Author Topic: Six-Tone Pitch Sets  (Read 8806 times)

dogbite

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Re: Six-Tone Pitch Sets
« Reply #30 on: July 07, 2010, 07:56:20 AM »
this really does have to do with the subject of sets (of tones) and music theory

the subject of teaching and pedagogy (method of teaching) is interesting to say the least. we instructors must balance the needs of a diverse pool of human students, many of whom are unaware of their real motivations regarding the study and practice of their instrument. i do in fact find myself swinging the pendulum between two poles:

1) the student's best friend: "i know this stuff is hard, so let's take our time and work through it together"

2) the student's worst nightmare: "why do you never do the homework i assigned yet expect to move on to some other level?"

i believe that we must swing this pendulum on purpose to give us the patience (let's take our time and have some fun) to work through difficult concepts yet at the same time kick ourselves in the butt (let's get this shit behind us already) in order to get some worthwhile yet realistic goals achieved.

now how to best do this? an assessment of our own personal learning style is in order. do what works for you, put that pendulum in the place that suits your needs best but be careful not to confuse wants with needs.

i wasn't always excited about set theory in music but i am today and will do my best to show you why. in order to do this i must ask that the reader make sure to:

* learn your major (diatonic, ionian, whatever you want to call them) scales as thoroughly as possible on your instrument in terms of 1) playing them on a whim (Db major, sixth fret position, you have two seconds, go...) and 2) name the notes by name (F# is the sixth tone of what scale, you have two seconds, go...)

* learn the circle of fifths by any means whatsoever (did you know that Five Cats Got Drunk At Eddie's Bar? what did they do? they Gbot Dbrunk Abt Ebddie's Bbar!

and freely associate the key signatures for all of the above scales to their correct designations (who has five sharps, you have two seconds, go...)

ever watch the BBC show "Hustle"? i saw it: five main characters in eddie's bar :)

anyhoo, i will follow up in logical order so stay tuned. i will also ask the observer to keep it simple; for example, i know to expect a student to respond with "what about the minor scales" and i will say, "don't do this to yourself." you have enough of a task already without burdening yourself further with "what ifs"...
« Last Edit: July 07, 2010, 08:23:27 AM by dogbite »
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dogbite

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set theory
« Reply #31 on: July 07, 2010, 09:17:31 AM »
so what is "set theory" anyway, as applied to music theory? i will assume that the onlooker has worked through the major scales, key signatures, and circle of fifths as previously suggested and feels comfortable enough to move on. remember that the major scales along the circle of fifths serve as the landscape with which you will perceive tonal music. although set theory is most often applied to atonal music, i'm using it to help me with tonal music because of its vast resources.

the chromatic scale has twelve tones. i am talking about twelve-tone equal-tempered notes here. the term pitch class refers to all octaves of one of the twelve tones; for example, which C# am i talking about here? all of them of course! there are therefore twelve pitch classes and i will most of the time refer to them as:

C Db D Eb E F Gb G Ab A Bb B

i know that some of them are better off notated as sharps sometimes (E Gb G? no: E F# G!) but F#/Gb is cumbersome so be able to freely switch a name from sharp to flat if what you're looking at is ambiguous or otherwise just plain wrong!

now replace the pitch classes with the numbers on a clock with C at noon:

C = 0 (12:00 = 0 hours, yes?)
Db = 1
D = 2
Eb = 3
E = 4
F = 5
Gb = 6
G = 7
Ab = 8
A = 9
Bb = 10
B = 11

and there are other ways to think of this: how about TABLATURE ON A STRING TUNED TO C!!!

it's easy, it really is. the initial value of this process becomes clear during transposition. intervals are distances. a major third is a distance of four (because C to E is 4 hours on our chromatic clock) so:

what's a major third above A? let's see: A = 9 and four hours after nine is one o'clock which is C#

now go paint over the numbers on your non-digital clocks with chromatic half-steps. "what time is it dear?" ten minutes shy of Bb honey!

the letter name system is a bit strange because of its affinity for the C scale's natural notes - you should see the students crumble when asked to do anything at all in Gb but it's really just six o'clock and uses the same fingerings!

now let's describe a C minor pentatonic scale - C Eb F G Bb becomes 0 3 5 7 10

here are the others:

C: 0 3 5 7 10
C#: 1 4 6 8 11
D: 2 5 7 9 0
Eb: 3 6 8 10 1
E: 4 7 9 11 2
F: 5 8 10 0 3
F#: 6 9 11 1 4
G: 7 10 0 2 5
G#: 8 11 1 3 6
A: 9 0 2 4 7
Bb: 10 1 3 5 8
B: 11 2 4 6 9

now look at the smallest set of numbers: 02479, which is the prime form or most simple description of the minor (major also by the way) pentatonic scale.

in other words, pentatonic scale = 02479

how are we doing?

more later :)
s/aka/db

millions

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Re: Six-Tone Pitch Sets
« Reply #32 on: July 07, 2010, 07:07:40 PM »
DB said: "...now look at the smallest set of numbers: 02479, which is the prime form or most simple description of the minor (major also by the way) pentatonic scale....in other words, pentatonic scale = 02479..."

