0}$
- | - | D8 | 1.00023 | 0.0146916 | 0.0976998 | 0.185623 | - | - | $10^{0}$
- | - | D6D8 | 0.987778 | 0.0147991 | 0.762986 | 0.490231 | - | - | $10^{0}$
- | - | D10 | 0.984127 | 0.0148064 | 1.02352 | 0.488319 | -1.16773 | 1.71363 | $10^{-2}$
5 | 30 | D6 | 0.806484 | 0.00885404 | - | - | - | - | $10^{3}$
- | - |  D62 | 0.992757 | 0.0123987 | - | - | - | - | $10^{1}$
- | - | D8 | 0.996115 | 0.0123682 | -0.125553 | 0.151125 | - | - | $10^{0}$
- | - | D6D8 | 0.977947 | 0.0124495 | 0.840751 | 0.460585 | - | - | $10^{0}$
- | - | D10 | 0.976295 | 0.0123872 | 0.980555 | 0.460251 | -1.95454 | 1.41855 | $10^{-2}$
6 | 3 | D6 | 1.029 | 1.41854 | - | - | - | - | $10^{-3}$
- | - |  D62 | 1.03012 | 1.42176 | - | - | - | - | $10^{-3}$
- | - | D8 | 1.00443 | 1.42172 | 0.99779 | 38.6663 | - | - | $10^{-5}$
- | - | D6D8 | 1.00436 | 1.42188 | 1.0049 | 38.8975 | - | - | $10^{-5}$
- | - | D10 | 1.00766 | 1.42184 | 0.670863 | 38.8982 | 3.91389 | 527.092 | $10^{-5}$
Table 12: As described in Tab. 11. $M_{X}$ | $\beta$ | dim | $c_{6}$ | $\delta c_{6}$ | $c_{8}$ | $\delta c_{8}$ | $c_{10}$ | $\delta c_{10}$ | $\chi^{2}_{\rm min}$
---|---|---|---|---|---|---|---|---|---
6 | 6 | D6 | 1.0231 | 0.353626 | - | - | - | - | $10^{-2}$
- | - |  D62 | 1.02757 | 0.356846 | - | - | - | - | $10^{-2}$
- | - | D8 | 1.0007 | 0.356806 | 1.03749 | 9.59054 | - | - | $10^{-5}$
- | - | D6D8 | 1.00041 | 0.356982 | 1.06816 | 9.82338 | - | - | $10^{-5}$
- | - | D10 | 1.00192 | 0.356962 | 0.915134 | 9.82374 | 1.75886 | 130.497 | $10^{-5}$
6 | 9 | D6 | 1.01645 | 0.156419 | - | - | - | - | $10^{-2}$
- | - |  D62 | 1.02658 | 0.159669 | - | - | - | - | $10^{-2}$
- | - | D8 | 1.00025 | 0.159631 | 1.00743 | 4.20693 | - | - | $10^{-5}$
- | - | D6D8 | 0.999566 | 0.159811 | 1.07715 | 4.44411 | - | - | $10^{-5}$
- | - | D10 | 1.0007 | 0.159795 | 0.963307 | 4.44439 | 1.26644 | 57.0715 | $10^{-5}$
6 | 12 | D6 | 1.00712 | 0.0873974 | - | - | - | - | $10^{-2}$
- | - |  D62 | 1.02536 | 0.0906925 | - | - | - | - | $10^{-1}$
- | - | D8 | 1.00017 | 0.0906579 | 0.952493 | 2.32371 | - | - | $10^{-5}$
- | - | D6D8 | 0.998933 | 0.0908387 | 1.07632 | 2.56762 | - | - | $10^{-5}$
- | - | D10 | 0.999839 | 0.0908262 | 0.985441 | 2.56787 | 0.964559 | 31.3958 | $10^{-5}$
6 | 15 | D6 | 0.994553 | 0.0554535 | - | - | - | - | $10^{-2}$
- | - |  D62 | 1.02351 | 0.0588048 | - | - | - | - | $10^{-1}$
- | - | D8 | 1.00002 | 0.0587756 | 0.87614 | 1.45338 | - | - | $10^{-3}$
- | - | D6D8 | 0.998072 | 0.0589564 | 1.06774 | 1.70696 | - | - | $10^{-3}$
- | - | D10 | 0.998801 | 0.0589462 | 0.995035 | 1.70717 | 0.724855 | 19.5399 | $10^{-5}$
6 | 18 | D6 | 0.978148 | 0.0381078 | - | - | - | - | $10^{-1}$
- | - |  D62 | 1.02061 | 0.0415239 | - | - | - | - | $10^{0}$
- | - | D8 | 0.999437 | 0.0415022 | 0.779703 | 0.982276 | - | - | $10^{-3}$
- | - | D6D8 | 0.996613 | 0.0416802 | 1.05023 | 1.24935 | - | - | $10^{-3}$
- | - | D10 | 0.99712 | 0.0416732 | 0.999976 | 1.24952 | 0.46265 | 13.134 | $10^{-5}$
6 | 21 | D6 | 0.957354 | 0.0276604 | - | - | - | - | $10^{0}$
- | - |  D62 | 1.01624 | 0.0311447 | - | - | - | - | $10^{0}$
- | - | D8 | 0.998114 | 0.0311326 | 0.661893 | 0.700304 | - | - | $10^{-2}$
- | - | D6D8 | 0.994301 | 0.0313028 | 1.01666 | 0.986123 | - | - | $10^{-2}$
- | - | D10 | 0.994494 | 0.0313001 | 0.997552 | 0.986199 | 0.159608 | 9.3128 | $10^{-4}$
6 | 24 | D6 | 0.931688 | 0.0208979 | - | - | - | - | $10^{1}$
- | - |  D62 | 1.00994 | 0.0244451 | - | - | - | - | $10^{0}$
- | - | D8 | 0.995695 | 0.0244438 | 0.521384 | 0.51987 | - | - | $10^{-1}$
- | - | D6D8 | 0.990869 | 0.0245973 | 0.954089 | 0.831728 | - | - | $10^{-1}$
- | - | D10 | 0.990557 | 0.0246013 | 0.984813 | 0.83158 | -0.229103 | 6.88186 | $10^{-4}$
6 | 27 | D6 | 0.90095 | 0.0162879 | - | - | - | - | $10^{1}$
- | - |  D62 | 1.00132 | 0.0198795 | - | - | - | - | $10^{0}$
- | - | D8 | 0.991787 | 0.0198876 | 0.356441 | 0.399246 | - | - | $10^{-1}$
- | - | D6D8 | 0.986123 | 0.0200113 | 0.838812 | 0.746804 | - | - | $10^{-1}$
- | - | D10 | 0.984951 | 0.020025 | 0.95527 | 0.746114 | -0.770891 | 5.27224 | $10^{-4}$
6 | 30 | D6 | 0.865429 | 0.0130248 | - | - | - | - | $10^{2}$
- | - |  D62 | 0.990113 | 0.0166263 | - | - | - | - | $10^{0}$
- | - | D8 | 0.985926 | 0.0166361 | 0.165411 | 0.316445 | - | - | $10^{-1}$
- | - | D6D8 | 0.979825 | 0.016718 | 0.640609 | 0.709609 | - | - | $10^{-1}$
- | - | D10 | 0.977269 | 0.0167405 | 0.901631 | 0.707769 | -1.59143 | 4.18391 | $10^{-3}$
7 | 3 | D6 | 1.02288 | 1.9313 | - | - | - | - | $10^{-4}$
- | - |  D62 | 1.0237 | 1.93451 | - | - | - | - | $10^{-4}$
- | - | D8 | 1.00665 | 1.93448 | 0.902131 | 71.6873 | - | - | $10^{-5}$
- | - | D6D8 | 1.00662 | 1.93459 | 0.906743 | 72.0024 | - | - | $10^{-5}$
- | - | D10 | 1.01023 | 1.93455 | 0.407461 | 72.0035 | 7.97618 | 1330.36 | $10^{-5}$
7 | 6 | D6 | 1.01721 | 0.481831 | - | - | - | - | $10^{-3}$
- | - |  D62 | 1.02049 | 0.485035 | - | - | - | - | $10^{-3}$
- | - | D8 | 1.00147 | 0.485006 | 1.00242 | 17.8204 | - | - | $10^{-5}$
- | - | D6D8 | 1.00131 | 0.48513 | 1.02397 | 18.1364 | - | - | $10^{-5}$
- | - | D10 | 1.00259 | 0.485113 | 0.847507 | 18.1368 | 2.7797 | 330.276 | $10^{-5}$
7 | 9 | D6 | 1.01213 | 0.21341 | - | - | - | - | $10^{-3}$
- | - |  D62 | 1.01955 | 0.216634 | - | - | - | - | $10^{-2}$
- | - | D8 | 1.00053 | 0.216606 | 0.995301 | 7.8458 | - | - | $10^{-5}$
- | - | D6D8 | 1.00017 | 0.216734 | 1.04504 | 8.16588 | - | - | $10^{-5}$
- | - | D10 | 1.00098 | 0.216723 | 0.933747 | 8.16615 | 1.71181 | 145.1 | $10^{-5}$
7 | 12 | D6 | 1.0053 | 0.119464 | - | - | - | - | $10^{-3}$
- | - |  D62 | 1.0186 | 0.122719 | - | - | - | - | $10^{-2}$
- | - | D8 | 1.00008 | 0.122693 | 0.960405 | 4.35565 | - | - | $10^{-5}$
- | - | D6D8 | 0.999435 | 0.122822 | 1.04923 | 4.68214 | - | - | $10^{-5}$
- | - | D10 | 1.00006 | 0.122813 | 0.964044 | 4.68236 | 1.2666 | 80.3171 | $10^{-5}$
7 | 15 | D6 | 0.996155 | 0.0759814 | - | - | - | - | $10^{-3}$
