Nando Florestan's blog (Posts about music)

Chopin, the king of the waltz

Nando Florestan — Sun, 13 Apr 2025 09:00:00 GMT

Chopin's opinion

When a young Frédéric Chopin left Poland to try and become a professional composer, he did not go to Paris directly. First, he spent many months in Vienna. I am not a music historian; a few years ago I was surprised by a letter, written during his stay in Vienna (1830-31), in which he wrote:

Here waltzes are called 'works'! and Strauss and Lanner, who play for them to dance, are called Kappelmeisters. It is, however, not the case that everyone here thinks that way; on the contrary, almost everyone laughs at it, but that is why they print only waltzes. (...)

(Fragment of a letter written by Chopin in Vienna on January 26th, 1831 to his teacher Józef Elsner in Warsaw.)

A great recording of Chopin's waltzes

Functional music

The distinction between absolute music and functional music has always been important.

Waltzes are functional music, they are for dancing. Marches are for processions, lullabies serve to soothe or lull, and modern film scores often refuse to say anything more than a very dispensable atmosphere setting. Since the era of the radio, music is everywhere, filling the silence, but that's only a function – music can be more important than that.

In contrast, absolute music is for pure listening, without another external practical function. It is in the foreground of one's attention. Examples are fugues, sonatas, quartets, symphonies etc.

Children of the menuet

In the classical period, inside a sonata or symphony, usually the second movement was a menuet. Its function was to balance the seriousness of the dissertation in the first movement with a little dance, relaxing the mind. Inside the symphony, the menuet was replaced with the scherzo, and thus the symphony was no longer for the aristocracy, but for the burgeoisie. This was done by Beethoven and after that, the next notable symphony that contained an aristocratic dance was Prokofiev's first, when again something political is happening to the burgeoisie.

That was in absolute music. In functional music, the menuet became the waltz.

Chronology

When I read the letter fragment, I was a little shocked that Chopin would have such a poor opinion of an important dance form. Indeed, he thought the public had a "corrupt taste" and he claimed to be unable to play those waltzes.

But my historic understanding was lacking. I had to realize that, today, when we think of waltzes, we think of Johann Strauss Jr, with his Blue Danube and hundreds of pearls like that – but Strauss is the next generation, he lived later than Chopin!

The Strauss mentioned in Chopin's letter has to be Senior, not the Junior:

Chopin lived from 1810 to 1849.
- His first waltz, op.18, was published in 1834.
- His 3 waltzes op.34 were published in 1838.
- His waltz op.42 was published in 1840.
Johann Strauss I lived 1804-1849.
Johann Strauss II lived from 1825 to 1899.
- His opus 1 was composed in 1844.
- The Radetzky March op.228 was premiered in 1848.
- The Blue Danube, op. 314, was composed in 1866.
- The Emperor Waltz (Kaiser-Walzer) op.437 was composed in 1889.

Before Chopin, I can remember little else in the genre besides some Schubert, whose waltzes are miniatures, and nowhere near the complexity of Chopin's.

The waltzes Chopin knew were functional music, waltzes we don't hear anymore, music that didn't become immortal. And then Chopin's editors wanted him to write waltzes, because that's what was selling. Especially in Vienna. And for sure he felt frustrated with the genre of the waltz until then – he wanted to write opuses. So he did.

Chopin's waltzes

Everyone knows Chopin's waltzes are more for listening than for dancing, because of the pull and push necessary in the tempo. But Chopin's waltzes also have these characteristics that turn them into important absolute music:

Counterpoint. Although the waltz traditionally is just a melody over a very straight and boring oom-pah-pah harmonic accompaniment, Chopin's accompaniment is so rich it becomes something else. There are secondary melodies and counterpoints in this accompaniment, the voicing and arrangement receives a lot of thought and experimentation. This is the main way in which he was able to turn a waltz into something worth listening to. Because in music, simultaneity is what makes something interesting.
Chopin's harmonies are much more sophisticated than your popular tonic-dominant-tonic waltz. Chopin's chords are often more complex, more dissonant, and he knows how to create that nostalgia from a tonic pedal near the end.
The forms of these waltzes are often simple but very well calculated, they never last longer than they are welcome, repetition is at the optimal point.
The expression is wider and more profound. You get a couple very sad waltzes, such as the op.34 n.2 in A minor. Some sections within a waltz can be slower or faster.
There are sometimes musical experiments in Chopin's waltzes.
- In waltz n.5 (op.42), the 1st section has a melody in 2 against the accompaniment in 3.
- In waltz n.4 (op.34 n.3), the main section has a dizzying stream of 8th notes instead of what the common man calls a melody.
- In op. 70 n.3, the main section has a Bachian quality in the way a single melody really contains 2 melodies on 2 separate instruments. Another section has 2 melodies in a different way.

The melodies are always incredible, but melodies in waltzes are commonly pretty. What sets Chopin's waltzes apart isn't the extraordinary quality of the melodies, I don't think. It's the quality of the overall composition – he had to forget the function, in order to create immortal music.

In short, Chopin's waltzes are Chopin.

The king

When Strauss Jr. was composing his great waltzes, he already had the example of Chopin. I think it is arguable that Chopin was the real king of the waltz in his time, because he was really the first composer to push the functional genre well into the realm of immortal absolute music.

Jaws is not a two-note theme

Nando Florestan — Fri, 23 Aug 2024 09:00:00 GMT

They recognize it as soon as it starts – on the first note, even, as the reaction of this crowd shows at 16 seconds.

We are talking about John William's main theme for Steven Spielberg's classic film Jaws (1975).

When the thing starts, it already has a certain personality, definite choices in the orchestration, that enable anyone to know what it is. It's like a tutti of the lowest orchestra. The contrabasses and cellos are helped, as the video immediately shows, by a contrabassoon and two bassoons. I think there is no bass clarinet, I can't hear it.

Actually, low strings doubled by bassoons... that's a very common orchestral sound. But I suppose if you know it's Williams, then on the first note you know what this is.

It is a truism that the theme has only two notes. I understand what people mean when they say that. Journalists have said that often when John Williams is interviewed, and he won't correct them, he accepts the number two. He says "yes, and".

But that's wrong. Of course the theme does not have only two notes, nor could it. It's a horror film, therefore it has all twelve notes.

As you probably know, there are normal people, and there are musicians. Normal people haven't counted the pitches. But any composer will – it's not a case of "ackchually", but a necessity for understanding the language and the material.

We have only examined the introduction so far, which is insinuating the theme. When it stops lurking and launches into the fast ostinato, at 40 seconds, two things happen. First, a piano joins the texture, probably to give it a very definite attack. But most importantly, you get the same 2 alternating notes, Mi and Fa – but also a third note in the accents: Re.

So the main idea, the main subject of the film, requires three notes, not two. And Re is the lowest – the note we would hear as the bass of any implied chord there. Re is used in the accents, you couldn't write the actual theme without this third note.

With 3 notes given so far, one could be forgiven for thinking we are probably in the key of D Minor. The composer is now quick to prove that wrong – at 48 seconds the texture becomes more complex. The 4-horn call is incredibly important, contributing only pitches that are a minor second apart from the 3 pitches of the ostinato. This means it could not sound more dissonant in relation to the ostinato, it could not be in a more foreign "key". The pitches are Mi Bemol, Sol Bemol, and Re Bemol.

One can play all the pitches mentioned so far by dropping one's closed hand on a piano, from C# to F#. This fact suggests a kind of dodecaphonic thinking. A composer realizes if you expand that with only one more neighbour note, you get the range of a tritone, and then you can apply dodecaphonic procedures to that, like transposing the whole thing up a tritone to use the rest of the 12-note scale.