Okay, I see. I never started from a tonal scale before. I'll explain how I see this, and maybe it will make sense to whoever reads this.

The "02479" is the "mode" of the C minor pent scale on A, listed as "90247." It's been put into prime form, smallest intervals to the left. So, it's been changed in a way; the minor third C-Eb is no longer on bottom. Is that correct, db?

"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
"The trouble with New Age music is that there's no evil in it."-Brian Eno

dogbite

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Re: Six-Tone Pitch Sets
« Reply #33 on: July 07, 2010, 08:21:37 PM »
DB said: "...now look at the smallest set of numbers: 02479, which is the prime form or most simple description of the minor (major also by the way) pentatonic scale....in other words, pentatonic scale = 02479..."

Okay, I see. I never started from a tonal scale before. I'll explain how I see this, and maybe it will make sense to whoever reads this.

The "02479" is the "mode" of the C minor pent scale on A, listed as "90247." It's been put into prime form, smallest intervals to the left. So, it's been changed in a way; the minor third C-Eb is no longer on bottom. Is that correct, db?



90247 = A C D E G

and

02479 = C D E G A

so, yes it's been changed in the form of rearranging the "set" as in a collection of objects - in set theory the order doesn't matter as opposed to a tone row where order is everything :)

did i understand your query correctly?
s/aka/db

dogbite

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Re: Six-Tone Pitch Sets
« Reply #34 on: July 07, 2010, 08:56:56 PM »
so, back to basics. look at the clock. a single tone is but a single hour on the clock. it has no relation to any other tone until we have:

two tones.

a minor second interval is represented by two points occupying any two consecutive hours on the clock, but:

one hour forward is eleven hours backwards, yes? and one hour backward is eleven hours forward.

therefore, we have a graphic description of the minor second interval and its inversion, the major seventh.

in set theory this situation is represented by the prime form:

01

to be read as "if the first tone was at noon and the second tone at one o'clock" or "a tone along with the tone one half-step above."

02 is a whole step (and its inversion, the minor seventh)

03 is a minor third (and its inversion, the major sixth)

04 is a major third (and its inversion, the minor sixth)

05 is a perfect fourth (and its inversion, the perfect fifth)

06 is a tritone, which is its own inversion

i like eliminating the 0, which seems redundant to me if we're always referencing the first tone at noon; therefore, i call the intervals represented by any dyad simply as:

1, 2, 3, 4, 5, or 6

while in set theory land, or "what is the closest distance on the clock between the two tones"

and no, i don't talk like this in front of my students - well, most of them anyway :)
« Last Edit: July 07, 2010, 10:40:01 PM by dogbite »
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millions

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Re: Six-Tone Pitch Sets
« Reply #35 on: July 08, 2010, 04:22:42 AM »
in set theory this situation is represented by the prime form:

i like eliminating the 0, which seems redundant to me if we're always referencing the first tone at noon; therefore, i call the intervals represented by any dyad simply as:
1, 2, 3, 4, 5, or 6
while in set theory land, or "what is the closest distance on the clock between the two tones"
and no, i don't talk like this in front of my students - well, most of them anyway :)
Dog, that's a good idea in eliminating zero to identify the six basic intervals, but I will emphasize that with sets, we keep the zero, because we are identifying distances, not note names. Am I correct?

For me, this was an important distinction, between using number as identity, and number as quantity.
It's not like the Sudoku game, which has nothing to do with math at all. There, the numbers are just place-holders or identifiers; names. They could have made them apples and oranges.
Whereas with a set, we are defining intervals, or quantities of pitch-distance. This is number as quantity.

So when we look at a set such as 02479, we should remember that we are looking at a series of relationships or distances, not "number-names."

Earlier, when you said "...so, yes it's been changed in the form of rearranging the "set" as in a collection of objects - in set theory the order doesn't matter as opposed to a tone row where order is everything," I can see how this might be misunderstood to mean "discrete objects" with "number names." Aren't they really a series of distances between things? Correct me if I err.
« Last Edit: July 08, 2010, 04:28:57 AM by millions »
"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
"The trouble with New Age music is that there's no evil in it."-Brian Eno

dogbite

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Re: Six-Tone Pitch Sets
« Reply #36 on: July 08, 2010, 09:50:56 AM »
in set theory this situation is represented by the prime form:

i like eliminating the 0, which seems redundant to me if we're always referencing the first tone at noon; therefore, i call the intervals represented by any dyad simply as:
1, 2, 3, 4, 5, or 6
while in set theory land, or "what is the closest distance on the clock between the two tones"
and no, i don't talk like this in front of my students - well, most of them anyway :)
Dog, that's a good idea in eliminating zero to identify the six basic intervals, but I will emphasize that with sets, we keep the zero, because we are identifying distances, not note names. Am I correct?

