Interval Projection

[dogbite]

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.

this is tough stuff. forte calls the analysis of intervals the "interval class vector" - yeah, i started scratching my head when i first saw this too.

<032140>

is the interval class vector of the pentatonic scale, read as:

<0 half-steps, 3 whole-steps, 2 minor thirds, 1 major third, 4 perfect fourths, 0 tritones>

this confirms for me that your description of hanson's interval analysis (is this what "interval projection" refers to?) is identical to forte's with a different nomenclature.

am i getting this right?

to onlookers: this kind of interval analysis tells us that the pentatonic scale is extremely consonant, as it contains no half-steps (or major sevenths) and no tritones. the analysis tells us of the "flavor" of any pitch set through the quantities of certain types of intervals in the combinations of tones...

every prime form (representing a specific pitch set) has a unique interval vector, but there are interval vectors that represent more than one pitch set

like i said, this is tough stuff. look up the paul nelson tutorial online or see the forte or rahn texts for more on this.

millions,

what is the hanson text called?

[millions]

Dog, I'm glad you like that! The Hanson book is "Harmonic Materials of Modern Music." I'm hoping to combine ideas from this with Forte and Schat's "Tone Clock," as soon as I grok it all.

"Interval projection" is simply "stacking" intervals, but rearranged as being in one octave, so that scales can be generated. This is how Pythagorus got our 12-note series.

Here's a graphic representation of "interval content" of a set. Basically, it shows every possible interval relation. There are ten connecting lines, if you count carefully. Can you see, also, how this might lead to seeing new "subsets" as shapes within?




Hanson discusses the projection of the minor second next, the only interval besides the fifth which yields all 12 chromatic notes when "projected" onto itself. I think I'll skip that one, and go on to the major second.

[millions]

As I said, the projection of the minor second has its points of interest, but the hexad produced by it is more melodically useful than harmonic. That would be C-C#-D-D#-E-F. Remember the principle of inversion; this explains why six-note sets (hexads) are the most varied & harmonically useful.

The interesting points are that, with the fifth, the minor second gives all 12 notes when projected.

Next interval is the major second, which, if you will recall, is produced by stacking two fifths (C-G-D, yields C-D), and also by stacking two minor seconds (C-C#-D yields C-D). It seems logically the next interval in line for projection. Also, it can yield a pure six-tone scale, whereas the minor third and major third can only be projected to four and three tones respectively.

The major second's projection limit is the hexad C-D-E-F#-G#-A#, or the whole-tone scale. The next note would be B#, the enharmonic equivalent of C.

The major-second hexad has a lot of symmetries; its form is the same clockwise or counter-clockwise, and its inversion produces the identical tones.

Left here, at the six-limit, the hexad has limited use. The lack of a fifth makes it sound harmonically ambiguous.

The other two interval-series scales we looked at, the perfect fifth and the minor second, can be transposed to eleven other notes, which allows for "modulation;" the whole-tone scale can only move to one note above or below; that is, C-D-E-F#-G#-A# can only go to Db-Eb-F-G-A-B. No "modes" are really practical, either, since whole-tone scales on C, D, E, etc. are essentially all the same.

Here's the diagram of the two whole-tone scales:








Next, we'll go beyond the hexad, and produce other scales derived from this basic schema.

[millions]

As you can see by the previous diagram, adding a seventh note to our "pure projection" hexad is going to be arbitrary, as we look further into it. Whatever note we add is chromatically adjacent to our existing six notes, and will end up creating essentially similar versions (or "modes") of the scale.
So, let's add "G" above C, in order to give us a fifth, and give this scale some stability. This gives us the seven-note scale C-D-E-F#-G-G#-A#. So we still have all the "goodies" of the whole-tone scale: lots of major seconds, major thirds and tritones, plus, now we have two fifths (C-G/G-D), two minor thirds, and two minor seconds.

Here's the complete projection of the major second, with interval analysis:

C D...................................................s
C D E................................................m s2
C D E F#...........................................m2 s3 t
C D E F# G#......................................m4 s4 t2
C D E F# G# A#.................................m6 s6 t3
C D E F# G  G# A#.............................p2 m6 n2 s6 d2 t3...this is the most interesting.
C D E F# G  G# A  A#.........................p4 m6 n4 s7 d4 t3
C C E F# G  G# A  A# B.......................p6 m7 n6 s8 d6 t3
C C# D E F# G  G# A  A# B..................p8 m8 n8 s9 d8 t4
C C# D D# E F# G  G# A  A# B............p10 m10 n10 s10 d10 t5
C C# D D# E E# F# G  G# A  A# B.......p12 m12 n12 s12 d12 t6


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

This is definitely an exception to the "hexad variety" rule, since in this case 7 notes adds three new intervals, and increases the variety of this scale. This is an extremely interesting scale. All of its "modes" are worth exploring, and result in a variety of different tonal effects, some "majorish," some "minorish."

