Author Topic: Interval Projection  (Read 13151 times)

dogbite

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Re: Interval Projection
« Reply #30 on: April 05, 2011, 10:08:03 PM »
Dawg, my next concern is with how, exactly (or if) these Hanson ideas differ (if at all) from the Forte sets.

As far as I can determine, they seem very similar, each one dealing with the 'interval content' of each set, or scale. I know that Forte arranges these sets in 'normal order' in order to eliminate redundancy; yet, this could possibly blind one to the possibilities inherent in the 'un-normalized' sets, if we approach all of them as scales, not sets.

The harmonic results might differ, as we saw in the videos on the George Russell thread; how starting on 'F' instead of 'C' changes the harmonic halo.

So, if we use the Forte sets as 'scales,' then this sets up a root, and an heirarchy; and that's what tonality does. We could further establish tonality by free repetition of the appropriate notes. This approach sees each 'set' as a scale, with a definite starting point, whether or not it has been 'normalized'.

The normalization process classifies similar sets as being equivalent in interval content; yet, the 'modes' of the normalized sets might yield quite different sonorities, just as D dorian differs from C major, yet they are members of the same 'interval set.' And when we begin constructing chords within these 'scales' or 'sets,' then the harmonic results become more apparent.

Am I on the right track here, or do I need to re-read Forte, or have I read it the wrong way?

millions,

you are absolutely correct here. you must allow each set to be transposed through its constituent modes in order to explore its diversity. the normalization process is merely for "inventory" purposes, so that some kind of logical ordering may take place. rahn's criteria is slightly different than forte's but this affects only a half-dozen of some 352 scale/arpeggio/set types. for the record, i use rahn's. the inventory problem cannot be understated: it is very difficult to keep track of 4096 pitch sets. i can barely remember the names of a half-dozen neighbors across the street, let alone 4096 of anything. here's how it breaks down:

...there are 2048 inversions (modes) of 352 pitch sets for a total of 4096 tonal combinations. the term inversion is a bit problematic in that inversions as applied to chords is different than what is used in the case of an inverted melody. for those who are more curious and don't yet have a headache or some form of chronic insomnia:

0   tones have   0   inversions / modes of   1   pitch set in   1   key for a total of   1   tonal combination
                           
1   tone has   1   inversion / mode of   1   pitch set in   12   keys for a total of   12   tonal combinations
                           
2   tones have   10   inversions / modes of   5   pitch sets in   12   keys with a total of   60   tonal combinations
   plus   1   inversion / mode of   1   pitch set in   6   keys with a total of   6   tonal combinations
   equals   11   inversions / modes of   6   pitch sets      for a total of   66   tonal combinations
                           
3   tones have   54   inversions / modes of   18   pitch sets in   12   keys with a total of   216   tonal combinations
   plus   1   inversion / mode of   1   pitch set in   4   keys with a total of   4   tonal combinations
   equals   55   inversions / modes of   19   pitch sets      for a total of   220   tonal combinations
                           
4   tones have   160   inversions / modes of   40   pitch sets in   12   keys with a total of   480   tonal combinations
   plus   4   inversions / modes of   2   pitch sets in   6   keys with a total of   12   tonal combinations
   plus   1   inversion / mode of   1   pitch set in   3   keys with a total of   3   tonal combinations
   equals   165   inversions / modes of   43   pitch sets      for a total of   495   tonal combinations
                           
5   tones have   330   inversions / modes of   66   pitch sets in   12   keys for a total of   792   tonal combinations
                           
6   tones have   450   inversions / modes of   75   pitch sets in   12   keys with a total of   900   tonal combinations
   plus   9   inversions / modes of   3   pitch sets in   6   keys with a total of   18   tonal combinations
   plus   2   inversions / modes of   1   pitch set in   4   keys with a total of   4   tonal combinations
   plus   1   inversion / mode of   1   pitch set in   2   keys with a total of   2   tonal combinations
   equals   462   inversions / modes of   80   pitch sets      for a total of   924   tonal combinations
                           
7   tones have   462   inversions / modes of   66   pitch sets in   12   keys for a total of   792   tonal combinations
                           
