Delta-rational triads in small edos

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Shown below are approximate delta-rational triads in edos < 24 with naive least-squares error < 0.00810. All delta signatures are in the 3-integer limit.

edo steps delta signature naive least-squares error symmetric least-squares error
6 0, 1, 2 +1+1 0.00671 0.05262
6 0, 2, 5 +1+2 0.00064 0.00255
6 0, 4, 5 +3+1 0.00084 0.00439
7 0, 1, 2 +1+1 0.00485 0.04494
7 0, 2, 5 +1+2 0.00518 0.02473
7 0, 4, 5 +3+1 0.00439 0.02767
7 0, 4, 6 +3+2 0.00075 0.00468
8 0, 1, 2 +1+1 0.00366 0.03923
8 0, 4, 7 +1+1 0.00250 0.00649
8 0, 1, 3 +1+2 0.00801 0.08197
8 0, 2, 5 +1+2 0.00804 0.04497
8 0, 2, 3 +2+1 0.00723 0.07438
8 0, 4, 5 +3+1 0.00604 0.04470
8 0, 4, 6 +3+2 0.00441 0.03250
9 0, 1, 2 +1+1 0.00287 0.03481
9 0, 4, 7 +1+1 0.00317 0.00941
9 0, 1, 3 +1+2 0.00624 0.07286
9 0, 2, 3 +2+1 0.00562 0.06586
9 0, 5, 7 +2+1 0.00548 0.02377
9 0, 4, 5 +3+1 0.00679 0.05773
9 0, 4, 6 +3+2 0.00716 0.06098
10 0, 1, 2 +1+1 0.00230 0.03128
10 0, 4, 7 +1+1 0.00649 0.02168
10 0, 1, 3 +1+2 0.00500 0.06557
10 0, 1, 4 +1+3 0.00786 0.09929
10 0, 2, 7 +1+3 0.00721 0.04684
10 0, 2, 3 +2+1 0.00449 0.05909
10 0, 5, 7 +2+1 0.00177 0.00868
10 0, 3, 4 +3+1 0.00679 0.08635
10 0, 4, 5 +3+1 0.00708 0.06802
10 0, 6, 9 +3+2 0.00336 0.01975
11 0, 1, 2 +1+1 0.00189 0.02841
11 0, 2, 4 +1+1 0.00807 0.05755
11 0, 1, 3 +1+2 0.00410 0.05961
11 0, 4, 10 +1+2 0.00565 0.02016
11 0, 1, 4 +1+3 0.00643 0.09038
11 0, 2, 7 +1+3 0.00416 0.03036
11 0, 2, 3 +2+1 0.00367 0.05359
11 0, 5, 7 +2+1 0.00062 0.00342
11 0, 3, 4 +3+1 0.00553 0.07822
11 0, 4, 5 +3+1 0.00712 0.07636
11 0, 8, 10 +3+1 0.00231 0.01080
11 0, 6, 9 +3+2 0.00135 0.00892
12 0, 1, 2 +1+1 0.00158 0.02602
12 0, 2, 4 +1+1 0.00671 0.05262
12 0, 5, 9 +1+1 0.00542 0.01684
12 0, 1, 3 +1+2 0.00342 0.05465
12 0, 3, 8 +1+2 0.00625 0.03266
12 0, 4, 10 +1+2 0.00064 0.00255
12 0, 1, 4 +1+3 0.00535 0.08294
12 0, 2, 7 +1+3 0.00205 0.01661
12 0, 3, 10 +1+3 0.00606 0.03163
12 0, 2, 3 +2+1 0.00306 0.04903
12 0, 5, 7 +2+1 0.00219 0.01335
12 0, 8, 11 +2+1 0.00369 0.01302
12 0, 3, 4 +3+1 0.00459 0.07149
12 0, 4, 5 +3+1 0.00703 0.08326
12 0, 8, 10 +3+1 0.00084 0.00439
12 0, 2, 3 +3+2 0.00766 0.19843
12 0, 6, 9 +3+2 0.00441 0.03250
