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Copyright by
Ayleen Hilt Burns
1979 AN EXPERIMENTAL INVESTIGATION OF MELODIC CONTOUR .
RECOGNITION IN BEETHOVEN THEME AND VARIATIONS
DISSERTATION
Presented in Partial Fulfillment of the Requirements for
the Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
Ayleen Hilt Burns, B.S., M.A,
******
The Ohio State University
1979
Reading Committee: Approved By
Henry L. Cady
A. Peter Costanza
Dean H. Owen School off Music William Poland \
Co-Adviser School of Music ACKNOWLEDGMENTS
I want to thank Dr. Henry Cady, my advisor, for the knowledge,
ideas, and guidance which he contributed to this study. His concern
and encouragement were instrumental in helping me complete this
dissertation which spanned my residencies in five states, three
teaching positions, and more years than were intended when it was
begun. I also want to thank him for making it possible for me to
conduct the experiment at the University of Delaware. The students,
faculty, and Plato Project staff were helpful in every way.
Dr, Dean Owen has my sincere thanks for giving generously of his
expertise and time which enabled me to do an interdisciplinary project
between psychology and music. Also, thank you to Dr. William Poland
for cultivating the "soil” from which grew the questions for this study
and to Dr. Peter Costan2a for his kindness and efficiency in directing
the completion of the dissertation project.
Jim Wilson of the Plato Project at the University of Delaware,
Dr. Frank Martin, Director, Statistical Center at the University of
Minnesota, and Julie Wild, a statistician at the University of
Minnesota, provided special assistance critical to the study. I want
to thank them for giving so ably and freely of their talents.
I want to thank my mother, Ayleen Hilt, for keeping things in a healthy perspective with her ever-present good humor and wit and for her constant support. Also, to Rosa Stolz, who provided insights and
ii encouragement at important junctures during this study, thank you for a special friendship.
Finally, my appreciation to my husband, Denver, for his constant encouragement and the endless hours which he gave unselfishly. His expertise and critical comments contributed significantly to this dissertation.
iii VITA
June 23, 1 9 4 1 . Born - Greeneville, Tennessee
1963 ...... B.S., Music Education, The Ohio State University, Columbus, Ohio
1964-1966 ...... Junior high school vocal music teacher, Bexley, Ohio
1967-1969 ...... Teaching Assistant, Class Piano, The Ohio State University
1969 ...... M.A., Piano Pedagogy, The Ohio State University
1969-1971 ...... Teaching Associate, Class Piano, The Ohio State University
1973-1974 ...... Instructor in Music Theory and Piano, Xavier University, New Orleans, Louisiana
1977-1978 , ...... Instructor in Music Theory, University of Minnesota, Minneapolis, Minnesota
1978 - present .... Instructor in Music (part-time faculty), Hamline University, St. Paul, Minnesota
PUBLICATIONS
1970. "Develop Musicianship through Improvisation." Clavier, IX, No. 4 (April, 1970), 49-46.
FIELDS OF STUDY
Major Field: Interdisciplinary— Music Education, Music Theory, and Psychology
Studies in Music Education: Professor Henry L. Cady
Studies in Music Theory and Psychology of Music: Professor William Poland
Studies in Psychology: Professor Dean H. Owen TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS...... ii
VITA ...... iv
LIST OF TABLES ...... vll
LIST OF FIGURES...... viii
LIST OF C H A R T S ...... xii
Chapter
I. INTRODUCTION...... 1
Problem...... 12 Objective...... 16 H y p o t h e s e s ...... 16
II. APPROACH TO THE PROBLEM...... 19
Operational Definitions of Terms...... 20 Varied Properties Selected from Beethoven ...... 26 Commentaryfor Justification of Hypotheses ...... 45
III. EXPERIMENT ...... 52
P r o t o t y p e s ...... 52 Variables...... 59 S u b j e c t s ...... 87 P r o c e d u r e s ...... 88
IV. RESULTS...... 98
V. DISCUSSION OF INTERACTIONS ...... 110
Interaction: Between Interpolation of Tones and Change in Order of Temporal Intervals ...... 116 Interaction: Between Octave Transposition of the First Part and Change in the Order of Temporal Intervals...... 122
v Chapter Page
Interactions: Among Octave Transposition of the First Part of the Tone Series, Octave Transposition of the Last Part of the Tone Series, and the Interpolation of T o n e s ...... 130 Interactions: Between Meter Change and the Interpolation of Tones ...... 143 Interactions: Among Meter Change, Interpolation of Tones, and Octave Transpositions of the Last Part of the Tone S e r i e s ...... 153
VI. SUMMARY AND CONCLUSIONS...... 166
Problem...... 166 P rocedures...... 169 Results...... 171 Discussion and Conclusions ...... 180 Suggestions for Further Research ...... 185
APPENDICES
A. List of Beethoven Theme and Variations Uses as a Basis for the Transformations...... 193
B. Notation of the 162 Transformations Used in the Experiment...... 195
C. Introductory Material Presented to Subjects ...... 223
BIBLIOGRAPHY ...... 24 3
MUSICAL SCORES ...... 244
ADDITIONAL SOURCES ...... 245
vi LIST OF TABLES
Table Page
1. Summary of four types of melodic variation in Op. 34 . . . 42
2. Summary of altered characteristics of prototypes ...... 83
3. Analysis of Variance of the differences in the mean percent correct recognitions resulting from the effects of the five v a r i a b l e s ...... 104
4. Data for the Analysis of Variance of differences in the mean percent correct recognitions among four groups of subjects with different amounts of eartraining experience...... 107
5. Analysis of Variance of differences in the mean percent correct recognitions among four groups of subjects with different amounts of eartraining experience...... 107
6 . Data for the Analysis of Variance of differences in the mean response times among four groups of subjects with different amounts of eartraining experience ...... 109
7. Analysis of Variance of differences in the mean response times among four groups of subjects with different amounts of eartraining experience ...... 109
vii LIST OF FIGURES
Figure Page
1. Octave designation of the pitches...... 21
2. Notation and symbols for indication of extent of time in the musical examples...... 22
3. Illustration of the independence of durations and temporal intervals ...... 23
4. Comparative lengths of consecutive temporal intervals in the Theme, Op. 3 4 ...... 23
5. The effect of changes in temporal intervals between pitches at the beginning and ending of changes in direction...... 25
6 . Octave registers in the Theme, Op. 3 4 ...... 28
7. Octave transposition in Variation one, Op. 3 4 ...... 29
8. Octave transposition in Variation two, Op. 3 4 ...... 29
9. Order of temporal intervals in Theme and Variation three, Op. 30, No. 1 ...... 32
10. Equal segments of time in the Theme, Op. 3 4 ...... 33
11. Subdivisions of equal segments in Variation two, Op. 34 . . 34
12. Subdivisions of equal segments in Variation Four, Op. 34 , 35
13. Sustained tones in the Theme, Op. 3 4 ...... 36
14. Interpolation of tones in Variation one, Op. 3 4 ...... 38
15. Interpolation of tones in Variation three, Op. 34 ...... 39
16. Directions of pitch intervals in the Theme, Variations one, and three, Op. 3 4 ...... 41
17. Notation of the Theme and Variations one, two, three, and four, Op. 3 4 ...... 44
viii Figure Page
18. Notation of Prototypes A and B ...... 52
19. Similarities in pitch and temporal characteristics of Prototypes A and B ...... 56
20. Differences in the order of the directions and sizes of pitch intervals between consecutive tones in Prototypes A and B ...... 58
21. Variable OF in conditions 1 and 2 with Variables OL, OR, LS, and I in unaltered s t a t e s ...... 63
22. Variable OL in conditions 1 and 2 with Variables OF, OR, LS, and I in unaltered states ...... 66
23. Variable OR in condition 1 with Variables OF, OL, LS, and I in unaltered s t a t e s ...... 69
24. Variable LS in conditions 1 and 2 with Variables OF, OL, OR, and I in unaltered s t a t e s ...... 72
25. Variable I in conditions 1 and 2 with Variables OF, OL, OR, and LS in unaltered states ...... 76
26. Variable I combined with Variable OR in its original c o n d i t i o n ...... 81
27. Variable I combined with Variable OP. in condition 1 . . . 82
28. Combinations of the five variables...... 85
29. Correspondence between the percent of correct recognitions and the number of altered conditions of the v a r i a b l e s ...... 102
30. Sample graph for illustration of interactions ...... 113
31. Graph of the interaction between interpolation of tones and change in the order of temporal intervals ......
32. Delineations of time in the original statement...... 118
33. Delineations of time when the order of temporal intervals is changed ...... 118
34. Delineations of time when tones are interpolated...... 119
ix Figure Page
35. Delineations of time when the order of temporal intervals is changed and tones are interpolated...... 220
36. Graph of the interactions between octave transposition of the first part and change in the order of temporal intervals...... 223
37. Division of the tone series by the repetition of temporal intervals in the original statement ...... 224
38. Division of the tone series when the octave of the first part of the tone series is raised...... 225
39. Division of the tone series when the octave of the first part of the tone series is l o w e r e d ...... 225
40. Division of the tone series by the repetition of temporal intervals when order of temporal intervals is c h a n g e d ...... 226
41. Division of the tone series when the octave of the first part is raised and the order of temporal intervals is c h a n g e d ...... 227
42. Division of the tone series when the octave of the first part is lowered and order of temporal intervals is changed ...... 127
43. Graphs of the interactions involving octave transpositions of the first and last parts and the interpolation of tones ...... 132
44. Change of simple interval in the original statement to a compound interval when the octave of the first part is raised...... 133
45. Change of the compound interval to a simple interval . . . 134
46. Change of the compound interval to a larger compound i n t e r v a l ...... 135
47. Change of a simple interval to a compound interval with the interpolation of t o n e s ...... 136
48. Change of a simple interval to a compound interval with the interpolation of t o n e s ...... 137
x Figure Page
49. Change of a compound interval to a simple interval with the interpolation of tones ...... 230
50. Change of a compound interval to a larger compound interval with the interpolation of tones .... 139
51. Graph of the interactions between meter change and the interpolation of t o n e s ...... 144
52. Total time in the original s t a t e m e n t ...... 143
53. Extension of total t i m e ...... 146
54. Number of tones in the total time when tones are i n t e r p o l a t e d ...... 147
55. Number of tones in the total time when tones are interpolated ...... 148
56. Graph of the interactions involving meter change, interpolation of tones, arjd octave transposition of the last part ...... 155
57. Simple intervals in the original statement ...... 156
58. Increase of intervals with a compound interval when tones are Interpolated ...... 156
59. Change to a compound Interval when the meter is changed to six-eight ...... 157
60. Change to a compound interval when the meter is changed to three-four ...... 158
61. Increase of pitch intervals when tones are interpolated ...... 159
62. Change to a compound interval when the meter is changed to six-eight and tones are interpolated ...... 160
63. Change to a compound interval when the meter is changed to three-four and tones are interpolated ...... 161
64. The effects of alterations of amount of time occupied by different pitch intervals In two different tone series on the identity of their contours ...... 189
xi LIST OF CHARTS
Chart Page
1. The Latin Square for Variable O R ...... 91
2. Order of variable combinations in the experiment ...... 92
xii CHAPTER I
INTRODUCTION
Musical scholars have long asserted that the varied restatement
of musical material is important as a device for maintaining a
listener's interest throughout a composition. In his discussion
about the theme and variation form, Berry (1966) said,
One of the most Interesting areas of analysis with respect to the theme with variations is the consideration of the over-all plan which gives a profile and cumulative effect to the total series. The composer may achieve this by gradual increase (and/or subsidence) of motion, of dynamic complexity, and of degrees of submergence of the most recognizable features of the theme. Control of such factors as the interrelationships among variations and theme is extremely important in the unity of the total composi tion. (p. 302)
In the above quotation no distinction is made between the organi zational effectiveness of a composition from the composer's point-of- view and the listener's point-of-view. The omission of this distinc tion is illustrative of an assumption that is implicit In much of the literature. The assumption is that the factors contributing to the compositional organization in the theme and variation form are the same ones that contribute to the listener's perceptual organiza tion. However, this assumption has never been experimentally tested.
The "theme" in the theme and variation form is generally conceived to be a musical passage that is a composite of elements
1 2
that can be restated In exact or varied form in subsequent passages
(Berry, 1966; Nelson, 1948}.
Melody is frequently cited as an element of the theme that is
retained in the variations. In discussions about the variation of
melody in the theme and variation form terms such as "melodic
outline," "melodic contour," and "melody-line" are often used (Berry,
1966; Nelson, 1962; Goetschius, 1915). These terms are introduced in
descriptions of melodic variation as different labels for a charac
teristic of melody that remains unaltered after other aspects of the
melody have undergone change in the theme and variation form (Berry,
1966; Nelson, 1962), Berry stated:
There are, of course, many other means of varying the theme melody. For instance, the line may be presented in altered rhythm or altered pitch succession,... or in augmentation or diminution, or it may be stated in mirror inversion....
..., a melodic variation in which broad original outlines are preserved but other basic changes take place— tempo, mode, meter,— may have the effect of new melody where in fact a profound transformation has occurred without abandonment of the bases of the original line.... Thematic transformation on the original melodic motives and profile is a very important resource, to be distinguished from ornamentation in which the original melody is simply adorned with figurations which subtracted, leave the original line essentially intact, (p. 315)
In the descriptions that are cited above the authors do not define or describe what characterizes the "melodic contour," "melodic outline," or "melody-line" of a theme. Also, the listener is not accounted for in these descriptions. In order to account for the listener in the definitions of these terms, the concept of "melodic outline," "melodic contour," and "melody-line" must be established experimentally as aurally perceivable.