This seven-note scale is listed as the IV "mode" of the "Neapolitan Major" scale, called "Lydian minor," on p. 102 in the Guitar Grimoire "A Notated Intervallic Study of Scales" (orange cover), as 1-2-3-#4-5-b6-b7.

[millions]

Before I move on to the next interval projection, I'd like to think out loud about some possibilities.
Pythagorus was the Greek who discovered a scale created by "stacking" fifths. Supposedly the Chinese did the same thing, and arrived at different conclusions; so the idea of "projecting" intervals is not exclusive to one culture, but seems to be more like a natural principle which was "discovered" by different cultures at different times.
Looking back at the hexad created by the fifths-projection, C-D-E-G-A-B, this hexad could be interpreted as a "C major pentatonic with a leading tone."
Its complement (the other six notes) is Gb-Ab-Bb-Db-Eb-F, or a "Gb major pentatonic with a leading tone."

There's that tritone/fifths connection again! This hexad also contains every interval EXCEPT the tritone.

I wonder if this has to do with the evolution of our Western system of music? We now have a six-note scale with a leading tone, which reinforces the I-V chords. The next fifth in line for the hexad C-D-E-G-A-B is F#, which creates our FIRST tritone (C-F#).

[millions]

Spring is here! I'll be posting some more info on interval projection shortly.

[millions]

Here comes the next interval-projection, the minor third. As you may recall, there are only six basic intervals, because they invert. The next largest interval after the minor third would be the major third; we'll get to that next. After that, the perfect fourth, and that's invertible as a fifth, so it's already been covered.
Tritones are redundant, inverting back on to themselves, never projecting past those two notes in their 'pure' form, so arbitrary tones must be added, and intervals projected further from those.

For the minor third projection, we'll start on C, project a minor third from that, yielding an Eb, then another minor third from there, giving us Gb. From Gb, up a minor 3rd is Bbb, or enharmonically, an A, giving us the familiar 'diminished seventh' chord. As you can see by the 'Bbb/A,' the glitch in our diatonic 7-letter-name scale system is revealed by this. A scale must consist of seven different letter-names. This is a good time to discuss this in more detail.

On a keyboard, Gb and F# are the same note, physically.

If one starts building fifths from a starting point of C, then going "forward" or clockwise around the "circle of fifths" would yield C-G-D-A-E-B-F#(C#-no need for D#).

If, on the other hand, you go in reverse (counter-clockwise), you travel the "circle of fourths", which yields C-F-Bb-Eb-Ab-Db-Gb (Cb).

As you can see, there are three keys which "overlap" under two different names: B (Cb), F# (Gb), and C# (Db). The reason it goes no further has to do with the physical layout of the keyboard itself (there are two semitone steps in the letter sequence), and the subsequent "letter-naming" of notes which results. To be a diatonic scale, you must have seven different letter names. You can also see from this, that the ''seven-letter do-re-mi' system of naming and notation is tied to the keyboard and its layout. For a guitarist, this can seem very arbitrary and confusing, because, well, it IS.

For example, there is no key of "Fb" because this is E, a sharp key; but if we named it anyway, we would get Fb-Gb-Ab-Bbb (you can't repeat A - there must be seven different letter names with no repeats), Cb-Db-Eb-Fb. This "repeating letter or double-flat" dilemma does not arise on the three "repeat" keys of B (Cb), F# (Gb), and C# (Db), because this is the "seven-letter limit".

This is worth pondering; I know the guy who wrote the Guitar Grimoire pondered it, because he lists all the 'impossible' scales, like Fb, just to demonstrate the idea. Next time, we'll finish up our minor-third scale projection. Can you guess which note could be added next? Think 'stability' and 'tonality.'

[millions]

Continuing with our discussion of interval projection, which is essentially just a way of creating scales which are based on the stacking of intervals, which is how Pythagorus created our 12-note-per-octave scale, we are now to the minor third.

For the minor third projection, we'll start on C, project a minor third from that, yielding an Eb, then another minor third from there, giving us Gb. From Gb, up a minor 3rd is Bbb, or enharmonically, an A, giving us the familiar 'diminished seventh' chord. The interval content of this is four minor thirds, and two tritones (C-Gb/Eb-A).