13 0, 1, 2 +1+1 0.00134 0.02400
13 0, 2, 4 +1+1 0.00566 0.04848
13 0, 5, 9 +1+1 0.00217 0.00737
13 0, 7, 12 +1+1 0.00389 0.00949
13 0, 1, 3 +1+2 0.00290 0.05045
13 0, 3, 8 +1+2 0.00366 0.02103
13 0, 4, 10 +1+2 0.00279 0.01219
13 0, 5, 12 +1+2 0.00644 0.02247
13 0, 1, 4 +1+3 0.00453 0.07663
13 0, 2, 7 +1+3 0.00056 0.00496
13 0, 3, 10 +1+3 0.00255 0.01472
13 0, 2, 3 +2+1 0.00259 0.04518
13 0, 5, 7 +2+1 0.00324 0.02166
13 0, 8, 11 +2+1 0.00014 0.00053
13 0, 3, 7 +2+3 0.00697 0.07780
13 0, 4, 9 +2+3 0.00800 0.06578
13 0, 3, 4 +3+1 0.00387 0.06582
13 0, 4, 5 +3+1 0.00687 0.08905
13 0, 7, 9 +3+1 0.00758 0.05018
13 0, 8, 10 +3+1 0.00296 0.01701
13 0, 2, 3 +3+2 0.00725 0.20459
13 0, 6, 9 +3+2 0.00642 0.05226
13 0, 8, 12 +3+2 0.00491 0.02791
14 0, 1, 2 +1+1 0.00115 0.02228
14 0, 2, 4 +1+1 0.00485 0.04494
14 0, 5, 9 +1+1 0.00016 0.00059
14 0, 7, 12 +1+1 0.00759 0.02013
14 0, 1, 3 +1+2 0.00249 0.04685
14 0, 3, 8 +1+2 0.00177 0.01110
14 0, 4, 10 +1+2 0.00518 0.02473
14 0, 1, 4 +1+3 0.00388 0.07122
14 0, 2, 7 +1+3 0.00052 0.00505
14 0, 3, 10 +1+3 0.00004 0.00023
14 0, 4, 13 +1+3 0.00663 0.03015
14 0, 2, 3 +2+1 0.00222 0.04190
14 0, 5, 7 +2+1 0.00395 0.02871
14 0, 8, 11 +2+1 0.00280 0.01191
14 0, 2, 5 +2+3 0.00767 0.13817
14 0, 3, 7 +2+3 0.00516 0.06277
14 0, 4, 9 +2+3 0.00516 0.04637
14 0, 5, 11 +2+3 0.00807 0.05654
14 0, 3, 4 +3+1 0.00330 0.06099
14 0, 4, 5 +3+1 0.00668 0.09400
14 0, 7, 9 +3+1 0.00548 0.03965
14 0, 8, 10 +3+1 0.00439 0.02767
14 0, 2, 3 +3+2 0.00687 0.20986
14 0, 3, 5 +3+2 0.00721 0.13014
14 0, 6, 9 +3+2 0.00776 0.06905
14 0, 8, 12 +3+2 0.00075 0.00468
15 0, 1, 2 +1+1 0.00100 0.02078
15 0, 2, 4 +1+1 0.00419 0.04189
15 0, 5, 9 +1+1 0.00184 0.00738
15 0, 8, 14 +1+1 0.00677 0.01640
15 0, 1, 3 +1+2 0.00216 0.04373
15 0, 3, 8 +1+2 0.00037 0.00252
15 0, 4, 10 +1+2 0.00685 0.03554
15 0, 1, 4 +1+3 0.00336 0.06652
15 0, 2, 7 +1+3 0.00131 0.01374
15 0, 3, 10 +1+3 0.00179 0.01233
15 0, 4, 13 +1+3 0.00275 0.01366
15 0, 2, 3 +2+1 0.00192 0.03906
15 0, 5, 7 +2+1 0.00442 0.03478
15 0, 7, 10 +2+1 0.00806 0.04226
15 0, 8, 11 +2+1 0.00468 0.02160
15 0, 1, 2 +2+3 0.00795 0.39746
15 0, 2, 5 +2+3 0.00663 0.12894
15 0, 3, 7 +2+3 0.00378 0.04976
15 0, 4, 9 +2+3 0.00303 0.02958
15 0, 5, 11 +2+3 0.00471 0.03593
15 0, 3, 4 +3+1 0.00286 0.05682