But that is not what Williams does here. I would say the thinking is more along the lines of pitch sets, combined with polytonality, because we easily hear the two simultaneous musical events (the ostinato and the horn call) in two different "keys".

So here a public that has already forgotten polytonality (except that jazz never forgot it) is exposed to an advanced instance of it. Darius Milhaud, author of the best polytonal music, "Saudades do Brasil", would be proud.

The accents in the trombones... there is something weird about them, in this recording they sound like La Bemol. This surprised me because, as I started writing this, I was pretty sure it was Si Bemol. This video soon becomes too audio-distorted to remain useful, we must switch to a studio recording now. In the studio recording, the trombone accents are Si Bemol, definitely. (0:30)

This kind of difference is something that always happens with Williams. For his most famous pieces, he always created a Boston Pops version, with a more normal set of instruments, such that it can be played by any orchestra without hiring many extras. Thus the original film music always sounds a bit more "custom" than the orchestral suites made for concerts. But he doesn't stop at orchestration – there are changes in the discourse, in the order of the sections, and in tiny details such as this, the pitch of the trombone accents.

Would you like me to speculate about why he would change B flat to A flat? OK. Both are possible as part of the horn call scale. But B flat sounds much more in harmony with the ostinato. In fact, B flat dangerously undermines the atonality because it "wants" to become the root under the ostinato, which would then become a simple B Flat Major chord (Si bemol, Re, Fa). A flat does not have that problem, being an "ugly" tritone away from the Re.

Now (0:36) the pitches Fa and La Bemol also become a secondary part of the horn call. The end of the horn call has longer notes in this version, while in the Boston Pops version the notes are very short.

Speaking as a composer, the horn call is the actual main theme of this piece. This is because the ostinato is not amenable to development – it is most useful if it remains there doing its thing without change.

A simpleton like Hans Zimmer could also compose the ostinato – anyone could. But to make the rest of the texture and the rest of the piece, John Williams' taste, knowledge, creativity and development skill are necessary. Therefore, the value of the piece isn't really in the ostinato, it's in everything else!

One wonders if normal people have heard the rest. I think they have, but they can't talk about it, they don't have the vocabulary for it. So they talk about the two-note theme. It's a sort of musical bikeshedding that happens. If I were to interview Williams, I wouldn't ask him about the same tired old things, I would make a technical interview, a composer's interview.

At 1:00 a huge tutti is building up, the violins carry the melody, and guess what, the melody is based on the horn call. There is still something oscillating in the background in the interval of a minor second (piano?), but that is "drowned" by the rest. It is also two times slower as the oscillation in the ostinato.

At 1:20, on top of a chaotic atonal texture, flutes play some kind of American march fragment. Or maybe it's not a march, maybe it's a children's song. Either way, we are reminded of the general public having fun in the beaches – those that the main character, a reluctant hero, needs to save. This song is again superposed "in a different key" from the rest. The polytonal procedure is important in the language here. It makes sense because these people are unaware of the danger – the dangerous shark and the people having fun are in two different realities.

These are the most interesting things I could say about the Jaws theme. To sum up:

It's never just 2 notes. In fact, it's 3. In fact, it's a horrible chromatic cluster. In fact, it's polytonal.
The polytonality is a musical way of expressing two simultaneous, conflicting realities. But it is a latent conflict: we can hear the conflict, but the two individual situations cannot, each being "happy" in its own key.
Anyone could have composed the ostinato. But only a jazz musician like Williams would have remembered to use polytonality to express the latent conflict.

The entire music of the film is genius of course. Not just the recognizable theme. Witness, for example, the "father and son" scene.

Here, I believe the piano and harp representing the child are in the scale of D Major, but really in G lydian. (The lydian scale is the brightest one available.) The most important thing in the melody is the C# resolving to B natural.

The innocent youth is incapable of understanding the depths of gravity that a responsible adult will have to go through. This scene drives home the simultaneous realities of the youth on vacation and the policeman on whose shoulders the responsibility for the dead girl weighs.