For me, this was an important distinction, between using number as identity, and number as quantity.
It's not like the Sudoku game, which has nothing to do with math at all. There, the numbers are just place-holders or identifiers; names. They could have made them apples and oranges.
Whereas with a set, we are defining intervals, or quantities of pitch-distance. This is number as quantity.

So when we look at a set such as 02479, we should remember that we are looking at a series of relationships or distances, not "number-names."

Earlier, when you said "...so, yes it's been changed in the form of rearranging the "set" as in a collection of objects - in set theory the order doesn't matter as opposed to a tone row where order is everything," I can see how this might be misunderstood to mean "discrete objects" with "number names." Aren't they really a series of distances between things? Correct me if I err.

i believe that it is important to have clear and concise definitions of terms. the elimination of the "0" may lead to ambiguity if the observer forgets that he's standing on the zero thus he cannot see it. then during a simple act of transposition, say up a minor third that "4" becomes a "7" and if we forget that there was a "0" involved, two notes (0 and 4) become one (7).

i just observed in the listing of the prime forms that every one of them started with a zero and it seemed redundant so i took 'em out in order to see what they looked like and here is what i discovered:

by using 1) the prime form itself, 2) the inverted prime form, 3) the complement of the prime form, and 4) the inverted complement of the prime form, all of the sets may be described with a maximum of five digits and by adding leading zeros, they all look like zip codes.

02479 represents not only the pentatonic scale but also the complementary major scale

00073 represents not only the minor triad but also its inversion, the major triad as well the complementary sets of two distinct nine tone scales, and

the blues scale is the complement of 13568...

but, what a minute! the reader is going to say, "why do we need to do all this? isn't music theory perfectly well described by the tools already given us???"

and he'd be right. a major scale is 1 2 3 4 5 6 7 and a dorian scale is 2 3 4 5 6 7 1 [2] or 1 2 b3 4 5 6 b7

and a major triad is 1 3 5 and minor is 1 b3 5, etc...

so why all the fuss? in an attempt to describe all possible sets the classical language of using the major scale to describe tones and intervals is just too cumbersome. there are just too many possibilities [something like 4096 total tonal combinations reduced to 2048 by establishing a reference tone (root) reduced to 352 scale/arpeggio types reduced to 200+ prime forms through inversion...]

argh!

i don't really want to think of 352 of anything let alone 4096, although i was surprised that forte was able to reduce the problem to just over two hundred...

but it's still too much. i wanted to see all the possibilites in as simple terms as possible and what i did was to use:

122 prime forms
inverted forms of the ones that turned out to be unique
complementary forms (again, the ones that turned out to be unique)
inverted complementary forms of the ones that turned out to be unique

in order to get all of them

forte's list already used inversions and at least listed them according to complementary sets of all except the six-tone groups and this is what i had to go back and figure out: the sixes.

so who are these studies designed for? those who have spent enough time with dyads and triads, seventh and ninth chords, major and minor scales (natural, harmonic and melodic too), diminished and whole-tone and says:

"i want more"

have i mastered them? nope. not by a long shot. these studies were written for me, out of curiosity, and fun, and because i could; and for anyone else who wants to see. i would not presume to replace the rich and diverse history of music practice, theory and instruction - all i can do is add my 2 cents worth (or 352, or 122, or however you want to count them).

but you are correct, millions, in that we must state whether we are writing of locations or distances when specifying a prime form or any other of the related forms, because there is danger of misunderstanding, confusion, and the occasional food fight!

ps - this reminds me of a conversation i had with a wonderful GIT instructor who convinced me that a b3 is not equivalent to a minor third because a b3 is a location and a minor third is a distance...

i hope that this sets well with y'all :)
s/aka/db

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Re: Six-Tone Pitch Sets
« Reply #37 on: July 08, 2010, 04:07:39 PM »
That's a good, clear answer, dog. I'm always interested in your thought process, of how you got from A to B.

I'm going to hang on to this pentatonic for a moment. Here's a graphic representation of it, which may be easier for visual thinkers to grasp.



Now the set has a shape which reveals symmetries and relationships. Bb-C is the bottom of a symmetrical pentagon. Now, below, we will see what happened when this set was put in "prime form."






Correct me if I'm wrong, db. Now we can see that the pentagon has been rotated, still retaining its shape, but now with a different "zero" point.