15 0, 4, 5 +3+1 0.00647 0.09826
15 0, 7, 9 +3+1 0.00390 0.03063
15 0, 8, 10 +3+1 0.00537 0.03679
15 0, 2, 3 +3+2 0.00652 0.21442
15 0, 3, 5 +3+2 0.00622 0.12129
15 0, 5, 8 +3+2 0.00724 0.08190
15 0, 8, 12 +3+2 0.00224 0.01523
16 0, 1, 2 +1+1 0.00088 0.01947
16 0, 2, 4 +1+1 0.00366 0.03923
16 0, 5, 9 +1+1 0.00308 0.01325
16 0, 6, 11 +1+1 0.00752 0.02588
16 0, 8, 14 +1+1 0.00250 0.00649
16 0, 1, 3 +1+2 0.00189 0.04101
16 0, 2, 6 +1+2 0.00801 0.08197
16 0, 3, 8 +1+2 0.00068 0.00497
16 0, 4, 10 +1+2 0.00804 0.04497
16 0, 6, 15 +1+2 0.00781 0.02682
16 0, 1, 4 +1+3 0.00294 0.06240
16 0, 2, 7 +1+3 0.00189 0.02135
16 0, 3, 10 +1+3 0.00314 0.02334
16 0, 4, 13 +1+3 0.00014 0.00075
16 0, 2, 3 +2+1 0.00168 0.03658
16 0, 4, 6 +2+1 0.00723 0.07438
16 0, 5, 7 +2+1 0.00473 0.04006
16 0, 7, 10 +2+1 0.00601 0.03397
16 0, 8, 11 +2+1 0.00601 0.02997
16 0, 11, 15 +2+1 0.00029 0.00101
16 0, 1, 2 +2+3 0.00749 0.40058
16 0, 2, 5 +2+3 0.00579 0.12087
16 0, 3, 7 +2+3 0.00271 0.03838
16 0, 4, 9 +2+3 0.00141 0.01491
16 0, 5, 11 +2+3 0.00217 0.01794
16 0, 6, 13 +2+3 0.00526 0.03524
16 0, 3, 4 +3+1 0.00249 0.05319
16 0, 4, 5 +3+1 0.00625 0.10198
16 0, 7, 9 +3+1 0.00269 0.02281
16 0, 8, 10 +3+1 0.00604 0.04470
16 0, 12, 15 +3+1 0.00369 0.01658
16 0, 2, 3 +3+2 0.00620 0.21840
16 0, 3, 5 +3+2 0.00542 0.11356
16 0, 5, 8 +3+2 0.00572 0.06977
16 0, 8, 12 +3+2 0.00441 0.03250
16 0, 10, 15 +3+2 0.00593 0.03303
17 0, 1, 2 +1+1 0.00077 0.01832
17 0, 2, 4 +1+1 0.00323 0.03688
17 0, 3, 6 +1+1 0.00757 0.05581
17 0, 5, 9 +1+1 0.00399 0.01838
17 0, 6, 11 +1+1 0.00521 0.01921
17 0, 8, 14 +1+1 0.00072 0.00201
17 0, 1, 3 +1+2 0.00167 0.03860
17 0, 2, 6 +1+2 0.00704 0.07714
17 0, 3, 8 +1+2 0.00148 0.01158
17 0, 5, 13 +1+2 0.00652 0.02881
17 0, 6, 15 +1+2 0.00376 0.01392
17 0, 1, 4 +1+3 0.00259 0.05876
17 0, 2, 7 +1+3 0.00231 0.02808
17 0, 3, 10 +1+3 0.00414 0.03306
17 0, 4, 13 +1+3 0.00232 0.01346
17 0, 5, 16 +1+3 0.00377 0.01684
17 0, 2, 3 +2+1 0.00148 0.03439
17 0, 4, 6 +2+1 0.00635 0.06986
17 0, 5, 7 +2+1 0.00493 0.04469
17 0, 7, 10 +2+1 0.00440 0.02673
17 0, 8, 11 +2+1 0.00696 0.03727
17 0, 11, 15 +2+1 0.00309 0.01141
17 0, 1, 2 +2+3 0.00708 0.40333
17 0, 2, 5 +2+3 0.00510 0.11375
17 0, 3, 7 +2+3 0.00187 0.02836
17 0, 4, 9 +2+3 0.00018 0.00199
