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three-division-tuning-method

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Use when calculating pentatonic scale pipe lengths or tuning traditional Chinese instruments. Applies the three-division method (三分损益法) to derive 宫商角徵羽 from base unit 81 using alternating 2/3 and 4/3 ratios.

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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

NoteLengthRelationship
宫81Base
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