- | - |  D62 | 1.01717 | 0.0792776 | - | - | - | - | $10^{-1}$
- | - | D8 | 0.999478 | 0.0792539 | 0.907748 | 2.74141 | - | - | $10^{-4}$
- | - | D6D8 | 0.998457 | 0.0793832 | 1.04596 | 3.0768 | - | - | $10^{-4}$
- | - | D10 | 0.998956 | 0.0793763 | 0.97794 | 3.07699 | 0.96701 | 50.3672 | $10^{-5}$
7 | 18 | D6 | 0.984248 | 0.0523652 | - | - | - | - | $10^{-1}$
- | - |  D62 | 1.01489 | 0.0557085 | - | - | - | - | $10^{-1}$
- | - | D8 | 0.998356 | 0.0556884 | 0.838465 | 1.86601 | - | - | $10^{-4}$
- | - | D6D8 | 0.996878 | 0.055817 | 1.0349 | 2.21329 | - | - | $10^{-4}$
- | - | D10 | 0.997257 | 0.0558117 | 0.98335 | 2.21344 | 0.692647 | 34.1397 | $10^{-5}$
7 | 21 | D6 | 0.969136 | 0.0381316 | - | - | - | - | $10^{0}$
- | - |  D62 | 1.01137 | 0.0415263 | - | - | - | - | $10^{-1}$
- | - | D8 | 0.996413 | 0.0415109 | 0.750673 | 1.33996 | - | - | $10^{-3}$
- | - | D6D8 | 0.994416 | 0.0416366 | 1.01074 | 1.70288 | - | - | $10^{-3}$
- | - | D10 | 0.99466 | 0.0416332 | 0.977706 | 1.70299 | 0.414387 | 24.4041 | $10^{-5}$
Table 13: As described in Tab. 11. $M_{X}$ | $\beta$ | dim | $c_{6}$ | $\delta c_{6}$ | $c_{8}$ | $\delta c_{8}$ | $c_{10}$ | $\delta c_{10}$ | $\chi^{2}_{\rm min}$
---|---|---|---|---|---|---|---|---|---
7 | 24 | D6 | 0.950354 | 0.0289032 | - | - | - | - | $10^{0}$
- | - |  D62 | 1.00617 | 0.0323494 | - | - | - | - | $10^{-1}$
- | - | D8 | 0.99327 | 0.0323398 | 0.64258 | 1.0007 | - | - | $10^{-2}$
- | - | D6D8 | 0.990728 | 0.0324588 | 0.965326 | 1.38406 | - | - | $10^{-2}$
- | - | D10 | 0.990763 | 0.0324583 | 0.960662 | 1.38407 | 0.0539281 | 18.1427 | $10^{-5}$
7 | 27 | D6 | 0.927504 | 0.0225904 | - | - | - | - | $10^{1}$
- | - |  D62 | 0.998806 | 0.0260823 | - | - | - | - | $10^{-1}$
- | - | D8 | 0.98855 | 0.0260791 | 0.510485 | 0.770686 | - | - | $10^{-2}$
- | - | D6D8 | 0.985523 | 0.0261851 | 0.882197 | 1.18069 | - | - | $10^{-2}$
- | - | D10 | 0.985194 | 0.0261896 | 0.926521 | 1.18049 | -0.467042 | 13.9165 | $10^{-5}$
7 | 30 | D6 | 0.900336 | 0.0180942 | - | - | - | - | $10^{1}$
- | - |  D62 | 0.988796 | 0.0216182 | - | - | - | - | $10^{-1}$
- | - | D8 | 0.981838 | 0.0216204 | 0.349993 | 0.609175 | - | - | $10^{-1}$
- | - | D6D8 | 0.978547 | 0.0217043 | 0.733969 | 1.05361 | - | - | $10^{-1}$
- | - | D10 | 0.977557 | 0.0217169 | 0.868176 | 1.0529 | -1.27983 | 10.9691 | $10^{-4}$
8 | 3 | D6 | 1.01969 | 2.52294 | - | - | - | - | $10^{-5}$
- | - |  D62 | 1.02032 | 2.52614 | - | - | - | - | $10^{-5}$
- | - | D8 | 1.0092 | 2.52613 | 0.768645 | 122.352 | - | - | $10^{-5}$
- | - | D6D8 | 1.00919 | 2.5262 | 0.771495 | 122.764 | - | - | $10^{-5}$
- | - | D10 | 1.01367 | 2.52614 | -0.0367678 | 122.766 | 16.8834 | 2965.98 | $10^{-5}$
8 | 6 | D6 | 1.01363 | 0.629749 | - | - | - | - | $10^{-3}$
- | - |  D62 | 1.01614 | 0.632943 | - | - | - | - | $10^{-3}$
- | - | D8 | 1.00215 | 0.632922 | 0.963676 | 30.4573 | - | - | $10^{-5}$
- | - | D6D8 | 1.00207 | 0.633013 | 0.979354 | 30.8692 | - | - | $10^{-5}$
- | - | D10 | 1.0034 | 0.632995 | 0.740265 | 30.8698 | 4.94083 | 737.606 | $10^{-5}$
8 | 9 | D6 | 1.00944 | 0.279159 | - | - | - | - | $10^{-3}$
- | - |  D62 | 1.01511 | 0.282366 | - | - | - | - | $10^{-3}$
- | - | D8 | 1.00081 | 0.282345 | 0.979773 | 13.4404 | - | - | $10^{-5}$
- | - | D6D8 | 1.00061 | 0.28244 | 1.01676 | 13.8562 | - | - | $10^{-5}$
- | - | D10 | 1.00134 | 0.28243 | 0.885373 | 13.8565 | 2.6664 | 324.973 | $10^{-5}$
8 | 12 | D6 | 1.00409 | 0.156453 | - | - | - | - | $10^{-3}$
- | - |  D62 | 1.01423 | 0.159683 | - | - | - | - | $10^{-2}$
- | - | D8 | 1.00011 | 0.159663 | 0.96136 | 7.48544 | - | - | $10^{-5}$
- | - | D6D8 | 0.99974 | 0.159759 | 1.028 | 7.90736 | - | - | $10^{-5}$
- | - | D10 | 1.00024 | 0.159752 | 0.93779 | 7.90758 | 1.78433 | 180.588 | $10^{-5}$
8 | 15 | D6 | 0.997018 | 0.0996584 | - | - | - | - | $10^{-3}$
- | - |  D62 | 1.01298 | 0.102919 | - | - | - | - | $10^{-2}$
- | - | D8 | 0.999289 | 0.1029 | 0.924236 | 4.73028 | - | - | $10^{-5}$
- | - | D6D8 | 0.998705 | 0.102997 | 1.02838 | 5.16059 | - | - | $10^{-5}$
- | - | D10 | 0.999094 | 0.102991 | 0.958947 | 5.16078 | 1.3277 | 113.802 | $10^{-5}$
8 | 18 | D6 | 0.987788 | 0.0688098 | - | - | - | - | $10^{-2}$
- | - |  D62 | 1.01098 | 0.0721055 | - | - | - | - | $10^{-1}$
- | - | D8 | 0.997935 | 0.0720883 | 0.871844 | 3.23503 | - | - | $10^{-4}$
- | - | D6D8 | 0.997091 | 0.072185 | 1.02034 | 3.67624 | - | - | $10^{-4}$
- | - | D10 | 0.997399 | 0.0721807 | 0.96541 | 3.6764 | 1.00712 | 77.5742 | $10^{-5}$
8 | 21 | D6 | 0.976005 | 0.0502132 | - | - | - | - | $10^{-1}$
- | - |  D62 | 1.00784 | 0.0535471 | - | - | - | - | $10^{-1}$
- | - | D8 | 0.995699 | 0.0535325 | 0.803608 | 2.3351 | - | - | $10^{-4}$
- | - | D6D8 | 0.994557 | 0.0536278 | 1.00129 | 2.79011 | - | - | $10^{-4}$
- | - | D10 | 0.99478 | 0.0536247 | 0.961571 | 2.79023 | 0.69189 | 55.7899 | $10^{-5}$
8 | 24 | D6 | 0.961253 | 0.0381497 | - | - | - | - | $10^{0}$
- | - |  D62 | 1.00314 | 0.0415225 | - | - | - | - | $10^{-1}$
- | - | D8 | 0.992216 | 0.0415112 | 0.716856 | 1.75299 | - | - | $10^{-3}$
- | - | D6D8 | 0.990758 | 0.041603 | 0.964379 | 2.22523 | - | - | $10^{-3}$
- | - | D10 | 0.990864 | 0.0416015 | 0.945701 | 2.2253 | 0.306376 | 41.7204 | $10^{-5}$
8 | 27 | D6 | 0.943141 | 0.0298882 | - | - | - | - | $10^{0}$
- | - |  D62 | 0.996439 | 0.0332972 | - | - | - | - | $10^{-1}$
- | - | D8 | 0.987127 | 0.0332897 | 0.606923 | 1.35623 | - | - | $10^{-3}$
- | - | D6D8 | 0.985379 | 0.0333745 | 0.896676 | 1.84988 | - | - | $10^{-3}$
- | - | D10 | 0.985302 | 0.0333756 | 0.910389 | 1.84983 | -0.209871 | 32.154 | $10^{-5}$
8 | 30 | D6 | 0.921307 | 0.0239913 | - | - | - | - | $10^{1}$
- | - |  D62 | 0.98722 | 0.0274291 | - | - | - | - | $10^{-1}$
- | - | D8 | 0.980012 | 0.0274258 | 0.46873 | 1.07517 | - | - | $10^{-2}$
- | - | D6D8 | 0.978078 | 0.0274981 | 0.77873 | 1.59523 | - | - | $10^{-2}$
- | - | D10 | 0.977671 | 0.0275037 | 0.85077 | 1.59492 | -1.0202 | 25.4018 | $10^{-5}$
Table 14: As described in Tab. 11.

