00001 #include "MusimatChapter9.h" 00002 MusimatChapter9Section(C091102) { 00003 Print("*** 9.11.2 The Set Complex ***"); 00004 /***************************************************************************** 00005 00006 9.11.2 The Set Complex 00007 00008 Using these tools, we can create a matrix containing the prime form, inverse, 00009 retrograde, and all transpositions of any row, called the set complex. The 00010 purpose of these transformations is to generate variants that are related to 00011 the original intervallic structure of the prime row, to be used as material 00012 in developing compositions. 00013 00014 Matrix is simply a two-dimensional grid, or list of lists, all of the same 00015 length. The individual elements of Matrix can be accessed by extending the 00016 index operator […]. The first operand is the matrix, the second is the row 00017 position, and the third is the column position. (Whether row or column comes 00018 first is arbitrary. The following order is called row/column order.) 00019 00020 0 1 00021 M = 0 A B 00022 1 C D 00023 00024 For example, for this matrix, M[0][0] == A, M[0][1] == B, M[1][0] == C, and 00025 M[1][1] == D. The following is a method for creating a set complex. It 00026 basically copies the prime form to the zeroth row, then copies the inverse 00027 form to the zeroth column, then for each other cell in the matrix sums the 00028 corresponding row and column value modulo the length of the prime form: 00029 *****************************************************************************/ 00030 para1(); // Step into this function to continue. 00031 para2(); // Step into this function to continue. 00032 } 00033 00034 IntegerMatrix setComplex(IntegerList prime) { 00035 Print("Prime row:", prime); 00036 Integer len = Length(prime); 00037 IntegerMatrix M(len, len); 00038 IntegerList inverted = invert(prime); 00039 Print("Inverted row:", inverted); 00040 For (Integer i = 0; i < len; i = i + 1) { 00041 M[0][i] = prime[i]; 00042 M[i][0] = inverted[i]; 00043 } 00044 For (Integer i = 1; i < len; i = i + 1) { 00045 For (Integer j = 1; j < len; j = j + 1) { 00046 Integer a = M[i][0]; 00047 Integer b = M[0][j]; 00048 // M(i,j) = Mod(M(i,0) + M(0,j), len); 00049 M[i][j] = Mod(a + b, len); 00050 } 00051 } 00052 Return(M); 00053 } 00054 00055 00056 Static Void para1() { 00057 /***************************************************************************** 00058 To demonstrate these tools, table 9.4 shows the set complex the following row: 00059 (G, As, B, F, E, Cs, C, A, Gs, D, Ds, Fs) 00060 o Prime rows are read left to right, 00061 o Retrograde rows, right to left 00062 o Inverse rows, top to bottom, 00063 o Retrograde inverse rows, bottom to top. 00064 This completes the precomposition phase. Now it's time to look at methods to 00065 traverse the rows created with the preceding techniques to generate a 00066 composition. 00067 00068 Table 9.4 Set Complex 00069 {0, 3, 4, 10, 9, 6, 5, 2, 1, 7, 8, 11} 00070 {9, 0, 1, 7, 6, 3, 2, 11, 10, 4, 5, 8} 00071 {8, 11, 0, 6, 5, 2, 1, 10, 9, 3, 4, 7} 00072 {2, 5, 6, 0, 11, 8, 7, 4, 3, 9, 10, 1} 00073 {3, 6, 7, 1, 0, 9, 8, 5, 4, 10, 11, 2} 00074 {6, 9, 10, 4, 3, 0, 11, 8, 7, 1, 2, 5} 00075 {7, 10, 11, 5, 4, 1, 0, 9, 8, 2, 3, 6} 00076 {10, 1, 2, 8, 7, 4, 3, 0, 11, 5, 6, 9} 00077 {11, 2, 3, 9, 8, 5, 4, 1, 0, 6, 7, 10} 00078 {5, 8, 9, 3, 2, 11, 10, 7, 6, 0, 1, 4} 00079 {4, 7, 8, 2, 1, 10, 9, 6, 5, 11, 0, 3} 00080 {1, 4, 5, 11, 10, 7, 6, 3, 2, 8, 9, 0} 00081 00082 The functions below perform steps in preparation for calling setComplex(). 00083 getPC() converts a PitchList to an IntegerList by summing the diatonic 00084 pitch class and accidental of the Pitch. 00085 generateSetComplex() first calls getPC() to convert the Pitch list to 00086 an IntegerList, then transposes the resulting set, then 00087 invokes setComplex() to generate the set complex Matrix. 00088 *****************************************************************************/ 00089 } 00090 00091 IntegerList getPC(PitchList p) { 00092 IntegerList x; 00093 00094 For (Integer i = 0; i < Length(p); i++) 00095 x[i] = PitchClass(p[i]) + Accidental( p[i] ); 00096 00097 Return( x ); 00098 } 00099 00100 00101 Void generateSetComplex(PitchList P) { 00102 IntegerList set = getPC(P); 00103 Print("Starting set:", set); 00104 Integer offset = set[0]; 00105 IntegerList tSet = transpose(set, -offset); 00106 Print("Transposed set:", tSet); 00107 IntegerMatrix sc = setComplex( tSet ); 00108 Print("Set complex:"); 00109 Print(sc); 00110 } 00111 00112 Static Void para2() { 00113 /***************************************************************************** 00114 Here we evaluate the set complex for our test set. 00115 *****************************************************************************/ 00116 PitchList test(G, As, B, F, E, Cs, C, A, Gs, D, Ds, Fs); 00117 Print("*** Pitch classes for test ***"); 00118 Print(test); 00119 generateSetComplex(test); 00120 00121 } 00122 00124 /* $Revision: 1.3 $ $Date: 2006/09/05 08:02:45 $ $Author: dgl $ $Name: $ $Id: C091102.cpp,v 1.3 2006/09/05 08:02:45 dgl Exp $ */ 00125 // The Musimat Tutorial © 2006 Gareth Loy 00126 // Derived from Chapter 9 and Appendix B of "Musimathics Vol. 1" © 2006 Gareth Loy 00127 // and published exclusively by The MIT Press. 00128 // This program is released WITHOUT ANY WARRANTY; without even the implied 00129 // warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. 00130 // For information on usage and redistribution, and for a DISCLAIMER OF ALL 00131 // WARRANTIES, see the file, "LICENSE.txt," in this distribution. 00132 // "Musimathics" is available here: http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10916 00133 // Gareth Loy's Musimathics website: http://www.musimathics.com/ 00134 // The Musimat website: http://www.musimat.com/ 00135 // This program is released under the terms of the GNU General Public License 00136 // available here: http://www.gnu.org/licenses/gpl.txt 00137
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