Chapter20 音乐

目录

音乐

琴键

哆来咪

振动与调式

利萨如曲线

和声与音调与和弦

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音乐

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        在音乐理论中，一个八音度（octave）是一个频率范围相差二倍的区间。在大多数西洋乐器中，一个八音度被分为12个频率比相等的半音程（semitone）。因为12个半音程构成了二倍的八音度频率范围，所以一个半音程的频率倍数为。 该数值在本节将会频繁出现，可以令

   sigma = 2^(1/12)

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琴键

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        钢琴键盘有两个八度音，白键标为 CDEFGAB，加上下一个C键。如果白键和黑键都计算的话，每一个八度音有12个键。每个键的频率比相邻的键高或低半音程。每一个黑键可以理解成上一个白键的升音（sharp，记为#），或下一个白键的降音（flat，b）

function pianoex(a)
% PIANOEX  A simple music synthesizer.
%   pianoex, with no arguments, presents a gui with 25 piano keys,
%   13 toggles to form chords, 3 toggles to choose intonation, and
%   an oscilloscope and staff to display the generated notes and chords.
%
%   pianoex(n) plays the n-th note above middle C.
%   pianoex(-n) plays the n-th note below middle C.
%   pianoex([n1 n2 ... ]) plays the chord specified by a vector.
%
%   pianoex(score) plays the cell array score constructed like
%   the one in vivaldi.m. 
%
%   for n = [0 2 4 5 7 9 11 12]
%      pianoex(n)
%   end
%   plays the C-major scale.
%
%   Requires sound card and, optionally, the Signal Processing Toolbox.if isempty(get(gca,'userdata'))make_keyboardendif nargin > 0T = 5/8;H = get(gca,'userdata');tv = get(H.voice,'value');voice = find([tv{:}]==1);set(H.togs(1),'value',1,'background','blue')for k = 1:size(a,1)if iscell(a)if size(a(k,:),2)==2[y,fs] = synthesize(a{k,1},a{k,2}*T,voice);else[y,fs] = synthesize(a{k,1},T,voice);endv = a{k,1};elseif isscalar(a)H.root = a;ch = get(H.togs,'value');a = a + find([ch{:}]) - 1;end[y,fs] = synthesize(a,T,voice);v = a;endsound(y,fs);for j = 1:25set(H.keys(j),'background',get(H.keys(j),'userdata'))endset(H.keys(v(v<13)+13),'background','blue')j = round(get(H.scope,'xdata')*fs);set(H.scope,'ydata',y(j))set(H.notes,'vis','off')set(H.sharps,'vis','off')delta = [0 0 1 1 2 3 3 4 4 5 5 6]/2;for n = 1:length(v)y = 3.5*floor(v(n)/12)+delta(mod(v(n),12)+1);set(H.notes(n),'ydata',y,'vis','on')if any(mod(v(n),12) == [1 3 6 8 10])set(H.sharps(n),'pos',[.25 y],'vis','on');endenddrawnowendset(gca,'userdata',H);end
end% ------------------------------------function [y,fs] = synthesize(chord,T,voice)
% SYNTHESIZE  Generate musical notes and cords.
% synthesize(note,T,voice)  Generates a note for T seconds with given tuning.
% synthesize([note1, ..., noten],T,voice)  Generates a chord of several notes.
% Notes are integers specifying half tones above and below middle C.
% voice = 1 for piano, 2 for just intonation, 3 for equal temperament.% Examples:
%   synthesize(0) generatees middle C (261.625 Hz) for 1/2 second.
%   synthesize(1) is C#.
%   synthesize(-1) is C-flat.
%   synthesize(12) is high C, one octave above middle C.
%   synthesize([0 4 7]) is the perfect fifth major triad C-E-G. 
%   A scale with one second per note:
%      for k = [0 2 4 5 7 9 11 12]
%         synthesize(k,1)
%      endswitch voicecase 1  % equal temperamentsigma = 2^(1/12);C4 = 440*sigma^(-9);fs = 44100;t = 0:1/fs:T;y = zeros(size(t));for n = chordhz = C4 * sigma^n;y = y + sin(2*pi*hz*t);endy = y/length(chord);case 2  % just intonationsigma = 2^(1/12);C4 = 440*sigma^(-9);fs = 44100;t = 0:1/fs:T;r = [1 16/15 9/8 6/5 5/4 4/3 7/5 3/2 8/5 5/3 7/4 15/8];r = [r/2 r 2*r 4];y = zeros(size(t));for n = chordhz = C4 * r(n+13);y = y + sin(2*pi*hz*t);endy = y/length(chord);case 3  % pianomiddle_c = get(gcf,'userdata');fs = 44100;t = 0:1/fs:T;y = zeros(size(t));for n = chordy = y + resamplex(middle_c,2^(n/12),length(y));endend
