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Tuning and Temperament

Tuning and Temperament. An overview. 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

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Tuning and Temperament

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  1. Tuning and Temperament An overview

  2. 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

  3. 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

  4. The problem… • C to E interval (81/64) 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

  5. 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! • Difference is 1.0136432 - called the Pythagorean comma

  6. Problem of how to manage pure intervals with bad ones (too wide or too narrow) • Bad interval called a “wolf” • Solution is that certain tones have to be adjusted higher or lower - this is called “tempering”

  7. Just Intonation • Preference given to pure triads built on I, IV, V - most common chords in a key

  8. Building a just scale… • Start with C = 1E is 5/4G is 3/2 • Up to F = 4/3A is 5/4 · 4/3 = 5/3C is 2/1 (octave) • Back to G=3/2B is 3/2 · 5/4 =15/8D is 3/2 · 3/2 = 9/4; drop octave to 9/8

  9. Just intonation scale

  10. More problems… • 2 different sizes of whole steps: 9/8 and 10/9 • Great for CEG, FAC, GBD, but others have wolves • Difficult to modulate to distant keys

  11. Meantone tuning • Take intervals which are too wide and temper them to the average, or mean • Example: four 5ths used to get from C to E (C - G - D - A - E) • Solution: shrink each 5th by 1/4 of the syntonic comma • Called “1/4 Comma Meantone Tuning”

  12. Well Temperament • Intervals are tempered and various mis-tunings are moved around • Intervals in certain keys are favored and left closer to pure; others are left more dissonant • Result: different keys have different colorations or characters; modulations to remote keys are more noticeable • Many different temperaments devised

  13. Equal Temperament • Each octave is divided into 12 equal semitones • Each semitone has same frequency ratio • Each 5th is equal in size • 12 5ths combined = perfect octave above starting place • Each 5th is shrunk by 1/12 of Pythagorean comma

  14. Mathematical basis • Octave ratio is 2:1 • Find some number, multiplied by itself 12 times = 2 • Semitone ratio = 1.05946 to 1

  15. Interval comparisons…

  16. Possible disadvantages of equal temperament? • Loss of key “color” and character; every one is the same • Every interval is slightly out of tune: no pure, beatless intervals • In practice, choral and instrumental groups will adjust tuning to reduce beats • Keyboard instruments are fixed and unchangeable

  17. Division of the semitone • Each semitone divided into 100 cents • A cent is a ratio, just like a semitone is • Octave is 1200 cents

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