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\usepackage[polish,ngerman,english]{babel}



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\begin{document}

\title{Bach'\selectlanguage{polish}ı\selectlanguage{english}n M\selectlanguage{ngerman}üzi\selectlanguage{polish}ğ\selectlanguage{english}i ve Matematik\selectlanguage{english}}



\author[1]{Melis Saylaç }%
\affil[1]{Affiliation not available}%


\vspace{-1em}



  
  \date{May 08, 2024}


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\sloppy


\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.91\columnwidth]{figures/image1/image1}
\caption{\selectlanguage{polish}{\textbf{{Bach'ın Matematikle Çevrili Büyülü Dünyası}}%
}}
\selectlanguage{english}\end{center}
\end{figure}

\par\null

\emph{~~~~~~~Johann Sebastian Bach}, 21 Mart 1685 do\selectlanguage{polish}ğ\selectlanguage{english}umlu Alman barok
m\selectlanguage{ngerman}üzik bestecisi ve orgcudur. Bach 19. yüzy\selectlanguage{polish}ı\selectlanguage{english}lda m\selectlanguage{ngerman}üzi\selectlanguage{polish}ğ\selectlanguage{english}inin tekrar
\selectlanguage{ngerman}çal\selectlanguage{polish}ı\selectlanguage{english}nmaya ba\selectlanguage{polish}ş\selectlanguage{english}lamas\selectlanguage{polish}ı \selectlanguage{english}ve ilginin tekrar canlanmas\selectlanguage{polish}ı\selectlanguage{english}na kadar kendi d\selectlanguage{ngerman}öneminde
büyük bir besteci olarak bilinmemi\selectlanguage{polish}ş \selectlanguage{english}ancak bug\selectlanguage{ngerman}ün en büyük bestecilerden
biri olarak kabul edilir.

Bilginler ve müzisyenler y\selectlanguage{polish}ı\selectlanguage{english}llarca matemati\selectlanguage{polish}ğ\selectlanguage{english}i kullanarak m\selectlanguage{ngerman}üzikleri analiz
ettiler, hatta yaratt\selectlanguage{polish}ı\selectlanguage{english}lar. Bir\selectlanguage{ngerman}çok iyi besteci müzi\selectlanguage{polish}ğ\selectlanguage{english}in alt\selectlanguage{polish}ı\selectlanguage{english}nda yatan
matematik ilkelerinin derin anlam\selectlanguage{polish}ı\selectlanguage{english}n\selectlanguage{polish}ı \selectlanguage{english}kavrayabildi. Bunlardan bir tanesi
ise g\selectlanguage{ngerman}örkemli Johann Sebastian Bach't\selectlanguage{polish}ı\selectlanguage{english}r. Bach'\selectlanguage{polish}ı\selectlanguage{english}n besteleri dahili\selectlanguage{polish}ğ\selectlanguage{english}inin
bir par\selectlanguage{ngerman}ças\selectlanguage{polish}ı \selectlanguage{english}olsa da asl\selectlanguage{polish}ı\selectlanguage{english}nda onlar\selectlanguage{polish}ı\selectlanguage{english}n temelinde de basit matematik vard\selectlanguage{polish}ı\selectlanguage{english}r.

Matematik\selectlanguage{ngerman}çiler çal\selectlanguage{polish}ış\selectlanguage{english}malar\selectlanguage{polish}ı\selectlanguage{english}n\selectlanguage{polish}ı\selectlanguage{english}n estetiksel y\selectlanguage{ngerman}önünü vurgulamaktan
çekinmezler. Hem matematik hem müzik ikisine de desenler hakimdir.
Matematikçinin kal\selectlanguage{polish}ı\selectlanguage{english}plar\selectlanguage{polish}ı, \selectlanguage{english}bestecinin kal\selectlanguage{polish}ı\selectlanguage{english}plar\selectlanguage{polish}ı \selectlanguage{english}gibi g\selectlanguage{ngerman}üzel olmal\selectlanguage{polish}ı;
\selectlanguage{english}fikirler uyumlu bir \selectlanguage{polish}ş\selectlanguage{english}ekilde bir araya gelmelidir.