17 0, 5, 11 +2+3 0.00024 0.00210
17 0, 6, 13 +2+3 0.00227 0.01643
17 0, 7, 15 +2+3 0.00654 0.03941
17 0, 3, 4 +3+1 0.00220 0.04999
17 0, 4, 5 +3+1 0.00604 0.10524
17 0, 7, 9 +3+1 0.00175 0.01597
17 0, 8, 10 +3+1 0.00650 0.05162
17 0, 12, 15 +3+1 0.00111 0.00540
17 0, 13, 16 +3+1 0.00716 0.03181
17 0, 1, 2 +3+2 0.00803 0.49486
17 0, 2, 3 +3+2 0.00591 0.22191
17 0, 3, 5 +3+2 0.00477 0.10676
17 0, 5, 8 +3+2 0.00452 0.05911
17 0, 7, 11 +3+2 0.00795 0.07119
17 0, 8, 12 +3+2 0.00599 0.04762
17 0, 10, 15 +3+2 0.00225 0.01353
18 0, 1, 2 +1+1 0.00069 0.01730
18 0, 2, 4 +1+1 0.00287 0.03481
18 0, 3, 6 +1+1 0.00671 0.05262
18 0, 5, 9 +1+1 0.00467 0.02291
18 0, 6, 11 +1+1 0.00340 0.01335
18 0, 8, 14 +1+1 0.00317 0.00941
18 0, 10, 17 +1+1 0.00672 0.01597
18 0, 1, 3 +1+2 0.00148 0.03646
18 0, 2, 6 +1+2 0.00624 0.07286
18 0, 3, 8 +1+2 0.00209 0.01744
18 0, 5, 13 +1+2 0.00403 0.01907
18 0, 6, 15 +1+2 0.00064 0.00255
18 0, 7, 17 +1+2 0.00117 0.00398
18 0, 1, 4 +1+3 0.00230 0.05553
18 0, 2, 7 +1+3 0.00263 0.03407
18 0, 3, 10 +1+3 0.00488 0.04172
18 0, 4, 13 +1+3 0.00398 0.02477
18 0, 5, 16 +1+3 0.00059 0.00285
18 0, 2, 3 +2+1 0.00132 0.03246
18 0, 4, 6 +2+1 0.00562 0.06586
18 0, 5, 7 +2+1 0.00505 0.04879
18 0, 7, 10 +2+1 0.00314 0.02036
18 0, 8, 11 +2+1 0.00763 0.04370
18 0, 10, 14 +2+1 0.00548 0.02377
18 0, 11, 15 +2+1 0.00519 0.02051
18 0, 1, 2 +2+3 0.00672 0.40577
18 0, 2, 5 +2+3 0.00452 0.10742
18 0, 3, 7 +2+3 0.00120 0.01945
18 0, 4, 9 +2+3 0.00078 0.00948
18 0, 5, 11 +2+3 0.00126 0.01196
18 0, 6, 13 +2+3 0.00003 0.00023
18 0, 7, 15 +2+3 0.00310 0.02007
18 0, 3, 4 +3+1 0.00195 0.04715
18 0, 4, 5 +3+1 0.00583 0.10814
18 0, 7, 9 +3+1 0.00102 0.00992
18 0, 8, 10 +3+1 0.00679 0.05773
18 0, 11, 14 +3+1 0.00674 0.03857
18 0, 12, 15 +3+1 0.00084 0.00439
18 0, 1, 2 +3+2 0.00753 0.49220
18 0, 2, 3 +3+2 0.00564 0.22502
18 0, 3, 5 +3+2 0.00423 0.10073
18 0, 5, 8 +3+2 0.00356 0.04967
18 0, 7, 11 +3+2 0.00607 0.05818
18 0, 8, 12 +3+2 0.00716 0.06098
18 0, 10, 15 +3+2 0.00056 0.00364
19 0, 1, 2 +1+1 0.00062 0.01638
19 0, 2, 4 +1+1 0.00256 0.03295
19 0, 3, 6 +1+1 0.00598 0.04979
19 0, 5, 9 +1+1 0.00517 0.02693
19 0, 6, 11 +1+1 0.00196 0.00817
19 0, 8, 14 +1+1 0.00505 0.01591
19 0, 9, 16 +1+1 0.00687 0.01871
19 0, 1, 3 +1+2 0.00133 0.03454
