The System of 1->3->7->12 (Pt.2): Kabbalah and Western Tonal Theory

Pt.2

 

 

 

Pt. 5: Unification And It’s Map of Unification

(Line)

So from these seven points if we connect them with lines I have observed three different types of relationships.

The first is the simplest, and there are 6 of them.  This is Unification or return back to Tonic or 1/I. Their order is representative of the order of cadential progressions, seen in pt. 9 & 10. I find the idea that there are six of these to be possibly reflective of the idea that in Kabbalah the seven is created by 6 things with one in the center.

 

 

Pt. 6: Exchanges of Non-Tonic Connections, And It’s Map of Exchange

(Lines)

Exchanges are defined as connections that are not of return to Tonic; the idea of returning being very important.

Their order again is dependent upon the order seen in pt. 9 & 10.  The ordering of these lines is also essential to the structure of pt. 7.

One thing to note about these was that I calculated the exchanges of non tonic last, and with the addition of parts 5, 6, & 7 I had reached 31 total with all of the lines I saw in the diagram. (6 from unification, 10 from Relationships, and 15 from Exchanges) This was 1 short of the number 32, which is the number of mystical passive wisdom described in the first line of the Sefer Yetzirah. I contemplated for a while what the 32nd might be, and after some time finally concluded that a note or chord progression from Tonic to Tonic would not be of return to Tonic, as it was there already. This not only shows the final of the 32 paths being of , but also alludes to the non-returning nature or state of those who from Oneness, go back to Oneness, as described in many philosophies. It was quite the beautiful realization that the nature of the path from 1 to 1 is non returning.

 

 

Pt. 7: Regions Between Lines And Points, And It’s Map of Relationships

(Area)

 

 

This is the section which I am the least confident and in my analysis of the 7. I realized very quickly that there were many possibilities on how to map out the way you would think about this idea of relationships or shapes between the 7 points.

The one I chose is to look at two points and to say what is the third thing opposite it is, whether that be a point or a line. If it is 1 or the Tonic center-point it is opposite it reflects on either side of the line that is created from the connection (such as R4), otherwise it creates a triangle.

There were some exceptions made to this when I got to the III chord, and I want to do some different perspectives on this concept of regions to further this understanding. I believe exploration into the 12 will be very useful in this Pursuit.

Again this was calculated using the interval class vector (ic vec) shown in pt 9.

From the 10 relationships from that the 6 from unifications and the 16 from connections not of return to Tonic, add to 32 paths. I don’t see much evidence besides the number itself as a connection to 32 mystical Paths of wisdom, but I do see an interesting pattern in the numbers are drawn. I find it poetic that the first one is 6 which is a line with a circle connected to it. The next one is 10 which is a circle with a line next to it. And the third is a line and circle connected next to a line. Even if it’s arbitrary I find this relationship of these three numbers to be quite interesting, although I am still exploring why I might feel that.

 

 

 

 

 

 

 

 

 

Pt. 8: 12 elementals, 12 chromatic notes, 12 possible keys {work in progress}

(12)

(A    A#/Bb    B    C    C#/Db    D    D#/Eb    E    F    F#/Gb    G    G#/Ab)

 

The second I looked at a diagram connecting 12 points, I immediately recognized as the circle of fifths. I also quickly saw that there were a ton of connecting points which made it very difficult to draw. In that I realized it is the largest realm of this 1-3-7-12 system that I have yet to explore.

I did see an interesting connection when we look at the pitch class inversions on the circle like a chromatic clock face, as its shape reflected that of the tree of life when the inversion axis was at 0/6.

 

 

 

 

 

 

 

 

 

 

I am still just starting to grasp this idea of a mod 12 system of music, as well as the approach to the elementals in Kabbalah, but I have found some diagrams I have found interesting which have inspired ideas, and can possibly act as a guide to understanding it:

Many of the diagrams I use here were created by Marshall Lefferts with the Cosmometry Project, and I will be quoting his work on Cosmometry of Music in this section some, as it has been my biggest source so far in trying to grasp the concept of 12. His work can be found here: www.cosmometry.net

This is called the “chromatic” scale — 12 equal-interval steps up the scale called half-steps. The seven letters comprise the 7-note “diatonic scale” we’re all familiar with as “do, re, mi, fa, sol, la, ti…do.” In the key of C, these are C D E F G A B…C (the ending C is the beginning of the next “octave” wherein the pattern repeats at double the pitch (frequency) of the previous scale). This 7-tone scale consists of a combination of whole steps (two-note intervals) and half steps (one-note intervals) along the 12-tone chromatic scale. This 7-tone pattern is the same in all twelve keys.

One of the primary interval relationships is called the tri-tone due to it being an interval of two notes that are 3 whole steps apart. Given that 3 whole steps is equal to 6 half steps, we find that these two notes exactly divide the 12-tone scale in half. Being that the tri-tone consists of two opposite notes in the 12-tone system, it’s obvious then that there would be six pairs of these tri-tone intervals. These are colored coded in the diagram above as A/D#, A#/E, B/F, C/F#, C#/G, D/G#.

 

A very basic harmonic relationship is called the fifth. Simply, it’s the interval between the first and fifth notes of a major scale. For example, in the key of A the fifth note is E. In the world of classical music theory, one of the ways to show fundamental relationships is to array the 12 notes around a circle following a sequence of fifths, as in the illustration below. This shows that there is a progression in music that naturally moves or cycles through all 12 keys in a way that is harmonically pleasing to our senses (it is also referred to as the cycle of fifths).

