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MacBinary | 2002-04-13 | 16.0 KB | [ TEXT/Bfly]
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MacBinary II, inited, Sat Apr 13 09:44:11 2002, modified Sat Apr 13 09:44:11 2002, creator 'Bfly', type ASCII, 13439 bytes "Intonation and Tuning" , at 0x34ff 2756 bytes resource
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id metadata key value macFileType [ TEXT] macFileCreator [ Bfly]
hex view +--------+-------------------------+-------------------------+--------+--------+ |00000000| 00 15 49 6e 74 6f 6e 61 | 74 69 6f 6e 20 61 6e 64 |..Intona|tion and| |00000010| 20 54 75 6e 69 6e 67 00 | 00 00 00 00 00 00 00 00 | Tuning.|........| |00000020| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........| |00000030| 00 00 00 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........| |00000040| 00 54 45 58 54 42 66 6c | 79 01 00 00 80 01 01 00 |.TEXTBfl|y.......| |00000050| 00 00 00 00 00 34 7f 00 | 00 0a c4 b8 dd e6 ab b8 |.....4..|........| |00000060| dd e6 ab 00 00 00 00 00 | 00 00 00 00 00 00 00 00 |........|........| |00000070| 00 00 00 00 00 00 00 00 | 00 00 81 81 fb 07 00 00 |........|........| |00000080| 20 0d 49 6e 74 6f 6e 61 | 74 69 6f 6e 20 61 6e 64 | .Intona|tion and| |00000090| 20 54 75 6e 69 6e 67 0d | 0d 50 6c 65 61 73 65 20 | Tuning.|.Please | |000000a0| 6e 6f 74 65 20 74 68 61 | 74 20 74 68 65 72 65 20 |note tha|t there | |000000b0| 69 73 20 61 20 62 75 67 | 20 69 6e 20 51 75 69 63 |is a bug| in Quic| |000000c0| 6b 74 69 6d 65 20 35 20 | 77 68 69 63 68 20 70 72 |ktime 5 |which pr| |000000d0| 65 76 65 6e 74 73 20 69 | 6e 74 6f 6e 61 74 69 6f |events i|ntonatio| |000000e0| 6e 73 20 6f 74 68 65 72 | 20 74 68 61 6e 20 65 71 |ns other| than eq| |000000f0| 75 61 6c 20 74 65 6d 70 | 65 72 61 6d 65 6e 74 20 |ual temp|erament | |00000100| 66 72 6f 6d 20 77 6f 72 | 6b 69 6e 67 2e 20 20 49 |from wor|king. I| |00000110| 66 20 79 6f 75 20 77 61 | 6e 74 20 74 6f 20 77 6f |f you wa|nt to wo| |00000120| 72 6b 20 77 69 74 68 20 | 69 6e 74 6f 6e 61 74 69 |rk with |intonati| |00000130| 6f 6e 73 20 6f 72 20 75 | 73 65 20 74 68 65 20 54 |ons or u|se the T| |00000140| 75 6e 65 20 64 69 72 65 | 63 74 69 76 65 20 69 6e |une dire|ctive in| |00000150| 20 61 62 63 73 20 79 6f | 75 20 6d 75 73 74 20 72 | abcs yo|u must r| |00000160| 65 76 65 72 74 20 74 6f | 20 51 75 69 63 6b 74 69 |evert to| Quickti| |00000170| 6d 65 20 34 2e 20 20 51 | 75 69 63 6b 74 69 6d 65 |me 4. Q|uicktime| |00000180| 