## References

  * [1] S. Weinberg,  _Phenomenological Lagrangians_ ,  Physica A 96(1-2), 327 (1979),  10.1016/0378-4371(79)90223-1.
  * [2] T. Corbett, M. J. Dolan, C. Englert and K. Nordström,  _Anomalous neutral gauge boson interactions and simplified models_ ,  Phys. Rev. D 97(11), 115040 (2018),  10.1103/PhysRevD.97.115040,  1710.07530.
  * [3] G. J. Gounaris, J. Layssac and F. M. Renard,  _Signatures of the anomalous $Z_{\gamma}$ and $ZZ$ production at the lepton and hadron colliders_,  Phys. Rev. D 61, 073013 (2000),  10.1103/PhysRevD.61.073013,  hep-ph/9910395.
  * [4] C. Degrande,  _A basis of dimension-eight operators for anomalous neutral triple gauge boson interactions_ ,  JHEP 02, 101 (2014),  10.1007/JHEP02(2014)101,  1308.6323.
  * [5] J. Ellis, H.-J. He and R.-Q. Xiao,  _Probing new physics in dimension-8 neutral gauge couplings at e +e? colliders_,  Sci. China Phys. Mech. Astron. 64(2), 221062 (2021),  10.1007/s11433-020-1617-3,  2008.04298.
  * [6] O. J. P. Éboli and M. C. Gonzalez-Garcia,  _Classifying the bosonic quartic couplings_ ,  Phys. Rev. D 93(9), 093013 (2016),  10.1103/PhysRevD.93.093013,  1604.03555.
  * [7] D. Liu, R.-Q. Xiao, S. Li, J. Ellis, H.-J. He and R. Yuan,  _Probing Neutral Triple Gauge Couplings via $\boldsymbol{Z\gamma\,(\ell^{+}\ell^{-}\gamma)}$ Production at $\boldsymbol{e^{+}e^{-}}$ Colliders_ (2024),  2404.15937.
  * [8] C. Zhang,  _SMEFTs living on the edge: determining the UV theories from positivity and extremality_ (2021),  2112.11665.
  * [9] T. Corbett,  _The Feynman rules for the SMEFT in the background field gauge_ ,  JHEP 03, 001 (2021),  10.1007/JHEP03(2021)001,  2010.15852.
  * [10] A. Helset, A. Martin and M. Trott,  _The Geometric Standard Model Effective Field Theory_ ,  JHEP 03, 163 (2020),  10.1007/JHEP03(2020)163,  2001.01453.
  * [11] C. W. Murphy,  _Dimension-8 operators in the Standard Model Eective Field Theory_ ,  JHEP 10, 174 (2020),  10.1007/JHEP10(2020)174,  2005.00059.
  * [12] H.-L. Li, Z. Ren, J. Shu, M.-L. Xiao, J.-H. Yu and Y.-H. Zheng,  _Complete set of dimension-eight operators in the standard model effective field theory_ ,  Phys. Rev. D 104(1), 015026 (2021),  10.1103/PhysRevD.104.015026,  2005.00008.
  * [13] T. Corbett, J. Desai, O. J. P. Eboli and M. C. Gonzalez-Garcia,  _Dimension-eight Operator Basis for Universal Standard Model Effective Field Theory_ (2024),  2404.03720.
  * [14] C. Hays, A. Martin, V. Sanz and J. Setford,  _On the impact of dimension-eight SMEFT operators on Higgs measurements_ ,  JHEP 02, 123 (2019),  10.1007/JHEP02(2019)123,  1808.00442.
  * [15] T. Corbett,  _The one-loop tadpole in the geoSMEFT_ ,  SciPost Phys. 11, 097 (2021),  2106.10284.
  * [16] R. Boughezal, E. Mereghetti and F. Petriello,  _Dilepton production in the SMEFT at O(1/ $\Lambda$4)_,  Phys. Rev. D 104(9), 095022 (2021),  10.1103/PhysRevD.104.095022,  2106.05337.
  * [17] T. Corbett, A. Martin and M. Trott,  _Consistent higher order $\sigma\left(\mathcal{GG}\to h\right)$, $\Gamma\left(h\to\mathcal{GG}\right)$ and $\Gamma(h\to\gamma\gamma)$ in geoSMEFT_,  JHEP 12, 147 (2021),  10.1007/JHEP12(2021)147,  2107.07470.
  * [18] S. Alioli, R. Boughezal, E. Mereghetti and F. Petriello,  _Novel angular dependence in Drell-Yan lepton production via dimension-8 operators_ ,  Phys. Lett. B 809, 135703 (2020),  10.1016/j.physletb.2020.135703,  2003.11615.
  * [19] R. Boughezal, Y. Huang and F. Petriello,  _Exploring the SMEFT at dimension eight with Drell-Yan transverse momentum measurements_ ,  Phys. Rev. D 106(3), 036020 (2022),  10.1103/PhysRevD.106.036020,  2207.01703.
  * [20] T. Kim and A. Martin,  _Monolepton production in SMEFT to $\mathcal{O}$(1/$\Lambda$4) and beyond_,  JHEP 09, 124 (2022),  10.1007/JHEP09(2022)124,  2203.11976.
  * [21] L. Allwicher, D. A. Faroughy, F. Jaffredo, O. Sumensari and F. Wilsch,  _Drell-Yan tails beyond the Standard Model_ ,  JHEP 03, 064 (2023),  10.1007/JHEP03(2023)064,  2207.10714.
  * [22] S. Dawson, S. Homiller and M. Sullivan,  _Impact of dimension-eight SMEFT contributions: A case study_ ,  Phys. Rev. D 104(11), 115013 (2021),  10.1103/PhysRevD.104.115013,  2110.06929.
  * [23] C. Degrande and H.-L. Li,  _Impact of Dimension-8 SMEFT operators on Diboson Productions_ (2023),  2303.10493.
  * [24] T. Corbett, A. Helset, A. Martin and M. Trott,  _EWPD in the SMEFT to dimension eight_ ,  JHEP 06, 076 (2021),  10.1007/JHEP06(2021)076,  2102.02819.
  * [25] T. Corbett, J. Desai, O. J. P. Éboli, M. C. Gonzalez-Garcia, M. Martines and P. Reimitz,  _Impact of dimension-eight SMEFT operators in the electroweak precision observables and triple gauge couplings analysis in universal SMEFT_ ,  Phys. Rev. D 107(11), 115013 (2023),  10.1103/PhysRevD.107.115013,  2304.03305.
  * [26] A. Martin and M. Trott,  _More accurate $\sigma(\mathcal{G}\,\mathcal{G}\rightarrow h)$, $\Gamma(h\rightarrow\mathcal{G}\,\mathcal{G},\mathcal{A}\mathcal{A},\bar{\Psi}{\Psi})$ and Higgs width results via the geoSMEFT_ (2023),  2305.05879.
  * [27] R. Boughezal, Y. Huang and F. Petriello,  _Impact of high invariant-mass Drell-Yan forward-backward asymmetry measurements on SMEFT fits_ ,  Phys. Rev. D 108(7), 076008 (2023),  10.1103/PhysRevD.108.076008,  2303.08257.
  * [28] A. Martin,  _A case study of SMEFT $\mathcal{O}(1/\Lambda^{4})$ effects in diboson processes: $pp\to W^{\pm}(\ell^{\pm}\nu)\gamma$_ (2023),  2312.09867.
  * [29] T. Corbett and A. Martin,  _Higgs associated production with a vector decaying to two fermions in the geoSMEFT_ (2023),  2306.00053.
  * [30] J. Ellis, K. Mimasu and F. Zampedri,  _Dimension-8 SMEFT analysis of minimal scalar field extensions of the Standard Model_ ,  JHEP 10, 051 (2023),  10.1007/JHEP10(2023)051,  2304.06663.
  * [31] A. Martin and M. Trott,  _$ggh$ variations_,  Phys. Rev. D 105(7), 076004 (2022),  10.1103/PhysRevD.105.076004,  2109.05595.
  * [32] J. Talbert and M. Trott,  _Dirac masses and mixings in the (geo)SM(EFT) and beyond_ ,  JHEP 11, 009 (2021),  10.1007/JHEP11(2021)009,  2107.03951.
  * [33] M. Trott,  _Methodology for theory uncertainties in the standard model effective field theory_ ,  Phys. Rev. D 104(9), 095023 (2021),  10.1103/PhysRevD.104.095023,  2106.13794.
  * [34] S. Dawson, M. Forslund and M. Schnubel,  _SMEFT Matching to $Z^{\prime}$ Models at Dimension-8_ (2024),  2404.01375.
  * [35] I. Brivio, Y. Jiang and M. Trott,  _The SMEFTsim package, theory and tools_ ,  JHEP 12, 070 (2017),  10.1007/JHEP12(2017)070,  1709.06492.
  * [36] G. D’Ambrosio, G. F. Giudice, G. Isidori and A. Strumia,  _Minimal flavor violation: An Effective field theory approach_ ,  Nucl. Phys. B 645, 155 (2002),  10.1016/S0550-3213(02)00836-2,  hep-ph/0207036.
  * [37] J. D. Wells and Z. Zhang,  _Effective theories of universal theories_ ,  JHEP 01, 123 (2016),  10.1007/JHEP01(2016)123,  1510.08462.
  * [38] M. Farina, G. Panico, D. Pappadopulo, J. T. Ruderman, R. Torre and A. Wulzer,  _Energy helps accuracy: electroweak precision tests at hadron colliders_ ,  Phys. Lett. B 772, 210 (2017),  10.1016/j.physletb.2017.06.043,  1609.08157.
  * [39] J. Brehmer, A. Freitas, D. Lopez-Val and T. Plehn,  _Pushing Higgs Effective Theory to its Limits_ ,  Phys. Rev. D 93(7), 075014 (2016),  10.1103/PhysRevD.93.075014,  1510.03443.
  * [40] J. Fuentes-Martín, M. König, J. Pagès, A. E. Thomsen and F. Wilsch,  _A Proof of Concept for Matchete: An Automated Tool for Matching Effective Theories_ (2022),  2212.04510.
  * [41] M. N. Costantini, E. Hammou, Z. Kassabov, M. Madigan, L. Mantani, M. Morales Alvarado, J. M. Moore and M. Ubiali,  _SIMUnet: an open-source tool for simultaneous global fits of EFT Wilson coefficients and PDFs_ (2024),  2402.03308.
  * [42] M. Madigan,  _Can PDFs absorb new physics?_ ,  PoS EPS-HEP2023, 479 (2024),  10.22323/1.449.0479.
  * [43] E. Hammou, Z. Kassabov, M. Madigan, M. L. Mangano, L. Mantani, J. Moore, M. M. Alvarado and M. Ubiali,  _Hide and seek: how PDFs can conceal new physics_ ,  JHEP 11, 090 (2023),  10.1007/JHEP11(2023)090,  2307.10370.
  * [44] S. Carrazza, C. Degrande, S. Iranipour, J. Rojo and M. Ubiali,  _Can New Physics hide inside the proton?_ ,  Phys. Rev. Lett. 123(13), 132001 (2019),  10.1103/PhysRevLett.123.132001,  1905.05215.
  * [45] A. Greljo, S. Iranipour, Z. Kassabov, M. Madigan, J. Moore, J. Rojo, M. Ubiali and C. Voisey,  _Parton distributions in the SMEFT from high-energy Drell-Yan tails_ ,  JHEP 07, 122 (2021),  10.1007/JHEP07(2021)122,  2104.02723.
  * [46] A. Alloul, N. D. Christensen, C. Degrande, C. Duhr and B. Fuks,  _FeynRules 2.0 - A complete toolbox for tree-level phenomenology_ ,  Comput. Phys. Commun. 185, 2250 (2014),  10.1016/j.cpc.2014.04.012,  1310.1921.
  * [47] T. Hahn,  _Generating Feynman diagrams and amplitudes with FeynArts 3_ ,  Comput. Phys. Commun. 140, 418 (2001),  10.1016/S0010-4655(01)00290-9,  hep-ph/0012260.
  * [48] T. Hahn and M. Perez-Victoria,  _Automatized one loop calculations in four-dimensions and D-dimensions_ ,  Comput. Phys. Commun. 118, 153 (1999),  10.1016/S0010-4655(98)00173-8,  hep-ph/9807565.
  * [49] A. M. Sirunyan _et al._ ,  _Search for resonant and nonresonant new phenomena in high-mass dilepton final states at $\sqrt{s}$ = 13 TeV_,  JHEP 07, 208 (2021),  10.1007/JHEP07(2021)208,  2103.02708.
  * [50] J. de Blas, J. C. Criado, M. Perez-Victoria and J. Santiago,  _Effective description of general extensions of the Standard Model: the complete tree-level dictionary_ ,  JHEP 03, 109 (2018),  10.1007/JHEP03(2018)109,  1711.10391.
  * [51] I. Banta, T. Cohen, N. Craig, X. Lu and D. Sutherland,  _Effective Field Theory of the Two Higgs Doublet Model_ (2023),  2304.09884.
  * [52] C. Hays, A. Helset, A. Martin and M. Trott,  _Exact SMEFT formulation and expansion to $\mathcal{O}(v^{4}/\Lambda^{4})$_,  JHEP 11, 087 (2020),  10.1007/JHEP11(2020)087,  2007.00565.
  * [53] S. Dawson, D. Fontes, S. Homiller and M. Sullivan,  _Role of dimension-eight operators in an EFT for the 2HDM_ ,  Phys. Rev. D 106(5), 055012 (2022),  10.1103/PhysRevD.106.055012,  2205.01561.
  * [54] B. Henning, X. Lu and H. Murayama,  _How to use the Standard Model effective field theory_ ,  JHEP 01, 023 (2016),  10.1007/JHEP01(2016)023,  1412.1837.
  * [55] T. Corbett, O. J. P. Eboli, J. Gonzalez-Fraile and M. C. Gonzalez-Garcia,  _Robust Determination of the Higgs Couplings: Power to the Data_ ,  Phys. Rev. D 87, 015022 (2013),  10.1103/PhysRevD.87.015022,  1211.4580.
  * [56] T. Corbett, O. J. P. Eboli, D. Goncalves, J. Gonzalez-Fraile, T. Plehn and M. Rauch,  _The Higgs Legacy of the LHC Run I_ ,  JHEP 08, 156 (2015),  10.1007/JHEP08(2015)156,  1505.05516.
  * [57] A. Biekötter, T. Corbett and T. Plehn,  _The Gauge-Higgs Legacy of the LHC Run II_ ,  SciPost Phys. 6(6), 064 (2019),  10.21468/SciPostPhys.6.6.064,  1812.07587.
  * [58] J. Campbell and T. Neumann,  _Precision Phenomenology with MCFM_ ,  JHEP 12, 034 (2019),  10.1007/JHEP12(2019)034,  1909.09117.
  * [59] S. Dawson, P. P. Giardino and A. Ismail,  _Standard model EFT and the Drell-Yan process at high energy_ ,  Phys. Rev. D 99(3), 035044 (2019),  10.1103/PhysRevD.99.035044,  1811.12260.
  * [60] S. Alioli, W. Dekens, M. Girard and E. Mereghetti,  _NLO QCD corrections to SM-EFT dilepton and electroweak Higgs boson production, matched to parton shower in POWHEG_ ,  JHEP 08, 205 (2018),  10.1007/JHEP08(2018)205,  1804.07407.
  * [61] S. Rappoccio,  _The experimental status of direct searches for exotic physics beyond the standard model at the Large Hadron Collider_ ,  Rev. Phys. 4, 100027 (2019),  10.1016/j.revip.2018.100027,  1810.10579.


