Explore a harmonic relationship
Add a birth profile or explore the example, then select a harmonic and pair in the tool.
Open the harmonic explorer →How positions transform
Input two or more ecliptic longitudes in one declared zodiac/reference frame and an integer factor h. For every longitude L, calculate H(L) = (h × L) mod 360°. Output the transformed longitude, transformed separation, natal separation, nearest rational fraction, reduced order, residual, and the chosen transformed orb. A displayed sign in Hh is a coordinate label of the transformed circle; it is not a second observed sky or a house placement.
Chosen factor and primitive relation
A selected factor can reveal a relation that belongs to a lower order. Example: longitudes 10° and 82° differ by 72° = 1/5 of a circle. H5 puts them together, and H10 also puts them together because 10 × 72° wraps to 0°. The inspector therefore shows nearest fraction 1/5 and reduced order 5; an H10 selection alone does not license an H10 interpretation.
Worked explanation by Astral Atlas, derived from the integer longitude transform and ordinary fraction reduction.
Residual and orb
Multiply the directed natal separation by h and wrap to the nearest distance from 0°. That distance is the transformed deviation. Compare it with the chosen harmonic orb; divide the orb by h for the local natal tolerance. The Atlas default is 8° transformed, and you can change it. At H5 this allows 1.6° natal deviation; at H10 it allows 0.8°. These are worked calculations, not a universal school threshold.
Use a named reading, then inspect the pattern
Cochrane’s 2012 qualitative study examines prominent multi-planet patterns, with unresolved and poor fits alongside proposed themes. It uses a 16° harmonic orb; H3 does not yield a settled simple theme. The study notes are offered for learning. This pair inspector establishes angular qualification and reduced order; it does not reproduce his prominence or symmetry selection.
David Cochrane, The First 32 Harmonics · abstract and results ↗
Keep related systems separate
A longitude harmonic operates on the selected ecliptic longitudes. A conventional aspect is an angular rule with its own history and orb practice. Zodiac signs label 30° coordinate sectors and do not supply rulerships to a transformed wheel unless a named school says so. Jyotish Navamsa is a ninth-divisional mapping with its own rashi, sidereal-reference and interpretive rules; sharing the number nine with H9 is not equivalence. Nakshatra is a 27-sector sidereal mansion system and is likewise neither a harmonic factor nor an H27 reading.
P. V. R. Narasimha Rao, Vedic Astrology, An Integrated Approach · Rasi and Navamsa ↗
What changes with a sidereal reference?
Subtracting the same origin offset c from every natal longitude shifts each transformed position by h × c. Pair differences stay the same, because the common offset cancels. So a tropical and a sidereal harmonic can have different displayed signs while preserving the same pair geometry.
Choose the input reference deliberately. The source-specific teaching remains named; selecting Lahiri does not turn a modern harmonic interpretation into a Jyotish varga interpretation.
Word origin · etymology
Mathematical and musical vocabulary borrowed into modern astrology.
Harmonic derives through harmony from Greek harmonia, joining or agreement. In chart practice the name points to integer proportional divisions of the circle; it does not identify a measured acoustic or electromagnetic frequency.
How we know · epistemology
Reproducible angular transformation plus school-specific interpretation
Integer multiplication and circular reduction are arithmetic. An interpretation requires a named school. The study themes offered here concern prominent multi-planet patterns; a qualifying pair alone does not establish the study’s pattern selection, a personality trait or a forecast.
David Cochrane, The First 32 Harmonics · abstract, method and theoretical framework ↗
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Sources & provenance
Use the citations beside each section to distinguish a calculation convention, a historical teaching and an editorial example.