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 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 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 (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 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

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 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

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

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

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

10. Just intonation scale CDEFGABC 1 9898 5454 4343 3232 5353 15 8 2 9898 10 9 16 15 9898 10 9 9898 16 15

11. 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 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

12. 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” 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”

13. 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 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

14. 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 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

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

16. Interval comparisons… Just scaleEqual temperament Major third A to C# 440 - 550440 - 554.37 Perfect fifth A to E 440 - 660440 - 659.26

17. Possible disadvantages of equal temperament? Loss of key “color” and character; every key is the same Every interval is slightly out of tune: no pure, beatless intervals Loss of key “color” and character; every key is the same Every interval is slightly out of tune: no pure, beatless intervals

18. Temperament applied to “real life” in music Keyboard instruments are fixed and unchangeable - other instruments have to adjust In practice, choral and instrumental groups will adjust tuning to reduce beats - they will create pure intervals in chords Keyboard instruments are fixed and unchangeable - other instruments have to adjust In practice, choral and instrumental groups will adjust tuning to reduce beats - they will create pure intervals in chords

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

20. Relation of cents and frequency Not the same! Each octave is 1200 cents, including: A 440 to A 880 A 880 to A 1760 A 1760 to A 3520, etc. Not the same! Each octave is 1200 cents, including: A 440 to A 880 A 880 to A 1760 A 1760 to A 3520, etc.