[{"data":1,"prerenderedAt":4318},["ShallowReactive",2],{"project-text-tuner-pt":3},{"id":4,"title":5,"body":6,"description":4310,"extension":4311,"meta":4312,"navigation":4313,"path":4314,"seo":4315,"stem":4316,"__hash__":4317},"projects_pt\u002Fpt\u002Fprojetos\u002Ftuner.md","Tuner",{"type":7,"value":8,"toc":4293},"minimark",[9,14,18,23,106,196,237,311,315,418,433,436,440,489,492,583,619,705,743,835,970,1099,1174,1322,1355,1531,1538,1648,1712,1728,1732,1738,1819,1829,1933,1959,2072,2153,2227,2383,2417,2507,2587,2650,2654,2685,2806,2874,3000,3012,3016,3070,3076,3256,3319,3396,3416,3525,3532,3557,3572,3659,3663,3693,3731,3734,3774,3790,3879,3886,3924,3928,3935,3949,4042,4058,4131,4206,4209,4236,4240,4281,4284],[10,11,13],"h2",{"id":12},"como-funciona-a-matemática-por-trás","Como funciona: a matemática por trás",[15,16,17],"p",{},"Aqui explico, com rigor mas sem mistério, todo o caminho do som até a nota. As fórmulas estão na notação de verdade, mas cada uma vem com a ideia intuitiva, um exemplo numérico e, em vários pontos, um experimento interativo para você mexer e ver acontecer.",[19,20,22],"h3",{"id":21},"_1-do-som-ao-sinal-amostragem","1. Do som ao sinal: amostragem",[15,24,25,26,51,52,56,57,68,69,86,87,105],{},"O som é uma variação contínua da pressão do ar no tempo, uma função ",[27,28,29],"math",{},[30,31,32,35],"mrow",{},[33,34,15],"mi",{},[30,36,37,44,47],{},[38,39,43],"mo",{"fence":40,"form":41,"stretchy":42},"true","prefix","false","(",[33,45,46],{},"t",[38,48,50],{"fence":40,"form":49,"stretchy":42},"postfix",")",". O microfone mede essa pressão muitas vezes por segundo. Essa taxa é a ",[53,54,55],"strong",{},"frequência de amostragem"," ",[27,58,59],{},[60,61,62,65],"msub",{},[33,63,64],{},"f",[33,66,67],{},"s",": no navegador, normalmente 44,1 kHz ou 48 kHz, conforme o aparelho (o afinador acima mostra a do seu enquanto o microfone está ligado). Nos exemplos deste texto, uso ",[27,70,71],{},[30,72,73,79,82],{},[60,74,75,77],{},[33,76,64],{},[33,78,67],{},[38,80,81],{},"=",[83,84,85],"mn",{},"44100"," Hz. O resultado é uma sequência de números ",[27,88,89],{},[30,90,91,94],{},[33,92,93],{},"x",[30,95,96,99,102],{},[38,97,98],{"fence":40,"form":41,"stretchy":42},"[",[33,100,101],{},"n",[38,103,104],{"fence":40,"form":49,"stretchy":42},"]",":",[27,107,112],{"display":108,"className":109,"style":111},"block",[110],"tml-display","display:block math;",[113,114,117,157],"mtable",{"displaystyle":40,"className":115},[116],"tml-jot",[118,119,120],"mtr",{},[121,122,124],"mtd",{"style":123},"padding-left:0em;padding-right:0em;",[30,125,126,128,136,138,140],{},[33,127,93],{},[30,129,130,132,134],{},[38,131,98],{"fence":40,"form":41,"stretchy":42},[33,133,101],{},[38,135,104],{"fence":40,"form":49,"stretchy":42},[38,137,81],{},[33,139,15],{},[30,141,142,144,155],{},[38,143,43],{"fence":40,"form":41,"stretchy":40},[145,146,147,149],"mfrac",{},[33,148,101],{},[60,150,151,153],{},[33,152,64],{},[33,154,67],{},[38,156,50],{"fence":40,"form":49,"stretchy":40},[118,158,159],{},[121,160,161],{"style":123},[30,162,163,165,167,170,173,176,178,181,183,186,188,191,194],{},[33,164,101],{},[38,166,81],{},[83,168,169],{},"0",[38,171,172],{"separator":40},",",[83,174,175],{},"1",[38,177,172],{"separator":40},[83,179,180],{},"2",[38,182,172],{"separator":40},[38,184,185],{},"…",[38,187,172],{"separator":40},[33,189,190],{},"N",[38,192,193],{},"−",[83,195,175],{},[15,197,198,199,210,211,232,233,236],{},"Trabalho com blocos de ",[27,200,201],{},[30,202,203,205,207],{},[33,204,190],{},[38,206,81],{},[83,208,209],{},"2048"," amostras. A 44,1 kHz, cada bloco cobre ",[27,212,213],{},[30,214,215,217,220,226,229],{},[33,216,190],{},[33,218,219],{},"\u002F",[60,221,222,224],{},[33,223,64],{},[33,225,67],{},[38,227,228],{},"≈",[83,230,231],{},"46"," ms de som, tempo suficiente para conter ",[53,234,235],{},"vários ciclos"," mesmo das notas mais graves do trompete.",[238,239,240],"callout",{},[15,241,242,245,246,249,250,282,283,287,288,291,292,306,307,310],{},[53,243,244],{},"Por que isso importa?"," Um som ",[53,247,248],{},"periódico"," obedece ",[27,251,252],{},[30,253,254,256,264,266,268],{},[33,255,93],{},[30,257,258,260,262],{},[38,259,43],{"fence":40,"form":41,"stretchy":42},[33,261,46],{},[38,263,50],{"fence":40,"form":49,"stretchy":42},[38,265,81],{},[33,267,93],{},[30,269,270,272,274,277,280],{},[38,271,43],{"fence":40,"form":41,"stretchy":42},[33,273,46],{},[38,275,276],{},"+",[33,278,279],{},"T",[38,281,50],{"fence":40,"form":49,"stretchy":42},", em que ",[27,284,285],{},[33,286,279],{}," é o ",[53,289,290],{},"período"," (segundos por ciclo). A frequência é só o seu inverso: ",[27,293,294],{},[30,295,296,298,300],{},[33,297,64],{},[38,299,81],{},[145,301,302,304],{},[83,303,175],{},[33,305,279],{},". Achar a nota é ",[53,308,309],{},"achar o período",".",[19,312,314],{"id":313},"_2-frequência-e-período-na-prática","2. Frequência e período, na prática",[15,316,317,318,357,358,361,362,372,373,376,377,381,382,396,397,417],{},"A onda mais simples é a senoide ",[27,319,320],{},[30,321,322,324,332,334,342],{},[33,323,93],{},[30,325,326,328,330],{},[38,327,43],{"fence":40,"form":41,"stretchy":42},[33,329,46],{},[38,331,50],{"fence":40,"form":49,"stretchy":42},[38,333,81],{},[30,335,336,339],{},[33,337,338],{},"sin",[38,340,341],{},"⁡",[30,343,344,346,348,351,353,355],{},[38,345,43],{"fence":40,"form":41,"stretchy":42},[83,347,180],{},[33,349,350],{},"π",[33,352,64],{},[33,354,46],{},[38,356,50],{"fence":40,"form":49,"stretchy":42},". Lendo a fórmula em palavras: o valor ",[53,359,360],{},"oscila entre −1 e +1",", e o termo ",[27,363,364],{},[30,365,366,368,370],{},[83,367,180],{},[33,369,350],{},[33,371,64],{}," diz ",[53,374,375],{},"quão rápido",". Quanto maior ",[27,378,379],{},[33,380,64],{},", mais ciclos por segundo. O período é o tempo de um ciclo, ",[27,383,384],{},[30,385,386,388,390],{},[33,387,279],{},[38,389,81],{},[145,391,392,394],{},[83,393,175],{},[33,395,64],{},". Por exemplo, um Lá de 440 Hz tem ",[27,398,399],{},[30,400,401,403,405,407,409,412,414],{},[33,402,279],{},[38,404,81],{},[83,406,175],{},[33,408,219],{},[83,410,411],{},"440",[38,413,228],{},[83,415,416],{},"2,3"," ms por ciclo.",[15,419,420,421,424,425,428,429,432],{},"No experimento abaixo, a ",[53,422,423],{},"janela de tempo é fixa em 20 ms",", como a tela de um osciloscópio. Por isso, ao ",[53,426,427],{},"aumentar a frequência",", mais ciclos se espremem na tela e a faixa destacada, que marca ",[53,430,431],{},"um período",", encolhe. Ao diminuir, acontece o contrário. Mexa e veja:",[434,435],"tuner-wave",{},[19,437,439],{"id":438},"_3-medindo-o-quanto-a-onda-se-parece-consigo-mesma","3. Medindo o quanto a onda se parece consigo mesma",[15,441,442,443,473,474,477,478,481,482,485,486,488],{},"O sinal é a lista de amostras ",[27,444,445],{},[30,446,447,453,455,461,463,469,471],{},[60,448,449,451],{},[33,450,93],{},[83,452,169],{},[38,454,172],{"separator":40},[60,456,457,459],{},[33,458,93],{},[83,460,175],{},[38,462,172],{"separator":40},[60,464,465,467],{},[33,466,93],{},[83,468,180],{},[38,470,172],{"separator":40},[38,472,185],{}," A ideia central do MPM é que um som afinado ",[53,475,476],{},"se repete",". Então, se eu pegar o sinal e uma ",[53,479,480],{},"cópia dele deslocada"," por uma certa quantidade de amostras, as duas versões ficam ",[53,483,484],{},"quase idênticas"," exatamente quando o deslocamento for igual a ",[53,487,431],{},". Achar o período é achar esse deslocamento.",[15,490,491],{},"Antes das fórmulas, o que cada símbolo significa:",[238,493,494],{},[495,496,497,514,525,541,576],"ul",{},[498,499,500,509,510,513],"li",{},[27,501,502],{},[60,503,504,506],{},[33,505,93],{},[33,507,508],{},"j",": o valor da ",[53,511,512],{},"j-ésima amostra"," (a altura da onda naquele instante, entre −1 e +1).",[498,515,516,520,521,524],{},[27,517,518],{},[33,519,508],{},": o ",[53,522,523],{},"índice"," da amostra (0, 1, 2, …), a posição no tempo.",[498,526,527,532,533,536,537,310],{},[27,528,529],{},[33,530,531],{},"τ"," (tau): o ",[53,534,535],{},"deslocamento"," em amostras entre a onda e a cópia. É o que estamos procurando; em inglês, ",[538,539,540],"em",{},"lag",[498,542,543,557,558,562,563,566,567,575],{},[27,544,545],{},[60,546,547,549],{},[33,548,93],{},[30,550,551,553,555],{},[33,552,508],{},[38,554,276],{},[33,556,531],{},": a