end% ------------------------------------function y = resamplex(y,factor,L)
% RESAMPLE 
% Requires Signal Processing Toolbox[p,q] = rat(factor,1.e-4);y = resample(y,q,p)';L = floor(L);if L < length(y)y = y(1:L);elsey(L) = 0;endr = 150:-1:0;y(L-r:L) = y(L-r:L).*r/150;
end% ------------------------------------function make_keyboard
% MAKE_KEYBOARD Create a piano keyboard.clfwhite = [-12:2:-8 -7:2:-1 0:2:4 5:2:11 12];black = setdiff(-11:11,white);dx = 1/16;H.keys = zeros(1,25);for k = [white black]if k == -12% Whitebw = [1 1 1];x = dx/2;y = .08;dy = .40;elseif k == -11% Blackbw = [0 0 0];x = dx;y = .23;dy = .25;endcallback = ['pianoex(' int2str(k) '), set(gcbo,''value'',0)'];H.keys(k+13) = uicontrol('units','normal','position',[x y dx dy], ...'style','toggle','background',bw,'userdata',bw,'callback',callback);x = x + dx;if k == -9 || k == -2 || k == 3x = x + dx;endend% Key Labelsy = .02;dy = .05;s = 'C';grayc = get(gcf,'color');for x = dx/2:dx:1-dxuicontrol('units','normal','position',[x y dx dy],'background',grayc, ...'style','text','string',s,'fontsize',14,'fontweight','bold')s = char(s+1);if s > 'G', s = 'A'; endend% Clefsaxes('position',[.03 .65 .07 .3])C = load('clefs.mat');image(C.clefs_pic)colormap(gray)axis off% Staffaxes('position',[.1 .65 .1 .3],'xticklabel','','yticklabel','', ...'xlim',[0 1],'ylim',[-7 7]);box onfor y = [-5:-1 1:5]line([0 1],[y y],'color','black')endline([.33 .67],[0 0],'color','black')% NotesH.notes = zeros(13,1);H.sharps = zeros(13,1);for n = 1:13H.notes(n) = line(.5,0,'marker','.','markersize',24,'vis','off');H.sharps(n) = text(.375,0,'#','fontsize',12,'color','blue','vis','off');end% Chord togglesH.togs = zeros(1,13);callback = ['if get(gcbo,''value'')==0,' ...'set(gcbo,''background'',[.94 .94 .94]),' ...'else, set(gcbo,''background'',''blue''), end,' ...'H=get(gca,''userdata''); pianoex(H.root)'];for k = 0:12H.togs(k+1) = uicontrol('style','toggle','units','normal', ...'pos',[.05+k*.05 .54 .04 .04],'value',k==0,'background',grayc, ...'callback',callback);uicontrol('style','text','units','normal','string',num2str(k), ...'fontsize',12,'fontweight','bold','pos',[.04+k*.05 .49 .05 .05], ...'background',grayc)endset(H.togs(1),'value',1,'background','blue')% TuningH.voice = zeros(1,3);voicestring = {'equal','just','piano'};callback = ['H=get(gca,''userdata'');' ...'set(H.voice,''background'',[.94 .94 .94],''value'',0),' ...'set(gcbo,''background'',''blue'',''value'',1),' ...'pianoex(H.root)'];% Do not show piano if Signal Processing Toolbox is not available.kmax = 2 + (exist('resample','file')==2);for k = 1:kmaxH.voice(k) = uicontrol('style','toggle','units','normal', ...'pos',[1.00-k*.08 .54 .04 .04],'value',0,'background',grayc, ...'callback',callback);uicontrol('style','text','units','normal','string',voicestring{k}, ...'fontsize',12,'fontweight','bold', ...'pos',[.98-k*.08 .49 .08 .05],'background',grayc)endset(H.voice(1),'value',1,'background','blue');% Scopetmin = .06;tmax = .12;s = tmin:4/44100:tmax;axes('position',[.30 .65 .65 .30],'xlim',[tmin tmax],'ylim',[-1 1])H.scope = line(s,0*s);box onset(zoom,'motion','horizontal','direction','in','enable','on')% Piano sampleS = load('piano_c.mat');middle_c = double(S.piano_c)/2^15;H.root = 0;set(gca,'userdata',H)set(gcf,'userdata',middle_c)shg
end

