#1
You modernist buffs, give me some direction on this.

I saw that jazz_rock_feel was talking about this in another thread, and it sounds really interesting. I feel like I could understand it, if I studied it. I deal with a lot of upper-level math and computer programming techniques in university. So, I don't think it'd be a huge issue if the material was a bit dense. (Hell, try Numerical Analysis some time. It's as dense as it gets, or so it seems on certain days.)

Anyway...suggestions? Books? Websites? Pieces using set theory?

GO!
Last edited by crazysam23_Atax at Mar 24, 2014,
#2
Surprisingly, Wikipedia is a great source for esoteric theory concepts. We're working with set theory and tone rows in my theory class currently, and the professor has assigned several readings from Wikipedia.

As far as the math goes, it's really very simple. If you can subtract and follow patterns, you can do it.

A good place to start is learning integer notation. With that system, C = 0, C#/Db = 1 etc, etc, B = 11, and then it comes back to C = 0.

The basic concept of composition using set theory is choosing a set of intervals (e.g. [0,1,4]) and manipulating that set.
Rather than focusing on using C, Db, and E in that set, you would be focusing on using the intervals of a half-step, a minor third, and a major third. For example, you could use the pitches [10,11,2] and it would still reduce down to [0,1,4], as it uses all the same intervals as I listed above.

There are three important categories with which you can analyze your set: Normal Order, Prime Order, and Interval Vector.

Normal Order: To find Normal Order, find the two pitches with the largest interval between them and start your set with the second one. For example, if you had the pitches C#, F, G#, the largest interval is between G# and C#, so you would start your Normal Order at C#, making your Normal Order [1,5,8]


Prime Form: Prime Form is essentially reducing the set down to its smallest form. This means getting all the smallest intervals to the left side of the set. If you look at the last set [1,5,8], you'll notice that there is a minor third between 5 and 8, which is a smaller interval than the major third between 1 and 5.
To get Prime Form, we need to reduce to zero. Reducing to zero in this case is very simple, we simply subtract 1 from the entire set, making it [0,4,7]. Unfortunately in this case, that does not solve our problem- we still are not in Prime Form.

This next part will be a lot easier if you have a piece of paper, so go ahead and grab one.

To start the next step, draw a clock face using interval notation- This means 0 will take the place of 12, the rest of the numbers remain the same as they would on a clock. When you have that drawn, circle the numbers of the set you're using ([1,5,8]). Now take those numbers, and mirror them horizontally across the clock face. So 1 becomes 11, 5 becomes 7, and 8 becomes 4.

Your set is now [4,7,11]. At this point, you can reduce to zero (subtract 4 from the set. Always subtract the first number when reducing). You should end up with [0,3,7]. If you were to begin composing using this set, the Prime Form would be the basic that you start from.


Interval Vector: Interval Vector serves to illustrate the types of intervals in a set. It has six integers- each starting at 0, and each representing an interval. Going left to right, the intervals represented are: half-step, whole-step, minor third, major third, perfect fourth, tritone. The reason 6ths, 7ths, and 5ths are not included is because 3rds, 2nds, and 4ths can be inverted into those intervals (e.g. E -> C is a minor sixth, but also a major third if you go backwards).

So in continuing with good 'ol [1,5,8], we would have our Interval Vector as 001110. This illustrates that the set has 0 half-steps, 0 whole-steps, 1 minor third, 1 major third, 1 fourth, and 0 tri-tones.


Hopefully this wasn't too confusing, if you have any more questions, I'd be happy to help!
Quote by stealstrings





C:
#3
^Couple of issues with the details. You conflated prime form and best normal order. Best normal order is the normal order with the smallest intervals to the left and prime form is best normal order reduced to zero.

