An Article To pUt To Rest Discussions of "Tonality" and its Condition

Great post Juji!

People aren't really taught to like tonality, though, or most other forms of music. They're taught to appreciate them. There's a difference.

Not a strict one. There's no such thing as a truly independent taste. It strongly depends on our environment, on what we are exposed to, how we are exposed to it, and how the people around us display their tastes.

We might not be explicitely taught to like tonality (maybe to appreciate it though, as you mentioned), but similarly to picking up a language simply by being surrounded by it, we pick up our own cocktails of tastes, which we later develop, modify, expand, or curtail.

It depends on how you are qualifying artificial. Any procedure of sound manipulation creates artificiality. The real question is what qualifies as "more" artificial. Tonality is a template for composers to intuitively create (assuming they have good ear training) without using procedural methods like math to produce a work. Atonality is more rooted in the mathematical process of creating relationships among different sonorities. What is the more artificial process? The humanistic, intuitive process, or the mathematically-based process?

Sorry, but that's nonsense. Atonality isn't any more mathematical than tonality. I have no clue where you got that idea from. A Palestrina motet, or even a Bach fugue is much more algorithmic and "mathematical" than almost all of Sch

I am afraid that "tonality" as we know it lasted a bit more than 300 years. That's a very short time span from the time we know the first instrument was made (about 10,000 bc) to today. And if you dare look at the amount of music written in the last 100 years, including the music before "tonality", I would say that most music written is in fact, non tonal. Furthermore, the concept of "tonality" itself changes in time. Monteverdi's "tonality" (or "modality") is different than Bach's, which is different than Mozart's, which is different than Brahms', which is different than Scriabin's, which is different than Berg's, which is different than John Adams', which is different than Laurence Crane's. So which "tonality" are we talking about?

So which Atonality are we talking about? Schoenbergs? Stravinsky? Webern? Berg? etc. etc.

I challenge you to find me ONE concert that is not dedicated to a particular composer/era or by an ensemble dedicated to a particular composer/era, that does NOT contain at LEAST one piece by a 20th/21st century composer, whether that is Bartok, Schoenberg, Shostakovich, Boulez, Messiaen, Birtwistle or whatever. Because I have personally been to none such concert.

Here ya go:

New York Philharmonic: Bach, Strauss and Pictures at an Exhibition

New York Philharmonic: Mussorgsky's Pictures at an Exhibition

New York Philharmonic: Muti, Uchida, Ravel and Schubert

And this is just ONE major orchestra in the US. You must not have been to many concerts.

Birtwistle, Boulez, Stockhausen, Ligeti, Xenakis, Laurence Crane, Elliot Carter, Paul Whitty - they all don't "appeal to the audience's needs". Yet they make a living out of composing.

No. They appealed to the Avant-Garde audience. That's how they became famous and earned respect. Today, that audience is no longer the mainstream being replaced by one that loves tonal music rather than atonal avant-garde.

Sound is an entity. Sound is waves, transferred in media such as water, air, or solids. Music is not an entity. Music is a concept, it's organised sound in time/space. What is pleasing to the ear is totally subjective, apart from some (perhaps) biologically predispositions we have, like not liking really high-pitched sounds, or really really loud and sharp sounds. I assure you most contemporary music sounds nothing like that. And if we are talking about music, we are talking about sound. Music is all about sounds. Music IS sounds, and that's a de facto definition.

I'm confused on what this has to do with anything. I was responding to something else, not making a statement.

Wrong. Again, it all depends on your definition of "tonal", because John Adams' music is "tonal", and so is Laurence Crane's, but they are not in any way "tonal" in the way Mozart's music is. And I doubt I'll hear a lot of works in the style of Mozart played in a "contemporary concert". Have you?

When I say tonal, I mean not atonal, which is the absence of a key center. Yes, there are gray areas and I certainly don't mean tonal as in Mozart. But for the sake of discussion, "tonal" means with a definable key center just like common-practice.

Um, wrong again. "Neo-romanticism" is simply a label applied to the small number of composers who have returned to writing in a more romantic idiom, yet still contemporary. Take Penderecki's later works. Take some of Rihm's works (like his "Mit Geschlossen Mund", among others). They are "neo-romantic". John Williams' works are not "neo-romantic". They are mere compositions "in the style of" (and in fact, a lot of it sounds like Holst, Sibelius and a few others), most of which are composed for the purpose of dressing a commercial (usually Hollywood) film in nice, cozy music that the people will have little trouble remembering, and will enhance their "emotional" approach of the movie by "enhancing" their emotions at the moment. Occasionally, use of leit-motifs might contribute to preparing moments, or having the music develop with the characters.