In a series of psychological experiments, Dowling (1971 * 1972) and
Dowling and Fujitani (1971) experimentally demonstrated that a
characteristic which they defined as the "contour” of a melody was
aurally perceivable. Dowling and Fujitani defined this property as
"... the sequence of ups and downs in a melody regardless of Interval
size." (p. 524) It is assumed that the melodic characteristic defined
as "contour" by Dowling and Fujitani is the characteristic that was
asserted by Berry, Nelson, and Goetschius to be constant between the
theme statement and subsequent variations in some compositions in the
classical theme and variation form.
Dowling and Fujitani (1971) conducted two experiments. In the
first experiment they were interested in the degree to which the
specific pitches in a tone series contributed to the recognizability
of a melody.
Subjects for the experiment were 49 undergraduates enrolled in
an introductory psychology course. They had a mean of 2.17 years of
musical training.
The stimuli were a series of five tones selected randomly from
fourteen semitones with the range of f to g^ (seven semitones above
and below middle C). All the tones were of equal duration. In each
trial the subjects heard a new tone series as the standard against which a subsequent tone series was to be compared. Half of the sets
of comparison tone series were transposed (began on a different pitch
from the standard tone series) and the other half of the sets were untransposed (began on the same pitch as the standard tone series). The task of the subjects was to determine whether the comparison
tone series were similar or dissimilar to the standard. Subjects used
a four-category scale as a basis for responding: "Sure Same," "Same,"
"Different," "Sure Different." The four category scale was used in
order to determine a sensory operating characteristic for each subject.
The subjects were subdivided into three groups according to the type
of comparison that they were to make. The first group heard a standard
tone series, followed by either an identical comparison tone series
(same interval sizes, contour, and pitches when untransposed) or a random collection of tones. This group was instructed to respond
"same" when the comparison tone series was exactly the same as the standard. The second group heard the standard tone series followed by either an identical comparison tone series or a tone series like the standard in contour only. This group was also instructed to respond
"same" only if the comparison tone series was identical to the standard. The third group heard a standard followed by either a comparison tone series like the standard in contour only or a random collection of tones.
Dowling and Fujitani considered as correct responses those which chose the comparison tone series that had been designated "same" for each comparison set.
The results showed that when the comparison tone series: were untransposed, the number of correct recognitions was greater for those comparisons that were exactly like the standard (same intervals, contour — as well as same pitches) than for the comparisons that were like the standard in contour only. When the comparison tone series were transposed the results showed that the number of correct recogni
tions for the comparison tone series that were like the standard in
contour only was about the same number correct recognitions for the
comparison tone series that were an exact transposition of the standard
tone series. They concluded that without transposition, pitches
provided the basis for recognition and that with transposition, contour
provided the basis for recognitions.
In the second experiment five familiar melodies were transformed.
The purpose of this experiment was to determine whether the presence
of relative sizes of successive pairs of intervals in a melody
contributed to the recognizability of the contour.
The subjects in this experiment were 28 undergraduates enrolled
in an introductory psychology course with a mean of almost 2 years of
musical training.
The stimuli were the first two phrases of the following melodies:
"Auld Lang Syne"; "Twinkle, Twinkle Little Star"; "Good King Vlencelas"
"Yankee Doodle"; and "Oh Susanna." All the melodies were in the pitch
range of c 1 to c 1 and were performed at the same tempo. The durations
of the tones of the first two phrases of "Twinkle, Twinkle, Little
Star" were assigned to the tones of the first two phrases of the other
four melodies. Any tones in these melodies that did not fit the
sequence of tonal durations of "Twinkle, Twinkle, Little Star" were
eliminated. This resulted in alteration of some melodies more than
others because the sequence of tonal durations of the original versions
of some of these melodies resembled "Twinkle, Twinkle, Little Star" more than others. The sequence of durations of the tones in all the 6
transformed versions of the five melodies were also the same as
"Twinkle, Twinkle, Little Star."
The transformations that Dowling and Fujitani used in their study
changed the interval sizes in the melodies so that the distributions of the intervals in the transformed versions were in accordance with
those distributions they considered to be characteristic of folk tunes
in Western and non-Western cultures. They considered these distribu tions to be characteristic based on studies by Fuchs (1962) and
Merriam (1964).
There were three types of transformations: 1) transformations in which the sizes of the intervals were changed but the relative sizes of successive pairs of intervals and the contours of the original melodies were preserved; 2) transformations in which the sizes of the intervals were changed so that the relative sizes of successive pairs of intervals were also altered but the contours of the melodies were preserved; 3) transformations in which the sizes of the intervals were changed so that both the relative sizes of successive pairs of intervals and the contours of the melodies were altered. In these transformations the tones that corresponded with the first notes in each measure and a characteristic, which Dowling and Fujitani labeled as the "implicit harmony," remained the same as the original version of the melodies.
In the following description of the third type of transformation, they discussed their procedures for preserving the "implicit harmony" of the melodies.
... The distorted version (c) preserved only the first notes of each measure and the implicit harmony of the original. Substitutions were made for notes on the three remaining beats of each measure. The directions of intervals were selected at random, and then the note in the chord implicitly underlying the beat which was closest to the preceding note going in that direction was chosen, provided it was different from the corresponding note in the original. (Only tonic, dominant, and subdominant triads were used.) (p. 530)
The task of the subjects after each presentation of a transforma
tion of one of the five melodies was to choose from a list of the
titles of the melodies the title of the melody which was transformed.
Dowling and Fujitani found that recognitions of the melodies for the transformations in which the contours plus relative interval sizes of the melodies were preserved were slightly better than for the transformations in which only the contours of the melodies were preserved. Recognitions of the melodies were not appreciably better than chance for the transformations in which the first tones of each measure and the implicit harmony of the melodies were preserved.
Dowling (1971) sought to determine if a melody could be recognized when the intervals have been inverted. In this study the subjects were 14 undergraduates with a mean of about 2 1/2 years of musical training. The construction of the tone series that were used as the standards in the comparison sets, method of presentation of stimuli, differences among the comparison tone series, and subjects' task were the same as in the first experiment in the study by Dowling and Fujitani
(1971). In Dowling's experiment the comparison tone series always began on a different pitch from the standard tone series and were presented as inverted and noninverted forms of the standard.
Dowling found that the inversion of the comparison tone series reduced the number of correct responses in all comparison sets. Correct responses were least for both Inverted and noninverted conditions in the set of comparisons In which the subjects were to label as "same’1 those comparison tone series that were like the standard tone series in exact interval size, and "different" those comparison tone series that were like the standard in contour only. The presence of the exact interval sizes did not improve performance of the sub jects over the presence of the contour in the noninverted condition, and did not improve performance at all in the inverted condition. The results showed that contour recognition was adversely affected by inversion and interval recognition was not.
From the results in this study Dowling concluded that "... melodic contour ard the set of interval sizes in a melody are separable features or dimensions of the melodic pattern. Contour and interval size are handled in different and largely independent ways in cognitive processing." (p. 349) He also concluded that inversion is a perceivable compositional device because the number of correct responses for the inverted forms of the standard tone series was better than chance.
In another study Dowling (197 2) was interested in determining whether a series of tones could be recognized under the conditions of inversion, retrograde, and retrograde inversion. This study was also like the previous studies by Dowling (1971) and Dowling and Fujitani
(1971) in construction of the standard tone series and differences among the comparison tone series.
In this experiment the comparison tone series were presented as inverted, retrograded, and retrograde-inverted forms of the standard tone series. When the tone series in the three types of comparison
sets underwent the three kinds of transformation the following
comparison tone series were constructed:
1. Tone series which were inversions, retrogrades, and
retrograde-inversions of the standard tone series with the
exact interval sizes of standard tone series preserved.
2. Tone series which were inverted, retrograded, and
retrograde-inverted forms of the contour of the standard tone
series with the interval sizes different from the standard.
3. Tone series which were different from the standard in at
least one interval direction in each of the transforms of
the standard as well as different interval sizes from the
standard tone series.
These comparison tone series were presented at two different rates
of tones per second. The first presentation of these comparison tone
series was at a rate of five tones per second and the second presentation was at a rate of two tones per second.
All subjects heard all three types of comparison sets in which the comparison tone series were the inverted, retrograded, retrograde-inverted forms of the standard tone series. The task of half of the subjects was to determine if the transformed versions of the standard contained the same contour or an inverted, retrograded, or retrograde-inverted form of the contour of the standards, regardless of interval sizes.
The tasks were easier at the slower rate of presentation of the comparison tone series than at the faster rate. This was the case 10 for all sets except for the one in which the exact intervals of the standard tone series were to be recognized in the retrograde-inverted form. The number of correct responses were the same for both rates of presentation of this transformation of the standard tone series.
Dowling found that the preservation of exact intervals of the standard tone series in the comparison tone series under the three conditions of transformation did not make the tasks easier than when only the contours of the standard tone series were preserved in the comparison tone series. The tasks were performed under all three conditions of transformation with better than chance accuracy. The list of the transformations in ascending order of their effect on the increase in difficulty of the tasks is: inversion, retrograde, retrograde-inversion.
White (1960) did not actually use the term "contour" to label the property which he thought the subjects used to identify ten familiar melodies under certain conditions of transformation in his study.
However, it is inferred from the following statement that he was dealing with the property of contour in his experiment.
... the fact that simple transposition has virtually no effect on the ease with which a melody is recognized suggests that it is perhaps the sequence of intervals between adjacent notes in a melody which carries the information, (p. 101)
The ten melodies that White used were: "Bicycle Built for Two";
"Deck the Halls with Boughs of Holly"; "Home Sweet Home";"Jeannie with the Light Brown Hair"; "Londonderry Air"; "Old Folks at Home";
"On Top of Old Smoky"; "Red River Valley"; "Star-Spangled Banner"; and "Yankee Doodle." 11
There were four sessions In the experiment in which the subjects
heard two different lengths of the melodies played forward and backward.
In session one they heard the first twenty-four tones of each of the
ten melodies piayed forward and in session two they heard the first
twenty-four tones of the ten melodies played backward. In session three
they heard the first six tones of each of the ten melodies played forward and in session four they heard the first six tones of the melodies played backward.
There were twelve transformations. Eleven of the twelve trans formations altered the sizes and/or directions of the intervals. White used a set of mathematical formulae for establishing the direction and the number of semitones by which he reduced or increased the sizes of successive intervals in the melodies. White pointed out that these transformations would have different degrees of effect on different melodies depending upon the sizes of the successive intervals of the original versions of the melodies. In these transformations the durations of the tones were the same as in the original versions of the melodies. In the remaining transformations the intervals of the melodies were not changed but the durations of the tones were changed so . that they all were equal to a quarter-note value.
The task of the subjects after each presentation of a transforma tion was to choose from a list of the titles of the melodies the title of the melody that was transformed.
White found that the melodies were as easily identified from the first six tones as from the first twenty-four, when played either forward or backward. In the forward versions the transformations with 12
the least effect were those in which relative sizes of consecutive
intervals as well as the sequence of ups and downs were preserved. The most disruptive of the forward versions of the transformations were
those in which the pitches were changed so that all the tones in each of the melodies were the same pitch and the sequence of durations of the tones was the only characteristic by which the melodies could be
identified. Temporal reversal of the melodies made the identification of the melodies extremely difficult.
Problem
The assertions in the literature about the recognizability of melodic contour under conditions of variation such as those in the classical theme and variation form have never been experimentally investigated. For this reason, it is not known whether melodic contour is an aurally perceivable constant between the theme and these variations.
The discussions in the literature about variation of the theme melody are inadequate for providing orderly information that can be used for such an investigation. There are three problems that are common to all the discussions.
First, all predictions about the effect of various alterations are expressed in terms of how these alterations will affect the recognizability of the melody of the theme but in these same statements the concept of melodic contour is frequently interchanged with melody.
Berry (1966) cited several characteristics of a melody which when altered would seem to become a new melody. Furthermore, he stated 13
that in fact the "line" of the original melody may be maintained in the presence of these alterations (supra, p. 2). Earlier in his-discussion, he described two ways of changing the theme melody as the presentation of the line in altered rhythm or altered pitch succession. It is not clear from this series of statements by Berry what he thought the relationship is between melody and melodic contour. It is assumed in this study that contour is related to melody but that it is not one and the same as melody. Experimental data are needed to determine what alterations, which are asserted to affect the recognizability of melody, also have an effect on the recognizability of the melody's contour.
Second, the descriptions and explanations of variation of the theme melodies in the theme and variation form offer few indications of exactly which characteristics are changed and what the procedures are for alteration. One reason for this problem is that the meanings for some of the terms that are used for the altered characteristics of the theme melodies are ambiguous. Berry (1966) identified tempo, mode, meter, and rhythm; and Nelson (1962) identified these along with tone color and dynamics as characteristics of the theme melodies that can be . varied and yet leave the contours intact. Others also include these terms in their lists of alterations of theme melodies that are commonly found in the theme and variation form (Goetschius, 1915; Prout, 1895;
Colies, 1970; Stanford, 1912; Stein, 1962; Apel, 1970; and MacPherson,
1915). However, despite the usage of the same terms for the altered characteristics of the theme melodies, some of them apply to concepts that do not have definitions generally accepted among musical scholars. 1A
This is true, in particular, for those terms pertaining to the temporal characteristics of melodies. For example, "rhythm" is a term frequently encountered in descriptions of varied characteristics of melodies in the theme and variations form. However, there is a great deal of diversity among the authors as to what was meant by this term.
