Sesqui-quadrate

The sesqui-quadrate, also called the sesquiquadrate or trioctile, is an aspect of one hundred and thirty-five degrees, formed when two planets stand three-eighths of the circle apart, a square plus a semi-square. It is a minor aspect of the hard or challenging kind, sharing the frictional character of the square, and it is given a narrow orb. The tradition reads it as a relation of accumulated tension and agitation that seeks release, akin to the semi-square with which it is paired as the two eighth-division hard aspects. Like the semi-square it was introduced by Kepler in his expansion of the aspects beyond the classical five. These are the interpretive conventions of the astrological tradition for this angular relation, one of the minor hard aspects that supplement the major square and opposition with further shadings of challenge between planets.

The sesqui-quadrate, one and a half squares, completes the eighth-harmonic family together with the semi-square, and both mark the forty-five degree points that fall between the major aspects rather than coinciding with them. It is generally counted, along with the biquintile, among the aspects Kepler treated in his Harmonices Mundi of 1619, where he set out a theory of aspects grounded in the geometry of constructible polygons rather than in the older, purely traditional five. Where the semi-square is usually read as a minor friction, the sesqui-quadrate, standing closer to the full ninety-degree square and to the opposition, is generally given a sharper reading, a pressure that forces an adjustment rather than merely announcing one, and it is often paired with the semi-square in a chart as evidence of a recurring pattern of strain across a planet's aspects.

A hundred and thirty five degrees is three eighths of the circle, which places the sesqui-quadrate in the eighth harmonic alongside the semi-square. The name says the same thing in older language: a square and a half, ninety degrees plus forty five. It is also called the sesquiquadrate, the sesquisquare and the trioctile, and the multiplicity of names is a fair indication of how unsettled its place has been.

Its standing follows its partner's exactly. Neither angle can be reached by counting signs, since eight does not divide twelve, so neither exists in sign-based doctrine and both are found only by degree. Both come into circulation through Kepler's harmonic argument and are taken up in earnest only from the nineteenth century. Both are given narrow orbs, usually about two degrees, and both are read as friction rather than crisis. Where writers distinguish them, the semi-square is described as internal or chronic and the sesqui-quadrate as breaking outward into events, but that distinction is a convention of the modern literature rather than a finding, and different authors reverse it.

One structural point is worth keeping. Because a hundred and thirty five degrees is not a multiple of thirty, the two planets involved can stand in signs that are in aversion to one another, meaning that a traditional reading would hold them unable to interact at all while a harmonic reading holds them in a tense relation. That is not a disagreement about strength. It is a disagreement about whether anything is there, and it is the same fault line that runs under every one of the minor aspects.

The sesqui-quadrate belongs to the eightfold family of divisions and is among the aspects most confidently credited to Johannes Kepler by name, set out in the Harmonices Mundi of 1619 together with the quintile and biquintile as angles derived from consonant musical ratios rather than from the older twelvefold sign scheme. Its glyph reflects its composite definition directly: the semi-square's open-angle sign drawn combined with the square's sign, so that the mark itself states the arithmetic, a square's ninety degrees plus a semi-square's forty-five.

Of the minor aspects it is one of the few to have been given a dedicated place in the Unicode Standard, SESQUIQUADRATE at U+26BC in the Miscellaneous Symbols block, sitting in the character table immediately after the quincunx, itself one place removed in kind though not in origin, since the quincunx's admission as an aspect is credited to the later harmonic tradition rather than to Kepler specifically. The narrow orb customarily given to the sesqui-quadrate reflects its status as one of the weaker, later-recognised relations even within Kepler's own scheme. Kepler set out his reasoning for admitting it, and for the quintile and biquintile alongside it, in the fourth of the five books that make up the Harmonices Mundi, the volume most directly concerned with astrology among a work that ranges across geometry, music theory and what became his third law of planetary motion.