音樂中泛音純律之研究----未來人類最完美之律 2003.8.22訂 1991 傅國權 著 古代的人使用五度相生律,中國人在管子一書中早已論及,比希臘哲學家畢達哥拉斯 也是使用五度相生律早200年,現代人用的十二平均律,明朝朱載堉已發明出來早德國 魏克瑪斯泰95年早法國40年..在樂律學的研究上中國人一向領先世界各國, 中華民族向稱禮樂之邦,早期音樂家王光祈博士,可為此一音樂文化的瀚海之中撒下種子, 悠悠千載的歲月中,當我們站在歷史巨人的身上向前看去,雖個人渺小但站立山頂我為峰, 宇宙真理歷歷顯現於前. 西元1596年(明朝萬曆24年)發明出來十二平均律 迄今已400多年.樂律學尚未有突破.1980年我提出大3度純律12音12律論文. 1991年研究出泛音純律(十二平均律加上大3度純律優點改進而成.)可為21世紀音律之用. 泛音律: 自然泛音次序為: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 - - - 臨音振動比值為: 1/1 2/1 3/2 4/3 5/4 6/5 7/6 8/7 9/8 10/9 11/10 12/11 13/12 14/13 15/14 16/15 17/16 - - - 以下泛音律以三為標準型A=440 A為黃鐘生律之首時得之各頻率. 以第一例為C=270 離12平均律稍遠. 以第二例當C=264 A=429 A不是440 故不適用此為舊式大3度純律之缺點. 從以上資料我們要作結論: 1.樂律是一組音內的音階比值大小的結構,公式及原理. 2.泛音 純律音之比值: C #C D #D E F #F G #G A 1/1 20/19 85/76 91/76 24/19 51/38 27/19 57/38 30/19 32/19 #A B 34/19 36/19 一.以第13泛音A=440.00 HZ 為標準音推算出各音頻律值: 取5位數四捨五入 泛音序 頻率值 音名 1 33.846 C1 公式(440/13)=基音x1 2 67.692 C2 x2 3 101.54 #G2 x3 4 135.38 C3 x4 5 169.23 E3 x5 6 203.08 #G3 x6 7 236.92 bB3 x7 8 270.77 C4 x8 9 304.62 D4 x9 10 338.46 E4 x10 11 372.31 #F4 x11 12 406.15 #G4 x12 13 440.00 A4 x13 14 473.85 bB4 x14 15 507.69 B4 x15 16 541.54 C5 x16 17 575.38 #C5 x17 18 609.23 D5 x18 19 643.08 #D5 x19 20 676.92 E5 x20 21 710.77 F5 x21 22 744.62 #F5 x22 23 778.46 G5 x23 24 812.30 #G5 x24 由上表推論可知: C= 第 8 分音 270.77x1= 270.76 #C= 第 17 分音 575.38/2= 287.69 因數 D= 第 9 分音 304.62x1= 304.62 #D= 第 19 分音 643.08/2= 321.54 E= 第 10 分音 338.46x1= 338.46 F= 第 21 分音 710.77/2= 355.38 因數 #F= 第 11 分音 372.31x1= 372.31 G= 第 3 分音 101.54x4= 406.16 倍數 #G= 第 23 分音 778.46/2= 389.23 A= 第 13 分音 440.00x1= 440.00 bB= 第 7 分音 236.92x2= 473.84 倍數 B= 第 15 分音 507.69x1= 507.69 二. 以基音 C = 264.00 HZ 為起音推算出各頻率值 ( 大三度+小三度之純律 ) 泛音序 頻率值 音名 1 264.00/1= 264.00 C 2 528.00/2= 264.00 C 3 792.00/2= 396.00 G 4 1056.0/4= 264.00 C 5 1320.0/4= 330.00 E 6 1584.0/4= 396.00 G 7 1848.0/4= 462.00 bB 8 2112.0/8= 264.00 C 9 2376.0/8= 297.00 D 10 2640.0/8= 330.00 E 11 2904.0/8= 363.00 #F 12 3168.0/8= 396.00 G 13 3432.0/8= 429.00 A 14 3696.0/8= 462.00 bB 15 3960.0/8= 495.00 B 16 4224.0/16=264.00 C 17 4488.0/16=280.50 #C 18 4752.0/16=297.00 D 19 5016.0/16=313.50 #C 20 5280.0/16=330.00 E 21 5544.0/16=346.50 F 22 5808.0/16=363.00 #F 23 6072.0/16=379.50 #F 24 6336.0/16=396.00 G 由以上推論而得: C=264.00 D=297.00 E=330.00 F=346.50 #F=363.00 G=396.00 A=429.00 bB=462.00 B=495.00 本泛音標準音 A=429 不合標準音故不適用. 三. 