Don't confuse commercial music, which by definition is music that must be appealing to as many people as possible, thus has no interest in continuing a tradition in artistic music but only tries to sell. Thus, commercial music is TRYING to go by people's tastes, instead of SHAPING them, which is the case with all of classical music up to the beginning of the 20th century (with the invention of recording media, and the division between "popular" and "concert" musics).

My point was that John Williams brought Romantic music to the mainstream, as in not constrained to classical music circles. And why would Mr. Lucas want Romantic-sounding music in the first place? Because people like it, prefer it to all the atonal music.

This also brings up another point on if it is the composer's purpose to shape other's opinions or not. One could argue both ways. I find myself on both sides of the fence. I think that a composer must acomplish both. They must speak a language that the audience understands, but make them feel something that they haven't felt before. True masterpieces can make this happen IMHO. Yes, Williams' music was commercial, but it was still able to shape what people felt. Everyone who watched Star Wars was able to realte to the music, and the series' themes became some of the most well-known themes of all time, even over classical themes. Romantic music, or at least the "style" became mainstream and it hasn't stopped since. Obviously, since this is a very small composer's circle, most will argue for the now old Avant-Garde way. But that's not what the audence want's to buy tickets for (except, of course, for the limited amount of people that are in those same Avant-garde circles).

BTW, (on a tangent), the division between "Popular" and "Proper" music has always existed. Always. It's just seems more apparent nowadays because Proper music is in decline. But know that in the 1800s, Chopin and Schubert where not the tunes common folk was humming in their heads. At least in the old days there was integration between folksong and classical. Now, this is sadly rare.

On the other hand, can you please provide evidence/proof/facts/support for these claims which you've made?

This is next on my to-do list.

I feel really sorry for you.

No you don't. Don't lie. BTW, what is with everyone calling me an immature teeanger just because I have a different point of view? Is thinking outside the confines of your little boxes so hard to do?

Quote:

Originally Posted by jujimufu

I challenge you to find me ONE concert that is not dedicated to a particular composer/era or by an ensemble dedicated to a particular composer/era, that does NOT contain at LEAST one piece by a 20th/21st century composer, whether that is Bartok, Schoenberg, Shostakovich, Boulez, Messiaen, Birtwistle or whatever. Because I have personally been to none such concert.

Tokke:

Here ya go:

New York Philharmonic: Bach, Strauss and Pictures at an Exhibition

Let me paste that concert's programme:

J.S. Bach:

Brandenburg Concerto No. 2

Szymanowski:

Symphony No. 4 for Piano and Orchestra

R. Strauss:

Burleske for Piano and Orchestra

Mussorgsky/ Ravel:

Pictures at an Exhibition

Richard Strauss? 20th c. Szymanovsky? 20th c. Ravel? 20th c.

Note: the Strauss piece was from before 1900, but he is definitely a 20th c. composer, which is what Juji was looking for.

And, for the record, this "neo-romantic" thing was not really the word I was looking for. Neo-romantic was the early 20th century composers like Barber, Holst, Britten, Copland, etc. I mean to say post-modernism, the movment away from modernist music starting around the 1970s until the present.

LOL Ultra fail. Mega fail.

Let me paste that concert's programme:

J.S. Bach:

Brandenburg Concerto No. 2

Szymanowski:

Symphony No. 4 for Piano and Orchestra

R. Strauss:

Burleske for Piano and Orchestra

Mussorgsky/ Ravel:

Pictures at an Exhibition

Richard Strauss? 20th c. Szymanovsky? 20th c. Ravel? 20th c.

Note: the Strauss piece was from before 1900, but he is definitely a 20th c. composer, which is what Juji was looking for.

Strauss was definately NOT a 20th century composer, at least in the style of one. He was the last great romantic, always was. And last I checked, Ravel did on write Pictures. As for Szymanowski? Ah, well. I tried.

EDIT:

before going on to my post, a swift correction to Justin: Richard Strauss was born 2 years after Debussy. Debussy is a 20th century composer. Strauss died just a few months shy of 1950. Strauss is most definitely a 20th century composer, regardless of whatever idiom he chose to compose in.