amostra ",[27,559,560],{},[33,561,531],{}," posições ",[53,564,565],{},"à frente"," de ",[27,568,569],{},[60,570,571,573],{},[33,572,93],{},[33,574,508],{},", a \"cópia deslocada\".",[498,577,578,582],{},[27,579,580],{},[33,581,190],{},": o tamanho do bloco (2048 amostras).",[15,584,585,586,589,590,593,594,618],{},"A primeira medida natural de quão ",[53,587,588],{},"diferentes"," a onda e sua cópia estão é a ",[53,591,592],{},"função de diferença quadrática"," (SDF, na sigla em inglês). Para cada amostra, calculamos a diferença ",[27,595,596],{},[30,597,598,604,606],{},[60,599,600,602],{},[33,601,93],{},[33,603,508],{},[38,605,193],{},[60,607,608,610],{},[33,609,93],{},[30,611,612,614,616],{},[33,613,508],{},[38,615,276],{},[33,617,531],{},", elevamos ao quadrado e somamos tudo:",[27,620,622],{"display":108,"className":621,"style":111},[110],[30,623,624,637,645,647,675],{},[625,626,627,630],"msup",{},[33,628,629],{},"d",[38,631,636],{"lspace":632,"rspace":632,"className":633},"0em",[634,635],"tml-prime","tml-lrg-pad","′",[30,638,639,641,643],{},[38,640,43],{"fence":40,"form":41,"stretchy":42},[33,642,531],{},[38,644,50],{"fence":40,"form":49,"stretchy":42},[38,646,81],{},[30,648,649],{},[650,651,652,655,663],"munderover",{},[38,653,654],{"movablelimits":42},"∑",[30,656,657,659,661],{},[33,658,508],{},[38,660,81],{},[83,662,169],{},[30,664,665,667,669,671,673],{},[33,666,190],{},[38,668,193],{},[83,670,175],{},[38,672,193],{},[33,674,531],{},[625,676,677,703],{},[30,678,679,681,687,689,701],{},[38,680,43],{"fence":40,"form":41,"stretchy":40},[60,682,683,685],{},[33,684,93],{},[33,686,508],{},[38,688,193],{},[60,690,691,693],{},[33,692,93],{},[30,694,695,697,699],{},[33,696,508],{},[38,698,276],{},[33,700,531],{},[38,702,50],{"fence":40,"form":49,"stretchy":40},[83,704,180],{},[15,706,707,710,711,714,715,718,719,738,739,742],{},[53,708,709],{},"Por que elevar ao quadrado?"," Por dois motivos. O quadrado torna toda diferença ",[53,712,713],{},"positiva"," (tanto faz se a cópia está acima ou abaixo: o desencaixe conta como erro de qualquer jeito) e ",[53,716,717],{},"penaliza mais"," os desencaixes grandes. Assim, ",[27,720,721],{},[30,722,723,730],{},[625,724,725,727],{},[33,726,629],{},[38,728,636],{"lspace":632,"rspace":632,"className":729},[634,635],[30,731,732,734,736],{},[38,733,43],{"fence":40,"form":41,"stretchy":42},[33,735,531],{},[38,737,50],{"fence":40,"form":49,"stretchy":42}," é sempre maior ou igual a zero e funciona como um ",[53,740,741],{},"placar de erro",": quanto menor, melhor o encaixe.",[15,744,745,746,756,757,771,772,782,783,56,797,800,801,811,812,56,815,819,820,823,824,834],{},"E por que a soma vai de ",[27,747,748],{},[30,749,750,752,754],{},[33,751,508],{},[38,753,81],{},[83,755,169],{}," só até ",[27,758,759],{},[30,760,761,763,765,767,769],{},[33,762,190],{},[38,764,193],{},[83,766,175],{},[38,768,193],{},[33,770,531],{},", e não até ",[27,773,774],{},[30,775,776,778,780],{},[33,777,190],{},[38,779,193],{},[83,781,175],{},"? Porque ",[27,784,785],{},[60,786,787,789],{},[33,788,93],{},[30,790,791,793,795],{},[33,792,508],{},[38,794,276],{},[33,796,531],{},[53,798,799],{},"precisa existir",": se ",[27,802,803],{},[30,804,805,807,809],{},[33,806,508],{},[38,808,276],{},[33,810,531],{}," passasse do fim do bloco, não haveria amostra para comparar. Consequência importante: quanto ",[53,813,814],{},"maior o deslocamento",[27,816,817],{},[33,818,531],{},", ",[53,821,822],{},"menos pares"," sobram para somar (apenas ",[27,825,826],{},[30,827,828,830,832],{},[33,829,190],{},[38,831,193],{},[33,833,531],{}," deles). Guarde isto, porque vai voltar já já.",[15,836,837,838,842,843,851,852,866,867,890,891,894,895,898,899,945,946,56,949,105],{},"Quando ",[27,839,840],{},[33,841,531],{}," chega ao período, cada ",[27,844,845],{},[60,846,847,849],{},[33,848,93],{},[33,850,508],{}," encontra um ",[27,853,854],{},[60,855,856,858],{},[33,857,93],{},[30,859,860,862,864],{},[33,861,508],{},[38,863,276],{},[33,865,531],{}," quase igual, as diferenças vão a zero e ",[27,868,869],{},[30,870,871,878,886,888],{},[625,872,873,875],{},[33,874,629],{},[38,876,636],{"lspace":632,"rspace":632,"className":877},[634,635],[30,879,880,882,884],{},[38,881,43],{"fence":40,"form":41,"stretchy":42},[33,883,531],{},[38,885,50],{"fence":40,"form":49,"stretchy":42},[38,887,228],{},[83,889,169],{},", formando um \"vale\". Para passar da ideia de ",[538,892,893],{},"diferença"," para a de ",[538,896,897],{},"semelhança",", abrimos o quadrado ",[27,900,901],{},[30,902,903,921,923,929,931,933,935,937,939],{},[625,904,905,919],{},[30,906,907,909,912,914,917],{},[38,908,43],{"fence":40,"form":41,"stretchy":42},[33,910,911],{},"a",[38,913,193],{},[33,915,916],{},"b",[38,918,50],{"fence":40,"form":49,"stretchy":42},[83,920,180],{},[38,922,81],{},[625,924,925,927],{},[33,926,911],{},[83,928,180],{},[38,930,193],{},[83,932,180],{},[33,934,911],{},[33,936,916],{},[38,938,276],{},[625,940,941,943],{},[33,942,916],{},[83,944,180],{}," em cada termo da soma. Surgem três pedaços, e o terceiro é o mais importante, a ",[53,947,948],{},"autocorrelação",[27,950,951],{},[30,952,953,962],{},[625,954,955,958],{},[33,956,957],{},"r",[38,959,636],{"lspace":632,"rspace":632,"className":960},[634,961],"tml-sml-pad",[30,963,964,966,968],{},[38,965,43],{"fence":40,"form":41,"stretchy":42},[33,967,531],{},[38,969,50],{"fence":40,"form":49,"stretchy":42},[27,971,973],{"display":108,"className":972,"style":111},[110],[113,974,976,1053],{"displaystyle":40,"className":975},[116],[118,977,978,1003],{},[121,979,982],{"className":980,"style":123},[981],"tml-right",[30,983,984,991,999,1001],{},[625,985,986,988],{},[33,987,629],{},[38,989,636],{"lspace":632,"rspace":632,"className":990},[634,635],[30,992,993,995,997],{},[38,994,43],{"fence":40,"form":41,"stretchy":42},[33,996,531],{},[38,998,50],{"fence":40,"form":49,"stretchy":42},[38,1000,81],{},[30,1002],{},[121,1004,1007],{"className":1005,"style":123},[1006],"tml-left",[30,1008,1009,1018,1028,1030,1038],{},[30,1010,1011],{},[1012,1013,1014,1016],"munder",{},[38,1015,654],{"movablelimits":42},[33,1017,508],{},[1019,1020,1021,1023,1025],"msubsup",{},[33,1022,93],{},[33,1024,508],{},[83,1026,180],{"className":1027},[961],[38,1029,276],{},[30,1031,1032],{},[1012,1033,1034,1036],{},[38,1035,654],{"movablelimits":42},[33,1037,508],{},[1019,1039,1040,1042,1050],{},[33,1041,93],{},[30,1043,1044,1046,1048],{},[33,1045,508],{},[38,1047,276],{},[33,1049,531],{},[83,1051,180],{"className":1052},[961],[118,1054,1055,1060],{},[121,1056,1058],{"className":1057,"style":123},[981],[30,1059],{},[121,1061,1063],{"className":1062,"style":123},[1006],[30,1064,1065,1067,1069,1077,1083,1087],{},[38,1066,193],{"form":41,"stretchy":42},[83,1068,180],{},[30,1070,1071],{},[1012,1072,1073,1075],{},[38,1074,654],{"movablelimits":42},[33,1076,508],{},[60,1078,1079,1081],{},[33,1080,93],{},[33,1082,508],{},[1084,1085],"mspace",{"width":1086},"0.1667em",[60,1088,1089,1091],{},[33,1090,93],{},[30,1092,1093,1095,1097],{},[33,1094,508],{},[38,1096,276],{},[33,1098,531],{},[27,1100,1102],{"display":108,"className":1101,"style":111},[110],[30,1103,1104,1108,1111,1118,1126,1128,1154,1160,1162],{},[1105,1106,1107],"mtext",{},"com",[1084,1109],{"width":1110},"1em",[625,1112,1113,1115],{},[33,1114,957],{},[38,1116,636],{"lspace":632,"rspace":632,"className":1117},[634,961],[30,1119,1120,1122,1124],{},[38,1121,43],{"fence":40,"form":41,"stretchy":42},[33,1123,531],{},[38,1125,50],{"fence":40,"form":49,"stretchy":42},[38,1127,81],{},[30,1129,1130],{},[650,1131,1132,1134,1142],{},[38,1133,654],{"movablelimits":42},[30,1135,1136,1138,1140],{},[33,1137,508],{},[38,1139,81],{},[83,1141,169],{},[30,1143,1144,1146,1148,1150,1152],{},[33,1145,190],{},[38,1147,193],{},[83,1149,175],{},[38,1151,193],{},[33,1153,531],{},[60,1155,1156,1158],{},[33,1157,93],{},[33,1159,508],{},[1084,1161],{"width":1086},[60,1163,1164,1166],{},[33,1165,93],{},[30,1167,1168,1170,1172],{},[33,1169,508],{},[38,1171,276],{},[33,1173,531],{},[15,1175,1176,1177,1196,1197,1222,1223,1226,1227,1274,1275,1278,1279,1298,1299,1318,1319,310],{},"Olhe os três pedaços. Os dois