20 34 2e 30 2e 33 20 69 | 73 20 73 74 69 6c 6c 20 | 4.0.3 i|s still | |00000190| 61 76 61 69 6c 61 62 6c | 65 20 66 72 6f 6d 20 41 |availabl|e from A| |000001a0| 70 70 6c 65 20 61 73 20 | 69 74 27 73 20 74 68 65 |pple as |it's the| |000001b0| 20 6c 61 73 74 20 51 75 | 69 63 6b 74 69 6d 65 20 | last Qu|icktime | |000001c0| 76 65 72 73 69 6f 6e 20 | 77 68 69 63 68 20 77 6f |version |which wo| |000001d0| 72 6b 73 20 6f 6e 20 74 | 68 65 20 6f 6c 64 65 72 |rks on t|he older| |000001e0| 20 6d 61 63 68 69 6e 65 | 73 20 77 69 74 68 20 36 | machine|s with 6| |000001f0| 38 30 78 30 20 70 72 6f | 63 65 73 73 6f 72 73 2e |80x0 pro|cessors.| |00000200| 0d 0d 53 69 6e 63 65 20 | 74 69 6d 65 20 69 6d 6d |..Since |time imm| |00000210| 65 6d 6f 72 69 61 6c 2c | 20 6d 75 73 69 63 69 61 |emorial,| musicia| |00000220| 6e 73 20 68 61 76 65 20 | 74 75 6e 65 64 20 74 68 |ns have |tuned th| |00000230| 65 69 72 20 69 6e 73 74 | 72 75 6d 65 6e 74 73 20 |eir inst|ruments | |00000240| 62 79 20 65 61 72 2c 20 | 73 69 6d 70 6c 79 20 63 |by ear, |simply c| |00000250| 68 6f 6f 73 69 6e 67 20 | 74 68 65 20 70 69 74 63 |hoosing |the pitc| |00000260| 68 65 73 20 77 68 69 63 | 68 20 73 6f 75 6e 64 20 |hes whic|h sound | |00000270| 72 69 67 68 74 20 66 6f | 72 20 74 68 65 20 74 75 |right fo|r the tu| |00000280| 6e 65 20 61 6e 64 20 74 | 68 65 20 6b 65 79 20 69 |ne and t|he key i| |00000290| 6e 20 77 68 69 63 68 20 | 74 68 65 79 20 61 72 65 |n which |they are| |000002a0| 20 67 6f 69 6e 67 20 74 | 6f 20 70 6c 61 79 2e 20 | going t|o play. | |000002b0| 20 49 74 27 73 20 73 74 | 69 6c 6c 20 74 68 65 20 | It's st|ill the | |000002c0| 62 65 73 74 20 77 61 79 | 2e 20 20 48 6f 77 65 76 |best way|. Howev| |000002d0| 65 72 2c 20 6d 75 73 69 | 63 61 6c 20 69 6e 73 74 |er, musi|cal inst| |000002e0| 72 75 6d 65 6e 74 20 6d | 61 6b 65 72 73 20 68 61 |rument m|akers ha| |000002f0| 76 65 20 61 20 6d 6f 72 | 65 20 64 69 66 66 69 63 |ve a mor|e diffic| |00000300| 75 6c 74 20 70 72 6f 62 | 6c 65 6d 20 74 6f 20 64 |ult prob|lem to d| |00000310| 65 61 6c 20 77 69 74 68 | 2c 20 73 69 6e 63 65 20 |eal with|, since | |00000320| 69 6e 20 6d 61 6e 79 20 | 63 61 73 65 73 20 74 68 |in many |cases th| |00000330| 65 79 20 6e 65 65 64 20 | 74 6f 20 66 69 78 20 74 |ey need |to fix t| |00000340| 68 65 20 70 69 74 63 68 | 65 73 20 77 68 69 63 68 |he pitch|es which| |00000350| 20 61 6e 20 69 6e 73 74 | 72 75 6d 65 6e 74 20 70 | an inst|rument p| |00000360| 6c 61 79 73 20 69 6e 20 | 61 64 76 61 6e 63 65 2c |lays in |advance,| |00000370| 20 77 69 74 68 6f 75 74 | 20 6b 6e 6f 77 69 6e 67 | without| knowing| |00000380| 20 77 68 61 74 20 74 68 | 65 20 70 6c 61 79 65 72 | what th|e player| |00000390| 20 77 69 6c 6c 20 75 73 | 65 20 69 74 20 66 6f 72 | will us|e it for| |000003a0| 2e 20 20 54 68 69 73 20 | 69 73 20 6e 6f 74 20 61 |. 