8   tones have   320   inversions / modes of   40   pitch sets in   12   keys with a total of   480   tonal combinations
   plus   8   inversions / modes of   2   pitch sets in   6   keys with a total of   12   tonal combinations
   plus   2   inversions / modes of   1   pitch set in   3   keys with a total of   3   tonal combinations
   equals   330   inversions / modes of   43   pitch sets      for a total of   495   tonal combinations
                           
9   tones have   162   inversions / modes of   18   pitch sets in   12   keys with a total of   216   tonal combinations
   plus   3   inversions / modes of   1   pitch set in   4   keys with a total of   4   tonal combinations
   equals   165   inversions / modes of   19   pitch sets      for a total of   220   tonal combinations
                           
10   tones have   50   inversions / modes of   5   pitch sets in   12   keys with a total of   60   tonal combinations
   plus   5   inversions / modes of   1   pitch set in   6   keys with a total of   6   tonal combinations
   equals   55   inversions / modes of   6   pitch sets      for a total of   66   tonal combinations
                           
11   tones have   11   inversions / modes of   1   pitch set in   12   keys for a total of   12   tonal combinations
                           
12   tones have   1   inversion / mode of   1   pitch set in   1   key for a total of   1   tonal combination
                           
therefore, there are   2048   inversions / modes of   352   pitch sets      for a total of   4096   tonal combinations

whew!
« Last Edit: April 05, 2011, 10:09:59 PM by dogbite »
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dogbite

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Re: Interval Projection
« Reply #31 on: April 05, 2011, 10:51:19 PM »
dawg sniffs a passerby, sleepily...

dreams of quartal triads manifest in the melodic minor scale permeate his memory:

Eb-----F-----G-----A------Bb-----C-----Db
Bb-----C-----Db---Eb-----F------G------A
F-------G-----A-----Bb-----C-----Db----Eb

trust me on this: guitarists, play any sequence of the above quartal triads from the Bb melodic minor scale, harmonized with A7#9:

C (natural)
G
C#
A

and resolve to any kind of Dm, Dm7, Dm9, etc...

miss me?

----

ps - and what's this about pentatonic subsets of seven tone scales?

the quartal triads shown above may be classified according to five types:

1) sus4 [example: C F G, voiced as G C F]

2) sus+4 [example: C F# G, voiced as G C F#]

3) sus+4+5 [example: C F# G#, voiced as G# C F#]

4) sus-5 [example: C F Gb, voiced as Gb C F]

5) sus-4-5 [example: C Fb Gb, voiced as Gb C Fb]

we are probably familiar with the tertiary triads according to this harmonic analysis:

I ii iii IV V vi vii°

or in C:

C Dm Em F G Am B°

therefore, the sus chords found in the quartal triads may be defined as:

Isus IIsus IIIsus IVsus+4 Vsus VIsus VIIsus-5

similarly, C melodic minor may be defined as:

Cm Dm Eb+ F G A° B° [note the bIII augmented triad as well as the diminished triads on VI and VII (vi° and vii°)]

(i ii bIII+ IV V vi° vii°)

and

Csus Dsus Ebsus+4+5 Fsus+4 Gsus Asus-5 Bsus-4-5

(Isus IIsus bIIIsus+4+5 IVsus+4 Vsus VIsus-5 VIIsus-4-5)

the quartal triads from melodic minor are what i've been working on lately. i can do them; however, i get confused when i try to apply them on the fly...

it's good stuff, providing compelling harmonies for chord progressions.
« Last Edit: April 05, 2011, 10:54:21 PM by dogbite »
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millions

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Re: Interval Projection
« Reply #32 on: April 06, 2011, 04:35:59 AM »
I haven't explored this fully, but the first thing I noticed was how similar the Bb melodic major scale is to a diminished scale, the only difference  being F/F#, which is good, because the F gives stability to the Bb, and the F# is that '6 or b6' scale factor which only seems to affect chords. Of course, I'm thinking in Bb, not A...
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dogbite

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Re: Interval Projection
« Reply #33 on: April 06, 2011, 09:59:59 AM »
I haven't explored this fully, but the first thing I noticed was how similar the Bb melodic major scale is to a diminished scale, the only difference  being F/F#, which is good, because the F gives stability to the Bb, and the F# is that '6 or b6' scale factor which only seems to affect chords. Of course, I'm thinking in Bb, not A...