primeiros, ",[27,1178,1179],{},[30,1180,1181,1187],{},[60,1182,1183,1185],{},[38,1184,654],{"movablelimits":42},[33,1186,508],{},[1019,1188,1189,1191,1193],{},[33,1190,93],{},[33,1192,508],{},[83,1194,180],{"className":1195},[961]," e ",[27,1198,1199],{},[30,1200,1201,1207],{},[60,1202,1203,1205],{},[38,1204,654],{"movablelimits":42},[33,1206,508],{},[1019,1208,1209,1211,1219],{},[33,1210,93],{},[30,1212,1213,1215,1217],{},[33,1214,508],{},[38,1216,276],{},[33,1218,531],{},[83,1220,180],{"className":1221},[961],", são a ",[53,1224,1225],{},"energia"," dos dois trechos (a soma dos quadrados das amostras): variam devagar e de forma parecida entre si. Toda a ação está no terceiro, ",[27,1228,1229],{},[30,1230,1231,1238,1246,1248,1254,1260,1262],{},[625,1232,1233,1235],{},[33,1234,957],{},[38,1236,636],{"lspace":632,"rspace":632,"className":1237},[634,961],[30,1239,1240,1242,1244],{},[38,1241,43],{"fence":40,"form":41,"stretchy":42},[33,1243,531],{},[38,1245,50],{"fence":40,"form":49,"stretchy":42},[38,1247,81],{},[60,1249,1250,1252],{},[38,1251,654],{"movablelimits":42},[33,1253,508],{},[60,1255,1256,1258],{},[33,1257,93],{},[33,1259,508],{},[1084,1261],{"width":1086},[60,1263,1264,1266],{},[33,1265,93],{},[30,1267,1268,1270,1272],{},[33,1269,508],{},[38,1271,276],{},[33,1273,531],{},". Como ele entra com sinal ",[53,1276,1277],{},"negativo"," na diferença, deixar ",[27,1280,1281],{},[30,1282,1283,1290],{},[625,1284,1285,1287],{},[33,1286,629],{},[38,1288,636],{"lspace":632,"rspace":632,"className":1289},[634,635],[30,1291,1292,1294,1296],{},[38,1293,43],{"fence":40,"form":41,"stretchy":42},[33,1295,531],{},[38,1297,50],{"fence":40,"form":49,"stretchy":42}," pequeno é o mesmo que deixar ",[27,1300,1301],{},[30,1302,1303,1310],{},[625,1304,1305,1307],{},[33,1306,957],{},[38,1308,636],{"lspace":632,"rspace":632,"className":1309},[634,961],[30,1311,1312,1314,1316],{},[38,1313,43],{"fence":40,"form":41,"stretchy":42},[33,1315,531],{},[38,1317,50],{"fence":40,"form":49,"stretchy":42}," grande. Ou seja: ",[53,1320,1321],{},"menor diferença ⟺ maior autocorrelação",[15,1323,1324,1327,1328,1331,1332,1335,1336,1339,1340,1343,1344,1346,1347,1350,1351,1354],{},[53,1325,1326],{},"Por que a autocorrelação cresce no encaixe?"," Ela ",[53,1329,1330],{},"multiplica"," cada amostra pela sua cópia. Dois números de ",[53,1333,1334],{},"mesmo sinal"," dão produto ",[53,1337,1338],{},"positivo"," (+×+ e −×−); de ",[53,1341,1342],{},"sinais opostos",", produto ",[53,1345,1277],{},". Quando a cópia está alinhada, picos batem com picos e vales com vales: quase todos os produtos ficam positivos e a soma ",[53,1348,1349],{},"dispara",". Desalinhada, ela mistura positivos e negativos que se ",[53,1352,1353],{},"cancelam",", e a soma cai para perto de zero.",[238,1356,1357],{},[15,1358,1359,1362,1363,1389,1390,1400,1401,1469,1470,1527,1528],{},[53,1360,1361],{},"Mini-exemplo concreto."," Pegue o trechinho ",[27,1364,1365],{},[30,1366,1367,1369,1371,1373,1375,1377,1379,1381,1383,1385,1387],{},[38,1368,98],{"fence":40,"form":41,"stretchy":42},[83,1370,175],{},[38,1372,172],{"separator":40},[38,1374,193],{"form":41,"stretchy":42},[83,1376,175],{},[38,1378,172],{"separator":40},[83,1380,175],{},[38,1382,172],{"separator":40},[38,1384,193],{"form":41,"stretchy":42},[83,1386,175],{},[38,1388,104],{"fence":40,"form":49,"stretchy":42},". Comparando com ele mesmo (",[27,1391,1392],{},[30,1393,1394,1396,1398],{},[33,1395,531],{},[38,1397,81],{},[83,1399,169],{},"): ",[27,1402,1403],{},[30,1404,1405,1407,1410,1412,1414,1424,1434,1436,1438,1440,1442,1444,1454,1464,1466],{},[83,1406,175],{},[38,1408,1409],{},"⋅",[83,1411,175],{},[38,1413,276],{},[30,1415,1416,1418,1420,1422],{},[38,1417,43],{"fence":40,"form":41,"stretchy":42},[38,1419,193],{"form":41,"stretchy":42},[83,1421,175],{},[38,1423,50],{"fence":40,"form":49,"stretchy":42},[30,1425,1426,1428,1430,1432],{},[38,1427,43],{"fence":40,"form":41,"stretchy":42},[38,1429,193],{"form":41,"stretchy":42},[83,1431,175],{},[38,1433,50],{"fence":40,"form":49,"stretchy":42},[38,1435,276],{},[83,1437,175],{},[38,1439,1409],{},[83,1441,175],{},[38,1443,276],{},[30,1445,1446,1448,1450,1452],{},[38,1447,43],{"fence":40,"form":41,"stretchy":42},[38,1449,193],{"form":41,"stretchy":42},[83,1451,175],{},[38,1453,50],{"fence":40,"form":49,"stretchy":42},[30,1455,1456,1458,1460,1462],{},[38,1457,43],{"fence":40,"form":41,"stretchy":42},[38,1459,193],{"form":41,"stretchy":42},[83,1461,175],{},[38,1463,50],{"fence":40,"form":49,"stretchy":42},[38,1465,81],{},[83,1467,1468],{},"4",", tudo positivo. Deslocando 1 amostra, os pares ficam ",[27,1471,1472],{},[30,1473,1474,1476,1478,1488,1490,1500,1502,1504,1506,1508,1510,1520,1522,1524],{},[83,1475,175],{},[38,1477,1409],{},[30,1479,1480,1482,1484,1486],{},[38,1481,43],{"fence":40,"form":41,"stretchy":42},[38,1483,193],{"form":41,"stretchy":42},[83,1485,175],{},[38,1487,50],{"fence":40,"form":49,"stretchy":42},[38,1489,276],{},[30,1491,1492,1494,1496,1498],{},[38,1493,43],{"fence":40,"form":41,"stretchy":42},[38,1495,193],{"form":41,"stretchy":42},[83,1497,175],{},[38,1499,50],{"fence":40,"form":49,"stretchy":42},[38,1501,1409],{},[83,1503,175],{},[38,1505,276],{},[83,1507,175],{},[38,1509,1409],{},[30,1511,1512,1514,1516,1518],{},[38,1513,43],{"fence":40,"form":41,"stretchy":42},[38,1515,193],{"form":41,"stretchy":42},[83,1517,175],{},[38,1519,50],{"fence":40,"form":49,"stretchy":42},[38,1521,81],{},[38,1523,193],{"form":41,"stretchy":42},[83,1525,1526],{},"3",": os sinais opostos se cancelam. ",[53,1529,1530],{},"Alinhado, grande; desalinhado, pequeno.",[15,1532,1533,1534,1537],{},"Essa autocorrelação \"crua\", porém, tem ",[53,1535,1536],{},"dois defeitos"," que impedem usá-la diretamente como medida de afinação:",[238,1539,1540],{},[15,1541,1542,1545,1546,1549,1550,1553,1554,1575,1576,1600,1601,1604,1605,1619,1620,1639,1640,1643,1644,1647],{},[53,1543,1544],{},"Defeito 1: depende do volume."," Toque a ",[538,1547,1548],{},"mesma"," nota ",[53,1551,1552],{},"mais forte"," e todas as amostras crescem na mesma proporção, digamos ",[27,1555,1556],{},[30,1557,1558,1564,1567,1569],{},[60,1559,1560,1562],{},[33,1561,93],{},[33,1563,508],{},[38,1565,1566],{},"→",[83,1568,180],{},[60,1570,1571,1573],{},[33,1572,93],{},[33,1574,508],{},". Aí cada produto ",[27,1577,1578],{},[30,1579,1580,1586,1588],{},[60,1581,1582,1584],{},[33,1583,93],{},[33,1585,508],{},[1084,1587],{"width":1086},[60,1589,1590,1592],{},[33,1591,93],{},[30,1593,1594,1596,1598],{},[33,1595,508],{},[38,1597,276],{},[33,1599,531],{}," fica ",[53,1602,1603],{},"4 vezes maior"," (",[27,1606,1607],{},[30,1608,1609,1611,1613,1615,1617],{},[83,1610,180],{},[38,1612,1409],{},[83,1614,180],{},[38,1616,81],{},[83,1618,1468],{},"), e ",[27,1621,1622],{},[30,1623,1624,1631],{},[625,1625,1626,1628],{},[33,1627,957],{},[38,1629,636],{"lspace":632,"rspace":632,"className":1630},[634,961],[30,1632,1633,1635,1637],{},[38,1634,43],{"fence":40,"form":41,"stretchy":42},[33,1636,531],{},[38,1638,50],{"fence":40,"form":49,"stretchy":42}," também. O número muda só porque o som está mais alto, mesmo com o ",[53,1641,1642],{},"encaixe idêntico",". Uma boa medida de semelhança ",[53,1645,1646],{},"não"," deveria depender do volume.",[238,1649,1650],{},[15,1651,1652,1660,1661,1671,1672,1676,1677,1680,1681,56,1700,1703,1704,1707,1708,1711],{},[53,1653,1654,1655,1659],{},"Defeito 2: encolhe conforme ",[27,1656,1657],{},[33,1658,531],{}," cresce."," Lembra que a soma tem só ",[27,1662,1663],{},[30,1664,1665,1667,1669],{},[33,1666,190],{},[38,1668,193],{},[33,1670,531],{}," parcelas? Com ",[27,1673,1674],{},[33,1675,531],{}," maior, há ",[53,1678,1679],{},"menos produtos"," sendo somados, então ",[27,1682,1683],{},[30,1684,1685,1692],{},[625,1686,1687,1689],{},[33,1688,957],{},[38,1690,636],{"lspace":632,"rspace":632,"className":1691},[634,961],[30,1693,1694,1696,1698],{},[38,1695,43],{"fence":40,"form":41,"stretchy":42},[33,1697,531],{},[38,1699,50],{"fence":40,"form":49,"stretchy":42},[53,1701,1702],{},"diminui sozinha",", mesmo num som perfeitamente periódico. Resultado: o pico no ",[53,1705,1706],{},"dobro"," do período nasce menor que o do período, formando uma \"ladeira\" que pode ",[53,1709,1710],{},"esconder"," o pico certo ou favorecer um errado.",[15,1713,1714,1715,1718,1719,1722,1723,1727],{},"Os dois defeitos têm a ",[53,1716,1717],{},"mesma