The Evolution of Common Western - PowerPoint Presentation

Slide1

The Evolution of Common Western

Tuning Systems

Wayne Wu

Slide2

Tuning System

Slide3

Octave

Ratio of

“Same note”

12 pitches/semitones

How do we get there?

 

Tuning System

Slide4

Three major tuning systems (temperaments) in Western Music:

Pythagorean Tuning

¼ Comma Meantone Temperament

12-tone Equal Temperament

Tuning System

Slide5

Integer multiplications to the base frequency

Intervals in between harmonics are called “just intervals”

Have a listen

 

Harmonic Series

Base Frequency

220 Hz

220 Hz

440 Hz

660 Hz

880 Hz

1100 Hz

Base Frequency

220 Hz

220 Hz

440 Hz

660 Hz

880 Hz

1100 Hz

Slide6

Octave:

12 semitones

Perfect Fifth:

7 semitones

Major Third:

4 semitones

Have a listen

 

Important Just Intervals

Slide7

Preserves Octaves (

) and Perfect Fifths (

)

Remember: a Perfect Fifth is 7 semitones and an Octave is 12 semitones

7 and 12 are coprime

If we stack the Perfect Fifths, we will eventually get all 12 pitches!

 

Pythagorean Tuning

Slide8

Start with a base note

Stack the

ratio to get new pitches that are a Perfect Fifth apart

Use the

ratio to bring the pitches to different Octaves

Repeat 6 times up and 5 times down

 

Pythagorean Tuning

Slide9

Pythagorean Tuning

 

Slide10

Pythagorean Tuning

 

Slide11

Pythagorean Tuning

 

Slide12

Pythagorean Tuning

 

Slide13

Pythagorean Tuning

E♭

B♭

F

C

G

D

A

E

B

F♯

C♯

G♯

E♭

B♭

F

C

G

D

A

E

B

F♯

C♯

G♯

D

E♭

E

F

F♯

G

G♯

A

B♭

B

C

C♯

D

E♭

E

F

F♯

G

G♯

A

B♭

B

C

C♯

Slide14

However…

 

If we step down one more perfect fifth from

E♭

, calculate the ratio of

A♭

as:

… but we calculated

G♯

as

!

 

Slide15

So What Exactly Is That Interval?

The ratio between

G♯

and

E♭

can be calculated by:

Whereas a Just Perfect Fifth have a ratio of

This is called the Wolf Interval

Compared to a Just Perfect Fifth

 

Slide16

Preserves Octaves (

) and Major Thirds (

)

Remember: a Major Third is 4 semitones and an Octave is 12 semitones

4 divides 12

If we stack the Major Thirds, we don’t get very far

 

¼ Comma Meantone Temperament

Slide17

Slide18

Calculate the Fifth according to the Major Third

Take 2 Octaves and a Major Third up:

semitones

Take 4 Fifths up:

semitones

Calculate the Fifths by:

Repeat the stacked Fifths procedure in Pythagorean Tuning

 

¼ Comma Meantone Temperament

Slide19

¼ Comma Meantone Temperament

E♭

B♭

F

C

G

D

A

E

B

F♯

C♯

G♯

E♭

B♭

F

C

G

D

A

E

B

F♯

C♯

G♯

D

E♭

E

F

F♯

G

G♯

A

B♭

B

C

C♯

D

E♭

E

F

F♯

G

G♯

A

B♭

B

C

C♯

Slide20

Sanity Check…

If we step down one more perfect fifth from

E♭

, calculate the ratio of

A♭

as:

… still not the same as the

for

G♯

!

