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Consonance in Music and Mathematics: Application to Temperaments and Orchestration

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Consonance in Music and Mathematics: Application to Temperaments and Orchestration Constança Martins de Castro Simas Thesis to obtain the Master of Science Degree in Mathematics and Applications Supervisor: Henrique Manuel dos Santos Silveira de Oliveira Examination Committee Chairperson: António Manuel Pacheco Pires Supervisor: Henrique Manuel dos Santos Silveira de Oliveira Member of the Committee: João Maria da Cruz Teixeira Pinto December, 2014 ii Acknowledgments I’m very glad that the process of writing this thesis led me to learn and share interesting opinions with people from different backgrounds. I want to thank my advisor for accepting and encouraging my eccentric ideas. Gladly, the brain- storming reunions contributed to this final result in the best way possible. My colleagues Ricardo Vieira Lisboa and João Carvalho were also a great help throughout this process. They gave their input while solving some mathematical problems, and sometimes just kept me company while i complained about what were the work’s issues at the moment. Carrying on to a little different background i want to thank all the musicians that helped me. Some of them by agreeing on spending their time and talent recording material and others just because of the interest they showed in helping me understand how their musical instruments worked. Namely Hen- rique Costa, Thierry Redondo, João Ferreira, Catarina Dinis, Rafaela Oliveira, Diana Santos, Natacha Fernandes, Joana Mendes, Ana Conceição, Sérgio Sousa, Aldara Medeiros e Sílvia Rocha. I want to thank specially Tiago Ramos and Rita Blanco who were always there to support me, also in the whole logistics required to record with a dozen instrumentalists in one afternoon. Great thanks also to my music analysis teacher, Pedro Figueiredo, who kept giving me some ideas to work with. Lastly, i want to thank my whole family. My cousin Tiago Simas Freire for supporting me in some musicology issues, but most of all my parents who also suffered when some things didn’t seem to be solvable, and helped me in everything they could to make me succeed. iii iv Resumo A música e a matemática são duas áreas que, ao longo do tempo, têm vindo a perder as suas ligações. Nesta tese são feitas várias abordagens à relação entre as mesmas. Uma delas mais relacionada com as afinações de escalas usadas na antiguidade, e outra com a acústica e consonância numa orquestra actual. Pitágoras estudou a relação entre os números racionais e os sons puros, procedendo depois à afinação de escalas de forma a preservá-los. Como consequência, obteve uma escala que não era uniforme em termos de consonância. Hoje em dia, os músicos convivem com uma afinação que, apesar de ser prática e extremamente uniforme, não dá valor aos sons puros. Neste estudo, serão procuradas as afinações capazes de optimizar ambos os aspectos. Para tal, criar-se-á um método computacional capaz de devolver a consonância de duas notas, num intervalo de zero a um. A segunda vertente explorada ao longo da tese, é relacionada com a orquestra e os diferentes sons que a constituem. As "cores" dos instrumentos na orquestra são diferenciadas de forma matemática. Um som produzido por um instrumento tem um espectro harmónico que lhe corresponde, definindo o seu timbre. Para obter estes espectros, serão utilizados conceitos da teoria de Fourier, mais especi- ficamente a Transformada de Fourier Discreta. Este processo parte da gravação dos instrumentos, passando pela análise das respectivas ondas sonoras na linguagem Mathematica. Palavras-chave: consonância, espectro harmónico, transformada de Fourier v vi Abstract Music and Mathematics are two fields that lost their link over time. In this thesis, we make two different approaches to the relation between them. One of the approaches relates with the tuning of scales performed in ancient times, and the other with acoustics and consonance in orchestra nowadays. Pythagoras studied the relation between rational numbers and pure sounds, tuning scales in a way that would preserve this perfect consonances. However, this turned out to have an irregular consonance in the different intervals of the scale. Nowadays, musicians deal with a tuning system that, despite being extremely practical and uniform, doesn’t value much the pure intervals. Our goal is to find a tuning sys- tem capable of optimizing the features mentioned above. Therefore, we shall develop a computational method which outputs the consonance between two musical notes, in a range from zero to one. The second matter developed along this thesis is about the orchestra and the different sounds in it. The “colors” of the instruments in the orchestra are distinguished through a mathematical procedure. A sound