Author Topic: Interval Projection  (Read 13144 times)

millions

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Interval Projection
« on: June 23, 2010, 03:12:20 PM »
The material being discussed here is from Howard Hanson's "Harmonic Materials of Modern Music".

The Projection of the Fifth

As we all know, going around the circle of fifths yields all twelve notes before repeating. Therefore, there is a progression into chromaticism that is visible in this process.

First, some 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

"Projection": the building of sonorities or scales by superimposing a series of similar intervals one above the other.

Beginning with C, we add G, then D, to produce the triad C-G-D, or reduced to an octave, or its "melodic projection", C-D-G. Numerically, in terms of 1/2 steps, 2-5. In terms of total interval content, using the nomenclature above: p2 s.

Next, we add A to the stack, forming the tetrad C-G-D-A, reduced melodically to C-D-G-A. Numerically, 2-5-2. Interval content: p3 n s2.
The minor third appears for the first time.

Next, pentad C-G-D-A-E, reduced to C-D-E-G-A, recognizable as the pentatonic scale. The major third appears for the first time. Numerically, 2-2-3-2. Interval analysis: p4 m n2 s3.

The hexad adds B, forming C-G-D-A-E-B, reduced to C-D-E-G-A-B. Numerically: 2-2-3-2-2. Interval content: p5 m2 n3 s4 d.
For the first time, the dissonant minor second (or major seventh) appears.

Continuing, we add F# to get the heptad C-G-D-A-E-B-F#, reduced as C-D-E-F#-G-A-B. Here the tritone appears; also, this is the first scale which in its melodic projection contains no interval larger than a major second; i.e., look, ma, no gaps. It contains all six basic intervals for the first time in our series.
Numerically: 1-1-2-2-1-2-2. Intervals: p6 m3 n4 s5 d2 t.

Octad: Add C#, yielding C-C#-D-E-F#-G-A-B. Numerically, 1-1-2-2-1-2-2. Intervals: p7 m4 n5 s6 d4 t2.

Nonad: Add G#: C-C#-D-E-F#-G-G#-A-B. Numerically, 1-1-2-2-1-1-1-2. Intervals: p8 m6 n6 s7 d6 t3.

The Decad adds D#, yielding C-C#-D-D#-E-F#-G-G#-A-B.
Numerically, 1-1-1-1-2-1-1-1-2. Intervals: p9 m8 n8 s8 d8 t4.

Undecad: Add A#. C-C#-D-D#-E-F#-G-G#-A-A#-B. In 1/2 steps, numerically, it is 1-1-1-1-2-1-1-1-1-1. Interval content: p10 m10 n10 s10 d10 t5.

The last one, the duodecad, adds the last note, E#. C-C#-D-D#-E-E#-F#-G-G#-A-A#-B.
Numerically: 1-1-1-1-1-1-1-1-1-1-1.
Interval content: p12 m12 n12 s12 d12 t6.

Note the overall progression:
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

What can be noted is the affinity of the perfect fifth and the major second, since the projection of one fifth upon another always produces the concomitant interval of a major second;
The relatively greater importance of the minor third over the major third; the late arrival of the minor second, and lastly, the tritone.

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.
In the octad, the same number of major thirds & minor seconds; In the nonad, same number of maj thirds, min thirds, and min seconds. In the decad, an equal number of maj/min thirds and seconds.
When 11 and 12 are reached, the only difference is the number of tritones.

So the sound of a sonority, whether it be harmony or melody, depends on what is present, but also on what is not present.
The pentatonic sounds as it does because it contains mainly perfect fifths, and also maj seconds, minor thirds, and one major third, but also because it does not contain the minor second or tritone.

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

This is probably the greatest argument against the rigorous use of atonal theory in which all 12 notes are used in a single melodic or harmonic pattern. These constructs begin to lose contrast, and a monochromatic effect emerges.

Each scale discussed here can have as many versions as there are notes in the scale. The seven-tone scale has seven versions, beginning on C, D, E, and so forth. These "versions" should not be confused with involutions of the same scale.

What has the projection of a fifth revealed to us?

Quoting 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."
"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 #1 on: June 23, 2010, 03:31:17 PM »
 :o ???
I'm starting to think I might not fit in amongst all you geniuses haha!

