Signiﬁcant Occurrence in Even Musical Texture in Bach’s...

Significant Occurrence in Even

Musical Texture in Bach’s Preludes

A Study Using Mathematical Tools

Dalia Cohen Idith Segev

Israel

14th Biennial Baroque ConferenceJuly 1, 2010

Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture

Curve of all notes of Do minor

-20

-10

0

10

20

0 50 100 150 200 250 300 350

No

. o

f h

alf-t

on

es

No. of semiquaver

time

All notes

Basic Definitions

1 “Learned” schemata

Collection of notesIntervalsHarmoniesTonal organization

Basic Definitions

1 “Learned” schemata

Collection of notesIntervalsHarmoniesTonal organization

2 “Natural” schemata

Range of occurrence related to normative rangeTypes of curves of change (in our study - only of pitch)Cognitive operations: augmentation/diminishing,contrast, shift, splitting and gathering, equivalenceConcurrence and non-concurrence, rarity

Definitions

1 The ’New Variable’

includes small, hidden, meaningful deviations of all thenatural schematarelates strongly to the tonal harmony

Definitions

1 The ’New Variable’

includes small, hidden, meaningful deviations of all thenatural schematarelates strongly to the tonal harmony

2 “Evenness” - a constant parameter or characteristic

DurationPitch curve of the pattern (up to small deviations)

Curve of all notes of Do minor

-20

-10

0

10

20

0 50 100 150 200 250 300 350

No

. of half-t

on

es

No. of semiquaver

time

All notes

First stage

Selecting Preludes from the WTC-I by Bachwith common characteristics

’even’ duration

First stage


’even’ duration

’even’ curves of pitch - patterns, up to smalldeviations

First stage


’even’ duration

’even’ curves of pitch - patterns, up to smalldeviations

similar structure

The Generic Model of an Even Prelude by Bach

OpeningSection

SequenceOrganPoint

Cadence



OpeningSection

SequenceOrganPoint

Cadence

1 I,II,V,I; or I, IV, VII, I;



OpeningSection

SequenceOrganPoint

Cadence


2 Bach is using one of his famous Figured Bass Formulase.g. (second, sext), 2,6,2,6,...



OpeningSection

SequenceOrganPoint

Cadence



4 The Cadence - ’Bach the Believer’



OpeningSection

SequenceOrganPoint

Cadence



3 The Organ Point - appears usually in the second half ofthe prelude

4 The Cadence - ’Bach the Believer’


The Location of the Organ points

Do M

Do m

Re M

Re m

Mi m

Fa M

0 10 20 30 40 50 60 70 80 90 100 %

Prelude

Basic Musical Units

1 Nuclei


Basic Musical Units

1 Nuclei

2 Patterns


Basic Musical Units

1 Nuclei

2 Patterns

3 Groups of patterns (higher levels)


Convex and Concave curves of Nuclei and Patterns

1 First pattern (repeated), of Do minor prelude (no. 2)



2 First pattern (repeated), of Mi minor prelude (no. 10)



2 First pattern (repeated), of Mi minor prelude (no. 10)

3 First and second patterns of Fa M prelude (no. 11)

Straight-Lines Curves in Different Directions

1 First pattern of Do M prelude (no. 1)



2 First pattern of Re M prelude (no. 5)



2 First pattern of Re M prelude (no. 5)

3 First pattern of Re minor prelude (no. 6)

Basic Graphical Presentation of Nuclei

Special Grouping of Nuclei in Re M

Second Bar of Re M Prelude (no. 5)

4 4 4 3

II V I

րրց

bar 2 → I; bar 5 → V

bar 10 = III→II; bar 12)=II→I

bar 18 = VI; bar 19 → V:IV

bar 21 →IV; bar 24 → I

Second step

1 Expressing curves of pitch by numbers, always related tothe first pattern - the reference pattern.[0, 4, 7, 12, 16], is the first nucleus of Do M.


Second step

1 Expressing curves of pitch by numbers, always related tothe first pattern - the reference pattern.[0, 4, 7, 12, 16], is the first nucleus of Do M.

2 Analyzing them by using Musical and Mathematicaltools


Mathematical Tools

1 Peak notes and low notes (of the upper voice)

Mathematical Tools


2 Median - ’center of gravity’ or an ’inner organ point’

Mathematical Tools



3 Linear Regression

Mathematical Tools



3 Linear Regression

4 Parabolic Regression

Curve of all Peak-notes of Do M

-10

0

10

20

30

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35

No

. o

f h

alf-t

on

es

No. of pattern

Max

I V I V V:IV

time

Curve of all Peak-notes of Re M

-20

-10

0

10

20

30

40

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69

No

. o

f h

alf-t

on

es

No. of pattern

Max

I V III II I VI IV I V

time

Curve of all Peak-notes of Re M

-20

-10

0

10

20

30

40

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69

No

. o

f h

alf-t

on

es

No. of pattern

Max


� �B B

time

Mathematical Tools - cont.