Justin, please, let us clear up a rather MAJOR error you keep repeating:

The VAST majority of John Williams' musical output is NOT tonal.

The VAST majority of John William's musical output is not "neo-romantic".

And MOST of John Williams' concert works (barring the various "fanfares" and pieces composed for official occassions) are far from being neo-romantic, and most are only quite loosely tonal.

As for your statement that

John Williams brought Romantic music to the mainstream

well, that's just plain silly. The score for Gone With the Wind was QUITE popular in its time.

You DO realize that maestro Williams had been writing music for film and television for more than 20 years prior to meeting George Lucas?

You DO realize that maestro Williams was carrying on a long-standing tradition of "romantic" music for film that has existed since the beginnings of film-scoring? In other words, he didn't create the "neo-romantic" film scoring movement.

Take a peek at John Williams' 1st Violin Concerto, or his Concerto for Flute strings and percussion. Or even at any of his recent works, like the Cello Concerto. They can HARDLY be described as "neo-romantic" works.

Even in his film music, Williams used immense quantities of thoroughly non-tonal (in other words "not neo-romantic") material. I can safely state this, since I have almost all of his musical output on CD. Yup, the main theme is often a lush, neo-romantic pastiche... but the rest of the score more often than not deviates from that main theme in rather drastic measure.

ALL of the above, dear Justin, is why people are telling you you know nothing. The world did not start with Star Wars: the Phantom Menace.

EDIT:

before going on to my post, a swift correction to Justin: Richard Strauss was born 2 years after Debussy. Debussy is a 20th century composer. Strauss died just a few months shy of 1950. Strauss is most definitely a 20th century composer, regardless of whatever idiom he chose to compose in.

Justin, please, let us clear up a rather MAJOR error you keep repeating:

The VAST majority of John Williams' musical output is NOT tonal.

The VAST majority of John William's musical output is not "neo-romantic".

And MOST of John Williams' concert works (barring the various "fanfares" and pieces composed for official occassions) are far from being neo-romantic, and most are only quite loosely tonal.

As for your statement that

well, that's just plain silly. The score for Gone With the Wind was QUITE popular in its time.

You DO realize that maestro Williams had been writing music for film and television for more than 20 years prior to meeting George Lucas?

You DO realize that maestro Williams was carrying on a long-standing tradition of "romantic" music for film that has existed since the beginnings of film-scoring? In other words, he didn't create the "neo-romantic" film scoring movement.

Take a peek at John Williams' 1st Violin Concerto, or his Concerto for Flute strings and percussion. Or even at any of his recent works, like the Cello Concerto. They can HARDLY be described as "neo-romantic" works.

Even in his film music, Williams used immense quantities of thoroughly non-tonal (in other words "not neo-romantic") material. I can safely state this, since I have almost all of his musical output on CD. Yup, the main theme is often a lush, neo-romantic pastiche... but the rest of the score more often than not deviates from that main theme in rather drastic measure.

ALL of the above, dear Justin, is why people are telling you you know nothing. The world did not start with Star Wars: the Phantom Menace.

Ah, yes. Well, errors be in my ways.

Anyways, I think Juji's point was rather this: Atonality is unspecific. All it means is the absence of tonality. By itself, it implies no rules, no character, no techniques. Common practice tonality however is something very specific, something which has been created and developed by men as a cultural phenomenon. That's why it's artificial. It is human-made. Atonality however isn't something that is "made" at all. It is simply a name of the removal of something that was made, i.e. tonality. It does not mean the avoidance of tonality (in which case it would be something artificial again), simply the absence of it. The fact that many "atonal" composers consciously avoided tonal impressions is irrelevant to this, as it has nothing to do with the nature of atonality, but with the historical context in which it first appeared.

You seem to be minimizing historical context in atonality while emphasizing it as relates to (common practice) tonality. Your argument on how atonality is not artificial because of its nature, insofar as deeming the process irrelevant, can as easily be applied to tonality.

...is thinking outside the confines of your little boxes so hard to do?

Wait....you're the one telling people to think outside the box?! :blink:

A BOX!! YOU LIVE IN ONE!!

You seem to be minimizing historical context in atonality while emphasizing it as relates to (common practice) tonality. Your argument on how atonality is not artificial because of its nature, insofar as to deeming the process irrelevant, can as easily be applied to tonality.