Halfdim7

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Re: Interval Projection
« Reply #2 on: June 23, 2010, 04:53:39 PM »
Okay, first off "Nonad"  :D :D :D
Words I had to look up in the Dictionary: "concomitant", "involutions"

Very interesting, millions.
So, basically the argument is that 6 tone sonorities provide the most creative material?
I think I understand the reasoning behind this(if you mix all the colors on the pallet, all you're left with is black), but It's probably gonna take a few reads before I'm certain I really grasp the whole concept.
 

....lame-ass, jive, pseudo bluesy, out-of-tune, noodling, wimped out, fucked up playing....

dogbite

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Re: Interval Projection
« Reply #3 on: June 23, 2010, 10:42:33 PM »
this is neat stuff.  although i don't have the book (hanson's) the language is familiar from the twelve-tone studies i've seen of forte and wuorinen.  in studying the six-tone groups i came to the conclusion that it is important to grasp several concepts of cyclic group theory:

1) inversion (melodic inversion, not the same as harmonic inversion)

2) complement

for #1 above, an inversion is a set of tones literally turned upside down:

C E G is a tone (C) followed by a second tone going up a major third (E) followed by a third tone up a minor third (from the second tone, G)

its inversion is a tone (C) followed by a second tone going down a major third (Ab) followed by a third tone down a minor third (from the second tone, F)

therefore, the melodic inversion of a C major triad (C E G) is an F minor triad (F Ab C)

this is an important concept because all sets of tones are inversions of either themselves or other sets and this will greatly simplify the study of especially the six-tone groups.

as to #2 above, the complement of a set of tones is defined as "all tones not included in the original set" such as the major scale (often defined as the white keys on the piano, the natural notes) and its complement, the pentatonic scale (often defined as the black keys on the piano, the non-natural notes) and why is this important?

because the six-tone groups are unique in that their complements are either themselves or other six-tone groups

next question:

how many (unique six-tone sets) are there?
« Last Edit: June 23, 2010, 10:44:46 PM by dogbite »
s/aka/db

millions

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Re: Interval Projection
« Reply #4 on: June 24, 2010, 04:39:09 PM »
"...next question: how many (unique six-tone sets) are there?"

Would it be cheating if I ran over and got my Forte book out & looked in the index? I know there's not that many; 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."

I'm going to fly by the seat of my pants & discuss fifths, cycles, and the "12 limit." As you may have guessed, when intervals other than the fifth are projected, they tend to "poop out" in a short time. The next interval Hanson discusses is the major third; it yields a cycle of only three notes before re-connecting with itself at the octave. There are four adjacent versions of this, hence the four augmented chords (3x4=12).

But we forget, don't we? that the "12 cycle" we take for granted is itself imperfect, the stacked circle of 3/2 fifths not really perfectly "reconnecting" with itself after 12 notes have been generated. Thus, Pythagorus coined the term "close enough for rock & roll."

What if we did the same thing with the major third, and used a stack of 'just' major thirds (5/4) to generate our cycle? Where will it stop, this Pythagoran Wheel of Fortune?

I decided to do it with cents. We know that an octave is 1200 cents, and that a 'just' major third is 14 cents flat of our equal-tempered third, which is 400 cents, making the 'just' M3rd about 386 cents (386.3 is closer).

So, how many times do we have to "stack" this 5/4 major third before it (apparently) comes back to its starting point? We're shooting for some multiple of 1200 cents, since that's the octave.

Okay, 1200 + 1200 is 2400, + 1200 is 3600, 4800, 6000, 7200, 9600, 10,800, 12,000, 13,200, 14,400, 15600, 16,800, 18,000, 19,200, 20,400, 21,600, 22,800, 24,000, 25200, 26400, 27,600, 28,800, 30,000, 31,200, 32,400, 33,600, 34,800, 36000,

Now, we start adding up the 5/4s, or 386.3s. That gives us 386.3, 772.6, 1158.9, 1545.2, 1931.5, 2317.8, 2704.1, 3090.4, 3476.7, 3863.0, 4249.3, 4635.6, 5021.9, 5408.2, 5794.5, 6180.8, 6567.1, 6953.4, 7339.7, 7726.0, 8112.3, 8498.6, 8884.9, 9271.2, 9657.5, 10043.8, 10430.1, 10816.4, 11202.7, 11589.0, 11975.3, 12361.6, 12747.9, 13134.2, 13520.5, 13906.8, 14293.1, 14679.4, 15065.7, 15452.0, 15838.3, 16224.6, 16610.9, 16997.2, 17383.5, 17769.98, 18156.1, 18542.4, 18928.7, 19315.0, 19701.3, 20087.6, 20473.9, 20860.2, 21246.5, 21632.8, 22019.1, 22405.4, 22791.7, 23178.0, 23564.3, 23950.6, 24336.9, 24723.2, 25109.5, 25495.8, 25882.1, 26268.4, 26654.7, 27041.0, 27427.3, 27813.6, 28199.9, 28586.2, 28972.5, 29358.8, 29745.1, 30131.4, 30517.7, 30904.0, 31290.3, 31676.6, 32062.9, 32449.2, 32835.5, 33221.8, 33608.1, Wait! Bingo! 33,608.1 is almost equal to 33, 600: only 8.1 cents off! I say, let's go for it, before our calculators run out of battery power.
So how many cycles of 5/4s is that? Let's see...33,608.1 divided by 386.3 is 87!

