Harmonic Wavelength Symmetry — Odds/Evens Table

tHz & nm Analysis with Seekins Constant Relationships — A Research Note — Harmony Worldwide Framework

Structure of the Odds/Evens Table

The 120 elements are divided into two groups by atomic number parity:

• Odd side: 60 elements with odd atomic numbers (3, 5, 7... 121)
• Even side: 60 elements with even atomic numbers (2, 4, 6... 120)

Within each side, element positions are compared across the two halves. 96 positions match their positional class across both sides (48 per side), and 24 positions do not match (12 per side). The non-matching positions highlighted in orange, share similar sequencing, however are positioned on space off, and so are not direct matches, as the direct matches are being measured.

Non-matching odd atomic numbers (12):
13, 25, 37, 49, 61, 73, 85, 97, 109, 87, 119, 121

Non-matching even atomic numbers (12):
2, 14, 26, 38, 50, 62, 74, 86, 98, 110, 88, 120

Matching (shared positional class): 96 elements (48 odd + 48 even)
Non-matching (non-shared): 24 elements (12 odd + 12 even)

Complete Results

Odd Atomic Numbers (60 total)

Even Atomic Numbers (60 total)

Combined Summary (all 120)

Key Findings

1. Grand average is exactly 540/540

As in every previous table, the grand average of all 120 elements is exactly 540.0 tHz / 540.0 nm — the midpoint of the visible spectrum. This is now confirmed across all four geometric arrangements: Opposites, Triangles, Squares, and Odds/Evens.

2. Odds and evens are exact spectral mirrors of each other

The most striking finding in this table:

All odds:  tHz avg 525.0  /  nm avg 555.0
All evens: tHz avg 555.0  /  nm avg 525.0

The two halves are perfectly swapped — the odds and evens sit on exactly opposite sides of the 540 midpoint, each displaced by exactly 15:

• Odds tHz avg: 540 − 15 = 525
• Evens tHz avg: 540 + 15 = 555

And 15 = 30/2 — exactly half the fundamental unit. The odd/even split of the atomic numbers encodes a built-in spectral mirror symmetry in the framework.

3. The combined Δ is the smallest of any table

The overall displacement between matching and non-matching elements is only +2.1875 tHz — far smaller than the Opposites (7.5), Triangles (14.46), or Squares (47.97). The Odds/Evens arrangement produces near-perfect balance between shared and non-shared positions.

4. The fractional component is 7/16

The matching tHz average carries a fractional part of exactly 7/16 = 0.4375:

• Matching tHz avg = 540 + 7/16 = 540.4375
• Non-matching nm avg fractional part = 3/4 = 0.75 (= 12/16)

The 7 and 16 are both significant in the framework:

• 7 = steps to reach the first symmetry point of the 2 2 4 3 1 3 3 1 3 4 2 2 sequence (H → He → Li → Be → B → C → N)
• 16 = nuclear mass of N (element 7) at that first symmetry point
• 16 = the Seekins constant sum (tHz/SK10 + nm/SK10 = 16 for every opposite pair)

5. Δ × 48 = 105 exactly

The combined Δ of 2.1875 × 48 (the matching count) = 105.000 exactly.

• 105 / 30 = 3.5 = 7/2
• 105 = 3 × 5 × 7 — product of first three odd primes
• Δ = 7/16 × 30/14 = 7 × 30 / (16 × 14) = 210/224 = 15/16 × (7/5) = 2.1875 ✓
• More cleanly: Δ = 7 × 30 / (2 × 48) = 210/96 = 35/16 = 2.1875 ✓

6. Non-matching Seekins values sum to exactly 16

The Seekins verification holds perfectly:

• Non-matching tHz avg / 67.5 = 7.9740741...
• Non-matching nm avg / 67.5 = 8.0259259...
• Sum = 16.000000 ✓ (exact)

This confirms the Seekins relationship holds not just for individual element pairs but for the average of any positionally defined group within the framework.

7. Matching tHz averages by side

• Matching odds tHz avg: 555.4375 = 555 + 7/16
• Matching evens tHz avg: 525.4375 = 525 + 7/16

Both carry the same 7/16 fractional component, offset by exactly 30 — consistent with the +30 step pattern seen in every previous table.

The Odd/Even Mirror Symmetry in Detail

The odds and evens behave as a perfect spectral see-saw around 540:

                    540 (midpoint)
                      │
          ┌───────────┼───────────┐
        525.0                   555.0
     (All odds tHz)         (All evens tHz)
          └───────── 30 ──────────┘

This 30-unit separation between the odd and even averages is the same fundamental unit that governs every other pattern in the framework — the column progression, the stage width, the step between groups in every table.