        在微型键盘上用for循环演奏出两个八音度的25个琴键半音音阶。

for n= -12:12, pianoex(n), end

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哆来咪

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        对于来说，C大调音阶为。

 在微型键盘上演奏：

for n =[0 2 4 5 7 9 11 12], pianoex(n), end

计算出音符间半音程个数：

diff([0 2 5 7 9 11 12])

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振动与调式

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        振动的模型可以称为由一些称为调式或特征函数的加权和来表示。不同的调式在不同的特征频率或特征值下振动。

        依赖于时间的模态振动为：

function vibrating_string
% VIBRATING_STRING   Wave equation in one dimension.
%   Solutions of the one-dimensional wave equation are expressed
%   as a time-varying weighted sums of the first nine modes.
%   The one-dimensional domain is an interval of length pi, so the k-th
%   frequency and mode are lambda(k) = k^2 and u(k) = sin(k*x).m = 11;   % Number of grid points
speed = 1;
bvals = [0; 0; 0; 0];
t = 0;while bvals(4) == 0% Initialize figureshgclfset(gcf,'colormap',hot(64))b = zeros(1,4);for k = 1:4b(k) = uicontrol('style','toggle','value',bvals(k), ...'units','normal','position',[.08+.15*k .03 .14 .05]);endset(b(1),'string','modes/wave')set(b(2),'string','slower')set(b(3),'string','faster')set(b(4),'string','stop')if bvals(2)==1speed = speed/sqrt(2);set(b(2),'value',0);endif bvals(3)==1speed = speed*sqrt(2);set(b(3),'value',0);endbvals = cell2mat(get(b,'value'));modes = bvals(1)==0;x = (0:4*m)/(4*m)*pi;orange = [1 1/3 0];gray = get(gcf,'color');if modes% Modesfor k = 1:9subplot(3,3,k)h(k) = plot(x,zeros(size(x)));axis([0 pi -3/2 3/2])set(h(k),'color',orange,'linewidth',3)set(gca,'color',gray','xtick',[],'ytick',[])enddelta = 0.005*speed;bvs = bvals;while all(bvs == bvals)t = t + delta;for k = 1:9u = sin(k*t)*sin(k*x);set(h(k),'ydata',u)enddrawnowbvs = cell2mat(get(b,'value'));endelse% Wave% The coefficients.a = 1./(1:9);h = plot(x,zeros(size(x)));axis([0 pi -9/4 9/4])set(h,'color',orange,'linewidth',3)set(gca,'color',gray','xtick',[],'ytick',[])delta = 0.005*speed;bvs = bvals;while all(bvs == bvals)t = t + delta;u = zeros(size(x));for k = 1:9u = u + a(k)*sin(k*t)*sin(k*x);endset(h,'ydata',u)drawnowbvs = cell2mat(get(b,'value'));endendbvals = bvs;
end
set(b(1:3),'vis','off')
set(b(4),'string','close','callback','close(gcf)')

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利萨如曲线

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        利萨如曲线给出了音乐和弦数学描述的一些内在内容。

