The Link Between Music And Mathematics...

A little formula: Art is harmony, harmony is mathematics, mathematics is music, music is art.

😵

Paola, I take the liberty of summarizing the contents of this first piece of your writing with 2 terms: exploration and complicity.

Is that its essence, or have I missed something?

suonileganti wrote:

to Stellina: it's true, classical music is also used, but certainly not for its perfection; it is done for the emotional message it can convey. For example, in cases of coma, you can imagine that it is not the virtuoso passages that communicate something to the patient....

one must consider that 99% of those undergoing treatment are not music connoisseurs, so the selection criteria follow other characteristics such as timbre and rhythm; thus pieces are divided, for example, into "contemplative", "evocative", "sound magma" ..... and other possibilities

to Frank: yes, in the sessions I conduct music is played; the only existing rule is "verbal language is not used". Patients produce sounds, sometimes music; we use Orff instruments (tambourines, xylophone, metallophone, maracas, rhythm sticks, etc., etc.). The goal is to connect with the other person, convey understanding, and establish trust. All of this is done by playing; the technique that allows this is very complex and is entirely a matter of empathy—perceiving what the other feels through their sounds—subsequently tuning in and communicating with sounds and rhythms of the same emotional depth.

This is possible by using very specific techniques learned in music therapy schools. I studied in Assisi in a four-year music therapy course, 21 exams, 250 hours of internship in an appropriate facility, and a final thesis; then I attended a one-year specialization course in Aosta, specializing in music therapy applied to dementia and Alzheimer's.

Returning to the session, let me give an example: the patient is welcomed into the prepared room with instruments placed in the center; no one speaks, you only observe in an attitude of listening. Anything can happen, and in the first few minutes, you are really playing the game. For example... the person does nothing, does not engage in choosing an instrument, makes no sounds; so you wait patiently for anything audible, even a cough or a foot dragging on the floor. If you perceive that it is the right moment, you respond using a suitable instrument by imitation. This is just to start; I could give you hundreds of examples of how a session begins. It is extremely interesting to hear what happens inside people. I also have videos that I made during sessions to write observation protocols (work done at a later stage), videos that I brought to various medical congresses on dementia and music therapy, which deeply moved the medical experts present. Unfortunately, they cannot be distributed due to privacy issues; we obtained permission from families only for medical research use in appropriate contexts.

If you are interested in deepening your knowledge about music therapy, I can recommend some excellent texts; otherwise, ask... and I will answer you

Thank you Paola, so much passion and such wonderful preparation. You are a goldmine 😉

I would truly appreciate the recommendation of texts, really.

I am new to the forum but not new to similar discussions 😎

The subject is so vast that talking about it ends up being banal.

"Musica est exercitium arithmeticae occultum nescientis se numerare animi". This is the definition of music given by Descartes in the Compendium Musicae. It translates more or less as "music is a hidden mathematical activity of the mind, a mind that does not know it is calculating". According to Descartes, and according to a tradition dating back to Pythagoras, musical perception and enjoyment are closely linked to mathematical operations. This is where the idea arises that a "well-made" composition, consonant and correct intervals, and regularity are all appreciable poetic elements. In fact, it is the same aesthetic concept as Greek and Renaissance art. The link between music and mathematics is philosophical, in this sense. But even more important is the PHYSICAL link between music and mathematics or, better yet, between sound and mathematics. Pythagoras was again the first to delve into the harmonic and interval relationships between sounds, starting that long road that would lead to the foundation of acoustics as a branch of physics that studies sound. Descartes, along with Galileo and others, would bring these studies back into vogue, also comparing them with the first "harmonic" treatises of the time, specifically those of Zarlino. We are at the beginnings of the foundation of harmonic practice (against modal and contrapuntal practice), namely the emancipation of imperfect consonances of thirds and sixths and the affirmation of chordal logic. Among the various epochal consequences, the need to solve the tuning problems of fixed instruments, i.e., the creation of various temperaments. Tuning an instrument means applying logarithms to the relationships between sounds in an octave. Quite complex calculations, in short 🥷

Before getting to Xenakis, there is a long way to go, so...