\par\null

\begin{quote}
G.H. Hardy'nin dedi\selectlanguage{polish}ğ\selectlanguage{english}i gibi, ''G\selectlanguage{ngerman}üzellik ilk testtir: dünyada çirkin
matematik için kal\selectlanguage{polish}ı\selectlanguage{english}c\selectlanguage{polish}ı \selectlanguage{english}bir yer yoktur.''
\end{quote}

\par\null

\subsection*{\texorpdfstring{{Simetri}}{Simetri}}

{\label{673277}}\par\null\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/tumblr-nfvqr1xirF1txe8seo1-500/tumblr-nfvqr1xirF1txe8seo1-500}

\end{center}
\end{figure}

~~~~Simetri asl\selectlanguage{polish}ı\selectlanguage{english}nda m\selectlanguage{ngerman}üzi\selectlanguage{polish}ğ\selectlanguage{english}in temel y\selectlanguage{ngerman}önüdür bu nedenle ,simetri için
matematiksel çerçeve olan, grup teorisinin müzikal analize uygulanmas\selectlanguage{polish}ı
ş\selectlanguage{english}a\selectlanguage{polish}şı\selectlanguage{english}rt\selectlanguage{polish}ı\selectlanguage{english}c\selectlanguage{polish}ı \selectlanguage{english}de\selectlanguage{polish}ğ\selectlanguage{english}ildir. Transpozisyon, inversiyon ve retrogression m\selectlanguage{ngerman}üzikal
teknikleri grup i\selectlanguage{polish}ş\selectlanguage{english}lemleri olarak yorumlanabilir ve bu bak\selectlanguage{polish}ış \selectlanguage{english}a\selectlanguage{ngerman}ç\selectlanguage{polish}ı\selectlanguage{english}s\selectlanguage{polish}ı \selectlanguage{english}eski
kompozisyonlar \selectlanguage{ngerman}üzerine yeni bir \selectlanguage{polish}ışı\selectlanguage{english}k tutmu\selectlanguage{polish}ş\selectlanguage{english}tur. Bir\selectlanguage{ngerman}çok besteci geometrik
dönü\selectlanguage{polish}ş\selectlanguage{ngerman}ümleri bir kompozisyon cihaz\selectlanguage{polish}ı \selectlanguage{english}olarak kullanm\selectlanguage{polish}ış\selectlanguage{english}t\selectlanguage{polish}ı\selectlanguage{english}r. D\selectlanguage{ngerman}önü\selectlanguage{polish}ş\selectlanguage{ngerman}ümler ve
simetriler, bestecilerin kreasyonlar\selectlanguage{polish}ı\selectlanguage{english}n\selectlanguage{polish}ı \selectlanguage{english}zenginle\selectlanguage{polish}ş\selectlanguage{english}tirmeleri i\selectlanguage{ngerman}çin bir dizi
araç olarak dü\selectlanguage{polish}ş\selectlanguage{ngerman}ünülebilir. Böylece, seçilen bir tema, varyasyonlar
üretmek için zaman içinde aktar\selectlanguage{polish}ı\selectlanguage{english}labilir, tersine \selectlanguage{ngerman}çevrilebilir veya
yans\selectlanguage{polish}ı\selectlanguage{english}t\selectlanguage{polish}ı\selectlanguage{english}labilir.(Lynch, 2017)

\par\null\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/pngflow-com/pngflow-com}

\end{center}
\end{figure}

Bir besteci kar\selectlanguage{polish}şı \selectlanguage{english}noktaya odakland\selectlanguage{polish}ığı\selectlanguage{english}nda, vurgu, m\selectlanguage{ngerman}üzi\selectlanguage{polish}ğ\selectlanguage{english}in dikey eksenine
(Akorlar ve uyum) de\selectlanguage{polish}ğ\selectlanguage{english}il, yatay eksene (melodi telleri) yerle\selectlanguage{polish}ş\selectlanguage{english}tirilir
b\selectlanguage{ngerman}öylece çal\selectlanguage{polish}ış\selectlanguage{english}man\selectlanguage{polish}ı\selectlanguage{english}n amac\selectlanguage{polish}ı, \selectlanguage{english}ayn\selectlanguage{polish}ı \selectlanguage{english}anda ger\selectlanguage{ngerman}çekle\selectlanguage{polish}ş\selectlanguage{english}en \selectlanguage{ngerman}çe\selectlanguage{polish}ş\selectlanguage{english}itli m\selectlanguage{ngerman}üzikal çizgiler
aras\selectlanguage{polish}ı\selectlanguage{english}ndaki etkile\selectlanguage{polish}ş\selectlanguage{english}im haline gelir.

Kanon*larla yarat\selectlanguage{polish}ı\selectlanguage{english}c\selectlanguage{polish}ı \selectlanguage{english}olman\selectlanguage{polish}ı\selectlanguage{english}n her t\selectlanguage{ngerman}ürlü yolu vard\selectlanguage{polish}ı\selectlanguage{english}r. Cevap, konunun
kesinlikle birebir bir kopyas\selectlanguage{polish}ı \selectlanguage{english}olmak yerine, asl\selectlanguage{polish}ı\selectlanguage{english}nda geriye do\selectlanguage{polish}ğ\selectlanguage{english}ru, ba\selectlanguage{polish}ş