Bb melodic major is Bb C D Eb F Gb Ab Bb, which is the fifth mode of Eb melodic minor, and

Bb melodic minor is Bb C Db Eb F G A Bb, which is the scale i'm pretty sure you're talking about.

one way to think of a diminished scale is as a melodic minor with a split fifth:

Bb C Db Eb F G A Bb

becomes

Bb C Db Eb E F# G A Bb

when its fifth, F, is "split" into a half step up and down at the same time...

the melodic minor scale can be thought of as a hybrid "diminished/wholetone" scale. for example, notice that

Bb C Db Eb F G A Bb

has a diminished fragment:

G A Bb C Db Eb

as well as a wholetone fragment:

Db Eb F G A

and note that also the altered dominant scale (A Bb C Db Eb F G A, the seventh mode of Bb melodic minor) is referred to (by jamey aebersold in particular) as a "diminished whole-tone" scale as well as "superlocrian" - because of this connection between melodic minor and diminished, i think of them as more or less, interchangeable:

if

E7#9 = E altered dominant (E F G Ab Bb C D E of F melodic minor)

then

F diminished (F melodic minor with a split fifth, F G Ab Bb B C# D E F) becomes fair game...

of course this "half-whole" diminished scale is also a standard choice for E7#9, but it is in my opinion useful to view the choices as related to each other in some kind of unified "whole"...

for example, for Am maj7 (A C E G#), A melodic minor is a choice, but so also is A diminished.

how 'bout this: for Bb9#11, use not only F melodic minor (Bb lydian dominant) but also F diminished:

F G Ab Bb C D E F

F G Ab Bb B C# D E F
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millions

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Re: Interval Projection
« Reply #34 on: April 06, 2011, 03:50:33 PM »

has a diminished fragment:

G A Bb C Db Eb

as well as a wholetone fragment:

Db Eb F G A

The whole tone fragment yields one of my favorite piano voicings, by placing a note from the Bb melodic minor scale under it as root, like you do with diminished b9s:

Db Eb F G A becomes, with Eb as root: Eb/9b5, but voiced real close, like Mal Waldron's "Duquility" from "the Quest."

The diminished fragment, as well yields another of my favorite piano voicings:
G A Bb C Db Eb becomes, with Eb as root: Eb/13 #11. In this case, I feel I must call A a #11, because Bb, the fifth, is still present, even though the A is voiced lower, right next to the fifth. Do you agree?
« Last Edit: April 06, 2011, 04:09:53 PM by millions »
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Re: Interval Projection
« Reply #35 on: April 06, 2011, 06:15:14 PM »

has a diminished fragment:

G A Bb C Db Eb

as well as a wholetone fragment:

Db Eb F G A

The whole tone fragment yields one of my favorite piano voicings, by placing a note from the Bb melodic minor scale under it as root, like you do with diminished b9s:

Db Eb F G A becomes, with Eb as root: Eb/9b5, but voiced real close, like Mal Waldron's "Duquility" from "the Quest."

The diminished fragment, as well yields another of my favorite piano voicings:
G A Bb C Db Eb becomes, with Eb as root: Eb/13 #11. In this case, I feel I must call A a #11, because Bb, the fifth, is still present, even though the A is voiced lower, right next to the fifth. Do you agree?

yes i do. in harmonizing dominant chords with the diminished scale, the fifth is often present; therefore, the #4 or #11 description is less amphibious, uh, ambi - uh - ambiguous*

begging the pardon of my left-handed frog-friends :)
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millions

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Re: Interval Projection
« Reply #36 on: April 06, 2011, 06:42:14 PM »
Quote
therefore, the sus chords found in the quartal triads may be defined as:

Isus IIsus IIIsus IVsus+4 Vsus VIsus VIIsus-5

similarly, C melodic minor may be defined as:

Cm Dm Eb+ F G A° B° [note the bIII augmented triad as well as the diminished triads on VI and VII (vi° and vii°)]

(i ii bIII+ IV V vi° vii°)

and

Csus Dsus Ebsus+4+5 Fsus+4 Gsus Asus-5 Bsus-4-5

(Isus IIsus bIIIsus+4+5 IVsus+4 Vsus VIsus-5 VIIsus-4-5)

the quartal triads from melodic minor are what i've been working on lately. i can do them; however, i get confused when i try to apply them on the fly...