Of course he can only think about his problem, nothing else. What is more insistent than a 2-note ostinato? A single held note. John Williams will not mention the shark musically here, since there is no danger. An even worse alternative would be to invent a theme for the politician. Let's not do those. Not when one note will suffice.

Therefore, polytonality is used again. Guess what note is in the bass? C natural. Why?

It's the lowest note cellos and contrabasses are able to emit.
It's right between the C# and B natural I just mentioned, just a semitone apart from both, therefore it maximizes dissonance to the child's melody.

If the shark isn't just 2 notes and actually requires 3... at the same time, John Williams can convey an idea with a single bass note if he wants to. That's amazing!

One character imitating another... that reminds me of another Spielberg movie with another perfect musical score. In that one, the imitation is the very first act that humanizes the hideous deformed monster who, by the end of the movie, will have become your best friend in the entire galaxy and leave you in tears for his absence. But in Jaws, the point is the contrast between their two mindsets, and Williams put that contrast in the music.

Usually film music will seem to be from the point of view of one character. This is a rare example of two POVs, simultaneous, accomplished through bitonality.

When Spielberg first heard the shark theme on the piano, he didn't seem to like it. If memory serves, it went more or less like this: "Really? That's all?" "You have to imagine the orchestra, of course." "Yeah but that simple?" "It's a very primal movie you've made here, Steven."

Williams' accusation that the movie is "primal" is not true. Let us just say his scores have more complexity and truth than his assertions. Or rather, in trying to defend himself, he went for the wrong argument. The correct response would have been, "of course that's not all, that is only the ostinato. On top of that we'll have modernist techniques, an entire orchestra playing in polytonal layers." Because that is what he actually did. The music for Jaws is not based on two notes, that's nonsense.

"Jaws" was a great success, and remains one of the greatest accomplishments in cinema, not because it was primal, but because it is sophisticated. Since all the sequels lack sophistication, they are failures. This sophistication is obviously in the music, but also in the director's technique. Watch this analysis of a single scene to understand how.

How to listen to classical music 1/99

Nando Florestan — Sat, 20 Apr 2024 13:42:08 GMT

Classical music is the best example of an acquired taste. It's something that requires an investment. But rest assured, you profit proportionally to how much you invest.

There are two kinds of people who don't like classical music:

A few seriously tried to learn to listen to classical music, but they really have a problem with it.
Most never tried it seriously, so in fact they could still learn. Usually they don't know how to start or how to continue.

If you want to start listening to classical music, this playlist of mine was created to be didactic. Mostly you want radio stations in which people talk about the music and play entire works (not fragments). In the UK that's BBC Radio 3. In Brazil that's Cultura FM de São Paulo. Elsewhere I don't know, it might not even exist. In Poland, for instance, there is more than one radio station for classical music but they are all square, with a very conservative and restricted repertoire, no comments on the music and they rarely play entire works. Just awful.

If you decide to learn to listen to classical music, what does that amount to? What lies ahead of you exactly? Let's answer this question. Sorry, the answer is long.

What it amounts to

More music than you expect

Most people today listen to less than a century of rock and pop music and little else. But classical music isn't just one more genre, it is a vast repertoire containing many, many genres. I mean, it has existed since the 5th Century. In practice we listen to about 8 centuries of music, with many revolutions in the genres and musical languages used.

That repertoire is probably larger than all non-classical musics combined. You'd need several lifetimes. But not all of it is good, no. The stuff that remains in the active repertoire tends to be the stuff that is really eternal.

It will be strange at first

You know the first 50 years of cinema history are full of masterpieces, naturally. But you feel it's harder to watch black and white films with mono audio (or even silent) and limited techniques than to watch contemporary high-res films with great sound and a contemporary cinematic language. However, the more you watch old Chaplin films, the less strange they seem to you. You can get used to them.

Something similar will happen when you listen to classical music. It will be quite uncomfortable at first. Each era will sound strange to you for different reasons. Each composer will feel strange for a while. In certain cases, individual pieces will cause bewilderment. But understanding will come. You must give it time.

