(*^ ::[paletteColors = 128; automaticGrouping; currentKernel; fontset = title, inactive, nohscroll, noKeepOnOnePage, preserveAspect, groupLikeTitle, center, M7, bold, L1, e8, 24, "Times"; ; fontset = subtitle, inactive, nohscroll, noKeepOnOnePage, preserveAspect, groupLikeTitle, center, M7, bold, L1, e6, 18, "Times"; ; fontset = subsubtitle, inactive, nohscroll, noKeepOnOnePage, preserveAspect, groupLikeTitle, center, M7, italic, L1, e6, 14, "Times"; ; fontset = section, inactive, nohscroll, noKeepOnOnePage, preserveAspect, groupLikeSection, grayBox, M22, bold, L1, a20, 24, "Helvetica"; ; fontset = subsection, inactive, nohscroll, noKeepOnOnePage, preserveAspect, groupLikeSection, blackBox, M19, bold, L1, a15, 18, "Helvetica"; ; fontset = subsubsection, inactive, nohscroll, noKeepOnOnePage, preserveAspect, groupLikeSection, whiteBox, M18, bold, L1, a12, 16, "Helvetica"; ; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 14, "Times"; ; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = input, nowordwrap, preserveAspect, groupLikeInput, M42, N23, bold, L1, 12, "Courier"; ; fontset = output, output, inactive, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L-5, 12, "Courier"; ; fontset = message, inactive, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = print, inactive, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = info, inactive, nowordwrap, preserveAspect, groupLikeOutput, M42, N23, L1, 12, "Courier"; ; fontset = postscript, PostScript, formatAsPostScript, output, inactive, nowordwrap, preserveAspect, groupLikeGraphics, M7, l34, w282, h287, L1, 12, "Courier"; ; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, italic, L1, 10, "Times"; ; fontset = header, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = Left Header, nohscroll, cellOutline, 12, "Palatino"; ; fontset = footer, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M7, L1, 12; fontset = Left Footer, cellOutline, blackBox, 12, "Palatino"; ; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 10, "Times"; ; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12, "Courier"; ; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M7, L1, 12; next21StandardFontEncoding; ] :[font = section; inactive; preserveAspect; plain; fontName = "Times"; startGroup; ] Interference ;[s] 1:0,0;12,-1; 1:1,22,17,Helvetica,1,24,0,0,0; :[font = subsection; inactive; preserveAspect; startGroup; ] Interference due to Phase Relationships :[font = text; inactive; preserveAspect; ] Here are two sine waves, both at 440 Hz, the second one shifted over by an amount that will cause the two waves to almost cancel each other out when they are played. :[font = input; preserveAspect; startGroup; ] Plot[ {Sin[2 Pi 440 t] ,Sin[ 2 Pi 440 t + 3 ]}, {t, 0, 1/440}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 35; pictureWidth = 229; pictureHeight = 141; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 419.047619 0.309017 0.294302 [ [(0.0005)] 0.23333 0.30902 0 2 Msboxa [(0.001)] 0.44286 0.30902 0 2 Msboxa [(0.0015)] 0.65238 0.30902 0 2 Msboxa [(0.002)] 0.8619 0.30902 0 2 Msboxa [(-1)] 0.01131 0.01472 1 0 Msboxa [(-0.5)] 0.01131 0.16187 1 0 Msboxa [(0.5)] 0.01131 0.45617 1 0 Msboxa [(1)] 0.01131 0.60332 1 0 Msboxa [ -0.001 -0.001 0 0 ] [ 1.001 0.61903 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave gsave 0.002 setlinewidth 0.23333 0.30902 moveto 0.23333 0.31527 lineto stroke grestore [(0.0005)] 0.23333 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.44286 0.30902 moveto 0.44286 0.31527 lineto stroke grestore [(0.001)] 0.44286 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.65238 0.30902 moveto 0.65238 0.31527 lineto stroke grestore [(0.0015)] 0.65238 0.30902 0 2 Mshowa gsave 0.002 setlinewidth 0.8619 0.30902 moveto 0.8619 0.31527 lineto stroke grestore [(0.002)] 0.8619 0.30902 0 2 Mshowa gsave 0.001 setlinewidth 0.06571 0.30902 moveto 0.06571 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.10762 0.30902 moveto 0.10762 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.14952 0.30902 moveto 0.14952 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.19143 