#1

All right. There seems to be some inexplicable interest in set theory all of a sudden so I thought it might be a good idea to put together a little lesson on it. Xiaoxi did something similar a long time ago, but I’m going to approach it a bit differently than he did so this will be more of a fresh start than a continuation.

The point of this thread is to give a little introduction to set theory and a bit of a taste of how it works. It’s also to gauge interest. If a lot of people seem into it there very well could be a Set Theory Part 1 (you’ll get the whole Part 0 thing in a bit).

There is quite a bit of history and preamble before we get to the money. I highly suggest you read it because it will give you a little bit of context for what this stuff is all about and in my opinion will make it easier to grasp.

I left this bit out of a spoiler tag because it's really important.

Before I tell you exactly what it is, I’ll tell you what it isn’t. Set theory is not math. Read that again.

SET THEORY IS NOT MATH.

In fact, barring the same name and a few similar terminologies, it has very little to do with mathematical set theory, even on a conceptual level. I swear to god if you come in here and go, “Ugh, music isn’t math bro,” I’m gonna cut you.

The point of this thread is to give a little introduction to set theory and a bit of a taste of how it works. It’s also to gauge interest. If a lot of people seem into it there very well could be a Set Theory Part 1 (you’ll get the whole Part 0 thing in a bit).

There is quite a bit of history and preamble before we get to the money. I highly suggest you read it because it will give you a little bit of context for what this stuff is all about and in my opinion will make it easier to grasp.

First of all, what is set theory, or more fully, Pitch Class Set Theory? Set theory is primarily an analytical tool. It was first put forth in the seventies by Allen Forte. Although Forte (pronounced like ‘fort’, not like the dynamic indication in case you were wondering) was building on previous concepts, people more or less attribute what we now think of as set theory to him. Like many theorists (I’m looking at you, Schenker) Forte can be very doctrinaire about his theories. Set theory itself is an attempt to organize the processes in atonal music in a way that was as elegant as the way we organize the processes in tonal music. In some ways it succeeds and in some ways it fails.

Forte would have you believe it’s perfect and would say that if there is a note or group of notes in a piece that you can’t account for by manipulating some sort of set logic, then it’s a comment on the quality of the composition (i.e. it’s bad). Seriously. He says this kind of shit. This is actually eerily like Schenker’s thoughts on pieces that don’t follow his concepts, but that's a story for another day. What’s my view? Set theory, much like harmonic analysis in a tonal piece, can tell us about an aspect of a piece, but it is by no means the be all end all of analysis.

Forte would have you believe it’s perfect and would say that if there is a note or group of notes in a piece that you can’t account for by manipulating some sort of set logic, then it’s a comment on the quality of the composition (i.e. it’s bad). Seriously. He says this kind of shit. This is actually eerily like Schenker’s thoughts on pieces that don’t follow his concepts, but that's a story for another day. What’s my view? Set theory, much like harmonic analysis in a tonal piece, can tell us about an aspect of a piece, but it is by no means the be all end all of analysis.

I left this bit out of a spoiler tag because it's really important.

Before I tell you exactly what it is, I’ll tell you what it isn’t. Set theory is not math. Read that again.

SET THEORY IS NOT MATH.

In fact, barring the same name and a few similar terminologies, it has very little to do with mathematical set theory, even on a conceptual level. I swear to god if you come in here and go, “Ugh, music isn’t math bro,” I’m gonna cut you.

Now that we know what it isn’t, here’s what it is. It’s a way of looking at a group of pitches and drawing comparisons between them. Yup, that’s it. Much like we can say that the notes CEG and FAC are related because they are both major chords, we can relate the notes CEG and BDF# by saying they’re both set class 3-11 (0 3 7) and we can relate the notes CC#FF# and CC#D#G by saying they share an interval class vector (aka Z-relation). Basically, set theory is comparing groups of notes and seeing what that can tell us about how a composer structured the pitch material of a piece (or perhaps how we can structure the pitch material of our own pieces).

So why set theory? What purpose does it serve? Well, in 1908 the world of music changed. A lot of people will say it was ruined, I say it became the most exciting it’s ever been. In 1908 Scheonberg wrote something called Das Buch der Hangenden Garten, which is a song cycle that had the first piece to have no reference to a key (since keys were a thing).

https://www.youtube.com/watch?v=NDkDU4jnzk8&t=20m30s,

From that point on this concept of “atonal” music pervaded much of the art music world. Without the principle of a tonal center to guide our pitch analysis of a piece, we need a new way to look at music and think about music. Enter set theory.

https://www.youtube.com/watch?v=NDkDU4jnzk8&t=20m30s,

From that point on this concept of “atonal” music pervaded much of the art music world. Without the principle of a tonal center to guide our pitch analysis of a piece, we need a new way to look at music and think about music. Enter set theory.

Let’s check out some of the nuts and bolts. The first steps in learning set theory is learning to think in integers, not notes and a touch of Mod12 arithmetic.

When we’re thinking about sets we’re working in what’s called “Pitch Class Space.” Pitch class space is a way of thinking about pitch in a circle, or a way that ignores register. In other words these two notes are exactly the same to us:

This is pitch class space:

In fact, a “pitch class” is all of the pitches of that note, regardless of octave. The pitch class of C includes every C ever from C-5 to C10 and beyond in both directions. Ultimately, this isn’t different from how we think about music theory in general. We can say GBD is a G major chord, regardless of how those notes are situated registrally.

The other thing about pitch class space that does make it different from traditional tonal thinking is that it makes no distinction between enharmonics. G# and Ab are the same note all day long. This is the big reason why integers are so useful to us. We can assign one integer for each chromatic note in the octave and not worry about whether it’s a sharp or a flat or anything else, it’s just that note.

When we’re thinking about sets we’re working in what’s called “Pitch Class Space.” Pitch class space is a way of thinking about pitch in a circle, or a way that ignores register. In other words these two notes are exactly the same to us:

This is pitch class space:

In fact, a “pitch class” is all of the pitches of that note, regardless of octave. The pitch class of C includes every C ever from C-5 to C10 and beyond in both directions. Ultimately, this isn’t different from how we think about music theory in general. We can say GBD is a G major chord, regardless of how those notes are situated registrally.

The other thing about pitch class space that does make it different from traditional tonal thinking is that it makes no distinction between enharmonics. G# and Ab are the same note all day long. This is the big reason why integers are so useful to us. We can assign one integer for each chromatic note in the octave and not worry about whether it’s a sharp or a flat or anything else, it’s just that note.

Here’s how the integers work:

And remember, 1 refers to both Db and C#, 3 refers to both Eb and D#, etc.

Let’s get some examples going. (Yeah that’s right, you gotta do some work.)

List the pitch class in integer notation for each of these (you can use the clock if you need):

a) A#

b) D

c) C

d) F#

e) Bb

Put your answers in spoiler tags like without the space.

Make some observations about which notes shared which numbers and make sure that how this works is clear in your head. If you don’t get this there’s no point in reading further. Ask me and I’ll try and explain it in a different way.

The second key aspect to understanding set theory is Mod12.

If I were to count from 0 to 15 normally I’d do this:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

If I we’re counting with Mod12 I’d do this:

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

As soon as I get to 12 I restart at 0. Sounds weird, but you do it every day when you tell time if you use am/pm. It’s 1pm now, what time will it be in 15 hours? 16 o’clock? Definitely not. It’ll be 4am. You got to 12 and started over again. The only difference between telling time and pitch class space is that in pitch class space we start at 0, not 12. We’ve already applied this concept when we talked about pitch class space. When you go around the circle to B natural or ‘11,’ you don’t call the next C ‘12,’ you just call it ‘0’ again.

This becomes really important when we’re talking about intervals, which in a way is what set theory is all about. This leads us to something called ordered pitch-class intervals. These are exactly what they sound like: the interval from the first pitch class up to the second pitch class.

If we have pitch class 1 up to pitch class 5, what’s the interval between them? Hopefully it’s pretty obvious that it’s 4. In case it’s not, you take the first number and subtract it from the second number (5 – 1 = 4). What we’re saying is that there is 4 semitones between pitch class 1 and 5. In note names we have C# and F which certainly are 4 semitones apart.

But what about pitch class 5 up to pitch class 2? How can we figure out the interval between those two? Second minus first gets us 2 – 5 = -3. That can’t be right. Mod12 to the rescue!

The baby way? Look at the clock above and count. You’ll see there are 9 semitones.

The big boy way? Add 12 to the second number before you subtract the first number (2 + 12) – 5 = 9.

The grown ass man way? Memorize it. But baby steps friends, baby steps.

Here are the pitch class intervals with their traditional counterparts:

0 P1

1 m2

2 M2

3 m3

4 M3

5 P4

6 A4, d5

7 P5

8 m6

9 M6

10 m7

11 M7

Now if you’re smart you’re thinking, “But Zach (that’s my real world name), you told me that register doesn’t matter, why can’t we just drop the top note down an octave and say the distance between pitch class 5 and pitch class 2 is 3?” Baby steps motherfu

EXAMPLES!

Name the pitch class interval between the two classes:

a) 5 2

b) 2 5

b) 4 6

c) 6 4

d) 1 11

e) 11 1

Notice what each pair of answers adds up to

And remember, 1 refers to both Db and C#, 3 refers to both Eb and D#, etc.

Let’s get some examples going. (Yeah that’s right, you gotta do some work.)

List the pitch class in integer notation for each of these (you can use the clock if you need):

a) A#

b) D

c) C

d) F#

e) Bb

Put your answers in spoiler tags like

Answers

Make some observations about which notes shared which numbers and make sure that how this works is clear in your head. If you don’t get this there’s no point in reading further. Ask me and I’ll try and explain it in a different way.

The second key aspect to understanding set theory is Mod12.

If I were to count from 0 to 15 normally I’d do this:

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

If I we’re counting with Mod12 I’d do this:

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

As soon as I get to 12 I restart at 0. Sounds weird, but you do it every day when you tell time if you use am/pm. It’s 1pm now, what time will it be in 15 hours? 16 o’clock? Definitely not. It’ll be 4am. You got to 12 and started over again. The only difference between telling time and pitch class space is that in pitch class space we start at 0, not 12. We’ve already applied this concept when we talked about pitch class space. When you go around the circle to B natural or ‘11,’ you don’t call the next C ‘12,’ you just call it ‘0’ again.

This becomes really important when we’re talking about intervals, which in a way is what set theory is all about. This leads us to something called ordered pitch-class intervals. These are exactly what they sound like: the interval from the first pitch class up to the second pitch class.

If we have pitch class 1 up to pitch class 5, what’s the interval between them? Hopefully it’s pretty obvious that it’s 4. In case it’s not, you take the first number and subtract it from the second number (5 – 1 = 4). What we’re saying is that there is 4 semitones between pitch class 1 and 5. In note names we have C# and F which certainly are 4 semitones apart.