Advice: quickly find something you like; stay with that composer, get to know him; start moving away from him both forwards and backwards in time, gradually.

My personal taste has gone through multiple revolutions. Musics that I detested on the first hearing, in time, became my favourite things in the world. This is a wonderful side of classical music: you do get smarter in your listening as you broaden your horizons.

You also lose a kind of temporal myopia. You gain perspective on your own time, by knowing what things were like in the past. For instance, children today are used to that robotic voice created by AutoTune, which enables people who can't sing to sell music, but sounds completely artificial. But they may react negatively to bel canto (opera singing), with its wide vibrato and focus on the loud side. But you see, there was never any whispering in opera, because the people at the back of the theatre paid for tickets too, so they deserve to hear something. Bel canto developed when electric amplification didn't exist, and it is the most natural way to sing such that everyone can hear you in the theatre. Those guys have to compete with an entire orchestra... For a while you may find it strange, but it's just one more thing that you can get used to, and then forever enjoy.

Listening is a skill to be developed

People used to know this: Listening is a skill that one develops through focused practice. People don't seem to know this anymore. That's because we live in a historically bizarre era: the era of entertainment, which only started in the 20th Century.

What is entertainment? Just cheap art. Leave your brain at the door, grab the popcorn, and forget the movie 5 minutes after it's over. Entertainment is the art that doesn't assume the responsibilities of art. Real art gets you thinking, it annoys you, it forces you to revisit the work, and often it forces you to change your opinion. Real art demands things of you, it gets you to do the work. Marketing departments feel that demanding things from the public is bad for sales, so they shun real art.

But how to develop your listening skill? Well, first of all, understand that...

Classical music is not background music

People use music as a sort of nice background noise for other noisy activities. Classical music is unfit for that purpose. It requires your attention and your silence.

When listening to classical music one absolutely must stop doing other activities that require the same neurons – the neurons used for language processing. No talking, no reading, no listening to audiobooks, no noisy environments, no programming, no scrolling on your phone. But yes, you can silently tidy up your room while you listen, it wouldn't likely interfere.

Why no noisy environments? Because classical music is normally expressive, in the sense that it has softer and louder parts. It is not abnormal, like the contemporary music that stays equally loud from beginning to end. Speaking softly versus screaming is an expressive device that should be used. The reason pop music doesn't... is because then pop music can be used as a distraction in noisy urban life: driving, commuting, working, etc.

If you use classical music like that, 2 things may happen:

An annoyance: you can only hear the loud parts, not the soft parts, because these get masked by environment noise.
A disease: by raising the volume you will absolutely, certainly, without any trace of doubt, lose your hearing. This is of course not limited to classical music. Any prolonged listening, to anything, at a high volume WILL cause hearing loss. It's no joke, just ask around yourself – you already know people who have hearing loss, but maybe they didn't tell you about it. Please do get informed about hearing loss.

Therefore, if you need background music, go ahead and listen to some sausage. Keep classical music for your personal temple.

Classical music is an abstract narrative

But why does classical music require focused attention? Because it's a plot.

Most classical music (but not all of it) is like a story. But the events in the story are abstract rather than specific. There are a number of stories that may fit the same piece of music. As if a story had been told, its events erased, but the feelings and atmospheres remained.

Classical music is also definitely not about lyrics, although these may be present. If you learn to listen to classical music, then instrumental music becomes meaningful to you. This is because music is the art of sound. Lyrics are an entirely separate art called poetry. If you currently dislike music without lyrics, then you'll be learning what the art of music is really about.

