Three Division Tuning Method

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