Alt\u0131n oran, do\u011fadaki \u00e7e\u015fitli varl\u0131klar\u0131n yap\u0131s\u0131nda bulunan, m\u00fckemmel uyumu temsil eden orand\u0131r. Pek \u00e7ok yerde kar\u015f\u0131m\u0131za \u00e7\u0131kabilir; \u00c7i\u00e7ek yapraklar\u0131, deniz kabuklar\u0131, \u00e7am a\u011fac\u0131 kozalaklar\u0131, insan v\u00fccudu. Alt\u0131n oran,do\u011fadaki g\u00fczelli\u011fi temsil eder. Y\u00fczy\u0131llard\u0131r sanat\u0131n pek \u00e7ok dal\u0131nda da kullan\u0131lm\u0131\u015f ve \u00fczerinde \u00e7e\u015fitli ara\u015ft\u0131rmalar yap\u0131lm\u0131\u015ft\u0131r. Do\u011fada, resimde, mimaride, m\u00fczikte kullan\u0131lan alt\u0131n oran asl\u0131nda Fibonacci adl\u0131 bir matematik\u00e7inin ortaya att\u0131\u011f\u0131 bir say\u0131 dizisinden do\u011fmu\u015ftur. Papatyalar\u0131n Fibonacci say\u0131lar\u0131 kadar ta\u00e7 yapraklar\u0131 vard\u0131r. Dolay\u0131s\u0131 ile \u201cSeviyor , sevmiyor\u201d fal\u0131 kula\u011fa pek romantik gelmese de, maalesef Fiboanacci say\u0131lar\u0131n\u0131n da\u011f\u0131l\u0131m\u0131n\u0131n istatisli\u011fine ba\u011fl\u0131d\u0131r. Ay\u00e7i\u00e7e\u011finin \u00e7i\u00e7ek k\u0131sm\u0131ndaki tohumlar her zaman bir Fiboanacci say\u0131s\u0131na kar\u015f\u0131l\u0131k gelir. Bu t\u00fcm \u00e7i\u00e7ekler i\u00e7in de ge\u00e7erlidir, ay\u00e7i\u00e7e\u011finin \u00f6zelli\u011fi tohumlar\u0131n\u0131n g\u00f6zle g\u00f6r\u00fcl\u00fcr bi\u00e7imde say\u0131labilmesindendir. Bir\u00e7ok \u00e7i\u00e7e\u011fin, tohum ba\u015f\u0131, bir k\u0131v\u0131rc\u0131\u011f\u0131n yapraklar\u0131, bir so\u011fan\u0131n katmanlar\u0131, ananas ve kozalaklar\u0131n kat kat kabuklar\u0131 gibi bitkisel \u015fekillerin bir\u00e7o\u011fu Fiboanacci sarmallar\u0131 i\u00e7erir. M\u0131s\u0131r\u2019daki piramitler kendi i\u00e7lerinde bu oran\u0131 kullanmakta ayr\u0131ca konu\u015fland\u0131klar\u0131 yerler itibar\u0131 ile bu orana uygun sipiral i\u00e7erisinde bulunmaktad\u0131rlar. Eski Yunan sanat\u0131 ve mimarisinde \u00e7ok\u00e7a kullan\u0131lm\u0131\u015ft\u0131r. R\u00f6nesans sanat\u00e7\u0131lar\u0131 Leonardo Da Vinci, Raphael, Rubens, Boticelli bu oran\u0131 kullananlar\u0131n ba\u015f\u0131nda gelirler.