... too many things to say and so little time ...

thallo wrote:

I am new to the forum but not new to similar discussions 😎

The subject is so vast that talking about it ends up being banal.

"Musica est exercitium arithmeticae occultum nescientis se numerare animi". This is the definition of music given by Descartes in the Compendium Musicae. It translates more or less as "music is a hidden mathematical activity of the mind, a mind that does not know it is calculating". According to Descartes, and according to a tradition dating back to Pythagoras, musical perception and enjoyment are closely linked to mathematical operations. And this is where the idea arises that a "well-made" composition, consonant and correct intervals, regularity are all appreciable poetic elements. In fact, it is the same aesthetic concept as Greek and Renaissance art. The link between music and mathematics is philosophical, in this sense. But even more important is the PHYSICAL link between music and mathematics or, better yet, between sound and mathematics. Pythagoras was again the first to delve into the harmonic and interval relationships between sounds, starting that long road that would lead to the foundation of acoustics as a branch of physics that studies sound. Descartes, along with Galileo and others, would again bring these studies back into vogue, comparing them also with the first "harmonic" treatises of the time, specifically those of Zarlino. We are at the beginnings of the foundation of harmonic practice (against modal and contrapuntal practice), namely the emancipation of imperfect consonances of third and sixth and the affirmation of chordal logic. Among the various epoch-making consequences, the need to solve the intonation problems of fixed instruments, i.e., the creation of various temperaments. Tuning an instrument means applying logarithms to the relationships between sounds in an octave. Quite complex calculations, in short 🥷

Before getting to Xenakis we have a long way to go, so...

... too many things to say and so little time ...

Thallo, very true.

But don't you think there is a difference between mathematics at the service of instruments/musical acoustics and mathematics taken as a compositional element, such as combinatorics, integral serialism, etc.?

Obviously, Dino. But for the purposes of our discussion, very broad and generic purposes, they are both connections between music and mathematics.

Furthermore, if you want to distinguish between an organological level and a certain technical-compositional level, I tell you right away that there are and have been many composers who created their own instruments and tuning systems. Henry Partch, La Monte Young, Riccardo Nova, are all composers who developed tuning criteria based on mathematical systems. As far as I know, they did so by adhering to the "rules" of natural tuning, i.e., using criteria of consonance and dissonance in a certain sense "classical" (which isn't actually true, but that would be a complex discussion). But nothing prevents some Joe Schmo tomorrow from deciding to tune a violin with relationships based on the number 666 and its multiples and writing a composition specifically for it 😳

This is also the beauty of applied mathematics as a compositional criterion. You can decide at what level to apply it. That is, dodecaphony applies commutative criteria to the 12 sounds, but since the 12 sounds are "discretizations" of a continuum, I could quite well apply serialism to cents, i.e., to the division of the octave into 1200 parts. Or, even better, I could take an oscillator and program it directly with parameters. In this sense, mathematical criteria are never OBLIGATIONS, but compositional choices.

I mean... even in this discussion someone has "naively" contrasted inspiration with logicism. But if I decide, I don't know, to write a canon, which is an absolutely "mathematical" process, I do it in light of a series of "inspirations" even before using mechanical criteria. I have to choose the notes, the tempo, the dynamics, the overall duration. There are processes, like serialism or the "additive" process of minimalist music, that seem entirely aseptic, but in reality they are not because they are applied to MATERIAL that is nonetheless chosen by the composer, "composed" in traditional terms.

And it's the same principle on which much musical alea is based. If you decide to apply a random criterion, at first glance you have the impression of totally losing control over the result. But if you carefully evaluate the material to which you apply that criterion, then the number of possible results decreases drastically.

Ultimately, if placed in the hands of good composers, even the most absurd compositional procedures, logical and illogical, can lead to beautiful compositions.