19 0, 2, 6 +1+2 0.00557 0.06902
19 0, 3, 8 +1+2 0.00255 0.02268
19 0, 4, 11 +1+2 0.00709 0.04386
19 0, 5, 13 +1+2 0.00206 0.01040
19 0, 6, 15 +1+2 0.00178 0.00755
19 0, 7, 17 +1+2 0.00428 0.01554
19 0, 1, 3 +1+3 0.00799 0.27146
19 0, 1, 4 +1+3 0.00206 0.05263
19 0, 2, 7 +1+3 0.00287 0.03943
19 0, 3, 10 +1+3 0.00543 0.04948
19 0, 3, 11 +1+3 0.00755 0.06184
19 0, 4, 13 +1+3 0.00524 0.03488
19 0, 5, 16 +1+3 0.00188 0.00966
19 0, 2, 3 +2+1 0.00118 0.03073
19 0, 4, 6 +2+1 0.00501 0.06229
19 0, 5, 7 +2+1 0.00511 0.05244
19 0, 7, 10 +2+1 0.00213 0.01470
19 0, 8, 11 +2+1 0.00810 0.04942
19 0, 10, 14 +2+1 0.00341 0.01578
19 0, 11, 15 +2+1 0.00676 0.02853
19 0, 1, 2 +2+3 0.00639 0.40796
19 0, 2, 5 +2+3 0.00404 0.10176
19 0, 3, 7 +2+3 0.00067 0.01148
19 0, 4, 9 +2+3 0.00153 0.01974
19 0, 5, 11 +2+3 0.00241 0.02451
19 0, 6, 13 +2+3 0.00182 0.01512
19 0, 7, 15 +2+3 0.00041 0.00282
19 0, 8, 17 +2+3 0.00446 0.02639
19 0, 3, 4 +3+1 0.00174 0.04462
19 0, 4, 5 +3+1 0.00563 0.11072
19 0, 6, 8 +3+1 0.00759 0.09109
19 0, 7, 9 +3+1 0.00044 0.00455
19 0, 8, 10 +3+1 0.00698 0.06316
19 0, 11, 14 +3+1 0.00491 0.02998
19 0, 12, 15 +3+1 0.00234 0.01304
19 0, 1, 2 +3+2 0.00708 0.48982
19 0, 2, 3 +3+2 0.00540 0.22780
19 0, 3, 5 +3+2 0.00378 0.09534
19 0, 5, 8 +3+2 0.00278 0.04125
19 0, 7, 11 +3+2 0.00456 0.04660
19 0, 8, 12 +3+2 0.00802 0.07286
19 0, 10, 15 +3+2 0.00273 0.01888
19 0, 12, 18 +3+2 0.00665 0.03654
20 0, 1, 2 +1+1 0.00056 0.01556
20 0, 2, 4 +1+1 0.00230 0.03128
20 0, 3, 6 +1+1 0.00537 0.04724
20 0, 5, 9 +1+1 0.00553 0.03054
20 0, 6, 11 +1+1 0.00080 0.00356
20 0, 8, 14 +1+1 0.00649 0.02168
20 0, 9, 16 +1+1 0.00403 0.01164
20 0, 11, 19 +1+1 0.00166 0.00392
20 0, 1, 3 +1+2 0.00119 0.03282
20 0, 2, 6 +1+2 0.00500 0.06557
20 0, 3, 8 +1+2 0.00291 0.02739
20 0, 4, 11 +1+2 0.00569 0.03735
20 0, 5, 13 +1+2 0.00049 0.00262
20 0, 6, 15 +1+2 0.00368 0.01658
20 0, 7, 17 +1+2 0.00670 0.02586
20 0, 1, 3 +1+3 0.00764 0.27369
20 0, 1, 4 +1+3 0.00185 0.05002
20 0, 2, 7 +1+3 0.00304 0.04427
20 0, 2, 8 +1+3 0.00786 0.09929
20 0, 3, 10 +1+3 0.00584 0.05647
20 0, 3, 11 +1+3 0.00626 0.05448
20 0, 4, 13 +1+3 0.00622 0.04400
20 0, 4, 14 +1+3 0.00721 0.04684
20 0, 5, 16 +1+3 0.00381 0.02091
20 0, 6, 19 +1+3 0.00177 0.00780
20 0, 2, 3 +2+1 0.00106 0.02917