\u201cSay\u0131\u201d kavram\u0131 \u00e7a\u011flar boyunca insanl\u0131\u011f\u0131n ilgisini \u00e7ekmi\u015ftir. Say\u0131 kavram\u0131 matemati\u011fin ve ayn\u0131 zamanda ya\u015fam\u0131n temel kayna\u011f\u0131d\u0131r. \u201cSay\u0131\u201d esas olarak somut bir kavramd\u0131r ama ya\u015fam\u0131n , do\u011fan\u0131n heryerinde yard\u0131r. \u0130nsanlar ilk\u00e7a\u011flardan itibaren say\u0131lar\u0131 ifade etmek i\u00e7in \u00e7e\u015fitli yollar denemi\u015fleridir. Ancak milattan 300 y\u0131l kadar \u00f6nce Hindistan say\u0131lar\u0131 rakamlamaya ba\u015flam\u0131\u015ft\u0131r. Bug\u00fcn kullan\u0131lan Hint-Arap say\u0131lar sistemindeki simgeler (0,1,2,3,…) baz\u0131 kitaplar\u0131n bas\u0131lmas\u0131 ile birlikte Avrupa\u2019ya da yay\u0131lm\u0131\u015ft\u0131r. Bu kitaplar aras\u0131nda da 13. Y\u00fczy\u0131l\u0131n ba\u015flar\u0131nda yay\u0131nlanan \u201cIl Liber Abbaci\u201d hesaplama y\u00f6ntemleri ile ilgili bir kitap vard\u0131r. Bu kitap orta\u00e7a\u011f\u0131n en yetenekli matematik\u00e7isi olarak kabul edilen \u0130talyan matematik\u00e7i Leonardo Fibonacci (Leonardo da Pisa) taraf\u0131ndan yaz\u0131lm\u0131\u015ft\u0131r.
Leonardo Fibonacci 1170 y\u0131l\u0131nda \u0130talya\u2019n\u0131n Pisa \u015fehrinde do\u011fmu\u015ftur. Annesini k\u00fc\u00e7\u00fck ya\u015fta kaybeden Fibonacci\u2019nin babas\u0131 Cezayir- \u0130talya aras\u0131ndaki bir ticaret postas\u0131n\u0131 idare etmektedir ve bu nedenle s\u0131k s\u0131k seyahat etmektedirler. Leonardo\u2019nun Hint-Arap say\u0131 sistemini \u00f6\u011frenmesi de bu sayede olmu\u015ftur. Bu seyahatlerde Fibonacci, \u00f6nemli Arap matematik\u00e7iler ile \u00e7al\u0131\u015fma f\u0131rsat\u0131na sahip olmu\u015ftur. Fiboacci , d\u00f6neminde Avrupa\u2019da Romen rakamlar\u0131 kullan\u0131lmakta ve s\u0131f\u0131r say\u0131s\u0131 bilinmemekte idi. Leonardo Fibonacci Hint-Arap say\u0131lar sistemini ve s\u0131f\u0131r\u0131 Avrupa\u2019ya getirmi\u015ftir. Bu seyahatler ve \u00e7al\u0131\u015fmalar sonras\u0131nda 1202 y\u0131l\u0131nda, 32 ya\u015f\u0131nda iken \u00fcnl\u00fc kitab\u0131n\u0131 yazm\u0131\u015ft\u0131r: Il Liber Abbaci. Bu kitap olduk\u00e7a \u00f6nemli bir kitapt\u0131r, o d\u00f6nemin matemati\u011finin b\u00fcy\u00fck k\u0131sm\u0131n\u0131n kay\u0131tlar\u0131n\u0131 i\u00e7ermektedir.,