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No| |000004c0| 6e 65 20 6f 66 20 50 79 | 74 68 61 67 6f 72 61 73 |ne of Py|thagoras| |000004d0| 27 20 6f 77 6e 20 77 72 | 69 74 69 6e 67 73 20 73 |' own wr|itings s| |000004e0| 75 72 76 69 76 65 2c 20 | 61 6e 64 20 77 65 20 6f |urvive, |and we o| |000004f0| 6e 6c 79 20 6b 6e 6f 77 | 20 61 62 6f 75 74 20 68 |nly know| about h| |00000500| 69 6d 20 66 72 6f 6d 20 | 74 68 65 20 77 72 69 74 |im from |the writ| |00000510| 69 6e 67 73 20 6f 66 20 | 6f 74 68 65 72 73 3b 20 |ings of |others; | |00000520| 20 69 74 20 73 65 65 6d | 73 20 74 68 61 74 20 68 | it seem|s that h| |00000530| 65 20 61 74 74 61 63 68 | 65 64 20 67 72 65 61 74 |e attach|ed great| |00000540| 20 6d 79 73 74 69 63 61 | 6c 20 73 69 67 6e 69 66 | mystica|l signif| |00000550| 69 63 61 6e 63 65 20 74 | 6f 20 74 68 65 20 6e 75 |icance t|o the nu| |00000560| 6d 62 65 72 20 66 69 76 | 65 2c 20 61 6e 64 20 73 |mber fiv|e, and s| |00000570| 70 65 6e 74 20 73 6f 6d | 65 20 74 69 6d 65 20 64 |pent som|e time d| |00000580| 65 76 65 6c 6f 70 69 6e | 67 20 61 20 74 68 65 6f |evelopin|g a theo| |00000590| 72 79 20 6f 66 20 6c 69 | 66 65 2c 20 74 68 65 20 |ry of li|fe, the | |000005a0| 75 6e 69 76 65 72 73 65 | 20 61 6e 64 20 65 76 65 |universe| and eve| |000005b0| 72 79 74 68 69 6e 67 20 | 62 61 73 65 64 20 6f 6e |rything |based on| |000005c0| 20 66 69 76 65 2d 66 6f | 6c 64 20 73 79 6d 6d 65 | five-fo|ld symme| |000005d0| 74 72 79 2e 20 20 48 65 | 20 70 72 6f 62 61 62 6c |try. He| probabl| |000005e0| 79 20 75 73 65 64 20 61 | 20 6d 6f 6e 6f 63 68 6f |y used a| monocho| |000005f0| 72 64 20 74 6f 20 69 6e | 76 65 73 74 69 67 61 74 |rd to in|vestigat| |00000600| 65 20 6d 75 73 69 63 61 | 6c 20 70 69 74 63 68 2c |e musica|l pitch,| |00000610| 20 61 6e 20 69 6e 73 74 | 72 75 6d 65 6e 74 20 77 | an inst|rument w| |00000620| 69 74 68 20 6f 6e 65 20 | 73 74 72 69 6e 67 20 61 |ith one |string a| |00000630| 6e 64 20 61 20 73 6c 69 | 64 69 6e 67 20 62 72 69 |nd a sli|ding bri| |00000640| 64 67 65 20 77 68 69 63 | 68 20 63 61 6e 20 62 65 |dge whic|h can be| |00000650| 20 6d 6f 76 65 64 20 74 | 6f 20 61 6e 79 20 70 6f | moved t|o