20 0, 4, 6 +2+1 0.00449 0.05909
20 0, 5, 7 +2+1 0.00513 0.05572
20 0, 7, 10 +2+1 0.00132 0.00964
20 0, 10, 14 +2+1 0.00177 0.00868
20 0, 11, 15 +2+1 0.00795 0.03566
20 0, 13, 18 +2+1 0.00683 0.02473
20 0, 14, 19 +2+1 0.00271 0.00912
20 0, 1, 2 +2+3 0.00609 0.40993
20 0, 1, 3 +2+3 0.00795 0.37204
20 0, 2, 5 +2+3 0.00363 0.09666
20 0, 3, 7 +2+3 0.00024 0.00432
20 0, 4, 9 +2+3 0.00212 0.02896
20 0, 5, 11 +2+3 0.00332 0.03580
20 0, 6, 13 +2+3 0.00323 0.02848
20 0, 7, 15 +2+3 0.00171 0.01266
20 0, 8, 17 +2+3 0.00139 0.00875
20 0, 9, 19 +2+3 0.00623 0.03408
20 0, 3, 4 +3+1 0.00156 0.04235
20 0, 4, 5 +3+1 0.00543 0.11304
20 0, 6, 8 +3+1 0.00679 0.08635
20 0, 7, 9 +3+1 0.00002 0.00027
20 0, 8, 10 +3+1 0.00708 0.06802
20 0, 11, 14 +3+1 0.00343 0.02234
20 0, 12, 15 +3+1 0.00350 0.02075
20 0, 1, 2 +3+2 0.00669 0.48768
20 0, 2, 3 +3+2 0.00517 0.23031
20 0, 3, 5 +3+2 0.00339 0.09050
20 0, 5, 8 +3+2 0.00214 0.03370
20 0, 7, 11 +3+2 0.00334 0.03623
20 0, 9, 14 +3+2 0.00743 0.06029
20 0, 10, 15 +3+2 0.00441 0.03250
20 0, 12, 18 +3+2 0.00336 0.01975
21 0, 1, 2 +1+1 0.00050 0.01482
21 0, 2, 4 +1+1 0.00208 0.02978
21 0, 3, 6 +1+1 0.00485 0.04494
21 0, 5, 9 +1+1 0.00580 0.03378
21 0, 6, 11 +1+1 0.00013 0.00059
21 0, 7, 13 +1+1 0.00716 0.02783
21 0, 8, 14 +1+1 0.00760 0.02684
21 0, 9, 16 +1+1 0.00176 0.00535
21 0, 11, 19 +1+1 0.00146 0.00364
21 0, 1, 3 +1+2 0.00108 0.03125
21 0, 2, 5 +1+2 0.00800 0.13198
21 0, 2, 6 +1+2 0.00452 0.06245
21 0, 3, 8 +1+2 0.00318 0.03166
21 0, 4, 11 +1+2 0.00453 0.03147
21 0, 5, 13 +1+2 0.00077 0.00439
21 0, 6, 15 +1+2 0.00518 0.02473
21 0, 1, 3 +1+3 0.00731 0.27570
21 0, 1, 4 +1+3 0.00168 0.04765
21 0, 2, 7 +1+3 0.00316 0.04865
21 0, 2, 8 +1+3 0.00709 0.09463
21 0, 3, 10 +1+3 0.00613 0.06280
21 0, 3, 11 +1+3 0.00519 0.04782
21 0, 4, 13 +1+3 0.00696 0.05226
21 0, 4, 14 +1+3 0.00554 0.03821
21 0, 5, 16 +1+3 0.00534 0.03109
21 0, 6, 19 +1+3 0.00093 0.00435
21 0, 2, 3 +2+1 0.00096 0.02777
21 0, 4, 6 +2+1 0.00405 0.05621
21 0, 5, 7 +2+1 0.00512 0.05867
21 0, 7, 10 +2+1 0.00066 0.00509
21 0, 10, 14 +2+1 0.00045 0.00232
21 0, 13, 18 +2+1 0.00425 0.01631
21 0, 14, 19 +2+1 0.00492 0.01756
21 0, 1, 2 +2+3 0.00581 0.41170
21 0, 1, 3 +2+3 0.00750 0.36939
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