0.30902 moveto 0.19143 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.27524 0.30902 moveto 0.27524 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.31714 0.30902 moveto 0.31714 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.35905 0.30902 moveto 0.35905 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.40095 0.30902 moveto 0.40095 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.48476 0.30902 moveto 0.48476 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.52667 0.30902 moveto 0.52667 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.56857 0.30902 moveto 0.56857 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.61048 0.30902 moveto 0.61048 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.69429 0.30902 moveto 0.69429 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.73619 0.30902 moveto 0.73619 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.7781 0.30902 moveto 0.7781 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.82 0.30902 moveto 0.82 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.90381 0.30902 moveto 0.90381 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.94571 0.30902 moveto 0.94571 0.31277 lineto stroke grestore gsave 0.001 setlinewidth 0.98762 0.30902 moveto 0.98762 0.31277 lineto stroke grestore gsave 0.002 setlinewidth 0 0.30902 moveto 1 0.30902 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0.01472 moveto 0.03006 0.01472 lineto stroke grestore [(-1)] 0.01131 0.01472 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.16187 moveto 0.03006 0.16187 lineto stroke grestore [(-0.5)] 0.01131 0.16187 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.45617 moveto 0.03006 0.45617 lineto stroke grestore [(0.5)] 0.01131 0.45617 1 0 Mshowa gsave 0.002 setlinewidth 0.02381 0.60332 moveto 0.03006 0.60332 lineto stroke grestore [(1)] 0.01131 0.60332 1 0 Mshowa gsave 0.001 setlinewidth 0.02381 0.04415 moveto 0.02756 0.04415 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.07358 moveto 0.02756 0.07358 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.10301 moveto 0.02756 0.10301 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.13244 moveto 0.02756 0.13244 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.1913 moveto 0.02756 0.1913 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.22073 moveto 0.02756 0.22073 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.25016 moveto 0.02756 0.25016 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.27959 moveto 0.02756 0.27959 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.33845 moveto 0.02756 0.33845 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.36788 moveto 0.02756 0.36788 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.39731 moveto 0.02756 0.39731 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.42674 moveto 0.02756 0.42674 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.4856 moveto 0.02756 0.4856 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.51503 moveto 0.02756 0.51503 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.54446 moveto 0.02756 0.54446 lineto stroke grestore gsave 0.001 setlinewidth 0.02381 0.57389 moveto 0.02756 0.57389 lineto stroke grestore gsave 0.002 setlinewidth 0.02381 0 moveto 0.02381 0.61803 lineto stroke grestore grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath gsave gsave gsave 0.004 setlinewidth 0.02381 0.30902 moveto 0.06349 0.38519 lineto 0.10317 0.45617 lineto 0.14286 0.51712 lineto 0.1627 0.5425 lineto 0.18254 0.56389 lineto 0.20238 0.58092 lineto 0.2123 0.5877 lineto 0.22222 0.59329 lineto 0.23214 0.59766 lineto 0.2371 0.59939 lineto 0.24206 0.6008 lineto 0.24702 0.6019 lineto 0.2495 0.60233 lineto 0.25198 0.60269 lineto 0.25446 0.60296 lineto 0.2557 0.60307 lineto 0.25694 0.60316 lineto 0.25818 0.60323 lineto 0.25942 0.60328 lineto 0.26066 0.60331 lineto 0.2619 0.60332 lineto 0.26314 0.60331 lineto 0.26438 0.60328 lineto 0.26563 0.60323 lineto 0.26687 0.60316 lineto 0.26935 0.60296 lineto 0.27183 0.60269 lineto 0.27431 0.60233 lineto 0.27679 0.6019 lineto 0.28175 0.6008 lineto 0.28671 0.59939 lineto 0.29167 0.59766 lineto 