Composing with Pitch Set Theory

I'm currently working on a pitch class set composition- I have my set chosen and I've mapped out several potential permutations that I could use. However, I'm confused about how to approach the material and how to effectively apply it. 

For example, I understand that, unlike 12 tone rows, you don't have to go through the entire set before repeating a note. If that's the case, couldn't you do just about anything you want, and then just retrospectively describe it using set theory? How would you go about making it aurally clear that it was written using pitch set theory? 

Right now I'm approaching it like a 12 tone composition- I'm just stringing together different permutations, going through all the notes in each one before starting a new one. But I get the feeling that's not necessary. 

Overall, I'm still a little confused as to the purpose behind this kind of theory. What's the point of using it to write music? I want to make sure that I'm not doing anything in the composition that would defeat the purpose of using pitch set theory. 

Thanks in advance!

I think that, perhaps, you are making a mistake believing that taking the theory and developing some examples is enough to make a convincent composition. Think about tonal harmony, what happens if you take a series of chords perfectly ordered by their function? Nothing, or almost nothing.

PC Set theory is a language, a musical idiom different from others, even 12 tone composition. The fact is that PC Set theory allows you to build a musical work where the most important is the intervalic relationship between the pitches.

First, the more you study the possibilities the better: transposition, inversion, rotation are the essentials, but also subsets, supersets, complementary set, etc... If you have more tools you can do better.

And most important is how to manipulate the material you have. For example, interval vectors are very useful.

If you use an interval vector like this: <222222>, or <323232> your music will be even. There will not be predominant intervals. If you use something like <404120> you know your music will have many minor seconds and major seconds (or b3) and it will have personality.

Interval vectors are also useful to know if a transposition of the set will have common tones with the one you are at a moment. If you subsequently use sets with common tones the result is smooth, if you have no common tones, your change will be strong. 

Well, it's imposible to explain everything here, but I'm sure you know where to find information. 

Practicing is the better option to improve. But, in a word, you must control the PC Set theory, not the opposite.

An example. Each change of color is a set of pitches.

Motiveness Partitura completa.pdf

Motiveness.mp3

Thank you so much for your reply!

2 hours ago, Luis Hernández said:

If you use an interval vector like this: <222222>, or <323232> your music will be even. There will not be predominant intervals. If you use something like <404120> you know your music will have many minor seconds and major seconds (or b3) and it will have personality.

I sort of understand what you're saying here, but would you mind elaborating a little further on this? For example, what do you mean by predominant intervals? If a certain interval vector is completely filled out like that (<222222>), wouldn't that mean that there would be at least one instance of every single type of interval? 

2 hours ago, Luis Hernández said:

Interval vectors are also useful to know if a transposition of the set will have common tones with the one you are at a moment. If you subsequently use sets with common tones the result is smooth, if you have no common tones, your change will be strong. 

I think I understand what you're saying here- in that you can pivot between two different permutations by using common tones shared by both of them(?)- but how would knowing the interval vector help you in finding permutations with common tones?

Finally, I really enjoyed the piece that you included. I love the combination of the bright saxophone with the muddled sound of the Rhodes. I noticed that for example, in the first measure of the sax, you're not using the pitches in order. When transposed to start on 0, you go [0,1,4,9,6], rather than [0,1,4,6,9]. Do the ordering of the pitches matter in set theory, or is that only in 12 tone theory? 

Thank you again for your advice, this has been a huge help!

Yes, the interval vector tells you exactly how many intervals you have in your set

The first number is the minor seconds/ major sevenths

The second number is the mayor seconds/minor sevenths

The third number is the minor thirds / major sixths

The fourth number is the major thirds / minor sixths

The fifth number is the perfect fourths / perfect fifths

The sixth number is the tritones

So the interval vector ALWAYS has six numbers.

Let's say our set is [0,1,4,6,9] = C - Db - Eb - F# - A

The interval vector is <113221>   So, in your set you hava

1 minor second / mayor seventh

1 major second / minor seventh

3 minor thirds / major sixths

2 major thirds / minor sixths

2 perfect 4ths / perfect 5ths

1 tritone

PC Set theory works with inversion equivalency. That's why an interval is considered equal to its inversion.

This concepts leads to surprising facts. Take this vector: [001110] It means the Set has 1 m3, 1 M3, 1 P4/P5. So it corresponds to the minor triad and, also, to the major triad:

Regarding the other issue, it works this way. Each number of the interval vector is the times every interval happens.

Let's take the set whose prime form is (0,1,2,4) = C - Db - D - E     Interval vector <221100>

The interval vector tells you that if you transpose your set a minor second (first number = 2) you will have 2 common pitches:  C - Db - D - E.   vs.   Db - D - Eb - F

Take the third value of the vector corresponding to minor third (1).... when transposed a minor third you will have 1 pitch in common: C - Db - D - E.   vs.    Eb - E - F - G

If you transpose a perfect fourth (0) or a tritone  you will have no pitches in common.

You can use this in your compositions: transposing to intervals with common pitches is smooth, transposing to intervals with no common pitches is rougher.

Lastly, yes.... I have no problem altering sometimes the strict order of the pitches. It happens also in atonality. As it happens in tonal harmony when you use a supposed avoid note where it is not awaited. That's why I say we have to control the  system. I believe in "rules", but I don't think they are sacred.

Hello Luis Hernandez, 
I am French and I am interested in Set Theory. By doing a search on the Net, I discovered your answer to millert1409 including your explanation on the relationship between the transposition of a PC-Set and the common notes from the Interval Vectors. To tell the truth, I had not paid attention to this particularity of the IV, simply noting the number of occurrences of the interval classes.
However, very interested in your explanation, I tried this one on several IV but I run into a small problem concerning the number of tritones.If I take as an example the interval vectors of the set {0, 1, 4, 6, 9} is <113221>, when I transpose successively from the second minor to the fourth, the number of common notes displayed by the IV is similar. On the other hand, if I transpose {0, 1, 4, 6, 9} to tritone I get {6, 7, 10, 0, 3} two (6 / f # and 0 / c) instead of a single tritone. Is it the fact in the count that we must remember that 6 + 6 = 12 and not 0 ?
Otherwise, thank you for this explanation on this feature of the Interval Vector which I had not paid attention. 

PS : Sorry for my bad english.