00001 #include "MusimatTutorial.h" 00002 MusimatTutorialSection(B0201b) { 00003 Print("*** Pythagorean Chromatic Scale ***"); 00004 /***************************************************************************** 00005 00006 Pythagorean Chromatic Scale 00007 00008 We can compute the frequency of a Pitch in Pythagorean 00009 chromatic tuning, assuming a reference such as A4 equals 440 Hz. We start by computing the fre- 00010 quency of Pythagorean middle C from the reference frequency, using equation (3.11). 00011 *****************************************************************************/ 00012 00013 para1(); // Step into this function to continue the tutorial 00014 para2(); // Step into this function to continue the tutorial 00015 } 00016 00017 00018 /***************************************************************************** 00019 Next, referring to figure 3.7, we tabulate the ratios of the Pythagorean chromatic scale in 00020 Musimat using a RealList: 00021 *****************************************************************************/ 00022 00023 RealList pythagoreanChromatic( 00024 1.0/1.0, 256.0/243.0, 9.0/8.0, 32.0/27.0, 00025 81.0/64.0, 4.0/3.0, 1024.0/729.0, 3.0/2.0, 00026 128.0/81.0, 27.0/16.0, 16.0/9.0, 243.0/128.0 00027 ); 00028 00029 Static Void para1() { 00030 00031 /***************************************************************************** 00032 Last, we define a variation of the pitchToHz() function. This version has the same name but 00033 takes three arguments instead of one. When supplied with a certain Pitch p, it returns the fre- 00034 quency corresponding to its Pythagorean intonation as a Real value in hertz. 00035 *****************************************************************************/ 00036 00037 para2(); 00038 } 00039 00040 Real pitchToHz( 00041 Pitch p, // pitch 00042 Real refC, // reference frequency 00043 RealList scale // ratios of scale degrees 00044 ) { 00045 Integer key = PitchClass(p) + Accidental(p); // get key from pitch 00046 Real oct = Octave(p); // get octave from pitch 00047 Return(refC * scale[key] * Pow(2.0, (oct - 4)));// compute frequency 00048 } 00049 00050 Static Void para2() { 00051 00052 /***************************************************************************** 00053 The Return() statement calculates the frequency of the key from the reference frequency 00054 times the ratio for that degree, then adjusts it for the proper octave. 00055 00056 We define the reference frequencies in Musimat as follows: 00057 *****************************************************************************/ 00058 00059 Real R = 440.0; 00060 Real cPi4 = R * 16.0/27.0; // Pythagorean middle C, 260.74 Hz 00061 00062 /***************************************************************************** 00063 Finally, we call pitchToHz() with the pitch we want, the Pythagorean reference 00064 frequency, and the ratios of the Pythagorean scale: 00065 *****************************************************************************/ 00066 00067 Print("Pythagorian A4=", pitchToHz(A4 , cPi4, pythagoreanChromatic)); 00068 00069 /***************************************************************************** 00070 This prints A4=440.0, and 00071 *****************************************************************************/ 00072 00073 Print("Pythagorian C4=", pitchToHz(C4 , cPi4, pythagoreanChromatic)); 00074 00075 /***************************************************************************** 00076 prints C4=260.74, as expected. 00077 *****************************************************************************/ 00078 } 00079 00081 /* $Revision: 1.3 $ $Date: 2006/09/05 08:03:08 $ $Author: dgl $ $Name: $ $Id: B0201b.cpp,v 1.3 2006/09/05 08:03:08 dgl Exp $ */ 00082 // The Musimat Tutorial � 2006 Gareth Loy 00083 // Derived from Chapter 9 and Appendix B of "Musimathics Vol. 1" � 2006 Gareth Loy 00084 // and published exclusively by The MIT Press. 00085 // This program is released WITHOUT ANY WARRANTY; without even the implied 00086 // warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. 00087 // For information on usage and redistribution, and for a DISCLAIMER OF ALL 00088 // WARRANTIES, see the file, "LICENSE.txt," in this distribution. 00089 // "Musimathics" is available here: http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10916 00090 // Gareth Loy's Musimathics website: http://www.musimathics.com/ 00091 // The Musimat website: http://www.musimat.com/ 00092 // This program is released under the terms of the GNU General Public License 00093 // available here: http://www.gnu.org/licenses/gpl.txt 00094
1.7.2