Three Division Tuning Method (三分损益法)

This ancient Chinese method calculates the lengths of musical pipes for the pentatonic scale (宫商角徵羽 - Gong, Shang, Jue, Zhi, Yu).

Overview

The method uses alternating division by 3 to generate successive notes, creating a mathematical relationship between pipe lengths.

Base Values

• Start with 81 (九九八十一) as the base unit for 宫

Calculation Steps

1. Calculate 徵

   • Take 宫 value: 81
   • Divide by 3, subtract 1/3: 81 × (2/3) = 54
   • Result: 徵 = 54

2. Calculate 商

   • Take 徵 value: 54
   • Divide by 3, add 1/3: 54 × (4/3) = 72
   • Result: 商 = 72

3. Calculate 羽

   • Take 商 value: 72
   • Divide by 3, subtract 1/3: 72 × (2/3) = 48
   • Result: 羽 = 48

4. Calculate 角

   • Take 羽 value: 48
   • Divide by 3, add 1/3: 48 × (4/3) = 64
   • Result: 角 = 64

Final Scale Lengths

Note	Length	Relationship
宫	81	Base
商	72	宫 × 8/9
角	64	宫 × 64/81
徵	54	宫 × 2/3
羽	48	宫 × 16/27

Principle

• "下生" (descending generation): multiply by 2/3
• "上生" (ascending generation): multiply by 4/3
• Alternating creates the pentatonic scale

Validation

1. Confirm the final scale lengths match: 宫=81, 商=72, 角=64, 徵=54, 羽=48
2. Verify each calculation alternates correctly between 下生 (×2/3) and 上生 (×4/3) operations
3. Check that the mathematical relationships in the summary table are consistent with the step-by-step calculations