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Symbols are important. They do not only convey signals whose meaning is understood by those taught of them; some symbols go deeper than that, their understanding being universal, an inherent part of our psyche as a species, of our genetically based mental organisation.

All of us deal with symbols all the time, whether we are aware of it or not. Words are symbols for what they mean, letters are symbols for sounds, and figures are symbols for numbers. We have traffic signs when we drive, and we have icons when using computers. A gesture is a symbol, and so is most art.

Music is different. Notes are symbols, but performed music has a reality of its own; we do not really need to understand a symbolism to experience its essence. It is in a strict meaning a completely sensual art, it requires no intellectual processing.

Its complementary opposite is pure mathematics, an art dealing only with symbols. A purely intellectual art that is so pure that the symbols are disconnected from a real meaning for a totally abstract one. The formal system that evolves by the interaction of these abstract symbols exists in its own right, completely separate from physical reality.

“Guided only by their feeling for symmetry, simplicity, and generality, and an indefinable sense of the fitness of things, creative mathematicians now, as in the past, are inspired by the art of mathematics rather than by any prospect of ultimate usefulness.“

Eric Temple Bell (1883-1960)

Music and mathematics are essentially the same; opposites and the same, as two sides of one coin. Mathematics is the intellectual expression of music; music is the sensual expression of mathematics. Both are based on balance and harmony and the use of contrast, as well as on the creation and release of tensions caused by the use of controlled imbalance and disharmony. To take examples entirely from the western classical epoch, this is illustrated by the binomial theorem and Laplace operators as well as Beethoven's piano sonatas and Bach's fugues; by as well abstract algebra and topology, as the Brandenburg concertos and Palestrina's vocal polyphony.

(This holds for all music and mathematics, irrespective of time and place. Yet I want to put some attention to the western classical mathematics and music (in the widest possible sense of the word “classical”), because in my opinion they constitute the peak of western civilisation, perhaps its only achievements of lasting value. I'm convinced that millennia in the future, when our time is forgotten and the western culture is long dead and gone, these two, but nothing more, will be remembered of it. That is, if there will still be humans to remember anything.)

Many thinkers during millennia have held music and mathematics as reflections of the order of cosmos, universe, reality, or the divine. I would say that we can never know that. We know only one thing: they reflect something within the human mind and how the human mind perceives order in cosmos, universe, reality or the divine. Perhaps the beautiful music we hear is just noise, and the beauty just a function of our ear or mental organisation. Perhaps the spider in the corner of my room hears another music, another beauty from another cosmos, a cosmos that doesn't exist in my reality, and where the square of the hypotenuse is not equal to the sum of the squares of the other two sides.

Incidentally, this sentence might raise the question whether spiders actually hear. It is not important for this context, just assume they do. The meaning of what I say is clear anyway. Consider that as poetic licence. Otherwise the question is interesting. Far from everything is known or understood about spiders' senses or nervous system, but they do have a sort of hearing. Trichobothria on their body, fine hairs, are very sensitive to air movements, and the latest research suggests that they operate similar to microphones.

“...indeed what reason may not go to Schoole to the wisdome of Bees, Aunts, and Spiders? what wise hand teacheth them to doe what reason cannot teach us? ruder heads stand amazed at those prodigious pieces of nature, Whales, Elephants, Dromidaries and Camels; these I confesse, are the Colossus and Majestick pieces of her hand; but in these narrow Engines there is more curious Mathematicks, and the civilitie of these little Citizens more neatly sets forth the wisedome of their Maker.”

(Sir Thomas Browne, 1605-1682, according to: J. R. Newman (ed.) “The World of Mathematics”, New York: Simon and Schuster, 1956, p. 1001.)

(The article is based on material previously published in Meriondho Leo, and in my e-book “From Vision to Visual Music”, 2017.)

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Interesting post! Firstly I really appreciate the author @mictorrani for sharing this good information about mathematics and music. I believe that mathematics can describe many concepts in music. Mathematics explains how strings vibrate at certain frequencies. Also, It uses mathematics to study elements of music such as tempo, chord progression, form, and meter. As a math student, I have learned some important things in this post. Recently, I have joined this math educational website- https://sks23cu.net/dwqa-question/dwqa-question_category/questions/algebra to learn algebra equations and other mathematical questions with answers that can help to boost my brain with math knowledge. Now I can solve my academic math problems by myself.