Berry (1966) stated that variation of the "rhythm" of a melody could be the result of lessening the differences in lengths of the durations of tones in a melody. Nelson (1962) described the "rhythmical recasting of a theme" as a change which resulted from alterations in
"meter"— such as a change from "outlining duple meter" to "outlining triple meter." Goetschius (1915), on the other hand, discussed changes that he labeled as "transformations of rhythm" which consisted of "shifting the tones in a measure [so] that accented and uncaccented points are exchanged or otherwise modified." (p. 7A)
Third, the musical elements that are altered are not described in terms that specify the physical conditions that characterize these elements. For example, in the discussions about variations of "rhythm" that were just cited, Nelson used terms such as "rhythmical recasting" and "outlining meters" and Goetschius described shifting tones in a
"measure" and exchanging "accented and unaccented" points as kinds of variation of musical elements. None of these terms specify what physical conditions would produce these variations in these musical elements. Berry did cite changing the differences in "long and short" durations in a melody as a type of rhythmic variation— which specified some physical conditions for the variation of "rhythm." But he did not specify any limits for these conditions so that they would result 15
In the alterations that he described.
Another aspect to the problem of the lack of orderly information
for the basis of experimental investigation is that differences among
variations within a composition and across different compositions with
respect to the preservation of characteristics of the theme is alluded
to but never explained. Berry (1966) implied that there is a range of
alterations that have a range of effects on the recognizability of
melody— from "profound transformation" to mere "ornamentation" (supra,
p. 2). In the following quotation Goetschius (1915) asserted an order
of occurrences of the variations with respect to the degree of
recognizability of the theme that is common in the cla 'cal theme and
variation form.
... It must be understood, however, that not every variation assumes broader proportions; the first few variations are always more directly related to the theme, and the impulse of freedom grows as the form advances... (p. 83)
Neither Berry nor Goetschius indicated in the statements cited
above, nor in any other part of their discussions, what characterizes
the differences in degree of effect among the alterations that they
were considering.
This study investigated the problem of the recognizability of melodic contour under conditions of variation in the classical theme
and variation form. As part of this problem this investigation sought
to establish experimentally the physical conditions for variations of
five musical elements and to compare the effects of these variations
singly and in combination on the recognizability of melodic contour. 16
Obj ectlve
The objective of this study was to determine experimentally the effects of the transformations that were derived from Beethoven's theme and variations on the listener's ability to recognize the contour of a series of tones.
To pursue this objective the following hypotheses were stated :
Hypotheses
1. When there is an increase in the number of alterations present at one time in a variation there will be a correspond ing increase in the difficulty of recognition of contour.
2. When the octave register of the first part of a tone series is transposed there will be an increase in the difficulty of recognition of the contour.
3. When the octave register of the last part of a tone series is transposed there will be an increase in the difficulty of recognition of the contour.
4. When the octave register of the first part of a tone series is transposed the increase in difficulty of recognition of the contour will be greater than when the octave register of the last part of a tone series is transposed.
5. When the octave register of both the first and last parts of a tone series are changed and the pitch interval size between the beginning and ending of the tone series is changed the increase in difficulty of recognition of the contour will be greater than when only the first part of the tone series is transposed.
6. When the octave register of part of a tone series is transposed there will be no difference between the directions of the transposition in the increase in the difficulty of recognition of the contour.
7. When the octave register of both the first and last parts of a tone series are transposed there will be an interaction between the directions of the octave transpositions.
a. When the direction of the octave transpositions of the first and last parts of a tone series results in a change of the pitch interval size between the beginning and ending of the tone series 17
there will be an increase in the difficulty of recognition of the contour that Is greater than the increase that would result from the summation of the effects of transposition of the octave of each part alone.
b. When the direction of the octave transpositions of the first and last parts of a tone series does not result in a change of the pitch interval size between the beginning and ending of the tone series there will be a decrease in the difficulty of recognition of the contour from when only the first or last part is transposed.
8. When the order of the temporal intervals that are delineated by the onsets of consecutive tones in a series is changed there will be an increase in the difficulty of recognition of the contour.
9. When the meter of a tone series is changed there will be no increase in difficulty of recognition of the contour.
10. When both the order of consecutive temporal intervals and the meter of the tone series are changed there will be an interac tion between these changes that will result in an increase in difficulty of recognition of the contour that is greater than the increase in difficulty in recognition that would result when only the order of the temporal intervals or only meter was changed.
11. When both the order of the temporal intervals and octave register of parts of a series of tones are changed there will be an increase in the difficulty of recognition of the contour that is greater than when the order of the temporal intervals and the meter of the tone series are changed.
12. When tones are interpolated in a tone series there will be an increase in the difficulty of recognition of the contour.
13. When tones are interpolated in a tone series the increase in difficulty in recognition of the contour will be greater for the variation in which there are more interpolated tones per durational unit.
14. When the interpolation of tones in the tone series is combined with each of the other alterations in the temporal and pitch characteristics of the tone series there will be an interaction between these changes that will result in an Increase in the difficulty of recognition of the contour that is greater than 18
the Increase that would result from the summation of the effects of the interpolation of tones and each of the altera tions in the temporal and pitch characteristics.
Justifications for these hypotheses follow in Chapter II:
Approach to the Problem. CHAPTER II
APPROACH TO THE PROBLEM
Sixteen Beethoven compositions in the theme and variation form were analyzed for the purpose of determining what characteristics of the theme melodies were varied and what procedures of alterations were used. These alterations were applied according to a set of transforma tion rules to two different tone series which were composed for use in the experimental portion of this study.
Colies (1970) considered, as indicated in the following statement, that Beethoven’s theme and variations were pivotal in the history of the technique of variation in the theme and variation form.
Beethoven’s work forms an era in the history of variation- making. In fundamentals Beethoven did not leave the line taken up by the composers of the sonata period, but he brought the old and new principles more to an equality than before and was also very much more daring in presenting his model in entirely new lights. The proportion of purely orna mental variation that follow the theme very closely are more conspicuous in the early part of his life than later; but even among such comparatively early examples ... there is a fertility of resource and imagination, and ... a daring inde pendence of style, which differ entirely from anything previously done in the same line. (p. 682)
Beethoven's variations were chosen as a basis for this study with the consideration that if Colles's assertion is correct then
Beethoven’s variations would provide a view of variation procedures that was broader in terms of historical perspective and other composers* work than just the techniques used by one composer. 19 20
Sixteen particular compositions were chosen because they were
among those that were cited as having some variations in which some
melodic characteristics of the theme were retained (Colles, 1970;
Prout, 1895; Tovey, 1935-1939, 1944). (See Appendix A for a list of
the compositions that were analyzed.)
The terms defined in the next section are used in the descriptions
of the alterations identified in the Beethoven compositions. The
vocabulary in musical and psychological discourse is replete with terms
for musical events that have definitions in which concepts about
subjective and objective phenomena are used interchangeably. It is
beyond the scope of this study to invent a new vocabulary in order to
have terms that are free of previous connotations in which objective
and subjective phenomena are confounded. The intent for the following
operational definitions is to establish terms as labels for phenomena
that are under consideration in this study. The loudness and timbre
of the tones in the series that are used in the experiment are constant throughout the transformations. Tor this reason, descriptions
of these properties of a tone are not included in the following operational definitions.
Operational Definitions of Terms
PITCH - a perceived periodic complex sound with a reference frequency of 440 c.p.s. for a^ in the equal-tempered scale.
The pitches that are used in this study are tuned to the equal- tempered tuning system. These pitches are labeled with the letter names that designate the pitches in one octave in the equal-tempered tuning system.
DURATION - the extent of time from the moment of onset to the moment of cessation of a pitch. 21
TONE - a musical event that articulates moments in time by the onset and cessation of one pitch.
OCTAVE REGISTER - the pitch level of a tone with respect to its occurrence in one of the octaves of the equal-tempered tuning system.
The term r,octave register" is used in order to make a distinction between the pitch level of a tone and the size of an interval which is labeled an "octave."
The following notation and symbols (Figure 1) illustrate the system and corresponding labels that will be used to indicate the octave designation of the pitches.
Fig. 1.— Notation and symbols for Octave designation of the pitches.
TEMPORAL INTERVAL - the extent of time from one moment to another moment of articulation in the passage of time.
The units of measurement of the extent of time in the musical passages are durational values that are represented by the note-values in the standard system of musical notation. These durational values are relative values only (e.g., the quarter note is twice as long as the eighth-note value, the eighth-note value is twice as long as the sixteenth-note value, etc.). The absolute values of these units of measurement are established by assigning one of them a fixed value in terms of clock-time.
The total time of a musical passage Is measured by progressively counting the designated durational unit from the first to the last event in the passage of time. The following notation and symbols (Figure 2) illustrate how the measurement of time in the musical examples will be indicated. The eighth note is used as the unit of measurement In all of the musical examples in this study. This is indicated by J*— at the place from which the ex tent of time in the musical example is measured. The number of eighth-note units contained in the examples is indicated in two 22
ways: 1) the number of eighth notes per measure is indicated by the upper line of numbers; 2) the number of eighth notes for the total time of the musical example is indicated by the lower line of numbers.
i—r*— _ — r - F f — — ff 1 — ~ A 3 { A 3 I A3 t A 3 -V JW a 3 s L n 9 f /«n a i}
Fig. 2.— Notation and symbols for indication of extent of time in the musical examples.
The temporal interval of one tone is the same length as the duration of that tone.
The temporal interval articulated by the onsets of two consecutive tones and the durations of the tones are independent only within certain limits of each other. In the variations the most common alteration of the temporal interval of the onsets of the two tones is the extension of the interval by the delay of the second tone. The temporal interval and durations of each of the two tones are independent when such an alteration occurs. In the variations the most common alteration of the durations of consecutive tones in which the temporal interval of the tones is not changed is the shortening of the durations of both tones. The temporal interval and the durations of the two tones are independent of each other when such an alteration occurs when the durations are shortened up to the moment of onset of each tone.
The following notation (Figure 3) illustrates how the durations and temporal interval of two consecutive tones can be changed independently of each other.
In Figure 3a the extent of time from the onset to the cessation of the pitch of each of the tones (i.e., the duration of each tone) is four eighth-note durational units long. In Figure 3b the duration of each of the tones is one eighth-note durational unit long. The extent of time from the onset of one tone to the onset of the other (i.e., the temporal interval delineated by the onsets of the two tones) is four eighth-note durational units long in both examples. 23
f* ' > )V f > ) V {* * * 3 * '»H K i ^ j y f ^ 7 | /■*£>*# 5111
<&
<£--d~ ?lJ ~ - H l w . . >,.,tTvriii )• I » ) 4 ft 1 I M m > -
Fig. 3.— Illustration of the independence of durations and temporal intervals.
In Figure 3c the temporal interval delineated by the onsets of the two tones is four eighth-note durational units long. In Figure 3d the temporal interval is six eighth-note durational units long. In both examples the duration of each of the tones is four eighth-note durational units long.
SERIES OF TONES - three or more tones that occur successively in the same voice.
The temporal intervals in a series of tones are described in relative terms as well as measured in durational units.
Figure 4 illustrates how the temporal Intervals delineated by the onsets of the tones in the melody of Op. 34 can be described in terms of the comparative lengths of consecutive intervals. When consecutive intervals are the same length the symbol "Sa" is used. When consecutive intervals are different lengths the longer interval is labeled with the symbol "I/* and the shorter interval is labeled with the symbol "S."
Fig. 4.— Comparative lengths of consecutive temporal intervals in the Theme, Op. 34. 24
CONTOUR - the succession of pitch intervals of a series of tones. This property has two characteristics: one characteristic is the succession of the direct ions of the pitch intervals; the other characteristic is the succession of sizes of the pitch intervals.
This definition is different from the Dowling and Fujitani (1971) definition in which the sizes of the successive intervals are not included as a characteristic of contour. In this study the size of the intervals in a series of tones is included because the sizes and directions of a series of tones are not always independent of each other. The size of pitch intervals of a series of three or more tones can be changed without changing the direction of the pitch intervals but the direction of the pitch intervals of three or more successive tones cannot be changed without changing the size of the successive intervals.
The succession of pitch interval directions is referred to in terms of "up," "down," and "same." "Up" will be symbolized by +,"down" will be symbolized by and "same" will be symbolized by o.
The concept of contour as it is generally used in musical discourse does not considet all pitch intervals in a series of tones of equal importance in determining similarities and differences among contours of melodies. For example, the two tone series— Tone Series A and Tone Series B that are illustrated in Figure 5a have the same highest and lowest pitches but are different from each other in their order of consecutive pitches and pitch intervals. However, when the two tone series occur with the temporal intervals as in Figure 5b most musicians would agree that the contours of the two tone series are identical. The temporal intervals between pitches at the beginning and ending of the changes in direction in the successions of pitch intervals (gl to g^; g” to b^; b* to g^) are the same so that the presence of the intervening pitch interval before b^ in Tone Series A and after b^ in Tone Series B does not seem very important to differentiating the two tone series, and thus, the Tone Series A and B might sound alike. The temporal intervals between the pitches at the beginning and ending of the changes in direction are different in Figure 5c. Also the intervening pitch intervals before and after b^ in the respective tone series occur through a larger proportion of the total time of the tone series in Figure 5c than in Figure 5b. It seems that the intervening pitch intervals are more important in differen tiating the contours of the tone series in Figure 5c than in Figure 5b, and thus, Tone Series A and B might sound different.
It is not known what the effects of time are in establishing which pitch intervals of a tone series are important to differentiating contours. For this reason, in this study, the consecutive inter vals of a tone series are accorded equal importance in differentiating the contours regardless of the temporal intervals in the tone series. 25
& .
+ -
b.
IIIIHK m i t'«rM W £ i *i s 61 9 4 - _ - 4- 4- - + +
c.
i... . _|f 7 p # -- 1»—------^ --- 9 --- #—— = < F ; r - r j ------A
J^I g. 2> *{ ? ie 1 8 & I A M s b n 9 4 - 4 4-
Fig. 5.— The effect of changes in temporal intervals between pitches at the beginning and ending of changes in direction. 26
TtELODY - an organized tone series that has at least three different pitches. Godwin (1972) discussed theoretical considerations for the concept of melody.
