View
217
Download
0
Category
Preview:
Citation preview
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
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
’even’ curves of pitch - patterns, up to smalldeviations
First stage
Selecting Preludes from the WTC-I by Bachwith common characteristics
’even’ duration
’even’ curves of pitch - patterns, up to smalldeviations
similar structure
The Generic Model of an Even Prelude by Bach
OpeningSection
SequenceOrganPoint
Cadence
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
The Generic Model of an Even Prelude by Bach
OpeningSection
SequenceOrganPoint
Cadence
1 I,II,V,I; or I, IV, VII, I;
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
The Generic Model of an Even Prelude by Bach
OpeningSection
SequenceOrganPoint
Cadence
1 I,II,V,I; or I, IV, VII, I;
2 Bach is using one of his famous Figured Bass Formulase.g. (second, sext), 2,6,2,6,...
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
The Generic Model of an Even Prelude by Bach
OpeningSection
SequenceOrganPoint
Cadence
1 I,II,V,I; or I, IV, VII, I;
2 Bach is using one of his famous Figured Bass Formulase.g. (second, sext), 2,6,2,6,...
4 The Cadence - ’Bach the Believer’
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
The Generic Model of an Even Prelude by Bach
OpeningSection
SequenceOrganPoint
Cadence
1 I,II,V,I; or I, IV, VII, I;
2 Bach is using one of his famous Figured Bass Formulase.g. (second, sext), 2,6,2,6,...
3 The Organ Point - appears usually in the second half ofthe prelude
4 The Cadence - ’Bach the Believer’
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
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
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
Basic Musical Units
1 Nuclei
2 Patterns
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
Basic Musical Units
1 Nuclei
2 Patterns
3 Groups of patterns (higher levels)
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
Convex and Concave curves of Nuclei and Patterns
1 First pattern (repeated), of Do minor prelude (no. 2)
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)
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)
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)
Straight-Lines Curves in Different Directions
1 First pattern of Do M prelude (no. 1)
2 First pattern of Re M prelude (no. 5)
3 First pattern of Re minor prelude (no. 6)
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.
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
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
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
Mathematical Tools
1 Peak notes and low notes (of the upper voice)
2 Median - ’center of gravity’ or an ’inner organ point’
Mathematical Tools
1 Peak notes and low notes (of the upper voice)
2 Median - ’center of gravity’ or an ’inner organ point’
3 Linear Regression
Mathematical Tools
1 Peak notes and low notes (of the upper voice)
2 Median - ’center of gravity’ or an ’inner organ point’
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 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
I V III II I VI IV I V
� �B B
time
Mathematical Tools - cont.
Statistical Averages
Median - ’center of gravity’ or an ’inner organ point’
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
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
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
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
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
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
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
I V III II I VI IV I V
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
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
Summary
1 Revealing rules of organization in Bach’s ’even’preludes using musical and mathematical tools
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.
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
Summary
1 Revealing rules of organization in Bach’s ’even’preludes using musical and mathematical tools
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.
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.
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
Summary
1 Revealing rules of organization in Bach’s ’even’preludes using musical and mathematical tools
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.
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.
Dalia Cohen, Idith Segev Significant Occurrence in Even Musical Texture
Random Matrix
randomCorrelation coefficients
5
10
15
20
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
Recommended