I think it's worth noting that the 'fourths' that are being used here are diatonic, derived from the scale; they are not 'perfect' fourths, but are fourths in a scalar or diatonic sense, which means four letter-names, as in Bb-C-Db-Eb, where Bb-Eb is the fourth (in this case, perfect). This changes when other fourth intervals are used, as with Db-G and Eb-A, which are 'augmented fourths' or 'diminished fifths' depending on the context. Intervallically, those two rogues are tritones, that interval which, when inverted, becomes itself again.
« Last Edit: April 06, 2011, 06:44:04 PM by millions »
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dogbite

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Re: Interval Projection
« Reply #37 on: April 06, 2011, 10:35:38 PM »
Quote
therefore, the sus chords found in the quartal triads may be defined as:

Isus IIsus IIIsus IVsus+4 Vsus VIsus VIIsus-5

similarly, C melodic minor may be defined as:

Cm Dm Eb+ F G A° B° [note the bIII augmented triad as well as the diminished triads on VI and VII (vi° and vii°)]

(i ii bIII+ IV V vi° vii°)

and

Csus Dsus Ebsus+4+5 Fsus+4 Gsus Asus-5 Bsus-4-5

(Isus IIsus bIIIsus+4+5 IVsus+4 Vsus VIsus-5 VIIsus-4-5)

the quartal triads from melodic minor are what i've been working on lately. i can do them; however, i get confused when i try to apply them on the fly...


I think it's worth noting that the 'fourths' that are being used here are diatonic, derived from the scale; they are not 'perfect' fourths, but are fourths in a scalar or diatonic sense, which means four letter-names, as in Bb-C-Db-Eb, where Bb-Eb is the fourth (in this case, perfect). This changes when other fourth intervals are used, as with Db-G and Eb-A, which are 'augmented fourths' or 'diminished fifths' depending on the context. Intervallically, those two rogues are tritones, that interval which, when inverted, becomes itself again.

imo, the term "diatonic fourths" is sufficient here - sufficient to define the happenstance of "imperfect" fourths within commonly used scales. for example, the major scale comprises both perfect fourths and a single augmented fourth. the melodic minor scale includes both of these as well as a diminished fourth (e.g. C# F and B Eb, enharmonic major thirds)...

the "quartal" triads include all of the possible successive fourths within the scales, giving rise to all of the (5) types listed earlier: sus4, sus+4, sus-5, sus+4+5, and sus-4-5. harmonic minor and harmonic major would necessitate the addition of yet two additional quartal triads: sus-4 (such as C Fb G) and sus+5 (such as C F G#)...

i am currently (still) working on solidifying my approach to quartals within both the diatonic (major) scale and melodic minor - three-string sets on either string 1, 2, and 3 or strings 2, 3, and 4.
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Re: Interval Projection
« Reply #38 on: May 01, 2011, 09:32:49 AM »
A diversion: I'm skimming through "Bartok: An Analysis of his Music" by Elliott Antokoletz, and there's an interesting chapter called 'Basic Principles of Symmetrical Pitch Construction."
It states, basically, that traditional Western music was based on an uneven division of the octave, namely the perfect fourth and fifth.
Look at all the intervals we have examined thus far: all of them have complementary intervals which add up to an octave (min. 3rd/maj. 6th, etc.), and the smaller of these two complements generates a cycle which divides the octave symmetrically: one cycle of m2, two cycles of M2, three of m3, four cycles of M3, and six cycles of tritones; except the p4 and p5: this complementary interval does not generate a cycle which divides the octave symmetrically, but must extend through many octaves in order to reach its initial starting point again. Thus, there is only one cycle of perfect fourths, or perfect fifths.