For this study melody is a feature of the theme that is repeated in exact or modified form in the variations.
THEME - a musical passage from which material is derived and repeated in exact or modified form in subsequent musical passages. For this study it is a musical passage that is designated by the composer as "theme." The melodic material in the highest pitch- line that occurred in the first period of the theme statements in the Beethoven compositions is referred to as the "theme melody.11
Varied Properties Selected from Beethoven
The variations of Op. 34 and Op. 30, Ho. 1, by Beethoven were used
as a basis for the transformation rules which prescribe the variables
and their range of variation in the experiment.
The purpose of the following descriptions is to specify the
physical conditions of the variations in the Beethoven compositions so
that they may be used for experimental investigation.
There are alterations of other characteristics of the themes and of the theme melodies that occur in the variations in Op. 34 and
Op. 30, No. 1 that are not under consideration in this study. For example, each variation in Op. 34 occurs in a different key from the one in which the theme originally occurred. Also, in both Op. 34 and Op. 30, No. 1 some of the variations contain a few harmonic changes and textural changes from the original theme statement. These alterations are omitted in the notated examples of the variations that follow. 27
There were three objectives for the analysis of melodic variation
in the Beethoven theme and variations:
1. Identify alterations of the theme melodies that were independent of each other and that could be Isolated from other kinds of changes between the theme statement and the variations.
2. Identify alterations of the theme melodies that can be described in physical terms.
3. Determine how the alterations of the theme melodies occurred in the compositions in terms of:
a. how different conditions of alteration of the same characteristic can be distinguished from each other.
b. what characteristics of the melody were altered in combination with alterations of other characteristics.
From among the kinds of melodic variation that occur in these
sixteen compositions, four were selected to be used in the experimental
portion of this study:
1. octave transposition of the pitches in the melody.
2. change in the order of the temporal intervals in the melody.
3. change in the length and subdivision of equal segments of time in the melody.
4. interpolation of new tones between the tones of the melody.
The tone series of the theme melodies in Op. 34 and Op. 30, No. 1
occur in the highest pitch-line in the theme statements and in the variations. In the notated examples of the variations that accompany
the following discussion, the notes that represent the tones of the
theme melody, or pitches that are substituted for the melody tones,
are circled and labeled with numbers above the notation. The tones are numbered progressively from the first tone to the last tone in
the theme. 28
Octave Transposition
In Op. 34 the octave register of the pitch of the first-tone in the theme melody is one octave register above the octave register of the pitch of the last tone. The pitch of the first tone is c^ and the pitch of the last tone is g1— tone no. 15. (This tone is considered as the last tone of the theme because tones nos. 16, 17, and 18 do not occur in all the variations} (Figure 6).
I X 3 H !> (> 7 f 9 it m tX & fij
) 3 U f <» 1 f 1
Fig. 6.— Octave registers in the Theme, Op. 34.
In Variation one the pitches of tones nos. 9-15 are transposed one octave higher than the octave in which they occurred in the theme statement (Figure 7). The octave transposition of these tones changes the difference in octave register between the pitch of tone no. 1 and tones nos. 9-15; the pitch of tone no. 15 is changed from one octave register below that of tone no. 1 to the same octave register of the pitch of tone no. 1. 29
l X 3 H S & ^ t<* a |t : j 4=
fJLj i t J j i; / ^ 3 l X
J*=l J3 1 S <• *1 9 9 to li /A. J3 /V /,>'
Fig. 7.— Octave transposition in Variation one, Op. 34.
In Variation two the pitches of the tones of the first part of the theme melody are transposed so that the pitches of tones nos. 1-9 are lowered two octaves (Figure 8). The octave transposition changes the difference in octave register between the pitches of the first and last tones; the pitch of tone no. 1 is changed from one octave register above the pitch of tone no. 18 to two octave registers below the octave register of the pitch of tone no. 18.
I A 3 5 h> IS 9 ton a « /V /*' / 8
I
#? to a m is it* n is n «^> ai aa as 3 5
Fig. 8.— Octave transposition in Variation two, Op. 34. 30
In the procedures for melodic ornamentation In the classical theme
and variatior form the durations of the melody tones are varied
independently of the temporal intervals that are delineated by the
onsets of the tones. This is accomplished by maintaining the temporal
intervals of the onsets of the tones of the original statement of the
melody while shortening the durations of the melody tones and
interpolating new tones in the interval between the melody tones.
The interpolation of new tones in the temporal intervals is one
of two ways in which sound is continued through the temporal intervals. The
other way that sound is continued through these temporal intervals is
by the sustaining of the durations of the tones of the melody through
the temporal intervals delineated by these tones. Therefore, the
durations of the melody tones are considered in this study as a
characteristic of continued sound in the temporal intervals that are
delineated by the onsets of the tones in the theme melody. And the
temporal intervals are considered as a characteristic of the tone
series that is independent of the durations of those tones. For this
reason, changing the order of temporal intervals, changing the length
and subdivision of temporal intervals, and changing how sound is
continued in the temporal intervals as they occur in Op. 34 and Op. 30,
No. 1 are treated separately in the following descriptions. These alterations are also independent of each other in the transformations
that are used in the experiment. 31
Change In the Order of the Temporal Intervals
The order of the temporal Intervals that are delineated.by the
onset of the consecutive tones in the theme melody of Op. 34 remain
unaltered throughout the variations. However, in Op. 30, No. 1 this
characteristic of the theme melody is varied. In the theme statement
of Op. 30, No. 1, there are three longer intervals in a succession of
eleven intervals. These longer temporal intervals occur at regular
intervals in the succession. In Variation three,there are four
longer temporal intervals that occur at regular intervals. The order
of the long and short intervals are changed so that the tones
succeeded by the longer temporal intervals in the theme melody are
succeeded by the shorter temporal intervals in the variation (except
for tone no. 3) and the tones succeeded by the shorter temporal
intervals in the theme statement are succeeded by the longer temporal
intervals In the variation (Figure 9).
Change in the Length and Subdivision of Equal Segments of Time
The order of the temporal intervals delineated by the onsets of
the tones in the theme melody of Op, 34 results in four longer temporal intervals occurring at regular Intervals (Figure 10).
These longer temporal intervals divide the total time of the theme melody of Op. 34 into four equal segments plus an eighth-note durational unit. (The four equal segments are bracketed with dashed lines in the notated examples of the Theme statement and Variations two and four.) Theme:
tfr i ~qf J* -1 -iif------*HV r i>CJ d Hr - f- M - Ji :*:
Analysis
ton«i: P 9 in 11 ' ( | t I _ ralatlva dtfr. Si Si S1 Sa I 5a 5a I batwsan tamper* I I . | I ln tarv ali 1 t 1 !l I U 2 3 't 5 t 7 6 1 2 3 *t 5 f> 7 P l) 2 3 ft 9 6 ? 6 * u P © 10 11 1? 13 t't 15 17 lfl 19 20 21 22 ?3 2ft0 2 6 27 2« 29 30 31 32
Variation three:
Analysis:
tonas: «310 ll
mlatlv* HKf. i Scl 1 5 bettfern t«-ipor»l I Intervals i >*! 1 1 1 < 1 h, 5 6 A I I 23*56? ^123(15 J»l '* © * 7 f- r - 9 10 11 12(g) 1ft 15 (6 17 1' 19 7 0 0 2? 23 21* 25 2 ( 2? 2^ ?g 30 j] ^2
Fig. 9.— Order of temporal intervals in Theme and Variation three, Op. 30, No. 1.
The circled numbers in the lower lines of the analyses indicate the moments of the onsets of the tones that are succeeded by the longer temporal intervals. 33
i "a. 3 i jc 7 * 1 fP fO- ft 14
Fig. 10.— Equal segments of time in the Theme, Op. 34.
The tones that are succeeded by the longer temporal intervals that occur at regular intervals are tones nos. 2, 5, 8 and 14. These tones occur at the beginning of the four equal segments. The fourth segment is articulated by the onset of tone no. 14 and the cessation of tone no. 18. The extent of time in these segments is four eighth- note durational units long.
These four equal segments are subdivided by intervening tones.
The intervening tones are tones nos. 3 and 4, 6, and 7, 9-13, and
15-18. In the first, second, and fourth equal segments,the intervening tones nos. 3, 6 and 15 subdivide the respective equal segments at the third eighth-note durational unit. The result is the subdivision of the time in these segments in half. The length of the subdivisions is two eighth—note durational units long.
In Variation two and Variation four the total time in which the tone series occurs is extended and the length of the four equal segments is increased proportionately. In both the variations, each of the four equal segments is extended from four eighth-note durational units to six eighth-note durational units long. The total time for the occurrence of the four equal segments is twenty-four eighth-note durational units long. 34
In Variation two, three of the four segments remain subdivided in
half by the same intervening tones, but the extent of the tipie of the
subdivisions is longer than in the theme statement (Figure 11).
j i. 5 H j s, 7 a 9 /•_// (a. a _i+ is it
Fig. 11.— Subdivisions of equal segments in Variation two, Op. 34.
In the first, second, and fourth equal segments tones nos. 3, 6 and
15 subdivide the time in the respective equal segments at the fourth eighth-note durational unit which results in the subdivision of these segments in half. In the third equal segment a series of ascending pitch interval directions occurs In the first half and a series of descending pitch interval directions occur in the second half. The change In direction between the two series of pitch intervals occurs at the fourth eighth-note durational unit in the third equal segment which results in the subdivision in half of that segment. The length of the subdivisions is three eighth-note durational units long.
In Variation four, three of the four equal segments are subdivided into thirds by the intervening tones, but the extent of time of the subdivisions is the same as in the theme statement (Figure 12). 35
i i j 4 S' i v r * ,* ,t n- si /* Ar/t/J/p
*- 4 *—
Fig. 12.— Subdivisions of equal segments in Variation four, Op. 34.
In the first equal interval the change in pitch (c^ to d^)of the tones with the equal durations occurs at the fifth eighth-note durational unit. In the second equal segment, tone no. 6 occurs at the third eighth-note durational unit. In the third equal segment there is a change in the direction of pitch intervals at the third eighth-note durational unit and a repetition of a pitch at the fifth eighth-note durational unit. These subdivide the first, second, and third equal segments into thirds. The length of the subdivisions is two eighth- note durational units long.
These alterations that have just been described are sometimes considered by musicians as alterations in "meter." From this point-of-view, the subdivisions of the equal segments into halves results in what would be labeled as "duple meter"; the subdivisions of the equal segments into thirds results in what would be labeled as "triple meter," The "meters" also are commonly labeled as
"two-four meter," "six-eight meter," and "three-four meter," for the Theme statement and Variations two and four, respectively, because of their time signatures. For clarity these conventional
terms will be used as labels for these changes in the hypotheses
statements, discussion of results, and conclusions in this study.
These changes will be referred to in terms of the "length and
subdivision of equal segments of time" and symbolized as LS in the
discussion about the experiment in order to provide precise descrip
tions of what was manipulated.
Interpolation of New Tones
In the theme statement of Op. 34 the tones in the melody are
sustained through the temporal intervals that are delineated by the
onsets of the tones in the series (Figure 13).
t t 'i ^ 5 ■ U 1 ? 9 jo n i3 /V , n tr | o +■ - +■ ■+* - - — — o . — y y | f ■■ f f i J —J - J r | t----1---
> . i A 34 S ^ 1 8 f 10 11 IX /S ts ft 11 * .i______I L ______j L______J 1_------1
Fig. 13.— Sustained tones in the Theme, Op. 34
The delineations of the time of the theme melody into four equal
segments have been described already. The total time of these four equal segments is divided into two halves that contain tones having similar temporal and pitch characteristics at the beginning of each of the halves. The changes that make these similarities apparent differentiate the halves. The pitches of tones nos. 2 and 3 are 37
repeated by tones nos. 8 and 10 at the same length of temporal
Interval. The series of pitch intervals of tones nos. 8, 9».10, 11,
12 and 13 includes more intervals, but is the same in the succession
of pitch interval directions as the series of pitch interval direc
tions of tones nos. 2, 3 and 4. Each of these halves also have
similarities in durational characteristics of the tones that occur in
the two equal segments into which the halves are subdivided. The
durations of the tones in the first equal segment are j n ; the
durations of the tones in the second segment are J J W . The
durations in the second segment are a modified repetition of those in
the first segment. The durations of the tones in the third equal
segment are n i m ; the durations of the fourth equal segment
are J J J J J, and are modified repetitions of the third segment.
(The four equal segments are bracketed by dashed lines and the two
halves, into which the total time of the four equal segments is divided, are bracketed with solid lines in the notated example of
the theme statement.)
Two alterations of the continuation of sound through these
intervals occur in the variations. In Variations one and three tones are interpolated between the melody tones so that the temporal
intervals and pitch intervals of the tone series in the melody are divided into smaller parts.
In Variation one some of the intervals between the melody tones are filled with new tones so that the majority of the melody tones and the interpolated tones have the same durations (Figure 14). 33
f, - . - , , . , 3 44 H 5 fc *1 8 ^ *" U h f e ft A + “■ + ♦—*+-»++++-^_*..1--- 0 ++WtV*4++._++..+*.,+.+.c+ _ ;+----- +44+-f t t i-V-~: 11 iu?~r~.~T~.^ ~~~.~; 1 1 . j
Fig. 14.— Interpolation of tones in Variation one, Op. 34.
There is a predominence of four tones per eighth-note durational unit.
The same pitch interval direction begins each succession of pitch intervals that occurs at the moments that correspond to the beginning of the first three of the four equal segments in the theme statement. The pitch interval directions occur as follows: at the second and third eighth-note durational units are + -; at the sixth and seventh eighth-note durational units are ++++-+-; at the tenth and eleventh eighth-note durational units are +++++++.