 

With the interconnecting lines of the 12-around-1 matrix as our guide, we can then see that there is a set of basic interval relationships that map out polygonal patterns corresponding to specific intervals. As seen below, these are:

 •  One 12-sided dodecagon ringing the outside… our circle of fifths

 •  Two 6-sided hexagons showing two sets of whole-tone (whole step) scales

 •  Three 4-sided squares showing three sets of diminished (minor third) chords

 •  Four 3-sided triangles showing four sets of augmented (major third) chords

 •  One 12-pointed dodecagram showing the chromatic sequence of intervals

 •  Six lines showing our tri-tone interval pairs

Each of these notes is the basis for a set of 3-note chords, or triads, that comprise the seven basic chords in a given key. There are three major triads (having a major third with a minor third stacked on top of it)), three minor triads (having a minor third with a major third stacked on top of it), and one diminished triad (having two minor thirds stacked on top of eachother). In the key of A these triads are:

A Major              1

B minor              2

C# minor            3

D Major              4

E Major              6

F# minor            6

G# diminished   7

Using our color-coding spectrum, we can map out the seven triads onto our circle of fifths matrix with one colored triangle per chord.

 

 

Also note that in the image above only one half of the matrix is filled in with chords. The lower half is completely empty, with five notes remaining unused. These notes are not in the key of A. They are, however in the key of D# (also known as Eb). This is the exact opposite key from A — the tri-tone key, so to speak. Eb shares two of its seven notes with A, though… the D and G# (G# is called Ab when in the key of Eb, but it’s the same note). These two notes are themselves tri-tones, as evidenced by being opposite each other in the matrix. In the key of A they are the 4th and 7th notes respectively. In the key of Eb they are swapped in their positions, being the 7th and 4th notes respectively.

Adding in the triads for Eb, we get this beautiful, balanced pattern showing how it takes two opposite keys to create the whole system of music.

– Tri-Tone Duality of Music   by. Marshall Lefferts

Here are some of the ideas I’m starting to formulate about the idea of 12:

I find this map interesting, specifically the expanding/imploding pattern that it follows from 5th to tritone interesting. It goes from a state of one at the center to a state of one of the edges, or vise versa depending on how you look at it.

The second is a map of polyrhythms, or different numbered rhythms playing at the same time. What’s interesting to me about this is the patterns the dots form are representative of the harmonic series, with the line formed on the radius being the fundamental, the diameter line being the 2nd partial, the 3 pointed star being the 3rd, etc. I’m not sure of the source on this one except for this link: https://imgur.com/tWq3D7l 

 

This musical composition and accompanying video also show this concept very beautifully:

 

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This next one is a 3D interpretation of the circle of fifths. I need to explore it more, but visually I see some similarities with the diagrams I drew in pt. 5, 6, & 7. I predict this geometric shape will be used more to describe the relationship of the 7 than the 12, but could be used as a model for both. 

The reflection on the x and y axis of a circle or reflecting on the 4 cardinal points of the circle is something that Bela Bartok explored, and is something I need to research further. It might be important to note the similarities of his compositional style and the Fibonacci sequence and ratio as well.

 

 

 

 

 

 

 

 

 

 

This final diagram attempts to loosely describe the multidimensional nature of the 12 system, although I don’t think imagining it as an image in a 2D space like this picture shows on a screen does it justice. Instead I imagine it sort like this interpolation between different perplexities of T-sne data…on in English is the movement between different perspectives an Artificial Intelligence program can have on a set of data; sort like moving between different brains’ perspectives on a set of objective data.

The paths that all the moving points follow can be thought of like the poly-rhythmic diagram, but on a much larger scale. Here are some other diagrams that also show this concept some:

This is a good diagram for showing the fact that different tone (notes) and rhythmic values (beats) are both simply different speeds of cycles that come back around in unison.

 

 

 

This next one is a good depiction of the reflecting property of the tritone or 6 within the circle. It starts with a perfectly oddly balanced rainbow at the top, which I think resembles the harmonic series, and if we go counterclockwise, it goes down the color spectrum, adding a line of the previous colors at each point on the clock-face. when it reaches 6 (the reflecting point) Red dominates the right side, with the rainbow being reflected over the left side of the this point; starting a reflection of it’s journey down the clock-face at the start of its journey back to the top

 

 

Further explorations into the 12

I mapped out all the intervals seen on the tone wheel or mandala above to clock faces, and put all the possible combinations next to each other. I then marked in color when the tones aligned (were the same), doubled, or tripled within a given point on the clock. Since 0 acts as both 0 and 12, when a tone had a relationship with 12 (like 4, which triples to make 12) I marked one half of the line with color, and the other half black, symbolizing paradoxical nature of the octave. I’m still doing work with this, and trying to discover a pattern within these tone’s relationships.

 

 

 

 

 

Pt. 9: Relationship ic vec

 

This part has mostly been explained in previous sections, and will be expanded upon in the next part.

 

 

 

Pt. 10: Moving Around the 7, While Minding the 5 Fold

 

From the small diagram in pt. 0, we can see the 3 circles forming 5 crossover points. I see a connection between these and the 5 cadential groups, which progress from (III)->(iv)->(IV/ii)->(V/vii)->(I)

I have seen these 5 crossover points described as the characteristics I wrote on the diagram sheet, but am unsure how these hold up, as I need to explore them more.

The chart in this part shows an example of a chord progression going from I->III->iv->IV->ii->vii->V->I. The arrows show how the diagram should be thought of as something that’s in motion, and the parts 5, 6, & 7 should be thought of as a sort of geometric model of music frozen in time.

 

 

 

Want more???

Continue to Pt.3

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