Definitions

tHz (Terahertz) — A unit of frequency. In this framework, each element is assigned a tHz wavelength derived from its position in the resonance matrix. tHz values decrease by exactly 3 per element moving down each column.

nm (Nanometres) — A unit of length used to measure wavelength. In this framework, nm and tHz are reverse scales: tHz + nm = 1080 for every element. As tHz decreases, nm increases by the same amount.

540 tHz / 540 nm — The midpoint of the visible light spectrum (360–720 nm / 360–720 tHz), and the spectral balance point of the entire resonance matrix. The grand average of all 120 elements equals exactly 540 in both tHz and nm.

Δ (Delta) — The Greek letter used in mathematics to mean "change" or "difference" between two values. For example, Δ tHz = −7.5 means the shared average minus the non-shared average equals −7.5.

Seekins Constant — 6.75 × 10⁻³⁴ joules/sec, identified by Christopher Seekins while refining the wavelength relationships of the elements. When any element's tHz wavelength is divided by 67.5 (Seekins Constant × 10) and added to its spectral opposite's value, the result is always exactly 16. This holds for every element pair and for the computed average of any positionally defined group in the framework.

SK — Abbreviation for Seekins Constant (6.75). SK×10 = 67.5, the scaling factor used in Seekins value calculations.

Resonance Matrix — The 12-column × 10-row arrangement of 120 elements (Hydrogen separate) that forms the foundation of the Harmonic Periodic System. Nuclear mass increases by exactly +30 per row down each column. Each column carries a uniform charge remainder (+, −, or N) determined by nuclear mass ÷ 3.

Charge (+, −, N) — Determined by dividing an element's nuclear mass by 3:

Since nuclear mass increases by +30 per row (and 30 ÷ 3 = 10, no remainder), every element in a column shares the same remainder — making charge a column-wide property.

Nuclear Mass (L) — The custom nuclear mass assigned to each element in the Harmonic Periodic System. This is the number of evenly spaced radial lines in that element's circle, and the starting line for the circle code traversal.

Circle Code — The geometric system in which each element n corresponds to a circle with L evenly spaced radial lines. Starting at line L, the element steps exactly n spaces around the circumference, touching every line exactly once while completing exactly n full revolutions.

n — The element's atomic number, also equal to the number of revolutions completed in the circle code traversal.

L — The element's nuclear mass; also the total number of lines in its circle code.

gcd(n, L) — Greatest Common Divisor of n and L: the largest number that divides evenly into both the atomic number and the nuclear mass. For most elements gcd(n, L) = 1, meaning the standard step of n spaces touches every line exactly once. For the Four Horsemen (elements 15, 45, 75, 105), gcd(n, L) = 5, which causes the standard step to retrace lines rather than visit each one uniquely. A special coprime step size is required for these four elements to complete the traversal correctly.

The Four Horsemen — Elements 15 (P), 45 (Rh), 75 (Re), and 105 (Db), for which gcd(n, L) = 5. These are the only elements requiring a special step size in the circle code. They occupy the −/− spoke of the 6-rod wheel.

OE / OO (Odd-Even / Odd-Odd) — A classification of each column based on the parity of its proton count and neutron count:

The 12 columns, viewed as 6 rods (opposite pairs), alternate perfectly: 3 rods of OE/OE (both ends odd/even — symmetric) and 3 rods of OO/OE (one odd/odd end, one odd/even end — asymmetric). The Four Horsemen column belongs to the only −/− spoke, which is OO/OE.

Shared Positional Class (Matching) — Elements whose positional class (Transition Metal, Other Metal, or Alkali/Alkali Earth/Rare Earth) matches the corresponding position across all groups simultaneously in a given geometric arrangement. Marked with black dots in the table images, on some graphics, found on this site.

Non-shared Positional Class (Non-matching) — Elements whose positional class does not match the corresponding position in other groups.

The 2 2 4 3 1 3 3 1 3 4 2 2 Sequence — The number of lines added from one circle to the next across one full 12-element row of the resonance matrix. This sequence sums to 31 (net +30 per stage, as the start and end lines overlap). The sequence is mirror-symmetric about its centre, and its first symmetry point is reached after 7 steps (H→He→Li→Be→B→C→N), where element 7 (N) has nuclear mass 16 — the values that appear in the 7/16 fractional component of the Odds/Evens spectral analysis.

Cycling through the 2 2 4 3 1 3 3 1 3 4 2 2 Sequence is referred to as a stage. Like 1 is not a prime number, hydrogen is not counted as part of the first stage, for some calculations. For example, adding up all of the lines in the circles (nuclear masses) of any stage, and subtracting the nuclear mass sum of all of the elements from the stage before, always equals 360. In this calculation, hydrogen is not counted. Without hydrogen's nuclear mass, all nuclear masses from stage 1 (He – Al) equal 207. All nuclear masses in stage 2 (Si – Mn) sum to 567. 567 − 207 = 360.

(Grail — from the Latin word graduale, meaning "in stages, by degree." On a side note, the oldest grail legends, according to the Templars, stem from Northern Scotland, where it was regarded as a bowl in which the elements were mixed.)

The Three Classes of 40 — The resonance matrix groups its 120 elements into three classes of 40:

Subclasses (The 50 Elements) — 50 elements carry an additional Roman numeral subclass label:

Columns with subclass elements: most columns have 4, two columns have 5 — all column count positions are symmetrical and in perfect harmony.

Column Seekins Sum — The sum of all 10 elements' nm wavelengths in a column, divided by 67.5 (SK×10). Every column's Seekins sum steps by exactly 40/9 = 4.4444... from column 1 to column 12, producing the sequence 500/9, 540/9, 580/9 ... 940/9. Two columns produce whole numbers: Column 2 (Li, −) = 60 and Column 11 (Mg, −) = 100, differing by exactly 40 (the fundamental class size). The grand total of all 12 column Seekins sums = 960 = 16 × 60, confirming the Seekins constant scales across the entire matrix.

960 = 16 × 60 — The grand total of all column Seekins sums. 16 is the Seekins constant pair sum; 60 is the number of odd or even atomic numbers in the matrix (and the number of elements per side in the Odds/Evens table).

40/9 — The step between consecutive column Seekins sums. 40 = the fundamental class size (each of the three classes of elements contains exactly 40 members). 9 = the denominator connecting the Seekins constant to the column structure.