Prime Harmonics and the Dimensions

Why does a universe have the dimensions it has, and could another have more? Part 3.6 leaves the binding rule that sets the saturation D⥣ open, and Part 4.1 builds the dimensional ladder on doublings, each new dimension at twice the pulses (2ⁿ). But a harmonic series is not a ladder of doublings. It is every whole multiple of a fundamental, 1, 2, 3, 4, 5, 6, 7 and on, and the doublings, 1, 2, 4, 8, the octaves, are only one thread through it. This part reads the dimensions from the whole series, and finds the four we live in at its head.

The Harmonic Series, Not the Octave

Harmonic Series G

f_h = h · f₁

Dimensional analysis:
Where (from the book): f_h [𝕋⁻¹], h [∅], f₁ [𝕋⁻¹]
[𝕋⁻¹]
[∅] × [𝕋⁻¹] = [𝕋⁻¹]
[𝕋⁻¹] = [𝕋⁻¹] ✓ Valid PulseCore Verified ✓

Every whole multiple of the fundamental, not only its doublings

Where:

• f_h [𝕋⁻¹] — the frequency of harmonic h
• h [∅] — the harmonic's number, any whole number
• f₁ [𝕋⁻¹] — the fundamental, one over a universe's Pulse Tempo (Part 4.9)

➢ The octaves are the powers of two, h = 1, 2, 4, 8, and repeat the fundamental's own pitch; every other harmonic brings a relation the octaves cannot make. Data Harmonics (Part 3.6) numbers a universe's layers the same way, Harmonic 1, 2, 3, 4 and on, whole numbers, not doublings.

Primes as Independent Axes

Every whole number is a product of primes in exactly one way, the fundamental theorem of arithmetic (Hardy & Wright, 2008). So every harmonic, and every ratio between two harmonics, is a point in a lattice with one independent axis for each prime: 2, 3, 5, 7, 11 and on. Euler drew this lattice in 1739 to map musical harmony (Euler, 1739); Partch built a music on its first primes (Partch, 1974); and Tenney made it a space with a measured distance (Tenney, 1983). A harmonic built from 2 and 3 alone, like 6 or 9 or 12, adds no new axis: it is a combination of the ones already there.

Harmonic Factorization G

h = 2^a · 3^b · 5^c · 7^d

Dimensional analysis:
Where (from the book): h [∅], a [∅], b [∅], c [∅], d [∅]
[∅]
[∅]^([∅]) × [∅]^([∅]) × [∅]^([∅]) × [∅]^([∅]) = [∅] × [∅]^([∅]) × [∅]^([∅]) × [∅]^([∅]) = [∅] × [∅] × [∅]^([∅]) × [∅]^([∅]) = [∅] × [∅]^([∅]) × [∅]^([∅]) = [∅] × [∅] × [∅]^([∅]) = [∅] × [∅]^([∅]) = [∅] × [∅] = [∅]
[∅] = [∅] ✓ Valid PulseCore Verified ✓

Every harmonic is a point on the prime axes

Where:

• h [∅] — a harmonic's number
• a [∅] — its steps along the axis of 2, the octave
• b [∅] — its steps along the axis of 3
• c [∅] — its steps along the axis of 5
• d [∅] — its steps along the axis of 7

➢ Written to the four primes that carry our universe's dimensions; a universe with more actual dimensions carries the next primes too, 11, 13 and on, each a further factor of the same form.

Fundamental and Etheric Dimensions

Binary Pulse Theory reads a universe's dimensions from this lattice. Its fundamental, or actual, dimensions are its prime harmonics, each a relation no other harmonic can make. The composite harmonics, 4, 6, 8, 9, 10 and on, are its Etheric Harmonics: real and present, but built on the primes, the harmonic reinforcement of Part 3.6 rather than independent dimensions; and an etheric harmonic exists only on fundamentals that carry. The actual dimensions are the primes a universe holds in tune, up to its prime limit p⥣, so D is the count of primes up to p⥣.

• 2, the octave: the pulse doubling itself, 0 → 1 → 0 repeated, the Binary Pulse; read as time, the one dimension made of the pulse's own repetition.
• 3, 5 and 7: the three odd primes, read as the three dimensions of space.
• 9 = 3 × 3: the first composite after them, the first etheric harmonic, reinforcing space rather than adding to it.

That is three and one, read from the head of the harmonic series. It is a reading, not yet a derivation: the theory has still to show why the prime 2 is time and the odd primes are space, though Part 4.9 finds the same split in the shape of the pulse itself, whose own overtones are the odd harmonics alone.

Where a Universe Stops Holding Its Harmonics

A harmonic counts as a dimension only while a universe holds it in tune with the rest. In music the same limit is heard: consonance falls as a ratio grows complex (Helmholtz, 1863; Plomp & Levelt, 1965), and the harmonics up to 7 are heard as consonant while 11 and 13 are heard as out of tune, which is why tuning theory speaks of the "7-limit". Tenney measured a ratio's complexity by its harmonic distance.

Harmonic Distance G

H = ln(p · q) / ln(2)

Dimensional analysis:
Where (from the book): H [∅], p [∅], q [∅]
[∅]
ln([∅] × [∅]) ÷ [∅] = ln([∅]) ÷ [∅] = [∅] ÷ [∅] = [∅]
[∅] = [∅] ✓ Valid PulseCore Verified ✓

How far a ratio p/q lies from the fundamental in the prime lattice

Where:

• H [∅] — the harmonic distance of the ratio p/q, in octaves
• p [∅] — the ratio's numerator, in lowest terms
• q [∅] — its denominator

➢ Tenney's height (Tenney, 1983): 3/2 lies at log₂ 6 ≈ 2.6, 5/4 at log₂ 20 ≈ 4.3, 7/4 at log₂ 28 ≈ 4.8, 11/8 at log₂ 88 ≈ 6.5. A universe that holds harmonics in tune to a greater distance holds more primes, and so more actual dimensions.

The binding rule is what fixes that distance for each level, and it is open. Three things it must give are known. The Zinf, level 0, holds two layers and no physical (the Layer Count Law), so it holds the primes 2 and 3 alone: D = 2. From level 1, the first universe to complete the fourfold scaffold, the limit is 7: D = 4. And at our level D is still 4, which our own sky measures (below). A larger universe could hold harmonics in tune further, to 11 or 13, and so have a fifth or sixth actual dimension; if the coherence requirement of Part 4.1 is the rule, the harmonic distance it allows must reach 7 but not 11 at level 202, and grow beyond.

The Sky Measures the Count

Part 3.9 derives the share of collapses that fail as 2^(−D), and Part 3.13 counts from it a universe's ratio of dark matter to ordinary matter, (1 − 2^(−D)) · 144/25. Read backwards, our sky's ratio, 5.36 ± 0.06, measures D = 3.9 (+0.3 / −0.2): four actual dimensions, the primes 2, 3, 5 and 7, with three ruled out at five standard deviations and five at more than three. The dark matter of a universe counts its dimensions, and ours counts four.

Links to Binary Pulse Theory

• Part 3.6: Data Harmonics numbers the layers as whole harmonics; the Dimensional Saturation Threshold is the prime limit, and the open binding rule is its cutoff.
• Part 3.9: the Dimensional Failure Law, P⦰ = 2^(−D), counts the actual dimensions.
• Part 3.13: the dark-matter ratio measures D.
• Part 4.1: the dimensional ladder of doublings is the octave thread of the series; its coherence requirement is the candidate for the cutoff.
• Part 4.9: the pulse's own waveform holds the odd harmonics alone, time and space in its spectrum.