So our Pythagoran "major third" scale is 87 notes per octave!
« Last Edit: June 30, 2015, 08:19:23 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 #5 on: June 24, 2010, 07:31:50 PM »
therefore, the melodic inversion of a C major triad (C E G) is an F minor triad (F Ab C)...this is an important concept because all sets of tones are inversions of either themselves or other sets and this will greatly simplify the study of especially the six-tone groups.

The consequence of this, doggie, is that interval relationships are preserved, but at the expense of pitch identity, as in tonality, where C-E-G retains its major quality regardless of the order, E-G-C, G-C-E, etc. Neither good nor bad...is it?
« Last Edit: June 30, 2015, 08:18:13 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 #6 on: June 24, 2010, 09:55:50 PM »
therefore, the melodic inversion of a C major triad (C E G) is an F minor triad (F Ab C)...this is an important concept because all sets of tones are inversions of either themselves or other sets and this will greatly simplify the study of especially the six-tone groups.

The consequence of this, doggie, is that interval relationships are preserved, but at the expense of pitch identity, as in tonality, where C-E-G retains its major quality regardless of the order, E-G-C, G-C-E, etc. Neither good nor bad...is it?


"at the expense of pitch identity"

i presume you mean function here and in the case of tonal structures (or those that are perceived as tonal) i would say so, but for the others i wouldn't be too sure.  i know that (to me anyway) many six-tone sets are perceived as subsets of identifiable tonal scales such as major and melodic minor as well as symmetrical (octatonic, diminished); however, many are not so the fact that application of complementary and inverted sets reduces forte's 50 (he actually accounts for inverted sets) into a mere 35 makes the list look a bit more manageable - and without sacrificing identity where none may have actually been perceived in the first place...

but not in search of lofty melodic materials implied by such musings; i merely use sets like this to warm up and discover sonorities not readily accessible through common scales.  and it's fun.

ps - your interval analysis reminds me of forte's "interval class vectors" - i trust you'll tell me if this is not so.
« Last Edit: June 24, 2010, 09:57:57 PM by dogbite »
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millions

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Re: Interval Projection
« Reply #7 on: June 25, 2010, 04:20:26 AM »

"at the expense of pitch identity"

i presume you mean function here and in the case of tonal structures (or those that are perceived as tonal) i would say so, but for the others i wouldn't be too sure.  
ps - your interval analysis reminds me of forte's "interval class vectors" - i trust you'll tell me if this is not so.

Dawg:  Of course, you could abstract "pitch" into function, as in I-III-V. Yes, "function" as in reference, hierarchy, stacking; not pure interval relations.

In Forte's sets, pitch identities are preserved; a C is a C is a C...the almighty 2:1 is preserved. Seems contradictory in a way. Pitch is a continuum, I thought...and even you admit to hearing some of the hexads as having tonal, scalar characteristics. Are these vestiges of "function?" Is "function" a built-in consequence of an octave divided into 12 parts, a 12-note system based on fifths?

I'm not sure what you are referring to with "interval analysis." You mean Arnold's invertible hexads?
« Last Edit: June 25, 2010, 04:36:29 AM by millions »
"In Spring! In the creation of art, it must be as it is in Spring!" -Arnold Schoenberg
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dogbite

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Re: Interval Projection
« Reply #8 on: June 25, 2010, 06:11:02 AM »
ps - your interval analysis reminds me of forte's "interval class vectors" - i trust you'll tell me if this is not so.
I'm not sure what you are referring to with "interval analysis." You mean Arnold's invertible hexads?

this:

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
...
p4 m n2 s3
...
p5 m2 n3 s4 d
...
p6 m3 n4 s5 d2 t
...
etc...
i think function is the result of expectation - common and familiar usage which leads us to anticipate what follows.  take the V7 I for example, the alpha cadence: what is most commonly (and correctly i believe) known as the authentic cadence.  because of the music i listen to, i don't necessarily see it as my "old" theory teacher would ("if the seventh is either improperly resolved or unresolved...") and am perfectly comfortable hearing the dominant seventh chord in either a tonic or subdominant role, as well as the "expected" dominant (V I) capacity.  i could literally sense the discomfort of the instructor when stepping outside of his forte which was the world of the fugue, piano invention, and bach chorale.  he knew zip about rock and pop, jazz or blues and i truly think that he struggled to "hear what i hear" - i mean the "improperly resolved 6/4 chord" which is common in say an elton john tune seemed to trouble him to no end.

i believe this to be part of the "nature vs. nurture" aspect of music listening.  is "hearing" pure physics or cultural norms?  50/50?

new thread on this perhaps?
« Last Edit: June 29, 2010, 09:39:19 AM by dogbite »
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millions

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Re: Interval Projection
« Reply #9 on: June 25, 2010, 03:29:01 PM »
From what I've read, Berg used rows which were rich in thirds, so he could build chords, or triads.
Even Schoenberg used rows with tonal implications.

Here's an old post of mine from the "Schoenberg's Op. 26 Wind Quintet" thread in Amazon Classical:

[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.]

I hope this regurgitation was helpful.
"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 #10 on: June 25, 2010, 06:25:23 PM »
Dogbite, I'm "probing around for insight" here, so please forgive me if I ramble on.

In reference to "function" through linear time, vs. relative simultaneous harmonic dissonance (tension, compared to what) & consonance (compared to what), here is an excerpt of an old conversation I had elsewhere:

Most dissonant intervals to most consonant intervals, within one octave:
1. minor seventh (C-Bb) 9:16
2. major seventh (C-B) 8:15
3. major second (C-D) 8:9
4. minor sixth (C-Ab) 5:8
5. minor third (C-Eb) 5:6
6. major third (C-E) 4:5
7. major sixth (C-A) 3:5
8. perfect fourth (C-F) 3:4
9. perfect fifth (C-G) 2:3
10. octave (C-C') 1:2
11. unison (C-C) 1:1

The steps of our scale, and the "functions" of the chords built thereon, are the direct result of interval ratios, all in relation to a "keynote" or unity of 1; the intervals not only have a dissonant/consonant quality determined by their ratio, but also are given a specific scale degree (function) and place in relation to "1" or the Tonic. This is where all "linear function" originated, and is still manifest as ratios (intervals), which are at the same time, physical harmonic phenomena.

"One (1:1) is the ultimate consonance. In the beginning was ONE. From this, sprang forth the universe.

"All musical understanding can be reduced to the understanding of one note."

The interval ratios in the chart above, to the right, are just a way of expressing the relationship of two notes. For example, 2:1 is the octave, or doubling of frequency; conversely, 1:2 halves it.

In the key of C, a simple 1-3-5 G triad is not identical to the simple 1-3-5 C triad, because of its position (functioning as V) in relation to the root. The D, resolving down, now becomes root, as well as being the top of a fourth G-C, which is heard as root on top.

Someone somewhere asked, "Why, universally/acoustically speaking, isn't the G triad identical to the C triad to anyone without absolute pitch? They are major triads and are equally dissonant. The functional difference is only apparent once tonic has been established. Tonic is established correctly once the listener has heard and connected (COGNITIVELY) the series of intervals that constitute the diatonic scale."
 
This is an irrational question. "G" must be seen in relation to its home key, whether that be "G" or "C", not in isolation. No chord exists in isolation, but all exist in relation to "1", unity, or tonic.

Implicit in any harmonic interval, whether it be 2:3 or 3:4, is an implicit relation, and specific note-position in the heirarchy, in relation to "1" or tonic, as well as its being more dissonant or more consonant in relation to "1" or the root.

I have been speaking of the V, V7, & vii chords in relation to Tonic, and their resolution to tonic. I never "isolated" the chords from their scale-steps; in talking about them as "I, V, and iii" chords, this is implicit.

"The functional difference is only apparent once tonic has been established. Tonic is established correctly once the listener has heard and connected (COGNITIVELY) the series of intervals that constitute the diatonic scale."

No, this happens simultaneously, as the result of a perceived relationship, or ratio. A ratio is not a fixed quantity, it is a relationship between two things.