But the scope of the feelings and atmospheres portrayed is much, much larger. You see, contemporary popular musics have rather narrow expressive ranges. For example:

punk rock and hip hop focus on male self-affirmation: "I might seem menacing but I am right, here are the reasons". This is something that loses all its wisdom as soon as a guy has sex a few times. I mean, self-affirmation is kinda ugly, egotistical and unwise.
bossa nova expresses an atmosphere of hanging out at the beach, just looking at the babes, being cool and feeling lonely. Again, rather narrow, you see.
"romantic" music was always a current genre, gently talking about love. Somehow, since the ascension of hip hop, love is NOT a cool subject in pop music – violence is. Definitely a historic aberration.
country music: I am not sure how to characterize, but definitely narrow in technique and emotional range.

Now depending on the symphony you hear, it might last an entire hour and contain an entire world of different feelings and atmospheres, sometimes in violent contradiction, so you don't get bored. It doesn't have to be a symphony; "Pictures at an Exhibition" would be a great example.

You need a time machine

Classical music is not music of the past. It continues to develop today. But to understand what happened in one era you have to know the previous one.

Rock and pop are not like that, not so much. They are sort of a reset in history, even though they do have their origins in other genres. The point is, they do not refer to their history, you don't have to know it to appreciate it. (In a few pieces you do.)

Classical music is much more historical. You have to use your time machine, place yourself at the time when the piece premiered. By knowing the music that came before, you know what was new about this piece, and what was old. You know what the innovation was, you can guess whether they were shocked or intrigued or whatever. The experts will find historic documents and tell you how the piece was received, what the critics wrote etc.

What if you don't time travel? Then classical music will be a shallow thing to you – you will think it is "beautiful". That's how most people hear classical pops today. "That was nice", they say. But "easy listening" is something else. Classical music was never beautiful; it was frequently shocking, "uglier" and "uglier". The music of one generation was never appropriate for the next generation (and vice-versa), exactly like today. Something always dramatically changed. What and why? That's what is interesting.

You will never understand classical music unless you time travel.

The genius composers of the past were tremendous inventors, and as such, they did quite strange things. You won't be able to understand what and why unless you get informed. I promise you, if you research the times, the composer's life and the piece, you will gain so much understanding and fruition. Figuring out the thing goes hand in hand with liking the thing.

You need proper equipment

I tried to listen to some Tchaikovsky on my iPad. I love Swan Lake, but on tiny speakers... it's annoying. In fact, it was unbearable. The reason is, the lower frequencies are missing. You get only the highs, so the sound is strident and aggressive and honestly, quite ugly. I am not speaking as an audiophile here, I am talking about an extremely palpable thing that everyone will feel.

If you listen to classical music on crummy speakers, I am sure you will give it up.

Classical music requires at least a good pair of headphones. Not those that come with your iPhone, no. Not JBL either, those are bad and expensive. Get good headphones. Or a good pair of speakers, that's even better.

Apple and other streaming companies have started to push surround sound for music in general onto their public. That's bull. It makes music worse, except for a very small portion of the repertoire, which was composed with surround sound in mind. Most music needs stereo, that's the best way to listen to it. The rest is a disservice started by marketing departments.

The golden age of equipment for listening to music was until the 80s, because they sold "Hi Fi" (high fidelity), which means, the sound that goes out the speakers is close to the sound that got into the microphone. That's the goal. Back then, people would listen to music together and discuss it passionately. But in the 90s, Sony, Philips and all those companies stopped talking about Hi Fi and started pushing "hyper bass". That's the beginning of the bullshit era, in which stereos were replaced by home theaters with 5 tiny speakers and one woofer, which by the way, create a nightmare of cables. Suddenly no home had a place for listening to music, that place was now for watching blockbusters. This is also a part of why you were never taught how to listen to classical music, either by your parents or by some friend.

Classical music doesn't want "hyper bass". The sub-bass you can hear in certain disco music (which sounds like a very low electric hum is singing) is not desired. That's the lowest sound humans can hear, there's no need to go that low. If the bass is too strong it will mask other frequencies, hiding detail, which makes things worse. For classical music, normal, common bass is fine, as long as it is present.