But what about pitch class 5 up to pitch class 2? How can we figure out the interval between those two? Second minus first gets us 2 – 5 = -3. That can’t be right. Mod12 to the rescue!

The baby way? Look at the clock above and count. You’ll see there are 9 semitones.

The big boy way? Add 12 to the second number before you subtract the first number (2 + 12) – 5 = 9.

The grown ass man way? Memorize it. But baby steps friends, baby steps.

Here are the pitch class intervals with their traditional counterparts:

0 P1

1 m2

2 M2

3 m3

4 M3

5 P4

6 A4, d5

7 P5

8 m6

9 M6

10 m7

11 M7

Now if you’re smart you’re thinking, “But Zach (that’s my real world name), you told me that register doesn’t matter, why can’t we just drop the top note down an octave and say the distance between pitch class 5 and pitch class 2 is 3?” Baby steps motherfu

*cker, baby steps.*EXAMPLES!

Name the pitch class interval between the two classes:

a) 5 2

b) 2 5

b) 4 6

c) 6 4

d) 1 11

e) 11 1

Notice what each pair of answers adds up to

#2

So now that we can think in integers and understand a bit of Mod12, let’s group some notes into sets. Basically, a set is just a list of the notes that are present. Try to write the sets that these notes would create:

Notation of sets is a bit of a tricky business. There are a few standards, but we’ll go with what I know. (Also, what I’m getting you to do right now isn’t really a thing that you would ever do, but I want you to see some stuff.) Write your sets with square brackets and commas between pitches. Order the integers from lowest to highest.

Here’s the first example done for you:

a) [0, 4, 7]

Notation of sets is a bit of a tricky business. There are a few standards, but we’ll go with what I know. (Also, what I’m getting you to do right now isn’t really a thing that you would ever do, but I want you to see some stuff.) Write your sets with square brackets and commas between pitches. Order the integers from lowest to highest.

Here’s the first example done for you:

a) [0, 4, 7]

Here’s the last thing I’ll talk about for now (I know, such a tease).

While having the notes in a set is great, it’s not all that useful to us. What you’ve done is just make a list of pitch classes. We want to be able to compare sets, which means we need to get them on the same playing field. The first step in doing that is something called normal order. Normal order is analogous to putting a triad in close position (and because, if you’ll recall, register doesn’t matter, it would also be in root position). Essentially, we want all of the notes packed into the smallest possible interval.

Finding the normal order is relatively simple. Here’s the step by step with an example set:

First list the pitches in a “scale.” It doesn’t matter what note you start on, but I tend to pick alphabetical order for simplicity’s sake.

So we have the pitches Bb, Db, G, B. In alphabetical order that would be Bb, B, Db, G. Now rewrite the Bb above and find the largest interval. The top note of that largest interval is the bottom note of your normal order.

Say what? Watch.

The largest interval when you have Bb, B, Db, G, Bb is between Db and G (an augmented 4th). If we put that G in the bottom and spell up from there we get G, Bb, B, Db, which only spans an augmented 4th and is the most compact spelling of that set. In integer notation we turned a set that looked like [1, 7, 10, 11] with our arbitrary lowest to highest ordering and made it look like [7, 10, 11, 1] with normal order. And remember that with our new found love of Mod12 arithmetic the interval from 7 to 1 is 6 (aka an A4).

This works because you’re inverting the largest interval. The inversion of the largest interval will always be the smallest interval. Ex) a major seventh (the largest possible interval inside an octave) inverts to a minor second (the smallest interval other than a unison).

Put all the sets that you did earlier into normal order. AND write the Adjacency Interval Series for each set. Oh fu

Here's the first set:

a) Normal order: [0, 4, 7] (it was already in normal order before)

AIS: 4-3

While having the notes in a set is great, it’s not all that useful to us. What you’ve done is just make a list of pitch classes. We want to be able to compare sets, which means we need to get them on the same playing field. The first step in doing that is something called normal order. Normal order is analogous to putting a triad in close position (and because, if you’ll recall, register doesn’t matter, it would also be in root position). Essentially, we want all of the notes packed into the smallest possible interval.

Finding the normal order is relatively simple. Here’s the step by step with an example set:

First list the pitches in a “scale.” It doesn’t matter what note you start on, but I tend to pick alphabetical order for simplicity’s sake.

So we have the pitches Bb, Db, G, B. In alphabetical order that would be Bb, B, Db, G. Now rewrite the Bb above and find the largest interval. The top note of that largest interval is the bottom note of your normal order.

Say what? Watch.

The largest interval when you have Bb, B, Db, G, Bb is between Db and G (an augmented 4th). If we put that G in the bottom and spell up from there we get G, Bb, B, Db, which only spans an augmented 4th and is the most compact spelling of that set. In integer notation we turned a set that looked like [1, 7, 10, 11] with our arbitrary lowest to highest ordering and made it look like [7, 10, 11, 1] with normal order. And remember that with our new found love of Mod12 arithmetic the interval from 7 to 1 is 6 (aka an A4).

This works because you’re inverting the largest interval. The inversion of the largest interval will always be the smallest interval. Ex) a major seventh (the largest possible interval inside an octave) inverts to a minor second (the smallest interval other than a unison).

Put all the sets that you did earlier into normal order. AND write the Adjacency Interval Series for each set. Oh fu

*ck. What's that? Exactly what it says. It's a list of the intervals of adjacent pitches in a set. Here's where your knowledge of pitch class intervals and Mod12 will come in real handy. (Haha, you thought you could escape without anymore work. Nope.)*Here's the first set:

a) Normal order: [0, 4, 7] (it was already in normal order before)

AIS: 4-3

Pardon all the image sizing weirdness, I was way too lazy to resize.

So that’s my little set theory primer. There’s a whole hell of a lot more to it than that, not least of all the whys. But I figure 2000+ words is enough for now. Again, if there’s legit interest I’ll do a Part 1. Peace out.

#3

Ugh... I just realized you might want some examples of pieces that uses this sort of logic. Reserving this post to add that stuff later.

#4

I think I'm gonna have to absorb this a bit. But expect my answers soon.

#5

This looks really interesting. I'll do it in the morning.

Maybe this will be an avenue for me to get into this kind of music.

Maybe this will be an avenue for me to get into this kind of music.

#6

Ugh...

how'd you put titles on your spoilers (all in the one post)... might make answer orientation easier??

#7

^

Put answers here.

Obviously, there shouldn't be a space between the / and spoiler in the last tag. Simple as that!

Put answers here.

Obviously, there shouldn't be a space between the / and spoiler in the last tag. Simple as that!

#8

answers[ /spolier]

EDIT: ^Motherfu...

EDIT: ^Motherfu...

#9

`a) A# 10`

b) D 2

c) C 0

d) F# 6

e) Bb 10

f) 7 G

g) 3 Eb

h) 11 B

i) 8 Ab

j) 4 Fb

k) 4 Fb

l) 4 E

`a) 5 2 = 9 `

b) 2 5 = 3

bii) 4 6 = 2

c) 6 4 = 10

d) 1 11 = 10

e) 11 1 = 2

Note: Each pair of answers adds up to 12

`a) [0, 4, 7] `

b) [0, 3, 7]

c) [0, 5, 9]

d) [0, 4, 7]

e) [2, 4, 7, 9]

f) [0, 9, 10, 11]

g) [0, 4, 7]

h) [0, 3, 4, 7, 11]

i) [0, 1, 5, 6, 7, 8]

`Normal Order AIS`

a) [0, 4, 7] 4,3

b) [0, 3, 7] 3,4

c) [5, 9, 0] 4,3

d) [0, 4, 7] 4,3

e) [2, 4, 7, 9] 2,3,2

f) [9, 10, 11, 0] 1,1,1

g) [0, 4, 7] 4,3

h) [11, 0, 3, 4, 7] 1, 3, 1, 3

i) [0, 1, 5, 6, 7, 8] 1, 4, 1, 1, 1

*Last edited by tonibet72 at Mar 28, 2014,*

#10

`a) A# 10`

b) D 2

c) C 0

d) F# 6

e) Bb 10

f) 7 G

g) 3 Eb [color="red"]Not quite.[/color]

h) 11 B

i) 8 Ab

j) 4 Fb

k) 4 Fb

l) 4 E`a) 5 2 = 9`

b) 2 5 = 3

b) 4 6 = 2

c) 6 4 = 10

d) 1 11 = 10

e) 11 1 = 2

Note: Each pair of answers adds up to 12

[color="red"]Right! Which is a great observation to make and will come in handy when we talk about inversion.[/color]`a) [0, 4, 7]`

b) [0, 3, 7]

c) [0, 5, 9]

d) [0, 4, 7]

e) [2, 4, 7, 9]

f) [0, 9, 10, 11]

g) [0, 4, 7]

h) [0, 3, 4, 7, 11] [color="red"]One too many notes here and a couple of wrong integers (hint: 4 and 11 don't belong)[/color]

i) [0, 1, 5, 6, 7, 8] [color="red"]One wrong integer

Also note that your normal orders and AIS for these will be wrong too.[/color]`Normal Order AIS`

a) [0, 4, 7] 4,3

b) [0, 3, 7] 3,4

c) [5, 9, 0] 4,3

d) [0, 4, 7] 4,3

e) [2, 4, 7, 9] 2,3,2

f) [9, 10, 11, 0] 1,1,1

g) [0, 4, 7] 4,3

h) [11, 0, 3, 4, 7] 1, 3, 1, 3

i) [0, 1, 5, 6, 7, 8] 1, 4, 1, 1, 1

Mostly looks pretty good. I've marked a couple of errors in your post. Just one integer wrong and some trouble with the two tricky sets, but over all really well done.

#11

Mostly looks pretty good. I've marked a couple of errors in your post. Just one integer wrong and some trouble with the two tricky sets, but over all really well done.

`Ex1.`

g) 2 Db (duh!)

Grouping into Sets:

h) G,Eb,G,C,Eb,G,Eb C,Eb,G [0, 3, 7,]

(yes, I mistook Eb for Cb, and Eb for E, sounds odd but true)

i) C,Db,G,Ab,F,D#,F Ab,C,Db,F,G [0, 1, 5, 7,8]

(No F#)

[color="Blue"](all in all, general Clef confusion!!!)[/COLOR]

Normal Order: AIS

h) [0, 3, 7] 3,4

i) i) [0, 1, 5, 7, 8] 1, 4, 2, 1

**Come on others... p-l-e-a-s-e!!!!**

we're

workin

on it so

shut up and

be patient mate...