Statistical Averages

Median - ’center of gravity’ or an ’inner organ point’


Median - a sort of mean of one pattern

First pattern of Do minor prelude (no. 2)

Median - a sort of mean of one pattern

First pattern of Do minor prelude (no. 2)

In order to obtain the Median, we arrange all pitchesaccording to increasing pitch.

The pitch that is located in the middle of the row, is themedian - Mi ♭.

Curve of all Medians of Do minor

-5

0

5

10

1 3 5 7 9 11 13 15 17 19 21 23

No

. o

f h

alf-t

on

es

No. of pattern

Median

I V III I V V

time

Mathematical Tools - cont.

Geometric Statistics

Linear Regression

Parabolic Regression


Regression line - a comparison between two

patterns

y = a · x + b

Find mina, b

S(a, b), where :

S(a, b) =n

∑

i=1

(

yi − (a · i + b))2

∂S

∂a= 0

∂S

∂b= 0

⇒ a, b = . . .

Creating a Regression Line

Patterns 1, 23, in Do m.

-15

-10

-5

0

5

10

15

No.

of h

alf-

tone

s

Do m Prelude, patterns 1,23

Parabolic Regression - a comparison between two

patterns

y = a · x2 + bx + c

Find mina, b, c

S(a, b, c), where :

S(a, b, c) =

n∑

i=1

(

yi − (a · i2 + b · i + c))2

∂S

∂a= 0

∂S

∂b= 0

∂S

∂c= 0

⇒ a, b, c = . . .

Creating a Parabolic Regression Curve - Do minor

-30

-25

-20

-15

-10

-5

0

5

10

15

1 2 3 4 5 6 7 8

No.

of h

alf-

tone

s

No. of semiquaver (time)

Prelude 2 Long, patterns 1,19

Linear Reg. of Do minor

-1

-0.5

0

0.5

1

1.5

2

2.5

3

1 3 5 7 9 11 13 15 17 19 21 23

Slope

No. of pattern time

Regression Slope

Zero

I V III I V V

Intersecting Linear and Parabolic Reg. of Do minor

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

1 3 5 7 9 11 13 15 17 19 21 23

De

gre

e 2

Slo

pe

No. of pattern

0

0.5

1

1.5

2

1 3 5 7 9 11 13 15 17 19 21 23

De

gre

e 1

Slo

pe I V III I V V

time

Linear Reg. of Do M

-0.5

0

0.5

1

1.5

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Line

ar s

lope

Do M Prelude - Linear versus parabolic slopes

I V I V V:IV

Intersecting Linear and Parabolic Reg. of Do M

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Par

abol

ic s

lope

No. of pattern (time)

-0.5

0

0.5

1

1.5

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31

Line

ar s

lope

Do M Prelude - Linear versus parabolic slopes

I V I V V:IV

Slopes of Re M

Linear Reg. of Re M

Intersecting Linear and Parabolic Reg. of Re M

-0.05

0

0.05

0.1

0.15

0.2

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57

No. of pattern

-1.5

-1

-0.5

0

0.5

1

1.5

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57


De

gre

e 1

Slo

pe

De

gre

e 2

Slo

pe

time

Summary

1 Revealing rules of organization in Bach’s ’even’preludes using musical and mathematical tools


Summary


2 Studying the ’New Variable’: small, hidden, meaningfuldeviations of Natural Schemata that shed new lighton the mutual relations between Natural and LearnedSchemata and on Bach’s style.


Summary



3 Defining the natural characteristics that support thetonal centers: 1. (4,4,4,3) pattern. 2. breaking asequence. 3. inverting a pitch curve. 4. inner organpoints.


Summary



3 Defining the natural characteristics that support thetonal centers: 1. (4,4,4,3) pattern. 2. breaking asequence. 3. inverting a pitch curve. 4. inner organpoints.

4 Constructing a generic model of an ’even’prelude byBach.


Random Matrix

randomCorrelation coefficients

5

10

15

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25

30

35

40

45

505 10 15 20 25 30 35 40 45 50

Do m Matrix of Correlation Coefficients SlopesPrelude 2 Long

Correlation coefficients

5

10

15

20

5 10 15 20

Do m Matrix of Parabolic Reg. CoefficientsPrelude 2 Long

Regression slopes of degree 2

5

10

15

20

5 10 15 20

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