Another observation that keeps cropping up is that the Dorian mode is symmetrical.
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"The trouble with New Age music is that there's no evil in it."-Brian Eno

dogbite

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Re: Interval Projection
« Reply #39 on: May 01, 2011, 10:21:02 PM »
A diversion: I'm skimming through "Bartok: An Analysis of his Music" by Elliott Antokoletz, and there's an interesting chapter called 'Basic Principles of Symmetrical Pitch Construction."
It states, basically, that traditional Western music was based on an uneven division of the octave, namely the perfect fourth and fifth.
Look at all the intervals we have examined thus far: all of them have complementary intervals which add up to an octave (min. 3rd/maj. 6th, etc.), and the smaller of these two complements generates a cycle which divides the octave symmetrically: one cycle of m2, two cycles of M2, three of m3, four cycles of M3, and six cycles of tritones; except the p4 and p5: this complementary interval does not generate a cycle which divides the octave symmetrically, but must extend through many octaves in order to reach its initial starting point again. Thus, there is only one cycle of perfect fourths, or perfect fifths.

Another observation that keeps cropping up is that the Dorian mode is symmetrical.

if i may add,

in terms of pure set theory, the reason that perfect fourths and fifths behave this way is that 5 (a perfect fourth is five half steps) and 7 (a perfect fifth is seven half steps) are not divisors of 12:

2 x 6 = 12
six tritones, each of which divide the octave into two equal parts:
C Gb, Db G, D Ab, Eb A, E Bb, F B

3 x 4 = 12
four augmented triads, each of which divide the octave into three equal parts:
C E Ab, Db F A, D Gb Bb, Eb G B

4 x 3 = 12
three diminished seventh tetrads, each of which divide the octave into four equal parts:
C Eb Gb A, Db E G Bb, D F Ab B

6 x 2 = 12
two wholetone scales, each of which divide the octave into six equal parts:
C D E Gb Ab Bb, Db Eb F G A B

however, neither 5 nor 7 go into 12; until
5 goes into 60, a multiple of 12 (circle of fourths, five octaves: C F Bb Eb Ab Db Gb B E A D G)
7 goes into 72, a multiple of 12 (circle of fifths, seven octaves: C G D A E B Gb Db Ab Eb Bb F)

the symmetry of the dorian mode is exemplified by the axis of symmetry of the major scale being the tone D:

A [1] B [½] C [1] D [1] E [½] F [1] G

in other words; from the starting tone D, the ascending pattern of half and whole steps is identical to descending:

D [1] E [½] F [1] G, etc...

D [1] C [½] B [1] A, etc...

this view of symmetry is central to my theoretical outlook, being the core of my conceptualization of not only the major scale, but all others including harmonic and melodic minor, diminished, whole tone and blues...
« Last Edit: May 01, 2011, 10:22:43 PM by dogbite »
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millions

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Re: Interval Projection
« Reply #40 on: May 02, 2011, 10:16:28 AM »
"...in terms of pure set theory, the reason that perfect fourths and fifths behave this way is that 5 (a perfect fourth is five half steps) and 7 (a perfect fifth is seven half steps) are not divisors of 12...neither 5 nor 7 go into 12; until
5 goes into 60, a multiple of 12 (circle of fourths, five octaves: C F Bb Eb Ab Db Gb B E A D G)
7 goes into 84, a multiple of 12 (circle of fifths, seven octaves: C G D A E B Gb Db Ab Eb Bb F)..."

The reason why this 'difference' of fourths and fifths was brought up is because the author of the Bartok book is saying that Bartok based his music on an even division of the octave, namely, the tritone.
From a perspective of pure arithmetic, the octave can be seen as 'unity.' The octave, without regard to register, in terms of pitch identity and relation to a 'root,' can be called '1' or unity. On a number line, anything less than one, proceeding back to zero (infinity), is fractional. Anything larger than one proceeds forward, into the 'other' infinity of octaves.
Perhaps this is why the 4th & 5th are different; instead of dividing the octave fractionally, they go 'outward' past one, past the octave, into other 'root' stations. Hence, the use of 4ths & 5ths to create root movement.

