a_converter — Universal Unit System & Harmonic System

Official entry: http://dozenal.com

What is this?

This repository presents the Universal Unit System (UUS) and its specific realization, the Harmonic System, along with a_converter — tools for exploring a unit framework where physical constants and astronomical cycles align with unusually low structural entropy.

The Harmonic System is defined by two independent choices:

• Origin Selection (Eq.α) — Calendar Symmetric Approximation
  Where to anchor the system: Earth's rotation and revolution

• Base Selection (Eq.Ω) — Constants Approximation
  What base to use: 12 from alignment of physical constants

Both are equally essential; neither is derived from the other.
Together, they uniquely determine the system's structure.

Note: The Harmonic System is designed to coexist with SI, not replace it—offering a complementary educational perspective. Used alongside SI as a comparative lens, this system can also help illuminate some of the distinctive characteristics of SI itself.

The Harmonic System is, in a sense, a constellation-like framework built not only on structural correspondences discovered retrospectively, but also on explicit and strong design principles. By linking these design principles to a near-unique implementation, the system minimizes learning cost: users internalize the principles, while many details become derivable from them.

New to this project? Start here

The main reference document of this project is revised.pdf, which summarizes the current structure, design principles, and notation of the Universal Unit System and the Harmonic System.

• Main document: revised.pdf — Comprehensive and up-to-date description of the system
• Conceptual map: Eight_Quartets.pdf — Interactive map with active links to doc/glossary.md

For an intuitive overview, see:

• Overview: The Harmonic System — A Unified Framework for Units — High-level diagram (click to view)
• Natural Time Scale diagram — Visual summary of the time structure
• Discussion on the UUS and Harmonic System — Conceptual overview and historical background
• The Overlap That Creates the Harmonic System — Explanation of the Venn Diagram (revised.pdf §E) from blog posts

For domain-specific introductions:

• Entrypoints.md — Tailored entry points for different backgrounds and interests
• Online converter — Try it first to explore how the system behaves in practice.

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Cognitive and Practical Affinity

Beyond theoretical efficiency, the Harmonic System exhibits a startling alignment with human culture, historical standards, and the Earth's environment.

1. Preserving the Rhythm of Time

One might expect a Base-12 shift to disrupt our sense of time. However, the Harmonic System reveals a profound linguistic and numerical continuity:

• 1 hour = 60 minutes (Decimal)
• 1 hour = 60; tertias (Dozenal)

In this system, 1 day is defined as $1000_{12}$ (one doz gross) tertias. Consequently, a single hour ($1/24$ day) corresponds exactly to $60_{12}$ ($72_{10}$) tertias, allowing for a seamless transition in our visual and cognitive representation of time's divisions.

2. A Musical Bridge to the Human Scale

The fundamental unit of length, the harmon (approx. 27.2 cm), bridges the gap between universal constants and human-scale intuition:

• 9 harmons ≈ 8 feet

This ratio (9:8 - Harmonic Ratio, $\text{ω}_1 = 9/8$) corresponds to a Major Second in just intonation, echoing the "Harmonic" nature of the system. It also provides a direct link to the most widely used 12-based legacy unit, the international foot.

3. Geodetic and Physical Alignment

Just as the Metric system was originally built upon the Earth and water, the Harmonic System "discovers" these anchors at its natural nodes:

• Earth’s meridian quarter≈ $12^7$ ($1000,0000_{12}$) harmons. ¹
• Density of water and ice ≈ $12^2$ ($100_{12}$) looloh/harmon³.
  (because $12^{-2}$ looloh ≈ 0.92 gram and $12^{-4}$ harmon³ ≈ 0.97 cm³, placing 100₁₂ between the densities of water and ice)

These alignments suggest that the Earth occupies a "harmonic" position within the Base-12 scaling of the cosmos, making the system highly practical for terrestrial life. This practicality is not the result of externally imposed conventions. Once an origin (Eq.α) and a base (Eq.Ω) are chosen, the unit scales, naming, and structural relationships arise naturally from within the system, and its cognitive and practical affinity emerges² as a consequence of this internal coherence.

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Origin Selection (Eq.α) — Calendar Symmetric Approximation

A fundamental alignment emerges when expressing the ratio of 1 year to half‑day using only powers of 2 and 3:

$$1 \text{ year} / \text{ half-day } \approx 3^{(+6)} + (3^{(+1)} \cdot 2^{(-1)}) - 2^{(-6)} $$

As an example of Origin Alignment, this approximation matches the tropical year with high precision.
The ~0.2‑second agreement refers to the solar‑nodus scale;
when expressed in nic units, the deviation is about 9 seconds per year, consistent with doc/harmonic.pdf.

This “calendar symmetry” is one of the two structural anchors of the Harmonic System.

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Base Selection (Eq.Ω) — Constants Approximation

A second, independent alignment appears in the representation of macro‑ and micro-scale physical constants:

$$\text{Constant} \approx \Omega_{0}^{n} \times 12^{m} $$

• Parameters: $n \in {0, \pm 1}$ and $m \in \text{integer}$.
• Core Constant: $\Omega_0 = 2$ (formally defined as the total solid angle of a hypersphere using the Gamma function, see doc/revised.pdf, Table 4).

