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A few days ago my instructor taught me the basis of the twelve-tone system and taught me how to write a dodecaphonic row and its retrograde, inversion, and the inversion of retrograde. Now I have no more classes with him for a few months and he is not reachable. One thing I am confused about is that he said in the basic row there shouldn't be a formation that represents any known chords. What does that mean? How can I write the simplest dodecaphonic row that follows this principle? Does that mean that each time I write a dodecaphonic row I should check each three adjacent rows and see whether they form a 'known' chord? What does that 'known' chord refer to?

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It sounds like your teacher is asking you to write a row without any inherent tonal qualities. We're not looking at three adjacent rows here, but rather tones that are next to each other within a single row. Take, for instance, this 12-tone row from Berg's Violin Concerto:

enter image description here

Notice that the first three pitches spell a G-minor triad; pitches 3--5 spell a D-major chord that becomes a D7 when you include the next pitch, and so the first 6 pitches spell i--V7 in G minor! (You can find more triads to, if you keep looking.)

So when your teacher tells you not to represent any known chords, he's telling you not to do what Berg did! Instead, he's asking you to do something more like Webern did in his Op. 30 Variations:

enter image description here

Note that no adjacent pitches create a triad; this is what you're being asked to do.

(PS: You can ignore the other markings on that Webern image!)

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There is nothing wrong with triads in a tone row, and it certainly is not against the rules. Having said that, perhaps your teacher wanted you to experiment with less standard melodies and harmonies. Keep in mind that because the tone row includes all twelve pitch classes, it can take a lot of work to avoid jarring harmonic transitions if you do employ the common minor and major triads. And yes, you should check each 3 adjacent rows (or columns, your choice!) and avoid standard major and minor chords, and perhaps diminished triads as well. Be careful of inversions, as well.

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