any po| |00000660| 73 69 74 69 6f 6e 2e 20 | 20 48 65 20 77 69 6c 6c |sition. | He will| |00000670| 20 68 61 76 65 20 6e 6f | 74 69 63 65 64 20 74 68 | have no|ticed th| |00000680| 61 74 20 69 66 20 74 68 | 65 20 62 72 69 64 67 65 |at if th|e bridge| |00000690| 20 69 73 20 70 6c 61 63 | 65 64 20 73 6f 20 74 68 | is plac|ed so th| |000006a0| 61 74 20 74 68 65 20 6c | 65 6e 67 74 68 73 20 6f |at the l|engths o| |000006b0| 66 20 73 74 72 69 6e 67 | 20 6f 6e 20 65 69 74 68 |f string| on eith| |000006c0| 65 72 20 73 69 64 65 20 | 61 72 65 20 69 6e 20 74 |er side |are in t| |000006d0| 68 65 20 72 61 74 69 6f | 20 32 20 3a 20 33 20 74 |he ratio| 2 : 3 t| |000006e0| 68 65 79 20 66 6f 72 6d | 20 74 68 65 20 73 77 65 |hey form| the swe| |000006f0| 65 74 65 73 74 20 73 6f | 75 6e 64 69 6e 67 20 6f |etest so|unding o| |00000700| 66 20 63 68 6f 72 64 73 | 20 77 68 65 6e 20 70 6c |f chords| when pl| |00000710| 61 79 65 64 20 74 6f 67 | 65 74 68 65 72 2e 20 20 |ayed tog|ether. | |00000720| 4e 6f 77 20 32 20 2b 20 | 33 20 3d 20 35 2c 20 61 |Now 2 + |3 = 5, a| |00000730| 6e 64 20 69 66 20 79 6f | 75 20 70 6c 61 79 20 61 |nd if yo|u play a| |00000740| 20 73 63 61 6c 65 20 73 | 74 61 72 74 69 6e 67 20 | scale s|tarting | |00000750| 6f 6e 20 74 68 65 20 6c | 6f 77 65 72 20 6e 6f 74 |on the l|ower not| |00000760| 65 20 79 6f 75 20 66 69 | 6e 64 20 74 68 61 74 20 |e you fi|nd that | |00000770| 74 68 65 20 68 69 67 68 | 65 72 20 6e 6f 74 65 20 |the high|er note | |00000780| 63 6f 72 72 65 73 70 6f | 6e 64 73 20 74 6f 20 74 |correspo|nds to t| |00000790| 68 65 20 66 69 66 74 68 | 20 6e 6f 74 65 20 6f 66 |he fifth| note of| |000007a0| 20 74 68 61 74 20 73 63 | 61 6c 65 20 28 74 68 65 | that sc|ale (the| |000007b0| 20 69 6e 74 65 72 76 61 | 6c 20 69 73 20 77 68 61 | interva|l is wha| |000007c0| 74 20 77 65 20 6e 6f 77 | 20 63 61 6c 6c 20 61 20 |t we now| call a | |000007d0| 70 65 72 66 65 63 74 20 | 66 69 66 74 68 29 2e 20 |perfect |fifth). | |000007e0| 20 50 79 74 68 61 67 6f | 72 61 73 20 77 61 73 20 | Pythago|ras was | |000007f0| 73 75 72 65 20 74 68 61 | 74 20 68 69 73 20 6d 61 |sure tha|t his ma| |00000800| 67 69 63 20 6e 75 6d 62 | 65 72 20 77 61 73 20 61 |gic numb|er was a| |00000810| 74 20 74 68 65 20 72 6f | 6f 74 20 6f 66 20 61 6c |t the ro|ot of al| |00000820| 6c 20 6d 75 73 69 63 2c | 20 61 6e 64 20 69 6e 76 |l music,| and inv| |00000830| 65 6e 74 65 64 20 61 20 | 77 61 79 20 6f 66 20 74 |ented a |way of t| |00000840| 75 6e 69 6e 67 20 73 75 | 63 68 20 