\selectlanguage{english}a\selectlanguage{polish}ş\selectlanguage{english}a\selectlanguage{polish}ğı, \selectlanguage{ngerman}çift h\selectlanguage{polish}ı\selectlanguage{english}zda ya da yar\selectlanguage{polish}ı\selectlanguage{english}m h\selectlanguage{polish}ı\selectlanguage{english}zda ya da bunlardan herhangi birinin bir
kombinasyonu olabilir. Kurallar\selectlanguage{polish}ı \selectlanguage{english}kat\selectlanguage{polish}ı\selectlanguage{english}d\selectlanguage{polish}ı\selectlanguage{english}r, ancak birka\selectlanguage{ngerman}ç önemli seçenek
sayesinde mevcut olas\selectlanguage{polish}ı\selectlanguage{english}l\selectlanguage{polish}ı\selectlanguage{english}klar sonsuzdur.

As\selectlanguage{polish}ı\selectlanguage{english}l zorluk, bu t\selectlanguage{ngerman}ür müzi\selectlanguage{polish}ğ\selectlanguage{english}in iyi bir ses \selectlanguage{ngerman}ç\selectlanguage{polish}ı\selectlanguage{english}karmas\selectlanguage{polish}ı\selectlanguage{english}d\selectlanguage{polish}ı\selectlanguage{english}r: sadece \selectlanguage{ngerman}öngörülen
kurallar\selectlanguage{polish}ı \selectlanguage{english}takip etmek, sonucun ho\selectlanguage{polish}ş \selectlanguage{english}veya ilgin\selectlanguage{ngerman}ç olaca\selectlanguage{polish}ğı\selectlanguage{english}na dair bir
garanti de\selectlanguage{polish}ğ\selectlanguage{english}ildir, bu nedenle hem uygun malzeme bulmak hem de hangi
kurallar\selectlanguage{polish}ı\selectlanguage{english}n en iyi \selectlanguage{polish}ş\selectlanguage{english}ekilde uygulanaca\selectlanguage{polish}ğı\selectlanguage{english}n\selectlanguage{polish}ı \selectlanguage{english}belirlemek i\selectlanguage{ngerman}çin büyük bir
estetik ar\selectlanguage{polish}ı\selectlanguage{english}tma gereklidir. Bu da Bach'\selectlanguage{polish}ı\selectlanguage{english}n m\selectlanguage{ngerman}ükemmel oldu\selectlanguage{polish}ğ\selectlanguage{english}u k\selectlanguage{polish}ı\selectlanguage{english}s\selectlanguage{polish}ı\selectlanguage{english}md\selectlanguage{polish}ı\selectlanguage{english}r.

Bach (1685-1750) hi\selectlanguage{ngerman}çbir \selectlanguage{polish}ş\selectlanguage{english}ekilde bu tarz bir kompozisyonun mucidi
de\selectlanguage{polish}ğ\selectlanguage{english}ildi. Buxtehude ve Sch\selectlanguage{ngerman}ütz gibi 17.yüzy\selectlanguage{polish}ı\selectlanguage{english}l Kuzey Alman \selectlanguage{ngerman}öncüllerinden
miras ald\selectlanguage{polish}ı \selectlanguage{english}ama kanonun kabul edilen b\selectlanguage{ngerman}üyük ustas\selectlanguage{polish}ı\selectlanguage{english}yd\selectlanguage{polish}ı \selectlanguage{english}(hala \selectlanguage{ngerman}öyle) ve
çal\selectlanguage{polish}ış\selectlanguage{english}malar\selectlanguage{polish}ı \selectlanguage{english}Bat\selectlanguage{polish}ı \selectlanguage{english}m\selectlanguage{ngerman}üzi\selectlanguage{polish}ğ\selectlanguage{english}indeki zirvesini temsil ediyor. Onun \selectlanguage{ngerman}özel dehas\selectlanguage{polish}ı
\selectlanguage{english}sadece kar\selectlanguage{polish}şı \selectlanguage{english}notay\selectlanguage{polish}ı \selectlanguage{english}ustal\selectlanguage{polish}ı\selectlanguage{english}kla ele alma meselesi de\selectlanguage{polish}ğ\selectlanguage{english}il, ayn\selectlanguage{polish}ı \selectlanguage{english}zamanda
b\selectlanguage{ngerman}üyük bir aral\selectlanguage{polish}ı\selectlanguage{english}k ve duygu derinli\selectlanguage{polish}ğ\selectlanguage{english}ini ifade etme ve bu matematik-y
tekniklerini sars\selectlanguage{polish}ı\selectlanguage{english}lmaz bir disiplinle kullanarak dengeli, g\selectlanguage{ngerman}üzel orant\selectlanguage{polish}ı\selectlanguage{english}l\selectlanguage{polish}ı
\selectlanguage{english}kompozisyonlar \selectlanguage{ngerman}üretme yetene\selectlanguage{polish}ğ\selectlanguage{english}iydi.(Lindsay, 2017)

\selectlanguage{polish}\textsubscript{Kanon*, taklit karşı noktanın temel tekniğidir. Kanon'da
bir melodi kurulur(konu) ve devam ettiği gibi, aynı melodinin ikinci bir
girişi gerçekleşir(cevap) ve iterasyonla etkileşime girerek melodinin
kendisini ~``taklit'' veya ``kovaladığı'' izlenimini verir.}\selectlanguage{english}

\par\null

\selectlanguage{polish}\subsection*{\texorpdfstring{{Goldberg
Varyasyonları}}{Goldberg Varyasyonları}}
\selectlanguage{english}
{\label{929095}}