Where to go next? When you examine the possibilities, this is not as arbitrary a process as it may seem at first. In fact, the choice of the next 'foreign' tone is not really that important, since any foreign tone added would produce either a different version, or the involution of the same scale.
An 'involution' is a mirror-image of the same intervals. There are three basic types of involution: simple, enharmonic, and isometric. I'll discuss the concept of involution in a later thread.

The next note added to our diminished seventh projection will be G, giving us a stable fifth above C, and making this scale more 'rooted' and less harmonically vague. This pentad is C-Eb-Gb-G-A. Interval content: p-m-n4-s d-t2. So, the addition of just one note has added to our earlier 4 min 3rds/2 tritones (n4-t2) new intervals of major second, minor second, and perfect fifth.

Next, we'll add a minor third above the G, giving us the six-note scale C-Eb-Gb-G-A-Bb. This note adds new intervals, a minor third, a perfect fifth, a major third, a major second, and a minor second, giving us the interval analysis p2-m2-n5-s2-d2-t2.

Try to find all the possible triads contained in this minor third hexad, and you will see a wealth of possibilities opening up. In the past, critics have always pressed me to show them 'how any of this matters,' so in future threads I will write some short pieces using these new scales.
 

[millions]

dogbite said earlier:
"..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..."
-------------------------------------------------------------------------------------------------------------------------------------------
In answer to dogbite, that makes sense, because 6+6=12.

Hanson calls 'inversion' (above) 'involution.' I'm not sure about the distinction between 'melodic inversion' and 'harmonic inversion.'

In Hanson's system, the involution of any major triad can be considered to be any minor triad, whether or not there is an axis of involution present.
B minor, Bb minor, G# minor, F# minor, Eb minor, and D minor triads are all considered as possible involutions of the C major triad, although there is no axis of involution as with F-Ab-C, where C is the 'pivot note' or axis of up/down.

1. B-C-D-E-F#-G (C major + B minor)
2. Bb-C-Db-E-F-G (C major + Bb minor)
3. C-D#-E-G-G#-B (C major + G# minor)
4. C#-E-F#-G-A-C (C major + F# minor)
5. C-Eb-E-Gb-G-Bb (C major + Eb minor)
6. F-G-A-C-D-E (C major + D minor)

Note that combining a sonority (in this case a major triad) with its involuted form always produces an isometric sonority, which means the interval formation is the same whether thought up (forward or clockwise on the circle) or down (backwards or counter-clockwise on the circle).

Example one, begun on B, is 1-2-2-2-1 (in half steps), whether considered from B to G or from G to B.

Example 2, C major and Bb minor, must be begun on Bb or E to make its isometric character clear. From Bb it is 2-1-3-1-2; from E it is 1-2-3-2-1. See the symmetry?

This cannot be achieved if we combine two major triads.

I guess the point to note here is that all of these 'sets' of notes we've been talking about are 'sonorities,' producing effects on the ear both harmonically and melodically. If the 'interval content' of two sets is identical, the sonorities are going to be the same, regardless of how we arrange them for symmetry as strings of intervals, or 'melodies,' or 'rows' of notes. For me, at this point, Hanson's ideas are more geared towards harmonic aspects of the sonorities or 'sets,' regardless of how they are ordered. I'm exploring this area in further depth, as I speak.

[Halfdim7]

By the way, how is dogbite these days? It's been quite a while sense he's shown up around here.

[millions]

I guess I'll have to e-mail the dawg and beg him to post. BTW, his mention of 'reciprocal scales' and sets gives me an idea for a new thread, of practical value, which can actually be applied to jazz! It involves pentatonics, not really as the reciprocals of various other scales, but as subsets of seven-note scales.

[dogbite]

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?

[millions]

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

I ran across this idea from Steve Khan's 'Pentatonic Khancepts' book & DVD. I didn't have the book out, and was just jamming to the tracks. The tracks are featured with or without Khan's lead examples. I was trying to play along with his leads, and just stumbled on to the concept behind it. Admittedly, I knew beforehand that it involved pentatonics.

It was basically a ii-V-I progression, as F#min7/B7alt/E maj 7.

The pentatonics are the familiar 'blues boxes' everyone knows. Just build the familiar pentatonic scales which span frets 2-5, 3-6, and 4-7, and move these up chromatically (2-3-4) (ii-V-I). That's it! It sounds instantly 'jazzy;' I was amazed.