Each of these successions of pitch intervals begin with the direction of "up." Also, the tones that occur at the moments corresponding to the beginnings of the first, third, and fourth equal segments all have durations and succeeding temporal intervals that are longer than those of the preceding and succeeding tones. (The pitch interval directions are labeled below the notation and the segments corresponding to the four equal segments in the theme statement are bracketed with dashed lines.)
The third equal segment in Variation one contains tones that have temporal and pitch characteristics that are similar to the temporal 39
and pitch characteristics of the tones that occur in the first equal
segment. These begin at the moments that correspond to the beginning
of the second half of the total time of the four equal segments in the
theme melody. The third equal segment is similar to the first equal
segment in the repetition of the pitch of melody tone no. 2 with the
pitch of melody tones no. 8. Also, the succession of pitch interval
directions that occur at the third and fourth eighth-note durational
units is repeated four times beginning at the twelfth eighth-note
durational unit. (The time in Variation one that corresponds to the
two halves of the total time of the four equal segments of the theme
melody is bracketed with solid lines in the notated example.)
In Variation three tones are interpolated so that all tones have
the same durations throughout the statement of the theme melody.
There are two tones per eighth-note durational unit (Figure 15).
' 1 3 ^ 5 <> i f 9 (0 ii u ij /v n ;g
- + - - *■ + + + + ,u_____ i \---- ______i, i_ .
Fig. 15.— Interpolation of tones in Variation three, Op. 34.
Successions of pitch interval directions and their repetitions—
in inverted form— occur in the time that corresponds to the beginnings of the four equal segments in the theme statement. The succession of pitch interval directions that occurs at the second, 40
third, fourth, and fifth eighth-note durational units is repeated in
inverted form at the sixth, seventh, eighth, and ninth eighth-note -
durational units. The succession of pitch interval directions that
occurs at the tenth, eleventh, twelfth, and thirteenth eighth-note durational units is repeated in inverted form at the fourteenth,
fifteenth, sixteenth, and seventeenth eighth-note durational units.
(The pitch interval directions are labeled below the notation. Also,
the segments in Variation three that correspond to the four equal
segments in the theme statement are bracketed with dashed lines.)
The first succession of pitch interval directions and its repetition in inverted form occur in the time corresponding to the first two of the four equal segments in the theme. The second succession that is repeated in inverted form occurs in the time corresponding to the second two of the four equal segments in the theme. The two repetitions differentiate the two successions that occur in the time corresponding to the two halves of the total time of the four equal segments in the theme. (The time in Variation three which corresponds to the two halves of the total time of the four equal segments in the theme are bracketed with solid lines in the notated Figure 15.)
Some of the directions of the pitch intervals in the theme melody are maintained in Variations one and three(Figure 16). The pitch interval directions between melody tones of the theme are duplicated by the pitch interval directions between the last interpolated tone and the succeeding melody tone (or the substituted pitch for that 41
I f a. i a i f I x & 4 Jki t j < l 5 £ 1 M /»II JiL f3 IV i s /i> f7 «--!------.„i------j ,------,,
' A 3 h _£ _4 « /i ;a- (i W tj' /u n
13 *f J fc 7 «
< 2 3 4 5 4 *7 t 1 i£> a i2 /j /v i* a* n &
i _____ I
Fig. 16.— Directions of Pitch Intervals in Theme, Variations one, and three, Op. 34.
tone) in the variations. For example, in Variation one the last
interpolated tone between the melody tones nos. 1 and 2 is c . (It
is marked by an in the example.) The pitch interval direction
between this tone and melody tone no. 2 is o, or no change. This is
the same as the pitch interval direction between melody tones nos. 1 and 2 in the theme statement.
Variations one and three differ in the number of pitch interval directions of the theme melody that are duplicated. There are fourteen 42 that are duplicated in Variation one, and eleven that are duplicated in Variation three. (The pitch interval directions between the last interpolated tone and each succeeding melody tone in Variations one and three are labeled above the notation in the examples. The pitch interval directions in the variations that are different from the corresponding pitch intervals in the theme melody are circled.)
The range of the pitches of the tones that are interpolated between the melody tones is the same in Variations one and three
(if the octave transposition in Variation one is eliminated.) In these two variations the pitches of the tones interpolated between the melody tones are generally an interval of a second or third above or below the pitch of the preceding melody tone. The one exception is the group of tones interpolated between the melody tones nos. 5, 6 and
7. Some of the pitches of the interpolated tones between these tones are an interval of a fifth above and an interval of a fourth below the melody tones which precede the interpolated tones.
In Table 1 below is a summary of the four types of melodic variation in differing conditions that occur singly and in combination in each of the variations in Op. 34. (See Figure 17, for a notated example of the theme and all four variations in Op. 34.) TABLE 1
SUMMARY OF FOUR TYPES OF MELODIC VARIATION IN OP. 34
Characteristics Octave Order of Length & Interpolation Total number of transposition temporal subdivision of tones altered intervals of equal characteristics segments(meter) in each Conditions First Last Duple triple One tone per Two tones variation part part meter meter durational per dura unit tional unit
Variations 1 X X two
2 X X X three
3 X one
4 X two X 44
Theme
a Var. 1
f i n
Var. 2
Var. 3
Var. 4
i[ p i
Fig. 17.— Notation of the Theme and Variations one, two, three, and four, Op. 34. 45
Commentary for Justification of Hypotheses
There seems to he an implicit assumption among musicians that the
more differences between the theme statement and a variation the more
difficult it will be for the listener to recognize the theme in the variation. This assumption seems to exist regardless of what elements
of the theme have been varied and of how the elements are varied. If
this assumption is true, then Variation two in Op. 34, having the
greatest number of alterations, would result in the greatest difficulty of recognition of any elements of the theme— such as contour— from among the four variations.
There is no distinct line of musical thought about the effects of alterations in the characteristics just described or other characteristics of melody on the recognizability of it and/or its contour- However, some predictions are made based on inferences developed from the literature in which other musical topics are discussed.
Deutsch (1972) found that subjects had a great deal of difficulty
In recognizing the tune of "Yankee Doodle" when each of the pitches of the tune occurred randomly in one of three octave registers. When the entire tune occurred as a whole in each of these three octaves recognitions were one hundred percent correct. The author concluded that "... tune recognition takes place along a channel which is independent from that which gives rise to octave generalization." (p. 41)
There are a number of differences besides just change in the register of the pitches between the conditions in which the pitches occurred randomly in the three different octaves and the conditions in which the entire tune occurred in each of the three octaves. These
differences suggest that Deutsch's conclusion was erroneous that
octave transposition was the only reason that the recognitions were
difficult in her experiment. In the conditions in which the isolated
pitches were transposed the alteration of the tune included changes in
the directional relationships of the consecutive pitch intervals,
changes in the sizes of consecutive pitch intervals, and changes in the
octave register of the whole tune. In the conditions in which the
entire tune was transposed to a different octave, the change in octave register of the tune was the only alteration while the directional
relationships and sizes of the pitch intervals of consecutive tones were maintained. The one hundred percent correct identifications, when the entire tune was transposed, would suggest that octave generaliza tion does occur with tune recognition as long as the size and direction of pitch intervals between consecutive tones remains unaltered.
When the octave register of parts of the tone series is transposed, as in the variations in Op. 34, the size of the pitch interval between the beginning and ending tones and between the tones at the point of the octave transposition are changed. It is possible that the octave registers of the beginning and ending pitches of the tone series form a frame of reference as a pitch interval size within which the directions and sizes of the contour of a series of tones are heard.
If this is true, then changing the octave register of part of a tone series so that there are changes in the pitch interval size between the first and last parts could disrupt the frame of reference and increase the difficulty of recognition of the contour. 47
Deutsch (1972) found that when the entire tune was transposed the number of correct identifications was the same regardless of* the direction of the octave transposition. Consideration of Deutsch's study and the assumption that a change in pitch interval size between the beginning and ending of a series of tones has an effect on the recognition of the contour leads to the following assertions:
1. It seems that when only one part is transposed the disruption of the frame of reference would be greater when the trans position occurs at the beginning— and thus, difficulty of recognition of contour would be greater when the first part is transposed than when the last part is transposed.
2. The direction of the octave transposition should have no effect on the recognition of contour when only part of the tone series is transposed— since the size of the interval at the place of transposition would be changed regardless of the direction of transposition.
3. When both the first and last parts are transposed there will be an interaction between the directions of the octave transpositions depending upon whether the pitch interval size between the first and last parts is changed.
a. When the octave transpositions of the first and last parts of the tone series result in a change in the pitch interval size between the beginning and ending, the interval size will be greater than when only one part is transposed. Thus, the increase in the difficulty of recognition should be greater than the increase in difficulty of recognition when the octave register of only the first part is changed.
b. When the octave transpositions of the first and last parts of the tone series result in no change in the pitch interval size between the first and last parts there will be a decrease in the difficulty of recognition of the contour from when only one part is transposed.
Eschman (1968) discussed the effect of temporal changes on the recognizability of a melody. It can be inferred from the following statement that changes in the order of the temporal intervals of the tones in a melody (as changes in the "bodily structure" of a melody) 48
were compared to changes in the subdivisions of equal segments of the
time in a melody (as changes in the "pattern within the measure ...
often expressed in historical types of dances....")
To be sure, there had been rhythmic-variations including some diminutions and augmentations in earlier sets, but these were usually changes in the pattern within the measure, in the lilt of the motive, often expressed in historical types of dances, rather than in the larger aspects of the rhythms. When augmented or diminished, if heard through the larger or smaller end of a musical "telescope," the figure was recognized as the same object. The theme, transformed as a waltz, mazurka, march, or gavot, was recognized as the same musical personality,,.. It is much more difficult to recognize a variation as such, when there is a radical change in the bodily structure. We associate such bodily changes with the term "Development" rather than Variation, since the latter has in the past been associated more frequently with superficial "Costuming." (pp. 112, 113)
Eschman's language is imprecise but his is the only statement
that appears to address two implicit assumptions in the literature
about the effects of altering these temporal characteristics on the
recognizability of a melody. One assumption is: the change in the
differences in lengths of temporal intervals of consecutive tones,
such as occurs when the order of the temporal intervals of tones in a melody are changed (as a change in the "bodily structure" of a series
of tones), increases the difficulty in recognition of the melody. The
second assumption is: changes in the subdivisions of the equal segments of time in a tone series of a melody— transformed as a waltz (subdivi
sion in thirds) or a march (subdivision in halves)— would not increase
the difficulty of recognition of a melody.
Nelson (1948, p. 100) classified an alteration of a temporal characteristic of melody, which he labeled as a change in "meter," as 49
one of the complex changes In a melody that has "impressive transform
ing" effects. When the examples that Nelson used to Illustrate this alteration are analyzed it Is found that the alteration which he
labeled as a change in "meter" is the combination of a change in the order of the temporal intervals and a change in the subdivisions of the equal segments of time in a tone series in a melody. Based on other examples used by Nelson (p. 46), it is inferred that Nelson considered the difficulty of recognition to increase when these two changes were made rather than only one.
There appears to be a contradiction between Eschman and Nelson with regard to their views on the effects of temporal changes on the recognizability of a melody. The contradiction is based on whether or not a change in the subdivision of equal segments of time in a melody increases the difficulty in recognizing that melody.
It is also possible that there is no contradiction between
Eschman and Nelson. Perhaps a change in the subdivisions of the equal segments of time in a melody results in an increase in the difficulty in recognizing that melody only when such a change is combined with a change in the order of the consecutive temporal intervals. If this is the case, it is possible to postulate an interaction between these changes that would result in a greater increase in difficulty of recognition of a melody than the increase in difficulty in recognition that would result from only a change in the order of temporal intervals. 50
A question that arises about this assertion is would these
alterations have the same effect on the recognizability of contour?
It seems that the order of the temporal intervals of the tones in
a series results in a temporal frame of reference against which the
successive occurrences of the pitch interval sizes and directions of a
tone series can be heard. If this is the case, it seems that changes
in two different characteristics that resulted in disruption of a
pitch frame of reference and a temporal frame of reference— such as
change in the order of temporal intervals and octave transposition—
•would make recognition of the contour of a melody more difficult than
two changes In the same characteristic that resulted in disruption in only one frame of reference— such as change in order of temporal
intervals and changes in meter.
Ortmann (1934) sought to identify factors affecting the performance of students in taking melodic dictation. He analyzed melodies and the error patterns made by the students when they notated the melodies. He found that the students made more errors on melodies that had more tones occurring during the same amount of time. Taking melodic dictation involves a number of skills. Therefore, the number of tones In the melody could affect the student's ability to do the task correctly for a number of reasons. One reason could be perceptual. If this is the case, then the interpolation of tones in the tone series of a melody— such as in Variations two and four in Op. 34— could increase the difficulty in correctly hearing the melody. Thus, difficulty of recognition of a melody could be increased when tones are interpolated. 51
If this were the case, then difficulty of recognition of melody would be greater for the variation in which there were -more interpolated
tones.
The discussion of the effect of all of the above alterations has been in terms of the recognizability of melody. A question that arises is would these alterations have the same effect on the recognizability of contour that is predicted for the recognizability of melody?
Preliminary investigation indicated that interactions may occur when each of the alterations in the pitch and temporal characteristics is combined with the interpolation of tones. When the interpolation of tones is combined with the alterations in the pitch and/or temporal characteristics, it appears that the interaction results in an increase in the difficulty of recognition of the contour that is greater than the increase that would result from the summation of the effects of the combination of the interpolation of tones and any one of these alterations. CHAPTER III
EXPERIMENT
The experiment was designed to produce comparisons among the
effects of five variables— two kinds of octave transposition, two
kinds of changes in the temporal intervals, and the interpolation of
of tones, singly and in all combinations, on the subjects' ability
to recognize the contours of two differing tone series.