# MelodyT5: A Unified Score-to-Score Transformer for Symbolic Music Processing

###### Abstract

In the domain of symbolic music research, the progress of developing scalable
systems has been notably hindered by the scarcity of available training data
and the demand for models tailored to specific tasks. To address these issues,
we propose MelodyT5, a novel unified framework that leverages an encoder-
decoder architecture tailored for symbolic music processing in ABC notation.
This framework challenges the conventional task-specific approach, considering
various symbolic music tasks as score-to-score transformations. Consequently,
it integrates seven melody-centric tasks, from generation to harmonization and
segmentation, within a single model. Pre-trained on MelodyHub, a newly curated
collection featuring over 261K unique melodies encoded in ABC notation and
encompassing more than one million task instances, MelodyT5 demonstrates
superior performance in symbolic music processing via multi-task transfer
learning. Our findings highlight the efficacy of multi-task transfer learning
in symbolic music processing, particularly for data-scarce tasks, challenging
the prevailing task-specific paradigms and offering a comprehensive dataset
and framework for future explorations in this domain.

##  1 Introduction

In the field of artificial intelligence, symbolic music processing—including
the analysis and generation of musical scores—presents a unique challenge that
merges musical creativity with computational complexity. Symbolic music, which
represents musical information with discrete symbols rather than continuous
audio signals, facilitates the precise manipulation and analysis of elements
such as melody, harmony, and rhythm. Historically, the application of AI in
this area has sought not only to mimic the creative process of human composers
[1, 2, 3, 4] but also to uncover the underlying patterns of musical
composition [5, 6, 7].

Despite significant progress, the field still faces persistent limitations.
One notable challenge is the prevalence of task-specific models [8, 9, 10,
11]. These models offer benefits for specific applications but lack
adaptability to the broader spectrum of symbolic music processing. This
fragmentation is further compounded by the scarcity of annotated datasets [12,
13, 14], which serve as the lifeblood of deep learning models. Unlike other
domains where data may be abundant and easy-to-collect, annotated symbolic
music datasets are both rare and costly to produce. Without access to ample
and diverse data, models struggle to generalize and may exhibit biases or
limitations [15] in their analysis and generation of symbolic music.

In addressing the challenges inherent to symbolic music processing, insights
from the Natural Language Processing (NLP) domain offer a promising avenue for
advancement. Techniques such as transfer learning [16, 17, 18] and multi-task
learning [19, 20, 21] have played a pivotal role in advancing NLP by promoting
the transfer of knowledge from pre-trained language models and exploiting
common patterns across various tasks. Prominent models like GPT [22], BERT
[23], and T5 [24] demonstrate the efficacy of these strategies in
understanding and generating language across diverse contexts. Notably, the T5
model, with its text-to-text framework, mirrors the conceptual shift necessary
for symbolic music by treating all tasks as variations of converting input
scores to output scores. By embracing such methodologies, which regard tasks
as facets of a unified problem, we seek to develop models for symbolic music
that not only excel in specific tasks but are also adaptable and proficient
across a wide range of tasks.

