The geometry behind astrological aspects.

Every planet has an ecliptic longitude λ — its position measured in degrees along the ecliptic from 0° (Aries point) to 360°. An aspect is the angular separation between any two of those longitudes.

The formula is simple: take the absolute difference, fold it into the range 0°–180° by taking the minimum of the angle and its supplement.

The major aspects aren't arbitrary. Each one divides 360° by a small integer — 1, 2, 3, 4, or 6. This is the harmonic series applied to geometry.

Notice that n = 5 is missing. Dividing the circle by 5 gives 72° — the quintile. It's real, it's used, but it took until the 17th century for someone to formalize it.

Johannes Kepler (1571–1630) extended the aspect system by applying the same logic to higher divisions. He was specifically interested in the connection to musical harmony — the ratios between string lengths that produce consonant intervals are the same ratios that divide the circle.

Kepler published this in Harmonices Mundi (1619) — the same book where he stated his Third Law of planetary motion. The harmonic framework wasn't a side project; it was the point.

An orb is the maximum allowed deviation from the exact angle. A conjunction with an 8° orb means any two planets within 8° of each other count as conjunct.

Aspect strength is often modeled as a linear falloff from exact:

An exact conjunction (0° deviation) scores 1.0. A conjunction at the edge of its 8° orb scores 0.0. Planets past the orb are not in aspect.

There is no consensus on orb sizes. Classical astrologers used planet-specific orbs (the Sun gets wider orbs than Mercury). Modern software typically uses aspect-specific flat orbs. Both approaches are defensible — neither is proven.