The other thing is there's no need to invert the pitches in pitch class space to see if a set is in prime form (I'm talking about doing the whole clock thing). For most practical things, you can look at inversion as interval mirror inversion, meaning to find out if you have your prime form all you have to do is reverse the adjacency interval series. In your example, [0, 4, 7] has an AIS of 4-3. Reverse it and you get 3-4, compare that to the original AIS and whichever has the smaller numbers to the left is the AIS for your set. Apply that to the set in your example and you get a prime form of (0 3 7). Seems like a lot of words, but it actually takes a fraction of the time of inverting all the pitches in a set and transposing to 0 again. And it'll REALLY save you time if you're dealing with larger sets. I've never actually run across anything where I've thought, 'Man, you know what would be useful? Inverting this set in pitch class space!' At the end of the day, reversing the intervals gets you the same results.


As for the OP, it's tricky. There's lots of places to learn the terms and basic operations of set theory (wiki has a decent article and I linked to this in the other thread: http://composertools.com/Theory/PCSets/ ) and for books you can go right to the source with Forte's The Structure of Atonal Music which is more or less where set theory was invented.

The tricky thing is the application of set theory, knowing when it's useful and when it's not and completely shifting the way you think about music.

If there's any interest from people I can do a write up with examples and shit.
#5
Quote by jazz_rock_feel
^Couple of issues with the details. You conflated prime form and best normal order. Best normal order is the normal order with the smallest intervals to the left and prime form is best normal order reduced to zero.

The other thing is there's no need to invert the pitches in pitch class space to see if a set is in prime form (I'm talking about doing the whole clock thing). For most practical things, you can look at inversion as interval mirror inversion, meaning to find out if you have your prime form all you have to do is reverse the adjacency interval series. In your example, [0, 4, 7] has an AIS of 4-3. Reverse it and you get 3-4, compare that to the original AIS and whichever has the smaller numbers to the left is the AIS for your set. Apply that to the set in your example and you get a prime form of (0 3 7). Seems like a lot of words, but it actually takes a fraction of the time of inverting all the pitches in a set and transposing to 0 again. And it'll REALLY save you time if you're dealing with larger sets. I've never actually run across anything where I've thought, 'Man, you know what would be useful? Inverting this set in pitch class space!' At the end of the day, reversing the intervals gets you the same results.


As for the OP, it's tricky. There's lots of places to learn the terms and basic operations of set theory (wiki has a decent article and I linked to this in the other thread: http://composertools.com/Theory/PCSets/ ) and for books you can go right to the source with Forte's The Structure of Atonal Music which is more or less where set theory was invented.

The tricky thing is the application of set theory, knowing when it's useful and when it's not and completely shifting the way you think about music.

If there's any interest from people I can do a write up with examples and shit.

I would be interested.
#7
If your set is 0, 4, 7 and you want to have the lowest interval first, couldn't you just look at this as a repeating set (since it is really 0, 4, 7, 0, 4, 7, 0, 4, 7) and then take the 4, 7, 0, and then this reduces to 0, 3, 8 ? That would have all the same intervals as in 0, 4, 7, 0, 4, 7.... except starting with the 2nd note.

Well, I don't know set theory, that just seemed like the logical solution to modify the set so it starts with the smaller interval.

Ken
Bernie Sanders for President!
#9
Quote by jazz_rock_feel
If there's any interest from people I can do a write up with examples and shit.

My favourite words at this forum "Examples and Shit"
only thing better... is well explained "Examples and Shit" as sometimes those who know, can tend to fly through things pretty quickly leaving us mere mortals wondering what's being said, of course i'm not suggesting being spoon fed, it just may take a few reads to get a few aspects "in tune" so to speak...

but yes I too would also be interested, if you do decide to jazz... thanks!
#10
To his credit, I've noticed jazz_rock_feel does a pretty good job of explaining and giving resources and so on.
#11
Quote by crazysam23_Atax
To his credit, I've noticed jazz_rock_feel does a pretty good job of explaining and giving resources and so on.


