Diving Deeper Intro Reverse Engineering the Zodiac

Here are some further thoughts on the harmonic patterns I outlined in Part 1. The inspiration here is my background in engineering mathematics, and the relationship between sine and cosine waves. In trigonometry, a circle is born from the interaction of sine and cosine waves (see my post “We shall not cease from exploration.”)

I have extended my harmonic analysis with both sine and cosine waves, which are natural siblings. A cosine wave is identical to a sine wave, but shifted in phase by 45º. This means that a sine wave’s zero points link up with a cosine wave’s maxima (peak or valley)

There is also a complementary relationship through the mathematical first derivative, where the cosine wave represents the rate of change of the sine wave (and vice versa, if you add a negative sign). From a physical perspective, this relationship is displacement vs. velocity, as well as velocity vs. acceleration.

1st & 2nd Harmonics: Quadrivial Deconstruction

Here I begin with the four quadrants of Wilber's Integral Theory, and deconstruct them with elements of the first and second harmonics.

Four Perspectival Lenses

In the language of Integral Philosophy, the four quadrants are equally valid perspectives on reality:

  • “I”: singular internal objectivity (UL),
  • “It”: singular external objectivity (UR)
  • “Its”: plural external interobjectivity (LR)
  • “We”: plural internal intersubjectivity (LL).


From Grunwald et al. 2016.

In Wilber’s theory these quadrants are not connected in a particular order. In applying them to astrology, I am equating them with the cycle of growth through an individual’s evolutionary life journey. Typically this cycle is most frequency related to the wheel of astrological houses, but there is a correlation with the zodiac as well. These four developmental lenses could be labeled core-intentional self, behavioral-environmental self, social-ecological self, and transpersonal-collective self.

We can deconstruct this grid with 1st harmonic waves, in which a single cycle traverses the entire circle. The 1st wave cycles between individualism and pluralism, or self and other. Its complementary wave cycles between interiority and exteriority.

I am still working out the nuances of this model with respect to sine waves, so for now it is simplest to consider each sector as all or nothing, rather than a smooth gradation.

Bringing in the second harmonic, which traverses two cycles around the circle, we have the oscillating pattern of exploration vs integration. This could also be seen as a form of extroversion vs. introversion, comparable with the 6th harmonic.

Harmonic core intention behavioral environment social ecology transpersonal collective
1st sine individual individual pluralistic pluralistic
1st cosine interior exterior exterior interior
2nd sine exploration integration exploration integration

3rd & 6th Harmonics: Elemental Deconstruction

Here I add the 3rd sine to complement the 3rd cosine from Part 1. It was born through contemplation of the similarities and differences of the four Elements. Boiling things down to a single word is quite difficult. Looking at what Fire and Earth share vs. Air and Water was a big challenge. At first I considered stable vs. reactive, but ultimately came up with concrete vs. abstract. Can you find a better word? Let me know!

Update: I was curious to see if ChatGPT could provide any insight, here’s what it said:

“Fire and Earth are considered to be more ‘material’ or ‘physical’ elements, while Air and Water are considered to be more ’ethereal’ or ’non-physical’ elements. Fire and Earth are both associated with the tangible and material aspects of life, such as the body, the senses, and physical sensations. They are both associated with qualities such as determination, practicality, and a focus on the present moment.

Air and Water, on the other hand, are associated with more abstract and intangible aspects of life, such as the mind, the emotions, and the imagination. They are both associated with qualities such as adaptability, flexibility, and a focus on the future.”

Harmonic Fire Earth Air Water
3rd sine concrete concrete abstract abstract
6th sine extroverted introverted extroverted introverted
3rd cosine feeling thinking thinking feeling

4th Harmonic: Modal Analysis

Similarly, I explored analysis-resynthesis by sine and cosine waves to extend our understanding of modality. Dividing a cosine wave into three parts (as below), we find that the first and last segments are two halves of the cycle peak, and the center segment is the valley.

Is there a common quality between cardinality and mutability than can be extracted? For lack of a better word I am labeling this quality as “change.” Cardinality begins with maximum change and accelerates towards waning change (aka initiating). Mutability begins with low change and accelerates towards growing change (aka adaptive). Fixity is oriented around the central point of minimum change, and has low velocity.

Above, I show an inverted sine wave as the mathematically correct derivative of a cosine wave. If the green wave represents quantity of change, then the red wave represents velocity of change. Decreasing velocity of change is a movement towards commitment. Increasing velocity of change is a movement towards variety. In the middle, there is a dynamic push/pull resulting in a zero sum.

The table below summarizes the red and green images above.

Harmonic Cardinal Fixed Mutable
4th cosine initiating (waning change) determined (minimal change) adaptive (growing change)
4th sine commitment (change deceleration) neutral (zero sum) variety (change acceleration)

Putting it All Together: Zodiacal Harmonic Resynthesis

Combining the above keynotes gives the following harmonic resynthesis:

Zodiac Modality Quadrant Element Ruler
Internal Subjectivity (Intentional / “I”)   See Part 1 for Yin/Yang planetary rulership
Aries Initiating commitment to personal exploration through concrete extroverted feeling. (Yang Mars)
Taurus Determined focus of personal exploration through concrete introverted thinking. (Yin Venus)
Gemini Adaptive variety of personal exploration through abstract extroverted thinking. (Yang Mercury)
External Objectivity (Behavioral / “It”)
Cancer Initiating commitment of behavioral/environmental integration through abstract introverted feeling. (Yin Moon)
Leo Determined focus of behavioral/environmental integration through concrete extroverted feeling. (Yang Sun)
Virgo Adaptive variety of behavioral/environmental integration through concrete introverted thinking. (Yin Mercury)
External Inter-Objectivity (Social / “Its”)
Libra Initiating commitment to social/ecological exploration through abstract extroverted thinking. (Yang Venus)
Scorpio Determined focus of social/ecological exploration through abstract introverted feeling. (Yin Mars / Pluto)
Sagittarius Adaptive variety of social.ecological exploration through concrete extroverted feeling. (Yang Jupiter)
Internal Inter-Subjectivity (Transpersonal / “We”)
Capricorn Initiating commitment to transpersonal/collective integration through concrete introverted thinking. (Yin Saturn)
Aquarius Determined focus of transpersonal/collective integration through abstract extroverted thinking. (Yang Saturn / Uranus)
Pisces Adaptive variety of transpersonal/collective integration through abstract introverted feeling. (Yin Jupiter / Neptune)

What do you think? Drop me an email if you have any corrections or elaborations.