Every interval has its complement. All the intervals except perfect fourths & fifths have a smaller number which divides the octave (12) symmetrically;

So each interval has 2 numbers which add up to an octave.
The m2 has itself 1 and 11;
M2 is 2 and 10;
m3 is 3 and 9;
M3 is 4 and 8;
p4 is 5 and 7;
tritone is 6 and 6;
p5 is 7 and 5;
m6 is 8 and 4;
M6 is 9 and 3;
m7 is 10 and 2;
and M7 is 11 and 1.

You can see the symmetry in this; and if we eliminate the redundancies, such as 10-2/2-10, we have 6 essential intervals.

As dawg said, "...neither 5 nor 7 go into 12; until
5 goes into 60, a multiple of 12 (circle of fourths, five octaves: C F Bb Eb Ab Db Gb B E A D G)
7 goes into 84, a multiple of 12 (circle of fifths, seven octaves: C G D A E B Gb Db Ab Eb Bb F)"

The fourth and fifth, as pointed out, cannot be used as divisors of 12 (the octave); therefore, they can be seen as "expanding" in nature, as they generate cycles of 12 notes (outside the octave). Remember, 60 and 84 had to be used as the common denominators for 5 and 7. These large numbers can be seen as 'outside the octave' or as a 'greater referential point.' Hence, the reason the 4th and 5th are the basis of traditional Western music; this facilitates movement outside the octave, to a new reference point or new key.

This means that 'modern' music, like Bartok's, is 'inward-going' or 'introspective' if you like to indulge in metaphor (after all, this is art, not science). This is what Marshall McLuhan was getting at in his book "Through the Vanishing Point," in which he explains how our perspective on things is literally reversed in modern art, putting us at the other end of the 'vanishing point.' Like looking down the wrong end of a telescope, or rather a microscope, the 'inner' world now becomes our universe, heading towards the 'other infinity' towards zero; just like our number line, where anything less than one, proceeding back to zero (infinity), is fractional, and anything larger than one proceeds forward, into the 'other' infinity of octaves.





« Last Edit: May 14, 2011, 05:35:08 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

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Re: Interval Projection
« Reply #41 on: May 10, 2011, 03:43:22 PM »
This is about music, and the nature of tonality. The ideas I put forth about intervals, although fairly simple, are laying the groundwork for a larger, more all-encompassing understanding of tonality and chromaticism. I see it as a necessary reference to the ideas which will follow. The simple ideas about intervals were necessary, in case some of these ideas about intervals & reciprocals might not be fully 'grokked' by some readers.

It's simple, and it's complicated, all at the same time; but after a thorough pondering and practical application (in composition) of Howard Hanson's ideas of interval projection, I decided it was time to tackle the ideas of another musical giant: Bartók, and what a revelation it has been! Especially the little book by Ernö Lendvai, which I highly recommend, that is, IF you are sufficiently prepared to read it. Some knowledge of intervals & reciprocals is necessary.
The Ernö Lendvai book deals a lot with the 'meta-concepts' of Bartók's methods. It generalizes to a great extent, and is not a very lengthy book, but it states the case elegantly, and it is a beautiful book. It divides Bartók's ideas into two main categories: the 'GS' approach, which has to do with the "Golden Section" and the Fibonacci series, and is also called his 'chromatic system'; and Bartók's 'diatonic system,' which is based on acoustic principles.

The beauty of all this is that the two approaches reflect each other in an inverse relationship.

In this quote by Ernö Lendvai, he reveals the most profound aspect of Bartók's system:

"A secret of Bartók's music, and perhaps the most profound, is that the 'closed' world of the GS (Golden Section) (1,2,3 and 6 being 'closed' or 'inward-directed' intervals, as opposed to 4ths and 5ths) is counterbalanced by the 'open' sphere of the acoustic system. The former always pre-supposes the presence of the complete system -- it is not accidental that we have always depicted chromatic formations in the closed circle of fifths. In the last, all relations are dependent on one tone since the natural sequence of overtones emerges from one single root: therefore it is open. Thus, the diatonic system has a fundamental 'root' note, and the chromatic system a 'central' note....Bartók's GS system always involves the concentric expansion or contraction of intervals..."
« Last Edit: May 14, 2011, 05:34:12 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