#12

List the pitch class in integer notation for each of these (you can use the clock if you need):

a) A#

b) D

c) C

d) F#

e) Bb

Name the pitch class interval between the two classes:

a) 5 2

b) 2 5

C) 4 6

D) 6 4

E) 1 11

F) 11 1

Notice what each pair of answers adds up to

Basically, a set is just a list of the notes that are present. Try to write the sets that these notes would create:

Normal order, Adjacency Interval series

I used my own method for the normal ordering. See if this makes sense:

E.g. un-normalised set = [1, 5, 7, 9, 11]

Calculate interval from first to last note, 11 – 1 = 10

Calculate interval from 2nd to 1st note = 8

Calculate interval from 3rd to 2nd note = 10

Calculate interval from 4th to 3rd note = 10

Calculate interval from 5th to 4th note = 10

Smallest interval across all notes = 8 when starting on 2nd note, so normalised set = [5, 7, 9, 11, 1]

a) A#

b) D

c) C

d) F#

e) Bb

a) 10

b) 2

c) 0

d) 6

e) 10

f) 7

g) 11

h) 4

i) 8

j) 4

k) 4

l) 4

b) 2

c) 0

d) 6

e) 10

f) 7

g) 11

h) 4

i) 8

j) 4

k) 4

l) 4

Name the pitch class interval between the two classes:

a) 5 2

b) 2 5

C) 4 6

D) 6 4

E) 1 11

F) 11 1

Notice what each pair of answers adds up to

a) 9

b) 3

C) 2

D) 10

E) 10

F) 2

Pairs add up to 12

b) 3

C) 2

D) 10

E) 10

F) 2

Pairs add up to 12

Basically, a set is just a list of the notes that are present. Try to write the sets that these notes would create:

a) CEG = [0, 4, 7]

b) CEbG = [0, 3, 7]

c) FAC = [0, 5, 9]

d) EGC = [0, 4, 7]

e) GDEA = [2, 4, 7, 9] don't know the bass clef, so these were painful

f) ABbCbDbb = [0, 9, 10, 11]

g) CEG = [0, 4, 7]

h) G Eb G C, Eb G Cb Eb Eb = [0, 3, 7, 11]

i) C Db G Ab, F D# F = [0, 1, 3, 5, 7, 8]

b) CEbG = [0, 3, 7]

c) FAC = [0, 5, 9]

d) EGC = [0, 4, 7]

e) GDEA = [2, 4, 7, 9] don't know the bass clef, so these were painful

f) ABbCbDbb = [0, 9, 10, 11]

g) CEG = [0, 4, 7]

h) G Eb G C, Eb G Cb Eb Eb = [0, 3, 7, 11]

i) C Db G Ab, F D# F = [0, 1, 3, 5, 7, 8]

Normal order, Adjacency Interval series

I used my own method for the normal ordering. See if this makes sense:

E.g. un-normalised set = [1, 5, 7, 9, 11]

Calculate interval from first to last note, 11 – 1 = 10

Calculate interval from 2nd to 1st note = 8

Calculate interval from 3rd to 2nd note = 10

Calculate interval from 4th to 3rd note = 10

Calculate interval from 5th to 4th note = 10

Smallest interval across all notes = 8 when starting on 2nd note, so normalised set = [5, 7, 9, 11, 1]

a) [0, 4, 7] -> [0, 4, 7] 4, 3

b) [0, 3, 7] -> [0, 3, 7] 3, 4

c) [0, 5, 9] -> [5, 9, 0] 4, 3

d) same as a)

e) [2, 4, 7, 9] -> [2, 4, 7, 9] 2, 3, 2

f) [0, 9, 10, 11] -> [9, 10, 11, 0] 1, 1, 1

g) same as a)

h) [0, 3, 7, 11] -> [7, 11, 0, 3] 4, 1, 3

i) [0, 1, 3, 5, 7, 8] -> [0, 1, 3, 5, 7, 8] 1, 2, 2, 2, 1

b) [0, 3, 7] -> [0, 3, 7] 3, 4

c) [0, 5, 9] -> [5, 9, 0] 4, 3

d) same as a)

e) [2, 4, 7, 9] -> [2, 4, 7, 9] 2, 3, 2

f) [0, 9, 10, 11] -> [9, 10, 11, 0] 1, 1, 1

g) same as a)

h) [0, 3, 7, 11] -> [7, 11, 0, 3] 4, 1, 3

i) [0, 1, 3, 5, 7, 8] -> [0, 1, 3, 5, 7, 8] 1, 2, 2, 2, 1

*Last edited by Jehannum at Mar 28, 2014,*

#13

Tip: An easy way for people used to playing guitar to think about integer notation is to imagine a guitar tuned to drop C. The fret number you would need to play to sound a note on the low C string corresponds to it's integer.

a) 10

b) 2

c) 0

d) 6

e) 10

f) 7

g) 1

h) 11

i) 8

j) 4

k) 4

l) 4

a) 9

b) 3

b) 2 (This was really mean, my brain won't let me have two b's, but it also won't let me give different letters to the ones in the original question, I hope you feel bad)

c) 10

d) 10

e) 2

Each pair of answers adds up to 12.

a) [0, 4, 7]

b) [0, 3, 7]

c) [0, 5, 9]

d) [0, 4, 7]

e) [2, 4, 7, 9]

f) [5, 6, 8, 9]

g) [0, 4, 9]

h) [0, 3, 7, 11]

i) [0, 1, 3, 5, 7, 8]

a) [0, 4, 7] AIS: 4-3

b) [0, 3, 7] AIS: 3-4

c) [5, 9, 0] AIS: 4-3

d) [0, 4, 7] AIS: 4-3

e) [2, 4, 7, 9] AIS: 2-3-2

f) [5, 6, 8, 9] AIS: 1-2-1

g) [9, 0, 4] AIS: 3-4

h) [7, 11, 0, 3] AIS: 4-1-3

i) [0, 1, 3, 5, 7, 8] AIS: 1-2-2-2-1

b) 2

c) 0

d) 6

e) 10

f) 7

g) 1

h) 11

i) 8

j) 4

k) 4

l) 4

a) 9

b) 3

b) 2 (This was really mean, my brain won't let me have two b's, but it also won't let me give different letters to the ones in the original question, I hope you feel bad)

c) 10

d) 10

e) 2

Each pair of answers adds up to 12.

a) [0, 4, 7]

b) [0, 3, 7]

c) [0, 5, 9]

d) [0, 4, 7]

e) [2, 4, 7, 9]

f) [5, 6, 8, 9]

g) [0, 4, 9]

h) [0, 3, 7, 11]

i) [0, 1, 3, 5, 7, 8]

a) [0, 4, 7] AIS: 4-3

b) [0, 3, 7] AIS: 3-4

c) [5, 9, 0] AIS: 4-3

d) [0, 4, 7] AIS: 4-3

e) [2, 4, 7, 9] AIS: 2-3-2

f) [5, 6, 8, 9] AIS: 1-2-1

g) [9, 0, 4] AIS: 3-4

h) [7, 11, 0, 3] AIS: 4-1-3

i) [0, 1, 3, 5, 7, 8] AIS: 1-2-2-2-1

#14

Okay so I don't grasp the concept.

I understand how it works, but in which way does this improve composing?

For example, no difference for enharmonic notes. Enharmonic is not merely a notational difference, but a way in how notes pull and push to each other.

Maybe it's merely my opinion, but this pull/push is not arbitrary, and a big part of how harmony develops through a piece. I'd go as far to say that the pitches themselves are not important, rather the way how you get from one note to the other is what music is all about.

I mean take an diminished chord and accent it or force it as a tonal center, and it would most likely sound out of place and/or jarring, but make it as a subtitute for a V in a classical 2-5-1 and it works all day long..even from time to time in simple pop songs.

We know that this works because of the concept of "2-5-1", yet an integer notation in itself would not give this information.

Also no difference for octaves in set theory? I'd say that especially with pieces with multiple (thought out) voices the octave space is very important.

Rules of counterpoint for instance have no voice crossing, because the octave ranges in which for example a soprano and alto voice stay in "their own" dictates how your ear perceives the musical motion.

I don't know, maybe I'm looking at it fom a wrong perspective. I did read it's from analysis point of view, but I find that working from chord to chord basis and understand it's place relative to the tonic is perfectly analyzed with accompanying sheet style writing.

I understand how it works, but in which way does this improve composing?

For example, no difference for enharmonic notes. Enharmonic is not merely a notational difference, but a way in how notes pull and push to each other.

Maybe it's merely my opinion, but this pull/push is not arbitrary, and a big part of how harmony develops through a piece. I'd go as far to say that the pitches themselves are not important, rather the way how you get from one note to the other is what music is all about.

I mean take an diminished chord and accent it or force it as a tonal center, and it would most likely sound out of place and/or jarring, but make it as a subtitute for a V in a classical 2-5-1 and it works all day long..even from time to time in simple pop songs.

We know that this works because of the concept of "2-5-1", yet an integer notation in itself would not give this information.

Also no difference for octaves in set theory? I'd say that especially with pieces with multiple (thought out) voices the octave space is very important.

Rules of counterpoint for instance have no voice crossing, because the octave ranges in which for example a soprano and alto voice stay in "their own" dictates how your ear perceives the musical motion.

I don't know, maybe I'm looking at it fom a wrong perspective. I did read it's from analysis point of view, but I find that working from chord to chord basis and understand it's place relative to the tonic is perfectly analyzed with accompanying sheet style writing.

*Last edited by xxdarrenxx at Mar 29, 2014,*

#15

Your problem is you're evaluating set theory in terms of it's power to explain a type of music it was never meant to explain. Set theory is supposed to help explain the work of post-tonal composers like Schoenberg, Berg, Webern, Boulez, Babbitt and similar. I've never seen anyone claim that it has particular insights with regards to how tonal music works.

#16

Your problem is you're evaluating set theory in terms of it's power to explain a type of music it was never meant to explain. Set theory is supposed to help explain the work of post-tonal composers like Schoenberg, Berg, Webern, Boulez, Babbitt and similar. I've never seen anyone claim that it has particular insights with regards to how tonal music works.

@xxdarrenxx: yeah it great that you have so many queries, jump on board, perhaps we can all learn something "new"... jazz is offering an introduction to help us understand that, lets show him a little respect and get involved... peace!

#17

A few corrections and then I'll get the Darren's post.

You've raised some legitimate concerns with set theory and some not so legitimate concerns. In general, what Nietsche said. Set theory is not an analytical tool for tonal music. You could argue some potential usefulness in analyzing horizontal structures like motives, but it's beyond useless for vertical structures. Look at minor chords and major chords. In terms of set theory, they're almost identical. Their set class is actually the exact same (0 3 7), which makes sense for set theory. If you have a major chord [0,4,7] and a minor chord [0,3,7] they're inversionally equivalent (haven't gone over that yet) and therefore the same set class. In tonal music you could never make that equivalence. In atonal music on the other hand, the two structures (major chord vs minor chord) aren't all that different in context.