I think we're getting into a confusion about the definition of "atonal" there. If you understand "atonal" as a historic term, i.e. as a description of a certain kind of music written by proponents the Second Viennese School and successors, sure, it's quite as specific as any tonality. In that case avoiding tonal implications -is- part of it.

If you merely see it as the absence of any specific tonality (as I did in my post), it's something entirely different. The problem is that those two aspects are often confused. That's why I mentioned "historical context" as an explanation why the proponents of the Second Viennese School and their successors consciously avoided tonal implications despite this being no part of atonality (in the sense of non-tonality) itself: Simply to clarify that I do not talk about atonality in the sense of this specific "style" here.

Avoiding tonal implications is not part of the process that led to non-tonality, but something which came on top of it, as a further emphasis. Even without avoiding tonal implications the music would be "non-tonal", simply for having no specific tonality. A selection of random pitches is "not tonal". Noises in nature are "not tonal".

I've never said that what Sch

The earliest atonal works by Schoenberg and others was largely a result of the massive chromaticism that was prevalent at the end of the 19th and start of the 20th centuries. I've seen some articles that would go so far as to say that the move to atonality and eventually serialism came about due to not being able to categorize a vast majority of chordal progression. Having said that, however, one could make a fair argument that the chief characteristics of atonality are - in fact - a clear reaction to the late Romantic and perhaps even the impressionist style.

These discussions of atonality vs tonality - and here I refer to the last 3 threads on this topic - are, in my opinion, rather ridiculous. I think time itself has shown that various styles of composition can and should exist simultaneously. Hell, look at the vast repertoire of classical music from the last 300 years - really look, you'll see things you never knew existed utilized in periods that these very techniques were said not to have been around. I can think of hundreds of pieces just from the baroque era alone that were not explicit in their tonal underpinning.

In the end, I think most important is that composers continue to write music - not just music that is pleasing to the ear... or music that tries to create a sense of being locked in hell and trying to find away out. After all, we are first and foremost writing music - let the music theorists do the analysis.

This is just to clarify Gardener's recent post, in which he made a very good point. Schoenberg, Boulez, Xenakis, Stockhausen, etc. DID in fact create music based on artificial rules. Nobody denies this. Clearly, the rules used in their mature compositions generally didn't imply tonality. However, atonality itself does NOT imply any rules at all. The point of this thread is just to say that the artificial rules of common-practice tonality are no better than any other rules! People such as JT are losing sight of the fact that this discussion is not about how atonality is better (because we all know it isn't), it's about how tonality is NOT better.

I think we're getting into a confusion about the definition of "atonal" there. If you understand "atonal" as a historic term, i.e. as a description of a certain kind of music written by proponents the Second Viennese School and successors, sure, it's quite as specific as any tonality. In that case avoiding tonal implications -is- part of it.

If you merely see it as the absence of any specific tonality (as I did in my post), it's something entirely different. The problem is that those two aspects are often confused. That's why I mentioned "historical context" as an explanation why the proponents of the Second Viennese School and their successors consciously avoided tonal implications despite this being no part of atonality (in the sense of non-tonality) itself: Simply to clarify that I do not talk about atonality in the sense of this specific "style" here.

Avoiding tonal implications is not part of the process that led to non-tonality, but something which came on top of it, as a further emphasis. Even without avoiding tonal implications the music would be "non-tonal", simply for having no specific tonality. A selection of random pitches is "not tonal". Noises in nature are "not tonal".

I've never said that what Sch

The earliest atonal works by Schoenberg and others was largely a result of the massive chromaticism that was prevalent at the end of the 19th and start of the 20th centuries. I've seen some articles that would go so far as to say that the move to atonality and eventually serialism came about due to not being able to categorize a vast majority of chordal progression. Having said that, however, one could make a fair argument that the chief characteristics of atonality are - in fact - a clear reaction to the late Romantic and perhaps even the impressionist style.

These discussions of atonality vs tonality - and here I refer to the last 3 threads on this topic - are, in my opinion, rather ridiculous. I think time itself has shown that various styles of composition can and should exist simultaneously. Hell, look at the vast repertoire of classical music from the last 300 years - really look, you'll see things you never knew existed utilized in periods that these very techniques were said not to have been around. I can think of hundreds of pieces just from the baroque era alone that were not explicit in their tonal underpinning.

In the end, I think most important is that composers continue to write music - not just music that is pleasing to the ear... or music that tries to create a sense of being locked in hell and trying to find away out. After all, we are first and foremost writing music - let the music theorists do the analysis.