Figure 1: The MelodyT5 framework employs a Transformer encoder-decoder
architecture with bar patching for music processing. It uses linear projection
of input bar patches, fed into a patch-level Transformer encoder. The encoder
output provides context for a patch-level Transformer decoder to
autoregressively produce target bar features. A character-level Transformer
decoder then uses these features to generate detailed characters for each bar,
forming the target musical score.

In this paper, we introduce MelodyT5, which leverages an encoder-decoder
framework to perform multiple symbolic music tasks as unified score-to-score
transformations. Pre-trained on the MelodyHub dataset, which contains over 1
million task instances across seven melody-centric tasks in ABC notation,
MelodyT5 overcomes the limitations of task-specific models and sparse data
availability in symbolic music processing. By implementing bar patching [7,
25], MelodyT5 can handle longer sequences effectively, expanding its
applicability to a wider range of tasks while maintaining computational
efficiency. Our results underscore the promise of employing multi-task
learning approaches in symbolic music processing, demonstrating the superior
performance of MelodyT5 across a spectrum of tasks and providing a rich
dataset for future research. The key contributions of our paper are as
follows:

  * •

MelodyT5, employing an encoder-decoder approach, redefines symbolic music
processing by including multiple tasks as unified score-to-score
transformations, demonstrating versatility and breaking traditional task-
specific constraints.

  * •

MelodyHub, a dataset comprising 261,900 unique melodies in ABC notation across
over 1 million task instances for seven different tasks, serves as the
cornerstone for effective pre-training of MelodyT5.

  * •

Our experimental results demonstrate the efficacy of multi-task transfer
learning in symbolic music, with models trained across multiple tasks
outperforming those trained in isolation.

##  2 Methodology

In this section, we delve into the methodology behind MelodyT5. We first
introduce the ABC notation and bar patching for music representation, then
present the architectural design of MelodyT5, and finally outline the pre-
training objective of our model, which focuses on score-to-score
transformations as the basis for multi-task learning.

###  2.1 Data Representation

We utilize ABC notation, a concise symbolic music format, for encoding musical
scores with ASCII characters. This text-based format elegantly represents
musical elements like notes, rhythms, and articulations in a human-readable
manner, thereby facilitating thorough music documentation. Additionally, it
promotes the utilization of NLP techniques for both music analysis and
generation, as evidenced by recent studies [4, 26, 27].

To process musical scores encoded in ABC notation more efficiently, we
implement the bar patching technique [7, 25]. Bar patching involves breaking
down musical sequences into units called bar patches. Each of these units
corresponds to either a bar or an information field (such as key and meter),
including a sequence of characters that represent musical symbols within that
patch. Unlike the conventional character-level or token-level tokenization of
ABC notation [28, 29], where individual characters or tokens are processed
independently, bar patching groups multiple characters into cohesive semantic
units. Typically, each patch comprises 10 or more tokens, thus effectively
reducing the overall sequence length of musical scores.

###  2.2 Model Architecture

As shown in Fig. 1, the MelodyT5 framework employs an encoder-decoder
architecture based on the Transformer network [30], tailored for symbolic
music processing. Integrating bar patching into MelodyT5 requires the
incorporation of two additional components: a linear projection layer and a
character-level decoder. Consequently, the model architecture encompasses the
following modules:

Table 1: The MelodyHub collection statistics include the number of instances
for each task along with the corresponding data sources. The JSB Chorales
dataset is augmented to 15 keys due to its small size and original data in the
C key.

Data Sources | Cataloging | Generation | Harmonization | Melodization | Segmentation | Transcription | Variation
---|---|---|---|---|---|---|---
ABC Notation [31] | 184,660 | 184,738 | 31,732 | 31,690 | —— | 174,779 | ——
FolkWiki [32] | 6,610 | 6,767 | 1,207 | 1,205 | —— | 6,218 | ——
JSB Chorales [33] | 4,980 | 4,980 | 4,950 | 4,950 | 19,125 | 4,980 | ——
KernScores [34] | 1,731 | 1,776 | —— | —— | 1,275 | 1,754 | ——
Meertens Tune Collections [35] | 16,662 | 16,662 | —— | —— | 16,660 | 16,297 | ——
Nottingham [36] | 1,031 | 1,031 | 1,014 | 1,014 | —— | 1,021 | ——
OpenScore Lieder [37] | 1,326 | 1,326 | —— | —— | —— | 1,255 | ——
The Session [38] | 44,620 | 44,620 | 3,081 | 3,078 | —— | 42,838 | 174,104
Total | 261,620 | 261,900 | 41,984 | 41,937 | 37,060 | 249,142 | 174,104

Linear Projection: This component converts each bar patch into a dense
embedding. It takes a multi-hot vector as input, formed by concatenating one-
hot vectors representing characters within the bar patch with the shape
$S\times V$, where $S$ represents the patch size (i.e., the maximum number of
characters in a patch) and $V$ represents the vocabulary size. If a patch
contains fewer than $S$ characters, it will be padded with a special token to
make it a $S$-character patch. The vector is then mapped to a dense embedding,
serving as input to the patch-level encoder or decoder.

Patch-level Encoder: It is responsible for generating contextualized
representations to understand the input musical score by operating on the
dense embeddings produced by the linear projection layer. Leveraging
mechanisms like self-attention and feed-forward neural networks, it captures
global dependencies within the input musical score.

Patch-level Decoder: Tasked with generating the dense representation of the
next bar patch, the patch-level decoder utilizes contextualized
representations from the encoder and patch embeddings of previously generated
content. It employs cross-attention and autoregressive generation mechanisms,
ensuring global coherence and continuity in the sequence of generated bar
patches.

Character-level Decoder: Operating in a step-by-step manner, the character-
level decoder produces characters within the next bar patch based on the dense
representation generated by the patch-level decoder. By utilizing the dense
representation as a context vector, it decodes each character within the bar,
focusing on local information, and sequentially reconstructs every bar patch
until it completes the generation of the target musical score.

The encoder-decoder architecture with bar patching in MelodyT5 enables
efficient score-to-score transformations by hierarchically modelling music at
both patch and character levels, capturing global structure and local details
inherent in compositions.

###  2.3 Pre-training Objective

The pre-training objective of MelodyT5 aims to optimize a unified encoder-
decoder framework for processing and generating symbolic music across a
variety of tasks, utilizing cross-entropy loss for next token prediction.

We consider a dataset $D$ consisting of pairs $(X,Y)$, where $X$ is an input
musical score and $Y$ is the target musical score. Each score is represented
as a sequence of bar patches $\\{B_{1},B_{2},\ldots,B_{n}\\}$, with each bar
patch $B_{i}$ further decomposed into a sequence of characters
$\\{c_{1},c_{2},\ldots,c_{m}\\}$. The model is trained to predict each token
(i.e., character) of the target score given the input score and the previously
generated tokens in an autoregressive manner.

Formally, the pre-training objective can be represented as minimizing the
cross-entropy loss across all tokens in the target sequence:

$\mathcal{L}(\theta)=-\sum_{(X,Y)\in D}\sum_{i=1}^{n}\sum_{j=1}^{m}\log
P_{\theta}(c_{j}^{i}|X,B_{<i},c_{<j}^{i})$ (1)

where $c_{j}^{i}$ is the $j$-th character in the $i$-th bar patch of score
$Y$, $B_{<i}$ includes all bar patches before the $i$-th, $c_{<j}^{i}$ are
characters before the $j$-th in the current patch, and $P_{\theta}$ is the
probability of the model, parameterized by $\theta$, of predicting the correct
character.

This objective incorporates the fundamental principle that the vast majority
of symbolic music tasks can be considered as transformations from score to
score, or, in other words, from an input musical score to a target musical
score. By pre-training on this objective, MelodyT5 acquires the ability to
understand and replicate a wide array of patterns and structures inherent to
different music tasks, which is pivotal for its success across various
applications within symbolic music processing.

##  3 Dataset

This section outlines the melody curation and task definition of the MelodyHub
dataset. MelodyHub, crucial for training MelodyT5, comprises seven melody-
centric tasks. This collection, sourced from sheet music datasets, includes
folk songs and other non-copyrighted musical scores from various traditions
and epochs.

###  3.1 Melody Curation

The MelodyHub dataset was curated using publicly available sheet music
datasets and online platforms, with original formats like ABC notation,
MusicXML, and Humdrum. The data curation process included several steps:

  1. 1.

Entries featuring explicit copyright indicators such as “copyright” or “©”
symbols were excluded.

  2. 2.

All data was converted to MusicXML format for standardization and subsequently
transformed into ABC notation to ensure format consistency.

  3. 3.

Melodies consisting of fewer than eight bars were omitted from the dataset to
maintain adequate complexity and musical richness.

  4. 4.

Removal of lyrics and non-musical content (e.g., contact information of
transcribers and URL links) aimed to focus solely on musical notation.

  5. 5.

Leading and trailing bars of complete rest were removed from each piece.

  6. 6.

Each piece underwent verification for the presence of a final barline, with
addition if absent.

  7. 7.

Entries were deduplicated to prevent redundancy.

By ensuring the quality and consistency of the MelodyHub dataset, these steps
led to a substantial collection of 261,900 melodies with uniform formatting,
making it suitable for training and evaluating symbolic music models like
MelodyT5.