millions

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Re: Interval Projection
« Reply #42 on: May 14, 2011, 05:33:30 AM »
So we can see from this exposition of the intervals that modern music started moving away from traditional tonality by way of exploiting the INHERENT SYMMETRIES in the 12-note scale.
Laymen must be careful not to confuse this approach with Schoenberg's 12-tone method. In fact, George Perle wrote an entire book on this subject, 'Twelve Tone Tonality,' and he is not referring to Schoenberg, but to a modern 'chromatic' approach to tonality. Even the Amazon reviews of the book reveal the basic miscomprehension on the part of the people who read the book! The book is NOT about dodecaphony or Schoenberg's 'Method of Composing Using 12 notes,' but you'd never know this from the customer reviews. Check out this brilliant one:
"...This book should really be titled "My method of composing with 12 tones". Because it is George Perle's account of how he composes serial music. The examples are taken from his music and a few other like minded folks, but it is not a general description of how composers such as Schoenberg, Stravinsky, or anyone else wrote serial music....So, if you really want to dig into one man's view of a way to work 12-tone sets into music, this could be an interesting book for you. If not, then turn to something else. Maybe Perle's "Serial Composition and Atonality" would be more appropriate and useful to you...."

Fine, except that the book is not concerned with serial music. Another customer 'expert' tells us:

"If you can slog your way through this one though, you'll have some really fresh and exciting compositional ideas as well as a renewed appreciation for the potential of the twelve-tone idiom."

It sounds suspiciously like this reviewer is confusing chromatic tonality with 12-tone or serial idioms. Bartók's 'chromatic' method was really only an 8-note scale, otherwise known as the diminished or whole-half tone scale. Apparently, this particular scale gave rise to an army of imitators, whose music exhibited symptoms of what I call 'diminished-itis.'

In the bigger picture, what these small, recursive intervals do is allow the creation of pitch cells; these are aggregates of notes which expand around an axis of symmetry. Thus, localized areas of tonal centricity can be created on any note. An analogy would be, traditional tonality is like a tree which grows up in one direction from one 'rooted' spot; in the chromatic approach, tonality becomes radiant 'flowers' of pitch, centering on any possible note in the vertical spectrum.
Another aspect of Bartók's approach which has puzzled many is the fact that he still uses the fifth & fourth as generators of traditional tonality, sometimes mixing the two approaches.
All of these ideas were 'in the air' so to speak, around the turn of the century, and were not unique to Bartók; examples of symmetry began showing up as early as R. Strauss, in his 'Elektra' and 'Metamorphosen,' before he retreated back into conservative classicism. Debussy, as most of us know, used the whole-tone scale in his music, most notably the prelude 'Voiles' from Book I. The 6-note whole-tone scale itself is a symmetrical projection of the major second, and there are only two of them; Debussy exploits this characteristic to create 2 areas of contrasting tonality. Schoenberg was influenced by this idea as well; in an old post of mine from an Amazon thread, "Schoenberg's Op. 26 Wind Quintet", I pointed this out:

[The row is (first hexad) Eb-G-A-B-C#-C, which gives an augmented/whole-tone scale feel, with a "resolution" to C at the end, then (second hexad) Bb-D-E-F#-G#-F, which is very similar in its augmented/whole-tone scale structure, which only makes sense: there are only two whole-tone scales in the chromatic collection, each a chromatic half-step away from the other. I've heard Debussy use the two whole-tone scales in this manner, moving down a half-step to gain entry to the new key area. This is why Schoenberg used a "C" in the first hexad, and the "F" in the second; these are "gateways" into the chromatically adjacent scale area. Chromatic half-step relations like these can also be seen as "V-I" relations, when used as dual-identity "tri-tone substitutions" as explained following.
Another characteristic of whole-tone scales is their use (as in Thelonious Monk's idiosyncratic whole-tone run) as an altered dominant, or V chord. There is a tritone present, which creates a b7/3-3/b7 ambiguity, exploited by jazz players as "tri-tone substitution". The tritone (if viewed as b7-3 rather than I-b5) creates a constant harmonic movement, which is what chromatic jazzers, as well as German expressionists, are after.
So Schoenberg had several ideas in mind of the tonal implications when he chose this row.]