As for the enharmonic thing, again, in a tonal setting you might have something there, but in an atonal setting that distinction is purely psychological. A composer might choose to write flats because he wants his performer to think 'down,' but that doesn't affect the concept of his pitch material.

The register thing is the one valid concern you raise. Set theory is dangerous in that sense because it disconnects the analysis from a very important aspect of the music. It's ironic that Webern is so commonly analyzed with set analysis, yet no composer has ever made register more important to their style. That is definitely one of the pitfalls, which is why I said this:

Set theory is one avenue into understanding a piece, but it's not the only avenue and can't be the only method of analysis to get an understanding of a piece. More often than not it's a starting point into understanding something deeper about what a composer is doing.

though. It's an important discussion to have.

`Ex1.`

g) 2 Db (duh!)

Grouping into Sets:

h) G,Eb,G,C,Eb,G,Eb C,Eb,G [0, 3, 7,] [color="red"]Still not quite. It's [0,3,7,11][/color]

(yes, I mistook Eb for Cb, and Eb for E, sounds odd but true)

i) C,Db,G,Ab,F,D#,F Ab,C,Db,F,G [0, 1, 5, 7,8] [color="red"]Close again, but missing an Eb. Should be [0,1,3,5,7,8][/color]

(No F#)

[color="Blue"](all in all, general Clef confusion!!!)[/COLOR]

Normal Order: AIS

h) [0, 3, 7] 3,4

i) i) [0, 1, [color="red"]3, [/color]5, 7, 8] 1, 4, 2, 1

a) 10

b) 2

c) 0

d) 6

e) 10

f) 7

g) 11 Nope

h) 4 And nope. Check again

i) 8

j) 4

k) 4

l) 4

a) 9

b) 3

C) 2

D) 10

E) 10

F) 2

Pairs add up to 12

a) CEG = [0, 4, 7]

b) CEbG = [0, 3, 7]

c) FAC = [0, 5, 9]

d) EGC = [0, 4, 7]

e) GDEA = [2, 4, 7, 9] don't know the bass clef, so these were painful It'll be k.

f) ABbCbDbb = [0, 9, 10, 11]

g) CEG = [0, 4, 7]

h) G Eb G C, Eb G Cb Eb Eb = [0, 3, 7, 11]

i) C Db G Ab, F D# F = [0, 1, 3, 5, 7, 8]

Normal order, Adjacency Interval series

I used my own method for the normal ordering. See if this makes sense:

E.g. un-normalised set = [1, 5, 7, 9, 11]

Calculate interval from first to last note, 11 – 1 = 10

Calculate interval from 2nd to 1st note = 8

Calculate interval from 3rd to 2nd note = 10

Calculate interval from 4th to 3rd note = 10

Calculate interval from 5th to 4th note = 10

Smallest interval across all notes = 8 when starting on 2nd note, so normalised set = [5, 7, 9, 11, 1]

This is EXACTLY how I calculate normal order and I think it's a faster method to do quickly in your head. I just find the method that I posted is more pedagogical. This way totally works though.a) [0, 4, 7] -> [0, 4, 7] 4, 3

b) [0, 3, 7] -> [0, 3, 7] 3, 4

c) [0, 5, 9] -> [5, 9, 0] 4, 3

d) same as a)

e) [2, 4, 7, 9] -> [2, 4, 7, 9] 2, 3, 2

f) [0, 9, 10, 11] -> [9, 10, 11, 0] 1, 1, 1

g) same as a)

h) [0, 3, 7, 11] -> [7, 11, 0, 3] 4, 1, 3 This is wrong, but that's my fault not yours. Stay tuned.

i) [0, 1, 3, 5, 7, 8] -> [0, 1, 3, 5, 7, 8] 1, 2, 2, 2, 1

Tip: An easy way for people used to playing guitar to think about integer notation is to imagine a guitar tuned to drop C. The fret number you would need to play to sound a note on the low C string corresponds to it's integer.a) 10

b) 2

c) 0

d) 6

e) 10

f) 7

g) 1

h) 11

i) 8

j) 4

k) 4

l) 4

a) 9

b) 3

b) 2 (This was really mean, my brain won't let me have two b's, but it also won't let me give different letters to the ones in the original question, I hope you feel bad) I don't. I noticed that before I posted it and was too lazy to fix it

c) 10

d) 10

e) 2

Each pair of answers adds up to 12.

a) [0, 4, 7]

b) [0, 3, 7]

c) [0, 5, 9]

d) [0, 4, 7]

e) [2, 4, 7, 9]

f) [5, 6, 8, 9] Wrong clef. In bass clef it's [0,9,10,11]

g) [0, 4, 9] [0,4,7]

h) [0, 3, 7, 11]

i) [0, 1, 3, 5, 7, 8]

a) [0, 4, 7] AIS: 4-3

b) [0, 3, 7] AIS: 3-4

c) [5, 9, 0] AIS: 4-3

d) [0, 4, 7] AIS: 4-3

e) [2, 4, 7, 9] AIS: 2-3-2

f) [5, 6, 8, 9] AIS: 1-2-1

g) [9, 0, 4] AIS: 3-4

h) [7, 11, 0, 3] AIS: 4-1-3 This isn't right, but that's my fault not yours. Stay tuned.

i) [0, 1, 3, 5, 7, 8] AIS: 1-2-2-2-1

Done really well, couple of silly mistakes, but I can see you get it. Your answers made me realize that I actually missed something in talking about normal order. I'll post an update.

Okay so I don't grasp the concept.

I understand how it works, but in which way does this improve composing?

For example, no difference for enharmonic notes. Enharmonic is not merely a notational difference, but a way in how notes pull and push to each other.

Maybe it's merely my opinion, but this pull/push is not arbitrary, and a big part of how harmony develops through a piece. I'd go as far to say that the pitches themselves are not important, rather the way how you get from one note to the other is what music is all about.

I mean take an diminished chord and accent it or force it as a tonal center, and it would most likely sound out of place and/or jarring, but make it as a subtitute for a V in a classical 2-5-1 and it works all day long..even from time to time in simple pop songs.

We know that this works because of the concept of "2-5-1", yet an integer notation in itself would not give this information.

Also no difference for octaves in set theory? I'd say that especially with pieces with multiple (thought out) voices the octave space is very important.

Rules of counterpoint for instance have no voice crossing, because the octave ranges in which for example a soprano and alto voice stay in "their own" dictates how your ear perceives the musical motion.

I don't know, maybe I'm looking at it fom a wrong perspective. I did read it's from analysis point of view, but I find that working from chord to chord basis and understand it's place relative to the tonic is perfectly analyzed with accompanying sheet style writing.

You've raised some legitimate concerns with set theory and some not so legitimate concerns. In general, what Nietsche said. Set theory is not an analytical tool for tonal music. You could argue some potential usefulness in analyzing horizontal structures like motives, but it's beyond useless for vertical structures. Look at minor chords and major chords. In terms of set theory, they're almost identical. Their set class is actually the exact same (0 3 7), which makes sense for set theory. If you have a major chord [0,4,7] and a minor chord [0,3,7] they're inversionally equivalent (haven't gone over that yet) and therefore the same set class. In tonal music you could never make that equivalence. In atonal music on the other hand, the two structures (major chord vs minor chord) aren't all that different in context.

As for the enharmonic thing, again, in a tonal setting you might have something there, but in an atonal setting that distinction is purely psychological. A composer might choose to write flats because he wants his performer to think 'down,' but that doesn't affect the concept of his pitch material.

The register thing is the one valid concern you raise. Set theory is dangerous in that sense because it disconnects the analysis from a very important aspect of the music. It's ironic that Webern is so commonly analyzed with set analysis, yet no composer has ever made register more important to their style. That is definitely one of the pitfalls, which is why I said this:

Set theory, much like harmonic analysis in a tonal piece, can tell us about an aspect of a piece, but it is by no means the be all end all of analysis.

Set theory is one avenue into understanding a piece, but it's not the only avenue and can't be the only method of analysis to get an understanding of a piece. More often than not it's a starting point into understanding something deeper about what a composer is doing.

though. It's an important discussion to have.

#18

`a) A# 10`

b) D 2

c) C 0

d) F# 6

e) Bb 10

f) 7 G

g) 3 Eb

h) 11 B

i) 8 Ab

j) 4 Fb

k) 4 Fb

l) 4 E

`a) 5 2 = 9 `

b) 2 5 = 3

bii) 4 6 = 2

c) 6 4 = 10

d) 1 11 = 10

e) 11 1 = 2

Note: Each pair of answers adds up to 12

`a) [0, 4, 7] `

b) [0, 3, 7]

c) [0, 5, 9]

d) [0, 4, 7]

e) [2, 4, 7, 9]

f) [0, 9, 10, 11]

g) [0, 4, 7]

h) [0, 3, 7]

i) [0, 1, 5, 6, 7, 8]

`Normal Order AIS`

a) [0, 4, 7] 4,3

b) [0, 3, 7] 3,4

c) [5, 9, 0] 4,3

d) [0, 4, 7] 4,3

e) [2, 4, 7, 9] 2,3,2

f) [9, 10, 11, 0] 1,1,1

g) [0, 4, 7] 4,3

h) [11, 0, 3, 4, 7] 1, 3, 1, 3

i) [0, 1, 5, 6, 7, 8] 1, 4, 1, 1, 1

Ok...I did this in a hurry. So, I hope I didn't screw up a lot. lol

#19

Reply to Jazzrock:

I see what you say about horizontal vs vertical.

I must say the distinction is almost a philosophical debate. On one hand you have something like Bach, which goes purely voice by voice basis, and it "works".

On the other hand, and espcially in modern music, a guitar player can play barre chords all day long on a song, and it could easily become a "hit song". So the horizontal aspect is largely disregarded.

Fun fact is that some (popular) electronic producers use voiceleading almost accidentally because of inputting notes in a piano roll. They can see individual bars of midi (multiple voices), and often work with layers up to 8 voices at once and input chords which are impossible to play, but work perfectly voice leading wise.

So is an inversion different or not? I'd say it does matter, because I heard pop songs that feauture inverted piano chords, and I heard them played by people who use merely root positions on the piano. The difference was noticeable and it felt not exactly right. They have their own sound albeit it being as minimal as decoration.

I now see that set theory might only be usefull for atonal pieces, and that I'm discussing perhaps a different matter.

On a small note.. Traditional theory does handle inversions as redundant where needed.

A major 7th interval is not done in counterpoint, because in that case the perceived sound is still a minor 2nd interval. So they do at times regard inversions as not making a difference in favour of the actual perceived sound.

Good posting however.

I see what you say about horizontal vs vertical.

I must say the distinction is almost a philosophical debate. On one hand you have something like Bach, which goes purely voice by voice basis, and it "works".