~~~~Genel bir arka plan olarak, Goldberg varyasyonlar\selectlanguage{polish}ı \selectlanguage{english}2 arya ve 30
varyasyondan olu\selectlanguage{polish}ş\selectlanguage{english}an klavsen i\selectlanguage{ngerman}çin yaz\selectlanguage{polish}ı\selectlanguage{english}lm\selectlanguage{polish}ış \selectlanguage{english}bir eserdi. Bach, Lutheran
olan son derece dindar bir adamd\selectlanguage{polish}ı \selectlanguage{english}ve bu par\selectlanguage{ngerman}çada kulland\selectlanguage{polish}ığı \selectlanguage{english}iki kavram
birlik ve \selectlanguage{ngerman}üçlü fikirlerdi. Genel gözlemler, her biri 32 çubuklu 3, 9, 21
ve 30 varyasyonlar\selectlanguage{polish}ı \selectlanguage{english}ve 95 barl\selectlanguage{polish}ı \selectlanguage{english}varyasyon 16 hari\selectlanguage{ngerman}ç olmak üzere,
varyasyonlar\selectlanguage{polish}ı\selectlanguage{english}n hepsinin 64 bara sahip oldu\selectlanguage{polish}ğ\selectlanguage{english}unu i\selectlanguage{ngerman}çerir.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.77\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}

Bu Cevapla, sayma Roma alfabesine (veya Latin alfabesine) dayan\selectlanguage{polish}ı\selectlanguage{english}r, bunun
gibi:

\begin{itemize}
\tightlist
\item
  Varyasyon 2, 129. barda ba\selectlanguage{polish}ş\selectlanguage{english}lar. 129 = 3 x 43 ve 43 = C+R+E+D+O. Credo,
  Latince H\selectlanguage{polish}ı\selectlanguage{english}ristiyan Kilisesi'nin bir inanc\selectlanguage{polish}ı\selectlanguage{english}d\selectlanguage{polish}ı\selectlanguage{english}r.
\item
  Varyasyon 3, bar 193 ile ba\selectlanguage{polish}ş\selectlanguage{english}lar. 1 birlik, 3 \selectlanguage{ngerman}üçlük, 9 üçlü\selectlanguage{polish}ğ\selectlanguage{ngerman}ün bir üç
  parçal\selectlanguage{polish}ı \selectlanguage{english}tablosu (3*3*3).
\item
  Varyasyon 6, \selectlanguage{ngerman}çubuk 353'te ba\selectlanguage{polish}ş\selectlanguage{english}lar. 6 ilk m\selectlanguage{ngerman}ükemmel say\selectlanguage{polish}ı\selectlanguage{english}d\selectlanguage{polish}ı\selectlanguage{english}r. Varyasyon 6,
  bar 416'da biten 64 bara sahiptir. 416 = 2\^{}5 x 13 (triunity
  tekrar). 41 = J+S+B+A+C+H. varyasyon 11, \selectlanguage{ngerman}önceki 416'ya kar\selectlanguage{polish}şı\selectlanguage{english}l\selectlanguage{polish}ı\selectlanguage{english}k gelen
  641 bar\selectlanguage{polish}ı\selectlanguage{english}nda ba\selectlanguage{polish}ş\selectlanguage{english}lar.
\item
  Varyasyon 15, bar 960 ile biter. 960 = 2 x 10 x 48, 48 = I + N + R +
  I. INRI, ``Yahudilerin Kral\selectlanguage{polish}ı İ\selectlanguage{english}sa '' anlam\selectlanguage{polish}ı\selectlanguage{english}na gelen Latince ifadenin
  k\selectlanguage{polish}ı\selectlanguage{english}saltmas\selectlanguage{polish}ı\selectlanguage{english}d\selectlanguage{polish}ı\selectlanguage{english}r.
\end{itemize}

~~~~\selectlanguage{polish}Ş\selectlanguage{english}imdi baz\selectlanguage{polish}ı\selectlanguage{english}lar\selectlanguage{polish}ı\selectlanguage{english}n\selectlanguage{polish}ı\selectlanguage{english}z t\selectlanguage{ngerman}üm bunlar\selectlanguage{polish}ı\selectlanguage{english}n sadece bizim a\selectlanguage{ngerman}ç\selectlanguage{polish}ı\selectlanguage{english}m\selectlanguage{polish}ı\selectlanguage{english}zdan bir analiz
olabilece\selectlanguage{polish}ğ\selectlanguage{english}ini veya bunlarla Bach'\selectlanguage{polish}ı\selectlanguage{english}n ger\selectlanguage{ngerman}çekten neyi kastetti\selectlanguage{polish}ğ\selectlanguage{english}ini
d\selectlanguage{ngerman}ü\selectlanguage{polish}ş\selectlanguage{ngerman}ünüyor olabilir. Bu parçadaki matematiksel olaylar\selectlanguage{polish}ı\selectlanguage{english}n tam olarak ne
kadar\selectlanguage{polish}ı\selectlanguage{english}n\selectlanguage{polish}ı\selectlanguage{english}n Bach taraf\selectlanguage{polish}ı\selectlanguage{english}ndan tasarland\selectlanguage{polish}ığı\selectlanguage{english}ndan ve bizimle ayn\selectlanguage{polish}ı ş\selectlanguage{english}ekilde
hesapland\selectlanguage{polish}ığı\selectlanguage{english}ndan emin olamay\selectlanguage{polish}ı\selectlanguage{english}z. Bununla birlikte, bu t\selectlanguage{ngerman}ür olaylar\selectlanguage{polish}ı\selectlanguage{english}n
say\selectlanguage{polish}ı\selectlanguage{english}s\selectlanguage{polish}ı, \selectlanguage{english}par\selectlanguage{ngerman}çaya hiçbir matematiksel dü\selectlanguage{polish}ş\selectlanguage{ngerman}ünce biçiminin konulmamas\selectlanguage{polish}ı\selectlanguage{english}n\selectlanguage{polish}ı \selectlanguage{english}son
derece olas\selectlanguage{polish}ı \selectlanguage{english}k\selectlanguage{polish}ı\selectlanguage{english}lmaktad\selectlanguage{polish}ı\selectlanguage{english}r.(Tay, 2016)

Bununla alakal\selectlanguage{polish}ı \selectlanguage{english}daha \selectlanguage{ngerman}çok \selectlanguage{polish}ş\selectlanguage{english}ey \selectlanguage{ngerman}ö\selectlanguage{polish}ğ\selectlanguage{english}renmek isterseniz \selectlanguage{polish}ş\selectlanguage{english}u sitelere de bir
bakman\selectlanguage{polish}ı\selectlanguage{english}z\selectlanguage{polish}ı \selectlanguage{ngerman}öneririm :

Music, Mathematics and
Bach:~\url{https://www.ias.ac.in/article/fulltext/reso/004/05/0061-0070}

Bach, Music+Math: Symmetry ~\url{https://youtu.be/Z8dKLbmGUbI}

Ayr\selectlanguage{polish}ı\selectlanguage{english}ca Quaternion Symmetry ile ilgili \selectlanguage{ngerman}çok güzel bir video da mevcut,
belki ilginizi çeker:
\url{https://www.youtube.com/watch?v=\_619nwr409o\&feature=share}

Kaynakça:~\url{https://www.irishtimes.com/news/science/the-mathematician-s-patterns-like-those-of-the-composer-must-be-beautiful-1.3096020}

\url{https://www.quora.com/What-is-the-relation-between-Bachs-music-and-math}

\selectlanguage{english}
\FloatBarrier
\end{document}