0.30159 0.59329 lineto 0.32143 0.58092 lineto 0.34127 0.56389 lineto 0.38095 0.51712 lineto 0.42063 0.45617 lineto 0.46032 0.38519 lineto 0.5 0.30902 lineto 0.53968 0.23285 lineto 0.57937 0.16187 lineto 0.61905 0.10091 lineto 0.63889 0.07553 lineto 0.65873 0.05414 lineto 0.67857 0.03712 lineto 0.68849 0.03033 lineto 0.69841 0.02474 lineto 0.70833 0.02037 lineto 0.71329 0.01865 lineto Mistroke 0.71825 0.01723 lineto 0.72321 0.01613 lineto 0.72569 0.0157 lineto 0.72817 0.01535 lineto 0.73065 0.01507 lineto 0.73189 0.01496 lineto 0.73313 0.01487 lineto 0.73438 0.0148 lineto 0.73562 0.01475 lineto 0.73686 0.01472 lineto 0.7381 0.01472 lineto 0.73934 0.01472 lineto 0.74058 0.01475 lineto 0.74182 0.0148 lineto 0.74306 0.01487 lineto 0.7443 0.01496 lineto 0.74554 0.01507 lineto 0.74802 0.01535 lineto 0.7505 0.0157 lineto 0.75298 0.01613 lineto 0.75794 0.01723 lineto 0.7629 0.01865 lineto 0.76786 0.02037 lineto 0.77778 0.02474 lineto 0.79762 0.03712 lineto 0.81746 0.05414 lineto 0.85714 0.10091 lineto 0.89683 0.16187 lineto 0.93651 0.23285 lineto 0.97619 0.30902 lineto Mfstroke grestore grestore gsave gsave 0.004 setlinewidth 0.02381 0.35055 moveto 0.06349 0.27373 lineto 0.10317 0.19931 lineto 0.14286 0.13236 lineto 0.18254 0.07746 lineto 0.20238 0.05573 lineto 0.22222 0.03834 lineto 0.23214 0.03136 lineto 0.24206 0.02557 lineto 0.25198 0.021 lineto 0.2619 0.01766 lineto 0.26687 0.01646 lineto 0.26935 0.01597 lineto 0.27183 0.01557 lineto 0.27431 0.01524 lineto 0.27555 0.01511 lineto 0.27679 0.01499 lineto 0.27803 0.0149 lineto 0.27927 0.01482 lineto 0.28051 0.01477 lineto 0.28175 0.01473 lineto 0.28299 0.01472 lineto 0.28423 0.01472 lineto 0.28547 0.01474 lineto 0.28671 0.01479 lineto 0.28919 0.01493 lineto 0.29043 0.01503 lineto 0.29167 0.01516 lineto 0.29415 0.01546 lineto 0.29663 0.01584 lineto 0.30159 0.01684 lineto 0.30655 0.01815 lineto 0.31151 0.01977 lineto 0.32143 0.02394 lineto 0.33135 0.02934 lineto 0.34127 0.03593 lineto 0.36111 0.05259 lineto 0.38095 0.07363 lineto 0.42063 0.12737 lineto 0.46032 0.19349 lineto 0.5 0.26749 lineto 0.53968 0.34431 lineto 0.57937 0.41873 lineto 0.61905 0.48567 lineto 0.65873 0.54057 lineto 0.67857 0.5623 lineto 0.69841 0.5797 lineto 0.71825 0.59246 lineto 0.72817 0.59703 lineto 0.73313 0.59886 lineto Mistroke 0.7381 0.60037 lineto 0.74306 0.60158 lineto 0.74554 0.60206 lineto 0.74802 0.60247 lineto 0.7505 0.60279 lineto 0.75174 0.60293 lineto 0.75298 0.60304 lineto 0.75422 0.60314 lineto 0.75546 0.60321 lineto 0.7567 0.60327 lineto 0.75794 0.6033 lineto 0.75918 0.60332 lineto 0.76042 0.60331 lineto 0.76166 0.60329 lineto 0.7629 0.60325 lineto 0.76414 0.60318 lineto 0.76538 0.6031 lineto 0.76786 0.60288 lineto 0.77034 0.60257 lineto 0.77282 0.60219 lineto 0.77778 0.6012 lineto 0.78274 0.59988 lineto 0.7877 0.59826 lineto 0.79762 0.59409 lineto 0.80754 0.5887 lineto 0.81746 0.58211 lineto 0.8373 0.56545 lineto 0.85714 0.5444 lineto 0.89683 0.49066 lineto 0.93651 0.42454 lineto 0.97619 0.35055 lineto Mfstroke grestore grestore grestore % End of Graphics MathPictureEnd :[font = text; inactive; preserveAspect; ] Here are the two waves played together, one through the right channel, the other through the left. Note: The following examples will only sound correctly on machines capable of stereo output. ;[s] 3:0,0;100,1;105,2;193,-1; 3:1,13,10,Times,0,14,0,0,0;1,12,9,Times,1,14,0,0,0;1,13,10,Times,0,14,0,0,0; :[font = input; Cclosed; preserveAspect; startGroup; ] Play[{Sin[2 Pi 440 t], Sin[2 Pi 440 t + 3]},{t, 0, 1}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 35; pictureWidth = 160; pictureHeight = 98; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 0.014715 0.588604 [ [ 0 0 0 0 ] [ 1 0.618034 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath % Start of sampled sound 1 string 22050 22050 16 { currentfile 1 index readhexstring pop } false 2 sampledsound 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(Sin[2*Pi*440*t] - 3.044895958370652*10^-6)* 1.000009388530494]]], CompiledFunction[{_Integer}, {0, 3, 14, 0}, {{1, 17}, {3, 1, 0}, {13, 0., 0}, {13, 0.0000453514739229025, 1}, {20, 0, 2}, {36, 1, 2, 