Bu kitab\u0131n i\u00e7erisinde bir problem vard\u0131r ki, i\u015fte bu problem \u201cFibonacci say\u0131lar\u0131\u201d veya \u201calt\u0131n oran\u201d olarak adland\u0131r\u0131lan, y\u00fczy\u0131llar boyu sanat ve pek \u00e7ok di\u011fer alanda ara\u015ft\u0131rmalara konu olan, \u00fczerinde hala tart\u0131\u015f\u0131lmaya devam edilen, hatta halen g\u00fcn\u00fcmde \u201c Fibonacci say\u0131lar\u0131\u201d ismi ile halen d\u00fczenli olarak yay\u0131nlanan dergiler bulunmas\u0131na sebep olan problemdir. \u0130\u015fte bu me\u015fhur problem:
Bir tav\u015fan \u00e7ifli\u011findeki bir \u00e7ift tav\u015fan, her ay yeni bir \u00e7ift tav\u015fan do\u011furmaktad\u0131r. Her yeni do\u011fan \u00e7ift, iki ay sonra bir \u00e7ift yavru yapmaya ba\u015flar ve bu b\u00f6ylece gider. Ka\u00e7 ay sonra ka\u00e7 \u00e7ift ka\u00e7 \u00e7ift tav\u015fan olur. (Tabii bu arada her yeni do\u011fan \u00e7iftin birinin di\u015fi di\u011ferinin erkek oldu\u011funu varsay\u0131yoruz. Ayr\u0131ca varsay\u0131mlar\u0131m\u0131z bununla da bitmiyor. Tahmin edece\u011finiz gibi, bu tav\u015fanlar\u0131n \u00f6l\u00fcms\u00fcz oldu\u011fu, adeta bilgisayar gibi programl\u0131 bir \u015fekilde do\u011furduklar\u0131 gibi ba\u015fka varsay\u0131mlar\u0131m\u0131zda var. Matematik problemlerinde genelde bu t\u00fcr varsay\u0131mlar \u00fczerinde durmak ihtiyac\u0131 hissedilmez).<\/p>\n\n\n\n
Bu problemin sonucunda \u015f\u00f6yle bir dizi kar\u015f\u0131m\u0131z \u00e7\u0131kar:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, \u2026. .<\/p>\n\n\n\n
Buradaki say\u0131lara dikat edecek olursak, dizide \u015f\u00f6yle bir kural oldu\u011funu g\u00f6r\u00fcr\u00fcz: son iki say\u0131n\u0131n toplam\u0131 bize bir sonraki say\u0131y\u0131 vermektedir.
1+1=2
1+2=3
2+3=5
3+5=8 … \u015feklinde.
Yine bu diziyi as\u0131l ilgin\u00e7 k\u0131lan durum dizideki say\u0131lar\u0131n oran\u0131d\u0131r. Dizideki iki ard\u0131\u015f\u0131k (arka arkaya gelen) say\u0131n\u0131n oran\u0131 ayn\u0131 say\u0131ya yak\u0131nsamaktad\u0131r:
0, 61803398\u2026\u2026 \u0130\u015fte bu oran, y\u00fczy\u0131llardan beri \u00f6zellikle mimaride ve resimde sonralar\u0131 foto\u011fraf\u00e7\u0131l\u0131kta ve m\u00fczikte \u00e7e\u015fitli d\u00f6nemlerde \u201cAlt\u0131n Oran\u201d veya \u201cM\u00fckemmel Oran\u201d olarak adland\u0131r\u0131lm\u0131\u015ft\u0131r.
Alt\u0131n oran\u0131 geometrik olarak ifade edecek olursak: a+b uzunlu\u011funda bir do\u011fru par\u00e7as\u0131 d\u00fc\u015f\u00fcnelim. a b a>b
a+b\/ a = a\/b
T\u00fcm do\u011fru par\u00e7as\u0131n\u0131n b\u00fcy\u00fck par\u00e7aya oran\u0131n\u0131n b\u00fcy\u00fck par\u00e7an\u0131n k\u00fc\u00e7\u00fck par\u00e7aya oran\u0131na e\u015fitli\u011fi bize alt\u0131n oran\u0131 vermektedir.
