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#1
Set theory is a new way to help us better understand the new music of our time, which traditional tonal theory is no longer adequate to do. In the "classical" world, we've exhausted tonal possibilities and are now exploring other ways to combine the 12 available notes that we have. From the renaissance up until the mid 20th century, we have always built the music on thirds (triadic harmony) which all serve to point to a central key (tonality), but it's time for a change. New music are often times not triadic, and even when they are, they are not being used in traditional ways that leads to a tonal point.

So how do we make sense of the presentation? We can't really analyze how everything serves to lead to a tonal center but we can analyze the relationship between each musical object. So in a way, our focus has shifted from one of centrality to one of dispersion. Of course, the masters of new music haven't lost sight of the importance of musical unity, and even though we are now more concerned with the individual parts, these parts have consistent relationships that sum up to a convincing whole.

Integer Notation: Using numbers to represent the 12 chromatic notes in our western system will help us clarify the many relationships that are possible. This basic idea is quite simple: starting on 0, which represents C, the next integer represents the next chromatic note. eg, 0=C, 1=C#/Db, 2=D, 3=D#/Eb, ...11=B. If you are familiar with fixed solfege, this is the same idea.

Integer notation also applies to intervals. In tonal music, we regard intervals in a diatonic sense, eg a (some kind of) 4th = distance between (some kind of) C and (some kind of) F. Augmented 4th = C# - Fx. In new music without this diatonic system, we need to be more careful and consider all the half steps in between. Now a 4th = 4 half steps, or from exactly C natural to E natural.

Before we go any further, we need to understand the "concept" of these notes. When we refer to a harmony such as Dmaj7 (jazz) or i 6-4 (figured bass), we mostly refer to the concept of that harmony (some combination of D, F#, A, C# for Dmaj7) rather than a specific configuration at a specific place on the staff. The individual notes can also be thought this way, eg C referring to the abstraction of C rather than a pitch with a specific frequency. This abstraction is known as pitch class. Therefore the numbers are also used as abstractions rather than, for example, "G above middle C".

Mod 12: We understand the intuitive concept of the octave: there are 12 pitches in an octave and anything reaching outside of it is a repeat in the pitch class sense. In dealing with numbers, we have to be consistent with this concept, meaning we must calculate numbers using modular arithmetic. This sounds fancy, but you do it all the time when figuring out time. If it's 1 o clock right now, it will be 2 o clock 13 hours later, not 14 o clock. Conveniently, just like the clock, our western system also has 12 integers. The difference is we start on 0 while the clock starts on 12.

Intervallic relationships become extremely important in new music for reasons stated in the intro, therefore we have to pay careful attention to them. There are 4 ways to count intervals, but the most useful method is known as interval class. In this method, we don't care whether we count up or down, but rather the absolute space between two pitch classes. Although we can reach an interval of 11 half steps from C up to B in an octave, the absolute distance between the pitch classes is just 1 half step. In this sense, there are only 6 interval classes (1 to 6). Everything else is a complement of these 6 absolutes. eg. 7 = 5, 8 = 4, 10 = 2, etc. Notice how they add up to 0 ("12").

Interval Class Vector: Although this will seem like a convoluted calculation right now, it will be very useful later. When we come across a collection of pitch classes in the music such as a chord, we can account for all intervallic relationships possible between all pitches in this collection. Keeping in mind the interval class method, we can set up a vector and tally up the occurrences of interval classes 1 to 6.



Here we are presented with a collection: C, C#, D#, E, G#, or 0, 1, 3, 4, 8. Using interval class counting, we've covered all the intervals between the 5 notes:
-there are 2 instances of interval class 1 (0 - 1, 3 - 4)
-there is 1 instance of interval class 2 (1 - 3)
-there are 2 instances of ic 3
-there are 3 instances of ic 4
-there are 2 instances of ic 5
-there are no instances of ic 6

The interval class vector for this particular collection of notes is thus 212320.

...modes and scales are still useless.


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Last edited by Xiaoxi at May 17, 2011,
#2
You have resurrected the nightmare that is my second year theory course in university
#3
Quote by pwrmax
You have resurrected the nightmare that is my second year theory course in university

If we had to suffer, everyone else must suffer as well!

...modes and scales are still useless.


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#4
Following you completely. But just one question: where would we use this? Anything outside of a theory class? Is this to effectively replace what is taught now?

I'm diggin, don't get me wrong. Just trying to find a use for it.

PS - Seems almost like a Nashville Numbers system for notes instead of chords.
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#5
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Following you completely. But just one question: where would we use this? Anything outside of a theory class? Is this to effectively replace what is taught now?