produced by an instrument is defined by its harmonic spectrum, representing its timbre. To derive these spectra, we shall use concepts from Fourier theory, specially the Discrete Fourier Transform. This process is initiated with the recording of instrument sounds in an anechoic chamber, and then completed with the analysis of the soundwaves using the Mathematica language. Keywords: consonance, harmonic spectrum, Fourier transform vii viii Contents Acknowledgments........................................... iii Resumo.................................................v Abstract................................................. vii List of Tables.............................................. xi List of Figures............................................. xiv 1 Introduction 1 1.1 Organization...........................................2 2 State-of-the-art 3 2.1 What is sound?..........................................3 2.2 What is consonance?......................................6 2.3 Soundwaves and Fourier Analysis...............................8 2.3.1 Wave equation and the Beats phenomenon......................8 2.3.2 Fourier Series......................................9 2.3.3 Fourier Transform.................................... 10 3 Computing consonance 12 3.1 Recording orchestral instruments................................ 13 3.2 Calculation of the frequency spectra.............................. 13 3.3 A program to compute consonance............................... 17 3.3.1 1st step: Grouping frequencies............................. 17 3.3.2 2nd step: Notion of timbre in the program....................... 17 3.3.3 3rd step: Critical Bandwidth............................... 18 4 Temperaments 22 4.1 Pythagorean Tuning....................................... 24 4.2 Just Intonation.......................................... 25 4.3 Meantone Temperaments.................................... 26 4.4 Equal Temperament....................................... 28 4.5 Consonance calculations and results for each Temperament................. 30 ix 5 Timbres and Consonance in the Orchestra 33 5.1 Strings............................................... 34 5.1.1 Violin........................................... 36 5.1.2 Viola............................................ 39 5.1.3 Cello............................................ 39 5.2 Woodwinds............................................ 41 5.2.1 Flute............................................ 43 5.2.2 Oboe........................................... 44 5.2.3 Clarinet.......................................... 46 5.2.4 Bassoon.......................................... 47 5.3 Brass............................................... 49 5.3.1 Trumpet.......................................... 51 5.3.2 French Horn....................................... 52 5.3.3 Trombone......................................... 53 5.3.4 Tuba............................................ 54 5.4 Percussion............................................ 56 5.5 Consonance analysis of orchestral excerpts.......................... 60 5.5.1 Weight function of a harmonic spectrum........................ 60 5.5.2 Wagner’s Tristan chord................................. 61 5.5.3 Beethoven’s fifths in the 9th symphony......................... 64 5.5.4 Clarinet and oboe, the most consonant dissonance.................. 65 6 Conclusions 66 6.1 Future Work............................................ 67 Bibliography 70 7 Appendix 71 x List of Tables 2.1 Ratios of the most important intervals.............................7 4.1 Ratios of the intervals in a Pythagorean chromatic scale................... 25 4.2 Ratios of the intervals in a Just chromatic scale........................ 26 4.3 Ratios of the intervals in an Equal chromatic scale...................... 29 4.4 Consonance in all the temperaments with weight g(x) .................... 31 4.5 Consonance in all the temperaments with weight from Figure 4.9.............. 32 5.1 Values of zeros for Bessel functions [Benson, 2007]...................... 58 5.2 Consonances in the notes of the Tristan chord ob-oboes/ cl-clarinets/ bas-bassoons/ cel-celli 62 5.3 Consonances of a minor second in the clarinet and the oboe................ 65 xi xii List of Figures 2.1 Rhythmic cells..........................................4 2.2 Beethoven’s 5th symphony...................................4 2.3 Beethoven’s 5th symphony...................................5 2.4 Violin’s sound wave.......................................5 2.5 Flute’s soundwave........................................6 2.6 Harmonic Spectrum of C....................................6 3.1 WAV sound on Mathematica .................................. 13 3.2 Function to create the list of points for plotting the frequency spectrum........... 15 3.3 ListPlot of the coordinates calculated for the Frequency Spectrum.............
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