以黃鐘 A4 音 頻率440.00 HZ 為律首起音推算出各頻率值 泛音序 頻率值 音名 1 440.00/1 =440.00 A 2 880.00/2 =440.00 A 3 1320.0/4 =330.00 E 4 1760.0/4 =440.00 A 5 2200.0/8 =275.00 #C 6 2640.0/8 =330.00 E 7 3080.0/8 =385.00 不採用 8 3520.0/8 =440.00 A 9 3960.0/8 =495.00 B 10 4400.0/16=275.00 #C 11 4840.0/16=302.50 不採用 12 5280.0/16=330.00 E 13 5720.0/16=357.50 不採用 14 6160.0/16=385.00 不採用 15 6600.0/16=412.50 #G 16 7040.0/16=440.00 A 17 7480.0/16=467.50 #A=bB 5cent 18 7920.0/16=495.00 B 19 8360.0/32=261.25 C ------------------ 20 8800.0/32=275.00 #C ------------------ 21 9240.0/32=288.75 D 不採用 22 9680.0/32=302.50 不採用 23 10120 /32=316.25 #D 不採用 24 10560 /32=330.00 E 12平均律 E=329.63 25 11000 /32=343.75 F=349.23 26 11440 /32=357.50 27 11880 /32=371.25 #F 28 12320 /32=385.00 G 29 12760 /32=398.75 30 13200 /32=412.50 #G 31 13640 /32=426.25 不採用 32 14080 /32=440.00 A 33 14520 /32=453.75 不採用 34 14960 /32=467.50 #A 35 15400 /32=481.25 不採用 36 15840 /32=495.00 B 37 16280 /64=254.38 38 16720 /64=261.25 39 17160 /64=268.13 40 17600 /64=275.00 41 18040 /64=281.88 42 18480 /64=288.75 43 18920 /64=295.62 44 19360 /64=302.50 45 19800 /64=309.375 #D 平均律311.13 -10cent 46 20240 /64=316.25 47 20680 /64=323.13 48 21120 /64=330.00 49 21560 /64=336.88 50 22000 /64=343.75 51 22440 /64=350.63 F 平均律349.23 52 22880 /64=357.50 53 23320 /64=364.38 54 23760 /64=371.25 #F 55 24200 /64=378.13 56 24640 /64=385.00 57 25080 /64=391.875 G 平均律392 G5泛音 261.25*3/2=391.88 58 25520 /64=398.75 59 25960 /64=405.63 60 26400 /64=412.50 61 26840 /64=419.38 62 27280 /64=426.25 63 27720 /64=433.13 64 28160 /64=440.00 65 28600 /64=446.88 66 29040 /64=453.75 67 29480 /64=460.63 68 29920 /64=467.50 #A 69 30360 /64=474.38 70 30800 /64=481.25 71 31240 /64=488.13 72 31680 /64=495.00 B 73 32120 /64=501.88 74 32560 /64=508.75 75 33000 /128=257.81 76 33440 /128=261.25 C 77 33880 /128=264.69 78 34320 /128=268.13 79 34760 /128=271.56 80 35200 /128=275.00 #C 81 35640 /128=278.44 82 36080 /128=281.88 83 36520 /128=285.31 84 36960 /128=288.75 85 37400 /128=292.19 D平均律293.66 86 37840 /128=295.63 87 38280 /128=299.06 88 38720 /128=302.50 89 39160 /128=305.94 90 39600 /128=309.38 -10cent 91 40040 /128=312.81 #D平均律311.13 9cent 92 40480 /128=316.25 93 40920 /128=319.69 94 41360 /128=323.13 95 41800 /128=326.56 96 42240 /128=330.00 E 97 42680 /128=333.44 98 43120 /128=336.88 99 43560 /128=340.31 100 44000 /128=343.75 101 44440 /128=347.19 102 44880 /128=350.63 F 由上表推論可知: 12平均律比較 C= 第 19 分音 8360 /32 = 261.25 比值 19/19 - 2 cent 261.63 #C= 第 5 分音 2200 / 8 = 275.00 比值 20/19 -12 cent 277.18 D= 第 85 分音 