Mathematics is in everything, but it seems to me I am observing the famous saying: "which came first, the egg or the chicken?". Just today with Paolo, who has just returned from the music fair in Frankfurt, we were talking about modern pianos and old pianos with reference to Steinway. I remember seeing a video about Kawai Grand Pianos in which halls appeared where they filled the piano soundboards with sensors, and by playing them, one could see where the sound waves clustered in the board, which parts of the instrument were set into vibration, the speed of wave propagation, etc., etc. Today, therefore, mathematical knowledge is exploited even in some musical fields to improve a product, but once upon a time? When this knowledge and these technologies did not exist, what made the difference was intuition. Just as with Steinway, who was the first manufacturer to build the piano frame in cast iron, or as with Stradivari, who intuited his famous chemical composition that gave this particular timbre to his instrument. This is to say that presumably even in composition, perhaps one has never reasoned mathematically, and what dictates instead is the poetic and artistic vein of the composer. There is a certain innate talent, where what is constructed and architected comes spontaneously, and only afterwards does one realize they have painted a Mona Lisa or built the Sistine Chapel.

TheSimon wrote:

Mathematics is everywhere, but it seems to me I am observing the famous saying: "which came first, the egg or the chicken?". Just today with Paolo, who has just returned from the music fair in Frankfurt, we were talking about modern pianos and old pianos, referring to Steinway. I remember seeing a video on Kawai Grand Pianos in which halls appeared where they filled the piano soundboards with sensors, and while playing them, one could see where the sound waves clustered in the board, which parts of the instrument were set into vibration, the speed of wave propagation, etc. etc. Today, therefore, mathematical knowledge is exploited even in some musical fields to improve a product, but once upon a time? When this knowledge and these technologies did not exist, what made the difference was intuition.

I disagree. But, even here, the discussion is complex.

1) the mathematical knowledge we rely on began to develop in the 1600s, in terms very close to those of today. Eighteenth-century instrument makers were not merely ignorant workers, quite the contrary! It is very naive to believe that organology is resolved as a matter of intuition or luck. There are different kinds of builders; obviously, organ builders are semi-scientists and luthiers are bunglers. But the "greats" in the history of intonation and temperaments knew how to do the calculations...

Just like Steinway, who was the first manufacturer to build the piano frame in cast iron, or like Stradivari, who intuited his famous chemical composition that gave this particular timbre to his instrument.

There are levels and levels, in my opinion. An organ builder or a bell founder could not afford to do such things. But it is one thing to say that violins can only be built with intuition, and another to say that thanks to intuition, one day a luthier improved violins. Because what I am saying is that a mathematical basis is required, not that intuition is useless. Great chefs are creative, but if they don't know how to debone a quail, then they can keep their creativity at home... in this sense, I suppose Mr. Steinway, before thinking about a cast iron cover, racked his brain to calibrate the strings.

This is to say that presumably even in composition, perhaps it has never been reasoned mathematically, and what dictates the law instead is the poetic and artistic vein of the composer. There is an innate talent underneath, where what is constructed and architected comes out spontaneously, and only afterwards does one realize they have drawn a Mona Lisa or built the Sistine Chapel.

the transition from construction to composition is wider than one might think. So, while being in disagreement with you on the first part, I might agree here.

But with the necessary specifications. I do NOT believe in innatism; for me, there always needs to be application in bringing out "inspiration," whatever that may be. The myth of the noble savage, of the pure man free from external influences, from technicalities, from teaching, who arrives and impresses us with perfection is, precisely, a myth. I continue to think that without years and years and years of study, Leonardo would never have painted the Mona Lisa. Not only that, I am always convinced that the image of the Mona Lisa tormented him for nights and days, that he didn't do it "on the fly," because he was the coolest guy in the coop. Great artists are meticulous, generally, unless they have particular ideas about intuition... and Leonardo, as far as I know, was not Pollock.