74 68 61 74 20 |uning su|ch that | |00000850| 61 6c 6c 20 6e 6f 74 65 | 73 20 6f 66 20 74 68 65 |all note|s of the| |00000860| 20 73 63 61 6c 65 20 77 | 65 72 65 20 72 65 6c 61 | scale w|ere rela| |00000870| 74 65 64 20 74 6f 20 74 | 68 65 20 74 6f 6e 69 63 |ted to t|he tonic| |00000880| 20 62 79 20 77 61 79 20 | 6f 66 20 74 68 65 20 70 | by way |of the p| |00000890| 65 72 66 65 63 74 20 66 | 69 66 74 68 2e 0d 0d 57 |erfect f|ifth...W| |000008a0| 68 65 6e 20 77 65 20 64 | 69 73 63 75 73 73 20 74 |hen we d|iscuss t| |000008b0| 68 65 20 70 69 74 63 68 | 20 6f 66 20 6e 6f 74 65 |he pitch| of note| |000008c0| 73 20 74 6f 64 61 79 2c | 20 77 65 20 6d 61 6b 65 |s today,| we make| |000008d0| 20 6d 65 61 73 75 72 65 | 6d 65 6e 74 73 20 6f 66 | measure|ments of| |000008e0| 20 74 68 65 20 66 72 65 | 71 75 65 6e 63 79 20 69 | the fre|quency i| |000008f0| 6e 20 63 79 63 6c 65 73 | 20 70 65 72 20 73 65 63 |n cycles| per sec| |00000900| 6f 6e 64 20 6f 72 20 68 | 65 72 74 7a 20 28 48 7a |ond or h|ertz (Hz| |00000910| 29 2e 20 20 4f 66 20 63 | 6f 75 72 73 65 20 50 79 |). Of c|ourse Py| |00000920| 74 68 61 67 6f 72 61 73 | 20 68 61 64 20 6e 6f 20 |thagoras| had no | |00000930| 73 75 63 68 20 63 6f 6e | 63 65 70 74 20 2d 20 69 |such con|cept - i| |00000940| 6e 73 74 65 61 64 20 68 | 65 20 77 6f 75 6c 64 20 |nstead h|e would | |00000950| 68 61 76 65 20 74 68 6f | 75 67 68 74 20 69 6e 20 |have tho|ught in | |00000960| 74 65 72 6d 73 20 6f 66 | 20 74 68 65 20 6c 65 6e |terms of| the len| |00000970| 67 74 68 20 6f 66 20 73 | 74 72 69 6e 67 20 6e 65 |gth of s|tring ne| |00000980| 63 65 73 73 61 72 79 20 | 74 6f 20 73 6f 75 6e 64 |cessary |to sound| |00000990| 20 61 20 70 61 72 74 69 | 63 75 6c 61 72 20 6e 6f | a parti|cular no| |000009a0| 74 65 2e 20 20 53 69 6e | 63 65 20 74 68 65 20 66 |te. Sin|ce the f| |000009b0| 72 65 71 75 65 6e 63 79 | 20 70 72 6f 64 75 63 65 |requency| produce| |000009c0| 64 20 62 79 20 61 20 73 | 74 72 69 6e 67 20 69 73 |d by a s|tring is| |000009d0| 20 69 6e 76 65 72 73 65 | 6c 79 20 70 72 6f 70 6f | inverse|ly propo| |000009e0| 72 74 69 6f 6e 61 6c 20 | 74 6f 20 69 74 73 20 6c |rtional |to its l| |000009f0| 65 6e 67 74 68 2c 20 77 | 65 20 63 61 6e 20 73 61 |ength, w|e can sa| |00000a00| 79 20 74 68 61 74 20 74 | 68 65 20 74 68 61 74 20 |y that t|he that | |00000a10| 74 68 65 20 73 74 72 69 | 6e 67 20 6c 65 6e 67 74 |the stri|ng lengt| |00000a20| 68 20 72 61 74 69 6f 20 | 28 32 2f 33 29 20 77 68 |h ratio |(2/3) wh| |00000a30| 69 63 