37400/128 = 292.19 比值 85/76 - 8 cent 293.66 #D= 第 91 分音 40040/128 = 312.81 比值 91/76 9 cent 311.13 E= 第 3 分音 1320 / 4 = 330.00 比值 24/19 2 cent 329.63 F= 第 51 分音 22440 /64 = 350.63 比值 51/38 7 cent 349.23 #F= 第 27 分音 11880 /32 = 371.25 比值 27/19 6 cent 369.99 G= 第 57 分音 25080 /64 = 391.88 比值 57/38 -0.5cent 392.00 #G= 第 15 分音 6600 /16 = 412.50 比值 30/19 -10 cent 415.30 A= 第 1 分音 440 / 1 = 440.00 比值 32/19 0 cent 440.00 #A= 第 17 分音 7480 /16 = 467.50 比值 34/19 5 cent 466.16 B= 第 9 分音 3960 / 8 = 495.00 比值 36/19 4 cent 493.89 四以基音D=Ax2/3=440x2/3 (293.33)HZ 為起音,推算各頻率值. 泛音序 頻率值 音名 1 440x2/3 x1 =293.33 D 2 440x2/3 x2 =586.66/2 =293.33 D 3 440x2/3 x3 =880.00/2 =440.00 A 4 440x2/3 x4 =1173.3/4 =293.33 D 5 440x2/3 x5 =1466.6/4 =366.66 #F 6 440x2/3 x6 =1760.0/4 =440.00 A 7 440x2/3 x7 =2053.3/4 =513.33 8 440x2/3 x8 =2346.6/8 =293.33 D 9 440x2/3 x9 =2640.0/8 =330.00 E 10 440x2/3 x10=2933.3/8 =366.66 #F 11 440x2/3 x11=3226.6/8 =403.33 12 440x2/3 x12=3520.0/8 =440.00 A 13 440x2/3 x13=3813.3/8 =476.66 14 440x2/3 x14=4106.6/16=256.66 15 440x2/3 x15=4400.0/16=275.00 #C 16 440x2/3 x16=4693.3/16=293.33 D 17 440x2/3 x17=4986.6/16=311.66 #D 18 440x2/3 x18=5280.0/16=330.00 E 19 440x2/3 x19=5573.3/16=348.33 F 20 440x2/3 x20=5866.6/16=366.66 #F 21 440x2/3 x21=6160.0/16=385.00 22 440x2/3 x22=6453.3/16=403.33 23 440x2/3 x23=6746.6/16=421.66 #G 24 440x2/3 x24=7040.0/16=440.00 A 25 440x2/3 x25=7333.3/16=458.33 26 440x2/3 x26=7626.6/16=476.66 27 440x2/3 x27=7920.0/16=495.00 B 28 440x2/3 x28=8213.3/32=256.66 29 440x2/3 x29=8506.6/32=265.83 30 440x2/3 x30=8800.0/32=275.00 31 440x2/3 x31=9093.3/32=284.16 32 440x2/3 x32=9386.6/32=293.33 33 440x2/3 x33=9680.0/32=302.50 34 440x2/3 x34=9973.3/32=311.66 35 440x2/3 x35=10266 /32=320.83 36 440x2/3 x36=10560 /32=330.00 37 440x2/3 x37=10853 /32=339.16 38 440x2/3 x38=11146 /32=348.33 39 440x2/3 x39=11440 /32=357.50 40 440x2/3 x40=11733 /32=366.66 41 440x2/3 x41=12026 /32=375.83 42 440x2/3 x42=12320 /32=385.00 43 440x2/3 x43=12613 /32=394.16 G 44 440x2/3 x44=12906 /32=403.33 45 440x2/3 x45=13200 /32=412.50 #G 46 440x2/3 x46=13493 /32=421.66 47 440x2/3 x47=13786 /32=430.83 48 440x2/3 x48=14080 /32=440.00 49 440x2/3 x49=14373 /32=449.16 50 440x2/3 x50=14666 /32=458.33 51 440x2/3 x51=14960 /32=467.50 #A 52 440x2/3 x52=15253 /32=476.66 53 440x2/3 x53=15546 /32=485.83 54 440x2/3 x54=15840 /32=495.00 55 440x2/3 x55=16133 /64=252.07 56 440x2/3 x56=16426 /64=256.65 57 440x2/3 x57=16720 /64=261.25 C 58 440x2/3 x58=17013 /64=265.82 59 440x2/3 x59=17306 /64=270.41 60 440x2/3 x60=17600 /64=275.00 61 440x2/3 x61=17893 /64=279.58 62 440x2/3 x62=18186 /64=284.16 63 440x2/3 x63=18480 /64=288.75 64 440x2/3 x64=18773 /64=293.33 D 65 440x2/3 x65=19066 /64=297.91 66 440x2/3 x66=19360 /64=302.50 67 440x2/3 x67=19653 /64=307.08 68 440x2/3 x68=19946 /64=311.66 69 440x2/3 x69=20240 /64=316.25 70 440x2/3 x70=20533 /64=320.83 71 440x2/3 x71=20826 /64=325.41 72 440x2/3 x72=21120 /64=330.00 E 73 440x2/3 x73=21413 /64=334.58 74 440x2/3 x74=21706 /64=339.16 75 440x2/3 x75=22000 /64=343.75 76 440x2/3 x76=22293 /64=348.33 77 440x2/3 x77=22586 /64=352.91 78 440x2/3 x78=22880 /64=357.50 79 440x2/3 x79=23173 /64=362.07 80 440x2/3 x80=23466 /64=366.66 81 440x2/3 x81=23760 /64=371.25 82 440x2/3 x82=24053 /64=375.83 83 440x2/3 x83=24346 /64=380.41 84 440x2/3 x84=24640 /64=385.00 85 440x2/3 x85=24933 /64=389.58 86 440x2/3 x86=25226 /64=394.16 87 440x2/3 x87=25520 /64=398.75 88 440x2/3 x88=25813 /64=403.33 89 440x2/3 x89=26106 /64=407.91 90 440x2/3 x90=26400 /64=412.50 91 440x2/3 x91=26693 /64=417.08 92 440x2/3 x92=26986 /64=421.66 93 440x2/3 x93=27280 /64=426.25 94 440x2/3 x94=27573 /64=430.83 95 440x2/3 x95=27866 /64=435.41 96 440x2/3 x96=28160 /64=440.00 97 440x2/3 x97=28453 /64=444.58 98 440x2/3 x98=28746 /64=449.16 99 440x2/3 x99=29040 /64=453.75 100 440x2/3x100=29333 /64=458.33 由上表推論可知: 12平均律 C=第57分音440x2/3 x57=16720 /64=261.25 比值 57/57 - 2 cent 261.63 #c=第60分音440x2/3 x60=17600 /64=275.00 比值 60/57 -14 cent 277.18 D=第64分音440x2/3 x64=18773 /64=293.33 比值 64/57 - 2 cent 293.66 #D=第17分音440x2/3 x17=4986.6/16=311.66 比值 68/57 - 3 cent 311.13 E=第 9分音440x2/3 x9 =2640.0/8 =330.00 比值 72/57 2 cent 329.63 F=第19分音440x2/3 x19=5573.3/16=348.33 比值 51/57 - 4 cent 349.23 #F=第 5分音440x2/3 x5 =1466.6/4 =366.66 比值 80/57 -16 cent 369.99 G=第43分音440x2/3 x43=12613 /32=394.16 比值 86/57 - 9 cent 392.00 #G=第45分音440x2/3 x45=13200 /32=412.50 比值 90/57 -12 cent 415.30 A=第 3分音440x2/3 x3 =880.00/2 =440.00 比值 96/57 0 cent 440.00 #A=第51分音440x2/3 x51=14960 /32=467.50 比值 5 cent 466.16 B=第27分音440x2/3 x27=7920.0/16=495.00 比值 4 cent 493.89 適合F A C E G(D=297) #D=311.66=-3cent 349.23-311.66=37.57 200/37.57=5.323 311.13-311.66=-0.53 -0.53x5.323=-2.82=-3cent F 392-349.23=42.77 200/42.77=4.67617 349.23-348.33=0.9 0.9x4.67617=4.2 F=348.33=-4cent 五.以基音F=Ax4/5=440x4/5 (352.00)HZ 為起音,推算各頻率值. 