Alt\u0131n oran yukar\u0131da da bahsetti\u011fimiz gibi sanat\u0131n pek \u00e7ok dal\u0131nda \u00e7e\u015fitli \u015fekillerde kullan\u0131lm\u0131\u015ft\u0131r. Sanat alan\u0131nda alt\u0131n oran\u0131n kullan\u0131m\u0131, y\u00fczy\u0131llard\u0131r yap\u0131lan \u00e7e\u015fitli ara\u015ft\u0131rmalar ile belli prensiplere oturtulmu\u015ftur. \u00d6rne\u011fin resimde, mimaride alt\u0131n oran\u0131n belirli kullan\u0131m y\u00f6ntemleri vard\u0131r. Ancak m\u00fczikte alt\u0131n oran\u0131n kullan\u0131m\u0131n\u0131n bu kadar eski bir tarihi yoktur. Hernekadar bu konu ile yap\u0131lm\u0131\u015f olan \u00e7e\u015fitli tart\u0131\u015fmalar olsa da bu tart\u0131\u015fmalar ancak belirli bir \u00e7er\u00e7evede kalabilmi\u015ftir. M\u00fczikte alt\u0131n oran\u0131n bilin\u00e7li olarak kullan\u0131m\u0131 ancak 20. y\u00fczy\u0131l ve sonras\u0131nda olmu\u015ftur.
M\u00fczikte baz\u0131 yirminci y\u00fczy\u0131l bestecileri Fiboanacci serisini bestelerinde uygulam\u0131\u015flard\u0131r. Luigi Nono (1924-90) I1 Canto sospeso (1955-56) adl\u0131 eserinde ve Karlheinz Stockhausen (1928-2007) Klavierst\u00fcck IX (1961) adl\u0131 eserinde Fibonacci serisini ses uzunluklar\u0131nda uygulayarak kullanm\u0131\u015flard\u0131r. Daha sonraki \u00e7al\u0131\u015fmalarda Stockhausen bu prensibi \u00f6l\u00e7\u00fc birimlerine uygulam\u0131\u015ft\u0131r. \u00d6rne\u011fin, 13 \u00f6l\u00e7\u00fcl\u00fck bir pasajda (ki buradaki 13 bir Fibonacci say\u0131s\u0131d\u0131r) a\u015fa\u011f\u0131daki \u00f6rnek1 de g\u00f6sterildi\u011fi gibi , her \u00f6l\u00e7\u00fcde yeni bir \u00f6l\u00e7\u00fc birimi kullan\u0131lm\u0131\u015ft\u0131r ve bu birimlerin hepsi birer Fibonnaci say\u0131s\u0131d\u0131r:
\u00d6rnek. 1 Stockhausen Klavierst\u00fcck IX teki \u00f6l\u00e7\u00fc birimleri.
13 2 21 8 1 3 8 1 5 13 2 5 3 8 8 8 8 8 8 8 8 8 8 8 8 8
Ayr\u0131ca, baz\u0131 m\u00fczik analizcileri, Fibonacci serilerini bu serilere hi\u00e7bir yak\u0131nl\u0131\u011f\u0131 veya yatk\u0131nl\u0131\u011f\u0131 olmayan bestecilerin eserlerinde g\u00f6zlemlemek istediler. En \u00e7ok merak edilen, form ve m\u00fczik tekni\u011finde ak\u0131c\u0131 ve yar\u0131 do\u011fa\u00e7lama bir stili olan Claude Debussy (1862-1918) idi. Debussy\u2019nin pek \u00e7ok eserinde, form a\u00e7\u0131s\u0131ndan \u00f6nemli noktalar, dinamik veya t\u0131n\u0131 farkl\u0131l\u0131klar\u0131 v.s. gibi, bir c\u00fcmle yap\u0131s\u0131 alt\u0131nda Fibonacci serisindeki oranlar ile uyumlu bir \u015fekilde ay\u0131r\u0131lm\u0131\u015ft\u0131r. Bu konu ile ilgili dikkat \u00e7eken bir \u00f6rnek, bestecinin en \u00fcnl\u00fc orkestral yap\u0131tlar\u0131ndan birisi olan ; \u2018Dialogue du vent et la mer\u2019in La Mer (1903-05) \u2018in 55 \u00f6l\u00e7\u00fcl\u00fck giri\u015f k\u0131sm\u0131nda g\u00f6r\u00fclmektedir. Bu 55 \u00f6l\u00e7\u00fc, 21,8,8,5 ve 13 \u00f6l\u00e7\u00fc olarak kendi i\u00e7erisinde b\u00f6l\u00fcnm\u00fc\u015ft\u00fcr ki bu say\u0131lar Fibonacci dizisine aittir. Bu giri\u015fin en \u00f6nemli k\u0131sm\u0131 34. \u00f6l\u00e7\u00fcde olur ve burada trombonlar\u0131n \u00f6zel t\u0131n\u0131lar\u0131 g\u00f6ze \u00e7arpar.
