00001 #include "MusimatTutorial.h" 00002 MusimatTutorialSection(B0204) { 00003 Print("*** B.2.4 Loudness ***"); 00004 /***************************************************************************** 00005 00006 B.2.4 Loudness 00007 00008 Loudness is expressed in common music notation using performance indications such as fortis- 00009 simo or piano (see section 2.7). But the performed intensity depends upon the acoustical power of 00010 the instrument and the interpretation of the performer. A better approach for the purpose here 00011 would be to define loudness in objective terms using decibels (see section 5.5.1). 00012 00013 Since microphones and loudspeakers measure and reproduce pressure waves, it is common to 00014 use dB SPL in audio work (see equation 5.32). It is also conventional in audio to take the loudest 00015 value that can be reproduced without distortion as a reference intensity of 0 dB (see section 4.24.2). 00016 Since measured intensities will be less intense than the reference, then by the definition of the deci- 00017 bel they will be expressed as negative decibel levels. We can write, for example, -6 dB to indicate 00018 an amplitude that is (very close to) one half of the amplitude of the 0 dB reference. 00019 00020 Restating (5.32), the equation for dB SPL, using standard mathematics symbols, is 00021 00022 y dB = 20 Log10(A'/A) 00023 00024 and simplifying by letting x=A'/A, we have y dB = 20 Log10(x). Solving for x, we have 00025 00026 x = 10^(y/20). (B.1) 00027 00028 (Note, in standard math, A' in English means "A prime", which means A' is like A, but is not 00029 necessarily the same. Also, the operator "^" in standard math means exponentiation, so that 00030 2^2=4. In modern programming languages, however, "^" means "exclusive-or".) 00031 00032 For example, setting y = -6 dB, we have x = 10^(-6/20) = 0.501. The value of x is the coefficient by 00033 which a signal must be multiplied to lower its amplitude by 6 dB. For another example, setting 00034 y = 0 dB, we have x = 10^(0/20) = 1. So multplying by 0 dB does not affect amplitude. Setting 00035 y = -120 dB, we have x = 10^(-120/20) = 0.000001, so multiplying a signal by -120 dB renders it vir- 00036 tually inaudible. Finally, if we wish to amplify a soft sound, scaling it by +6 dB makes it twice as 00037 loud. Thus, scaling sounds with decibel coefficients allows us to achieve arbitrary loudness levels 00038 for waveforms. So we define 00039 *****************************************************************************/ 00040 00041 para1(); // Step into this function to continue the tutorial 00042 } 00043 00044 Real dB(Real y){ 00045 Return(Pow(10.0, y/20.0)); 00046 } 00047 00048 Static Void para1() { 00049 00050 /***************************************************************************** 00051 For example, Print(dB(-6)) prints 0.501187, Print(dB(0)) prints 1.0, and 00052 Print(dB(-120)) prints 0.000001. 00053 00054 Suppose we have the following audio samples for a sound: 00055 *****************************************************************************/ 00056 00057 RealList mySound(0, 0.16, 0.192, -0.37, -0.45, -0.245, -0.43, 0.09); 00058 00059 /***************************************************************************** 00060 We wish to halve the sound's amplitude. Then 00061 *****************************************************************************/ 00062 00063 RealList scaledSound = mySound * dB(-6); 00064 Print(scaledSound); 00065 00066 /***************************************************************************** 00067 prints {0.02, 0.08, 0.10, -0.19, -0.23, -0.12, -0.22, 0.05, . . .}. 00068 00069 See volume 2, chapter 1, for more about sampled signals. 00070 00071 Musimat provides built-in definitions for standard music dynamics levels based on figure 4.7. 00072 *****************************************************************************/ 00073 00074 Real ffff = dB(0), fff = dB(-10), ff = dB(-18), f = dB(-24), 00075 mf = dB(-32), mp = dB(-40), p = dB(-48), pp = dB(-56), 00076 ppp = dB(-64); 00077 00078 Print( ffff ); Print( fff ); Print(ff); Print(f); Print(mf); Print(mp); Print(mp); Print(p); Print(pp); Print(ppp); 00079 00080 /***************************************************************************** 00081 Thus ffff does not change the amplitude of the signal, but all others attenuate it to varying 00082 degrees. 00083 *****************************************************************************/ 00084 } 00085 00087 /* $Revision: 1.3 $ $Date: 2006/09/05 08:03:08 $ $Author: dgl $ $Name: $ $Id: B0204.cpp,v 1.3 2006/09/05 08:03:08 dgl Exp $ */ 00088 // The Musimat Tutorial © 2006 Gareth Loy 00089 // Derived from Chapter 9 and Appendix B of "Musimathics Vol. 1" © 2006 Gareth Loy 00090 // and published exclusively by The MIT Press. 00091 // This program is released WITHOUT ANY WARRANTY; without even the implied 00092 // warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. 00093 // For information on usage and redistribution, and for a DISCLAIMER OF ALL 00094 // WARRANTIES, see the file, "LICENSE.txt," in this distribution. 00095 // "Musimathics" is available here: http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10916 00096 // Gareth Loy's Musimathics website: http://www.musimathics.com/ 00097 // The Musimat website: http://www.musimat.com/ 00098 // This program is released under the terms of the GNU General Public License 00099 // available here: http://www.gnu.org/licenses/gpl.txt 00100
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