On the other hand, and espcially in modern music, a guitar player can play barre chords all day long on a song, and it could easily become a "hit song". So the horizontal aspect is largely disregarded.

Fun fact is that some (popular) electronic producers use voiceleading almost accidentally because of inputting notes in a piano roll. They can see individual bars of midi (multiple voices), and often work with layers up to 8 voices at once and input chords which are impossible to play, but work perfectly voice leading wise.

So is an inversion different or not? I'd say it does matter, because I heard pop songs that feauture inverted piano chords, and I heard them played by people who use merely root positions on the piano. The difference was noticeable and it felt not exactly right. They have their own sound albeit it being as minimal as decoration.

I now see that set theory might only be usefull for atonal pieces, and that I'm discussing perhaps a different matter.

On a small note.. Traditional theory does handle inversions as redundant where needed.

A major 7th interval is not done in counterpoint, because in that case the perceived sound is still a minor 2nd interval. So they do at times regard inversions as not making a difference in favour of the actual perceived sound.

Good posting however.

*Last edited by xxdarrenxx at Mar 30, 2014,*

#20

Grouping into Sets:

h) G,Eb,G,C,Eb,G,Eb C,Eb,G [0, 3, 7,]

Still not quite. It's [0,3,7,

Upper Cb! how'd that get in there? Ooops

i) C,Db,G,Ab,F,D#,F Ab,C,Db,F,G [0, 1, 5, 7,8]

Close again, but missing an

2 D's?? toni what?.. seriously I dunno man? lol

h) G,Eb,G,C,Eb,G,Eb C,Eb,G [0, 3, 7,]

Still not quite. It's [0,3,7,

**11**]Upper Cb! how'd that get in there? Ooops

i) C,Db,G,Ab,F,D#,F Ab,C,Db,F,G [0, 1, 5, 7,8]

Close again, but missing an

**Eb**. Should be [0,1,3,5,7,8]2 D's?? toni what?.. seriously I dunno man? lol

#21

Integers.

- 10
- 2
- 0
- 6
- 10
- 7
- 1
- 11
- 8
- 4
- 4
- 4

Mod12

- 9
- 3
- 2
- 10
- 1
- 11

Sets

- [037]
- [059]
- [047]
- [2479]
- [5789]
- [049]
- [3 7 11]
- [35]
- [037]
- [0178]

I'll do the last one later. Im sick on the couch atm (so I finally had time to do this) and I'm tired of thinking.

*Last edited by Duaneclapdrix at Apr 4, 2014,*

#22

I'll get around to correcting and replying on the weekend. My graduation recital is tomorrow so I've been going mental preparing for that.

#23

Ugh, this makes me wish I'd spent more time getting good at reading sheet music. I'm understanding the concepts and really interested in what you could use them for, but to do the exercises would take forever because I have the extra step of figuring out every single note.

I can name any note on a guitar neck instantly and tell you the interval between any two notes just by looking at them (my ear still needs some work on this, but it's getting there). I really wish I could do the same when looking at a staff

I know I'm kinda going off topic, but can you see intervals in sheet music in a way that's similar to seeing them on the fretboard? Like, I know what a m3 looks like on the fretboard, or a P5 or M7 etc. I think if I could get used to just seeing the intervals then the note names would come a lot easier and wouldn't really matter anyway.

Anywho, I'm definitely interested in this subject and want to see where it goes. I might go ahead and spend

I can name any note on a guitar neck instantly and tell you the interval between any two notes just by looking at them (my ear still needs some work on this, but it's getting there). I really wish I could do the same when looking at a staff

I know I'm kinda going off topic, but can you see intervals in sheet music in a way that's similar to seeing them on the fretboard? Like, I know what a m3 looks like on the fretboard, or a P5 or M7 etc. I think if I could get used to just seeing the intervals then the note names would come a lot easier and wouldn't really matter anyway.

Anywho, I'm definitely interested in this subject and want to see where it goes. I might go ahead and spend

*way*too much time trying to do the work later. I guess that would count as practice for reading sheet music*and*this set theory stuff.
#24

I went ahead and did it, and I'm glad I did. After posting the above and then working through this I did start to see the intervals, which prompted me know the pitch without having to go all Every Good Boy Does Fine and feel like a 4th grader.

I feel like I caught on pretty quick and can't wait til your next one. I want to see where this goes.

Edit: I went and double checked my work and saw where I was wrong on a few things. I think this is right:

a) 10

b) 2

c) 0

d) 6

e) 10

f) 7

g) 1

h) 11

i) 8

j) 4

k) 4

l) 2?

a)9

b)3

b)2

c)10

d)10

e)2

They're inversions, they add up to an octave. I also see an easier way (to me). If the second number is smaller, just subtract it from the first number, then subtract that from 12... which is really the exact same thing as your way, but makes more sense in my head.

a) [0, 4, 7]

b) [0, 3, 7]

c) [0, 5, 9]

d) [0, 4, 7]

e) [2, 4, 7, 9]

f) [0, 9, 10, 11]

g) [0, 4, 7]

h) [0, 3, 7, 11]

i) [0, 1, 3, 5, 7, 8]

a) [0, 4, 7] AIS: 4-3

b) [0, 3, 7] AIS: 3-4

c) [5, 9, 0] AIS: 4-3

d) [0, 4, 7] AIS: 4-3

e) [2, 4, 7, 9] AIS: 2-3-2

f) [9, 10, 11, 0] AIS: 1-1-1

g) [0, 4, 7] AIS: 4-3

h) [11, 0, 3, 7] AIS: 1-3-4

i) [0, 1, 3, 5, 7, 8] AIS: 1-2-2-2-1

a) 10

b) 2

c) 0

d) 6

e) 10

f) 7

g) 3

h) 11

i) 8

j) 4

k) 4

l) 2

a)9

b)3

b)2

c)10

d)10

e)2

They're inversions, they add up to an octave. I also see an easier way (to me). If the second number is smaller, just subtract it from the first number, then subtract that from 12... which is really the exact same thing as your way, but makes more sense in my head.

a) [0, 4, 7]

b) [0, 3, 7]

c) [0, 5, 9]

d) [0, 4, 7]

e) [2, 4, 7, 9]

f) [1, 9, 10, 11]

g) [0, 4, 7]

h) [0, 3, 4, 7, 11]

i) [0, 1, 7, 8]

a) [0, 4, 7] AIS: 4-3

b) [0, 3, 7] AIS: 3-4

c) [5, 9, 0] AIS: 4-3

d) [0, 4, 7] AIS: 4-3

e) [2, 4, 7, 9] AIS: 2-3-2

f) [9, 10, 11, 1] AIS: 1-1-2

g) [0, 4, 7] AIS: 4-3

h) [11, 0, 3, 4, 7] AIS: 1-3-1-3

i) [7, 8, 0, 1] AIS: 1-5-1

I feel like I caught on pretty quick and can't wait til your next one. I want to see where this goes.

Edit: I went and double checked my work and saw where I was wrong on a few things. I think this is right:

a) 10

b) 2

c) 0

d) 6

e) 10

f) 7

g) 1

h) 11

i) 8

j) 4

k) 4

l) 2?

a)9

b)3

b)2

c)10

d)10

e)2

They're inversions, they add up to an octave. I also see an easier way (to me). If the second number is smaller, just subtract it from the first number, then subtract that from 12... which is really the exact same thing as your way, but makes more sense in my head.

a) [0, 4, 7]

b) [0, 3, 7]

c) [0, 5, 9]

d) [0, 4, 7]

e) [2, 4, 7, 9]

f) [0, 9, 10, 11]

g) [0, 4, 7]

h) [0, 3, 7, 11]

i) [0, 1, 3, 5, 7, 8]

a) [0, 4, 7] AIS: 4-3

b) [0, 3, 7] AIS: 3-4

c) [5, 9, 0] AIS: 4-3

d) [0, 4, 7] AIS: 4-3

e) [2, 4, 7, 9] AIS: 2-3-2

f) [9, 10, 11, 0] AIS: 1-1-1

g) [0, 4, 7] AIS: 4-3

h) [11, 0, 3, 7] AIS: 1-3-4

i) [0, 1, 3, 5, 7, 8] AIS: 1-2-2-2-1

a) 10

b) 2

c) 0

d) 6

e) 10

f) 7

g) 3

h) 11

i) 8

j) 4

k) 4

l) 2

a)9

b)3

b)2

c)10

d)10

e)2

They're inversions, they add up to an octave. I also see an easier way (to me). If the second number is smaller, just subtract it from the first number, then subtract that from 12... which is really the exact same thing as your way, but makes more sense in my head.

a) [0, 4, 7]

b) [0, 3, 7]

c) [0, 5, 9]

d) [0, 4, 7]

e) [2, 4, 7, 9]

f) [1, 9, 10, 11]

g) [0, 4, 7]

h) [0, 3, 4, 7, 11]

i) [0, 1, 7, 8]

a) [0, 4, 7] AIS: 4-3

b) [0, 3, 7] AIS: 3-4

c) [5, 9, 0] AIS: 4-3

d) [0, 4, 7] AIS: 4-3

e) [2, 4, 7, 9] AIS: 2-3-2

f) [9, 10, 11, 1] AIS: 1-1-2

g) [0, 4, 7] AIS: 4-3

h) [11, 0, 3, 4, 7] AIS: 1-3-1-3

i) [7, 8, 0, 1] AIS: 1-5-1

*Last edited by The4thHorsemen at Apr 4, 2014,*

#25

Ugh, this makes me wish I'd spent more time getting good at reading sheet music.. ..but to do the exercises would take forever because I have the extra step of figuring out every single note.

I can name any note on a guitar neck... I really wish I could do the same when looking at a staff .

**...I just use this...**(i'm pleased this thread's still going)

#26

Ah, I see I messed up my answer for that note on the tenor clef. I'd never tried to read it before, but figured "Well, treble clef has a stylized G pinpointing the G and Bass clef has a stylized F marking the F... this thing looks like a B so I guess that line must be B" but apparently it's a stylized C.

#27

...bump?

How'd the recital go?

How'd the recital go?

#28

...bump?

How'd the recital go?

I thought everyone had forgotten about this. Recital went okay, solid 6/10. The pieces I was hoping would go really well went really well and the rest... well, they were a thing I guess. Got an A so I can't complain too much.

Here's the answer key, I won't go through everyone's answers so you'll have to check yourself and ask if you don't get how I arrived at an answer.

Integers

a) 10

b) 2

c) 0

d) 6

e) 10

f) 7

g) 1

h) 11

i) 8

j) 4

k) 4

l) 4

Mod 12 Arithmetic

a) 9

b) 3

c) 2

d) 10

e) 10

f) 2

All the answers add up to 12. That's because these intervals are inversions of one another. We'll get to more inversion stuff later.