Fiboanacci serisi ile ili\u015fkili olan bir ba\u015fka besteci B\u00e9la Bart\u00f3k \u2018tur (1881-1945). Debussy gibi, Bartok\u2019ta toplum i\u00e7inde hi\u00e7bir zaman m\u00fczi\u011finde Fiboanacci say\u0131lar\u0131n\u0131 kulland\u0131\u011f\u0131ndan bahsetmemi\u015ftir. Ancak besteci ile ilgili \u00e7e\u015fitli anektotlarda \u00e7am kozala\u011f\u0131, ay\u00e7i\u00e7e\u011fi (ay\u00e7i\u00e7e\u011fi Bartok\u2019un en sevdi\u011fi \u00e7i\u00e7ektir) gibi \u201cdo\u011fal\u201d formlardan ho\u015fland\u0131\u011f\u0131ndan bahsedilmektedir. T\u00fcm bu \u00f6rnekler de Fibonacci serilerinin do\u011fadaki \u00f6rnekleridir. Ern\u00f6 Lendvai, Bartok ile ilgili bir seminerinde bestecinin \u00e7e\u015fitli eserlerindeki Fibonacci serilerinden ve \u00f6zellikle, Music for Strings, Percussion and Celeste (1936) adl\u0131 eserinden bahseder. A\u015fa\u011f\u0131da bestecinin bu eserin birinci b\u00f6l\u00fcm\u00fc ile ilgili olarak Lendvai\u2019nin k\u0131sa bir \u00f6zet halinde form analizi sunulmu\u015ftur:
Ex. 2 Bart\u00f3k\u2019un Music for Strings, Percussion and Celeste eserinin formu (numaralar \u00f6l\u00e7\u00fcleri ifade etmektedir):
Sergileme sonu S\u00fcrdinsiz Zirve S\u00fcrdinli Yap\u0131 de\u011fi\u015fikli\u011fi
F\u00fcg sergilemesi F\u00fcg geli\u015fimi Kurulum \u00c7evirim Koda 21 12 22 13 13 8 33 22 13 21 55 34 89
Yukar\u0131daki \u00f6rne\u011fe bakt\u0131\u011f\u0131m\u0131zda, par\u00e7adaki b\u00f6l\u00fcmlerin, Fibonacci serisindeki say\u0131lar\u0131n tam olarak kar\u015f\u0131l\u0131\u011f\u0131 olmasa da bu say\u0131lara \u00e7ok yak\u0131n de\u011ferler oldu\u011fu g\u00f6ze \u00e7arpmaktad\u0131r. \u00d6rne\u011fin f\u00fcg geli\u015fimi, sadece 12 \u00f6l\u00e7\u00fc s\u00fcrm\u00fc\u015ft\u00fcr (13 de\u011fil), zirve noktas\u0131 beklenece\u011fi gibi 55. \u00f6l\u00e7\u00fcde de\u011fil 56. \u00f6l\u00e7\u00fcdedir. Analizindeki \u00e7ok \u00f6nemli bir ba\u015fka durum da Lendvai, 88. \u00f6l\u00e7\u00fcden sonraki beklenmedik duraklaman\u0131n fazlandan bir \u00f6l\u00e7\u00fc anlam\u0131na geldi\u011fini yani bunun da bir Fibonacci say\u0131s\u0131 olan 89. say\u0131s\u0131 oldu\u011funu d\u00fc\u015f\u00fcnmektedir.