I couldn't agree more!!!
#12
Yeah, by "examples and shit" I mean putting together sample sets for you guys to identify and manipulate and showing some pieces where set theory has some traction as an analysis tool and maybe even doing some segmentation (i.e., set analysis). Basically start from the ground up and work with you guys to get some understanding. That's why I want to make sure there's interest, because if I make all this shit I want to make sure that there are people that are willing to put a little time into it.
#13
Quote by jazz_rock_feel
^Couple of issues with the details. You conflated prime form and best normal order. Best normal order is the normal order with the smallest intervals to the left and prime form is best normal order reduced to zero...
...If there's any interest from people I can do a write up with examples and shit.

A.I.S Method???
Here you write:

"...to find out if you have your prime form all you have to do is reverse the Adjacency Interval Series.
In your example, [0, 4, 7] has an AIS of 4-3.
okay so...
[0, 4, has a space/distance/chromatic interval of 4...
and... 4, 7] has a space/etc... of 3...
Hence 4-3

Reverse it and you get 3-4,
cool, understood!
compare that to the original AIS and whichever has the smaller numbers to the left is the AIS for your set.
okay 3 is less than 4, so we're staying with the 3-4 eg. here...
Apply that to the set in your example and you get a prime form of (0 3 7).
ummm... how'd we get (0 3 7) using AIS???
Seems like a lot of words, but it actually takes a fraction of the time...
I could actually follow it using Instant's Clock theory, but with your method having the added benefit of saving time and extra work, I am having trouble connecting the last couple threads (no pun intended here, and no offense meant InstantMo!)
any chance you might clarify this a little more for me please?
...At the end of the day, reversing the intervals gets you the same results."

NB: I do believe InstantMo had the Normal/Prime stuff in the right order, perhaps he might have grammered (is that even a word?) his Prime Form: paragraph a little different, but nonetheless understood.

...one last thing:
Lastly you wrote:
As for the OP, it's tricky...
What is OP? having read back through the conversation, the closest I could derive was Other Part?, thinking it might've been an acronym like omg, imo, atm etc...

Thanks Jazz, and here's hoping a few more people elect to come on board.
Come on guys - show some love - albeit a little mind-bending.

Unlike crazysam I am not studying Numerical Analysis, so I guess that leaves me as just plain dense! (read sam's initial thread if you didn't just get the joke!)

hopefully blue/green are better font colours this time, as opposed to yellow, (I did a quick switch back to UG Classic and *ouch*... yeah yellow text sure hurts in Classic), but sorry i'm keeping UG Black. Have quick switch to UG-Black and read this reply... ahh isn't that better? lols
Last edited by tonibet72 at Mar 26, 2014,
#14

What is OP? having read back through the conversation, the closest I could derive was Other Part?, thinking it might've been an acronym like omg, imo, atm etc...y]ng.
OP stands for "Original Post"

Quote by tonibet72
Unlike crazysam I am not studying Numerical Analysis, so I guess that leaves me as just plain dense! (read sam's initial thread if you didn't just get the joke!)

That made me chuckle.
Last edited by crazysam23_Atax at Mar 26, 2014,
#15
Quote by crazysam23_Atax
OP stands for "Original Post"


That made me chuckle.


Thanks Sam
#16
Quote by tonibet72
A.I.S Method???
Here you write:

"...to find out if you have your prime form all you have to do is reverse the Adjacency Interval Series.
In your example, [0, 4, 7] has an AIS of 4-3.
okay so...
[0, 4, has a space/distance/chromatic interval of 4...
and... 4, 7] has a space/etc... of 3...
Hence 4-3

Reverse it and you get 3-4,
cool, understood!
compare that to the original AIS and whichever has the smaller numbers to the left is the AIS for your set.
okay 3 is less than 4, so we're staying with the 3-4 eg. here...
Apply that to the set in your example and you get a prime form of (0 3 7).
ummm... how'd we get (0 3 7) using AIS???
Seems like a lot of words, but it actually takes a fraction of the time...
I could actually follow it using Instant's Clock theory, but with your method having the added benefit of saving time and extra work, I am having trouble connecting the last couple threads (no pun intended here, and no offense meant InstantMo!)
any chance you might clarify this a little more for me please?
...At the end of the day, reversing the intervals gets you the same results."

