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Exploring Tuning and Temperament: A Comprehensive Study

Dive into the world of Pythagorean tuning, temperaments, scales, and intervals. Learn about Just Intonation, Meantone Tuning, Equal Temperament, and more to understand the complexities of musical tuning systems.

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Exploring Tuning and Temperament: A Comprehensive Study

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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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