\u00d6l\u00e7\u00fc numaralar\u0131n\u0131n hepsi tam olarak Fibonacci say\u0131lar\u0131 ile \u00f6rt\u00fc\u015fmese de, oranlar ve Lendvai\u2019nin tespitleri olduk\u00e7a ikna edici niteliktedir. Lendvai\u2019nin bu \u00e7al\u0131\u015fmas\u0131ndaki temel g\u00f6r\u00fc\u015f\u00fc \u00f6zellikle Fibonacci serisinin Bartok\u2019un m\u00fczi\u011finde : matematikesl bir yakla\u015f\u0131mdan \u00e7ok m\u00fczik ve do\u011fa aras\u0131ndaki temel ili\u015fki olarak anla\u015f\u0131lmas\u0131 gerekti\u011fidir.
G\u00fcn\u00fcm\u00fczde m\u00fczi\u011fin tarihsel, k\u00fclt\u00fcrel fakt\u00f6rler ile \u015fekillenmi\u015f ve sosyal olgular ile de tamamlanan k\u00fclt\u00fcrel bir \u00fcr\u00fcn olarak g\u00f6r\u00fcld\u00fc\u011f\u00fc bir ger\u00e7ektir. M\u00fczik ve do\u011fa aras\u0131ndaki ili\u015fki temel bir ili\u015fkidir. Sanat\u0131n temel ama\u00e7lar\u0131ndan birisinin do\u011fay\u0131 taklit etme oldu\u011fu g\u00f6r\u00fc\u015f\u00fc \u2013 sanat\u0131n taklit teorisi- Aristo\u2019ya kadar uzanan bir tart\u0131\u015fmad\u0131r ve klasik bat\u0131 m\u00fczi\u011finde de 18.yy sonlar\u0131na kadar yayg\u0131nl\u0131\u011f\u0131n\u0131 korumu\u015ftur. Bu teoriyi ba\u011flam\u0131nda ortaya at\u0131lan g\u00f6r\u00fc\u015f, do\u011fa ve sanat aras\u0131ndaki ili\u015fkinin varl\u0131\u011f\u0131n\u0131n m\u00fczik arac\u0131l\u0131\u011f\u0131 ile evrensel ger\u00e7ekleri anlamam\u0131za m\u00fcmk\u00fcn k\u0131l\u0131nmas\u0131d\u0131r. Bu teori ve bu teorinin \u00e7e\u015fitli varyasyonlar\u0131 bat\u0131 klasik m\u00fczi\u011finin geli\u015fime etki etmi\u015ftir. \u201cDo\u011fal\u201d oldu\u011funa inan\u0131lan m\u00fczik kaliteli ve g\u00fczel, bunun tersi olarak, do\u011faya ayk\u0131r\u0131 olan m\u00fczik ise k\u00f6t\u00fc m\u00fczik olarak alg\u0131lanm\u0131\u015ft\u0131r.
18. y\u00fczy\u0131l\u0131n sonlar\u0131nda bu fikir de\u011fi\u015fikli\u011fe u\u011fram\u0131\u015f ve daha \u00e7ok tutkunun taklidi, d\u0131\u015f d\u00fcnyadan \u00e7ok insan\u0131n \u201cdo\u011fal\u201d olan i\u00e7 d\u00fcnyas\u0131, insan\u0131n do\u011fas\u0131 \u00f6nem kazanm\u0131\u015ft\u0131r. Bunula birlikte, taklit teorisi her ne kadar \u00f6nemini kaybetse de, Romantizm geli\u015ftik\u00e7e bu fikir yeni bir \u015fekle b\u00fcr\u00fcnd\u00fc. Romantik d\u00f6nemin anlay\u0131\u015f\u0131na g\u00f6re, iyi bir romantic d\u00f6nem bestecisi sadece insan\u0131n i\u00e7 d\u00fcnyas\u0131n\u0131 anlatmakla kalmamal\u0131 ayn\u0131 zamanda do\u011fadaki insan ruhunun do\u011fa \u00fcst\u00fc g\u00fc\u00e7lerini de aktarmal\u0131 idi. Bu sanat g\u00f6r\u00fc\u015f\u00fc \u2013 do\u011fa ve insan ruhu aras\u0131ndaki \u201cetkile\u015fim\u201d in do\u011fa \u00fcst\u00fc durumu \u2013 19. Y\u00fczy\u0131l sanat olu\u015fumunu belirgin olarak belirlemi\u015ftir.