cura",": ",[53,1720,1721],{},"normalizar",", ou seja, dividir pela energia dos trechos comparados, para o resultado virar um número entre −1 e 1 que não depende do volume nem de ",[27,1724,1725],{},[33,1726,531],{},". É exatamente o que a NSDF faz, no próximo passo.",[19,1729,1731],{"id":1730},"_4-a-nsdf-o-coração-do-mpm","4. A NSDF: o coração do MPM",[15,1733,1734,1735,1737],{},"McLeod e Wyvill resolvem os dois defeitos dividindo a autocorrelação pela ",[53,1736,1225],{}," dos trechos comparados. Definindo",[27,1739,1741],{"display":108,"className":1740,"style":111},[110],[30,1742,1743,1751,1759,1761,1787],{},[625,1744,1745,1748],{},[33,1746,1747],{},"m",[38,1749,636],{"lspace":632,"rspace":632,"className":1750},[634],[30,1752,1753,1755,1757],{},[38,1754,43],{"fence":40,"form":41,"stretchy":42},[33,1756,531],{},[38,1758,50],{"fence":40,"form":49,"stretchy":42},[38,1760,81],{},[30,1762,1763],{},[650,1764,1765,1767,1775],{},[38,1766,654],{"movablelimits":42},[30,1768,1769,1771,1773],{},[33,1770,508],{},[38,1772,81],{},[83,1774,169],{},[30,1776,1777,1779,1781,1783,1785],{},[33,1778,190],{},[38,1780,193],{},[83,1782,175],{},[38,1784,193],{},[33,1786,531],{},[30,1788,1789,1791,1800,1802,1817],{},[38,1790,43],{"fence":40,"form":41,"stretchy":40},[1019,1792,1793,1795,1797],{},[33,1794,93],{},[33,1796,508],{},[83,1798,180],{"className":1799},[961],[38,1801,276],{},[1019,1803,1804,1806,1814],{},[33,1805,93],{},[30,1807,1808,1810,1812],{},[33,1809,508],{},[38,1811,276],{},[33,1813,531],{},[83,1815,180],{"className":1816},[961],[38,1818,50],{"fence":40,"form":49,"stretchy":40},[15,1820,1821,1822,1604,1825,1828],{},"obtemos a ",[53,1823,1824],{},"NSDF",[538,1826,1827],{},"normalized square difference function",", ou função de diferença quadrática normalizada):",[1830,1831,1832],"key-formula",{},[27,1833,1835],{"display":108,"className":1834,"style":111},[110],[113,1836,1838,1901],{"displaystyle":40,"className":1837},[116],[118,1839,1840],{},[121,1841,1842],{"style":123},[30,1843,1844,1851,1859,1861],{},[625,1845,1846,1848],{},[33,1847,101],{},[38,1849,636],{"lspace":632,"rspace":632,"className":1850},[634],[30,1852,1853,1855,1857],{},[38,1854,43],{"fence":40,"form":41,"stretchy":42},[33,1856,531],{},[38,1858,50],{"fence":40,"form":49,"stretchy":42},[38,1860,81],{},[145,1862,1863,1884],{},[30,1864,1865,1867,1869,1876],{},[83,1866,180],{},[1084,1868],{"width":1086},[625,1870,1871,1873],{},[33,1872,957],{},[38,1874,636],{"lspace":632,"rspace":632,"className":1875},[634,961],[30,1877,1878,1880,1882],{},[38,1879,43],{"fence":40,"form":41,"stretchy":42},[33,1881,531],{},[38,1883,50],{"fence":40,"form":49,"stretchy":42},[30,1885,1886,1893],{},[625,1887,1888,1890],{},[33,1889,1747],{},[38,1891,636],{"lspace":632,"rspace":632,"className":1892},[634],[30,1894,1895,1897,1899],{},[38,1896,43],{"fence":40,"form":41,"stretchy":42},[33,1898,531],{},[38,1900,50],{"fence":40,"form":49,"stretchy":42},[118,1902,1903],{},[121,1904,1905],{"style":123},[30,1906,1907,1909,1911,1914,1921,1929,1931],{},[38,1908,193],{"form":41,"stretchy":42},[83,1910,175],{},[38,1912,1913],{},"≤",[625,1915,1916,1918],{},[33,1917,101],{},[38,1919,636],{"lspace":632,"rspace":632,"className":1920},[634],[30,1922,1923,1925,1927],{},[38,1924,43],{"fence":40,"form":41,"stretchy":42},[33,1926,531],{},[38,1928,50],{"fence":40,"form":49,"stretchy":42},[38,1930,1913],{},[83,1932,175],{},[15,1934,1935,1936,1955,1956,1958],{},"Dividir pela energia ",[27,1937,1938],{},[30,1939,1940,1947],{},[625,1941,1942,1944],{},[33,1943,1747],{},[38,1945,636],{"lspace":632,"rspace":632,"className":1946},[634],[30,1948,1949,1951,1953],{},[38,1950,43],{"fence":40,"form":41,"stretchy":42},[33,1952,531],{},[38,1954,50],{"fence":40,"form":49,"stretchy":42}," conserta os ",[53,1957,1536],{}," do passo anterior de uma só vez:",[495,1960,1961,2036],{},[498,1962,1963,1966,1967,1987,1988,2011,2012,2031,2032,2035],{},[53,1964,1965],{},"Volume:"," se o som dobra (",[27,1968,1969],{},[30,1970,1971,1977,1979,1981],{},[60,1972,1973,1975],{},[33,1974,93],{},[33,1976,508],{},[38,1978,1566],{},[83,1980,180],{},[60,1982,1983,1985],{},[33,1984,93],{},[33,1986,508],{},"), o numerador ",[27,1989,1990],{},[30,1991,1992,1994,1996,2003],{},[83,1993,180],{},[1084,1995],{"width":1086},[625,1997,1998,2000],{},[33,1999,957],{},[38,2001,636],{"lspace":632,"rspace":632,"className":2002},[634,961],[30,2004,2005,2007,2009],{},[38,2006,43],{"fence":40,"form":41,"stretchy":42},[33,2008,531],{},[38,2010,50],{"fence":40,"form":49,"stretchy":42}," fica 4 vezes maior, mas a energia ",[27,2013,2014],{},[30,2015,2016,2023],{},[625,2017,2018,2020],{},[33,2019,1747],{},[38,2021,636],{"lspace":632,"rspace":632,"className":2022},[634],[30,2024,2025,2027,2029],{},[38,2026,43],{"fence":40,"form":41,"stretchy":42},[33,2028,531],{},[38,2030,50],{"fence":40,"form":49,"stretchy":42}," também, porque as duas são feitas de produtos das mesmas amostras. Os dois fatores ",[53,2033,2034],{},"se cancelam"," na divisão e o resultado não muda.",[498,2037,2038,2045,2046,1604,2049,2059,2060,2064,2065,2068,2069,310],{},[53,2039,2040,2041,105],{},"Encolhimento com ",[27,2042,2043],{},[33,2044,531],{}," numerador e denominador têm o ",[53,2047,2048],{},"mesmo número de parcelas",[27,2050,2051],{},[30,2052,2053,2055,2057],{},[33,2054,190],{},[38,2056,193],{},[33,2058,531],{},"). Quando ",[27,2061,2062],{},[33,2063,531],{}," cresce e sobram menos termos, os dois encolhem ",[53,2066,2067],{},"juntos",", na mesma proporção, então a razão entre eles ",[53,2070,2071],{},"continua justa",[15,2073,2074,2075,2078,2079,2102,2103,2126,2127,2152],{},"O resultado é ",[53,2076,2077],{},"adimensional"," e fica sempre entre −1 e +1: ",[27,2080,2081],{},[30,2082,2083,2090,2098,2100],{},[625,2084,2085,2087],{},[33,2086,101],{},[38,2088,636],{"lspace":632,"rspace":632,"className":2089},[634],[30,2091,2092,2094,2096],{},[38,2093,43],{"fence":40,"form":41,"stretchy":42},[33,2095,531],{},[38,2097,50],{"fence":40,"form":49,"stretchy":42},[38,2099,81],{},[83,2101,175],{}," é o encaixe perfeito (cópia idêntica), ",[27,2104,2105],{},[30,2106,2107,2114,2122,2124],{},[625,2108,2109,2111],{},[33,2110,101],{},[38,2112,636],{"lspace":632,"rspace":632,"className":2113},[634],[30,2115,2116,2118,2120],{},[38,2117,43],{"fence":40,"form":41,"stretchy":42},[33,2119,531],{},[38,2121,50],{"fence":40,"form":49,"stretchy":42},[38,2123,81],{},[83,2125,169],{}," indica que não há relação e ",[27,2128,2129],{},[30,2130,2131,2138,2146,2148,2150],{},[625,2132,2133,2135],{},[33,2134,101],{},[38,2136,636],{"lspace":632,"rspace":632,"className":2137},[634],[30,2139,2140,2142,2144],{},[38,2141,43],{"fence":40,"form":41,"stretchy":42},[33,2143,531],{},[38,2145,50],{"fence":40,"form":49,"stretchy":42},[38,2147,81],{},[38,2149,193],{"form":41,"stretchy":42},[83,2151,175],{}," seria o oposto exato (onda espelhada).",[15,2154,2155,2158,2159,2162,2163,2166,2167,56,2175,2178,2179,819,2191,819,2203,2215,2216,2219,2220,2223,2224,105],{},[53,2156,2157],{},"Um aparte importante: o que são harmônicos?"," Uma ",[53,2160,2161],{},"senoide pura"," é o som mais \"limpo\" que existe (uma flauta chega perto). Mas quase todo instrumento real produz, ",[53,2164,2165],{},"ao mesmo tempo",", a frequência fundamental ",[27,2168,2169],{},[60,2170,2171,2173],{},[33,2172,64],{},[83,2174,169],{},[53,2176,2177],{},"e"," múltiplos inteiros dela: ",[27,2180,2181],{},[30,2182,2183,2185],{},[83,2184,180],{},[60,2186,2187,2189],{},[33,2188,64],{},[83,2190,169],{},[27,2192,2193],{},[30,2194,2195,2197],{},[83,2196,1526],{},[60,2198,2199,2201],{},[33,2200,64],{},[83,2202,169],{},[27,2204,2205],{},[30,2206,2207,2209],{},[83,2208,1468],{},[60,2210,2211,2213],{},[33,2212,64],{},[83,2214,169],{},"… Essas frequências extras são os ",[53,2217,2218],{},"harmônicos",", e a \"receita\" deles (quais são fortes, quais são fracos) é o que dá o ",[53,2221,2222],{},"timbre",", o que faz um trompete soar diferente de um clarinete tocando a mesma nota. O sinal vira uma ",[53,2225,2226],{},"soma de