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#=
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 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 = 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

9. Just intonation scale C D F G A B 1 9/8 5/ 4 4/ 3 3/ 2 5/ 3 2 15/ 8
9 8
10 9
16/ 15
9/ 8
10/ 9

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 = to 1

15. Interval comparisons…
Just scale
Equal temperament
Major third A to C#
Perfect fifth A to E

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