Here, here! :thumbsup:

*Justin Tokke leaves to work on his Symphony*

Clearly there are many definitions of 'atonality', and from your admission that artificial devices are an integral part of the process, it seems that distinction between tonality vs. atonality does not lie in the artificial nature of music making.

It certainly doesn't, since making music is by definition artificial, not matter what music it is. But that doesn't mean that atonality itself is artificial.

By the way, I never argued that the main distinction between tonality and atonality is that one is artificial while the other isn't. That's certainly not the main point of this distinction. But there was a claim of naturality in tonal music, which was disputed by Jujimufu by stating that if anything should be considered more natural/neutral then it would be atonality. I merely supported this proposition.

Personally I don't care in the least whether a specific attribute of art is natural, artificial or whatever. It doesn't mean anything, and it especially doesn't mean anything qualitative. Writing one specific kind of music doesn't make your process less artificial, nor the music that results out of it, nor is this artificiality anything bad. It was merely an argument to point out the absurdity of claiming "tonality is natural, atonality is a synthetic aberration".

"Behaviors related to [the Superiority Complex] may include an exaggeratedly positive opinion of one

This could be applied to almost every person on this forum including you and I.

Haha.

Point proven.

It certainly doesn't, since making music is by definition artificial, not matter what music it is. But that doesn't mean that atonality itself is artificial.

Musical tone in itself is not artificial. Musical tone is evident all around the natural world: Birds chirping, sea waves pounding the shore, wind rustling through the trees, objects screeching through the atmosphere upon reentry (witnessed in numerous reports of various shooting stars), and even the processes of plate tectonics produce very low resonances that are detectable on today's modern seismographs. The only thing that is artificial in music is man's means to organize and harness the use of tone and sound (composition).

TO go further on this, and bring this towards the discussion... it is one's cultural taste and discovery that determines what one likes or dislikes. If one has a very adventurous taste.. there is the more avant-garde in all fields of art (from viewing urinal art to listening to a person bowing a bicycle wheel) - to each his own.

Anyways, I think Juji's point was rather this: Atonality is unspecific. All it means is the absence of tonality. By itself, it implies no rules, no character, no techniques. Common practice tonality however is something very specific, something which has been created and developed by men as a cultural phenomenon. That's why it's artificial. It is human-made. Atonality however isn't something that is "made" at all. It is simply a name of the removal of something that was made, i.e. tonality. It does not mean the avoidance of tonality (in which case it would be something artificial again), simply the absence of it. The fact that many "atonal" composers consciously avoided tonal impressions is irrelevant to this, as it has nothing to do with the nature of atonality, but with the historical context in which it first appeared.

See, this is a bullshit argument at its core. Sorry Gardener, but it is...

First and foremost, the tonal language makes it possible for a composer to cohesively combine timbre, melody, harmony, rhythm, and formal elements of music without an instrument or orchestra in front of them to guide them in the process. Before electronic media, it was impossible for composers to hear all of these elements as they composed, so they HAD to have a method that would allow them to intuitively create music without a full ensemble present. The piano was a tool for some (Beethoven). Others only needed a quiet place by the fire (Mozart). So, your previous statement that composers were more strictly committed to prescribed procedures for creating music is just patently false. They composed tonal music in a tonal system they knew would aurally reinforce their ideas, and they composed mostly from an intuitive process knowing specifically what it would sound like before writing the work (if not in part than as a whole work, as sketches tend to explain this).

Essentially, what Juji is saying is that Atonality is natural or organic, where Tonality is artificial, because no pre-designed template exists that would influence the creation of the music. The problem is that almost all atonal works have been composed with the specific intent of avoiding tonality, which is reflected not only in the methodology but also in the very origins of the style. You'd like to discount that as not relevant, but it all comes back to the interference of the composer on sound - the manipulation of sound to create art. In tonality, the process was based on an aesthetic of dissonance resolving to consonance. In atonality, the process is based on anything but the aesthetic of dissonance resolving to consonance.

Juji is trying to create a catch-all for atonality that he applies very, very narrowly when it technically applies, inversely, to all music - no work of music is completely organic or even close to it. The absence of tonality does not change the fact that a manipulation of sound is taking place. Either way, the composition of a work of music, unless aleatory (which is still questionable due to queues and other such directions of the composer - still a manipulation of sound, at least at the temporal level), is a process of manipulating what is otherwise organic sound. Whether tonal or atonal, neither method is any more or less organic.