68 20 68 65 20 6f | 62 73 65 72 76 65 64 20 |ich he o|bserved | |00000a40| 63 6f 72 72 65 73 70 6f | 6e 64 73 20 74 6f 20 61 |correspo|nds to a| |00000a50| 20 66 72 65 71 75 65 6e | 63 79 20 72 61 74 69 6f | frequen|cy ratio| |00000a60| 20 6f 66 20 33 2f 32 20 | 69 2e 65 2e 20 74 68 65 | of 3/2 |i.e. the| |00000a70| 20 73 68 6f 72 74 65 72 | 20 73 74 72 69 6e 67 20 | shorter| string | |00000a80| 69 73 20 76 69 62 72 61 | 74 69 6e 67 20 65 78 61 |is vibra|ting exa| |00000a90| 63 74 6c 79 20 6f 6e 65 | 20 61 6e 64 20 61 20 68 |ctly one| and a h| |00000aa0| 61 6c 66 20 74 69 6d 65 | 73 20 66 61 73 74 65 72 |alf time|s faster| |00000ab0| 2e 20 20 54 6f 20 63 6f | 6e 73 74 72 75 63 74 20 |. To co|nstruct | |00000ac0| 61 20 70 79 74 68 61 67 | 6f 72 65 61 6e 20 73 63 |a pythag|orean sc| |00000ad0| 61 6c 65 2c 20 77 65 20 | 73 74 61 72 74 20 77 69 |ale, we |start wi| |00000ae0| 74 68 20 61 20 67 69 76 | 65 6e 20 74 6f 6e 69 63 |th a giv|en tonic| |00000af0| 20 28 73 61 79 20 41 20 | 3d 20 34 34 30 2e 30 20 | (say A |= 440.0 | |00000b00| 48 7a 2c 20 73 69 6e 63 | 65 20 74 68 61 74 20 69 |Hz, sinc|e that i| |00000b10| 73 20 6f 75 72 20 6d 6f | 64 65 72 6e 20 70 69 74 |s our mo|dern pit| |00000b20| 63 68 20 73 74 61 6e 64 | 61 72 64 29 2c 20 6d 75 |ch stand|ard), mu| |00000b30| 6c 74 69 70 6c 79 20 62 | 79 20 74 68 72 65 65 20 |ltiply b|y three | |00000b40| 61 6e 64 20 64 69 76 69 | 64 65 20 62 79 20 74 77 |and divi|de by tw| |00000b50| 6f 20 74 6f 20 67 65 74 | 20 74 68 65 20 70 69 74 |o to get| the pit| |00000b60| 63 68 20 6f 66 20 74 68 | 65 20 45 20 61 62 6f 76 |ch of th|e E abov| |00000b70| 65 3a 0d 0d 41 20 3d 20 | 34 34 30 2e 30 0d 45 20 |e:..A = |440.0.E | |00000b80| 3d 20 34 34 30 2e 30 20 | 2a 20 33 2f 32 20 3d 20 |= 440.0 |* 3/2 = | |00000b90| 36 36 30 2e 30 0d 0d 54 | 6f 20 67 65 74 20 74 68 |660.0..T|o get th| |00000ba0| 65 20 6e 65 78 74 20 6e | 6f 74 65 2c 20 42 2c 20 |e next n|ote, B, | |00000bb0| 77 65 20 6d 75 6c 74 69 | 70 6c 79 20 74 68 69 73 |we multi|ply this| |00000bc0| 20 61 67 61 69 6e 20 62 | 79 20 33 2f 32 2c 20 62 | again b|y 3/2, b| |00000bd0| 75 74 20 74 68 69 73 20 | 74 61 6b 65 73 20 75 73 |ut this |takes us| |00000be0| 20 6f 75 74 20 6f 66 20 | 74 68 65 20 6f 63 74 61 | out of |the octa| |00000bf0| 76 65 2c 20 73 6f 20 77 | 65 20 64 69 76 69 64 65 |ve, so w|e divide| |00000c00| 20 62 79 20 61 6e 20 65 | 78 74 72 61 20 74 77 6f | by an e|xtra two| |00000c10| 20 74 6f 20 67 65 74 20 | 75 73 20 62 61 63 6b 20 | to get |us back | |00000c20| 69 6e 20 74 68 65 20 73 | 61 6d 65 20 6f 63 74 61 |in the s|ame octa| |00000c30| 76 65 3a 0d 0d 42 20 3d | 20 36 36 30 20 2a 20 33 |ve:..B =| 660 * 3| |00000c40| 2f 34 20 3d 20 34 39 35 | 2e 30 0d 0d 49 66 20 77 |/4 = 495|.0..If w| |00000c50| 65 20 63 61 72 72 79 20 | 74 68 69 73 20 70 72 6f |e carry |this pro| |00000c60| 63 65 73 73 20 6f 6e 20 | 66 6f 72 20 6c 6f 6e 67 |cess on |for long| |00000c70| 20 65 6e 6f 75 67 68 2c | 20 6d 75 6c 74 69 70 6c | enough,| multipl| |00000c80| 79 69 6e 67 20 62 79 20 | 74 68 72 65 65 20 61 6e |ying by |three an| |00000c90| 64 20 64 69 76 69 64 69 | 6e 67 20 62 79 20 74 77 |d dividi|ng by tw| |00000ca0| 6f 20 6f 72 20 66 6f 75 | 72 20 61 73 20 6e 65 63 |o or fou|r as nec| |00000cb0| 65 73 73 61 72 79 20 74 | 6f 20 73 74 61 79 20 69 |essary t|o stay i| |00000cc0| 6e 20 74 68 65 20 73 61 | 6d 65 20 6f 63 74 61 76 |n the sa|me octav| |00000cd0| 65 2c 20 77 65 20 77 69 | 6c 6c 20 67 65 6e 65 72 |e, we wi|ll gener| |00000ce0| 61 74 65 20 66 72 65 71 | 75 65 6e 63 69 65 73 20 |ate freq|uencies | |00000cf0| 66 6f 72 20 61 6c 6c 20 | 74 77 65 6c 76 65 20 6e |for all |twelve n| |00000d00| 6f 74 65 73 20 6f 66 20 | 74 68 65 20 63 68 72 6f |otes of |the chro| |00000d10| 6d 61 74 69 63 20 73 63 | 61 6c 65 3a 0d 0d 41 09 |matic sc|ale:..A.| |00000d20| 34 34 30 2e 30 0d 41 23 | 09 34 36 39 2e 39 0d 42 |440.0.A#|.469.9.B| |00000d30| 09 34 39 35 2e 30 0d 43 | 09 35 32 38 2e 36 0d 43 |.495.0.C|.528.6.C| |00000d40| 23 09 35 35 36 2e 39 0d | 44 09 35 39 34 2e 37 0d |#.556.9.|D.594.7.| |00000d50| 44 23 09 36 32 36 2e 35 | 0d 45 09 36 36 30 2e 30 |D#.626.5|.E.660.0| |00000d60| 0d 46 09 37 30 34 2e 38 | 0d 46 23 09 37 34 32 2e |.F.704.8|.F#.742.| |00000d70| 35 0d 47 09 37 39 32 2e | 39 0d 47 23 09 38 33 35 |5.G.792.|9.G#.835| |00000d80| 2e 33 0d 0d 54 68 69 73 | 20 69 73 20 66 69 6e 65 |.3..This| is fine| |00000d90| 20 61 73 20 66 61 72 20 | 61 73 20 69 74 20 67 6f | as far |as it go| |00000da0| 65 73 3b 20 20 74 68 65 | 20 70 72 6f 62 6c 65 6d |es; the| problem| |00000db0| 20 77 69 74 68 20 69 74 | 20 62 65 63 6f 6d 65 73 | with it| becomes| |00000dc0| 20 61 70 70 61 72 65 6e | 74 20 77 68 65 6e 20 79 | apparen|t when y| |00000dd0| 6f 75 20 63 61 6c 63 75 | 6c 61 74 65 20 74 68 65 |ou calcu|late the| |00000de0| 20 6e 65 78 74 20 66 72 | 65 71 75 65 6e 63 79 2c | next fr|equency,| |00000df0| 20 74 68 65 20 6f 63 74 | 61 76 65 20 41 27 2e 20 | the oct|ave A'. | |00000e00| 20 54 68 65 20 6c 61 73 | 74 20 6e 6f 74 65 20 63 | The las|t note c| |00000e10| 61 6c 63 75 6c 61 74 65 | 64 20 61 62 6f 76 65 20 |alculate|d above | |00000e20| 69 73 20 44 20 3d 20 35 | 39 34 2e 37 2c 20 73 6f |is D = 5|94.7, so| |00000e30| 20 41 27 20 3d 20 35 39 | 34 2e 37 20 2a 20 33 2f | A' = 59|4.7 * 3/| |00000e40| 32 20 3d 20 38 39 32 2e | 30 20 77 68 69 63 68 20 |2 = 892.