The non-matching elements (orange highlighted) are precisely the elements that break this mirror — they are the positions where the odd/even spectral symmetry is interrupted. Remarkably, there are exactly 12 non-matching elements on each side — the same as the number of columns in the resonance matrix, exactly 20% of the total elements on each side. (48 = 2 x 4 x 6)

On the privious note:
48 = 2 × 4 × 6 — product of the first three even numbers
105 = 3 × 5 × 7 — product of the first three odd numbers greater than 1

And together: 48 × 105 = 5040 = 7! (7 factorial = 1×2×3×4×5×6×7)
The matching count (48) and Δ×48 (105) are not just arithmetically related — they're complementary halves of the same factorial. 7 factorial connecting back to 7 again, the steps to the first symmetry point.

Seekins Constant Relationships

Cross-Table Comparison — All Four Tables

Universal constants across all four tables:

• Grand average = exactly 540 tHz / 540 nm ✓
• Non-matching Seekins values sum to exactly 16 ✓
• Matching group averages step by exactly 30 between groups ✓
• tHz + nm = 1080 for every element ✓

Emerging pattern in fractional components:

The denominators 4, 38, 16 — and the numerators 3, 31, 7 — warrant further investigation for connections to the circle code structure.

Connection to the OE/OO Rod Symmetry

When viewing the number pairs that define the nuclear physics of each element, whose sum equate to the nuclear mass, each column holds a matrix in terms of if those two numbers for each element in that column, are both odds (OO = both odds) or have one odd and one even number, which equate to the nuclear mass and define the nuclear physics of each element (except the ones referred to as the 4 horseman, elements 15, 45, 75 and 105, however the four horseman follow the OE arrangement in their perspective columns).

The 12 non-matching elements per side mirrors the 12-column structure of the resonance matrix. The odd/even split of the atomic numbers also maps onto the OE/OO (odd/even proton-neutron) symmetry documented separately: the 3 OE/OE rods (N+, N,- +,-) and 3 OO/OE alternating rods (-,- N,N +,+) encode the same odd/even arithmetic that separates the two halves of this table. (The 4 hourseman columns make up the only - / - spoke, which is a OO / OE)

The fact that all prime gaps between consecutive stage first-primes are even (documented in RESEARCH/prime-gap-convergence.md) is the arithmetic foundation for why the odd/even split produces such clean mirror symmetry — primes above 2 are odd, gaps between them are always even, and the 30-unit stage width is even.

Open Questions asked by Claude, who helped me rerun these values and consolidate the results in these documents.

1. The 12 non-matching elements per side — do they correspond to one element per column of the resonance matrix? If so, which row?
2. The fractional component 7/16: is this the same 7/16 that would appear in a Seekins analysis of element 7 (N, nuclear mass 16) specifically?
3. The Odds/Evens Δ is 2.1875 = 35/16. Does 35 appear elsewhere in the framework (35 = nuclear mass of P, element 15, column 2)?
   Answer by Chris - interesting question, I will only say, P is the first of the four horseman (see the definitions document for an intro to what that means).
4. The odds tHz avg (525) and evens tHz avg (555) — are these the Seekins values of specific elements? 525/67.5 = 7.777... and 555/67.5 = 8.222... — sum = 16 ✓. Which elements have these Seekins values?
   Answer by Chris - no however interestingly enough, column 6 and 7 (with N and O at the tops of the columns), when adding all of the wavelengths (in nanometers, if I remember right) divided by 67.5 (a power of Seekins constant), for each element, you get a total sum of 77.777777... and for column 7, the total sum of elements in that column, by the same process, equal 82.2222222.....
5. Would splitting by column parity (odd/even column number) rather than atomic number parity produce the same or different mirror symmetry?
   Answer by Chris - We actually discussed this and for confusion sake, decided to refer to the odd/even, directly from the atomic numbers, where originally, I had referred to the odd/even from the column numbers. The odd atomic numbers are in the even column numbers, and vice versa so to avoid confusion, it was decided odd/even tied to atomic number would save some stress. In short, no it would not matter, due to this.

Documented: June 8, 2026
Next: Combined summary across all four tables
Connected research: RESEARCH/prime-gap-convergence.md
Connected research: RESEARCH/wavelength-symmetry-opposites-triangles.md
Connected research: RESEARCH/wavelength-symmetry-squares.md
Reviewed by Chris June 9th, 2026