Do\u011faya olan bu ilgi, belirgin olsa da olmasa da, m\u00fczik ve sanat alan\u0131nda Fibonacci serisinin organik bir y\u00f6n\u00fcn\u00fc ortaya koymaktad\u0131r. Yakla\u015f\u0131k son 200 senedir, m\u00fczik alan\u0131nda yap\u0131lan besteler, bestecilerin do\u011fa ve ya\u015fam\u0131n kendisi ile ilgili duygular\u0131ndan ortaya \u00e7\u0131km\u0131\u015ft\u0131r. 19. Y\u00fczy\u0131l sonlar\u0131 ve sonras\u0131nda, m\u00fczikteki bask\u0131n olan \u201ctonalite\u201d kavram\u0131 ciddi olarak zay\u0131flam\u0131\u015ft\u0131r.
Fibonacci serisinin, 20. Y\u00fczy\u0131ldan itibaren besteciler taraf\u0131ndan bilin\u00e7li olarak kullan\u0131lmas\u0131 ile birlikte, dah \u00f6nce yaz\u0131lan eserler de belli prensiplere dayand\u0131r\u0131labildi. Bunun d\u0131\u015f\u0131nda, Fibonacci say\u0131lar\u0131n\u0131n besteciler taraf\u0131ndan bilin\u00e7li olarak kullan\u0131m\u0131 ve bu konu ile ilgili analizcilerin ortaya att\u0131klar\u0131, bir \u00e7e\u015fit, b\u00fct\u00fcnle\u015ftirici bir m\u00fczik gelene\u011finin olmamas\u0131ndan kaynaklanan bir savunma olarak ta g\u00f6r\u00fclebilir. Ger\u00e7ekten de, Arnold Schoenberg (1874-1951) ve Anton Webern (1883-1945) gibi 20. Y\u00fczy\u0131l\u0131n once gelen bestecileri kompozisyonlar\u0131ndaki yeni yakla\u015f\u0131mlar\u0131ndan bahsederken, yapt\u0131klar\u0131 m\u00fczik ile ilgili olarak do\u011fa d\u0131\u015f\u0131 olman\u0131n \u00f6nemini \u00fczerine basarak vurgulam\u0131\u015flar\u0131d\u0131r. Sonu\u00e7 olarak, dinleyici i\u00e7in, bir eserin, Fibonacci serilerine uygunlu\u011fundan, veya eserin i\u00e7erisindeki gizli birtak\u0131m kurallar\u0131n ke\u015ffinden \u00e7ok, ku\u015fkusuz, eseri duydu\u011fundaki sonu\u00e7, eserin ba\u015far\u0131l\u0131 olup olmad\u0131\u011f\u0131 \u00f6nem ta\u015f\u0131maktad\u0131r.<\/p>\n\n\n\n
* [Marmara \u00dcniversitesi G\u00fczel Sanatlar Fak\u00fcltesi M\u00fczik B\u00f6l\u00fcm\u00fc \u00d6\u011fretim \u00fcyeleri.]
Lines, Malcolm \u2018Bir say\u0131 tut\u2019 T\u00fcbitak yay\u0131nlar\u0131,7. Bas\u0131m. s. 13.