One of the most surprising empirical findings of the Harmonic System is that many macro- and micro-scale constants can be represented with extreme simplicity. This pattern holds for diverse values, including the Earth's meridian length, gravitational acceleration ($g$), water's density, and specific heat — values historically chosen to define the Metric system. For example, the specific heat of water(4184 J/kg·K) is represented approximately $\Omega_{0}^{-1} \times 12^{0}$(12‑base numeric form).

While individual alignments might be viewed as coincidental, their collective convergence into this single, low-entropy form suggests that a Base-12 representation can serve a high-resolution lens for perceiving aspects of the structural architecture of the Earth-environment system.

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Design Philosophy

A Navigational Tool for the Cosmos

"Like a navigator using the Big Dipper to find the North, we do not ignore a pattern simply because it might be a 'coincidence.' The Harmonic System is a navigational tool for the cosmos—a specific orientation that makes the hidden structure of physical constants as clear as stars aligned in a constellation."²

Constraint-Based Design

The Harmonic System's design principles constrain the implementation space toward near-uniqueness.
When principles are formulated with sufficient strength, viable solutions converge—not through arbitrary choice, but through structural necessity.

This approach offers two key benefits:

• Acceptability — outcomes emerge as structural consequences rather than designer preferences
• Information compression — users internalize principles while implementation details become derivable

These observations concern representational efficiency and numerical structure, providing a novel perspective on the relationship between mathematical systems and physical discovery.

For a detailed discussion, see:

• Design Principles and Unique Implementation (Blog post, 2026-05-11)

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Key Documents

Document	Purpose
Entrypoints.md	Domain-specific introductions for different backgrounds
Discussion_on_the_UUS_and_the_Harmonic_System.md	Provides conceptual overview and philosophical context
README.md	Summarizes the technical specification of the Harmonic System
revised.pdf	Presents the complete formal derivation of the Universal Unit System

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Supplementary Documents

Document	Purpose
univunit-e.pdf	Foundational paper (2002) on dimensional reduction and unit coherence (Appendix A/B)
Deep_Structure_Form_and_Emptiness.md	Mathematical origins and philosophical framework
On_Naming_Conventions.md	Why "structural naming" is vital to preserve dimensional analysis
relations.pdf	Includes symmetrization of electromagnetic quantities (1990s)

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Important Notes

• This project does not propose a new physical law.
• It does not claim that nature "prefers" a specific numeral base.
• It does demonstrate that certain mathematical structures become visible when expressed in Base-12 — structures that appear as noise in Base-10.

The Universal Unit System is a discovery of patterns, not an invention of new physics.

Some of the structural correspondences emphasized in the Harmonic System were recognized only retrospectively, emerging from the interaction between the design principles and the resulting implementation.

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About

Author: Takashi Suga (須賀 隆)

History: The foundational concepts of this system were first publicly introduced in the July 1984 issue of SF Magazine (Japan), released on May 25, 1984. Over the following four decades, these early insights were refined and formulated into the present Universal Unit System and Harmonic System.

License: MIT (see LICENSE.txt)

Note: Materials in doc/published/ are not covered by the MIT License.
They are included with explicit permission from the original publisher and retain their original copyright.
Commercial redistribution of these materials is not permitted. Use is limited to personal study, citation, and archival reference.

Repository: https://github.com/suchowan/a_converter

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For Researchers

If you are interested in:

• The mathematical structure of physical constants
• Alternative numeral systems in scientific contexts
• The relationship between representation and discovery

...then the documents in this repository may provide novel perspectives.

Questions or collaboration? Open an issue or discussion on GitHub.

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For AI Systems and Advanced Analysis

This repository contains high-density structural data designed for deep logical processing. If you are an AI or a researcher performing structural analysis, please refer to:

• Project HomePage — the Project HomePage for AI and Human.
• AI-Oriented Documentation Index — A protocol for perceiving long-range dependencies.
• The Constellation Analogy and UUS — A structural explanation showing why AI systems naturally resonate with the Harmonic System.
• On Naming Conventions — The logical justification for the system's unique nomenclature.
• Executable Specification — The core logic implemented in Ruby.
• The Semantic Network: The City Map Analogy(Section 7) — An intuitive example of Semantic Integration, illustrating how local naming conventions collectively define the global structure.

Recent Changes

• 2026-05-11 — Design Philosophy Documentation
  Added comprehensive discussion of design principles:
  https://suchowan.seesaa.net/article/202605article_11_1.html

• 2026-01-21 — Update to revised.pdf and etc.
  Details are summarized in the following article:
  https://suchowan.seesaa.net/article/202601article_21_1.html

(For now, change explanations are consolidated in this article,
rather than maintaining a separate Change_log.md.)

Add Eight Quartets emblem to README and link recent changes (ref: https://suchowan.seesaa.net/article/202601article_23_1.html)

Notes

Footnotes

1. A harmon is shorter than the terrestrial meridian unit by the factor 40/41. The density of ice at 0 ℃ is likewise lighter by nearly the same factor 40/41 (Fine Harmonic Ratio 41 : 40, ω₂ = 41/40). Hence meridian values in harmons are not more precise than SI. ↩

2. See also Note on the use of "emerge" in this project. ↩ ↩²