An overview of Tuning and Temperament

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 (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

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”

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

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

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”

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

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

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

16. Interval comparisons…
Just scale Equal temperament
Major third
A to C#
440 - 550 440 - 554.37
Perfect fifth
A to E
440 - 660 440 - 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

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

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

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.