|0 which | |00000e50| 69 73 20 71 75 69 74 65 | 20 77 72 6f 6e 67 2c 20 |is quite| wrong, | |00000e60| 61 73 20 74 68 65 20 6f | 63 74 61 76 65 20 73 68 |as the o|ctave sh| |00000e70| 6f 75 6c 64 20 65 78 61 | 63 74 6c 79 20 64 6f 75 |ould exa|ctly dou| |00000e80| 62 6c 65 20 74 68 65 20 | 66 72 65 71 75 65 6e 63 |ble the |frequenc| |00000e90| 79 20 74 6f 20 38 38 30 | 2e 30 2e 20 20 54 68 69 |y to 880|.0. Thi| |00000ea0| 73 20 77 69 6c 6c 20 6e | 6f 74 20 68 61 76 65 20 |s will n|ot have | |00000eb0| 73 65 72 69 6f 75 73 6c | 79 20 62 6f 74 68 65 72 |seriousl|y bother| |00000ec0| 65 64 20 50 79 74 68 61 | 67 6f 72 61 73 2c 20 73 |ed Pytha|goras, s| |00000ed0| 69 6e 63 65 20 61 6e 63 | 69 65 6e 74 20 47 72 65 |ince anc|ient Gre| |00000ee0| 65 6b 20 6d 75 73 69 63 | 20 77 61 73 20 6c 69 6d |ek music| was lim| |00000ef0| 69 74 65 64 20 74 6f 20 | 6c 65 73 73 20 74 68 61 |ited to |less tha| |00000f00| 6e 20 61 6e 20 6f 63 74 | 61 76 65 20 69 6e 20 72 |n an oct|ave in r| |00000f10| 61 6e 67 65 2c 20 61 6e | 64 20 68 69 73 20 6c 79 |ange, an|d his ly| |00000f20| 72 65 20 70 72 6f 62 61 | 62 6c 79 20 6f 6e 6c 79 |re proba|bly only| |00000f30| 20 68 61 64 20 73 65 76 | 65 6e 20 73 74 72 69 6e | had sev|en strin| |00000f40| 67 73 2e 20 20 54 68 65 | 20 50 79 74 68 61 67 6f |gs. The| Pythago| |00000f50| 72 65 61 6e 20 73 63 61 | 6c 65 20 69 73 20 6e 6f |rean sca|le is no| |00000f60| 74 20 63 69 72 63 75 6c | 61 72 20 2d 20 74 68 61 |t circul|ar - tha| |00000f70| 74 20 69 73 2c 20 69 74 | 20 64 6f 65 73 6e 27 74 |t is, it| doesn't| |00000f80| 20 67 65 74 20 79 6f 75 | 20 62 61 63 6b 20 74 6f | get you| back to| |00000f90| 20 77 68 65 72 65 20 79 | 6f 75 20 73 74 61 72 74 | where y|ou start| |00000fa0| 65 64 2e 20 20 54 68 65 | 20 72 65 61 73 6f 6e 20 |ed. 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20 |and we c|an play | |000021a0| 6d 75 73 69 63 20 77 68 | 69 63 68 20 6d 6f 64 75 |music wh|ich modu| |000021b0| 6c 61 74 65 73 20 69 6e | 74 6f 20 61 6e 79 20 6b |lates in|to any k| |000021c0| 65 79 2e 20 20 45 66 66 | 65 63 74 69 76 65 6c 79 |ey. 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