"For a musician, visualization is the process of picturing in our minds eye what we hear in our mind's ear. Visualization is something we all do. In fact, putting a visual form before the mind's eye or forming a mental image is something that precedes most things that we do." (Jerry Bergonzi, "Melodic Structures", 1992)
"To me, that the distribution of prime numbers can be so accurately represented in a harmonic analysis is absolutely amazing and incredibly beautiful. It tells of an arcane music and a secret harmony composed by the prime numbers." (Enrico Bombieri, "PrimeTerritory", The Sciences, 1992)
"Mathematicians tells us that it is easy to invent mathematical theorems which are true, but that it is hard to find interesting ones. In analyzing music or writing its history, we meet the same difficulty, and it is compounded by another." (Charles Rosen, "The Frontiers of Meaning: Three Informal Lectures on Music", 1994)
"Just as music comes alive in the performance of it, the same is true of mathematics. The symbols on the page have no more to do with mathematics than the notes on a page of music. They simply represent the experience." (Keith Devlin, "Mathematics: The Science of Patterns", 1994)
"Mathematicians tells us that it is easy to invent mathematical theorems which are true, but that it is hard to find interesting ones. In analyzing music or writing its history, we meet the same difficulty, and it is compounded by another." (Charles Rosen, "The Frontiers of Meaning: Three Informal Lectures on Music", 1994)
"Music and math together satisfied a sort of abstract 'appetite', a desire that was partly intellectual, partly aesthetic, partly emotional, partly, even, physical." (Edward Rothstein,"Emblems of Mind: The Inner Life of Music and Mathematics", 1995)
"Music and higher mathematics share some obvious kinship. The practice of both requires a lengthy apprenticeship, talent, and no small amount of grace. Both seem to spring from some mysterious workings of the mind. Logic and system are essential for both, and yet each can reach a height of creativity beyond the merely mechanical." (Frederick Pratter, "How Music and Math Seek Truth in Beauty", Christian Science Monitor, 1995)
"I see some parallels between the shifts of fashion in mathematics and in music. In music, the popular new styles of jazz and rock became fashionable a little earlier than the new mathematical styles of chaos and complexity theory. Jazz and rock were long despised by classical musicians, but have emerged as art-forms more accessible than classical music to a wide section of the public. Jazz and rock are no longer to be despised as passing fads. Neither are chaos and complexity theory. But still, classical music and classical mathematics are not dead. Mozart lives, and so does Euler. When the wheel of fashion turns once more, quantum mechanics and hard analysis will once again be in style." (Freeman J Dyson, "Book Review of ‘Nature’s Numbers’", The American Mathematical Monthly, Vol. 103" (7), 1996)
"The mystery of sound is mysticism; the harmony of life is religion. The knowledge of vibrations is metaphysics, the analysis of atoms is science, and their harmonious grouping is art. The rhythm of form is poetry, and the rhythm of sound is music. This shows that music is the art of arts and the science of all sciences; and it contains the fountain of all knowledge within itself." (Inayat Khan, "The Mysticism of Sound and Music", 1996)
"How deep truths can be defined as invariants – things that do not change no matter what; how invariants are defined by symmetries, which in turn define which properties of nature are conserved, no matter what. These are the selfsame symmetries that appeal to the senses in art and music and natural forms like snowflakes and galaxies. The fundamental truths are based on symmetry, and there’s a deep kind of beauty in that." (K C Cole, The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997)
"Mathematical notation is for the scientist what musical notation is for the composer." (John Holland, "Emergence: From Chaos to Order", 1998)
"Fractals are patterns which occur on many levels. This concept can be applied to any musical parameter. I make melodic fractals, where the pitches of a theme I dream up are used to determine a melodic shape on several levels, in space and time. I make rhythmic fractals, where a set of durations associated with a motive get stretched and compressed and maybe layered on top of each other. I make loudness fractals, where the characteristic loudness of a sound, its envelope shape, is found on several time scales. I even make fractals with the form of a piece, its instrumentation, density, range, and so on. Here I’ve separated the parameters of music, but in a real piece, all of these things are combined, so you might call it a fractal of fractals." (Györgi Ligeti, [interview] 1999)
"Physicists are more like avant-garde composers, willing to bend traditional rules and brush the edge of acceptability in the search for solutions. Mathematicians are more like classical composers, typically working within a much tighter framework, reluctant to go to the next step until all previous ones have been established with due rigor. Each approach has its advantages as well as drawbacks; each provides a unique outlet for creative discovery. Like modern and classical music, it’s not that one approach is right and the other wrong – the methods one chooses to use are largely a matter of taste and training." (Brian Greene, "The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory", 1999)
"That randomness gives rise to innovation and diversity in nature is echoed by the notion that chance is also the source of invention in the arts and everyday affairs in which naturally occurring processes are balanced between tight organization, where redundancy is paramount, and volatility, in which little order is possible. One can argue that there is a difference in kind between the unconscious, and sometimes conscious, choices made by a writer or artist in creating a string of words or musical notes and the accidental succession of events taking place in the natural world. However, it is the perception of ambiguity in a string that matters, and not the process that generated it, whether it be man-made or from nature at large." (Edward Beltrami, "What is Random?: Chaos and Order in Mathematics and Life", 1999)
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