As I said in the preface for this, this is mostly useful to composers of new classical music, which seeks to do things that don't rely on tonal conventions. But I think if you grasp everything here it can be useful to advance any music today.

...modes and scales are still useless.


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#6
To internalize these concepts, you should do a short analysis below. Try not to look at other people's answers if you really want to try.


Translate the pitches in this thematic melody in figure A to integer notation. Analyze the interval classes between them (ignore repeats of a note).

How does figure B relate to figure A? How does this tie into the concepts we covered?



Listen to the complete movement here

...modes and scales are still useless.


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#7
That's very clever Xioaoxi. I never thought of orchestration that way... I'll be listening to that later.

Highlight to see my answer (UG Black): They're, barring repeated notes and rhythms, consist of the same notes in the same order. It just jumps between the Violin and Cello as well as different octaves in the second example. Effectively creating a pitch class much akin to Serialism and tone rows (in this particular instance).
#8
^that's good! And if you can be bothered, put the notes into integers (and their interval classes).

...modes and scales are still useless.


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#10
The answer to that is pretty clear once you've read it through.

...modes and scales are still useless.


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#12
Quote by pwrmax

That could be the beginning to a great piece but 2 things:

-you've created a nice row, (using derived trichords there?), but it would be nice to have more rhythmic variety in the collective melody.

-consider the possible transformations of the row.

...modes and scales are still useless.


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#13
Oh that's not mine, but I agree with the rhythmic variety comment.
#15
You said that you just take the shortest distance from the note, but how do you know which direction it would be going in?
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#17
A:
{2-1-9-10-5-3-4-0-8-7-6-11}

5 instances of ic1 (2-1;9-10;3-4;8-7;7-6)
1 instance of ic2 (5-3)
0 instances of ic3
1 instance of ic4 (4-0;0-8;1-9)
2 instances of ic5 (10-5;6-11)
0 instances of ic6

The interval class vector is 510120

B:
{2-1-5-6-5-3-4-0-8-7-6-10}

5 instances of ic1 (2-1;5-6;3-4;8-7;7-6)
1 instance of ic2 (5-3)
0 instances of ic3
2 instances of ic4 (1-5;4-0;0-8;6-10)
0 instances of ic5
0 instances of ic6

The interval class vector is 510200


This is probably completely wrong but hey, at least I tried. Very interesting topic so far.
#18
Quote by Venice King
You said that you just take the shortest distance from the note, but how do you know which direction it would be going in?

What we essentially have with the interval class method is an unordered form that doesn't address the directions. I mentioned there are 3 other which I didn't talk about, but they do address the exact directions and/or half steps. The reason why I didn't mention them is because they can complicate the relationships that we'll need simplified later.

Also keep in mind the concept of pitch class, in which the exact location of the pitch doesn't matter. If you look at the Schoenberg example, you can see how fig. B retains the same pitch classes but vary in linear direction from fig. A. That's why interval class is most useful for this kind of music.

Quote by jazz_rock_feel
For analysis, does 0 always relate to C, or is it the first note of the line, or does it matter?

0 is C. It's not "related to C". We're just replacing the abstraction of C with 0, C#/Db with 1, and so on.

...modes and scales are still useless.


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Last edited by Xiaoxi at May 17, 2011,
#19
Quote by Xiaoxi
What we essentially have with the interval class method is an unordered form that doesn't address the directions. I mentioned there are 3 other which I didn't talk about, but they do address the exact directions and/or half steps. The reason why I didn't mention them is because they can complicate the relationships that we'll need simplified later.

Also keep in mind the concept of pitch class, in which the exact location of the pitch doesn't matter. If you look at the Schoenberg example, you can see how fig. B retains the same pitch classes but vary in linear direction from fig. A. That's why interval class is most useful for this kind of music.


0 is C. It's not "related to C". We're just replacing the abstraction of C with 0, C#/Db with 1, and so on.


Question, are you going to comment on my "analysis" of it or do you want to refrain from that until more people have tried? Or did you just not notice it?
#20
Sok: your integer notation is correct, however I didn't ask for an interval vector, merely the interval class between each adjacent pitch class.

An interval vector here won't do much because this isn't an isolated collection. This is the entire 12 pitch classes (12 tone row), and the interval vector for all 12 notes, no matter what arrangement/order, is always going to be the same.

So with that in mind and not even calculating anything else, something isn't right because you have 2 different interval vectors for A and B. Also, there would be MANY instances of each interval class because each of the 12 pitch classes will relate to all the other 11 pitch classes in ways that addresses all 6 ic's.

I will put up an interval vector challenge later today.


Quote by Sóknardalr
Question, are you going to comment on my "analysis" of it or do you want to refrain from that until more people have tried? Or did you just not notice it?