Also, from this we can see that, historically, it was the tritone (in both V7-I's and in diminished seventh chords) which was the first emergent symmetry which led to the expansion of tonality; this interval was the color tone in the V7-I progression, being the major third and flat-seven, which would then exchange places for the next cycle. This gave rise to new roots, moving chromatically instead of by fifths. This was tied-in (as mentioned above) with 'flat-nine' dominant altered chords, which are closely related to the diminished seventh. The use of 'flat-nine dominants' as true V chords appears as early as Beethoven and Bach. The vii degree of the major scale, a diminished triad, has always been treated as an incomplete dominant ninth with G as the 'imaginary' root, and resolved as a V7 chord would be (to C).
So, it can be seen from all this that 'tonality' underwent great changes around the dawn of the 20th century; and one should not confuse this expanded chromatic version of tonality with Schoenberg's 12-tone method, which just confuses the issue.

In fact, I am more critical of Schoenberg than I ever was before; his method treated dissonances like consonances, and renounced a tonal center. But dissonance is not the same as consonance; it has different acoustical and physiological effects. Therefore, dissonance ought not be treated as if it were identical with consonance. Plus, Schoenberg's renunciation of a tonal center does not follow from any previously stated proposition, and is merely an assertion of his dogmatic belief that the negation of tonality was 'historically inevitable.'
"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

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Re: Interval Projection
« Reply #43 on: May 18, 2011, 10:50:51 PM »
re: "an assertion of his dogmatic belief that the negation of tonality was 'historically inevitable.'"

i agree with what you appear to be getting at. charles wuorinen states something similar in the opening pages of "simple composition," a primer to serialism and other atonal devices - that "all serious composers" would embrace such things in the future, and...

it never happened.

most music listeners don't listen to atonal or serial music and why is that? well, as much as us theorists would like to turn music into some kind of sonic experiment, most people would rather have us play a song that they like and who could blame them? i mean, the performer is not there for an expose into their inner psyche without regard for the pleasure of the listener, are they? perhaps an existentialist argument (about which i personally have zero interest in following through with) but this is something to which theorists best pay attention. the audience wants results, and i have exchanged correspondence with more than a couple of music writers who are very distrustful of unproven academic methods - and i know that schoenberg was a working composer, but honestly, my students have never heard of him and would clearly rather listen to john williams.

i certainly don't mean to begin an argument on these lines of thought but this is my current thinking, at the moment, right now, and may change by the time the sun comes up. i like studying all aspects of composition but it just seems overwhelming and perhaps even unnecessary to leave the time honored traditions of this thing called tonality. my theory teacher in college seemed to view this topic similarly - basically, he said if you're interested, go check this out (the wuorinen text) and left it at that, after of course the obligatory second-year study of tone rows :)

don't get me wrong - i am thoroughly gratified that i took the time to assimilate the language of not so much serialism but set theory and use this language to different ends; however, my compositional and improvisational style is clearly tonally based upon this rich chromatic landscape.

anyone's thoughts on this would be welcome.
« Last Edit: May 18, 2011, 10:54:18 PM by dogbite »
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Re: Interval Projection
« Reply #44 on: May 19, 2011, 03:47:27 PM »
Interesting, Dawg...Good to hear from you.
Those are the same lines I'm thinking along...really the difference in 'chromatic tonality' and Schoenberg's method is that he used an 'ordered' set. A scale, by contrast, is an 'unordered set,' which acts as an 'index' of tones from which to choose.
This simple difference makes Schoenberg's tone-rows essentially melodic in nature, not harmonic. So, not only is his method 'non-tonal,' but it is also 'non-harmonic.' So his music is very deterministic, and very contrapuntal by nature. Perhaps he was looking to Bach and before, and wanted to take music back to a place where only single-lines existed, like Gregorian chant, and harmony was simply the result of how these melodic threads coincided. The division of the 12-note row into two hexads is also reminiscent of Gregorian chant, which used tetrachords to create variety. The tetrachords were fixed entities, like a tone-row, which were strung-together to create new combinations; but there was no 'harmonic' thinking going on.
"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