Harmonic astrology began its development by astrologer John Addey during the late 1950s, and the variant known as vibrational astrology was developed by Santino Mancuso (aka David Cochrane) several decades later, but the roots of both in Vedic astrology have been around for hundreds of years. During this time, many rules for computation and interpretation have been developed, but the one thing that has frequently been lacking is a set of rigorous mathematical proofs of the essentials of this highly mathematical branch of astrology. This is unfortunate as good mathematical models are often essential for a discipline to advance. Hence, in this work proofs are presented for several essential theorems that apply to harmonic astrology, and by doing so greater clarity can be brought to the underlying structure of this branch of astrology. Furthermore, this paper is also designed to be two papers in one. In other words, those who are comfortable with mathematical rigor can read the theorems and proofs in full detail while those who are not used to such rigor can read just the examples and other surrounding content to understand the results contained herein. Furthermore, a link to videos that provide additional explanation of harmonics and the proofs below can be found in the references at the end of this article (Benton, 2024).