Pitch class set multiplication

Hi,

could someone explain what the point of Pitch class set multiplication is and how one should use it? I know that it is the cartesian product of two pitch sets, but what's exactly the point of this? Is the relationship of the product to the original sets even audible?

I hope someone could enlighten me

Hi,

could someone explain what the point of Pitch class set multiplication is and how one should use it? I know that it is the cartesian product of two pitch sets, but what's exactly the point of this? Is the relationship of the product to the original sets even audible?

I hope someone could enlighten me

Generally, these techniques are used just to give structure and form to atonal music. With practice, perhaps you could pick out that type of relationship, but in reality I think that it is just another interesting tool that should be used as the composer sees fit, like serialism.

In serialism, it really takes effort to hear each row as it goes, but you aren't really supposed to listen to the rows, just listen to the final product, the matrix is just a compositional tool.

http://en.wikipedia.org/wiki/Multiplication_(music)

I would say that even a trained listener couldn't follow it easily, but it's a method to pitch organization.

Yeah, I know that the sets are not exactly audible, but is there any logic behind the multiplication?For example,what would be the point in multiplicating CEG with DA. In which way would the result be relatet with the original sets?

How do I choose the sets I would like to multiplicate?

The point is to combine the intervallic characteristics of two sets to create new pitches, which still share many acustic similarities with the original sets. The chord CEG appears to the listener as a "major triad", the notes DA as a "major second". The resulting set contains both two major triads and three major seconds, giving a strong emphasis on the intervals of the original sets.

You might describe it as a way to combine sets to new sets that both keep as much of the original intervals as possible, while obtaining a multitude of new tones to work with. It also has the side effect of balancing out the number of times an interval set appears in the final result with the complexity or size of the interval: The "simpler" an original set is, i.e. the fewer different pitches it has, the more often will it appear in the resulting set.

In our example, the set CD is very simple, consisting of only one interval, which makes it appear more often than the somewhat more complex (bigger) set CEG.

Also note that while this strict, technical description wasn't used before serialism, the principle this comes from appears in quite a lot of music. Whenever one interval constellation appears on two or more basic pitches at the same time, you have a "pitch multiplication". This appears especially often in music that seeks to combine the horizontal harmony (melody/mode/etc.) with the vertical harmony (chords), such as in Debussy's music. Here you often have one kind of chord used on different basic pitches at the same time, or being moved in parallel (while possibly the last chord is still audible).

Another example is music that is played in a room with strong reverberation. Take a motet or choral sung in a cathedral for example. Since tonal music often contains certain chords that appear on various positions on the scale (such as major chords) and since this music generally makes use of a more or less fixed scale, due to the reverberation often several identical chords will be audible at the same time, but on different scale degrees. This is already a form of pitch multiplication.

And of course you'll find -lots- more it if you go into polytonal music or music with extended tonality, be that Liszt, Scriabin, Bartok, or Stravinsky.

Building chords out of repeating certain intervals starting from different notes has been one of the most common tools of creating harmonic cohesion of non-traditionally tonal (or atonal) works, since it's very effective at both creating a certain intervallic "colour", but still giving you many pitches to work with.

Whether the sets are audible depends on the complexity and size of the sets, amongst other things. In the example here, I think it's quite audible. Combining CEG with CD gives you this major-second sound that sounds totally different than when you combine CEG with C-Db.

Another form of multiplication that is mentioned in that Wikipedia article too is ring modulation. Commonly it's just an electronic effect, but there are composers (such as Hans Zender) who use it as a harmonic principle in instrumental music too, which can have quite fascinating results.

Thank you,Gardener, your post was very helpful!