###  3.2 Task Definition

Following the curation of melody data, the MelodyHub dataset was segmented
into seven tasks, as summarized in Table 1, presented in a score-to-score
format with input-output pairs. In MelodyHub, every input data includes a task
identifier (e.g., %%harmonization) at the outset to specify the intended task.
Below are the definitions of these tasks:

Cataloging: This task selects melodies with music-related metadata like
titles, composers, and geographical origins (e.g., C:J.S. Bach, O:Germany).
The input data includes information fields with these attributes, while
specific information is removed and the order is randomized. The output
includes the corresponding metadata without the musical score.

Generation: Here, the input solely consists of a task identifier (i.e.,
%%generation), while the output comprises comprehensive musical scores.
Following TunesFormer [25], control codes are affixed to all melodies as
information fields to denote musical structure information. These codes,
namely S:, B:, and E:, signify the number of sections, bars per section, and
edit distance similarity between every pair of sections within the tune.

Harmonization: This task involves melodies containing chord symbols. The chord
symbols are removed from the input, while the original data is retained as the
output. An additional information field denoting edit distance similarity (E:)
is appended to the output, indicating the similarity between the input and
output, ranging from 0 to 10 (no match at all to exact match). Lower
similarity values suggest the need for more chord symbols.

Melodization: In contrast to harmonization, this task operates inversely and
also employs melodies containing chord symbols. The notes in the original
score are replaced with rests, and adjacent rest durations are combined. The
resultant score, comprising rests and chord symbols, serves as the input.
Similar to harmonization, an E: field is added at the outset of the output,
with lower values facilitating the generation of more intricate melodies.

Segmentation: Melodies in Humdrum format (i.e., KernScores and Meertens Tune
Collections) containing curly braces indicating segmentation or voices from
the JSB Chorales dataset (four-part compositions) with fermatas are chosen.
These markers are transformed into breath marks. The input data omits all
breath marks, while the output introduces an E: field at the beginning to aid
the generation of breath marks, with lower values implying the need for more
breath marks to be added.

Transcription: ABC notation is initially converted to MIDI, then reconverted
back to ABC. The resultant ABC from the MIDI conversion loses substantial
score information, such as distinguishing enharmonic equivalents and missing
musical ornaments (e.g., trill). The MIDI-converted ABC serves as the input,
while the original ABC, appended with an added E: field, constitutes the
output. Lower E: values denote greater discrepancies between the transcribed
and input scores, particularly due to absent repeat symbols.

Variation: This task centres on data from The Session, wherein each ABC
notation file may contain multiple variants of the same tune. Tunes with two
or more variations are selected, with every possible pair of variants utilized
as both input and output. The output initiates with an E: field signifying the
extent of disparities between the input and output scores, with lower values
suggesting substantial variations in the musical scores.

Together, resulting in 1,067,747 task instances in total, these tasks include
various MIR challenges from analytical to generative, providing a
comprehensive resource111https://huggingface.co/datasets/sander-wood/melodyhub
for developing symbolic music models like MelodyT5.

##  4 Experiments

This section evaluates the effectiveness of MelodyT5 in symbolic music
processing through a series of experiments. It outlines experimental settings,
conducts ablation studies on multi-task learning impact, and compares MelodyT5
with baseline models in various tasks.

###  4.1 Settings

The experiments are structured to systematically assess the capabilities of
MelodyT5 for diverse symbolic music tasks. We utilize the MelodyHub dataset,
which is randomly split into 99% for training and 1% for validation.

MelodyT5 features a 9-layer patch-level encoder and decoder with shared
weights, a 3-layer character-level decoder, and a hidden size of 768,
amounting to 113 million parameters. This configuration processes ABC
sequences up to 16,384 characters, with a 256 patch length and a 64 patch
size. It employs a 128-size ASCII-based vocabulary, using characters 0-2 for
special tokens (pad, bos, and eos).

The AdamW optimizer [39] is used, setting a learning rate of 2e-4. The process
includes a 3-epoch warmup, a constant learning rate over 32 epochs, and a
batch size of 10 for each GPU, ensuring consistency in hyperparameter settings
across all tasks. It took approximately 2 days to complete the pre-training
using 6 RTX 3090 GPUs.

In ablation studies, we investigate the effects of multi-task learning on
MelodyT5, considering three settings: 1) omitting pre-training, 2) using only
the downstream task-specific data from MelodyHub, or 3) utilizing the entire
MelodyHub dataset, which includes all tasks.

In terms of comparative evaluations, we select open-source models that excel
in their respective domains for benchmarking. MelodyT5 is fine-tuned on
identical datasets to these models, ensuring fairness in comparison. For
models trained on proprietary datasets, we retrain them using accessible
datasets to ensure reproducibility.

Our objective evaluation strategy includes ablation studies focused on bits-
per-byte (BPB) for consistent measurement, alongside task-specific metrics for
comparative evaluations. Additionally, A/B tests are conducted for the
subjective evaluation against baseline models.

Table 2: Experimental results from ablation studies illustrate the impact of
multi-task learning on diverse symbolic music tasks, evaluated through BPB
(bits-per-byte) to compare performance across various pre-training settings.

Pre-training | Cataloging | Generation | Harmonization | Melodization | Segmentation | Transcription | Variation
---|---|---|---|---|---|---|---
WikiMT [40] | Wikifonia [41] | CMD [42] | EWLD [43] | Essen [44] | Liederschatz [45] | The Session [38]
None | 0.0376 | 1.2382 | 0.5680 | 0.7949 | 0.0272 | 1.1938 | 0.4932
Task-Specific | 0.0379 | 0.8850 | 0.3393 | 0.6322 | 0.0224 | 0.3432 | –
Multi-Task | 0.0350 | 0.8472 | 0.2925 | 0.5067 | 0.0119 | 0.2969 | 0.3949

###  4.2 Ablation Studies

In our ablation studies, MelodyT5 was evaluated on the test sets of various
symbolic music benchmarks. Due to the lack of a directly suitable external
dataset for the variation task, we chose to evaluate using the validation set
of The Session. As a result, there was no task-specific pre-training for the
variation task.

The ablation studies aim to explore two aspects: 1) the overall efficacy of
pre-training, particularly in the context of multi-task pre-training versus
task-specific pre-training, and 2) the extent to which performance gains from
multi-task pre-training vary among different tasks, especially considering
differences in the available volume of pre-training data across these tasks.

The ablation studies, as depicted in Table 2, show that pre-training is
crucial for improving the performance of symbolic music tasks. Models trained
with pre-training consistently outperform those without, indicating that pre-
training enhances model generalization and performance. Multi-task pre-
training is also superior to task-specific pre-training, as models trained
with multi-task pre-training show lower BPB scores. This highlights the
importance of leveraging multi-task pre-training to effectively capture shared
patterns and structures in symbolic music data, enabling MelodyT5 to
generalize better to downstream tasks.

Furthermore, it is noteworthy that while multi-task pre-training consistently
yields performance gains across most tasks, the extent of improvement varies,
which significantly correlates with the volume of task-specific data available
for pre-training. Specifically, tasks with less data, such as segmentation and
melodization, showcase more substantial performance gains from multi-task
learning. On the other hand, tasks with more data, like generation and
cataloging, though still benefiting from multi-task pre-training, show
relatively smaller improvements. This observation suggests that while multi-
task learning enhances model performance across the board, its impact is
especially notable in data-constrained scenarios.

In summary, the ablation studies demonstrate the effectiveness of multi-task
learning and underscore the impact of data volume on the benefits derived from
such an approach. Multi-task learning boosts model performance across symbolic
music tasks and provides notable advantages for tasks with limited data by
leveraging shared knowledge across tasks.

###  4.3 Comparative Evaluations

For comparative evaluations, we compare MelodyT5, which is multi-task pre-
trained on MelodyHub, with several task-specific baseline models, focusing on
melody generation, harmonization, melodization, and segmentation. These tasks
are well-established and have open-source models as competitive baselines. The
following baseline models have been selected for comparison:

  * •

TunesFormer [25] is applied for melody generation, featuring a Transformer-
based architecture with bar patching and control codes. This approach aims to
refine the efficiency of the generation process and ensure adherence to
musical forms.

  * •

STHarm [46] is utilized as the baseline in melody harmonization, employing a
Transformer framework to convert melodies into chords. Its primary focus is on
creating harmonies that preserve the structural integrity of the original
melody.

  * •

CMT [9] is chosen for melodization, which involves generating melodies based
on chord progressions. It employs a phased training approach, conditioning the
generation of rhythm and pitch on the chords to produce dynamic and coherent
musical outputs.

  * •

Bi-LSTM-CRF [8] is used for melody segmentation, integrating Bi-LSTM and CRF
to effectively identify and segment melodic phrases for music structure
analysis.

Table 3: Comparative objective evaluation of the MelodyT5 model against task-
specific baselines across various symbolic music tasks, utilizing task-related
metrics previously established. The baselines include TunesFormer [25] for
generation, STHarm [46] for harmonization, CMT [9] for melodization, and Bi-
LSTM-CRF [8] for segmentation.

Models | Generation | Harmonization | Melodization | Segmentation
---|---|---|---|---
CTRL↑ | CTnCTR↑ | PCS↑ | MCTD↓ | CTnCTR↑ | PCS↑ | MCTD↓ | F1 Score↑
MelodyT5 | 0.8664 | 0.7108 | 0.3274 | 1.2080 | 0.8438 | 0.5084 | 1.0320 | 0.9055
Baselines | 0.8162 | 0.5963 | 0.2343 | 1.3125 | 0.8607 | 0.4863 | 1.0610 | 0.8400

For an objective and quantifiable performance assessment that ensures
reproducibility, we leverage previously established task-specific metrics. The
selected metrics for our assessment include:

  * •

CTRL (Controllability) [25]: Evaluates the precision of generation control
through edit distance similarity between intended and actual control codes.