Sets (arbitrary order)

a) [0, 4, 7]

b) [0, 3 ,7]

c) [0, 5, 9]

d) [0, 4, 7]

e) [2, 4, 7, 9]

f) [0, 9 ,10 , 11]

g) [0, 4, 7]

h) [0, 3, 7 11]

i) [0, 1, 3, 5, 7, 8]

Sets (normal order)

I wish I had thought these through more and made it so the arbitrary order and normal order were different more often.

a) [0, 4, 7] AIS: 4-3

b) [0, 3 ,7] AIS: 3-4

c) [5, 9, 0] AIS: 4-3

d) [0, 4, 7] AIS: 4-3

e) [2, 4, 7, 9] AIS: 2-3-2

f) [9, 10 , 11, 0] AIS: 1-1-1

g) [0, 4, 7] AIS: 4-3

h) [11, 0, 3, 7] AIS: 1-3-4

i) [0, 1, 3, 5, 7, 8] AIS: 1-2-2-2-1

NOTE THAT h) IS NOT [7, 11, 0, 3].

There's a reason for that.

I'll get some more stuff up maybe today or tomorrow. I should really finish/start that last essay though...

*Last edited by jazz_rock_feel at Apr 16, 2014,*

#29

Cool cool cool.

#30

So hopefully you guys have gotten what I’ve said so far (if not ask!) To continue on, I have to complete an incomplete thought about normal order. Take a look at example h) from putting the sets into normal order. I said the answer isn't [7, 11, 0, 3] it's [11, 0, 3, 7]. But check it out. The distance between 11 and 7 and 7 and 3 is the same (pitch class interval 8). So why is [11, 0, 3, 7] the normal order and not [7, 11, 0, 3]? Because there’s a tie and to break this tie we need to look at the first note and NEXT TO LAST note. In this case we have 7 -> 0 and 11 -> 3. 7 -> 0 has a pitch class interval of 5 and 11 -> has a PCI of 4. The smaller interval is the one we want to go with because this means that the set is in it's most compact version and more tightly packed to the left, which is "better."

Seems arbitrary and it really is, along with a lot of things about set theory. The key is that it doesn't matter. Both are the same set, but what we're trying to do with set theory is get sets into a standard configuration so we can compare them. So what we need is consistency, which this provides.

Moral of the story? You can't get lazy. You have to check every interval and make sure to do any necessary tie breaking.

Do the normal order for both these sets:

Also, sometimes there’s no way to break the tie. We call these sets “transpositionally symmetrical,” which basically means that the same set of notes is reproduced under one or more transpositions. In other words, a feature of a transpositionally symmetrical set is that it will have more than one valid normal order.

If you know tonal music you probably know about the augmented triad and the fully diminished seventh chord. The augmented triad on C has a normal order of [0, 4, 8]. When you transpose it up a major third (PCI 4) you get a normal order of [4, 8, 0] and when you transpose it up PCI 8 you get [8, 0, 4]. Same set, three different, valid normal orders. The augmented triad is a transpositionally symmetrical set, as is the fully diminished seventh chord.

Put this set into normal order (bonus point if you can name the chord type):

So now we can move on from corrections. We know that with set theory we have octave equivalence and we also know that we have transpositional equivalence, if I didn’t mention it explicitly hopefully you figured that out. In case you didn’t, transpositional equivalence exists in tonal music as well. Just look at the idea of triads. All triads, regardless of which pitches exist in them, that are made of stacking a major third and a minor third are called major triads. Same deal goes for sets.

Sets with the same AIS are transpositionally equivalent and we can look at them as being “the same.” But set theory says, “fu

Also important to note: inversion in set theory IS NOT THE SAME as inversion in tonal music. In set theory inversion means mirror inversion. So what is mirror inversion? Well, first we’ll do the shitty slow way that you’ll never do again when I show you the good fast way.

For a “true” inversion you’ll mirror the notes in the set around an axis (namely the axis of 0). This is where that idea of inversions adding up to twelve will come in handy. This is harder to explain than it is to show so here it is:

0 inverts to 0

1 inverts to 11

2 inverts to 10

3 inverts to 9

4 inverts to 8

5 inverts to 7

6 inverts to 6

7 inverts to 5

8 inverts to 4

9 inverts to 3

10 inverts to 2

11 inverts to 1

Think about folding the clock in half. Here’s a hastily edited illustration:

So invert these sets (all are in normal order) and put them into normal order if the inversions aren’t already there (hint: subtract each pitch class from 12):

a) [0, 4, 7] (major triad)

b) [0, 3, 7] (minor triad)

Inversionally equivalent!!!!

c) [0, 2, 4, 6, 8, 10]

d) [0, 3, 6, 9]

e) [0, 3, 4, 7, 8, 11]

f) [5, 6, 8]

g) [1, 3, 7, 8]

Now, there’s some notational things like TnI that go along with inverting sets, but frankly I’ve never run into a use for how much nomenclature there is for half this stuff. If you want to know more then you can research it yourself. The important concept is inversion.

Now for the promised quick way. Don’t be a lazy turd and skip to here without doing the work above. The quick way, and why the idea of AIS was introduced, is that if you reverse the order of the intervals in a set you get the inversion of that set. So for example, the set [0, 1, 6] has an AIS of 1-5. If you reverse that to 5-1 and apply it to the original set you get [0, 5, 6] which just so happens to be that set inverted and transposed back to the original starting note. That is about a million times faster and more convenient that inverting to some arbitrary transposition with mirror inversion. So now that you know the good way, invert these sets:

h) [4, 7, 11]

i) [5, 8, 10]

j) [2, 3, 6, 7, 9]

k) [8, 11, 0, 2]

l) [4, 7, 8, 0] (is your inversion normal order????

Hint: nope. Can you think of why? Think about the start of this lesson.)

Get it? Good.

So now we know how to invert. At this point we can take one more step to finding the ever-elusive meaning of “set class.” Because sets are inversionally equivalent, to find the best version of the normal order for a set we have to not only find its most compact ordering, but the most compact ordering of its inversion and see which one is more compact (we like things being compact).

This final maximum compact ordering of the set is called (you guessed it) the “best normal order.” To find the best normal order of a set first find the normal order, invert it (the quick way), put the inversion in normal order and compare the interval between the first and next to last note (the interval between the first and last will be the same). Whichever is smaller is your best normal order and just like before, if there’s a tie keep going to the first and third to last note and keep going until you break the tie. If there truly is a tie, which is not uncommon, then that’s cool. We call sets like this inversionally symmetrical. The minor 7th chord is an example of a set that is inversionally symmetrical.

Here’s an example:

NOTE: The 6 between the E and the Ab in the first measure should be a 4

So the normal order is [4, 7, 8, 0], but the best normal order is [8, 9, 0, 4] (remember, two sets that are inversionally equivalent are said to be in the same set class).

Put these sets into their best normal order:

Are any of the sets members of the same set class?

Recap!

We consider all notes of the same pitch class to be equivalent (i.e. all E’s are E’s regardless of what octave they’re in). This is called octave equivalence.

We consider all sets that are related by transposition to be equivalent. This is called transpositional equivalence.

We consider all sets that are related by inversion to be equivalent. This is called inversional equivalent.

A set class is the group of all sets that are related by transposition, inversion or both.

Best normal order is the most compact ordering of a set and its inversion with the smallest intervals grouped to the left.

Now one last thing. To truly represent transpositional equivalence we create what’s called a prime form. A prime form is simply the best normal order transposed to start on 0. So for our example above, where the best normal order was [8, 9, 0, 4], the prime form of that set is (0148). Prime forms are notated in round brackets without commas or spaces. In you need to use a 10 or 11 that could get awkward, so instead use T and E, for example (02468T).

Put all of the above best normal orders into prime forms.

Phew! Are we sick of this yet?

Seems arbitrary and it really is, along with a lot of things about set theory. The key is that it doesn't matter. Both are the same set, but what we're trying to do with set theory is get sets into a standard configuration so we can compare them. So what we need is consistency, which this provides.

Moral of the story? You can't get lazy. You have to check every interval and make sure to do any necessary tie breaking.

Do the normal order for both these sets:

Also, sometimes there’s no way to break the tie. We call these sets “transpositionally symmetrical,” which basically means that the same set of notes is reproduced under one or more transpositions. In other words, a feature of a transpositionally symmetrical set is that it will have more than one valid normal order.

If you know tonal music you probably know about the augmented triad and the fully diminished seventh chord. The augmented triad on C has a normal order of [0, 4, 8]. When you transpose it up a major third (PCI 4) you get a normal order of [4, 8, 0] and when you transpose it up PCI 8 you get [8, 0, 4]. Same set, three different, valid normal orders. The augmented triad is a transpositionally symmetrical set, as is the fully diminished seventh chord.

Put this set into normal order (bonus point if you can name the chord type):

So now we can move on from corrections. We know that with set theory we have octave equivalence and we also know that we have transpositional equivalence, if I didn’t mention it explicitly hopefully you figured that out. In case you didn’t, transpositional equivalence exists in tonal music as well. Just look at the idea of triads. All triads, regardless of which pitches exist in them, that are made of stacking a major third and a minor third are called major triads. Same deal goes for sets.

Sets with the same AIS are transpositionally equivalent and we can look at them as being “the same.” But set theory says, “fu

*ck it.” Not only does it not matter which octave a pitch exists in, not only does it not matter which transposition a series of intervals exists in, but it doesn’t matter whether that series of pitches is an inverted version of another. A set and its inversion, plus all of its transpositions and transpositions of its inversion belong to the same “set class.” REMEMBER THAT. Set class. That’s a reeeeeaaaaalllly important concept. But first we have to figure out inversion.*Also important to note: inversion in set theory IS NOT THE SAME as inversion in tonal music. In set theory inversion means mirror inversion. So what is mirror inversion? Well, first we’ll do the shitty slow way that you’ll never do again when I show you the good fast way.

For a “true” inversion you’ll mirror the notes in the set around an axis (namely the axis of 0). This is where that idea of inversions adding up to twelve will come in handy. This is harder to explain than it is to show so here it is:

0 inverts to 0

1 inverts to 11

2 inverts to 10

3 inverts to 9

4 inverts to 8

5 inverts to 7

6 inverts to 6

7 inverts to 5

8 inverts to 4

9 inverts to 3

10 inverts to 2

11 inverts to 1

Think about folding the clock in half. Here’s a hastily edited illustration:

So invert these sets (all are in normal order) and put them into normal order if the inversions aren’t already there (hint: subtract each pitch class from 12):

a) [0, 4, 7] (major triad)

b) [0, 3, 7] (minor triad)

Inversionally equivalent!!!!

c) [0, 2, 4, 6, 8, 10]

d) [0, 3, 6, 9]

e) [0, 3, 4, 7, 8, 11]

f) [5, 6, 8]

g) [1, 3, 7, 8]

Now, there’s some notational things like TnI that go along with inverting sets, but frankly I’ve never run into a use for how much nomenclature there is for half this stuff. If you want to know more then you can research it yourself. The important concept is inversion.