You guys are too fast

...modes and scales are still useless.


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Last edited by Xiaoxi at May 17, 2011,
#21
Quote by Xiaoxi
Sok: your integer notation is correct, however I didn't ask for an interval vector, merely the interval class between each adjacent pitch class.

An interval vector here won't do much because this isn't an isolated collection. This is the entire 12 pitch classes (12 tone row), and the interval vector for all 12 notes, no matter what arrangement/order, is always going to be the same.

So with that in mind and not even calculating anything else, something isn't right because you have 2 different interval vectors for A and B. Also, there would be MANY instances of each interval class because each of the 12 pitch classes will relate to all the other 11 pitch classes in ways that addresses all 6 ic's.

I will put up an interval vector challenge later today.


You guys are too fast


I thought it was pretty weird that you would make us calculate an interval vector for something that uses all 12 tones too. My original guess was 24 instances of ic1 until I figured I may have been doing something wrong and thought of it another way. Is that at least correct? If not, then I'm going to wait for the next lesson and forget about interval vectors.
#22
Quote by Sóknardalr
I thought it was pretty weird that you would make us calculate an interval vector for something that uses all 12 tones too. My original guess was 24 instances of ic1 until I figured I may have been doing something wrong and thought of it another way. Is that at least correct? If not, then I'm going to wait for the next lesson and forget about interval vectors.

There would be 12 instances of ic 1. Check back later when I've put up an interval vector problem that'll be simpler but help you get a better idea.


Also, I just noticed that your integer notation for B is wrong. Pay attention to the clef in the cello. The rest of the integers are also wrong...

...modes and scales are still useless.


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#23
Quote by Xiaoxi
There would be 12 instances of ic 1. Check back later when I've put up an interval vector problem that'll be simpler but help you get a better idea.


Also, I just noticed that your integer notation for B is wrong. Pay attention to the clef in the cello. The rest of the integers are also wrong...


{2-1-9-10-5-3-4-0-8-7-6-11}

Same as A. I guess I did it too quickly; ignoring the bass clef and putting B as 10 instead of 11. It's pretty cool that both parts use the exact some progression of notes, though I haven't heard the B part yet as I just started listening now.

EDIT: But I still don't understand why the cello part is simply shoehorned into the integer notation. What if it's a really busy contrapuntal part?
Last edited by Sóknardalr at May 17, 2011,
#24
Quote by Sóknardalr
EDIT: But I still don't understand why the cello part is simply shoehorned into the integer notation. What if it's a really busy contrapuntal part?

The coincidence would be outstanding if that's what's happening. The fact of the matter is that the composer was going for a theme (the original pitch class) and variation (example B) and he felt the cello should play those notes.
#25
Quote by Sóknardalr

EDIT: But I still don't understand why the cello part is simply shoehorned into the integer notation. What if it's a really busy contrapuntal part?


{2-1-9-10-5-3-4-0-8-7-6-11} is essentially the thematic tone row of this movement. As we'll see later on, you'll find that serial composers rarely relegate the row to just one voice/instrument. It's a technique called klangfarbenmelodie, or tone color melody. It adds a whole new dimension to what we can do. One instance of the melody can be carried over multiple instruments and that can be its own motif or continuity or variation.

The cello could have been doing something else, and maybe even stating a related row, but they overlap here in such a way that fills out the thematic row.

...modes and scales are still useless.


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#26
Quote by Xiaoxi
{2-1-9-10-5-3-4-0-8-7-6-11} is essentially the thematic tone row of this movement. As we'll see later on, you'll find that serial composers rarely relegate the row to just one voice/instrument. It's a technique called klangfarbenmelodie, or tone color melody. It adds a whole new dimension to what we can do. One instance of the melody can be carried over multiple instruments and that can be its own motif or continuity or variation.

The cello could have been doing something else, and maybe even stating a related row, but they overlap here in such a way that fills out the thematic row.


What I mean is what if all of the instruments are playing their own parts or playing chords collectively? How do you put that into integer notation? I would think you would just create a row for each instrument, but why is that not done here then? Or doesn't it matter and just depends on the context?
#27
Quote by Sóknardalr
What I mean is what if all of the instruments are playing their own parts or playing chords collectively? How do you put that into integer notation? I would think you would just create a row for each instrument, but why is that not done here then? Or doesn't it matter and just depends on the context?

Integer notation is simply just saying 0 instead of C, 8 instead of G#/Ab, etc.

So the entire music can be translated to integers. If we come across a chord, regardless of whether it's built by the ensemble or on 1 instrument, {2 5 9} is the same thing as identifyign the chord as D F A, or a D minor triad.