Now, let's look at what happpens in "inversion," which mirror-images or "flops" the image over, like a pancake:



Notice that the pentagon is exactly the same when flopped. This means it retains its symmetry under inversion. Now that dog mentioned it, I want to see the complement (the diatonic scale) and see how it behaves. It's gotta exhibit symmetry too.
Maybe this way of thinking about these sets will help some people. I know that Pat Martino uses such shapes, and even has three-dimensional models! See his website.

Once again, dog, let me know if I'm on the right track. I'm going to eat some pizza now; all these circles are making me hungry.
« Last Edit: July 08, 2010, 08:30:33 PM by millions »
"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
"The trouble with New Age music is that there's no evil in it."-Brian Eno

dogbite

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Re: Six-Tone Pitch Sets
« Reply #38 on: July 09, 2010, 06:06:50 AM »
"Notice that the pentagon is exactly the same when flopped. This means it retains its symmetry under inversion. Now that dog mentioned it, I want to see the complement (the diatonic scale) and see how it behaves. It's gotta exhibit symmetry too."

yes, complements will show the exact symmetry as their counterparts. think of it like the negative of a photograph, or "what is there is defined by what isn't there" - definitely on the right track :)

That's a good, clear answer, dog. I'm always interested in your thought process, of how you got from A to B.

I'm going to hang on to this pentatonic for a moment. Here's a graphic representation of it, which may be easier for visual thinkers to grasp.



Now the set has a shape which reveals symmetries and relationships. Bb-C is the bottom of a symmetrical pentagon. Now, below, we will see what happened when this set was put in "prime form."






Correct me if I'm wrong, db. Now we can see that the pentagon has been rotated, still retaining its shape, but now with a different "zero" point.

Now, let's look at what happpens in "inversion," which mirror-images or "flops" the image over, like a pancake:



Notice that the pentagon is exactly the same when flopped. This means it retains its symmetry under inversion. Now that dog mentioned it, I want to see the complement (the diatonic scale) and see how it behaves. It's gotta exhibit symmetry too.
Maybe this way of thinking about these sets will help some people. I know that Pat Martino uses such shapes, and even has three-dimensional models! See his website.

Once again, dog, let me know if I'm on the right track. I'm going to eat some pizza now; all these circles are making me hungry.
s/aka/db

millions

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Re: Six-Tone Pitch Sets
« Reply #39 on: July 09, 2010, 06:39:07 PM »
Now, here's what the C major scale looks like:





I wonder why the axis of symmetry is the G#-D line? What's that got to do with C major? I guess G# is not really in there; "D" is the common pivot-point.

...And with its complement, the F# pentatonic major scale:


Hmmm...will an "axis of symmetry" always be a tritone?
« Last Edit: July 09, 2010, 06:45:23 PM by millions »
"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
"The trouble with New Age music is that there's no evil in it."-Brian Eno

millions

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Re: Six-Tone Pitch Sets
« Reply #40 on: July 13, 2010, 08:23:54 PM »
The more I look at Holdsworth's book (Just For the Curoius) the more convinced I am that he uses some version of Forte's set theory, or similar principles.

The particulars of such a method have their idiosyncracies. For example, Holdsworth does not think of scales as having "roots" or starting notes; rather, he sees the whole scale-shape laid out on the fingerboard as certain interval sequences. A major scale is thus seen as a unique set of intervals, spread over the entire neck. No modes, just one big shape. Depending on what root is played, or what chord, determines how the scale is seen: as major or minor, or rather, if it has a "flat third" or a "major third."

But, apparently, Holdsworth has thrown out most, but not all, of the "root" and "note" thinking, as well as "scale degree" thinking (b3, b5, #6, etc.) although he does mention this in the book, as certain scales being "jazz minor."

To my mind, Holdsworth is dealing with "visual shapes" mostly; he knows his scales so well, from the first fret to the twelfth, and beyond, that he just sees them as shapes, and knows how to use each shape under different root conditions. No more thinking in scale roots or chords; just shapes.

What does anybody else think about this? Do you think this is the way Holdsworth's main approach works?

"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
"The trouble with New Age music is that there's no evil in it."-Brian Eno

Halfdim7

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Re: Six-Tone Pitch Sets
« Reply #41 on: August 13, 2010, 06:57:30 PM »
This is obviously way outta my league, but I have something to add which you might find useful, mill.
I think I've mentioned before that I noticed, in a clip from Holdsworth's REH video, that his eyes moved in a peculiar side-to-side motion when he would recall a scale and play it on the guitar. He wasn't looking at the instrument, he was still facing the camera, but his eyes moved as though he were reading across the page of some invisible book that only he could see. This might lend credence to your "visual" hypothesis.
Or maybe he just has trouble making eye contact :-\
....lame-ass, jive, pseudo bluesy, out-of-tune, noodling, wimped out, fucked up playing....