Bailey, Kathryn \u2018\u2018Work in Progress\u2019: Analysing Nono\u2019s \u2018Il Canto Sospeso\u2019\u2019 Music Analysis, Vol. 11, No. 2\/3, Alexander Goehr 60th-Birthday Issue (Temmuz – Ekim., 1992) s. 290.
Kramer, Jonathan \u2018The Fibonacci Series in Twentieth-Century Music\u2019 Journal of Music Theory, Vol. 17, No. 1. (Spring, 1973) s.121.
Kramer, Jonathan \u2018The Fibonacci Series in Twentieth-Century Music\u2019 Journal of Music Theory, Vol. 17, No. 1. (Spring, 1973) s. 121.
Cited in the entry on \u2018Debussy\u2019 on Wikipedia: Howat, Roy Debussy in Proportion: A musical analysis (Cambridge: Cambridge University Press, 1983).
Akt: Lendvai, Ern\u00f6 Bela Bart\u00f3k: an Analysis of His Style (Londra: Kahn and Averill, 1971), s. 29.
Lendvai, Ern\u00f6 Bela Bart\u00f3k: an Analysis of His Style (Londra: Kahn and Averill, 1971).
Kramer, Jonathan \u2018The Fibonacci Series in Twentieth-Century Music\u2019 Journal of Music Theory, Vol. 17, No. 1. (Spring, 1973) s.122.
Howat, Roy \u2018Bart\u00f3k, Lendvai and the Principles of Proportional Analysis\u2019 Music Analysis, Vol. 2, No. 1 (Mart, 1983) s. 72.
Baker, Nancy Kovaleff \u2018Expression (I, 1) The New Grove Dictionary of Music and Musicians 2. bask\u0131, editor: Stanley Sadie (Londra: MacMillan, 2001).
Goehr, Lydia The Imaginary Museum of Musical Works: An Essay in the Philosophy of Music 1994, s. 163.
Baker, Nancy Kovaleff \u2018Expression (I, 1)\u2019 in The New Grove Dictionary of Music and Musicians 2.. bask\u0131, editor: Stanley Sadie (Londra: MacMillan, 2001).
Goehr, Lydia The Imaginary Museum of Musical Works: An Essay in the Philosophy of Music 1994, ss. 160-61.
See: Solie, Ruth A. \u2018The Living Work: Organicism and Musical Analysis\u2019 19th-Century Music, Vol. 4, No. 2 (Autumn, 1980) s. 149.
Schoenberg , kendisinin uygulad\u0131\u011f\u0131 disonans m\u00fczi\u011fi \u201cdo\u011faya ayk\u0131r\u0131l\u0131k\u201d olarak m\u00fcdafaa etmi\u015ftir. Tarih boyunca bestecilerin giderek daha disonans armonik seriler kulland\u0131klar\u0131n\u0131, zaman i\u00e7erisinde bu disonans kabul edilen aral\u0131klar\u0131n daha kabul g\u00f6r\u00fcr oldu\u011funu belirtmi\u015f, bu nedenle de Schoenberg kendi m\u00fczi\u011finin de bu \u201cdo\u011fal\u201d s\u00fcrecin bir par\u00e7as\u0131 oldu\u011funu ifade etmi\u015ftir.<\/p>\n\n\n\n
Do\u00e7.Dr. Ece KAR\u015eAL
Yrd.Do\u00e7. Dr. David WALTERS<\/p>\n\n\n\n
“Orkestra Dergisi, A\u011fustos – Ekim 2013, 431. Say\u0131. s: 15-24”<\/p>\n","protected":false},"excerpt":{"rendered":"
Alt\u0131n oran, do\u011fadaki \u00e7e\u015fitli varl\u0131klar\u0131n yap\u0131s\u0131nda bulunan, m\u00fckemmel uyumu temsil eden orand\u0131r. Pek \u00e7ok yerde…<\/p>\n