泛音序 頻率值 音名 1 352x1 =352 F 2 352x2 =704 /2=352 F 3 352x3 =1056 /4=264 C 4 352x4 =1408 /4=352 F 5 352x5 =1760 /4=440 A 6 352x6 =2112 /8=264 C 7 352x7 =2464 /8=308 8 352x8 =2816 /8=352 F 9 352x9 =3168 /8=396 G 10 352x10=3520 /8=440 A 11 352x11=3872 /8=484 12 352x12=4224/16=264 C 13 352x13=4576/16=286 14 352x14=4928/16=308 15 352x15=5280/16=330 E 16 352x16=5632/16=352 F 17 352x17=5984/16=374 #F 18 352x18=6336/16=396 G 19 352x19=6688/16=418 20 352x20=7040/16=440 A 21 352x21=7392/16=462 bB 22 352x22=7744/16=484 23 352x23=6746/16=421 bA 24 352x24=7040/16=440 A 25 352x25=8800/32=275 #C 26 352x26=9152/32=286 27 352x27=9504/32=297 D 不採用 28 352x28=9856/32=308 29 352x29=10208/32=319 30 352x30=10560/32=330 E 31 352x31=10912/32=341 32 352x32=11264/32=352 F 33 352x33=11616/32=363 34 352x34=11968/32=374 35 352x35=12320/32=385 36 352x36=12672/32=396 G 37 352x37=13024/32=407 38 352x38=13376/32=410 39 352x39=13728/32=429 40 352x40=14080/32=440 41 352x41=14432/32=451 42 352x42=14784/32=462 43 352x43=15136/32=473 44 352x44=15488/32=484 45 352x45=15840/32=495 B 46 352x46=16192/64=253 47 352x47=16544/64=258.5 48 352x48=16896/64=264 C 49 352x49=17248/64=269.5 50 352x50=17600/64=275 #C 51 352x51=17952/64=280.5 52 352x52=18304/64=286 53 352x53=18656/64=291.5 54 352x54=19008/64=297 55 352x55=19360/64=302.5 56 352x56=19712/64=308 57 352x57=20064/64=313.5 58 352x58=20416/64=319 59 352x59=20768/64=324.5 60 352x60=21120/64=330 61 352x61=21472/64=335.5 62 352x62=21824/64=341 63 352x63=22176/64=346.5 64 352x64=22528/64=352 65 352x65=22880/64=357.5 66 352x66=23232/64=363 67 352x67=23584/64=368.5 68 352x68=21120/64=330 69 352x69=24288/64=379.5 70 352x70=24640/64=385 71 352x71=24992/64=390.5 72 352x72=25344/64=396 73 352x73=25696/64=401.5 74 352x74=26048/64=407 75 352x75=26400/64=412.5 #G 76 352x76=26752/64=418 77 352x77=27104/64=423.5 78 352x78=27456/64=429 79 352x79=27808/64=434.5 80 352x80=28160/64=440 81 352x81=28512/64=445.5 82 352x82=28864/64=451 83 352x83=29216/64=456.5 84 352x84=29568/64=462 85 352x85=29920/64=467.5 86 352x86=30272/64=473 87 352x87=30624/64=478.5 88 352x88=30976/64=484 89 352x89=31328/64=489.5 90 352x90=31680/64=495 91 352x91=32032/64=500.5 92 352x92=32384/64=506 93 352x93=32736/64=511.5 94 352x94=33088/64=517 95 352x95=33440/64=522.5 96 352x96=33792/128=264 C 97 352x97=34144/128=266.75 98 352x98=34496/128=269.5 99 352x99=34848/128=272.25 100 352x100=35200/128=275 #C 由上表推論可知: 比數 比值 12平均律 C= F 第 3 分音 = 264 1.00 1/1 16 cent 261.63 #C= F 第 25 分音 = 275 1.04 25/24 -14 cent 277.18 D= 440x2/3 A之下屬音= 293.33 1.11 10/9 -2 cent 293.66 bE= F 第 14 分音 = 308 1.16 14/12 -14 cent 311.13 E= F 第 15 分音 = 330 1.25 15/12 2 cent 329.63 F= F 第 1 分音 = 352 1.33 4/3 13 cent 349.23 #F= F 第 17 分音 = 374 1.42 17/12 18 cent 369.99 G= F 第 9 分音 = 396 1.50 3/2 18 cent 392.00 #G= F 第 75 分音 = 412.5 1.59 25/16 -12 cent 415.30 A= F 第 5 分音 = 440 1.66 5/3 0 cent 440.00 bB= F 第 21 分音 = 462 1.75 21/12 -13 cent 466.16 B= F 第 45 分音 = 495 1.87 45/12 4 cent 493.89 bE 308 =-14cent bB 462 =-13cent #F374=18cent 352-308=44 528-462=66 415.3-369.99=45.31 200/44=4.55 200/66=3.03 200/45.31=4.41 (308-311.13)x4.55=-14.24 (462-466.16)x3.03=-12.6 (374-369.99)x4.41=17.68 六.以基音C=261.63 HZ 為起音,推算各頻率值. 泛音序 頻率值 音名 12平均律 1 261.63x1=261.63 C 261.63 2 261.63x2=523.26 /2=261.63 C 3 261.63x3=784.89 /2=392.445 G 392.00 4 261.63x4=1046.52 /4=261.63 C 5 261.63x5=1308.15 /4=327.0375 E 329.63 6 261.63x6=1569.78 /4=392.445 G 7 261.63x7=1831.41 /4=457.8525 bB 466.16 8 261.63x8=2093.04 /8=261.63 C 9 261.63x9=2354.67 /8=294.3375 D 293.66 10 261.63x10=2616.3 /8=327.0375 E 11 261.63x11=2877.93/8=359.74125 #F 349.23 12 261.63x12=3139.56/8=392.445 G 392.00 13 261.63x13=3401.19/8=425.14875 14 261.63x14=3662.82/8=457.8525 15 261.63x15=3924.45/8=490.55625 16 261.63x16=4186.08/16=261.63 17 261.63x17=4447.71/16=277.981875 18 261.63x18=4709.34/16=294.33375 19 261.63x19=4970.97/16=310.685625 20 261.63x20=5232.6 /16=327.0375 21 261.63x21=5494.23/16=343.389375 22 261.63x22=5755.86/16=359.74125 23 261.63x23=6017.49/16=376.093125 24 261.63x24=6279.12/16=392.445 G 25 261.63x25=6540.75/16=408.796875 26 261.63x26=6802.38/16=425.14875 27 261.63x27=7064.01/16=441.500625 A 440 28 261.63x28=7325.64/16=457.8525 29 261.63x29=7587.27/16=474.204375 30 261.63x30=7848.9 /16=490.55625 七.以基音C=261.25 HZ 為起音,推算各頻率值. 泛音列 頻率值 音名 12平均律 1 261.25x 1 261.25 C 261.63 2 261.25x 2 522.50/2=261.25 C 3 261.25x 3 783.75/2=391.875 G 392.00 4 261.25x 4 1045.00/4=261.25 C 5 261.25x 5 1306.25/4=326.5625 E 329.63 6 261.25x 6 1567.50/4=391.875 G 7 261.25x 7 1828.75/4=457.1875 bB 466.16 8 261.25x 8 2090.00/8=261.25 c 9 261.25x 9 2351.25/8=293.90625 D 293.66 10 261.25x 10 2612.50/8=326.5625 E 11 261.25x 11 2873.75/8=359.21875 #F 349.23 12 261.25x 12 3135.00/8=391.875 G 392.00 13 261.25x 13 3396.25/8=424.53125 14 261.25x 14 3657.50/8=457.1875 15 261.25x 15 3918.75/8=489.84375 16 261.25x 16 4180.00/16=261.25 17 261.25x 17 4441.25/16=277.478125 18 