Best to buy CDs

These days, on streaming platforms such as YouTube, even classical music often has its dynamic range compressed until it looks like a sausage and sounds at the same loudness level all the time. This is horrible. I am seeing videos so compressed that the softest passages actually sound louder than the loudest passages – this inverts the original meaning. Why are they compressing? Presumably so you can listen to the thing on the speakers of your phone, or in a noisy environment. We already established you must do neither.

You should be the happy owner of quite a few CDs, if only so you can feel the difference. CDs do not use that compression, in CDs you hear loudness differences pretty much like in the theater. The sound quality is also higher, unless you pay for a really good streaming service.

CDs often come with a booklet which, if well written, can be very informative about the repertoire.

If you buy a CD, you can own it for many decades. If you pay for a streaming service, you own nothing, and your favorite music can suddenly disappear.

Do not buy compilations. Ignore "Best of Mozart", that's nonsense. A Beethoven symphony is equally good from beginning to end, and you need to hear the entire thing because the entire thing has a certain meaning. Buy only complete works, never just extracts. Listen to complete works, from beginning to end.

Summing up this post

Here is the advice you learned in this article:

Preserve your hearing for the future.
The repertoire is more vast than you imagine, but the best stuff is well-known.
You will find each thing strange at first. That's normal, happens to everyone. Insist and pass the initial hurdle.
Listening is a skill to be gradually developed.
Classical needs your complete attention, it's not suitable for the background.
It usually tells an abstract story. Keep track of the plot.
Listen to complete works, not extracts or compilations.
You need to research time, place, composer and composition.
You need proper reproduction equipment.
Buy some albums.
Do not listen to any music in noisy environments.

That's a lot of advice, but not enough. More is coming.

No, music is not math

Nando Florestan — Sun, 14 Apr 2024 18:36:17 GMT

Stop saying "music is math".

How much mathematics was needed to develop musical instruments? Too much math, for sure! However, the same thing happened to cooking utensils, but no one says that "cooking is mathematics", or that "driving a vehicle is mathematics"...

Mathematics is the basis of all sciences. There isn't a single subject that doesn't have anything to do with math. Everything needs numbers.

When people think of the information in music, they basically think:

That the music runs through time, and time is measured, and most music uses fractions to specify durations -- a musical score is primarily a function of notes over time, and the duration of all notes is planned.
That at the same time, the melodic lines or musical events (tones) have another dimension: pitch, which is specified by the musical scale (do re mi fa sol la ti) or measured in Herz; for example, the most usual pitch of La is 440 Hz.
That tones simultaneously have intensity, which means that sounds can be accentuated or hidden, and they can be loud or soft. Intensity is also dynamic, meaning that a single sustained note from a brass instrument can, for example, start with an accent, drop quickly to a low volume, and then grow louder again.
That tones at the same time express timbre (sound color), which mainly depends on the musical instrument and the way it is played. Timbre is the most mysterious and difficult parameter to measure, but science does measure it... using mathematics.

In short, musical notes have 4 parameters: duration, pitch, intensity and timbre. As a result, music also has more or less the same 4 parameters. During a piece of music, these parameters change independently and constantly, and great composers establish relationships between all these parts.

But, so far, we have only concluded that the technique of music depends on numbers. Yes, the basic parameters can be expressed in numbers -- you count the beats while you play: 1, 2, 3, 4, repeat. However, the message is always human and often takes place in another dimension.

Hmm, a very mathematical composer... Have you ever heard the music of Iannis Xenakis? He developed a compositional technique called stochastic music, which uses statistics during composition. He was an architect and he wrote computer programs to help him compose. In this case, I agree that music has a deep connection to mathematics. But the computer didn't compose -- Xenakis did.

By the way, you've probably never heard his music; you probably even dislike such musical modernism; so why do you keep saying that music is math?

Most composers aren't mathematicians, they do little more than counting.