Now for the promised quick way. Don’t be a lazy turd and skip to here without doing the work above. The quick way, and why the idea of AIS was introduced, is that if you reverse the order of the intervals in a set you get the inversion of that set. So for example, the set [0, 1, 6] has an AIS of 1-5. If you reverse that to 5-1 and apply it to the original set you get [0, 5, 6] which just so happens to be that set inverted and transposed back to the original starting note. That is about a million times faster and more convenient that inverting to some arbitrary transposition with mirror inversion. So now that you know the good way, invert these sets:

h) [4, 7, 11]

i) [5, 8, 10]

j) [2, 3, 6, 7, 9]

k) [8, 11, 0, 2]

l) [4, 7, 8, 0] (is your inversion normal order????

Hint: nope. Can you think of why? Think about the start of this lesson.)

Get it? Good.

So now we know how to invert. At this point we can take one more step to finding the ever-elusive meaning of “set class.” Because sets are inversionally equivalent, to find the best version of the normal order for a set we have to not only find its most compact ordering, but the most compact ordering of its inversion and see which one is more compact (we like things being compact).

This final maximum compact ordering of the set is called (you guessed it) the “best normal order.” To find the best normal order of a set first find the normal order, invert it (the quick way), put the inversion in normal order and compare the interval between the first and next to last note (the interval between the first and last will be the same). Whichever is smaller is your best normal order and just like before, if there’s a tie keep going to the first and third to last note and keep going until you break the tie. If there truly is a tie, which is not uncommon, then that’s cool. We call sets like this inversionally symmetrical. The minor 7th chord is an example of a set that is inversionally symmetrical.

Here’s an example:

NOTE: The 6 between the E and the Ab in the first measure should be a 4

So the normal order is [4, 7, 8, 0], but the best normal order is [8, 9, 0, 4] (remember, two sets that are inversionally equivalent are said to be in the same set class).

Put these sets into their best normal order:

Are any of the sets members of the same set class?

Recap!

We consider all notes of the same pitch class to be equivalent (i.e. all E’s are E’s regardless of what octave they’re in). This is called octave equivalence.

We consider all sets that are related by transposition to be equivalent. This is called transpositional equivalence.

We consider all sets that are related by inversion to be equivalent. This is called inversional equivalent.

A set class is the group of all sets that are related by transposition, inversion or both.

Best normal order is the most compact ordering of a set and its inversion with the smallest intervals grouped to the left.

Now one last thing. To truly represent transpositional equivalence we create what’s called a prime form. A prime form is simply the best normal order transposed to start on 0. So for our example above, where the best normal order was [8, 9, 0, 4], the prime form of that set is (0148). Prime forms are notated in round brackets without commas or spaces. In you need to use a 10 or 11 that could get awkward, so instead use T and E, for example (02468T).

Put all of the above best normal orders into prime forms.

Phew! Are we sick of this yet?

*Last edited by jazz_rock_feel at Apr 22, 2014,*

#31

umm... I am up to Question h) and completely confused, and I don't think I could put it into just words alone so I have included an illustration to hope better explain?

but leave that for now and please read on:

perhaps I have got things wrong further back so I have posted my answers up to this point, I can provide examples of my woking if required (so it would make correcting any errors easier), but for now lets see how I went:

Edit: now someone else has posted their answers I can see the4thHorseman answers for h), i), j), k)... are the same as I was getting using the AIS Good Fast Way... some what of a relief (unless we're both wrong?) but my I would still like some explanation as to why/how my understanding of the slow shitty way doesn't match up with the AIS method... still confused!

Edit: nonetheless here are the rest of my answers without using the shitty slow way (and sorry about the small text to much work to change it now)

but leave that for now and please read on:

perhaps I have got things wrong further back so I have posted my answers up to this point, I can provide examples of my woking if required (so it would make correcting any errors easier), but for now lets see how I went:

Do the normal order for both these sets:

a): [8, 11, 0, 4]

b): [8, 1, 2, 3, 5] Edit: A is 9 [1, 2, 3, 5, 9] (then 1->5 is smallest PCI)

Put this set into normal order (bonus point if you can name the chord type):

Chord starts out as CMaj7b5 in Root position

From E[4] = 1st inversion

From A#[10] = 3rd inversion

[10, 0, 4, 6] covering a four notes within an 8 note span

NB: [4, 6, 10, 0] achieves this also

Invert these sets (all are in normal order) and put them into normal order if the inversions aren’t already there (hint: subtract each pitch class from 12):

a)

[0, 4, 7] (major triad)

[0, 8, 5] (inverted)

[5, 8, 0] (normal order)

b)

[0, 3, 7] (minor triad)

[0, 9, 5] (inverted)

[5, 9, 0] (normal order)

Inversionally equivalent!!!!

c)

[0, 2, 4, 6, 8, 10]

[0, 10, 8, 6, 4, 2] (inverted)

[0, 2, 4, 6, 8, 10] (normal order)

d)

[0, 3, 6, 9]

[0, 9, 6, 3] (inverted)

[0, 3, 6, 9] (normal order)

e)

[0, 3, 4, 7, 8, 11]

[0, 9, 8, 5, 4, 1] (inverted)

[0, 1, 4, 5, 8, 9] (normal order)

f)

[5, 6, 8]

[7, 6, 4] (inverted)

[4, 6, 7] (normal order)

g)

[1, 3, 7, 8]

[11, 9, 5, 4] (inverted)

[4, 5, 9, 11] (normal order)

a): [8, 11, 0, 4]

b): [8, 1, 2, 3, 5] Edit: A is 9 [1, 2, 3, 5, 9] (then 1->5 is smallest PCI)

Put this set into normal order (bonus point if you can name the chord type):

Chord starts out as CMaj7b5 in Root position

From E[4] = 1st inversion

From A#[10] = 3rd inversion

[10, 0, 4, 6] covering a four notes within an 8 note span

NB: [4, 6, 10, 0] achieves this also

Invert these sets (all are in normal order) and put them into normal order if the inversions aren’t already there (hint: subtract each pitch class from 12):

a)

[0, 4, 7] (major triad)

[0, 8, 5] (inverted)

[5, 8, 0] (normal order)

b)

[0, 3, 7] (minor triad)

[0, 9, 5] (inverted)

[5, 9, 0] (normal order)

Inversionally equivalent!!!!

c)

[0, 2, 4, 6, 8, 10]

[0, 10, 8, 6, 4, 2] (inverted)

[0, 2, 4, 6, 8, 10] (normal order)

d)

[0, 3, 6, 9]

[0, 9, 6, 3] (inverted)

[0, 3, 6, 9] (normal order)

e)

[0, 3, 4, 7, 8, 11]

[0, 9, 8, 5, 4, 1] (inverted)

[0, 1, 4, 5, 8, 9] (normal order)

f)

[5, 6, 8]

[7, 6, 4] (inverted)

[4, 6, 7] (normal order)

g)

[1, 3, 7, 8]

[11, 9, 5, 4] (inverted)

[4, 5, 9, 11] (normal order)

Edit: now someone else has posted their answers I can see the4thHorseman answers for h), i), j), k)... are the same as I was getting using the AIS Good Fast Way... some what of a relief (unless we're both wrong?) but my I would still like some explanation as to why/how my understanding of the slow shitty way doesn't match up with the AIS method... still confused!

Edit: nonetheless here are the rest of my answers without using the shitty slow way (and sorry about the small text to much work to change it now)

`[font="Arial Narrow"]So now that you know the good way, invert these sets:`

AIS AIS (invert) Inverted

h) [4, 7, 11] h) [3-4] [4-3] [4,8,11]

i) [5, 8, 10] i) [3-2] [2-3] [5,7,10]

j) [2, 3, 6, 7, 9] j) [1-3-1-2] [2-1-3-1] [2,4,5,8,9]

k) [8, 11, 0, 2] k) [3-1-2] [2-1-3] [8,10,11,2]

l) [4, 7, 8, 0] l) [3-1-4] [4-1-3] [4,8,9,0] --> Normal.Order [color="Blue"][8, 9, 0, 4][/COLOR]

Qis your inversion normal order????

Hint: nope. Can you think of why? Think about the start of this lesson.)

A: because the [color="Red"]PCI from 8->0 is 4[/COLOR]

-----------------------------------------------------------

normal order best normal Order (I) inverted best normal order (II)

(before inversion) (after inversion)

a) [10, 0, 3, 7] [color="Blue"][7, 10, 0, 3 ][/COLOR] [10, 2, 5, 7] [2, 5, 7, 10]

b) [9, 2, 3, 8] [color="Blue"][2, 3, 8, 9 ][/COLOR] same same

[color="Blue"][8, 9, 2, 3 ][/COLOR]

c) [9, 10, 4, 6] [4, 6, 9, 10] [9, 11, 5, 6] [color="Blue"][5, 6, 9, 11][/COLOR] [color="Red"](5->9= 4PCI)[/COLOR]

d) [9, 2, 3, 7] [2, 3, 7, 9 ] [9, 1, 2, 7] [color="Blue"][7, 9, 1, 2 ][/COLOR]

e) [8, 1, 3, 7] [color="Blue"][1, 3, 7, 8 ][/COLOR] [8, 0, 2, 7] [0, 2, 7, 8 ]

Q: Are any of the sets members of the same set class?