The following section describes the two tone series, the variables, the subjects, the procedures for the experiment, and the
collection and analysis of the data.
Prototypes
The two tone series, discussed below as Prototypes A and B, were constructed so that they represent the Beethoven theme melodies in Op. 30, No. 1, and Op. 34 but do not duplicate any specific series of tones in these melodies (Figure 18).
PROTOTYPE A w . |# Q — p —- r v--^
m m **-4— I p — - f -- m -- - P — i- PROTOTYPE B -* — i— 1-
Fig. 18.— Notation of Prototypes A and B.
52 53
Prototypes A and B consist of a series of eight tones, each with ? one of four different pitches that encompass the octave g 1 to g . The
pitches occur at variable temporal intervals that are filled by the
sustained pitches of the tones. The total time in which a prototype occurs is sixteen eighth-note durational units long. The absolute value of the eighth-note durational unit in this study is equal to one-half second, resulting in a total absolute value of eight seconds for the total time of each of the prototypes. In musical terms, the tempo of the prototypes is J = 60 M.M. Prototypes A and B differ from one another only in the order of the sizes and directions of the pitch intervals between consecutive tones. The characteristics of the prototypes are discussed below in terms of their similarities and differences.
Similarities (Figure 19)
A. Pitch Characteristics:
1. The pitches that occur in the prototypes are the 1st, 3rd, and 5th scale degrees in G major.
2. No pitches are consecutively repeated.
3. No series of pitches within the eight is repeated.
4. In the first part of the prototypes there are two octave registers represented by the lowest and highest pitches (gl, g^); similarly, in the last part of the prototypes there are two octave registers represented by the highest and lowest pitches (bl, g^).
a. The octave register of the pitch of the first tone is g1.
b. The octave register of the pitch of the last tone is g^. 54
B. Temporal Characteristics:
1. The succession of variable lengths of the temporal intervals that are delineated by the onsets of the tones in the series in the prototypes is as follows: tones: 1 2 3 4 5 6 78 1 ' i lit II relative differences 1 I L '| SS| L '* S *Sa*1 Sa L ! SS| ' L in temporal intervals: i i 1 l i t i i l i l l I I 234123412341234
>-i 3 4 5 6 7 8 9 10 11 12 13 14 15 16
a. The order of the temporal intervals is such that the four tones in each of the prototypes that are succeeded by the longer temporal intervals are: tones nos. 1,3, 6 and 8.
b. The four tones that are succeeded by the shorter temporal intervals are: tones nos. 2, 4, 5 and 7.
c. The longest temporal interval between the prototype tones is equal to one-fourth (four eighth-note durational units) the total time of the prototypes. (The propor tional value of the longest temporal interval— in terms of the total time of the prototype— "remains constant when the length and subdivision of the intervals are changed IVLS]; however, the actual value— number of eighth-note durational units— is changed in the presence of VLS.)
d. The shortest temporal interval between the prototype tones is equal to one eighth-note durational unit. (one sixteenth of the total time of the prototypes.) (The actual value of the shortest temporal interval is constant in the presence of VLS but the proportional value is changed in the presence of VLS.)
2. The four tones that are succeeded by the longer temporal intervals occur at regular intervals and divide the time into four equal segments. (These four equal segments are bracketed with dashed lines in the notated illustrations of the two prototypes in Figure 19.)
a. The extent of time of the first three of the four equal segments is defined by the onsets of the four tones that 55
are succeeded by the longer temporal intervals. The fourth equal segment is the extent of time that is defined by the onset and cessation of tone no. 8.
b. The extent of time in each of these equal segments is four eighth-note durational units long.
c. The total time of the statement of the prototypes is sixteen eighth-note durational units long.
3. The intervening tones (nos. 2, A, 5, 7) between the tones (nos. 1, 3, 6, 8) which occur at the beginning of each equal segment subdivide the time of the four equal segments.
a. In two of the equal segments the intervening tones subdivide the time unequally. In the first and third equal segments the intervening tones occur at the fourth eighth-note durational unit in the segment.
b. In the second equal segment the intervening tone (no. A) occurs at the third eighth-note durational unit of the interval and subdivides that temporal interval in half.
c. The length of the subdivision is two eighth-note durational units long.
A. A succession of temporal intervals that occur in the first part of the prototypes is repeated in the last part.
a. The temporal intervals by which tones nos. 6, 7 and 8 are separated are the same lengths as the temporal intervals by which tones nos. 1, 2 and 3 are separated.
b. The repetition of these temporal intervals of tones nos. 1, 2 and 3 by tones nos. 6, 7 and 8 divide the total time of the four equal segments into two parts which begin with the same temporal intervals. (The two parts are bracketed by solid lines in the notated illustrations of the prototypes in Figure 17.)
5. Sound is continued through the total time of the statement of the prototypes.
a. The pitches of the tones of the prototypes are sustained through the temporal intervals delineated by the onsets of the tones in the prototypes. 56
b. The length of the sustained pitch of tone no. 8 is equal to the time of the preceding three equal segments.
c. The total time of the statements of the prototypes is filled with unequal durations.
m _ 0 PROTOTYPE A f "H ) 1J — f -* — f r - =?Tt = t . - * . t A 3 t A ^ *1 1 A 3 -V A 3V S' 4 1 f «« *3. rt Ht/S" J4
PROTOTYPE B J £ t A 3" ^ -7
Fig. 19.— Similarities in pitch and temporal characteristics of Prototypes A and B.
Differences (Figure 20)
Prototypes A and B differ from one another in the order of the directions and sizes of the pitch intervals between consecutive tones, (i.e., they differ in their contours). Specifically, the pitches of tones nos. 3 and 4 and tones nos. 6 and 7 are in reverse order in the two tone series.
A. The succession of pitches of each prototype is:
Prototype A, succession of pitches:
Tones: 1 2 3 4 5 6 7 8
g1 g2 d2 g2 _bl d2 g2
Prototype B, succession of pitches:
Tones: 12345678
g1 g2 _b-*- <32 g2 d 2 _b^ g2
(The names of the pitches that are in different order in the respective prototypes are underlined.) 57
B. The succession of pitch interval directions of each prototype is:
Prototype A, succession of pitch interval directions:
Tones: 12345678 + - - + - + +
Prototype B, succession of pitch interval directions:
Tones: 12345678 + - + + - - +
C. The succession of pitch interval sizes of each prototype is:
Prototype A, succession of pitch interval sizes:
Tones: 12345678
octave,fourth,third, sixth, sixth, third, fourth
Prototype B, succession of pitch interval sizes:
Tones: 12345678
octave,sixth, third, fourth,fourth,third, sixth
The differences in pitch interval directions between Prototypes A and B occur between tones nos. 3 and 4 and tones nos. 6 and 7.
The differences in pitch interval sizes between Prototypes A and B occur between tones nos. 2 and 3, tones nos. 4 and 5, tones nos. 5 and 6, and tones nos. 7 and 8. 58
PROTOTYPE A
1 4 8
$
+ - + - + + octave fourth third sixth sixth third fourth
PROTOTYPE B
I ' J L 4 S 2
T#-- & * % f 4 + octave sixth third fourth fourth third sixth Fig. 20.— Differences in the order of the directions and sizes of the pitch intervals between consecutive tones in Prototypes A and B.
Selection of Characteristics
The choices of pitch and temporal characteristics of the proto types were affected by the following considerations:
A. The listener was to be encouraged to attend to the prototypes as wholes. For this reason—
1, The temporal intervals and pitch invervals of the highest and lowest pitches are the same for both prototypes.
2. The number and order of the pitches were chosen so that there are differences in only two pitch interval directions between the prototypes. These differences are not consecutive and do not occur at the first or the last interval in the prototypes. (The dependence of interval direction and interval size in establishing the contour of a tone series results in the two tone series differing in the size of the last interval.)
B. The number of changes in direction could make a difference in the difficulty of recognition of the contours of the tone series. Therefore, the prototypes had to be alike in the number of changes. 59
C. The interval sizes and the variety of interval sizes in a tone series could make a difference in the difficulty of recognition of the prototypes. For both of the above rea sons—
1. eight tones were used.
2. the changes in direction in the first part of one tone series are the same as those in the last part of the other tone series.
D. A common assumption among musicians is that perceived emphases known as "agogic accents" accompany longer temporal intervals in a series of tones. For this reason—
1. The order of the temporal intervals was chosen so that the second and third of the longer temporal intervals would coincide with the beginning of the differences in pitch interval directions between Prototypes A and B that occur between tones nos. 3 and 4 and 6 and 7. When the order of the temporal intervals is changed in Variable OR the longer temporal intervals divide these pitch interval directions by which the prototypes differ. Therefore, change in perceived emphasis on one of the contour characteristics might occur with Variable OR.
2. The pitches of the prototypes are restricted to the pitches of the G major triad in order to avoid a possible change in harmonic emphasis which might occur if other scale degrees in G major were succeeded by the longer temporal intervals when the order of the longer temporal intervals is changed in Variable OR.
Variables
The five variables that alter the pitch and temporal character istics of the two different tone series (Prototypes A and B) are:
1. the octave transposition of the pitches of the tones of the first part of the series (designated OF);
2. the octave transposition of the pitches of the tones of the last part of the series (designated OL);
3. the change in the order of the temporal intervals that are delineated by the onsets of the tones in the series (designated OR); 60
4. the change in the lengths and subdivisions of the four equal segments of time in the tone series (designated LS);
5. The interpolation of tones in the temporal intervals of the tone series (designated I).
A set of transformation rules are used to apply these five variables, singly and in combination, to Prototypes A and B. These rules specify:
1. the physical characteristics of the variables.
2. the limits within which the variables are Independent.
3. the processes which differentiate conditions of alteration in the same variable.
4. how to alter the same characteristics of the two tone series in the same way.
The transformations do not change the contour of one prototype into the other. The application of these rules results in several conditions of alteration for each variable. The variables are independent from each other in all conditions in the transformations.
The conditions of the characteristics in the original statement of the prototypes are the initial states of the respective variables.
Modular arithmetic is used in the statistical design for this study. A base 3 numerical system designates the different conditions of the variables in the design. In the following descriptions the numbers that label the conditions of the variables correspond to the symbols in the (mod 3) numerical system of the statistical design.
Therefore, the initial condition of each variable is designated by the symbol 0; an altered condition is designated by 1; another altered condition is designated by 2. 6* VARIABLE OF
Variable OF is the octave transposition of the pitches of the
tones of the first part of the series in the prototypes. The octave
register of the pitch of the first tone and a group of tones that
succeed it occurs in three conditions in the transformations. In con dition 0 the octave register of the pitches in the first part of the tone series is the same as in the original statement. In condition 1 the octave register of the pitches in the first part of the tone series is raised one octave. In condition 2 the octave register of the pitches of the first part of the tone series is lowered one octave.
The number of tones that are transposed Includes those pitches from the first tone and those that follow it to as close to the midpoint of the series as can be transposed up or down one octave without changing the pitch interval directions of the tone series.
(The number of pitches that are transposed in conditions 1 and 2 is the same for each prototype.) 62
Variable OF
Condit ions
0 1 2
The octave register of The octave register of The octave register of the pitch of the first the pitch of the first the pitch of the first tone is which is tone is g^. The tone is g. The pitches the same as in the pitches of tones nos. of tones nos. 1,2,3 and original statement. 1,2,3,4 and 5 are 4 are lowered one raised one octave. octave.
The pitches of tones nos. 1,2,3,4, and 5 are lowered one octave when Variable OF at condition 2 is combined with Variable OL at condition 2.
(Only when variables OF and OL are combined each at condition 2 can the pitch of tone no. 3 be lowered one octave without changing the pitch interval direc tions of the proto types. )
Figure 21 illustrates Variable OF in the two conditions of alteration (VOF-^ and VOF2) as it occurs with Variables OL, OR, LS, and
I each in the unaltered condition. 63
PROTOTYPE A
PROTOTYPE B
Fig, 21,— Variable OF In conditions 1 and 2 with Variables OL, OR, LS, and I in unaltered states. VARIABLE OL
Variable OL is the octave transposition of the pitches of the tones of the last part of the series in the prototypes. The octave register of the pitch of the last tone and a group of tones that precede it occurs in three conditions in the transformations. In condition 0 the octave register of the pitches in the last part of the tone series is the same as in the original statement. In condition 1 the octave register of the pitches in the last part of the tone series is raised one octave. In condition 2 the octave register of the pitches of the last part of the tone series is lowered one octave.
The number of tones that are transposed includes those pitches from the last tone and those that precede it to as close to the mid point in the series as can be transposed up or down one octave without changing the pitch interval directions of the tone series.
(The number of pitches transposed in conditions 1 and 2 is the same for each prototype.) 65 Variable OL
Conditions
0 1
The octave register of The octave register The octave register of the pitch of the last of the pitch of the the pitch of the last tone is which is last tone is g3. tone is gl. The pitches the same as in the The pitches of tones of tones nos, 6,7 and 8 original statement. nos. 5,6,7 and 8 are are lowered one octave. raised one octave. The pitches of tones nos. 5,6,7 and 8 are lowered one octave when Variable OL at condi tion 2 is combined with Variable OF at condition 2.