There are many musicological studies on musical innatism, on the famous "grammar" of music (meaning grammar as a consequence of Chomsky's famous generative grammar, i.e., a series of basic knowledge present in our minds, without which we would not be able to develop languages). These studies have managed to demonstrate very little. Everything that we consider "universal" in music, things like the alternation of consonance and dissonance, the preeminence of melody, the driving value of rhythm, are STILL debated issues. The most probable reality is that EVERY musical knowledge is the union of a sort of predisposition (unverifiable) and, above all, of much practice (and I am not talking about exercise, I am also talking about listening, socialization, reflection). Within this "practice" lies study. Even theoretical study, which is practice 😉 because understanding what a cadence IS to the core is as laborious as learning how to play it.

These practices give you a vocabulary with which to express yourself, they teach you musical "language". They teach you intuition! In the sense that they make you aware of which fields to apply it to. That is, Stradivari didn't say "damn! Now I'll apply a coffee maker to the violin so when musicians are tired they can make themselves a coffee....". If you don't know the problems of a certain musical (or compositional) field, you don't know where to apply your intuition. And I assure you that for every "useful" stroke of genius in music history, there have been at least 10 that were completely useless...

In conclusion... intuition yes, but not innate.

I am only saying that every string of a grand piano, when under tension, develops a certain force... the sum of the force of all the strings (plus the safety margin) determines how strong the soundboard must be.

This is mathematics.

Regarding composition, I relatively agree with Thallo; the fugue (let's say baroque, since Bach was mentioned) is like sudoku... intuition always hits between tonic and dominant, but what intuition is there in the canonical procedure? The procedures of diminution, augmentation, inversion, and retrograde... are they not mathematical calculations?

Of course, all aimed at beauty, but without mathematics there would be no music... even the pitch of a sound is given by a number of vibrations per second, harmonics are a logarithmic calculation, etc.

For me, music is profound; then at the top of the pyramid you find stochastic music, algorithmic composition, fractals, etc.

Pablo wrote:

A little formula: Art is harmony, harmony is mathematics, mathematics is music, music is art.

😵

Pablo, I'm starting to think you are absolutely right 😉

thallo wrote:

I disagree. But, even here, the issue is complex.

1) the mathematical knowledge we rely on began to develop in the 1600s, in terms very close to those of today. The builders of eighteenth-century instruments were not likeable ignorant workers, quite the opposite! It is very naive to believe that organology is resolved as a matter of intuition or luck. There are different kinds of builders, obviously; organ builders are semi-scientists and luthiers are charlatans.

"On which we rely..." I remind you that in the 1600s they were stuck at trigonometry, far from being very close terms... It seems like a little bit to design instruments that instead need formulas based on acoustics that use infinitesimal calculus, "discovered" only 250 years later in the second half of the 1800s by Cauchy, who coined the meaning of derivative as the limit of the incremental ratio, and later Riemann with his famous "Integration". Acoustics, more than other physical disciplines, uses this branch of mathematics since it reasons on a non-linear and non-canonical reality (the reason why most of the time it is impossible to use canonical formulas that do not require integration and derivation processes). We must always see what instruments we are talking about; infinitesimal calculus is not necessary for everyone. Certainly, pianos before the advent of modern mathematics were what they were, and in music history many composers were not impressed at all. The sounds had a very fast decay and the sound intensity was not suitable to fill concert halls. Current models easily fill the entire concert hall. My appeal was also directed toward the impossibility of being able to perform certain measurements because technological instruments capable of such a task were not available. Example... The most renowned piano companies in the world all use wood obtained from spruce from the Val di Fiemme grown at a very specific altitude. Why? Simply because instrumentation has demonstrated that the speed of sound propagation in these particular woods approaches 3600 meters per second. This kind of consideration was impossible to make at the time. We must then distinguish the term intuition from luck, since it seems to me that you group them together. Well, I did not speak of luck but of intuition; there is a big difference. Intuition is dictated by intelligence. You understand something without having to perform calculations and you intuit that it works, so you put it into practice and experimentally verify the validity of your intuition. Luck in this field means making a mistake in a process and therefore committing an unexpected error that ultimately leads to results in the opposite direction. 😮

But the "greats" of history regarding tuning and temperaments knew how to do calculations...