But we translate the notes into integers primarily when we suspect or identify some sort of motivic continuity in an isolated part of the music so that we can use the numbers to assess how it relates to what has happened before. I think you'll get a better idea later on.

...modes and scales are still useless.


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Last edited by Xiaoxi at May 17, 2011,
#28
pcs (both A and B): 2,1,9,10,5,3,4,0,8,7,6,11

ics: 1, 4, 1, 5, 2, 1, 4, 4, 1, 1, 5

If that was what you asked for.


Are you going to cover the practical application of set theory to harmony (obviously not functional harmony, but you get the point)? I've dabbled a bit in 12-tone serialism this spring (a very chaotic first attempt here ), and while an approach with one or other form of the tone row in each part, or spread over two parts as in the B example has been somewhat managable to work with, any attempt with to even a brief, more harmonic approach (ie quartal harmony) leaves me pretty stumped.
#29
Quote by descara

Are you going to cover the practical application of set theory to harmony (obviously not functional harmony, but you get the point)?
Yea, later.

I wouldn't say that harmony as its own concept ties very well with set theory. You can use serial ideas to generate harmony for continuity (as we will see later), but if we're talking about harmony in the triadic, quartal, etc, color sense, then it really is all up to your creativity and intuition. There are no more functional conventions but it's good to be consistent and logical in your voice leading and harmonic make up.

Good job on the intervals

...modes and scales are still useless.


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#30
Quote by Xiaoxi
Yea, later.

I wouldn't say that harmony as its own concept ties very well with set theory. You can use serial ideas to generate harmony for continuity (as we will see later), but if we're talking about harmony in the triadic, quartal, etc, color sense, then it really is all up to your creativity and intuition. There are no more functional conventions but it's good to be consistent and logical in your voice leading and harmonic make up.

Good job on the intervals


Neat!

I guess I'm mostly interested in how to, as you say, use serial ideas to generate harmony. It just feels somewhat ungraspable, not having any framework at all to work within, but then again, as you say, there's creativity and intuition.
#31
Quote by descara
Neat!

I guess I'm mostly interested in how to, as you say, use serial ideas to generate harmony. It just feels somewhat ungraspable, not having any framework at all to work within, but then again, as you say, there's creativity and intuition.

They're both cool ways to write new music. I think the difference is whereas one is more absolute in pitch class logic, the other one is more consistent in harmonic texture. That's not to say you can't have both.

Either way, I mostly don't treat harmony as a vertical concept. I rely on counterpoint to make up the harmony, so in a way the harmony is out of my control.

...modes and scales are still useless.


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#33
^Maybe you can develop some new theories with those... be my guest

...modes and scales are still useless.


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#35
I think it's simpler and more commonly used to just use A & B or T & E to refer to 10 and 11. But 10 and 11 is fine as well.

Interval classes only go from 1 to 6, so there is no need for elaborate nomenclature.

...modes and scales are still useless.


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#36
Quote by Xiaoxi
I think it's simpler and more commonly used to just use A & B or T & E to refer to 10 and 11. But 10 and 11 is fine as well.


One of the problems with B and T is they can't be easily represented on a seven-segment display (looking at it from a practical standpoint with an eye toward the implementation of dozenal numeration). Another problem is one of aesthetics; A, B, T and E clash with the numerals 0-9.

Interval classes only go from 1 to 6, so there is no need for elaborate nomenclature.


Right, I should've said the intervals themselves, as interval classes make no distinction between inverses. Still, it's nice to use dozenal, which makes the octave equal to 10. It's more simplifying than it is elaborate.
Last edited by Dodeka at May 17, 2011,
#37
^This is pretty deep into unnecessary semantics for what we're trying to cover. You can do it whichever way you like but I will stick to 10, 11 and later A B. There's already too much to cover and this is really besides the points.

We have no need to consider seven segment displays or dozenal numeration, which based on your post seems to be more concerned with frequency information rather than the abstractions of the 12 notes.

...modes and scales are still useless.


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#38
Quote by Xiaoxi
We have no need to consider seven segment displays or dozenal numeration, which based on your post seems to be more concerned with frequency information rather than the abstractions of the 12 notes.


Yes, but it seemed to lend itself to this kind of application as well. Guess no one cares. Good enough, carry on...
#39
Quote by Zen Skin
I studied mathematics and I dig this.

When is going to get hard??

You know -- homotopy groups and kernels spaces ...

I study maths, and to be honest maths being applied to music doesn't really work.

I'm intrigued by this though.
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#40
Quote by Venice King
I study maths, and to be honest maths being applied to music doesn't really work.

I'm intrigued by this though.

This isn't really math though. This is a way to establish patterns and consistencies using the 12 notes that we have.

...modes and scales are still useless.


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