A: No

-----------------------------------------------------------

Put all of the above best normal orders into prime forms.

a) (0358)

b) (0167)

c) (0146)

d) (0267)

e) (0267)[/FONT]

*Last edited by tonibet72 at Apr 19, 2014,*

#32

*stands on the sideline while sipping a beer*

#33

My answers

Edit: Oh, and this is getting pretty cool, but I'm still not really seeing how to apply this, will you go into actual uses for there concepts soon? This is a pretty interesting way of grouping and comparing groups of notes, I just don't see what to do with this information.

a) [8, 11, 0, 4]

b) [1, 2, 3, 5, 9]

either way I do it the AIS is 4-2-4 I can't identify that chord, but it's interesting.

a) [5, 8, 0]

b) [5, 9, 0]

c) [0, 2, 4, 6, 8, 10]

d) [0, 3, 6, 9]

e) [0, 1, 4, 5, 8, 9]

f) [4, 6, 7]

g) [4, 5, 9, 11]

h) [4, 8, 11]

i) [5, 7, 10]

j) [2, 4, 5, 8, 9]

k) [8, 10, 11, 2]

l) [4, 8, 9, 0]

a) [7, 10, 0, 3] 3-2-3

b) [8, 9, 2, 3] 1-5-1

c) [4, 5, 8, 10] 1-3-2

d) [2, 3, 7, 9] 1-4-2

e) [1, 2, 6, 8] 1-4-2

d) and e) are both 1-4-2

a) (0358)

b) (0167)

c) (0146)

d) (0157)

e) (0157)

b) [1, 2, 3, 5, 9]

either way I do it the AIS is 4-2-4 I can't identify that chord, but it's interesting.

a) [5, 8, 0]

b) [5, 9, 0]

c) [0, 2, 4, 6, 8, 10]

d) [0, 3, 6, 9]

e) [0, 1, 4, 5, 8, 9]

f) [4, 6, 7]

g) [4, 5, 9, 11]

h) [4, 8, 11]

i) [5, 7, 10]

j) [2, 4, 5, 8, 9]

k) [8, 10, 11, 2]

l) [4, 8, 9, 0]

a) [7, 10, 0, 3] 3-2-3

b) [8, 9, 2, 3] 1-5-1

c) [4, 5, 8, 10] 1-3-2

d) [2, 3, 7, 9] 1-4-2

e) [1, 2, 6, 8] 1-4-2

d) and e) are both 1-4-2

a) (0358)

b) (0167)

c) (0146)

d) (0157)

e) (0157)

Edit: Oh, and this is getting pretty cool, but I'm still not really seeing how to apply this, will you go into actual uses for there concepts soon? This is a pretty interesting way of grouping and comparing groups of notes, I just don't see what to do with this information.

*Last edited by The4thHorsemen at Apr 19, 2014,*

#34

umm... I am up to Question h) and completely confused, and I don't think I could put it into just words alone so I have included an illustration to hope better explain?

but leave that for now and please read on:

perhaps I have got things wrong further back so I have posted my answers up to this point, I can provide examples of my woking if required (so it would make correcting any errors easier), but for now lets see how I went:Do the normal order for both these sets:

a): [8, 11, 0, 4]

b): [8, 1, 2, 3, 5] Edit: A is 9 [1, 2, 3, 5, 9] (then 1->5 is smallest PCI)

Put this set into normal order (bonus point if you can name the chord type):

Chord starts out as CMaj7b5 in Root position

From E[4] = 1st inversion

From A#[10] = 3rd inversion

[10, 0, 4, 6] covering a four notes within an 8 note span

NB: [4, 6, 10, 0] achieves this also

Invert these sets (all are in normal order) and put them into normal order if the inversions aren’t already there (hint: subtract each pitch class from 12):

a)

[0, 4, 7] (major triad)

[0, 8, 5] (inverted)

[5, 8, 0] (normal order)

b)

[0, 3, 7] (minor triad)

[0, 9, 5] (inverted)

[5, 9, 0] (normal order)

Inversionally equivalent!!!!

c)

[0, 2, 4, 6, 8, 10]

[0, 10, 8, 6, 4, 2] (inverted)

[0, 2, 4, 6, 8, 10] (normal order)

d)

[0, 3, 6, 9]

[0, 9, 6, 3] (inverted)

[0, 3, 6, 9] (normal order)

e)

[0, 3, 4, 7, 8, 11]

[0, 9, 8, 5, 4, 1] (inverted)

[0, 1, 4, 5, 8, 9] (normal order)

f)

[5, 6, 8]

[7, 6, 4] (inverted)

[4, 6, 7] (normal order)

g)

[1, 3, 7, 8]

[11, 9, 5, 4] (inverted)

[4, 5, 9, 11] (normal order)

Edit: now someone else has posted their answers I can see the4thHorseman answers for h), i), j), k)... are the same as I was getting using the AIS Good Fast Way... some what of a relief (unless we're both wrong?) but my I would still like some explanation as to why/how my understanding of the slow shitty way doesn't match up with the AIS method... still confused!

All your answers are correct so far.

As for your confusion, remember with the slow way that you have to put your inversion into normal order to get a valid set. With your example you inverted [0, 1, 6] into [0, 11, 6] which are the right numbers, but the order doesn't make any sense. If you put your inversion into normal order you get [6, 11, 0] which you'll note has an AIS of 5-1, the reverse of the AIS of the original set.

As requested here's the answer to h)

`Original set: [4, 7, 11]`

Inversion (slow): [8, 5, 1]

Inversion (slow) in normal order: [1, 5, 8]

Inversion fast: AIS of 3-4 reverse to 4-3 and you get [4, 8, 11]

Comparing the fast and the slow way:

Fast: [4, 8, 11]

Slow: [1, 5, 8] transposed up 3 is [4, 8, 11]

My answers

Perfect.

As for usage... yeah that's the tough bit. Set theory is pretty dynamic in how it can be applied. I have a few plans for that though. I think I'll have one more small lesson on technical stuff, maybe with some more challenging examples and then I'll do a lesson with a composition exercise (meaning I give you instructions and you compose something within that framework) and an analysis (probably of a piece that I write so I can make the relationships fairly obvious/simple). MAYBE we'll get into real world analysis, but we'll have to see. That's hard and I can't even really claim to be an expert on using set theory extensively in an analysis.

*stands on the sideline while sipping a beer*

I guarantee it's PBR

#35

All your answers are correct so far.

As for your confusion, remember with the slow way that you have to put your inversion into normal order to get a valid set. With your example you inverted [0, 1, 6] into [0, 11, 6] which are the right numbers, but the order doesn't make any sense. If you put your inversion into normal order you get [6, 11, 0] which you'll note has an AIS of 5-1, the reverse of the AIS of the original set.

As requested here's the answer to h)

Okay thanks I believe that helps, I have posted the rest of my answers now anyway...

As for your confusion, remember with the slow way that you have to put your inversion into normal order to get a valid set. With your example you inverted [0, 1, 6] into [0, 11, 6] which are the right numbers, but the order doesn't make any sense. If you put your inversion into normal order you get [6, 11, 0] which you'll note has an AIS of 5-1, the reverse of the AIS of the original set.

As requested here's the answer to h)

`Original set: [4, 7, 11]`

Inversion (slow): [8, 5, 1]

Inversion (slow) in normal order: [1, 5, 8]

Inversion fast: AIS of 3-4 reverse to 4-3 and you get [4, 8, 11]

Comparing the fast and the slow way:

Fast: [4, 8, 11]

Slow: [1, 5, 8] transposed up 3 is [4, 8, 11]

Okay thanks I believe that helps, I have posted the rest of my answers now anyway...

#36

Perfect.

As for usage... yeah that's the tough bit. Set theory is pretty dynamic in how it can be applied. I have a few plans for that though. I think I'll have one more small lesson on technical stuff, maybe with some more challenging examples and then I'll do a lesson with a composition exercise (meaning I give you instructions and you compose something within that framework) and an analysis (probably of a piece that I write so I can make the relationships fairly obvious/simple). MAYBE we'll get into real world analysis, but we'll have to see. That's hard and I can't even really claim to be an expert on using set theory extensively in an analysis.

I guarantee it's PBR

I'm really enjoying working through these exercises, and my sight reading is even getting better just from this little bit. Not enough that I could read something and play it at the same time, but I'm getting more comfortable. I'm also enjoying seeing this really bizarre way of looking at notes. Maybe I'll get more into atonal music because of this. I've always thought some of it sounds pretty cool, but it's so weird that I kinda get lost and don't really understand what's happening. I can't wait til your next lessons.

Side note: I actually had PBR for the first time about a week ago and loved it, which surprised me. I'm usually a fan of beers that are a little darker (nothing super stout, but something with a decently strong flavor) but PBR was pretty damn good for a lighter tasting beer.

#37

Okay thanks I believe that helps, I have posted the rest of my answers now anyway...

Okay you're a bit confused about normal order vs. best normal order. Normal order is the most compact ordering of the notes, best normal order is the more compact ordering out of the normal order and the inversion of the normal order.

What you labelled as best normal order was actually just the normal order. You would then invert THAT and compare the two to see which is the best normal order. After making sure the inversion is in normal order, of course!

I'm really enjoying working through these exercises, and my sight reading is even getting better just from this little bit. Not enough that I could read something and play it at the same time, but I'm getting more comfortable. I'm also enjoying seeing this really bizarre way of looking at notes. Maybe I'll get more into atonal music because of this. I've always thought some of it sounds pretty cool, but it's so weird that I kinda get lost and don't really understand what's happening. I can't wait til your next lessons.

I'm glad you're getting something out of it. Part of my hope is that some of you will be able to look at atonal/post tonal music in a bit of a different light and understand a way to approach it in analysis and listening. Not that I'm expecting anyone to turn into new music specialists or anything, but baby steps, you know? Getting better at reading notes is just a bonus!

And PBR is vile.

*Last edited by jazz_rock_feel at Apr 19, 2014,*

#38

I guarantee it's PBR

BLECH never!

How could I channel all the Germans drinking that crap???

Side note: I actually had PBR for the first time about a week ago and loved it, which surprised me. I'm usually a fan of beers that are a little darker (nothing super stout, but something with a decently strong flavor) but PBR was pretty damn good for a lighter tasting beer.

You'd get a better understanding of set theory if you drank some decent beer

*Last edited by Xiaoxi at Apr 19, 2014,*

#39

a) [8, 11, 0, 4]

b) [1, 2, 3, 5, 9]

c) [4, 6, 10, 0]

a) [5, 8, 0]

b) [5, 9, 0]

c) [0, 2, 4, 6, 8, 10]

d) [0, 3, 6, 9]

e) [0, 1, 4, 5, 8, 9]

f) [4, 6, 7]

g) [1, 4, 5, 9]

h) [4, 8, 11]

i) [5, 7, 10]

j) [2, 4, 5, 8, 9]

k) [8, 10, 11, 1]

l) [8, 9, 0, 4]

a) [7, 10, 0, 3]

b) [2, 3, 8, 9]

c) [4, 5, 8, 10]

d) [2, 3, 7, 9]

e) [7, 9, 1, 2]

a) (0358)

b) (0167)

c) (0145)

d) (0157)

e) (0267)

#40

a) [8, 11, 0, 4]

b) [1, 2, 3, 5, 9]

c) [4, 6, 10, 0]

a) [5, 8, 0]

b) [5, 9, 0]

c) [0, 2, 4, 6, 8, 10]

d) [0, 3, 6, 9]

e) [0, 1, 4, 5, 8, 9]

f) [4, 6, 7]

g) [1, 4, 5, 9] x

h) [4, 8, 11]

i) [5, 7, 10]

j) [2, 4, 5, 8, 9]

k) [8, 10, 11, 1]

l) [8, 9, 0, 4]

a) [7, 10, 0, 3]

b) [2, 3, 8, 9]

c) [4, 5, 8, 10]

d) [2, 3, 7, 9]

e) [7, 9, 1, 2] x

a) (0358)

b) (0167)

c) (0145)

d) (0157)

e) (0267) x

Few mistakes.

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