I agree, but this is a field where mathematics is necessarily required, and let's remember that calculating a non-linear equation is not the same thing as calculating the twelfth root of 2. 😛

There are levels to this, in my opinion. An organ builder or a bell founder could not afford to do such things. But it is one thing to say that only with intuition can violins be built, and another thing to say that thanks to intuition one day a luthier improved violins. Because what I am saying is that a mathematical basis is needed, not that intuition is useless. Great chefs are creative, but if they don't know how to debone a quail, then they can keep their creativity at home... in this sense, I suppose Mr. Steinway, before thinking about a cast-iron cover, racked his brain to calibrate the strings

I agree. 😉 Mr. Steinway racked his brain to calibrate strings and therefore he intuited that perhaps the problem was poor frame stability. Let's remember, however, that at the base of everything there is always an intuition and never mathematics. We use mathematics to give explanations to the phenomena that surround us, and if we verify some laws, then we reuse them when it is convenient for us. It has never happened to me to see (except in quantum mechanics where intuition often, if not always, leads to completely unexpected results) that mathematics came before an intuition, and we have proofs every day from all the scientists who discover something: they intuit, theorize, research, experiment, confirm or refute.

the transition from construction to composition is wider than one might think. So, while I disagree with you on the first part, I might agree here.

But with the necessary caveats. I do NOT believe in innatism; for me, it always requires application to bring out "inspiration," whatever that may be. The myth of the noble savage, of the pure man free from external influences, technicalities, and teaching methods, who arrives and impresses us with perfection is, precisely, a myth. I continue to think that without years and years and years of study, Leonardo would never have painted the Mona Lisa. Not only that, I am always convinced that the image of the Mona Lisa tormented him for nights and days, and that he didn't just do it "on the fly" because he was the coolest guy in the coop. Great artists are generally meticulous, unless they have particular ideas about intuition... and Leonardo, as far as I know, was not Pollock.

There are many musicological studies on musical innatism, on the famous "grammar" of music (using grammar to mean a consequence of Chomsky's famous generative grammar, i.e., a series of basic knowledge present in our minds, without which we would be unable to develop languages). These studies have managed to demonstrate very little. Everything that we consider "universal" in music, things like the alternation of consonance and dissonance, the preeminence of melody, the driving value of rhythm, are STILL debated issues. The most likely reality is that EVERY musical knowledge is the union of a sort of (assessable) predisposition and, above all, a lot of practice (and I'm not talking about exercise; I'm also talking about listening, socialization, reflection). Within this "practice" lies study. Even theoretical study, which is practice 😉 because understanding deeply what a cadence IS is as laborious as learning how to play it.

These practices give you a vocabulary with which to express yourself; they teach you musical "language." They teach you intuition! In the sense that they make you aware of which fields to apply it to. That is, Stradivari didn't say "Damn! Now I'll apply a coffee maker to the violin so when musicians are tired they can make coffee....". If you don't know the problems of a certain musical (or compositional) field, you don't know where to apply your intuition. And I assure you that for every "useful" stroke of genius, in the history of music there have been at least 10 that were completely useless...

In conclusion... intuition yes, but not innate.

I seem to remember a child who wrote his first string quartet at age 5 (an age when mathematics is unknown and everything created is pure inspiration); his name was Mozart 😉

Of course, all aimed at beauty, but without mathematics there would be no music... even the pitch of a sound is given by a number of vibrations per second, harmonics are a logarithmic calculation, etc.

I am convinced of the contrary. Mathematics explains and investigates what we cannot explain and therefore understand with words; it is a tool that we use: passively to investigate, actively to create, based on the laws we have experienced and on what we have investigated in the passive way...

With mathematics, you know that by striking a taut string it emits a sound because it goes into vibration (you have investigated and experimented to understand an effect and explain it formally), but whether there is mathematics or not, by striking that string the sound would be emitted anyway... 😉

Then there are various levels, and on this I agree with Thallo. I am quite certain that the first forms of music existed without mathematics... The same cannot be said of temperaments, where mathematics had the function of standardizing things.

The goal of research is precisely this, and science moves forward out of the necessity to explain natural phenomena that, at the current state, we simply do not understand.