I started adding other notes, and realized that the pent scale spanning frets 4-7 was really part of the E major scale in the same place, the one that starts on E, fifth string, seventh fret, pinky.

The same for the others. It seems to sound better if you use a b9 voicing for the chord accompaniment on the V.

So, in applying this concept, you are actually leaving notes out of the regular 7-note scales, and emphasizing notes that you might not normally emphasize. Plus, since it's a pentatonic scale, it's a no-brainer.

[millions]

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?

[dogbite]

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

I ran across this idea from Steve Khan's 'Pentatonic Khancepts' book & DVD. I didn't have the book out, and was just jamming to the tracks. The tracks are featured with or without Khan's lead examples. I was trying to play along with his leads, and just stumbled on to the concept behind it. Admittedly, I knew beforehand that it involved pentatonics.

It was basically a ii-V-I progression, as F#min7/B7alt/E maj 7.

The pentatonics are the familiar 'blues boxes' everyone knows. Just build the familiar pentatonic scales which span frets 2-5, 3-6, and 4-7, and move these up chromatically (2-3-4) (ii-V-I). That's it! It sounds instantly 'jazzy;' I was amazed.

I started adding other notes, and realized that the pent scale spanning frets 4-7 was really part of the E major scale in the same place, the one that starts on E, fifth string, seventh fret, pinky.

The same for the others. It seems to sound better if you use a b9 voicing for the chord accompaniment on the V.

So, in applying this concept, you are actually leaving notes out of the regular 7-note scales, and emphasizing notes that you might not normally emphasize. Plus, since it's a pentatonic scale, it's a no-brainer.

what you have done here is II minor pentatonic bIII pentatonic III pentatonic subbed for the ii V7 I, or:

F#m7/11 Gm7/11 G#m7/11

for

F#m7 B7(b9) Emaj7

i learned this from vic trigger (of GIT faculty) and i would also add:

C#m7/11 Dm7/11 D#m7/11

because of the fusiony sound of D#m pentatonic over the Emaj7 for a maj7/6/9/#11 sound:

D# F# G# A# C#

or

7---9---3---#11---13

of

E...

i ripped this from the forum of doom, similarly discussing pentatonic subs:

pentatonic subs

learn to play pentatonic scales in groups of three along the circle of fifths, such as

Am Em, and Bm pentatonic...

A C D E G

E G A B D

B D E F# A

the three above pentatonics work great over:

Cmaj7
Am7
F#m7b5
D7

weave in and out of each in such a way as to not be stuck in any of the three pentatonics, thus adding a sense of harmonic texture and motion.

quote from other jazz guy:

"Doggy...That pentatonic submarine is great for the first 4 bars of Joe Henderson's "Inner Urge"...I am gonna practice that one as well as everything you post when I can....My dogs are taking up a lot of my time this weekend....I have two "Golden Doodles" named Jimi and Ella........"

end quote from other jazz guy

i especially like centering my focus on the V minor pentatonic, while weaving into the I (more inside) and the V/V (more outside) and check this out - for some tunes a fourth pentatonic may be appropriate:

A7, A7sus, or A bass

A C D E G A [A minor pentationic]

A B D E G A [E (or V) minor pentatonic]

A B D E F# A [B (V/V) minor pentatonic]

A B C# E F# A [F# (V/V/V) minor or A major pentatonic]

there is a dorian/mixolydian/bebop eight tone pc being implied here for a great fusion approach.  it is difficult to explain to non-guitarists how easy it is for us to apply pentatonic subs.  the guitar is actually tuned to a pentatonic scale (E A D G B E = E G A B D E = E minor pentatonic) which lets these scales simply roll off of the fingers.  i often view a pentatonic as the staring point to either:

1) omit two tones for triad fingerings

2) add two tones for "modal" fingerings


example on E minor pentatonic

E G A B D E

omit A and D for Em triad
add F# and C# for E dorian
add F# and C for E aeolian
add F and C for E phrygian
add Bb for E blues

example on G major pentatonic

G A B D E G

omit A and E for G triad
add F# and C# for G lydian
add F# and C for G ionian
add F and C for G mixolydian
add Bb for G major pentatonic blues

see how easy this is?  as always, season to taste.

we all live in a pentatonic submarine

hot dog - led zeppelin

the dogs of war - pink floyd

walkin' the dog - rufus thomas

who let the dogs out - baha men

i didn't really mean to hijack the thread or anything like that - hey, give me a spell before responding to that last one okay millions? i need to look it over real close like...