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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 音乐 在音乐理论中#xff0c;一个八音度#xff08;octave#xff09;是一个频率范围相差二倍的区间。在大多数西洋乐器中#xff0c;一个八音度被分为12个频率比相等的半音程#xff08;semitone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) 琴键 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flatb 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(v13)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(n13);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(k13) 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(s1);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, ...Hget(gca,userdata); pianoex(H.root)];for k 0:12H.togs(k1) uicontrol(style,toggle,units,normal, ...pos,[.05k*.05 .54 .04 .04],value,k0,background,grayc, ...callback,callback);uicontrol(style,text,units,normal,string,num2str(k), ...fontsize,12,fontweight,bold,pos,[.04k*.05 .49 .05 .05], ...background,grayc)endset(H.togs(1),value,1,background,blue)% TuningH.voice zeros(1,3);voicestring {equal,just,piano};callback [Hget(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 哆来咪 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]) 振动与调式 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))利萨如曲线 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) || mode0plot3(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和声与音调与和弦 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]
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