  * •

CTnCTR & PCS & MCTD [47]: These chord/melody harmonicity metrics evaluate
harmonization and melodization tasks by assessing harmonic and melodic
compatibility between melodies and chords.

  * •

F1 Score: Measures the balance between precision and recall in identifying
correctly segmented melodic phrases.

Table 3 shows that MelodyT5 outperforms task-specific baselines in all tasks.
It surpasses the specialized baseline TunesFormer in melody generation,
demonstrating enhanced control and precision in generating melodies according
to specific musical forms. MelodyT5 leads in harmonization, producing chords
that are harmonically compatible with the given melodies while maintaining
structural coherence. Although slightly trailing CMT in CTnCTR, it still shows
robust performance in other metrics for melodization, demonstrating its
ability to generate melodies well integrated with chord progressions. Its
performance in melody segmentation is significant, indicating its ability to
accurately discern and segment melodic phrases. This performance, achieved
without task-specific modifications, highlights the effectiveness of multi-
task transfer learning combined with unified score-to-score transformations in
symbolic music processing.

In addition to the objective metrics presented in Table 3, we recognize the
limitations of solely relying on such measures to evaluate the quality of
generated music. Thus, we further explored these areas (i.e., generation,
harmonization, and melodization) through subjective experiments to capture
listener preferences.

For our subjective evaluation, we randomly chose 30 pieces from the test set
for each task and conducted blind A/B testing. Participants were presented
with one randomly chosen pair from each of these 30 pairs to compare musical
scores generated by MelodyT5 and baseline models under identical conditions.
They were asked to choose between MelodyT5, the baseline, or no preference.
Each comparison was included in two videos, showcasing both the audio and the
Sibelius-rendered musical scores.

For generation, we compared the quality of melodies generated by MelodyT5 and
TunesFormer, given the same control codes and information fields. In
harmonization and melodization, comparisons were made against baselines given
identical melodies or chords, respectively. To ensure fairness, especially
considering the baseline model for melodization was limited to generating
outputs of only 8 bars, we trimmed the MelodyT5-generated scores to match the
output length of this baseline model.

The study involved 155 responses from students and educators with music
specializations, ensuring deep understanding of melody and harmony. To secure
data reliability, submissions were filtered out of those completed in less
than half the overall average duration of 4 minutes and 39 seconds, i.e.,
those under 2 minutes and 20 seconds. This resulted in a final tally of 124
valid questionnaires.

Figure 2: Comparative subjective evaluation of MelodyT5 against task-specific
baselines in symbolic music tasks, showing vote counts for each model.

Based on the subjective evaluation in Fig. 2, we observe a notable preference
for the MelodyT5 model over the baseline in the tasks of melody generation and
harmonization, with MelodyT5 receiving a higher number of votes. However, the
preferences reverse in the task of melodization, where the baseline model
receives a greater number of votes compared to MelodyT5. This indicates that
the baseline model CMT, which employs a two-phase training process focusing
separately on rhythm and pitch conditioned on chord progressions, may align
more closely with human rhythmic tendencies in melodization, leading to a
preference for its outputs in the subjective evaluation.

Overall, MelodyT5 excels in symbolic music processing, outperforming task-
specific models in most tasks and demonstrating the effectiveness of multi-
task transfer learning in this domain, despite occasional shortcomings.

##  5 Conclusions

This study presents MelodyT5, a model addressing challenges in symbolic music
processing by providing a unified framework for diverse tasks. By treating
music tasks as score-to-score transformations, MelodyT5 significantly improves
symbolic music processing through multi-task transfer learning. Objective and
subjective evaluations demonstrate that MelodyT5 generally outperforms or
matches task-specific baseline models without modification. The MelodyHub
dataset, with over one million task instances, offers a rich resource for
training and evaluating models. While excelling in melody-centric tasks,
further optimization is required to tackle more complex musical compositions,
such as polyphonic arrangements.

##  6 Ethics Statement

In our research, we are committed to upholding ethical standards regarding
data privacy and copyright protection. We diligently adhere to these
principles throughout our data collection and processing procedures. All data
utilized in our study is sourced from publicly accessible repositories, and we
have taken measures to exclude any known copyrighted materials.

##  7 Acknowlegdements

We would like to express our sincere gratitude to Associate Professor Bob L.
T. Sturm from KTH Royal Institute of Technology for his valuable discussions
and insights into the early stages of the project. We also acknowledge
Yuanliang Dong and Jiafeng Liu from the Central Conservatory of Music for
their assistance, and Leqi Peng from Fuyin Technology for her support with
subjective experiments.

This work was supported by the following funding sources: Special Program of
National Natural Science Foundation of China (Grant No. T2341003), Advanced
Discipline Construction Project of Beijing Universities, Major Program of
National Social Science Fund of China (Grant No. 21ZD19), and the National
Culture and Tourism Technological Innovation Engineering Project (Research and
Application of 3D Music).