senoides",[27,2228,2230],{"display":108,"className":2229,"style":111},[110],[113,2231,2233,2283,2331],{"displaystyle":40,"className":2232},[116],[118,2234,2235,2254],{},[121,2236,2238],{"className":2237,"style":123},[981],[30,2239,2240,2242,2250,2252],{},[33,2241,93],{},[30,2243,2244,2246,2248],{},[38,2245,43],{"fence":40,"form":41,"stretchy":42},[33,2247,46],{},[38,2249,50],{"fence":40,"form":49,"stretchy":42},[38,2251,81],{},[30,2253],{},[121,2255,2257],{"className":2256,"style":123},[1006],[30,2258,2259,2265],{},[30,2260,2261,2263],{},[33,2262,338],{},[38,2264,341],{},[30,2266,2267,2269,2271,2273,2279,2281],{},[38,2268,43],{"fence":40,"form":41,"stretchy":42},[83,2270,180],{},[33,2272,350],{},[60,2274,2275,2277],{},[33,2276,64],{},[83,2278,169],{},[33,2280,46],{},[38,2282,50],{"fence":40,"form":49,"stretchy":42},[118,2284,2285,2290],{},[121,2286,2288],{"className":2287,"style":123},[981],[30,2289],{},[121,2291,2293],{"className":2292,"style":123},[1006],[30,2294,2295,2297,2303,2309],{},[38,2296,276],{"form":41,"stretchy":42},[60,2298,2299,2301],{},[33,2300,911],{},[83,2302,180],{},[30,2304,2305,2307],{},[33,2306,338],{},[38,2308,341],{},[30,2310,2311,2313,2315,2317,2319,2321,2327,2329],{},[38,2312,43],{"fence":40,"form":41,"stretchy":42},[83,2314,180],{},[33,2316,350],{},[38,2318,1409],{},[83,2320,180],{},[60,2322,2323,2325],{},[33,2324,64],{},[83,2326,169],{},[33,2328,46],{},[38,2330,50],{"fence":40,"form":49,"stretchy":42},[118,2332,2333,2338],{},[121,2334,2336],{"className":2335,"style":123},[981],[30,2337],{},[121,2339,2341],{"className":2340,"style":123},[1006],[30,2342,2343,2345,2351,2357,2379,2381],{},[38,2344,276],{"form":41,"stretchy":42},[60,2346,2347,2349],{},[33,2348,911],{},[83,2350,1526],{},[30,2352,2353,2355],{},[33,2354,338],{},[38,2356,341],{},[30,2358,2359,2361,2363,2365,2367,2369,2375,2377],{},[38,2360,43],{"fence":40,"form":41,"stretchy":42},[83,2362,180],{},[33,2364,350],{},[38,2366,1409],{},[83,2368,1526],{},[60,2370,2371,2373],{},[33,2372,64],{},[83,2374,169],{},[33,2376,46],{},[38,2378,50],{"fence":40,"form":49,"stretchy":42},[38,2380,276],{},[38,2382,185],{},[15,2384,2385,2386,819,2394,2402,2403,2406,2407,2416],{},"Aqui ",[27,2387,2388],{},[60,2389,2390,2392],{},[33,2391,911],{},[83,2393,180],{},[27,2395,2396],{},[60,2397,2398,2400],{},[33,2399,911],{},[83,2401,1526],{},", … são as ",[53,2404,2405],{},"amplitudes"," de cada harmônico, a tal \"receita\": quanto maior ",[27,2408,2409],{},[60,2410,2411,2413],{},[33,2412,911],{},[33,2414,2415],{},"k",", mais forte aquele harmônico soa (o fundamental entra com amplitude 1). É essa mistura que define o timbre.",[15,2418,2419,2420,56,2423,2441,2442,819,2454,2466,2467,2470,2471,2475,2476,2479,2480,2483,2484,2492,2493,1196,2496,2499,2500,2503,2504,310],{},"O detalhe crucial: todos esses harmônicos compartilham o ",[53,2421,2422],{},"mesmo período",[27,2424,2425],{},[30,2426,2427,2429,2431,2433,2435],{},[33,2428,279],{},[38,2430,81],{},[83,2432,175],{},[33,2434,219],{},[60,2436,2437,2439],{},[33,2438,64],{},[83,2440,169],{},", porque ",[27,2443,2444],{},[30,2445,2446,2448],{},[83,2447,180],{},[60,2449,2450,2452],{},[33,2451,64],{},[83,2453,169],{},[27,2455,2456],{},[30,2457,2458,2460],{},[83,2459,1526],{},[60,2461,2462,2464],{},[33,2463,64],{},[83,2465,169],{},"… completam ciclos ",[53,2468,2469],{},"inteiros"," dentro de ",[27,2472,2473],{},[33,2474,279],{},". Somá-los muda só o ",[53,2477,2478],{},"formato"," de cada período (a onda fica mais pontuda e cheia de detalhes), mas ",[53,2481,2482],{},"não muda o período",". Por isso, a altura da nota continua sendo ",[27,2485,2486],{},[60,2487,2488,2490],{},[33,2489,64],{},[83,2491,169],{},", não importa o timbre. No experimento, troque entre ",[538,2494,2495],{},"Senoide pura",[538,2497,2498],{},"Trompete",": a ",[53,2501,2502],{},"forma"," da onda muda muito, mas os picos da NSDF ficam ",[53,2505,2506],{},"nos mesmos lugares",[238,2508,2509],{},[15,2510,2511,2514,2515,2542,2543,2547,2548,2551,2552,819,2560,2568,2569,2571,2572,2575,2576,2578,2579,2582,2583,2586],{},[53,2512,2513],{},"\"Então os picos repetidos são os harmônicos?\" Não!"," Essa é a confusão clássica. Os picos da NSDF em ",[27,2516,2517],{},[30,2518,2519,2521,2523,2526,2528,2530,2532,2534,2536,2538,2540],{},[33,2520,531],{},[38,2522,81],{},[33,2524,2525],{},"P",[38,2527,172],{"separator":40},[83,2529,180],{},[33,2531,2525],{},[38,2533,172],{"separator":40},[83,2535,1526],{},[33,2537,2525],{},[38,2539,172],{"separator":40},[38,2541,185],{}," aparecem simplesmente porque um sinal que se repete a cada ",[27,2544,2545],{},[33,2546,2525],{}," amostras ",[53,2549,2550],{},"também"," se repete a cada ",[27,2553,2554],{},[30,2555,2556,2558],{},[83,2557,180],{},[33,2559,2525],{},[27,2561,2562],{},[30,2563,2564,2566],{},[83,2565,1526],{},[33,2567,2525],{},"…: deslocar a cópia por 2 períodos a realinha tão bem quanto por 1. A prova é que a ",[53,2570,2161],{},", sem harmônicos, tem ",[53,2573,2574],{},"exatamente os mesmos"," picos repetidos. Selecione ",[538,2577,2495],{}," e veja! Ou seja, os picos repetidos são ",[53,2580,2581],{},"o sinal se repetindo",", não os harmônicos. É justamente isso que torna o passo 5 necessário: entre vários picos quase perfeitos, qual é o ",[53,2584,2585],{},"período fundamental"," (o primeiro), e não o dobro dele (uma oitava abaixo)?",[2588,2589,2591],"tuner-nsdf",{"title":2590},"Experimento: deslize a cópia e veja a NSDF",[15,2592,2593,2594,2598,2599,2618,2619,2621,2622,2624,2625,2629,2630,2639,2640,2643,2644,2646,2647,2649],{},"Em cima, a onda original (linha contínua) e a cópia deslocada por ",[27,2595,2596],{},[33,2597,531],{}," (linha tracejada). Embaixo, a curva ",[27,2600,2601],{},[30,2602,2603,2610],{},[625,2604,2605,2607],{},[33,2606,101],{},[38,2608,636],{"lspace":632,"rspace":632,"className":2609},[634],[30,2611,2612,2614,2616],{},[38,2613,43],{"fence":40,"form":41,"stretchy":42},[33,2615,531],{},[38,2617,50],{"fence":40,"form":49,"stretchy":42}," inteira, com guias no ",[53,2620,290],{}," e no seu ",[53,2623,1706],{},". Arraste ",[27,2626,2627],{},[33,2628,531],{},": ao chegar ao período, a cópia se sobrepõe à original e ",[27,2631,2632],{},[625,2633,2634,2636],{},[33,2635,101],{},[38,2637,636],{"lspace":632,"rspace":632,"className":2638},[634]," sobe para perto de 1; no dobro, sobe de novo. Troque o ",[53,2641,2642],{},"sinal",": mudar os harmônicos muda o ",[53,2645,2478],{}," da onda, mas ",[53,2648,1646],{}," a posição dos picos. Até a senoide pura tem os mesmos.",[19,2651,2653],{"id":2652},"_5-escolha-do-pico-certo","5. Escolha do pico certo",[15,2655,2656,2657,2676,2677,2680,2681,2684],{},"Do passo 4 saiu a curva ",[27,2658,2659],{},[30,2660,2661,2668],{},[625,2662,2663,2665],{},[33,2664,101],{},[38,2666,636],{"lspace":632,"rspace":632,"className":2667},[634],[30,2669,2670,2672,2674],{},[38,2671,43],{"fence":40,"form":41,"stretchy":42},[33,2673,531],{},[38,2675,50],{"fence":40,"form":49,"stretchy":42}," inteira, cheia de picos. Agora precisamos ",[53,2678,2679],{},"escolher um",": o que corresponde ao período fundamental. Não basta pegar o mais alto, e aqui está o porquê, em etapas (em inglês, essa escolha se chama ",[538,2682,2683],{},"peak picking","):",[2686,2687,2688,2729,2739,2762],"ol",{},[498,2689,2690,2693,2694,2704,2705,2728],{},[53,2691,2692],{},"Ignorar o pico trivial."," Em ",[27,2695,2696],{},[30,2697,2698,2700,2702],{},[33,2699,531],{},[38,2701,81],{},[83,2703,169],{},", a onda é comparada com ela mesma, então ",[27,2706,2707],{},[30,2708,2709,2716,2724,2726],{},[625,2710,2711,2713],{},[33,2712,101],{},[38,2714,636],{"lspace":632,"rspace":632,"className":2715},[634],[30,2717,2718,2720,2722],{},[38,2719,43],{"fence":40,"form":41,"stretchy":42},[83,2721,169],{},[38,2723,50],{"fence":40,"form":49,"stretchy":42},[38,2725,81],{},[83,2727,175],{}," sempre, o que não diz nada sobre o período. Pulamos esse primeiro \"morro\" da curva, até ela cruzar o zero pela primeira vez.",[498,2730,2731,2734,2735,2738],{},[53,2732,2733],{},"Achar os máximos-chave."," A curva sobe e desce, cruzando o zero várias vezes. Entre uma subida que cruza o zero e a descida seguinte há uma \"corcova\", e o ponto mais alto de cada corcova é um ",[53,2736,2737],{},"máximo-chave",". Olhar só para esses pontos evita confundir pequenas ondulações com picos de verdade.",[498,2740,2741,56,2744,2757,2758,2761],{},[53,2742,2743],{},"Medir o mais alto de todos,",[27,2745,2746],{},[60,2747,2748,2750],{},[33,2749,101],{},[30,2751,2752,2755],{},[33,2753,2754],{},"max",[38,2756,341],{},". Esse valor é também a ",[53,2759,2760],{},"clareza"," do som, de 0 a 1: perto