261.25x 18 4702.50/16=293.90625 19 261.25x 19 4963.75/16=310.234375 #D=bE 20 261.25x 20 5225.00/16=326.5625 21 261.25x 21 5486.25/16=342.890325 22 261.25x 22 5747.50/16=359.21875 23 261.25x 23 6008.75/16=375.546875 24 261.25x 24 6270.00/16=391.875 25 261.25x 25 6531.25/16=408.203125 26 261.25x 26 6792.50/16=424.53125 27 261.25x 27 7053.75/16=440.859375 28 261.25x 28 7315.00/16=457.1875 29 261.25x 29 7576.25/16=473.515625 30 261.25x 30 7837.50/16=489.84375 31 261.25x 31 8098.75/32=506.171875 32 261.25x 32 8360.00/32=261.25 33 261.25x 33 8621.25/32=269.4140625 34 261.25x 34 8882.50/32=277.578125 35 261.25x 35 9143.75/32=285.7421875 36 261.25x 36 9405.00/32=293.90625 37 261.25x 37 9666.25/32=302.0703125 38 261.25x 38 9927.50/32=310.234375 39 261.25x 39 10188.75/32=318.3984375 40 261.25x 40 10450.00/32=326.5625 41 261.25x 41 10711.25/32=334.7265625 42 261.25x 42 10972.50/32=342.890625 43 261.25x 43 11233.75/32=351.0546875 44 261.25x 44 11495.00/32=359.21875 45 261.25x 45 11756.25/32=367.3828125 46 261.25x 46 12017.50/32=375.546875 47 261.25x 47 12278.75/32=383.7109375 48 261.25x 48 12540.00/32=391.875 49 261.25x 49 12801.25/32=400.0390625 50 261.25x 50 13062.50/32=408.203125 51 261.25x 51 13323.75/32=416.3671875 52 261.25x 52 13585.00/32=424.53125 53 261.25x 53 13846.25/32=432.6953125 54 261.25x 54 14107.50/32=440.859375 55 261.25x 55 14368.75/32=449.0234375 56 261.25x 56 14630.00/32=457.1875 57 261.25x 57 14891.25/32=465.3515625 58 261.25x 58 15152.50/32=473.515625 59 261.25x 59 15413.75/32=481.6796875 60 261.25x 60 15675.00/32=489.84375 61 261.25x 61 15936.25/32=498.0078125 62 261.25x 62 16197.50/64=253.0859375 63 261.25x 63 16458.75/64=257.15625 64 261.25x 64 16720.00/64=261.25 65 261.25x 65 16981.25/64=265.33203125 66 261.25x 66 17242.50/64=269.4140625 67 261.25x 67 17503.75/64=273.49609375 68 261.25x 68 17765.00/64=277.578125 69 261.25x 69 18026.25/64=281.66015625 70 261.25x 70 18287.50/64=285.7421875 71 261.25x 71 18548.75/64=289.82421875 72 261.25x 72 18810.00/64=293.90625 73 261.25x 73 19071.25/64=297.98828125 74 261.25x 74 19332.50/64=302.0703125 75 261.25x 75 19593.75/64=306.15234375 76 261.25x 76 19855.00/64=310.234375 77 261.25x 77 20116.25/64=314.31640625 78 261.25x 78 20377.50/64=318.3984375 79 261.25x 79 20638.75/64=322.48046875 80 261.25x 80 20900.00/64=326.5625 81 261.25x 81 21161.25/64=330.64453125 82 261.25x 82 21422.50/64=334.7265625 83 261.25x 83 21683.75/64=338.80859375 