(Only when variables OF and OL are combined each at condition 2 can the pitch of tone no. 5 be lowered one octave without changing the pitch interval direc tions of the proto types. )
Figure 22 illustrates Variable OL in the two conditions of alteration (VOL^ and VOL2) as it occurs with Variables OF, OR, LS, and I, each in the unaltered conditions. 66
PROTOTYPE A
PROTOTYPE B
Y«-I
6. *7 iZ fi. Ill
Pig. 22.— Variable OL in conditions 1 and 2 with Variables OF, OR, LS, and I in unaltered states. 67
VARIABLE OR
Variable OR is the change in the order of the temporal intervals that are delineated by the onsets of the tones in the series in the prototypes. The order of the relative differences in length of time between temporal intervals that are delineated by the occurrences of consecutive tones in the prototypes occurs at two conditions in the transformations. In condition 0 this characteristic is the same as in the original statement. In condition 1 the order of the temporal intervals is changed so that the tones that were succeeded by the longer temporal Intervals in the original statement are changed so that they are succeeded by the shorter temporal intervals; and the tones in the original statement that are succeeded by the shorter temporal intervals are changed so that they are succeeded by the longer temporal intervals. 68
The order in the original statement of the prototypes:
Condition 0
tones: 1 % 1 4 5 6 7 8 I I ' l i t I I relative diff. j L 1 S * L | s l s l L I S I L between temporal . I 1 | I I | | intervals: J 11 i l l | ) 1234123412341234 moments of onset: @ 2 3 4 © 6 7 8 @ 10 11 12 @ 14 15 16
Cond it ion 1 tones: 1 2 3 4 5 6 7 8 IT ] | | III relative diff. ISl L 1 S I L | L I S 1 L l S between temporal | I I I | | I 1 intervals: I I I I | 111 1234123412341234 moments of onset: 1 ( 5 ) 3 4 5 (b) 7 8 9 ^ ^ 1 1 12 13 15 16
The circled numbers in the lower line Indicate the moments of onset of the tones that are succeeded by the longer temporal intervals,
Variable OR
Conditions
0 1 a. the four tones that are a. The four tones that are succeeded by the longer temporal succeeded by the longer temporal intervals are tones nos. 1, 3, 6 intervals are tones nos. 2, 4, 5 and 8. and 7. b. The four tones that are b. The four tones that are succeeded by the shorter succeeded by the shorter temporal intervals are tones temporal intervals are tones nos. 2, 4, 5 and 7. nos. 1, 3, 6 and 8.
Figure 23 illustrates Variable OR in the altered condition (VOR^) as it occurs with the Variables OF, OL, LS, and I, each in their unaltered conditions. 69
PROTOTYPE A
< 9. 3
VeRj It' A— m f H / -i ( 3l 3 *1 I 3 - 3 } 3- 3 i a- 3 >- 1 SL 3 H 5 4 n g <1 to II /3u 13 /*f IS' /£, PROTOTYPE B
I 3 1 3 q i a 3 h 13 ft IS" IC
i= I t SL -3 I SL 3 / a 3 ^ IS- 3
Fig, 23.— Variable OR in condition 1, with Variables OF, OL, LS, and I in unaltered states. 70
VARIABLE LS
Variable LS is the change in the length and subdivisions of the four equal segments of time in the tone series in the prototypes. The length and subdivision of the four equal segments into which the total time of the prototypes is divided occurs in three conditions in the transformations. In condition 0 the length and subdivision of the four equal segments is the same as in the original statement of the prototypes. In condition 1 the length of each of the equal segments is extended to six eighth-note durational units and the second equal segment is subdivided in half. In condition 2 the length of the four equal segments is extended to six eighth-note durational units and each of three of the four equal segments Is subdivided into thirds. 71
Variable LS
Conditions
0 1 2
a. The length of time a. The length of time a. The length of time in each of these equal in each of the these in each of these equal segments is four equal segments is six segments is six eighth- eighth-note durational eighth-note durational note durational units units long. units long. long.
b. The total time of b. The total time of b. The total time of the statement of the the statement of the the statement of the prototypes is sixteen prototypes is twenty- prototypes is twenty- eighth-note durational four eighth-note four eighth-note units long. durational units long. durational units long.
c. In the second equal c. In the second equal c. In three of the four segment, tone no. 4 segment, tone no, 4 equal segments, tones occurs at the third occurs at the fourth nos. 2, A and 7 occur eighth-note durational eighth-note durational at the fifth eighth- unit of the segment unit in the segment note durational unit in and subdivides that and subdivides the their respective segment segment in half. segment in half. and subdivide that segment in thirds. * d. The length of the d. The length of the d. The length of the subdivision is two subdivision is three subdivision is two eighth-note durational eighth-note durational eighth-note durational units long. units long. units long.
Figure 24 illustrates Variable LS in conditions 1 and 2 as it occurs with the Variables OF, OL, OR, and I each in their unaltered
states. 72
PROTOTYPE A
A i A 3 f A 3 / 3. 3 H AI 2 .3 4 <7 /O /I /A 13 H /J* 16 VIS, ■* J
* * f* n f M " I I Ai 3v r t t a 3 H s I X 2 q S L A 3 * 5" (o
31 3 4 5 6 I * 3 4 5 t A 3 ^ .r 6 t a 3 4 S’ & At PROTOTYPE B / a 3 */ /o // /A /3 /v /r /£ J'f 2L-B<# S Cl 2 3 * 5* C 117I A7VS-C/g3fs-C J^f a-B^ s ^ T S 1 to ff (SL w (Sf&i7 /?rt ao a/aaajaV Vt* t I A 3 ? 5 ^ 7 g /,= I 2- 3 <* .f C ta3^5‘ 6'/Z3Y5iC’/23yj't A / a 3 V f c» 7 a 7 /© /i n ftt*/s ic n *9 n ju>&f aj4?a Fig, 24,— Variable LS in conditions 1 and 2, with Variables OF, OL, OR and I in unaltered states. 73 VARIABLE I Variable I is the interpolation of tones in the temporal intervals of the tone series in the prototypes. Sound is continued through the total time of the statement of the prototypes at three conditions in the transformations. In condition 0 sound is continued by the unequal durations of the tones of the prototypes. In condition 1 sound is continued by equal durations of tones at one tone per eighth-note durational unit. In condition 2 sound is continued by equal durations of tones at two tones per eighth-note durational unit. Variable I Conditions 0 1 2 a. The tones of the a. Tones are inter- a. Tones are interpolated prototypes fill the polated between the between the prototype total time. prototype tones so that tones so that the total the total time is time is filled with tones filled with tones of of the prototypes and the prototypes and with with new tones so that new tones so that there there are two tones per is one tone per eighth-note durational eighth-note durational unit. unit. b. The durations of b. The durations of b. The durations of the the tones of the the tones of the tones of the prototypes prototypes range from prototypes all equal all equal a slxteenth- an eighth-note dura- an eighth-note dura note durational unit. tional unit to one- tional unit, fourth the total time of the statement of the prototypes. c. The total time is c. The total time is c. The total time is filled with tones of filled with tones of filled with tones of variable durations. the same duration. the same duration. 74 The choice of pitches for the tones that are interpolated is constrained by six considerations: 1. The range of the pitches of the interpolated tones is within a third above or below the pitch of the prototype tone which precedes the interpolated tones. Conditions 1 and 2 of Variable I differ only in the number of interpolated tones. Therefore, the highest and lowest pitches of the interpolated tones are the same between respective prototype tones in conditions 1 and 2. 2. The pitch interval directions of the prototypes are maintained. This is accomplished in one of two ways: a. The pitch interval direction between the last interpolated tone and the succeeding prototype tone is a duplication of the pitch interval direction between the prototype tones that delineate the interval in which the tones are interpolated. (See eighth-note durational units 9-12 in Figure 25.) The pitch interval direction between the tones that occur at the moments that correspond to the moments of equal subdivision duplicate the pitch interval direction between the prototype tones delineating the interval in which the tones are interpolated. (See eighth-note durational units 9-12 in Figure 25 with Prototype B.) (In Figure 25 the pitch interval directions of the prototypes and the pitch interval directions of the interpolated tones that duplicate the directions of the prototypes are labeled above the musical notation.) 3. The successions of pitch interval directions and pitch interval sizes of the interpolated tones are the same for each prototype. The successions that occur at corresponding moments in each prototype differ as to their pitch level depending upon the pitch level of the prototype tones and upon the interval directions of the prototypes to be duplicated. 4. One or a succession of pitch interval directions is repeated at regular intervals through the entire statement of the prototypes. The length of the intervals corresponds to the length of the four equal segments in the original statement and to their length when they are extended in Variable LS. (The succession of pitch interval directions that is repeated at regular intervals is indicated by the brackets below the symbols for the directions in Figure 25.) 75 5. In the repetition of the succession, pitch interval directions different from the original succession occur at moments that correspond to moments of some of the subdivisions of the equal segments. (In the repetitions the pitch interval directions that differ from those in the original succession are indicated by an in Figure 25.) The constraints imposed by the first, second, and third considerations that were just cited have priority over the constraints imposed by the fifth consideration just discussed. When Variable I in condition 1 is applied to Prototype B and condition 2 is applied to both prototypes (with the other variables in their unaltered conditions) the second repetition of the succession contains several pitch interval directions that are different from the original. These differences occur at moments other than those that correspond to the subdivisions. This is because there is a conflict among the constraints just cited. 6. All the pitches of the interpolated tones are diatonic to the key of G major. (Diatonic pitches were chosen to avoid differences among transformations due to the presence or absence of the "color" effects of chromatic tones.) Figure 25 illustrates Variable I in conditions 1 and 2 as it occurs with the Variables OF, OL, LS, and OR, each in the unaltered cond ition. PROTOTYPE A J-. i 3 4 S - 4 4- s L _ J V, 4 + +-+ lI r -J Figure 25.— Vairable I in conditions 1 and 2, with Variables OR, and LS in unaltered states. PROTOTYPE B i 3 * 7 8 “h -+ 4 - si f 'i u i r ~ ^ 1 3 4 a 3 4/ t 2 £ V 1 1 ? 4 ^ t a V I &. 7 « 1 to n a iZ H f s 16 i______j < _ / \ f ~r, , 3 — +■ - + * Lr_+_i. i - T - _ _ l a o i n n r ( ^ v i a 3 9 /D // /x /3 w /s /5 z J i i t z l j * c t + r - r . Fig. 25. continued 73 Interpolation of Tones, Other Considerations Preliminary investigation indicated that some repetitions of pitch interval directions in the series of interpolated tones were necessary for a listener to determine when the prototype tones occurred. The regularity of the repetitions that occur in Variable I that was just described was necessary so that the procedures for repetition could be specified. Also, this regularity of repetitions of successions of pitch interval directions is evident in variations in which there are interpolated tones in some of the Beethoven compositions that were analyzed. (The regularity of repetitions of inverted forms of successions of pitch interval directions in Variation three, Op. 34, has already been discussed (supra, pp. 38 and 39). Also, refer to the variations in the second movement of Symphony No. 5.) Preliminary investigation also indicated that the subdivisions of the equal segments were perceiveable in these transformations in which there were interpolated tones. It is doubtful that the procedures used in this study are the only ones that are effective in assuring the perceivability of the segments of time and their subdivisions in transformations in which tones are interpolated between the prototype tones. The purpose of the descrip tions is to specify what procedures were used in this study to produce the desired results. In the transformations in which Variable I is combined with Variable OR in its original condition, there are no obvious factors that differentiate the first part of the total time from the second part like the repetition of the temporal intervals that divided the 79 total time of the original statement. The only differentiating factor is the first pitch interval in the repeated succession of pitch interval directions which is different in the second repetition from the size of the first pitch interval in the other occurrences of the succession. The first pitch interval is a third in the second repetition of the succession; the first pitch interval is a second in the other occurrences of the succession. The pitch interval size that is different occurs at a moment that corresponds to the moment of the beginning of the second part of the total time of the equal segments in the original statement of the prototypes. In the transformations in which Variable I is combined with Variable OR in condition 1 the pitches of the tones that occur in the first statement of the repeated succession of pitch interval directions are repeated with the beginning of the second repetition of the succession. The beginning of the repetition of these pitches corresponds to the moments of the beginning of the repetition of the temporal intervals in the original statement of the prototypes that divided the total time of the four equal segments into two parts. The examples in Figure 26 illustrate some of the transformations of Prototype B in which Variable I is combined with Variable OR in its original condition. The examples in Figure 27 illustrate some of the transformations of Prototype B in which the Variable I is combined with Variable OR in condition 1. In Figure 26 illustrating the combination of Variable I and Variable OR the sizes of the first pitch interval in the succession that is repeated are labeled at each occurrence of this succession. 