In the case of simple pop song progressions, using static chord exchanges of say, C-F, it might be ambiguous whether "C" is I and "F" is IV, or if "C" is V and "F" is I; in fact, many pop songs play on this ambiguity.

In this age of equal-temperament, these relations are harder to identify than from a mean-tone tuning in which each scale degree has a more colored, distinct relation to I.

So what you are saying, is that all functions relate to the Tonic, or unity. This is exactly the way interval ratios work, also.

From Harry Partch, "Genesis of a New Music:"

[A ratio represents a tone and an interval at one and the same time; in its capacity as the symbol of a tone it is the over number that is nominally representative (in the upward manner), but since the over number exists only in relation to the under number, the ratio acquits its second function, as representative of an interval;

...conventional musical example: 3/2 represents "D" in the "key of G" - upward from "G"; it is thus simultaneously a representative of a tone and an implicit relationship to a "keynote" - or unity.]

Thus, it is seen that the steps of our scale, and the "functions" of the chords built thereon, are the direct result of interval ratios, all in relation to a "keynote" or unity of 1; the intervals not only have a dissonant/consonant quality determined by its ratio, but also is given a place in relation to "1" or the Tonic. This is where all "linear function" originated, and is still manifest as...ratios.

[...the identities of a tonality (tonal polarity around a 1 identity) can of course be deliberately placed to simulate the series of partials, but in this sense they are not partials.]

[The scale of musical intervals begins with absolute consonance (1 to 1), and gradually progresses into an infinitude of dissonance, the consonance of the intervals decreasing as the odd numbers of their ratios increase.]

So, how does the "time line" figure into the cognition of "function" of chords? Is it an afterthought? Is it essential? Does cognition of function always involve hearing a sequence of events, or can the functions be inherent in the chords themselves?
"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: Interval Projection
« Reply #11 on: June 25, 2010, 09:00:32 PM »
millions,

first, thanks for the ramble.  i do very much enjoy your ramblings.  tell me you haven't read george russell, who also speaks of "unity" in the hierarchy of tonal materials.

i believe the time line to be "key" to understanding several concepts of tonality, and your "C to F" example is perfect:

a thought experiment

play the C chord today (all day) and tomorrow play G (all day)

what key are you in today?  C, of course!

what key are you in tomorrow?  G, you say, and i would not disagree...

now let us progressively shorten the harmonic rhythm of our thought experiment to 12 hrs each, then 6 hrs each, and so forth until we reach two beats each at say 84 bpm...

at some point, one chord or the other will be perceived as the tonic.  at what point i do not know.  which chord i cannot say.  hence the concept of:

1) horizontal melody, and

2) vertical melody

more to follow, and thanks for the segue!

back soon
s/aka/db

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Re: Interval Projection
« Reply #12 on: June 28, 2010, 10:21:38 PM »
Which came first, the IV or the V?

dogbite

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Re: Interval Projection
« Reply #13 on: June 29, 2010, 12:57:56 AM »
Which came first, the IV or the V?

chicken or the egg?

i do not know but my understanding is that (a chord) a fifth up comes along with (a chord) a fifth down, so you really cannot have one without the other.  how's that for a non-answer :)
s/aka/db

millions

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Re: Interval Projection
« Reply #14 on: July 07, 2010, 07:13:31 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.

In Howard Hanson's terms, a pentad C-Eb-F-G-Bb like this is shown as a series of semitones: 3-2-2-3 (C-Eb-F-G-Bb).
Then, every internal interval relation between every note is found:
1. C to Eb,
2. C to F,
3. C to G,
4. C to Bb;
then
5. Eb to F,
6. Eb to G,
7. Eb to Bb;
then
8. F to G,
9. F to Bb,
F to E...whoops, already got that one as E to F, same thing...
F to C...whoops, already got that one as C to F, same thing...
then, finally
10. G-Bb
G-C...got it...
G-F...got it....that's all.

These "whoops" show you how intervals invert, so they are equivalent.

In Hanson's terms, our interval content for the C min pent set is p4 m n2 s3 (four perfect fifths/fourths (p4), one major third (m), two minor thirds (n2), and three major seconds (n3). This adds up to ten intervals.

So really, what the Forte set system does is converts everything into the least redundant, most unique set of interval relations. For instance, a C major scale is seen a a unique set of interval-spaces, like a moveable template-shape, no matter what note it starts on, or what mode it is, because all the notes are in a repeating circle (pitch classes). In fact, it might be easier to show some of these things graphically, on a circle.

« Last Edit: July 07, 2010, 07:18:12 PM by millions »
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