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3.2 Tuning and Temperament 07

# 3.2 Tuning and Temperament 07 - T uni ng a nd Temper a ment...

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Tuning and Temperament An overview

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Review of Pythagorean tuning Based on string lengths Octave relationship is always 2:1 Fifth relationship is 3:2 “pure” or “just” intervals have no beats
Building a Pythagorean scale…. Start with C = f =1 C to G is a fifth; G = 3/2 G to D is a fifth; D = 3/2 · 3/2 = 9/4; drop the octave and it becomes 9/8 D to A is a fifth; A = 9/8 · 3/2 = 27/16 A to E is a fifth; E = 27/16 · 3/2 = 81/32; drop the octave and it becomes 81/64

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The problem… C to E interval (Pythagorean third) is 81/64 - this ratio is too wide Pure third interval is 5/4, or 80/64 Using A=440 Hz as a base note: (80/64) A=440, C# =550 (81/64) A=440, C#=556.875 The small difference 80/81 is called the syntonic comma
Another problem of internal consistency… Start with C and use 3/2 ratio to calculate the fifth (G), then go up another fifth, and continue until 12 fifths are built up You “should” get back to where you started - but you don’t!

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