Let's take another example: Frédéric Chopin. The first thing about Chopin is that his music contains a sentimentality that is undeniable. Some Chopin music conveys the sadness of sadness, the misery of misery. To achieve his unique nostalgic effect, he developed a combination of existing techniques (such as pedal tones, chromatic harmonic changes etc.). But even in the happy pieces there is a tenderness that is not found in other composers.

Of the four parameters, the fourth is straightforward in Chopin: he always uses the piano. Timbre is only a consideration in piano solo music inasmuch as the piano is a symphonic instrument, capable of expression comparable to an entire orchestra, under just two hands.

Chopin liked 4 things: the piano, Bach, opera, and Polish music.

Chopin developed the piano technique, but he did not do it alone. He knew pianists and composers of the time, who also demanded more than Beethoven in their scores.
From Bach he learned that grace in the art of music lies in simultaneity -- simultaneous melodies and the relationships between them. The polyphony of the most brilliant composers is often more complex. This is the case in Chopin. But some listeners don't realize that -- they are not aware of the complex polyphony hidden in Chopin's "accompaniment", which is much more than accompaniment. A good pianist will make this polyphony clear and a good listener will know how to hear it.
From opera he loved bel canto melodies. He especially appreciated Vincenzo Bellini. Opera is the most popular genre of classical music. If polyphony is a complex aspect, the melody of the opera is what every music lover does understand. It is what the people take away with their whistles. Melodies on the piano must be played giving the impression of a singer singing. And the listener can help with his imagination...
Chopin was a nationalist and in exile he expressed his love for his country in his polonaises and mazurkas, which actually contain an influence of the Polish music he heard as a child.

The rhythm in Chopin is completely new. If you think that time in music is exact like in mathematics, you are squarely mistaken. The pianist must play romantic music like a human, not like a robot. This means that time is not as written, it is divined by the interpreter, who speeds up the beginnings of phrases and lengthens the ends of phrases, creating a kind of breathing between sentences, about which mathematics has no idea. A rhythmically straightforward rendering would strike anyone as false and quite unbearable.

And that is how playing Chopin can become a national sport. But they play Chopin from memory. Above I said one can count the beats when playing music. Well, maybe pianists counted a bit while learning certain pieces... but not when they finally play it live from memory. In that moment, they make sure they are free. So the notated durations are an instance of Wittgenstein's ladder. You use the ladder to climb the wall, but when you get up there, you throw the ladder away and become free.

In the mazurkas, time is very important. The rhythm of the mazurkas is strange because the second beat often lasts a little longer, creating a feeling of limping. In fact, changes in the rhythm are much more complex than the generalization I just made. And this is not written in the scores. Even Chopin's waltzes need this freedom in the rhythm, which is greater than in other waltzes. If you gave the sheet music to an alien, he wouldn't understand it. Music is part of cultural and human tradition.

How many times did I need numbers to talk about the most basic traits of Chopin's art? Zero. He counted triolets and polyrhythms like any composer, but that's not important. In both Xenakis and Chopin, what we perceive are two very different worldviews. Technique is very important in art and it always depends on mathematics, but worldview is just as important. They had sincere, new, important things to say.

The content of music is certainly not mathematics. The languages of music are many, so they cannot be mathematical either. They are systems developed by entire peoples, each in a different culture.

The art of music is the most profound and powerful art that man has devised. It can tell abstract stories and will move you to tears if you pay attention to it. Chopin and Xenakis are two of the most innovative composers. Chopin's entire second sonata talks about death, without a single word, of course. The fifth Polonaise is a nightmare of war with an idyll in the middle to give you a sense of what the war destroyed. But being an abstract story, maybe it's something analogous to war. The Berceuse is hypnotically tender, with flourishes like fireworks made of petals… I could go on. If an alien asked me what it is like to be human, I would give him Adam Harasiewicz's recordings of Chopin.

When you say that "music is mathematics", I wonder if you have had any contact with its true power. You should listen to composers who create whole worlds, like Mahler...

Music is the most striking testimony to the creative freedom of man; in that city, math is only one of the bricks.