## References

  * [1] J. Liu, Y. Dong, Z. Cheng, X. Zhang, X. Li, F. Yu, and M. Sun, “Symphony generation with permutation invariant language model,” in _Proceedings of the 23rd International Society for Music Information Retrieval Conference, ISMIR 2022, Bengaluru, India, December 4-8, 2022_ , 2022, pp. 551–558. [Online]. Available: https://archives.ismir.net/ismir2022/paper/000066.pdf
  * [2] P. Lu, X. Tan, B. Yu, T. Qin, S. Zhao, and T. Liu, “Meloform: Generating melody with musical form based on expert systems and neural networks,” in _Proceedings of the 23rd International Society for Music Information Retrieval Conference, ISMIR 2022, Bengaluru, India, December 4-8, 2022_ , 2022, pp. 567–574. [Online]. Available: https://archives.ismir.net/ismir2022/paper/000068.pdf
  * [3] L. Min, J. Jiang, G. Xia, and J. Zhao, “Polyffusion: A diffusion model for polyphonic score generation with internal and external controls,” in _Proceedings of the 24th International Society for Music Information Retrieval Conference, ISMIR 2023, Milan, Italy, November 5-9, 2023_ , 2023, pp. 231–238. [Online]. Available: https://doi.org/10.5281/zenodo.10265265
  * [4] S. Wu and M. Sun, “Exploring the efficacy of pre-trained checkpoints in text-to-music generation task,” in _The AAAI-23 Workshop on Creative AI Across Modalities_ , 2023. [Online]. Available: https://openreview.net/forum?id=QmWXskBhesn
  * [5] M. Zeng, X. Tan, R. Wang, Z. Ju, T. Qin, and T. Liu, “Musicbert: Symbolic music understanding with large-scale pre-training,” in _Findings of the Association for Computational Linguistics: ACL/IJCNLP 2021, Online Event, August 1-6, 2021_ , ser. Findings of ACL, vol. ACL/IJCNLP 2021. Association for Computational Linguistics, 2021, pp. 791–800. [Online]. Available: https://doi.org/10.18653/v1/2021.findings-acl.70
  * [6] Z. Wang and G. Xia, “Musebert: Pre-training music representation for music understanding and controllable generation,” in _Proceedings of the 22nd International Society for Music Information Retrieval Conference, ISMIR 2021, Online, November 7-12, 2021_ , 2021, pp. 722–729. [Online]. Available: https://archives.ismir.net/ismir2021/paper/000090.pdf
  * [7] S. Wu, D. Yu, X. Tan, and M. Sun, “Clamp: Contrastive language-music pre-training for cross-modal symbolic music information retrieval,” in _Proceedings of the 24th International Society for Music Information Retrieval Conference, ISMIR 2023, Milan, Italy, November 5-9, 2023_ , 2023, pp. 157–165. [Online]. Available: https://doi.org/10.5281/zenodo.10265247
  * [8] Y. Zhang and G. Xia, “Symbolic melody phrase segmentation using neural network with conditional random field,” in _Proceedings of the 8th Conference on Sound and Music Technology: Selected Papers from CSMT_. Springer, 2021, pp. 55–65.
  * [9] K. Choi, J. Park, W. Heo, S. Jeon, and J. Park, “Chord conditioned melody generation with transformer based decoders,” _IEEE Access_ , vol. 9, pp. 42 071–42 080, 2021. [Online]. Available: https://doi.org/10.1109/ACCESS.2021.3065831
  * [10] S. Wu, X. Li, and M. Sun, “Chord-conditioned melody harmonization with controllable harmonicity,” in _IEEE International Conference on Acoustics, Speech and Signal Processing ICASSP 2023, Rhodes Island, Greece, June 4-10, 2023_. IEEE, 2023, pp. 1–5. [Online]. Available: https://doi.org/10.1109/ICASSP49357.2023.10096398
  * [11] S. Wu, Y. Yang, Z. Wang, X. Li, and M. Sun, “Generating chord progression from melody with flexible harmonic rhythm and controllable harmonic density,” _EURASIP J. Audio Speech Music. Process._ , vol. 2024, no. 1, p. 4, 2024. [Online]. Available: https://doi.org/10.1186/s13636-023-00314-6
  * [12] Z. Wang, K. Chen, J. Jiang, Y. Zhang, M. Xu, S. Dai, and G. Xia, “POP909: A pop-song dataset for music arrangement generation,” in _Proceedings of the 21th International Society for Music Information Retrieval Conference, ISMIR 2020, Montreal, Canada, October 11-16, 2020_ , 2020, pp. 38–45. [Online]. Available: http://archives.ismir.net/ismir2020/paper/000089.pdf
  * [13] Y. Hsiao, T. Hung, T. Chen, and L. Su, “Bps-motif: A dataset for repeated pattern discovery of polyphonic symbolic music,” in _Proceedings of the 24th International Society for Music Information Retrieval Conference, ISMIR 2023, Milan, Italy, November 5-9, 2023_ , 2023, pp. 281–288. [Online]. Available: https://doi.org/10.5281/zenodo.10265277
  * [14] Y. Zhang, Z. Zhou, X. Li, F. Yu, and M. Sun, “Ccom-huqin: An annotated multimodal chinese fiddle performance dataset,” _Trans. Int. Soc. Music. Inf. Retr._ , vol. 6, no. 1, pp. 60–74, 2023. [Online]. Available: https://doi.org/10.5334/tismir.146
  * [15] A. Holzapfel, B. L. Sturm, and M. Coeckelbergh, “Ethical dimensions of music information retrieval technology,” _Trans. Int. Soc. Music. Inf. Retr._ , vol. 1, no. 1, pp. 44–55, 2018. [Online]. Available: https://doi.org/10.5334/tismir.13
  * [16] J. Howard and S. Ruder, “Universal language model fine-tuning for text classification,” in _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics, ACL 2018, Melbourne, Australia, July 15-20, 2018, Volume 1: Long Papers_. Association for Computational Linguistics, 2018, pp. 328–339. [Online]. Available: https://aclanthology.org/P18-1031/
  * [17] S. Ruder, M. E. Peters, S. Swayamdipta, and T. Wolf, “Transfer learning in natural language processing,” in _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, NAACL-HLT 2019, Minneapolis, MN, USA, June 2, 2019, Tutorial Abstracts_. Association for Computational Linguistics, 2019, pp. 15–18. [Online]. Available: https://doi.org/10.18653/v1/n19-5004
  * [18] M. Artetxe and H. Schwenk, “Massively multilingual sentence embeddings for zero-shot cross-lingual transfer and beyond,” _Trans. Assoc. Comput. Linguistics_ , vol. 7, pp. 597–610, 2019. [Online]. Available: https://doi.org/10.1162/tacl\\_a\\_00288
  * [19] X. Liu, P. He, W. Chen, and J. Gao, “Multi-task deep neural networks for natural language understanding,” in _Proceedings of the 57th Conference of the Association for Computational Linguistics, ACL 2019, Florence, Italy, July 28- August 2, 2019, Volume 1: Long Papers_. Association for Computational Linguistics, 2019, pp. 4487–4496. [Online]. Available: https://doi.org/10.18653/v1/p19-1441
  * [20] K. Song, X. Tan, T. Qin, J. Lu, and T. Liu, “MASS: masked sequence to sequence pre-training for language generation,” in _Proceedings of the 36th International Conference on Machine Learning, ICML 2019, 9-15 June 2019, Long Beach, California, USA_ , ser. Proceedings of Machine Learning Research, vol. 97. PMLR, 2019, pp. 5926–5936. [Online]. Available: http://proceedings.mlr.press/v97/song19d.html
  * [21] Z. Zhang, W. Yu, M. Yu, Z. Guo, and M. Jiang, “A survey of multi-task learning in natural language processing: Regarding task relatedness and training methods,” in _Proceedings of the 17th Conference of the European Chapter of the Association for Computational Linguistics, EACL 2023, Dubrovnik, Croatia, May 2-6, 2023_. Association for Computational Linguistics, 2023, pp. 943–956. [Online]. Available: https://doi.org/10.18653/v1/2023.eacl-main.66
  * [22] A. Radford, K. Narasimhan, T. Salimans, I. Sutskever _et al._ , “Improving language understanding by generative pre-training,” 2018.
  * [23] J. Devlin, M. Chang, K. Lee, and K. Toutanova, “BERT: pre-training of deep bidirectional transformers for language understanding,” in _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, NAACL-HLT 2019, Minneapolis, MN, USA, June 2-7, 2019, Volume 1 (Long and Short Papers)_. Association for Computational Linguistics, 2019, pp. 4171–4186. [Online]. Available: https://doi.org/10.18653/v1/n19-1423
  * [24] C. Raffel, N. Shazeer, A. Roberts, K. Lee, S. Narang, M. Matena, Y. Zhou, W. Li, and P. J. Liu, “Exploring the limits of transfer learning with a unified text-to-text transformer,” _J. Mach. Learn. Res._ , vol. 21, pp. 140:1–140:67, 2020. [Online]. Available: http://jmlr.org/papers/v21/20-074.html
  * [25] S. Wu, X. Li, F. Yu, and M. Sun, “Tunesformer: Forming irish tunes with control codes by bar patching,” in _Proceedings of the 2nd Workshop on Human-Centric Music Information Retrieval 2023 co-located with the 24th International Society for Music Information Retrieval Conference (ISMIR 2023), Milan, Italy, November 10, 2023_ , ser. CEUR Workshop Proceedings, vol. 3528. CEUR-WS.org, 2023. [Online]. Available: https://ceur-ws.org/Vol-3528/paper1.pdf
  * [26] L. Casini, N. Jonason, and B. L. Sturm, “Generating folk-like music in abc-notation with masked language models,” in _Ismir 2023 Hybrid Conference_ , 2023.
  * [27] R. Yuan, H. Lin, Y. Wang, Z. Tian, S. Wu, T. Shen, G. Zhang, Y. Wu, C. Liu, Z. Zhou, Z. Ma, L. Xue, Z. Wang, Q. Liu, T. Zheng, Y. Li, Y. Ma, Y. Liang, X. Chi, R. Liu, Z. Wang, P. Li, J. Wu, C. Lin, Q. Liu, T. Jiang, W. Huang, W. Chen, E. Benetos, J. Fu, G. Xia, R. B. Dannenberg, W. Xue, S. Kang, and Y. Guo, “Chatmusician: Understanding and generating music intrinsically with LLM,” _CoRR_ , vol. abs/2402.16153, 2024. [Online]. Available: https://doi.org/10.48550/arXiv.2402.16153
  * [28] B. L. Sturm, J. F. Santos, O. Ben-Tal, and I. Korshunova, “Music transcription modelling and composition using deep learning,” _CoRR_ , vol. abs/1604.08723, 2016. [Online]. Available: http://arxiv.org/abs/1604.08723
  * [29] C. Geerlings and A. Meroño-Peñuela, “Interacting with gpt-2 to generate controlled and believable musical sequences in abc notation,” in _NLP4MUSA_ , 2020. [Online]. Available: https://api.semanticscholar.org/CorpusID:227217204
  * [30] A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, L. Kaiser, and I. Polosukhin, “Attention is all you need,” in _Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA_ , 2017, pp. 5998–6008. [Online]. Available: https://proceedings.neurips.cc/paper/2017/hash/3f5ee243547dee91fbd053c1c4a845aa-Abstract.html
  * [31] “Abc notation,” http://abcnotation.com/, accessed: 2024-04-12.
  * [32] “Folkwiki,” http://www.folkwiki.se/, accessed: 2024-04-12.
  * [33] “Chord-conditioned melody harmonization with controllable harmonicity [icassp 2023],” https://github.com/sander-wood/deepchoir, accessed: 2024-04-12.
  * [34] “Kernscores,” http://kern.ccarh.org/, accessed: 2024-04-12.
  * [35] “The meertens tune collections,” https://www.liederenbank.nl/mtc/, accessed: 2024-04-12.
  * [36] “The nottingham music database,” https://ifdo.ca/~seymour/nottingham/nottingham.html, accessed: 2024-04-12.
  * [37] “Openscore lieder corpus,” https://musescore.com/openscore-lieder-corpus, accessed: 2024-04-12.
  * [38] “The session,” https://thesession.org/, accessed: 2024-04-12.
  * [39] I. Loshchilov and F. Hutter, “Decoupled weight decay regularization,” in _7th International Conference on Learning Representations, ICLR 2019, New Orleans, LA, USA, May 6-9, 2019_. OpenReview.net, 2019. [Online]. Available: https://openreview.net/forum?id=Bkg6RiCqY7
  * [40] “wikimusictext Dataset on Hugging Face Datasets,” https://huggingface.co/datasets/sander-wood/wikimusictext, accessed: 2024-04-01.
  * [41] “Download for Wikifonia all 6,675 Lead Sheets - Synth Zone Forum,” http://www.synthzone.com/forum/ubbthreads.php/topics/384909/Download_for_Wikifonia_all_6,6, accessed: 2024-04-01.
  * [42] “chord-melody-dataset on GitHub,” https://github.com/shiehn/chord-melody-dataset, accessed: 2024-04-01.
  * [43] “OpenEWLD on GitHub,” https://github.com/00sapo/OpenEWLD, accessed: 2024-04-01.
  * [44] “KernScores: Essen Folksong Collection,” http://kern.ccarh.org/cgi-bin/ksbrowse?l=/essen, accessed: 2024-04-01.
  * [45] “KernScores: Erk’s Liederschatz,” https://kern.humdrum.org/cgi-bin/browse?l=users/craig/songs/erk/liederschatz, accessed: 2024-04-01.
  * [46] S. Rhyu, H. Choi, S. Kim, and K. Lee, “Translating melody to chord: Structured and flexible harmonization of melody with transformer,” _IEEE Access_ , vol. 10, pp. 28 261–28 273, 2022. [Online]. Available: https://doi.org/10.1109/ACCESS.2022.3155467
  * [47] Y. Yeh, W. Hsiao, S. Fukayama, T. Kitahara, B. Genchel, H. Liu, H. Dong, Y. Chen, T. Leong, and Y. Yang, “Automatic melody harmonization with triad chords: A comparative study,” _CoRR_ , vol. abs/2001.02360, 2020. [Online]. Available: http://arxiv.org/abs/2001.02360