The Overlap That Creates the Harmonic System
The Harmonic System is often described as a structure that “emerges naturally”
from two foundational choices: a temporal origin and a numerical base.
However, the deeper reason for its coherence becomes clear only when we examine
how three independently constructed systems — the Earth Local System,
the Universal Unit System, and the Gravitic System — overlap in an extremely narrow region.
This article explains the Venn diagram in revised.pdf §E,
which visualizes this remarkable convergence.
The Venn diagram shown at the end of §E (p.21) in revised.pdf
illustrates how the Harmonic System emerges
as the overlap of three independently constructed systems.
● Red (right): Earth Local System — GCD System
A calendar time‑unit system derived from Earth’s rotation and revolution.
In the article from 2012‑02‑12:
“A Beautiful Symmetrical Relationship: The Tropical Year and the Half‑Day”
https://suchowan.seesaa.net/article/201202article_01.html
the following relation holds
(apparently established precisely by the end of the 20th century):
| tropical year / half‑day ＝ 3+6 ＋ 3+1×2-1 － 2-6 .
The GCD of the tropical year and the half‑day
(the largest quantity dividing both)
is taken as the fundamental unit of calendar time nodus.
Moreover, the following remarkable identity holds:
LCM(hexon) / GCD(nodus) = 100362812 (≈126) ,
where the LCM is the smallest quantity divisible by both.
This relationship is astonishingly elegant.
● Blue (top): Universal Unit System
A physical unit system generated from c0,ℏ,kB, and plus one.
plus one = R∞ → physical time unit = 0.390268 s [1]
plus one = aB → physical time unit = 0.391617 s
The blue region consists of values whose 12‑power multiples lie between these two.
From Table 6 of univunit-e.pdf:
λ = α-1×4π aB ,
where λ is one wavelength corresponding to one cycle of R∞.
Since
α-1 ≈ 137.036 ≈ 122,
4π ≈ 12.56637 ≈ 121,
we obtain:
α-1 × 4π ≈ 1722.045 ≈ 123 = 172810.
Thus the allowable region is extremely narrow.
solar nodus = 11 min 15 s = 675 s.
Its 1/12³ fraction, solar ternon = 0.390625 s,
lies inside this narrow interval of the blue region.
Since the blue region exists at every 12‑power scale, solar nodus itself
is also included as a 12³‑scaled member of the same region.
This alignment is a truly remarkable inclusion.
● Green (left): Gravitic System
A gravitational unit system generated from c0, ℏ, kB, and plus one,
where plus one is either:
- (a) n G
- (b) 4π n G
with integer n.
The union of these two sequences forms the green region.
For the (a)‑series, choosing n = 35[2] places the physical time unit
inside the common region shared by red, blue, and green.
In this case the physical time unit becomes 0.390654 s.
This is another striking coincidence.
● White (center): Harmonic System
The Harmonic System corresponds to the region where all three systems overlap:
- calendar time (Earth Local System)
- physical constants (Universal Unit System)
- gravitational system (Gravitic System)
These three independently constructed structures coincide within
an extremely narrow interval [3].
The Harmonic System exists precisely because these coincidences align.
The following diagram shows that the "Natural Time Scale" is established
as a result of the overlap of the three systems.
Notes
[1] 2026‑01‑30
“Synchronization of the Universal Unit System’s Time Unit with Earth’s Rotation”
https://suchowan.seesaa.net/article/202601article_30.html#:~:text=0.390268
[2]
This number 35 is easy to understand if you think of it as the product of 5 and 7,
which are prime numbers following 2 and 3. see doc/README.md §3.3
[3]
The articles from 2017‑12‑25
https://suchowan.seesaa.net/article/201712article_25.html
and
doc/README.md
describe the system from the perspective of the inclusion
green ⊃ white.
-> Japanese
The Harmonic System is not intended as a replacement for SI units.
Rather, it serves as a structural lens that reveals unexpected alignments
between astronomical cycles, physical constants, and gravitational parameters.