de 1, a nota está bem definida; baixo, o som é ruidoso ou não existe. É com esse número que o afinador decide se há nota para mostrar, e é ele que aparece como \"clareza\" nas mensagens do afinador.",[498,2763,2764,2767,2768,2786,2787,2798,2799,2802,2803,2805],{},[53,2765,2766],{},"Escolher o primeiro pico bom o bastante."," Definimos um limiar ",[27,2769,2770],{},[30,2771,2772,2774,2776],{},[33,2773,2415],{},[38,2775,1409],{},[60,2777,2778,2780],{},[33,2779,101],{},[30,2781,2782,2784],{},[33,2783,2754],{},[38,2785,341],{},", com ",[27,2788,2789],{},[30,2790,2791,2793,2795],{},[33,2792,2415],{},[38,2794,81],{},[83,2796,2797],{},"0,9",", e pegamos, varrendo da esquerda para a direita, o ",[53,2800,2801],{},"primeiro"," máximo-chave que o ultrapassa, e ",[53,2804,1646],{}," o mais alto.",[238,2807,2808],{},[15,2809,2810,2813,2814,2840,2841,2849,2850,2853,2854,2857,2858,2861,2862,2870,2871,2873],{},[53,2811,2812],{},"Por que o primeiro, e não o mais alto?"," Lembre do passo 4: os picos se repetem em ",[27,2815,2816],{},[30,2817,2818,2820,2822,2824,2826,2828,2830,2832,2834,2836,2838],{},[33,2819,531],{},[38,2821,81],{},[33,2823,2525],{},[38,2825,172],{"separator":40},[83,2827,180],{},[33,2829,2525],{},[38,2831,172],{"separator":40},[83,2833,1526],{},[33,2835,2525],{},[38,2837,172],{"separator":40},[38,2839,185],{}," e todos têm altura parecida. O pico em ",[27,2842,2843],{},[30,2844,2845,2847],{},[83,2846,180],{},[33,2848,2525],{}," corresponde a um período ",[53,2851,2852],{},"duas vezes maior",", ou seja, a uma frequência ",[53,2855,2856],{},"pela metade",": uma ",[53,2859,2860],{},"oitava abaixo",". Se pegássemos o mais alto e ele fosse o de ",[27,2863,2864],{},[30,2865,2866,2868],{},[83,2867,180],{},[33,2869,2525],{},", o afinador mostraria a nota uma oitava errada. O ",[53,2872,2801],{}," pico forte é sempre o período fundamental: o mais curto, a frequência mais alta, a nota certa.",[238,2875,2876],{},[15,2877,2878,2881,2882,2893,2894,2897,2898,2909,2910,2921,2922,2932,2933,2951,2952,2967,2968,2971,2972,2982,2983,2985,2986,2988,2989,2999],{},[53,2879,2880],{},"Exemplo passo a passo",", justamente o caso traiçoeiro. Suponha máximos-chave de altura 0,93 (em ",[27,2883,2884],{},[30,2885,2886,2888,2890],{},[33,2887,531],{},[38,2889,81],{},[83,2891,2892],{},"150","), ",[53,2895,2896],{},"0,97"," (em ",[27,2899,2900],{},[30,2901,2902,2904,2906],{},[33,2903,531],{},[38,2905,81],{},[83,2907,2908],{},"300",") e 0,90 (em ",[27,2911,2912],{},[30,2913,2914,2916,2918],{},[33,2915,531],{},[38,2917,81],{},[83,2919,2920],{},"450","). O mais alto é o de ",[27,2923,2924],{},[30,2925,2926,2928,2930],{},[33,2927,531],{},[38,2929,81],{},[83,2931,2908],{},", então ",[27,2934,2935],{},[30,2936,2937,2947,2949],{},[60,2938,2939,2941],{},[33,2940,101],{},[30,2942,2943,2945],{},[33,2944,2754],{},[38,2946,341],{},[38,2948,81],{},[83,2950,2896],{}," e o limiar é ",[27,2953,2954],{},[30,2955,2956,2958,2960,2962,2964],{},[83,2957,2797],{},[38,2959,1409],{},[83,2961,2896],{},[38,2963,81],{},[83,2965,2966],{},"0,873",". Se pegássemos o ",[53,2969,2970],{},"mais alto",", escolheríamos ",[27,2973,2974],{},[30,2975,2976,2978,2980],{},[33,2977,531],{},[38,2979,81],{},[83,2981,2908],{},", o dobro do período e uma ",[53,2984,2860],{},": erro! Mas, varrendo da esquerda, o ",[53,2987,2801],{}," a passar de 0,873 é o de ",[27,2990,2991],{},[30,2992,2993,2995,2997],{},[33,2994,531],{},[38,2996,81],{},[83,2998,2892],{}," (0,93 ≥ 0,873), e paramos ali. Período de 150 amostras: o fundamental correto.",[15,3001,3002,3003,3007,3008,3011],{},"O ",[27,3004,3005],{},[33,3006,531],{}," do pico escolhido é o nosso período, ainda em número ",[53,3009,3010],{},"inteiro"," de amostras. No próximo passo, refinamos isso para frações de amostra.",[19,3013,3015],{"id":3014},"_6-precisão-de-subamostra-interpolação-parabólica","6. Precisão de subamostra: interpolação parabólica",[15,3017,3018,3019,566,3021,3025,3026,3029,3030,3049,3050,3066,3067,310],{},"A NSDF só é calculada em valores ",[53,3020,2469],{},[27,3022,3023],{},[33,3024,531],{}," (amostra a amostra), mas o período real quase nunca cai exatamente sobre uma amostra: ele fica ",[53,3027,3028],{},"entre dois pontos",". A 44,1 kHz, um Lá de 440 Hz tem período ",[27,3031,3032],{},[30,3033,3034,3036,3038,3040,3042,3044,3046],{},[33,3035,531],{},[38,3037,81],{},[83,3039,85],{},[33,3041,219],{},[83,3043,411],{},[38,3045,228],{},[83,3047,3048],{},"100,2"," amostras. Arredondar para 100 daria ",[27,3051,3052],{},[30,3053,3054,3056,3058,3061,3063],{},[83,3055,85],{},[33,3057,219],{},[83,3059,3060],{},"100",[38,3062,81],{},[83,3064,3065],{},"441"," Hz, cerca de 4 cents de erro só de arredondamento. Precisamos do pico com precisão de ",[53,3068,3069],{},"fração de amostra",[15,3071,3072,3075],{},[53,3073,3074],{},"Por que justamente uma parábola?"," Dois fatos se encaixam:",[2686,3077,3078,3195],{},[498,3079,3080,3083,3084,3092,3093,3178,3179,3194],{},[53,3081,3082],{},"Todo pico suave, visto de bem perto, é uma parábola."," É o que diz a série de Taylor: perto de um máximo ",[27,3085,3086],{},[60,3087,3088,3090],{},[33,3089,93],{},[83,3091,169],{},", a inclinação é zero, então ",[27,3094,3095],{},[30,3096,3097,3099,3107,3109,3111,3123,3125,3131,3144,3156,3158],{},[33,3098,64],{},[30,3100,3101,3103,3105],{},[38,3102,43],{"fence":40,"form":41,"stretchy":42},[33,3104,93],{},[38,3106,50],{"fence":40,"form":49,"stretchy":42},[38,3108,228],{},[33,3110,64],{},[30,3112,3113,3115,3121],{},[38,3114,43],{"fence":40,"form":41,"stretchy":42},[60,3116,3117,3119],{},[33,3118,93],{},[83,3120,169],{},[38,3122,50],{"fence":40,"form":49,"stretchy":42},[38,3124,276],{},[145,3126,3127,3129],{},[83,3128,175],{},[83,3130,180],{},[625,3132,3133,3135],{},[33,3134,64],{},[30,3136,3138,3141],{"className":3137},[635],[38,3139,636],{"lspace":632,"rspace":632,"className":3140},[634],[38,3142,636],{"lspace":632,"rspace":632,"className":3143},[634],[30,3145,3146,3148,3154],{},[38,3147,43],{"fence":40,"form":41,"stretchy":42},[60,3149,3150,3152],{},[33,3151,93],{},[83,3153,169],{},[38,3155,50],{"fence":40,"form":49,"stretchy":42},[1084,3157],{"width":1086},[625,3159,3160,3176],{},[30,3161,3162,3164,3166,3168,3174],{},[38,3163,43],{"fence":40,"form":41,"stretchy":42},[33,3165,93],{},[38,3167,193],{},[60,3169,3170,3172],{},[33,3171,93],{},[83,3173,169],{},[38,3175,50],{"fence":40,"form":49,"stretchy":42},[83,3177,180],{},", que é a equação de uma parábola (com a boca para baixo, pois num máximo a curvatura ",[27,3180,3181],{},[625,3182,3183,3185],{},[33,3184,64],{},[30,3186,3188,3191],{"className":3187},[635],[38,3189,636],{"lspace":632,"rspace":632,"className":3190},[634],[38,3192,636],{"lspace":632,"rspace":632,"className":3193},[634]," é negativa).",[498,3196,3197,3200,3201,3230,3231,819,3241,1196,3245,3255],{},[53,3198,3199],{},"Três pontos definem uma única parábola."," A parábola ",[27,3202,3203],{},[30,3204,3205,3208,3210,3212,3219,3221,3223,3225,3227],{},[33,3206,3207],{},"y",[38,3209,81],{},[33,3211,911],{},[625,3213,3214,3216],{},[33,3215,93],{},[83,3217,180],{"className":3218},[961],[38,3220,276],{},[33,3222,916],{},[33,3224,93],{},[38,3226,276],{},[33,3228,3229],{},"c"," tem 3 incógnitas, e temos 3 pontos: ",[27,3232,3233],{},[30,3234,3235,3237,3239],{},[33,3236,531],{},[38,3238,193],{},[83,3240,175],{},[27,3242,3243],{},[33,3244,531],{},[27,3246,3247],{},[30,3248,3249,3251,3253],{},[33,3250,531],{},[38,3252,276],{},[83,3254,175],{},". Existe uma e só uma, com fórmula fechada. Uma reta nem teria pico, e um grau maior poderia ondular e inventar picos falsos.",[15,3257,3002,3258,3261,3262,819,3274,1196,3282,3294,3295,3313,3314,105],{},[53,3259,3260],{},"vértice"," dessa parábola é o palpite refinado do pico. Chamando de ",[27,3263,3264],{},[60,3265,3266,3268],{},[33,3267,101],{},[30,3269,3270,3272],{},[38,3271,193],{"form":41,"stretchy":42,"lspace":632,"rspace":632},[83,3273,175],{},[27,3275,3276],{},[60,3277,3278,3280],{},[33,3279,101],{},[83,3281,169],{},[27,3283,3284],{},[60,3285,3286,3288],{},[33,3287,101],{},[30,3289,3290,3292],{},[38,3291,276],{"form":41,"stretchy":42,"lspace":632,"rspace":632},[83,3293,175],{}," os três valores da NSDF, o vértice (em ",[27,3296,3297],{},[30,3298,3299,3301,3303,3305,3307,3309,3311],{},[33,3300,93],{},[38,3302,81],{},[38,3304,193],{"form":41,"stretchy":42},[33,3306,916],{},[33,3308,219],{},[83,3310,180],{},[33,3312,911],{},") dá