Your framework presents an interesting natural unit system based on a fundamental time parameter T and the speed of light c = 299,792,458 m/s, with base-3 exponential scaling. This resembles Planck units but uses powers of 3 as scaling factors instead of combinations of fundamental constants.

Analysis of Your Given Framework

From your definitions:

• Unique Size Measure: L = c × T (meters)

• Color Wavelength: λ_color = (c × T)/(3^35) meters

• Audible Frequencies: f_audio = (1/T) × 3^[0...13] Hz

Additional Physical Characteristics Derivable from Your Unique Time T

Using dimensional analysis principles, your unique time T can generate an entire hierarchy of physical quantities:

1. Mechanical Quantities

• Velocity Scale: v = c (naturally fixed)
• Acceleration Scale: a = c/T (m/s²)
• Jerk Scale: j = c/T² (m/s³)
• Length Scales: L_n = (c × T)/3^n for different hierarchical levels

2. Wave and Oscillation Characteristics

• Fundamental Frequency: f₀ = 1/T Hz
• Frequency Hierarchy: f_n = (1/T) × 3^n Hz
• Wavelength Hierarchy: λ_n = (c × T)/3^n meters
• Angular Frequency: ω = 2π/T rad/s
• Wave Number: k = 2π/(c × T) m⁻¹

3. Electromagnetic Spectrum Mapping

Your color measure λ_color = (c × T)/(3^35) suggests a specific wavelength in the electromagnetic spectrum. Given that visible light ranges from 380-700 nm, this could map to:

• Visible Light: If λ_color ≈ 500 nm, then T ≈ 5.6 × 10⁻¹⁶ seconds

• Infrared/Microwave: For larger T values

• Ultraviolet/X-ray: For smaller T values

4. Acoustic Characteristics

Your audible frequency range f_audio = (1/T) × 3^[0...13] spans 14 orders of magnitude. Since human hearing ranges from 20 Hz to 20,000 Hz:

• Base Audio Frequency: f₀ = 1/T

• Harmonic Series: 3^n × f₀ creates a ternary harmonic progression

• Frequency Bandwidth: From f₀ to f₀ × 3^13 ≈ 1.6 × 10⁶ × f₀