Author Topic: Six-Tone Pitch Sets  (Read 8805 times)

dogbite

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Six-Tone Pitch Sets
« on: June 26, 2010, 04:21:21 PM »
ripped from another thread, in a sense, for your viewing enjoyment

in my studies compiling all possible pitch sets (which i have since published in the form of guitar fingering diagrams) the six-tone groups stood out as unique in that:

1(a) some of them are inversions of each other
1(b) all of the others are inversions of themselves

2(a) some of them are complements of each other
2(b) all of the others are complements of themselves

3) some of them are inverted complements of themselves

i wouldn't mind a discussion of six-tone scales (or chords) whether or not they are viewed as fragments of familiar scales.

millions staed something on another thread, that
Quote
I think Schoenberg and his crew used up all of the good ones. Ha ha; Schoenberg was interested in hexads which had certain constant characteristics under inversion. One of these he called "The Miracle Row."

me being schooled in only the surface mechanics of twelve-tone theory, i do not know this first hand, so if i may beg your indulgence, i wish to explore any of the six-tone groups that any of us has experienced.

the obvious ones:

blues scale [1 b3 4 #4(b5) 5 b7]
whole-tone scale [1 2 3 #4(b5) #5(b6) #6(b7)]
augmented scale [1 #2(b3) 3 5 #5(b6) 7]
minor pentatonic add 2 (one of my favorites) [1 2 b3 4 5 b7]

in order to deal with the more unusual ones, it may be useful to use allen forte's "prime form" notation which reduces tones to more of a mathematical description: "how many semitones above the reference tone" such as "013467" to describe C Db Eb E Gb G or any of its modes.

to what end, the purpose of this discussion?  i do not know, but i am curious and i have learned much from others in this regard and wish to learn more :)
« Last Edit: June 26, 2010, 11:42:33 PM by dogbite »
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Halfdim7

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Re: Six-Tone Pitch Sets
« Reply #1 on: June 26, 2010, 05:09:00 PM »
Well, all I can add is that I really like the minor pentatonic add 2, as well :)
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dogbite

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Re: Six-Tone Pitch Sets
« Reply #2 on: June 26, 2010, 07:28:11 PM »
Well, all I can add is that I really like the minor pentatonic add 2, as well :)

may i point out that this scale is

1) an inversion of itself because:

1 2 b3 4 5 b7 8

is

whole half whole whole m3 whole

and if the pattern is reversed:

whole m3 whole whole half whole

a mode of the same scale is produced:

4 5 b7 1 2 b3 4

and

2) a complement of itself because:

A B C D E G

is the complement of:

Bb Db Eb F Gb Ab

which is the same type of scale...

another way of looking at this unique pitch set is that it is produced by a series of fifths:

C G D A E B

as is its complement:

Gb Db Ab Eb Bb F

minor pentatonic subs work great for this:

A minor pentatonic
A C D E G

and

E minor pentatonic
E G A B D

easy rule:

along with any minor pentatonic,
go up a fifth and play same...

works great for minor and even dominant bluesy grooves.
s/aka/db

millions

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Re: Six-Tone Pitch Sets
« Reply #3 on: June 27, 2010, 06:06:01 AM »
Dog, here's a copy of an old post of mine in Amazon Classical which is my take on Schoenberg's Op. 26 Wind Quintet:

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 (as quasi-dominant sonorities), 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 "dominant 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.

In Forte nomenclature, what would this be?
« Last Edit: June 27, 2010, 06:43:47 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: Six-Tone Pitch Sets
« Reply #4 on: June 27, 2010, 06:21:15 AM »
Doggie, I have a question. First, the background:

The nomenclature:
p=perfect fifth (or fourth)
m=major third (minor sixth)
n=minor third (major sixth)
s=major second (minor seventh)
d=minor second (major seventh)
t=augmented fourth, diminished fifth

Note the overall progression of the "fifths projection," which I am using as the chromatic scale model:
doad: p
triad: p2 s
tetrad: p3 n s2
pentad: p4 m n2 s3
hexad: p5 m2 n3 s4 d
heptad: p6 m3 n4 s5 d2 t
octad: p7 m4 n5 s6 d4 t2
nonad: p8 m6 n6 s7 d6 t3
decad: p9 m8 n8 s8 d8 t4
undecad: p10 m10 n10 s10 d10 t5
duodecad: p12 m12 n12 s12 d12 t6


Each new progression adds one new interval, plus adding one more to those already present; but beyond seven tones, no new intervals can be added. In addition to this loss of new material, there is also a gradual decrease in the difference of the quantitative formation.

As sonorities get projected beyond the six-range, they tend to lose their individuality.