3}, {33, 0, 3, 4}, {12, 2, 1}, {13, 3.141592653589793, 5}, {12, 440, 2}, {20, 1, 6}, {20, 2, 7}, {36, 6, 5, 7, 4, 8}, {49, 8, 9}, {13, -0.00002841884626286761, 10}, {33, 9, 10, 11}, {13, 1.000028673410517, 12}, {36, 11, 12, 13}, {8, 13}}, Function[{Play`Time7}, Block[{t = 0. + 0.0000453514739229025*Play`Time7}, (Sin[2*Pi*440*t] - 0.00002841884626286761)* 1.000028673410517]]]}, 22050, {22050, 16}]] ;[o] -Sound- :[font = text; inactive; preserveAspect; ] Note: The difference in volume is not due to the amplitude settings of the two sine waves, but due to the interference they cause each other when coming out of the speaker. Here is an example in which one of the sine waves is phase-shifted by the value Sin[t] * 3. Effectually, the amount of shift in the phase is moving from -3 to +3, which generates the changing magnitude of the change in amplitude. ;[s] 4:0,0;5,1;34,2;37,3;405,-1; 4:1,12,9,Times,1,14,0,0,0;1,13,10,Times,0,14,0,0,0;1,13,10,Times,2,14,0,0,0;1,13,10,Times,0,14,0,0,0; :[font = input; Cclosed; preserveAspect; startGroup; ] Play[{Sin[2 Pi 440 t],Sin[2 Pi 440 t + Sin[t] * 3]}, {t,0,Pi},SampleRate->44100] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 34; pictureWidth = 282; pictureHeight = 174; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 0.952381 0.014715 0.588604 [ [ 0 0 0 0 ] [ 1 0.618034 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath %%Object: Graphics [ ] 0 setdash 0 setgray gsave grestore 0 0 moveto 1 0 lineto 1 0.61803 lineto 0 0.61803 lineto closepath clip newpath % Start of sampled sound 1 string 44100 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sampled sound % End of Graphics MathPictureEnd :[font = output; output; inactive; preserveAspect; endGroup; endGroup; ] The Unformatted text for this cell was not generated. Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Sound- :[font = subsection; inactive; preserveAspect; startGroup; ] Interference due to Beating :[font = text; inactive; preserveAspect; ] Beats occur when two sounds are close to each other in frequency. The resulting sound periodically waxes and wanes because the two waves are periodically in and out of phase, hence, they periodically reinforce and interfere with each other. The frequency of the beats can be calculated from the following formula: beat frequency = freq2 - freq1 if freq2 - freq1 <= 15, beating will be heard if freq2 - freq1 > 10 to 15, we hear the two as separate frequencies In music, the tones we hear are complex, so many possibilities for beating arise, for example among the upper partials of notes played by instruments. Beating between pitches is frequently used for tuning instruments (the guitar), and for setting certain organ stops (the voix celeste). Beating also helps explain the timbre of massed strings, choral groups, and others, that not all tuned to exactly the same pitch. Here is a plot of the relationship between the frequencies 440 and 455 Hz, showing them moving apart. Later, they come back together. ;[s] 12:0,0;318,1;332,2;335,3;348,4;355,5;368,6;402,7;415,8;744,9;756,10;965,11;1029,-1; 12:1,13,10,Times,0,14,0,0,0;1,13,10,Times,2,14,0,0,0;1,13,10,Times,0,14,0,0,0;1,13,10,Times,2,14,0,0,0;1,13,10,Times,0,14,0,0,0;1,13,10,Times,2,14,0,0,0;1,13,10,Times,0,14,0,0,0;1,13,10,Times,2,14,0,0,0;1,13,10,Times,0,14,0,0,0;1,13,10,Times,2,14,0,0,0;1,13,10,Times,0,14,0,0,0;1,13,10,Times,0,14,0,0,0; :[font = input; preserveAspect; startGroup; ] Plot[{Sin[2 Pi 440 t],Sin[2 Pi 450 t]}, {t, 0, 15/450}] :[font = postscript; PostScript; formatAsPostScript; output; inactive; preserveAspect; pictureLeft = 40; pictureTop = 8; pictureWidth = 201; pictureHeight = 123; endGroup; ] %! %%Creator: Mathematica %%AspectRatio: 0.61803 MathPictureStart /Courier findfont 10 scalefont setfont % Scaling calculations 0.02381 28.571429 0.309017 0.294302 [ [(0.005)] 0.16667 0.30902 0 2 Msboxa [(0.01)] 0.30952 0.30902 0 2 Msboxa [(0.015)] 0.45238 0.30902 0 2 Msboxa [(0.02)] 0.59524 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Use options in the Actions Settings dialog box to control when Unformatted text is generated. ;[o] -Sound- ^*)