84 261.25x 84 21945.00/64=342.890625 85 261.25x 85 22206.25/64=346.97265625 86 261.25x 86 22467.50/64=351.0546875 87 261.25x 87 22728.75/64=355.13671875 88 261.25x 88 22990.00/64=359.21875 89 261.25x 89 23251.25/64=363.30078125 90 261.25x 90 23512.50/64=367.3828125 91 261.25x 91 23773.75/64=371.45703125 92 261.25x 92 24035.00/64=375.546875 93 261.25x 93 24296.25/64=379.62890625 94 261.25x 94 24557.50/64=383.7109375 95 261.25x 95 24818.75/64=387.79296875 96 261.25x 96 25080.00/64=391.875 97 261.25x 97 25341.25/64=395.95703125 98 261.25x 98 25602.50/64=400.0390625 99 261.25x 99 25863.75/64=404.12109375 100 261.25x 100 26125.00/64=408.203125 八.以基音bB=440x2/3x4/5 x2=469.33HZ 為起音,推算各頻率值. 440x16/15 =469.33HZ 泛音列 頻率值 音名 12平均律 1 440x16/15x 1 469.33 bB 466.16 2 440x16/15x 2 469.33 bB 466.16 3 440x16/15x 3 352 F 349.23 4 440x16/15x 4 469.33 bB 466.16 5 440x16/15x 5 293.33 D 293.66 6 440x16/15x 6 352 F 7 440x16/15x 7 410.66 8 440x16/15x 8 469.33 bB 466.16 9 440x16/15x 9 264 C 261.63 10 440x16/15x10 293.33 D 293.66 11 440x16/15x11 322.66 12 440x16/15x12 352 F A 泛音純律與十二平均律之比較 HARMORICA-PURE TEMPERAMENT & EQUAL TEMPERAMENT RATIO C 19泛音261.25 #F 27泛音371.25 293.33-261.25=32.08 412.75-371.25=41.25 200/32.08=6.2344 200/41.25=4.84 (261.25-261.63)=-0.38 (371.25-369.99)=1.26 -0.38x6.2344=-2.36cent 4.84x1.26=6.09 cent C 261.25 = -2 cent #F 371.25 = 6 cent ─────────── ─────────── #C 5泛音275 * G 57泛音391.88 312.81-275=37.81 440-391.88=48.12 200/37.81=5.29 200/48.12=4.1562 (275-277.18)=-2.18 (391.88-392)=-0.12 -2.18x5.29=-11.53 cent -0.12x4.1562=0.4987 cent #C 275 = -12 cent G 391.86 = -0.5 cent ─────────── ─────────── D 為A下屬音 293.33 #G 15泛音412.5 330-293.33=36.67 467.5-412.5=55 200/36.67=5.4540 200/55=3.63 (293.33-293.66)=-0.33 (412.5-415.3)=-2.8 -0.33x5.4540=-1.7998ent -2.8x3.63=-10.16 cent D 293.33 = - 2 cent #G 412.5 = -10 cent ─────────── ─────────── #D 91泛音312.81 A 349.23-311.13=38.1 200/38.1 =5.28 312.81-311.13 = 1.68 Ａ＝４４０ 1.68 x5.25= 8.82 cent #D 312.81 = 9 cent A 440 = 0 cent ─────────── ─────────── E 6泛音330 #A 17泛音467.5 371.25-330=41.25 522.5-467.5=55 200/41.25=4.84 200/55=3.63 (330-329.63)=0.37 (467.5-466.16)=1.34 0.37x4.84=1.79 cent 1.34x3.63=4.86 cent E 330 = 2 cent #A 467.5 = 5 cent ─────────── ─────────── F 51泛音350.63 B 9泛音495 391.86-350.63=41.23 550-495=55 200/41.23=4.85 200/55=3.63 (350.63-349.23)=1.4 (495-493.88)=1.12 1.4x4.85=6.79 cent 1.12x3.63=4.07 cent F 350.63=7 cent B 495 =4 cent