And forget about mathematics in atonal music, even though numerology and simple mathematical organizing methods play a HUGE role in composing Atonal works (Matrices, Set Theory, Hansen Analysis, and so on). I'll try to keep it as basic as possible. The absence of tonality does not indicate an absence of superficiality or the presence of organic, unfiltered sound. The very use of notes, rhythms, textures, and time itself imposes artificiality on sound. No system should be claimed to create more or less artificiality, lest we bring in a metaphysical debate about what is "real" and what is "artificial."

Like I said, it's a catch-all position with a paradox waiting for you around the corner. Who has the time for that, anyway?? :cool:

P.S. It appears others have responded and this post might appear to be cumulative. Oh well.

Feldman :X

Feldman :X

pardon my newbness on this, but... wut!?!

Nirvana - good catch on my title. Still think this is a good article. Hmm, interesting stuff about atonality versus tonlity - wonder if anyone has heard the work of the Rennaissance composer Solage who was under the patronage of Duc de Berry - would easily be at home in Wesrtern European contemporary classical music circa 1965. Also, Machaut - is that tonal? I read somewhere late 19th century music theorists derided Machaut as inept, primitive music.

Ah well, as expected, we've fallen into the quagmire of polemic. I thought this would just generate 2 or 3 pages and be done. Well, I appreciate everyone's passion - that is a healthy sign for us all despite our disagreements.

pardon my newbness on this, but... wut!?!

That's in reply to Antiatonality's post.

Feldman composed music that cohesively combines timbre, melody, rhythm and formal elements without an instrument or orchestra in front of him to guide him. And his music is neither tonal nor "atonal" in the Schoenberg/Webern sense. It is music which focuses on sounds. Feldman finds a few sounds he likes, he plays around with them, he revisits them, he places them in different context, he places them in different places in the manuscript paper. He squeezes them in in measures of different time signatures.

I never claimed that "atonality" is natural or organic.

Tonality is artificial by its definition. If there were no humans, there wouldn't be any tonality. Arguably, there wouldn't be any music either, but that's another topic.

Still, "atonality" is a state. "Tonality" is an artificial system which helps organise sounds in time/space according to various rules, thought to be derived by divine means and aesthetics, which hold none to little value over time. A read of James Tenney's "A History of Consonance and Dissonance" shows very simply how these two notions which seem to define "tonality" have changed dramatically over 100 or 200 years in history, and how what people once thought to be absolutely divinely perfect consonance became dissonance, and how dissonances became consonances, how their importance switched places etc.

Above all, you're contradicting yourself:

Juji is saying is that Atonality is natural or organic, where Tonality is artificial, because no pre-designed template exists that would influence the creation of the music.

aaaaaaand...

Juji is trying to create a catch-all for atonality that he applies very, very narrowly when it technically applies, inversely, to all music

So which one of the two do I do? Do I try to imply that "atonality" is natural/organic (which I never claimed, but assuming that that's your interpretation of my words), a state in which no pre-designed template exists that would influence the creation of music? Or do I try to imply that atonality is only Schoenberg's few "atonal" works and I don't try to extend this to a larger degree when it can actually be applied to all music? (which I seem to be doing above).

As far as I know, Schoenberg didn't use anything closely related to Matrices, Set Theory, Hansen Analysis, Imaginary Numbers or Googol's when it came to composing his Pierrot Lunaire, or his piano pieces Op.19 . Cage never did anything even remotely related to mathematics like that. Neither did Feldman. Or Earle Brown. Now that I think of it, very rarely are Boulez's later works related to mathematics like that. Or wait, are Elliot Carter's works mathematically driven? Or Laurence Crane's? Or Birtwistle's? Maybe I'm wrong, and Ligeti did calculate every single one of his notes with a pocket calculator after all.

That association with maths confuses me a bit. It seems more of a thing of people like Xenakis, Stockhausen and spectral composers (although I wouldn't group all of them in the same group). Please don't try to apply this to as wide a group of compositions of "atonal", because as you said above, there's no point if trying to prove that much "atonal music" is written using mathematics, simply because atonal music is a LOT of things.

I therefore disagree with you and agree wholeheartedly with Gardener.

Ah, thanks. I wasn't sure what you were referring to... For some reason, though, my mind went back to a line from a TV show (lol).