o deslocamento ",[27,3315,3316],{},[33,3317,3318],{},"δ",[27,3320,3322],{"display":108,"className":3321,"style":111},[110],[30,3323,3324,3326,3328,3334,3336],{},[33,3325,3318],{},[38,3327,81],{},[145,3329,3330,3332],{},[83,3331,175],{},[83,3333,180],{},[38,3335,1409],{},[145,3337,3338,3362],{},[30,3339,3340,3350,3352],{},[60,3341,3342,3344],{},[33,3343,101],{},[30,3345,3346,3348],{},[38,3347,193],{"form":41,"stretchy":42,"lspace":632,"rspace":632},[83,3349,175],{},[38,3351,193],{},[60,3353,3354,3356],{},[33,3355,101],{},[30,3357,3358,3360],{},[38,3359,276],{"form":41,"stretchy":42,"lspace":632,"rspace":632},[83,3361,175],{},[30,3363,3364,3374,3376,3378,3384,3386],{},[60,3365,3366,3368],{},[33,3367,101],{},[30,3369,3370,3372],{},[38,3371,193],{"form":41,"stretchy":42,"lspace":632,"rspace":632},[83,3373,175],{},[38,3375,193],{},[83,3377,180],{},[60,3379,3380,3382],{},[33,3381,101],{},[83,3383,169],{},[38,3385,276],{},[60,3387,3388,3390],{},[33,3389,101],{},[30,3391,3392,3394],{},[38,3393,276],{"form":41,"stretchy":42,"lspace":632,"rspace":632},[83,3395,175],{},[27,3397,3399],{"display":108,"className":3398,"style":111},[110],[30,3400,3401,3408,3410,3412,3414],{},[625,3402,3403,3405],{},[33,3404,531],{},[38,3406,3407],{"form":41,"stretchy":42},"⋆",[38,3409,81],{},[33,3411,531],{},[38,3413,276],{},[33,3415,3318],{},[238,3417,3418],{},[15,3419,3420,3423,3424,56,3427,3463,3464,3467,3468,56,3471,3497,3498,3501,3502,3513,3514,3524],{},[53,3421,3422],{},"A fórmula conta uma história."," O ",[53,3425,3426],{},"denominador",[27,3428,3429],{},[30,3430,3431,3441,3443,3445,3451,3453],{},[60,3432,3433,3435],{},[33,3434,101],{},[30,3436,3437,3439],{},[38,3438,193],{"form":41,"stretchy":42,"lspace":632,"rspace":632},[83,3440,175],{},[38,3442,193],{},[83,3444,180],{},[60,3446,3447,3449],{},[33,3448,101],{},[83,3450,169],{},[38,3452,276],{},[60,3454,3455,3457],{},[33,3456,101],{},[30,3458,3459,3461],{},[38,3460,276],{"form":41,"stretchy":42,"lspace":632,"rspace":632},[83,3462,175],{}," é a ",[53,3465,3466],{},"curvatura",", o quão pontudo é o pico. O ",[53,3469,3470],{},"numerador",[27,3472,3473],{},[30,3474,3475,3485,3487],{},[60,3476,3477,3479],{},[33,3478,101],{},[30,3480,3481,3483],{},[38,3482,193],{"form":41,"stretchy":42,"lspace":632,"rspace":632},[83,3484,175],{},[38,3486,193],{},[60,3488,3489,3491],{},[33,3490,101],{},[30,3492,3493,3495],{},[38,3494,276],{"form":41,"stretchy":42,"lspace":632,"rspace":632},[83,3496,175],{}," mede a ",[53,3499,3500],{},"assimetria",": se o vizinho da esquerda é mais alto, o pico real puxa para a esquerda (",[27,3503,3504],{},[30,3505,3506,3508,3511],{},[33,3507,3318],{},[38,3509,3510],{},"\u003C",[83,3512,169],{},"); se os vizinhos são iguais, ",[27,3515,3516],{},[30,3517,3518,3520,3522],{},[33,3519,3318],{},[38,3521,81],{},[83,3523,169],{}," e o pico cai bem na amostra.",[15,3526,3002,3527,3531],{},[27,3528,3529],{},[33,3530,3318],{}," é uma fração de amostra, entre −0,5 e +0,5. Aí a frequência sai direto:",[1830,3533,3534],{},[27,3535,3537],{"display":108,"className":3536,"style":111},[110],[30,3538,3539,3541,3543],{},[33,3540,64],{},[38,3542,81],{},[145,3544,3545,3551],{},[60,3546,3547,3549],{},[33,3548,64],{},[33,3550,67],{},[625,3552,3553,3555],{},[33,3554,531],{},[38,3556,3407],{"form":41,"stretchy":42},[3558,3559,3561],"tuner-parabola",{"title":3560},"A interpolação em ação",[15,3562,3563,3564,3567,3568,3571],{},"A curva real tem o pico num ponto ",[538,3565,3566],{},"fracionário"," (linha vertical tracejada), mas só medimos os ",[53,3569,3570],{},"3 pontos inteiros"," em volta. A parábola ajustada passa por esses 3 pontos, e o vértice dela é a estimativa. Arraste o período real: repare como o vértice acompanha o valor verdadeiro, enquanto só arredondar erraria.",[238,3573,3574],{},[15,3575,3576,3579,3580,3594,3595,3605,3606,2932,3619,1196,3634,3653,3654,3658],{},[53,3577,3578],{},"Exemplo numérico:"," com ",[27,3581,3582],{},[30,3583,3584,3590,3592],{},[60,3585,3586,3588],{},[33,3587,64],{},[33,3589,67],{},[38,3591,81],{},[83,3593,85],{}," Hz, se o pico está em ",[27,3596,3597],{},[30,3598,3599,3601,3603],{},[33,3600,531],{},[38,3602,81],{},[83,3604,3060],{}," e a parábola dá ",[27,3607,3608],{},[30,3609,3610,3612,3614,3616],{},[33,3611,3318],{},[38,3613,81],{},[38,3615,193],{"form":41,"stretchy":42},[83,3617,3618],{},"0,18",[27,3620,3621],{},[30,3622,3623,3629,3631],{},[625,3624,3625,3627],{},[33,3626,531],{},[38,3628,3407],{"form":41,"stretchy":42},[38,3630,81],{},[83,3632,3633],{},"99,82",[27,3635,3636],{},[30,3637,3638,3640,3642,3644,3646,3648,3650],{},[33,3639,64],{},[38,3641,81],{},[83,3643,85],{},[33,3645,219],{},[83,3647,3633],{},[38,3649,228],{},[83,3651,3652],{},"441,8"," Hz. Sem o ",[27,3655,3656],{},[33,3657,3318],{},", daria 441,0 Hz, uma diferença de cerca de 3 cents.",[19,3660,3662],{"id":3661},"_7-da-frequência-à-nota-e-aos-cents","7. Da frequência à nota e aos cents",[15,3664,3665,3666,3669,3670,3673,3674,3683,3684,3687,3688,3692],{},"No ",[53,3667,3668],{},"temperamento igual",", a oitava (dobrar a frequência) é dividida em 12 semitons iguais, em escala ",[53,3671,3672],{},"logarítmica",". Partindo do Lá4 de referência ",[27,3675,3676],{},[60,3677,3678,3680],{},[33,3679,64],{},[1105,3681,3682],{},"ref",", que você ajusta no controle ",[53,3685,3686],{},"Referência"," do afinador (440 Hz, 442 Hz…), a frequência da nota a ",[27,3689,3690],{},[33,3691,101],{}," semitons de distância é:",[27,3694,3696],{"display":108,"className":3695,"style":111},[110],[30,3697,3698,3700,3708,3710,3716,3718],{},[33,3699,64],{},[30,3701,3702,3704,3706],{},[38,3703,43],{"fence":40,"form":41,"stretchy":42},[33,3705,101],{},[38,3707,50],{"fence":40,"form":49,"stretchy":42},[38,3709,81],{},[60,3711,3712,3714],{},[33,3713,64],{},[1105,3715,3682],{},[38,3717,1409],{},[625,3719,3720,3722],{},[83,3721,180],{},[30,3723,3724,3726,3728],{},[33,3725,101],{},[33,3727,219],{},[83,3729,3730],{},"12",[15,3732,3733],{},"Invertendo com logaritmo na base 2, descobrimos a quantos semitons a nossa frequência está:",[1830,3735,3736],{},[27,3737,3739],{"display":108,"className":3738,"style":111},[110],[30,3740,3741,3743,3745,3747,3749,3756,3758],{},[33,3742,101],{},[38,3744,81],{},[83,3746,3730],{},[1084,3748],{"width":1086},[60,3750,3751,3754],{},[33,3752,3753],{},"log",[83,3755,180],{},[38,3757,341],{},[30,3759,3760,3762,3772],{},[38,3761,43],{"fence":40,"form":41,"stretchy":40},[145,3763,3764,3766],{},[33,3765,64],{},[60,3767,3768,3770],{},[33,3769,64],{},[1105,3771,3682],{},[38,3773,50],{"fence":40,"form":49,"stretchy":40},[15,3775,3776,3777,3781,3782,3785,3786,3789],{},"Arredondando ",[27,3778,3779],{},[33,3780,101],{},", achamos a ",[53,3783,3784],{},"nota mais próxima",". O que sobra, multiplicado por 100, é o ",[53,3787,3788],{},"desvio em cents"," (100 cents = 1 semitom):",[27,3791,3793],{"display":108,"className":3792,"style":111},[110],[113,3794,3796,3835],{"displaystyle":40,"className":3795},[116],[118,3797,3798,3804],{},[121,3799,3801],{"className":3800,"style":123},[981],[1105,3802,3803],{},"cents",[121,3805,3807],{"className":3806,"style":123},[1006],[30,3808,3809,3811,3814,3816,3818,3821,3823,3825,3827,3829,3831,3833],{},[38,3810,81],{},[38,3812,43],{"fence":42,"symmetric":40,"minsize":3813,"maxsize":3813},"1.2em",[33,3815,101],{},[38,3817,193],{},[33,3819,3820],{},"round",[38,3822,341],{},[38,3824,43],{"form":41,"stretchy":42},[33,3826,101],{},[38,3828,50],{"form":49,"stretchy":42},[38,3830,50],{"fence":42,"symmetric":40,"minsize":3813,"maxsize":3813},[38,3832,1409],{},[83,3834,3060],{},[118,3836,3837,3842],{},[121,3838,3840],{"className":3839,"style":123},[981],[30,3841],{},[121,3843,3845],{"className":3844,"style":123},[1006],[30,3846,3847,3849,3852,3854,3860,3862],{},[38,3848,81],{},[83,3850,3851],{},"1200",[1084,3853],{"width":1086},[60,3855,3856,3858],{},[33,3857,3753],{},[83,3859,180],{},[38,3861,341],{},[30,3863,3864,3866,3877],{},[38,3865,43],{"fence":40,"form":41,"stretchy":40},[145,3867,3868,3870],{},[33,3869,64],{},[60,3871,3872,3874],{},[33,3873,64],{},[1105,3875,3876],{},"mais próxima",[38,3878,50],{"fence":40,"form":49,"stretchy":40},[3880,3881,3883],"tuner-calculator",{"title":3882},"Calculadora: frequência → nota",[15,3884,3885],{},"Digite uma frequência e a referência para ver a fórmula sendo calculada e a nota resultante. Experimente 446 Hz com referência de 440 Hz: dá Lá4, +23 cents.",[238,3887,3888],{},[15,3889,3890,3893,3894,3897,3898,3901,3902,3905,3906,3916,3917,3919,3920,3923],{},[53,3891,3892],{},"Trompete em Si♭ (transposição):"," ao ler um ",[53,3895,3896],{},"Dó"," escrito, o instrumento soa um ",[53,3899,3900],{},"Si♭"," de concerto, 2 semitons abaixo. Para mostrar o nome da nota ",[538,3903,3904],{},"escrita",", somamos um deslocamento inteiro ao índice da nota, ",[27,3907,3908],{},[30,3909,3910,3912,3914],{},[33,3911,101],{},[38,3913,276],{},[83,3915,180],{},". Como é um número ",[53,3918,3010],{}," de semitons, o desvio em cents ",[53,3921,3922],{},"não muda",". Só muda o nome da nota.",[19,3925,3927],{"id":3926},"_8-estabilidade-mediana-e-filtro-exponencial","8. Estabilidade: mediana e filtro exponencial",[15,3929,3930,3931,3934],{},"A análise roda cerca de 60 vezes por segundo, e cada leitura tem um pequeno erro aleatório: a frequência estimada \"treme\" alguns décimos de Hz a cada quadro. Ligar a agulha direto nisso a deixaria vibrando. Por isso, aplico ",[53,3932,3933],{},"dois filtros em sequência",", cada um resolvendo um problema diferente.",[15,3936,3937,3940,3941,3944,3945,3948],{},[53,3938,3939],{},"Primeiro filtro, a mediana: elimina as leituras fora da curva."," Guardo as últimas ",[53,3942,3943],{},"7"," frequências e uso a ",[53,3946,3947],{},"mediana"," delas, o valor do meio depois de ordenar a lista. A mediana é imune a um valor absurdo isolado, coisa que a média comum não é.",[238,3950,3951],{},[15,3952,3953,3956,3957,3985,3986,3989,3990,4028,4029,4031,4032,4034,4035,4037,4038,4041],{},[53,3954,3955],{},"Mediana ou média, na prática."," Suponha as leituras ",[27,3958,3959],{},[30,3960,3961,3963,3965,3967,3969,3971,3974,3976,3979,3981,3983],{},[38,3962,98],{"fence":40,"form":41,"stretchy":42},[83,3964,411],{},[38,3966,172],{"separator":40},[83,3968,3065],{},[38,3970,172],{"separator":40},[83,3972,3973],{},"439",[38,3975,172],{"separator":40},[83,3977,3978],{},"880",[38,3980,172],{"separator":40},[83,3982,411],{},[38,3984,104],{"fence":40,"form":49,"stretchy":42},", em que 880 é uma leitura errada (um salto de oitava em um quadro). A ",[53,3987,3988],{},"média"," daria ",[27,3991,3992],{},[30,3993,3994,4018,4020,4023,4025],{},[30,3995,3996,3998,4000,4002,4004,4006,4008,4010,4012,4014,4016],{},[38,3997,43],{"fence":40,"form":41,"stretchy":42},[83,3999,411],{},[38,4001,276],{},[83,4003,3065],{},[38,4005,276],{},[83,4007,3973],{},[38,4009,276],{},[83,4011,3978],{},[38,4013,276],{},[83,4015,411],{},[38,4017,50],{"fence":40,"form":49,"stretchy":42},[33,4019,219],{},[83,4021,4022],{},"5",[38,4024,81],{},[83,4026,4027],{},"528"," Hz, puxada pelo erro. Já a ",[53,4030,3947],{}," (ordenando: 439, 440, ",[53,4033,411],{},", 441, 880) é ",[53,4036,411],{}," Hz: a leitura errada vira só \"a maior da lista\" e é descartada. A janela é ",[53,4039,4040],{},"ímpar"," (7) justamente para sempre haver um elemento exatamente no meio.",[15,4043,4044,4047,4048,4051,4052,4057],{},[53,4045,4046],{},"Segundo filtro, a média móvel exponencial: alisa o tremor que resta."," Este é aplicado já à ",[53,4049,4050],{},"posição da agulha",", o desvio em cents. Em vez de saltar para a leitura nova, misturo uma fração ",[27,4053,4054],{},[33,4055,4056],{},"α"," dela com o valor que já estava na tela:",[1830,4059,4060],{},[27,4061,4063],{"display":108,"className":4062,"style":111},[110],[113,4064,4066,4118],{"displaystyle":40,"className":4065},[116],[118,4067,4068],{},[121,4069,4070],{"style":123},[30,4071,4072,4078,4080,4082,4084,4090,4092,4104,4106],{},[60,4073,4074,4076],{},[33,4075,67],{},[33,4077,46],{},[38,4079,81],{},[33,4081,4056],{},[38,4083,1409],{},[60,4085,4086,4088],{},[33,4087,93],{},[33,4089,46],{},[38,4091,276],{},[30,4093,4094,4096,4098,4100,4102],{},[38,4095,43],{"fence":40,"form":41,"stretchy":42},[83,4097,175],{},[38,4099,193],{},[33,4101,4056],{},[38,4103,50],{"fence":40,"form":49,"stretchy":42},[38,4105,1409],{},[60,4107,4108,4110],{},[33,4109,67],{},[30,4111,4112,4114,4116],{},[33,4113,46],{},[38,4115,193],{},[83,4117,175],{},[118,4119,4120],{},[121,4121,4122],{"style":123},[30,4123,4124,4126,4128],{},[33,4125,4056],{},[38,4127,81],{},[83,4129,4130],{},"0,12",[15,4132,4133,4134,4142,4143,4153,4154,4162,4163,4177,4178,4181,4182,4185,4186,4189,4190,4205],{},"Lendo a fórmula: o novo valor exibido ",[27,4135,4136],{},[60,4137,4138,4140],{},[33,4139,67],{},[33,4141,46],{}," é ",[27,4144,4145],{},[30,4146,4147,4149,4151],{},[33,4148,4056],{},[38,4150,81],{},[83,4152,4130],{}," (12%) da leitura nova ",[27,4155,4156],{},[60,4157,4158,4160],{},[33,4159,93],{},[33,4161,46],{}," somado a 88% do valor anterior ",[27,4164,4165],{},[60,4166,4167,4169],{},[33,4168,67],{},[30,4170,4171,4173,4175],{},[33,4172,46],{},[38,4174,193],{},[83,4176,175],{},". Isso é um ",[53,4179,4180],{},"filtro passa-baixas",": deixa passar mudanças ",[53,4183,4184],{},"lentas"," (você realmente subindo a nota) e barra as ",[53,4187,4188],{},"rápidas"," (o tremor de um quadro para o outro). Como cada leitura empurra a agulha só 12% rumo ao novo valor, ela leva cerca de ",[27,4191,4192],{},[30,4193,4194,4196,4198,4200,4202],{},[83,4195,175],{},[33,4197,219],{},[33,4199,4056],{},[38,4201,228],{},[83,4203,4204],{},"8"," quadros (uns 130 ms) para assentar: rápido para o olho e lento o bastante para não tremer.",[15,4207,4208],{},"Quando a nota muda, a média recomeça. Sem isso, ao passar de +49 cents numa nota para −49 cents na seguinte, a agulha atravessaria o \"afinado\" no caminho.",[238,4210,4211],{},[15,4212,4213,4220,4221,4224,4225,4228,4229,4224,4232,4235],{},[53,4214,4215,4216,105],{},"O compromisso do ",[27,4217,4218],{},[33,4219,4056],{}," um valor menor (por exemplo, 0,05) deixa a agulha mais ",[53,4222,4223],{},"calma",", porém mais ",[53,4226,4227],{},"lenta"," para reagir; um maior (por exemplo, 0,30) a deixa mais ",[53,4230,4231],{},"responsiva",[53,4233,4234],{},"nervosa",". O 0,12 é um meio-termo confortável, e foi exatamente a esse número que cheguei quando a agulha estava nervosa demais.",[19,4237,4239],{"id":4238},"juntando-tudo","Juntando tudo",[15,4241,4242,4243,4246,4247,4261,4262,4264,4265,4268,4269,4272,4273,4276,4277,4280],{},"Esse é o caminho completo, repetido ",[53,4244,4245],{},"cerca de 60 vezes por segundo",": o microfone vira amostras ",[27,4248,4249],{},[30,4250,4251,4253],{},[33,4252,93],{},[30,4254,4255,4257,4259],{},[38,4256,98],{"fence":40,"form":41,"stretchy":42},[33,4258,101],{},[38,4260,104],{"fence":40,"form":49,"stretchy":42}," (passos 1 e 2); a ",[53,4263,1824],{}," mede o quanto a onda se parece com uma cópia deslocada dela mesma (passos 3 e 4); a ",[53,4266,4267],{},"escolha do pico"," encontra o período fundamental, fugindo do erro de oitava (passo 5); a ",[53,4270,4271],{},"parábola"," refina o período para frações de amostra (passo 6); a frequência vira ",[53,4274,4275],{},"nome de nota e cents"," (passo 7); e a ",[53,4278,4279],{},"mediana com o filtro exponencial"," deixa a agulha estável (passo 8). Do ar à nota afinada, em poucos milissegundos.",[19,4282,3686],{"id":4283},"referência",[495,4285,4286],{},[498,4287,4288,4289,4292],{},"Philip McLeod e Geoff Wyvill, ",[538,4290,4291],{},"A Smarter Way to Find Pitch",", Proceedings of the International Computer Music Conference (ICMC), 2005.",{"title":4294,"searchDepth":4295,"depth":4295,"links":4296},"",2,[4297],{"id":12,"depth":4295,"text":13,"children":4298},[4299,4301,4302,4303,4304,4305,4306,4307,4308,4309],{"id":21,"depth":4300,"text":22},3,{"id":313,"depth":4300,"text":314},{"id":438,"depth":4300,"text":439},{"id":1730,"depth":4300,"text":1731},{"id":2652,"depth":4300,"text":2653},{"id":3014,"depth":4300,"text":3015},{"id":3661,"depth":4300,"text":3662},{"id":3926,"depth":4300,"text":3927},{"id":4238,"depth":4300,"text":4239},{"id":4283,"depth":4300,"text":3686},"Afinador de trompete que roda no navegador. Identifica a nota pelo microfone com o método de McLeod (MPM) e explica, passo a passo, a matemática por trás.","md",{},true,"\u002Fpt\u002Fprojetos\u002Ftuner",{"title":5,"description":4310},"pt\u002Fprojetos\u002Ftuner","KZYhpsmM0nlholnsn2e25VaayLI5mg20lVlTvB9aXqI",1792101206848]