Hanson: "Since, as has been previously stated, all seven-tone scales contain all of the six basic intervals, and since, as additional tones are added, the resulting scales become increasingly similar in their component parts, the student's best opportunity for the study of different types of tone relationship lies in the six-tone combinations, which offer the greatest number of scale types."

Here's my question: the hexads contain every interval except the tritone, which is the only self-inverting interval. The heptad (7-note) does contain the tritone; why is this not considered to have "more variety" than the 6-note groups?
Is the tritone redundant in this sense? Does the presence of the tritone create some sort of redundancy or repetition? Can you show any examples of this? How does the tritone figure in to your invertible hexads?

Does it matter how the hexad is generated? It seems that by fifths, the whole chromatic collection is given, so this seems to me the ideal, most complete way.
« Last Edit: June 27, 2010, 06:39:34 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 #5 on: June 27, 2010, 07:56:18 AM »
Here's my question: the hexads contain every interval except the tritone, which is the only self-inverting interval. The heptad (7-note) does contain the tritone; why is this not considered to have "more variety" than the 6-note groups?
Is the tritone redundant in this sense? Does the presence of the tritone create some sort of redundancy or repetition? Can you show any examples of this? How does the tritone figure in to your invertible hexads?

Does it matter how the hexad is generated? It seems that by fifths, the whole chromatic collection is given, so this seems to me the ideal, most complete way.

thank you for your response; this is exactly what i wanted to throw into the idea pool.  my review of forte's pitch sets reveals exactly five (out of 50, remember that forte includes inversions as equal here, but not complements) hexads that contain no tritone and one of them is, of course, your hexad generated by stacked fifths (further, your hexad is identical to [or at least a mode of] the minor pentatonic add2 referred to above) - as to why hanson states that the hexads have "the greatest number of scale types" i think he's referring to the fact that forte's criteria generates 50 hexads but only 38 heptads.

do also remember that the hexad generated by perfect fifths is an inversion and a complememt of itself.  i discovered this:

forte's listing of hexads (which seems to be a complete list with nothing left out) includes 50; however, if one considers complements as identical this list is reduced to 35 - and guess how many heptads there are: 38 (of course since the complement of a heptad is a pentad, none of the heptads are complements of each other)

this is an example of what i call an "inventory problem."  how many are there?  it depends on how you count them or more to the point, the criteria by which one chooses in order to count them and since the hexads are "special" (in that some of them are complements of each other) much hinges on the criteria.

your question as to how the hexad is generated is interesting because although the "hexad of fifths" generates all possible intervals (except as you have pointed out, the tritone) i don't think that hanson is limiting his comment to a specific method of generation.

i don't know if i'm hitting your questions squarely on or flying off a tangent so feel free to throw another bone at the ol' dog.

edit - forte relates the complements of the hexads by "z-relation" which is an interval analysis similar to what "millions" has done here - these terms may seem foreign to many; therefore, readers may wish to brush up on set theory by looking here:

http://www.composertools.com/

set theory notion is used to reduce the large number of possibilities (for tonal combinations) to something more manageable.  the main problem is that most of these tone groups don't have established names, so the set theory notation gives us a list - something to put on the desktop.  the 4096 total tonal combinations reduces (through transposition) to 352 scale/arpeggio types (from 0 through 12 tones) and i'm not going to try to name 300+ of anything :)
« Last Edit: June 27, 2010, 03:58:22 PM by dogbite »
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millions

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Re: Six-Tone Pitch Sets
« Reply #6 on: June 27, 2010, 10:27:38 AM »
Just in case anyone is wondering what the bleep we're talking about, I'm at the limits of my brain capacity myself...

I posted that video on Forte, and here's what his book looks like. I also included the Rahn book, which for me, explains Forte's ideas better than he does himself.



There's indexes in the back of both books, which list every possible combination of 3 notes, 4 notes, 5 notes, and 6 notes. The reason it only goes up to 6 is because there are only 6 interval types, although there are 12 notes. This is because intervals are invertible; fifth/fourth, M3rd/min 6, m3rd/M7, m2/M7, M2/m7, and tritone with itself.

What Doggy has done is apply all these to the guitar. So it's like a dictionary of possibilities. What you do with them is a whole 'nubba lebbel.
« Last Edit: June 27, 2010, 01:53:51 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 #7 on: June 27, 2010, 02:26:33 PM »
Here's the "Miracle Set" I mentioned. On the left is the chromatic scale, and you read the row by reading the dots from left to right. An "axis of symmetry" is the dotted line. The 45º line is the overall axis. I was doing graphs like this in order to identify certain symmetries in the rows, which would be enclosed with a soilid line.