        二维利萨如曲线是由下面的参数方程绘制的：

        三维利萨如曲线由以下参数方程绘制：

        绘图命令可简化如下：

function lissajous(a,b)
% LISSAJOUS  2D and 3D Lissajous curves.
% x = sin(t), y = sin(a*t) or 
% x = sin(t), y = sin(a*t), z = sin(b*t).
% Usage:
%    lissajous(a,b)
%    lissajous('a','b'), to retain symbolic representation.
%    lissajous, to open gui with default values.
% Enter rational and irrational values of a and b in the edit boxes.
% Enter b = 0 for 2D curves.
%    eg:
%    a = 3/2 , b = 5/4
%    a = 2^(7/12), b = 2^(4/12)
%    a = 5/4 , b = 0
%
% See Wikipedia article: http://en.wikipedia.org/wiki/Lissajous_curve % Default value of the parameters.if nargin == 0a = '3/2';b = '5/4';elseif nargin == 1b = 0;end% Retain both character and numeric representation of the parameters.if ischar(a)sa = a;a = str2num(a);elsesa = rats(a);endif ischar(b)sb = b;b = str2num(b);elsesb = rats(b);end% Initializeloop = true;t = 0;dt = pi/1024;initialize_graphics;% Run the loop until the 'X' toggle is pushed.while loopt = t + dt;x = sin(t);y = sin(a*t);z = sin(b*t);plotdot(x,y,z)endclose% ------------------------------------------------function initialize_graphicsclfshgset(gcf,'color','white')initialize_plotuicontrol('string','a = ','style','edit','units','normalized', ...'position',[.25,.05,.05,.05],'background','white')uicontrol('tag','a','style','edit','units','normalized', ...'position',[.30,.05,.20,.05],'background','white', ...'string',sa,'callback',@abcallback);uicontrol('string','b = ','style','edit','units','normalized', ...'position',[.55,.05,.05,.05],'background','white')uicontrol('tag','b','style','edit','units','normalized', ...'position',[.60,.05,.20,.05],'background','white', ...'string',sb,'callback',@abcallback);uicontrol('string','line','style','toggle','units','normalized', ...'tag','mode','position',[.06,.48,.05,.05],'background','white', ...'foreground','blue','callback',@modecallback);uicontrol('string','X','style','toggle','units','normalized', ...'position',[.95,.95,.04,.04],'background','white', ...'callback',@xcallback);end% ------------------------------------------------function initialize_plotmode = get(findobj('tag','mode'),'value');if isempty(mode) || mode==0plot3(0,0,0,'.','erasemode','none')elseplot3(0,0,0,'.','markersize',24,'color',[0 2/3 0])endaxis([-1 1 -1 1 -1 1])set(gca,'xtick',[],'ytick',[],'ztick',[])axis squarebox onswitch bcase 0view(2)title('sin(t), sin(a*t)')case 1view(3)title('sin(t), sin(a*t), sin(b*t)')endt = 0;end% ------------------------------------------------function plotdot(x,y,z)% Move dot on plot.set(get(gca,'child'),'xdata',x,'ydata',y,'zdata',z)drawnowend% ------------------------------------------------function abcallback(h,~)% Callback for ab controls.switch get(h,'tag')case 'a', a = str2num(get(h,'string'));case 'b', b = str2num(get(h,'string'));endinitialize_plotend% ------------------------------------------------function modecallback(h,~)% Callback for mode control.switch get(h,'value')case 0, set(h,'string','line','foreground','blue')dt = dt/4;  % Slowercase 1, set(h,'string','dot','foreground',[0 2/3 0])dt = 4*dt;  % Fasterendinitialize_plotend% ------------------------------------------------function xcallback(~,~)% Callback for X control.loop = false;endend % Lissajous

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和声与音调与和弦

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        平均乐律和纯律。平均乐律由反复求的幂次定义，纯律是由一系列分数定义的。

%% Equal temperament and just intonation[sigma.^(0:12)1 16/15 9/8 6/5 5/4 4/3 7/5 3/2 8/5 5/3 7/4 15/8 2]' 

        和弦是两个或两个以上的音符同时发声的演奏方式。在pianoex程序界面上可以选中1到12的双态开关来产生和弦，标注为0的开关总处于选中的状态。

%% C major fifth chord, equal temperament and just temperament[sigma.^[0 4 7]1 5/4 3/2]'