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1
Temperament

• Introduction
• Musical Intervals
• Why do we need temperaments ?
• Some temperaments

2
Music to the Ear
Oval Window
Stirrup
Round Window
Basilar Membrane
Octaves f ? f x 2

• Equal musical intervals correspond to equal
  multiples in frequency
• These translate into equal distances along the
  basilar membrane

3
Music to the Ear
Oval Window
Stirrup
Round Window
Basilar Membrane
Fifths f ? f x 3/2 (approximately!)

• Equal musical intervals correspond to equal
  multiples in frequency
• These translate into equal distances along the
  basilar membrane

4
Ideal Musical Intervals

• Octave f ? f x 2
• Perfect 5th f ? f x 3/2
• Perfect 4th f ? f x 4/3
• Major 3rd f ? f x 5/4
• Minor 3rd f ? f x 6/5

These can be fined-tuned by beats
5
The Notes
C/Db
D/Eb
D/Eb
D/Eb
D/Eb
A B C D E F G
A B C D E F G
C D E F G
A
6
Musical Intervals
C/Db
D/Eb
D/Eb
D/Eb
D/Eb
A B C D E F G
A B C D E F G
C D E F G
A
Octave 12 half-tones ( x 2)
7
Musical Intervals
C/Db
D/Eb
D/Eb
D/Eb
D/Eb
A B C D E F G
A B C D E F G
C D E F G
A
Fifth 7 half-tones ( x 3/2)
8
Musical Intervals
C/Db
D/Eb
D/Eb
D/Eb
D/Eb
A B C D E F G
A B C D E F G
C D E F G
A
Fourth 5 half-tones ( x 4/3)
9
Musical Intervals
C/Db
D/Eb
D/Eb
D/Eb
D/Eb
A B C D E F G
A B C D E F G
C D E F G
A
Major 3rd 4 half-tones ( x 5/4)
10
Musical Intervals
C/Db
D/Eb
D/Eb
D/Eb
D/Eb
A B C D E F G
A B C D E F G
C D E F G
A
Minor 3rd 3 half-tones ( x 6/5)
11
Why Do We Need Temperaments ?

• If we keep octaves to be x 2s, then some 5ths
  will not match exactly

Progress up from C1/D1b in 5ths to C8 ? x
(3/2)7 129.75 Then descend back down to D1b in
octaves ? x (1/2)7 128 Total change is ? x
1.0136 BUT it is the same note !!
12
Pythagorean Temperament

• General idea
• Make all Octaves perfect
• Make all 5ths perfect too
• Except for ONE
• (Wolf Fifth)

13
Pythagorean Temperament

• To get from C4 to E4
• Go up (in 5ths) for 4 steps
• C4 ? G4 ? D5 ? A5 ? E6
• x 3/2 x 3/2 x 3/2 x 3/2
• Then descend (in octaves) for 2 steps
• E6 ? E5 ? E4
• x 1/2 x 1/2
• So frequency C4 is multiplied by
• (3/2)4 x (1/2)2 1.2656
• Therefore E4 C4 1.2656 1

Compute ratios Choose, for example, the
ratio of E4 C4
A3 B3 C4 D4 E4 F4 G4 A4 B4
etc.
14
Pythagorean Temperament
Compute ratios Choose, for example, the
ratio of E4 C4
A3 B3 C4 D4 E4 F4 G4 A4 B4
15
Pythagorean Temperament

• How do we tune it ?
• Do one Octave, then all others are easy
• Using C4 as reference, tune G4 (5th higher) by
  beats
• Then use G4 to tune D5
  
• Then do D4 (one octave lower) etc.

E5
D5
C5
C5
Bb4
Ab4
B4
G4
A4
F4
F4
C4
E4
Eb4
C4
D4
B3
C4
Bb3
16
Pythagorean Temperament

• How do we tune it ?

3
4
5
17
Just Temperament

• Principal intervals are I-IV-V or
  Tonic-Subdominant-Dominant in simple ratios
• 1 4/3 3/2 (e.g. C_4
  E_4 G_4 256 341 384)
• NOTE that
• 5/4 x 6/5 3/2
• So, we can make all these ratios perfect !
• BUT only in one key (e.g. C) where we make C4
  E4 F4 perfect.
• If you try to do this also for key of F, then C
  is not on octave.

Major 3rd
Minor 3rd
Perfect 5th
18
Equal Temperament

• We can make all half-tones the same
• This shares the pain !!
• Since there are 12 half-tones in an octave, then
  ratio is R where
• R12 2 ? R 1.05946
• (Check that 1.0594612 2 !)
• We can, for instance, then compute all the notes.
• E.g. if C_4 is 1.0, then D_4 is (1.05946,)2,
  etc..

19
Equal Temperament

• Here is how some of the ratios work out
• Only the octave is actually right, but others are
  not so bad.

OK! of course!
NOT 1.50000
NOT 1.33333
NOT 1.25000
20
These and Other Temperaments
Ratios
21
These and Other Temperaments
Frequencies (Hz)