Ok, let's agree to disagree

Eheheh, of course! Everyone has their own vision. Wouldn't have it any other way... As someone used to say: "I don't share your thought, but I will defend it to the death!" 😉

You may well stick to your own mutual and respectable idea, but I would like to provide some input

TheSimon wrote:

I seem to remember a child who wrote his first string quartet at age 5 (an age when one does not know mathematics and everything created is pure inspiration); his name was Mozart

...the same child moved within the following harmonic syntax VI II V I 😉

You know that the first symphonies (see KV 16 and following) were nonetheless influenced by someone who knew mathematics (was not 4 years old and was named JC Bach), and perhaps his father also laid his hands on those same "inspired" symphonies?

All this to say that the modal, tonal, serial, dodecaphonic system, etc., is not written in the ear... but the ear is trained to what we hear... and what if what we hear was based on mathematical criteria?

I repeat, VI II V I is mathematics, triads are mathematics... and everything is based on the system of harmonics; the rules regarding doubling find their foundation in the law of harmonics, which are as artificial as anything can exist 😉

Having said that, I meant something else, which is why I indicated Bach and Xenakis... I was talking specifically about the construction of a piece following mathematical principles in a strict sense and applying them to form, note choice, variation methods, etc... taste, intuition, and inspiration serve to prevent certain calculations from feeling heavy... and anyway, the rest is also calculation.

What is the circle of fifths? And the progression? And yet Mozart used them by the quintals 😉

Preventing mathematics from feeling heavy is a gift; finalizing it toward an aesthetic result is a necessity.

Does anyone have an idea of how Mozart wrote operas? He wrote two lines: bass and main melody... the bicinium, and then he filled in everything, all proven by the analysis of manuscripts and the types of ink used and prepared artisanally in different ways "from time to time," with the harmony.

And what is harmony if not a vertical superposition of sounds at a certain distance?

A very common technique that allowed for fast writing in the context of basso continuo... which in turn, what is it?

If not the development of the harmony contained within the "NUMBERED" bass?

Damn!, you went deep into it, I'm getting confused... 😳

I would like to make a small reflection, summarizing a bit for the sake of brevity:

the development of mathematics occurs alongside the development of physics, of science, that is, the attempt to understand the world and to use it.

Aside from speculative branches, therefore, mathematics is the result of man's approach to the world and is thus not a pre-existing object outside of man, on its own.

In my opinion, we could say that intuition guides science and certainly does not replace it! It is as if it were the head of a snake.

To say that there is mathematics in music because, for example, the string vibrates following mathematical laws seems useless and wrong to me. Or rather, it's like saying that the laws of physics are modelable to a certain extent with mathematics, but we already knew that! Mathematics is the synthesis of this understanding; we assembled it for this purpose.

It seems more interesting to me to understand how sensible it might be to transfer mathematical relationships, probably extrapolated from a cognitive process of nature, to a compositional process.

The matter is complex and therefore requires time and space...

until next time!

I am also a bit confused, understanding mathematics is more difficult than understanding music, because the former is abstract, while the latter is marvelously concrete and why not, also pleasant!

The problem of music and sound as "physical" elements analyzable and malleable according to physico-mathematical models is a historical problem, as well as a philosophical one. That is, I am talking about it because the history of music has talked about it 🙄 we should blame Rameau and even before him all the medieval philosophers who considered music as part of a COSMOLOGICAL system. When it is said that music is based on "natural" laws (like the naturalistic theories of harmonics which say, essentially, that consonances are natural and dissonances are unnatural), it is placed within a world that is not only artistic but also physical.

It has been done; many have done it. Even if we don't agree with seeing it that way, well, everyone has their own little musical philosophies. But in a discussion about mathematics and music, it seems excessive to ignore them...

On compositional techniques that apply somewhat "mathematical" processes.

There are many of them...

