Mathematical analysis of the relationship between jazz and fractal geometry over the piano

Fractal Geometry defines the algorithm of the natural phenomenon that exhibits a repeating pattern which displays at every scale. Fractal algorithm helps analyzing the complex and chaotic forms. The algorithmic ratios of fractal geometry that are defined in Architecture is related to harmonic ratios in aesthetics. For that reason it is also used to analyze harmonic ratios of aesthetic design. The connection between Architecture and Music is remarkably weak. Analyzing the harmonic ratios in music, implicitly establishes the foundations of the relationship between Architecture. The complex forms of nature stands for the complex form of music, jazz. Jazz is considered as the most sophisticated form of music and it's way beyond the classical music theory. Analyzing the form of jazz gives some harmonic ratios. Using fractal geometry algorithm the jazz harmonic ratios are converted to visual forms to identify the relationship between architecture and music. First chapter of the study includes the aim of the thesis and its scope. The aim of the thesis considers finding a connection between the fractal geometry and modal jazz chords. It may not be seen as directly connected to architecture at the first sight but it is related to one of its disciplines, geometry. Geometry is one of the most important issues in architectural design. Whether a building is aesthetic or not, it is directly related to some geometric ratios that is hidden in the form. The same ratios also exist in music which makes it understandable, loveable therefore aesthetic for the listeners. Both auditorial and visual disciplines, music and geometry have some harmonic ratios hidden that creates the most aesthetic form out of them. First of all, to create a connection between to different perceptions one instrument is chosen, piano. The reason of the choice is that piano has the whole harmonic scale that can be used to create chords. The frequencies in the third and fourth scale of the piano is used in modelling. Secondly, the Koch Curve fractal algorithm is used as the logic in modelling jazz chords. Only tonality is considered while modeling. The reason for that is the rhythm creates more complicated musical forms that needs to be analyzed with advanced mathmatics. Over the tonality, the modal jazz chords are converted to tree fractal models. Twelwe tree models are shown to 119 people with a sample survey. The purpose of the survey is finding the emotional impression of the models without listening to their chords. After that, the emotional meanings of the the jazz chords and the results of the survey are compared to find a connection between auditorial and visual perceptrion. Lastly, modal jazz chords and their fractal modals are matched up over their emotional impressions. It shows that both visual and auditorial perception has a hidden connection. The aim of the thesis is to find a connection exists and to use that connection to create architectural design over rhythmic and tonal, orchestral, more complicated jazz music. Second chapter includes the theoretical and practical fields. The relationship between architecture & geometry and architecture & music is explained. Understanding the basic foundation of the concept of the study affects the creativity in a positive way. Aesthetic in architecture is directly related to some harmonic ratios in geometry. The most common form of aesthetic is named as the golden ratio. Golden ratio is a hidden ratio that exists in every part of the nature. It is also defined by the Fibonacci series. Finding the golden ratio in architectural design gives an aesthetic value. In music the making of the instruments, toning, playing and composing have some ratios that makes it more aesthetic. In both disciplines geometry and music, the first intension is to find aesthetics whether it is visual of auditorial. In chapter three, the basic music studies are discussed. The definition and the history of music are explained. Basics of music are given to explain the jazz music. Basic classical music theory is the fundemantal of creating the modal jazz chords. In this study only C scale is used to create jazz modes. To create variety of modal chords, C major and C melodic minor scales are used. Over the major and melodic minor scales, it has ben generated twelve different modal jazz chords. The modal jazz chords which are the inputs of the fractal modeling are defined in this chapter. In jazz theory there are seven different modes ine every tone. The first mode is Ionian, second mode is Dorian, third mode is Phrygian, fourth mode is Lydian, fifth mode is Mixolydian, sixth mode is Aeolian and seventh mode is Locrian. Twelve different jazz chords are generated over seven jazz modes. To categorize the modal jazz chords, the basics of piano are explained and the frequencies of the notes are given in the third and fourth octave in piano. As the last part of chapther three, the music perception is discussed. The importance of the auditorial and the visual perception is that they are the key part that connects the different discplines by their emotional impressions. Chapter four includes the basic fractal geometry. Feedback mechanism, algorithm of the basic fractal models is explained. The logic of the fractal geometry relies on the feedback mechanism. Feedback mechanism uses the output as an input in every step of the process. That is the most important part of fractal that generates itself in every scale. Video feedback is the visual form of the feedback algorithm and help us to see the visual results while creating fractals. Chaos game is explained to define the complex form of a simple rule. After explaining the logic of fractal generation the basic fractal models are explained, Cantor Set, Sierpinski Triangle, Koch Curve, Peano Curve, Hilbert Space. Nature form of fractal geometry is the inspiration of the creation for the fractal models. The complexity of the nature could be explained by fractal geometry and its definitions. In chapter five, the practical method is defined to transform the modal jazz chords (that is created in chapter three) to fractal models. The modal jazz chords are defined note by note and their frequencies and their ratios are calculated. Ten four note chord and two five note chord is created. The fractal algorithm is generated by the frequency ratios of the modal jazz chords. For these modal jazz chords, the frequency ratios are calculated. These ratios is used to create a fractal tree. Two octave frequency value is appointed as the stem of the tree. And the notes in the chord are appointed as the branches of a tree. The algorithm is generated for five steps in each fractal model for chords. The steps and the main model is generated. The fractal models are created using Mathlab. Chapter six includes the survey results, the characteristic and similarities of the generated fractal models and the spesific comparison of jazz the harmonies. The the survey the models are shown to random people mixed as gender and age. The emotional impressions of the models are asked. The results are compared to the studies that had made about the emotional meanings of the jazz harmony. In addition to this, the fractals are compared inside. The same note chords and the same kind notes ara specifically compared. The characteristics of major scale and melodic minor scale are analyzed. Chapter seven includes the relationship between all the harmonic fractal models is analyzed. The results obtained points out that a certain relationship between music and architecture exists. The study approach could be improved by analyzing rhythmic patterns, compositions of jazz. The thesis includes a multidisciplinary study to create a connection between jazz and geometry. Music, Geometry, Mathmatics and Psychology are the main subjects of the study. With classical music theory jazz model chords are created, with mathmatics the jazz chords are analyzed, with geometry the modal jazz chords are converted to tree fractal models and with psychology the relation between jazz and fractal geometry is analyzed. The relationship between jazz and geometry shows that there is a certain emotional structure underneath the both of the disciplines. This relationship could be evolved to the relationship between more complex music, jazz compositions and more complex geometry, nature. Nature is the biggest inspiration of architecture. Architectural design is affected from natural forms. Understanding the harmony in both visual and auditorial forms in nature gives us a chance to reverse the connection and create it from music, especially jazz which is the exact equivalent of complexity in music. Using different models than Koch Curve, also using complex numbers and Julia Set, Mandelbrot Fractal in modelling would create architectural forms out of music. The prevision of this thesis is, improvement of the jazz chords and fractal models that are used, could provide a design system which relies on music. Jazz can create hospitals, concert halls, highways etc. The relationship between music and geometry could create a new field of multidisciplinary design system.