 

Slide21

Wolf Interval in Meantone

The ratio between

G♯

and

E♭

can be calculated by:

Just Perfect Fifth have a ratio of

Meantone Perfect Fifth have a ratio of

 

Slide22

Even Worse…

D

E♭

E

F

F♯

G

G♯

A

B♭

B

C

C♯

D

E♭

E

F

F♯

G

G♯

A

B♭

B

C

C♯

D-E♭

E♭-E

E-F

F-F♯

F♯-G

G-G♯

G♯-A

A-B♭

B♭-B

B-C

C-C♯

C♯-D

D

C

D

C

D

C

D

D

C

D

C

D

D-E♭

E♭-E

E-F

F-F♯

F♯-G

G-G♯

G♯-A

A-B♭

B♭-B

B-C

C-C♯

C♯-D

D

C

D

C

D

C

D

D

C

D

C

D

Every key sounds different!

Slide23

Characterization of the Keys

Some examples from Christian Friedrich Daniel

Schubart’s

Ideas Towards an Aesthetic of Music

:

C major is quite pure. Its character is innocence, simplicity, [and] baby-talk.

G minor, displeasure, uneasiness, worry about a failed scheme; discontent gnashing at the bit; in a word, anger and disgust.

B major, strongly colored, announcing wild passions, made up of the crudest colors. Anger, rage, jealousy, fury, desperation, and every burden of the heart lies in its sphere.

Slide24

The Key of

F♯

Major

Christian Friedrich Daniel

Schubart

skipped the key of

F♯

major because it’s useless under ¼ Comma Meantone Temperament.

When You Wish Upon A Star

in the key of

F♯

major in today’s tuning system:

When You Wish Upon A Star

in the key of

F♯

major in ¼ Comma Meantone Temperament :

Slide25

Keep the same ratio for all 12 semitones

Calculate the ratio of each semitone by:

 

12-tone Equal Temperament

A

A♯/B♭

B

C

C♯/D♭

D

D♯/E♭

E

F

F♯/G♭

G

G♯/A♭

A

A♯/B♭

B

C

C♯/D♭

D

D♯/E♭

E

F

F♯/G♭

G

G♯/A♭

Slide26

Consistency between enharmonic equivalents

Consistency between intervals

Consistency between Keys

What We Achieved

A

A♯/B♭

B

C

C♯/D♭

D

D♯/E♭

E

F

F♯/G♭

G

G♯/A♭

A

A♯/B♭

B

C

C♯/D♭

D

D♯/E♭

E

F

F♯/G♭

G

G♯/A♭

Slide27

Any Just Interval other than the Octaves

Any other interval ratio is irrational

Not much difference, though

What We Lost

12 ET Perfect Fifth:

12 ET Major Third:

 

Just Perfect Fifth:

Just Major Third:

 

Slide28

Flavor and characterization of the keys!

What We Lost

When You Wish Upon A Star

in the key of

G

major in Meantone (gentle and serene motion of the heart):

When You Wish Upon A Star

in the key of

A

major in Meantone (innocent love, contentment):

When You Wish Upon A Star

in the key of

B

major in Meantone (strongly colored, wild passions):

Slide29

Perfection?

Degeneracy?

Compromise.

It is a story about how we gave up our obsession with the “purity” of mathematical form or interval ratio to arrive at a compromised yet consistent tuning system.

… Moral of the Story?

Slide30

References

Schubart

, Christian F. D.

,

Ideen zu einer Ästhetik der Tonkunst: an

annotated

translation

.

Translated by Ted Alan DuBois, University of Southern California, 1983, pp. 433-436. digitallibrary.usc.edu/cdm/ref/collection/p15799coll3/id/262906

Durfee, Dallin S.; Colton, John S., “The physics of musical scales: Theory and experiment.”

American journal of physics

,

Vol.83 (10),

2015, pp. 835-842

. aapt.scitation.org/doi/full/10.1119/1.4926956

Neely, Adam, “Which key is the saddest?”

YouTube

, 9 January 2017, www.youtube.com/watch?v=6c_LeIXrzAk&t=229s.

Huang, Andrew, “The most mind-blowing concept in music (Harmonic Series).”

YouTube

, 7 May 2020, www.youtube.com/watch?v=Wx_kugSemfY