NB: I do believe InstantMo had the Normal/Prime stuff in the right order, perhaps he might have grammered (is that even a word?) his Prime Form: paragraph a little different, but nonetheless understood.

...one last thing:
Lastly you wrote:
As for the OP, it's tricky...
What is OP? having read back through the conversation, the closest I could derive was Other Part?, thinking it might've been an acronym like omg, imo, atm etc...

Man, you go through a lot of effort to colour your posts lol.

As for the AIS thing, I'm loathe to piecemeal information like this. It would be much better to just start from the beginning, but...

As the name implies, the AIS is the series of adjacent intervals. So if you have an AIS of 1-1 it means that your set would have three notes each a halfstep apart (0 1 2). In other words, go up 1 then go up 1. In my example, an AIS of 3-4 means go up 3 then go up 4. 0 + 3 is 3 and then 3+4 is 7 so you end up with (0 3 7).

Again, AIS is a little tidbit that doesn't mean anything without the whole picture.

You know... There seems to be a bit of interest and I've actually got some time tonight so maybe I'll write up a little intro lesson.
#17
Quote by tonibet72
Thanks Sam

You're welcome!

Quote by jazz_rock_feel
You know... There seems to be a bit of interest and I've actually got some time tonight so maybe I'll write up a little intro lesson.

Looking forward to this. Thanks beforehand.
#18
Quote by jazz_rock_feel
Man, you go through a lot of effort to colour your posts lol.

Not really... I type, highlight, button, button, push, done!
As for the AIS thing, I'm loathe to piecemeal information like this. It would be much better to just start from the beginning, but...

As the name implies, the AIS is the series of adjacent intervals. So if you have an AIS of 1-1 it means that your set would have three notes each a halfstep apart (0 1 2). In other words, go up 1 then go up 1. In my example, an AIS of 3-4 means go up 3 then go up 4. 0 + 3 is 3 and then 3+4 is 7 so you end up with (0 3 7).


okay capisce...thanks for that, and thanks in advance!
#19
Quote by jazz_rock_feel
That's why I want to make sure there's interest, because if I make all this shit I want to make sure that there are people that are willing to put a little time into it.


Although the majority of it would probably go over my head, I'd be interested in reading it!
#20
Quote by Tommat
Although the majority of it would probably go over my head, I'd be interested in reading it!

No, that's the idea! I'm going to start right from the beginning and try and make it as clear as possible.
#21
Quote by jazz_rock_feel
I'm going to start right from the beginning and try and make it as clear as possible.
woo hoo 'clink' yippie yay yea-yeah yah maan alrite

how the bloody hell did this llama get in here!!! oh that was me... sorry!
#22
Quote by jazz_rock_feel
AIS


We didn't discuss this in my class, and it sounds very useful, definitely something I'll ask about, thank you!

I didn't mean to say that the only way to find Prime Order is to do the clock thing, I suppose I was trying to imply that if simply reducing the original set to zero doesn't get the Prime Order, then mirroring then reducing generally will. As you said though, with larger sets that would quickly get very confusing and tedious.

Thanks for the corrections, I'm pretty new to these concepts, so it's great to learn additional helpful information.
Quote by stealstrings





C:
#23
Quote by InstantMustache
We didn't discuss this in my class, and it sounds very useful, definitely something I'll ask about, thank you!

I didn't mean to say that the only way to find Prime Order is to do the clock thing, I suppose I was trying to imply that if simply reducing the original set to zero doesn't get the Prime Order, then mirroring then reducing generally will. As you said though, with larger sets that would quickly get very confusing and tedious.

Thanks for the corrections, I'm pretty new to these concepts, so it's great to learn additional helpful information.

I mean, the clock thing totally works and is a way to invert that's maybe more "in the spirit" of set theory. It's just an annoyingly long way to do it