Dear Andy, Bill, Jim, and Colby, Please let me share with you some information and ideas about the valid tuning range (VTR) for has been termed the "schismatic" mapping, and two related mappings for intervals defined by chains of 7-10 generators. These three mappings together cover the full regular diatonic continuum from 7-TET to 5-TET. In fact, the concept of intervals from chains of "7-10 generators" might be the best general and descriptive way to categorize these diverse mappings. From this viewpoint, we might speak simply of a "lower" mapping (7-TET to 19-TET); a "middle" mapping (19-TET to 17-TET); and an "upper" mapping (17-TET to 5-TET). The transitions are points where the ordering of interval sizes changes. The first table shows the "Middle" or "Schismatic" mapping: ------------------------------------------------------------------------------ Middle or "Schismatic" mapping, 7-10 generators (19-TET to 17-TET) ------------------------------------------------------------------------------ Intvl Example 19-TET 696 697 698 699 700 701 702 703 704 705 17-TET ------------------------------------------------------------------------------ Apot C-C# 63 72 79 86 93 100 107 114 121 128 135 141 dim3 C#-Eb 252 240 230 220 210 200 190 180 170 160 150 141 Aug2 Eb-F# 252 264 273 282 291 300 309 318 327 336 345 353 dim4 C#-F 442 432 424 416 408 400 392 384 376 368 360 353 Aug5 F-C# 758 766 776 784 792 800 808 816 824 832 840 847 dim7 F#-Eb 948 936 927 918 909 900 891 882 873 864 855 847 Aug6 Eb-C# 948 960 970 980 990 1000 1010 1020 1030 1040 1050 1059 dim8 C#-C 1137 1128 1121 1114 1107 1100 1093 1086 1079 1072 1065 1059 ------------------------------------------------------------------------------ The next shows the "Upper" or "Ultraschismatic" mapping: ----------------------------------------------------------------------------- Upper or "Ultraschismatic" mapping, 7-10 generators (17-TET to 5-TET) ------------------------------------------------------------------------------> Intvl Example 17-TET 707 708 22-TET 710 711 712 713 714 715 716 717 ------------------------------------------------------------------------------> dim3 C#-Eb 141 130 120 109 100 90 80 70 60 50 40 30 Apot C-C# 141 149 156 164 170 177 184 191 198 205 212 219 dim4 C#-F 353 344 336 327 320 312 304 296 288 280 272 264 Aug2 Eb-F# 353 363 372 382 390 399 408 417 426 435 444 453 dim7 F#-Eb 847 837 828 818 810 801 792 783 774 765 756 747 Aug5 F-C# 847 856 864 873 880 888 896 904 912 920 928 936 dim8 C#-C 1059 1051 1044 1036 1030 1023 1016 1009 1002 995 988 981 Aug6 Eb-C# 1059 1070 1080 1091 1100 1110 1120 1130 1140 1150 1160 1170 ------------------------------------------------------------------------------> ------------------------------- Intvl Example 718 719 720 ------------------------------- dim3 C#-Eb 20 10 0 Apot C-C# 226 233 240 dim4 C#-F 256 248 240 Aug2 Eb-F# 462 471 480 dim7 F#-Eb 738 729 728 Aug5 F-C# 944 952 960 dim8 C#-C 974 967 960 Aug6 Eb-C# 1180 1190 1280 ------------------------------- The final table shows the "Lower" or "Infraschismatic" mapping. ----------------------------------------------------------------------- Lower or "Infraschimatic" mapping, 7-10 generators (7-TET to 19-TET) ----------------------------------------------------------------------- Intvl Example 7-TET 687 688 689 690 691 692 693 694 19-TET ----------------------------------------------------------------------- Apot C-C# 0 9 16 23 38 37 44 51 58 63 Aug2 Eb-F# 171 183 192 201 210 219 228 237 246 252 dim3 C#-Eb 343 330 320 310 300 290 280 270 260 252 dim4 C#-F 514 504 496 488 480 472 464 456 448 442 Aug5 F-C# 686 696 704 712 720 728 736 744 752 758 Aug6 Eb-C# 857 870 880 890 900 910 920 930 940 948 dim7 F#-Eb 1029 1017 1008 999 990 981 972 963 954 948 dim8 C#-C 1200 1191 1184 1177 1170 1163 1156 1149 1142 1137 ----------------------------------------------------------------------- In a much longer article, I've analyzed at length some of the characteristics and uses of these mappings. My purpose here is mainly to demonstrate the VTR of each, and also the wealth of interval sizes and categories supported by each of the mappings. With many thanks, Margo