« Last Edit: June 27, 2010, 02:31:41 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

Halfdim7

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Re: Six-Tone Pitch Sets
« Reply #8 on: June 28, 2010, 02:56:41 PM »
Well, all this has certainly piqued my interest(and eluded my understanding). I have a pretty anemic background in theory, really. I downloaded the Forte film, maybe it'll give me a better idea about the subject of this thread. Thanks also for the Composer Tools link, it seems like a good resource.
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dogbite

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Re: Six-Tone Pitch Sets
« Reply #9 on: June 29, 2010, 09:44:48 PM »
here's an interesting six-tone pitch group that i like:

C Db E F# G Bb

which may be played as two distinct triads (C E G and Gb Bb Db)

or arpeggiated as a C7b9#11:

C E G Bb Db F#

and is a subset of the octatonic "half-whole diminished" scale:

C Db D# E F# G A Bb C

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Swain

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Re: Six-Tone Pitch Sets
« Reply #10 on: June 29, 2010, 11:03:36 PM »
    This Thread is pretty interesting, and I'm not really sure of what I'm reading!  :-[

    I hope to have the time to read through it fully, soon.

    But, I did want to mention my affinity for pairing up a couple of Triads. So many cool options, and easy to put into practice.
   

    Here's a few favorites:

 

G + Am

G B D + A C E

Works great over Maj. or Dom. Chords, as it contains no 7th. But, it has the R, 3, 5, 2/9, 4/11, 6/13.

 

Or,

G + D

G B D + D F# A

R 3 5 M7 2/9

Or,

G + A

G B D + A C# E

R 3 5 2/9 #4/#11 13

G13#11(no7)

Or,


Gm + Am

G Bb D + A C E

R b3 5 + 2/9 4/11 6/13

I haven't tried running any of these combinations through any type of cycles, or any real systematic organizational process. I just tend to gravitate towaeds these particular pairs.




dogbite

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Re: Six-Tone Pitch Sets
« Reply #12 on: June 30, 2010, 08:12:10 AM »
the "triadic" approach that swain is referring to (using two distinct triads with no shared tones) seems to be related to bergonzi's but since i do not possess the book(s) in question, i cannot know for sure.  i know that trombonist ed byrne has written etudes on this (in the form of his bichordal triad pitch collections) but:

although the consonant triads used as examples by swain are a subset of all possible hexads, or hexatonics my original query was not limited to those.  although i am not as excited about atonal music (schoenberg, etc...) as "millions," the set theory treatises of allen forte, john rahn and charles wuorinen (and a host of others i'm sure) have provided for us a complete list of all possible hexatonics (as well as collections of tones numbering greater or lesser than six) and it is this vast six-tone universe that i am exploring at the moment.  it is a bit overwhelming; however, i have recently produced guitar fingering studies of all the hexatonics and it is the warmup from hell (and no i don't play every one of them every day).

in producing these studies, i was struck by the "uniqueness" of the six-tone groups in how they were related to each other.  i'll produce a complete list here if anyone's interested.

thanks for the links phil!
« Last Edit: June 30, 2010, 10:54:22 AM by dogbite »
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dogbite

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Re: Six-Tone Pitch Sets
« Reply #13 on: June 30, 2010, 08:37:32 AM »
Dog, here's a copy of an old post of mine in Amazon Classical which is my take on Schoenberg's Op. 26 Wind Quintet:

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.
...
So Schoenberg had several ideas in mind of the tonal implications when he chose this row.

In Forte nomenclature, what would this be?

millions,

the first row Eb G A B C C# is 04689A (remember that in forte's list the order doesn't matter, so the C# and C have been switched) which reduces (through harmonic inversion or mode shift) to 024568 and further (through melodic inversion or reversal of the intervals) to 023468 which is forte's hexad number 6-21.  my numbering of the list is different but i know you have forte's book so you can look it up this way...

the second one Bb D E F F# G# is 04678A and also reduces to 023468; therefore it is also listed in forte as 6-21

the interval structure from the hexads in schoenberg's op26 are 2 1 1 ½ ½ 1 and 2 1 ½ ½ 1 1 respectively which are inversions of each other and this is why forte listed them as the same

they appear through my "tonal lenses" as a whole-tone scale where one of the tones have been moved thus generating the major third interval as well as the chromatic (sequential) half-steps.
« Last Edit: June 30, 2010, 08:40:27 AM by dogbite »
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millions

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Re: Six-Tone Pitch Sets
« Reply #14 on: June 30, 2010, 10:32:42 AM »
Thanks for the reply, dogbite. I need to go brush-up on these Forte set ideas. It's good that you have found a way to apply these practically to the guitar.
"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