Frank introduced the situations of counterpoint. Well, in a technical world full of constraints, numerical relationships, numerology, give space to new expressive modes. Mensural and enigmatic canons are based on mathematical and metaphorical procedures; I am not an expert in this repertoire, but I remember at least one motet by Dufay, Nuper Rosarum Flores, in which sections, prolations, and various other numerical elements had clear Christian symbolism (the prolations were the measurements of Solomon's temple, to name one). Number is part of musical symbology, as it is of theological and philosophical symbology.

I skip tonality entirely since, as stated, the entire harmonic theory is partly a mathematical theory...

In the 1900s, mathematics returns to prominence forcefully. Obviously, when I speak of mathematics, I speak in a generic way... that is, things like radical serialism, dodecaphony, stochastic music, use very different approaches, sometimes geometric, sometimes pseudo-physical. For me, it is all expressible in mathematical terms, so it falls under the "mathematics" umbrella.

Schoenberg is the cause of everything (perhaps). Dodecaphony makes the series necessary, which is a logical-mathematical concept (it is a meaningful string, a polynomial). The treatments of the series are geometric (it is broken, inverted, reversed, etc.). And the symbolisms are numerological through and through.

The matter becomes radicalized with Webern, as we know, and with Boulez one moves to integral serialism, to the serialization not only of all musical material (pitches, rhythms, dynamics...) but of musical DIMENSIONS, in terms of space and time, from which the fundamental theorizations of smooth and striated space and smooth and striated time arise. Similarly, Stockhausen tries to "serialize" rhythm and pitches into a single continuum, conceiving a substantial coherence of time and sound (see "...wie die Zeit vergeht", a famous essay, repeatedly denied by real scientists, which considers rhythm as arising from a periodic wave too slow to be perceived as sound). Xenakis, as has already been said more or less, elaborates statistical models of sound articulation and diffusion—pseudo-scientific models, let's say.

Other sides... minimalists also introduce "additive" rhythmic conceptions to the West, typical of Indian music. Rhythm is considered simply a sum of beats, no longer the constant divided model of a cell. And as much arithmetic as one can imagine: melodic lines that in their repetitions always add more notes, more beats. As a direct consequence of the temporal theorizations of Boulez and Xenakis, moreover, the complexity of the divisibility of Western metric cells becomes insane; note durations can now be easily calculated in roots, complex fractions, fanciful expressions (see Ferneyhough).

... for now nothing else comes to mind, but I have certainly forgotten a lot of things

it is obvious that music and mathematics are related if we consider that the former lives in time and space, all forms of art "use" mathematics (painting, sculpture, architecture etc....) but there is something else in music that does not belong to the world of mathematics, namely being art, a form of human expression in the context of its era; music follows instinct, emotions, fashion, social contrasts, and human needs in repeating it infinitely as if it were the first time, influencing mood and thought (it is skillfully used, for example, in commercials to persuade people to buy)

Excuse me, but perhaps I let the topic of the discussion slip... That there is mathematics in music seems obvious to me. What I am instead attacking is a formalism. We made mathematics and we gave it meaning. We are the ones who tailor it to a purpose. It's fine with me if Mozart used it, but it's one thing to use it consciously and quite another to use it because the ear has become accustomed to listening to Bach. This is the formalism that I am discussing. To think that there is mathematics in music seems even trivial to me; after all, mathematics is everywhere. There are composers who perhaps consciously base their composition on a pre-established mathematical architecture, and others who write for their own taste. While it is true that a mathematical architecture can be seen even in the latter, the point is to evaluate the awareness of its use rather than not evaluating it. I find myself very much in agreement with GnazzJazz, who perfectly captured the meaning of what I was trying to say. I think I know mathematics well enough, since at university I only had 4 exams: analysis, physics 3, statistics, algebra, geometry, etc., etc. Mathematics is born precisely as support for man's desire to understand and explain the phenomena surrounding him, giving wide scope to other sciences such as physics, astronomy, etc., which use it to verify theories. The scientific method is based on theorization - research - data collection - testing the theory; after which, if the data confirm the theory, it becomes a law, otherwise the theory was wrong, and mathematics performs the work of supporting the development of the data. These are not things I am saying; these are things written in all physics books...