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I haven't quite understood a few things about twelve-tone compositions and I hope somebody here might shed some light on this subject.

As per wikipedia, these are the postulates:

1 - The row is a specific ordering of all twelve notes of the chromatic scale (without regard to octave placement).
2 - No note is repeated within the row.
3 - The row may be subjected to interval-preserving transformations—that is, it may appear in inversion (denoted I), retrograde (R), or retrograde-inversion (RI), in addition to its "original" or prime form (P).
4 - The row in any of its four transformations may begin on any degree of the chromatic scale; (...)

First of all, how does it work when there are multiple instruments? As per postulate 2, is one instrument supposed not to play a note if another instrument has already played it, or does each player have his own row to care about, without regard to the other players? Or only certain instruments are subject to respecting the row while other instruments can accompany with normal harmony?

Secondly, wikipedia also states that "postulate 2 does not mean, contrary to common belief, that no note in a twelve-tone work can be repeated until all twelve have been sounded"- therefore, what does postulate 2 mean? I suppose this might refer to polyphony as per my first question, so different rows are layered accross different instruments: two musicians are playing different rows at the same time, so the same note might appear twice in total before the end of the row, although only once in each row...

Third: the four postulates don't seem to imply that the notes should not be ordered in such a way to suggest a certain chord, so it wouldn't be a problem at all to start a row playing a C major arpeggio followed by a D minor arpeggio, is that correct?

Final point: wikipedia says "Although usually atonal, twelve tone music need not be". I thought twelve-tone serialism had been invented with the purpose to give a formal framework to create atonal works. If that's not the case, what was it created for? And how does one go about creating "real" atonal pieces? Just whole tone scale, diminished and augmented chords from start to finish?

I hope I didn't drag on too much... Thanks in advance to anybody who might help!

It was developed for the purpose of creating atonal music, and it would violate that spirit to create a tone row that used a major or minor arpeggio. Unisons, semitones, tritones, and major 7th intervals are consistent with the paradigm. Octaves, 5ths, major 3rds, and major 6ths are not. Minor 3rds and whole tones would be used sparingly.

Berg of course ignored this - the row for his violin concerto has G minor, D major, A minor and E major triads, finishing with 3 whole-tones.

Yes and other composers used something
like a musical theme as the series.Schoenberg in his 12 tone music used it strictly. (probably Webern as well)
However Schoenberg's incredible rhythmic sense related to his structures are probably what can make his music interesting .🤔
(well bearable then)

First i think wiki is here talking about classical serialism as designed by Schoenberg. Berg was a little more loose on these principles. Triads or other usual tonal melodic patterns are to be avoided but it is not necessarily always the case. Tonal harmony implies much more than just occasional triads. For example Berg used triads in his row in the lyric suite. Atonal simply means there is no set tonal center, so one should avoid cadences and harmonies built on triads such as I and V. In effect serialism does lead to atonal compositions.

For your other questions, the row once defined can be used in any form using transposition to one of the 12 tones and the prime, inverted, retrograde and retrograde inverted form. Thus in total there are 48 forms that can be used for a given row, often times it is plotted into the twelve tone matrix. Note also that a row can be written in its enharmonic form as well.

Serial music is usually analyzed by looking at the structure of the row and its intervals. For example one specific row form is the one that is called the all interval row such as the one used again by Berg in the same lyric suite. A further analysis can be carried by looking at subsets pitch class sets.

When writing for multiple instruments, each one is a voice (or several in the case of piano). Each voice can use any of the 48 forms starting at any given moment. You can thus create imitative or counterpointed music.

The rest of the musical components are free including rythmic patterns, accentuation, ....unlike what you would in in integral serialism with people like Boulez.

thank you very much all for your contributions. I'm very satisfied with all the feedback I received. I've actually also opened a discussion on reddit with a couple of VERY thorough responses, which more or less go in the direction of your replies. You might have a look here if you're interested. Cheers!
https://www.reddit.com/r/musictheory/comments/k4vvvp/about_twelve_tone_serialism/

One thing is that some composers do not necessarily comply strictly with all the rules, just like Berg. The example of Luigi Dallapiccola is not strictly compliant with serialism as you should not go back repeating a note after you already played the next note in the serie, but you can indeed repeat the same note as many times as you want, until you move to the next note.

Where it gets more complicated is when you have at the piano multiple voices, some of them appearing only to play 1 or 2 notes occasionally.

Then serialism with twelve tones is atonal in nature (at least i dont know any composition of usual complexity that would be strictly tonal per classical definition and would be also strictly compliant with the serial rules) but you can certainly write tonal music with non serial methods using the twelve tones in a free manner.

Best way to familiarize yourself with serial music is to analyze the suite opus 25 by Schoenberg. You will see by yourself how it works. I have played it, even though i am less than a fan of dodecaphonism.

It is probably not a surprise to some that I'm not much of a fan of 12-tone music either, but once upon a time I was assigned a project for a music composition class to compose a 12-tone piano solo piece. It is an interesting exercise to construct a tone row with enough melodic strength to use as the basis for a composition.

A mathematician would note that there are 12! (12 factorial) = 479,001,600 possible tone rows, including those that have tonal intervals embedded.

I've always been a fan of Webern and Berg, and I love Schoenberg's piano concerto.

Like tonal music, it depends on the composer and the piece.

Is serialism music? Is serialism garbage? The answers to both questions are the same.

Replace "serialism" with "Chopin", and the sentence would be equally true.

lol