The narrow overlap shown in the Venn diagram is
one of the clearest demonstrations of this structural harmony.
from two foundational choices: a temporal origin and a numerical base.
However, the deeper reason for its coherence becomes clear only when we examine
how three independently constructed systems — the Earth Local System,
the Universal Unit System, and the Gravitic System — overlap in an extremely narrow region.
This article explains the Venn diagram in revised.pdf §E,
which visualizes this remarkable convergence.
The Venn diagram shown at the end of §E (p.21) in revised.pdf
illustrates how the Harmonic System emerges
as the overlap of three independently constructed systems.
● Red (right): Earth Local System — GCD System
A calendar time‑unit system derived from Earth’s rotation and revolution.
In the article from 2012‑02‑12:
“A Beautiful Symmetrical Relationship: The Tropical Year and the Half‑Day”
https://suchowan.seesaa.net/article/201202article_01.html
the following relation holds
(apparently established precisely by the end of the 20th century):
| tropical year / half‑day ＝ 3+6 ＋ 3+1×2-1 － 2-6 .
The GCD of the tropical year and the half‑day
(the largest quantity dividing both)
is taken as the fundamental unit of calendar time nodus.
Moreover, the following remarkable identity holds:
LCM(hexon) / GCD(nodus) = 100362812 (≈126) ,
where the LCM is the smallest quantity divisible by both.
This relationship is astonishingly elegant.
● Blue (top): Universal Unit System
A physical unit system generated from c0,ℏ,kB, and plus one.
plus one = R∞ → physical time unit = 0.390268 s [1]
plus one = aB → physical time unit = 0.391617 s
The blue region consists of values whose 12‑power multiples lie between these two.
From Table 6 of univunit-e.pdf:
λ = α-1×4π aB ,
where λ is one wavelength corresponding to one cycle of R∞.
Since
α-1 ≈ 137.036 ≈ 122,
4π ≈ 12.56637 ≈ 121,
we obtain:
α-1 × 4π ≈ 1722.045 ≈ 123 = 172810.
Thus the allowable region is extremely narrow.
solar nodus = 11 min 15 s = 675 s.
Its 1/12³ fraction, solar ternon = 0.390625 s,
lies inside this narrow interval of the blue region.
Since the blue region exists at every 12‑power scale, solar nodus itself
is also included as a 12³‑scaled member of the same region.
This alignment is a truly remarkable inclusion.
● Green (left): Gravitic System
A gravitational unit system generated from c0, ℏ, kB, and plus one,
where plus one is either:
- (a) n G
- (b) 4π n G
with integer n.
The union of these two sequences forms the green region.
For the (a)‑series, choosing n = 35[2] places the physical time unit
inside the common region shared by red, blue, and green.
In this case the physical time unit becomes 0.390654 s.
This is another striking coincidence.
● White (center): Harmonic System
The Harmonic System corresponds to the region where all three systems overlap:
- calendar time (Earth Local System)
- physical constants (Universal Unit System)
- gravitational system (Gravitic System)
These three independently constructed structures coincide within
an extremely narrow interval [3].
The Harmonic System exists precisely because these coincidences align.
The following diagram shows that the "Natural Time Scale" is established
as a result of the overlap of the three systems.
Notes
[1] 2026‑01‑30
“Synchronization of the Universal Unit System’s Time Unit with Earth’s Rotation”
https://suchowan.seesaa.net/article/202601article_30.html#:~:text=0.390268
[2]
This number 35 is easy to understand if you think of it as the product of 5 and 7,
which are prime numbers following 2 and 3. see doc/README.md §3.3
[3]
The articles from 2017‑12‑25
https://suchowan.seesaa.net/article/201712article_25.html
and
doc/README.md
describe the system from the perspective of the inclusion
green ⊃ white.
-> Japanese
The Harmonic System is not intended as a replacement for SI units.
Rather, it serves as a structural lens that reveals unexpected alignments
between astronomical cycles, physical constants, and gravitational parameters.
The narrow overlap shown in the Venn diagram is
one of the clearest demonstrations of this structural harmony.
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