80 In Figure 27 illustrating Variable OR in condition 1 the pitches that are repeated with the beginning of the second repetition of the succession of pitch interval directions are marked with an asterisk. In Figures 26 and 27 the successions of pitch interval directions that are repeated at regular intervals are indicated by the brackets of dashed lines below the symbols for the directions. The two parts of the total time in the original statement of Prototype B and in the transformations are bracketed with solid lines below the musical notation. Summary of Altered Characteristics of Prototypes A summary of the characteristics of the prototypes that are changed by each of the variables is presented in Table 2. The following examples in Figure 28 illustrate some of the com binations involving all five variables when each transformation results in changing one of the variables. For each successive example an additional variable is changed. Example 1 - the octave register of the first part of the tone series is transposed one octave. Example 2 - the octave register of the last part of the tone series is transposed one octave higher. Example 3 - the order of the temporal intervals in the tone series is changed. the octave register of the first part of the tone series is transposed one octave lower. the octave register of the last part of the tone series is transposed one octave higher. 81 s 9 Q - I F = ! F i I _f L +- + + f - + + +■ + + S 1 & , . • f 7 s ' ' f '-=fl € i U L J'*0 Li LU i LIL_i i 1 L_1 1— I l J U - 4 - 4 ------1- - 4 4 ------+- > _ * -f- _ 4 _ + ( ______j i l?!7L ( 1 _ \ «------ t * C, - f 4 4 4 -- 4 - 4- 44~4”-+— 4-“ "44-4- - —4- + + “44-4—4-4 ** ______id r i______- \ £ i______j JL. J Fig. 26.— Variable I combined with Variable OR in its original condition. 82 --- 1 j_ u a r _ i j — - J \± f ______1 1______I t— I __ I JL Figure 27.— Variable I combined with Variable OR in condition 1, TABLE 2 SUMMARY OF ALTERED CHARACTERISTICS OF PROTOTYPES < < o r - Vi V0F o 50 v L s Condil ions 1 2 1 2 1 1 2 1 2 1st, 3rd, 5th scale degrees in G major No pitches consecutively repeated No series of pitches repeated Two octave registers in each part Pitch of the first tone is gl XX Pitch of the last tone is g* XX Succession of temporal intervals Longer temporal intervals succeed tones #1,3,6,8 X Shorter temporal intervals succeed tones #2,4,5,7 X Longest temporal interval between prototype tones is one-fourth the total time Shortest temporal interval between prototype tones is one eighth-note durational unit Division of total time into four equal segments Delineation of four equal segments by longer temporal intervals XX Extent of time in equal segments is four eighth-note durational units XX Total time of statement of prototypes is sixteen eighth-note durational units XX Intervening tones subdivide the four equal segments 1 First and third equal segments unequally subdivided i- • • X Second equal segment subdivided in half * X Length of subdivision is two eighth-note durational units 1 X Repetition of temporal intervals t Tones #6,7,8 succeeded by same temporal intervals as tones #1,2,3 . ■ X J Division of total time of equal segments into two parts TABLE 2 Continued V0F V0L V0R VLS VI G inditions 1 2 1 2 1 1 2 1 2 Sound continued through the total time of statement Pitches of prototype tones sustained through temporal intervals XX Length of pitch of tone #8 equal to time of equal segments X X Time filled with tones of unequal durations X X 85 PROTOTYPE A F f — B ? P X 3 H- j 2 SL 3 X 3 4 E= l j_ 3 4 J" ■'O // /3- 13 X / r /t u /i a >•* > s Lk f / / . .** ac x ^xa :i ay z l s u J? 1* Z*. JJ >f_ *f ► j f c i iT '-- -■- fi~" I •--r---.------=-- 1- f•\------1ii---- 1^— If----- L . — i\ _ . - f — l----- = V* ~r - ^ : * - 4 x) ^ 1 f a } « r * 7 ? f /O y/ /J /3 t4 IS ft, f : ± Li f **/o it IX Mil iS/L '■7 fg J« *C 2JJ12 ZiiJ>iS*A J7 -f t - * i ^ 4 - t i —— ]■■ n r ------r = } — ------1h - c = ^ = i — ------1 f ~ ~ ~ — H f e ^ - - * 'ii ------1 f ' a J V J* 4 7 , ? T 4 /f '4 /I */ IS it, n it If 16 XI J» Ji 3* •* i a j . *{.£*. u .T tj u n iitL ur/kiiit /latai ixnMstLHLastf sxxLUlUti * * i t 3 i i r i t * j y y £ * a 3 E aim p lt 5 V , a a -7 -f 6 t t f (I « a /A /* Jt 17 if J7 ju j/ Oi jj Jf J?/ Ji4f47f«*f ilV itfffiilT ff/fJB J/JL lia iH U 'Ji 11J#JJJ#N Ji33 3 0 f Ikfl 3IJ440 1> 11 <*1*4 *£ «4*1 d4 Fig. 28.— Combinations of the five variables. PROTOTYPE B 7 9 9 * i> /i 13 # /r ,t ‘I lM '£ '£ O >t Jf AA *i ii j- ar^t -i- tr id *b i; ji E.*a.**p[a > n s ~r f $ h ‘ ~ 3 <•f r 7 f “ , % .f /4 I : -i * r it 1 ?« IX ii ^ cT lit 't <$ J? ^ «4l£i. j Z l a m : . e 2 ** u ~ _____ r i i i T 3 « 4 Jt f a 3 * r tr 1 f f te *t *1 I■* >1 ' A J_ ~ r J "JJL ■* \ 'J 4 m i t£ A * * . 1^:3 21^' ^ JA- w^Hmplo. ii f " *i -t ' A 3 * / ‘6 ! ’ a* /i" /* ‘>' x. ^ *■ .T j ■ ; * ; i x ~ x ;w t At i i l t li i . r. % ir* a a . h z £ S aL - f e l I / i f i 3 -v £ * T r f -a /< /i /I -» .,' i. na. m a I*!t £ | 4 g | fe & &P i i t J v s' i 7 f ? * " a a n sr :i >~ t Fig. 28 Continued 87 Example 4 - the length of the equal segments is extended and subdivided into thirds. the octave register of the first part of the tone series is transposed one octave lower. the octave register of the last part of the tone series is transposed one octave higher. the order of the temporal intervals in the tone series is changed. Example 5 - tones are interpolated between the prototype tones so that all tones are of equal duration and occur at two tones per eighth-note durational unit. the octave register of the first part of the tone series is transposed one octave lower, the octave register of the last part of the tone series is transposed one octave higher. the order of the temporal intervals in the tone series is changed. the length of the equal segments is extended and subdivided into thirds. Subjects The sixty-six subjects who participated in the experiment were students at the University of Delaware. They were grouped according to whether they were music majors or nonmusic majors. There were twelve subjects who were nonmusic majors. The music major group was subdivided into three groups according to the level of eartraining course in which the students were presently enrolled, or if not presently enrolled, the last eartraining course in which they had enrolled. There were twenty-four subjects who were enrolled in the freshman level eartraining course; twenty-four subjects who were enrolled in the sophomore level eartraining course; and six subjects 88 were in a group in which different levels of eartraining (freshmen through graduating seniors— "mixed") were represented. Procedures Assignment of Subjects Subjects were assigned to the experiment in groups of six. In the group of "mixed" levels of eartraining experience there were not enough subjects at any one level of eartraining to form a group of six so that they could be assigned to the already existing groups or so that they could form a new group of subjects who had completed both freshman and sophomore levels of the eartraining courses. Therefore, a separate group of subjects was formed in the experiment which consisted of subjects who differed according to the eartraining course in which they were presently enrolled or in the last eartraining course in which they had enrolled. Statistical Design Combinations of all conditions of the five variables results in a total of 162 combinations. The application of the 162 combinations to both prototypes results in a total of 324 transformations. A random numbers table was used to choose 162 of the 324 transformations and assign which transformations were alterations of Prototype A and which were alterations of Prototype B that occur in the experiment. (See Appendix B for illustrations of the 162 trnasformations that occur in the experiment.) In order to reduce the number of combinations presented to any one subject a 3^ factorial design was constructed in which two degrees 89 of freedom for the four-way interaction are confounded. In this design each subject heard 54 of the 162 transformations. In the confounding scheme a block is assigned to one subject (or one group of subjects). Each block contains fifty-four different combinations of the five variables, which is one-third of all 162 possible combinations of the five variables. Within these blocks the variables occur in particular combinations in a particular order in their respective conditions. The combinations of the five variables are balanced between the blocks with respect to all main effects, and two - and three - factor interactions. Modular arithmetic is used to define the balanced sets of variable combinations in the blocks. In the following formula that is used for balancing the sets of the variable combinations the factors that correspond to the variables are: Variable OR = Factor X^; Variable LS = Factor X2 ; Variable OF = Factor X3 ; Variable OL = Factor X^; Variable I = Factor X5. The formula for defining the sets of variable combinations is: ^2 + ^3 + X4 + X5 = 0, or 1, or 2 (mod 3). Latin squares are used to balance for possible learning effects due to the order and sequence of the presentation of the variable combinations. The order of the combinations of Variables LS, OF, OL, and I in each block yield a Latin square across the three blocks in testing Session I or testing Session II. Each subject is presented an original condition (level 0) and an altered condition (level 1) of 90 Variable OR that is balanced for listening Sessions I and II. Three subjects In Session I and Session II yield a Latin square on Variable OR. Chart 1 illustrates the Latin square for Variable OR. The Latin squares blocks that balance the combination of Variables LS, OF, OL, and I and the Latin square with Variable OR result in Latin squares embedded in a Latin square. One full replication of the experiment, which consists of two single replications of the 162 variable combinations, requires six subjects, hence, the grouping of subjects in groups of six. Chart 2 illustrates the order of the combinations of the variables in the respective blocks for testing Sessions I and II. Experiment Administration A pilot study was conducted in which five students and the instructor from a composition and analysis class participated. All of the students were seniors majoring in music. The procedures were the same as those used in the experiment. An introduction preceded the first testing session and was available at the beginning of the second session for any subject who chose to review that information. This introduction included descriptions of the prototypes, practice sets of prototypes and transformations, explanations of the response categories, and instructions. (See Appendix C for the introductory material presented to the subjects.) The purpose of the descriptions of the prototypes in the introduction was to encourage the subjects to attend to differences 91 Session I Session II Subj ects a > H > 1 O VoRj* transformat ions 1 transformations 28-54 2 V0Rx v ° « o transformations 28-54 transformat ions 1-27 3 V0R0 V0R^ transformations 55-81 transformations 82-107 4 v0Ri V 0 R q transformat ions 82-107 transforraat ions 55-81 5 V°R0 VORx transformat ions 108-135 transformat ions 136-162 6 V0Ri V0R0 transformations 136-162 transformations 108-135 Chart 1.— The Latin Square for Variable OR. *Vor0 Indicates when Variable OR occurs In the original condition (level 0) and indicates when it occurs in an altered condition (level 1). The numbers designate which specific transformations that each subject (or group of subjects) hears in each of the testing sessions. Sanslon I = Session II 27 26 25 2 23 2 2 0 2 21 IQ ie 17 l£ 15 Ik 13 12 11 10 k U 0 0 c * e 0 2 3 5 1 hr 2—Odro aibe obntos inExperiment. Combinations ofVariable Order 2.— Chart C VLS VCS 2 2 0 2 2 2 2 2 0 0 A 2 0 2 2 'i ry 0 0 0 0 A 0 0 0 0 a 0 0 0 0 n 0 0 0 3 0 n 0 ujc Sbet Sbet 6 Subject k Subject 2 Subject ujc 1 ujc 3 ujc f Subject 3 Subject 1Subject 2 2 2 . 4 A 4 4 1 1 1 n i 1 3 c o 0 0 0 0 VCF 2 2 1 • A A 4 0 T - f 1 2 2 2 0 n 2 0 1 1 0 0 2 2 2 1 A 4 2 2 4 A t 4 0 2 0 2 0 '"I 2 I 0 ■4 1 c, v 0 2 A 2 4 * 1 0 A i a J 2 2 2 4 n 1 2 j 0 i : v s p O H 9 7 -?e 77 ■^e 7* 72 —»A, '’3 6 7 6 a -2 'll ^3 £3 i cp 4 ? e VC? A f 1*1 "ft A A 0 0 n 7 9 a 2 A 0 y ■ft ■ft 7 ft 0 2 7 A A, A ■*4 0 1 2 0 VLS 0 2 2 2 i. 0 'ft 2 2 A "j 2 0 0 2 0 2 2 ±. a 1 1 i. 1 I \ \ *^CF 0 4 ; 4 A 2 ft 1 : 4 2 2 3 0 - 2 <5 2 2 3 2 1 0 0 4 1 ?Cl 2 2 2 2 4 A 4 2 a 2 2 9 « ,A « 2 I £. -ft 4 4 A 3 1 -ft 4 3 J : v 2 2 ft 2 A 2 0 2 : - 4 2( ) 3 ) 0 12f( 3 1 2 2 t * A 12: 1 1 3 A 1 ■J 3 1 | f 4 ,- ■ y 4 *33 120 2 1 2 1 :z*fi : : :z*fi 1-3| 1-3| 122 121 12 4 11* 1 112 11^ 11* * 1 * * 0 111 0 0 1 1 0 2 ! * - ♦ - ^ A, -^2 < -a s s 4 c 3 j 3 f j 3 bj : ■ Ca v 7 3 A A 9 0 A J 0 3 A 3 A 3 2 2 0 2 A 0 2 0 3 y J ! 1 j 1 « ■ VLS .■ 1 M • 1 12 -1 0 4 - 3 2 2 2 A 2 3 A A A A A A 2 2 -f 7 A _L l VCF 2 2 2 3 2 2 2 A A \ * e * 2 4 9 a 3 0 1 4 4 1 4 4 *l VCL 2 2 4 A 2 A 2 1 1 A 2 2 * 2 2 A ; A 1 A J J 2 A A : 2 2 - A i 4 A ft A 2 2 -y 0 4 7 ^ 4 * 0 1 1 J _ i 92 93 Subject 1 Subject 3 Subject 5 Subject 2 Subject 4 Subject 6 T p.* 7 m 7 r ’.V s VLS V LE VCF +/CL v : / A^ VLS T c ? 7 CL V- 1 i 2 2P I 9 0 0 9 92 1 ■3 i 13* 1 2 9 2 4 A i i 2 4 | 1 2 9 1 9 0 4 ? 3 0 1 3 ? 2 | 1 1 1 i 2 4 1 1 : 3 9 9 2 r t 1 i 2 2 2 2 1 ! 2 1 A 4 |A 4 4 31 1 2 ^ a 1 ! ' 3 0 9 | 1 3 b * u w1 4 4 4 4 4 A 4 4 A 32 1 0 £ 6 1 2 1 ’ bO 2 | 2 I a 4 - A 1 2 * | J i: A 1 2 3 3 ' ’ j 2 i 4 4 A i 2 1 ■ 4 1 2 ! *.-? 1 3 * - 1 A -> * A 4 . 1 . 4 :-_3 4 4 3 ' 1 t o 1 - 1 A 2 oa 4 A 9 F \kLI 4 3« 1 2 2 ' ! ! 1 4 2 I 9 2 •• I r r-- ■ 4 1 2 a 2 *1 2 - 9 1 l b f A A 3 7 * ! z ■ 1 4 ♦ - 4 - i * j 2 ! 2 4 2 | - t * 3 : - 3 ° 1 *5 4 n . 3 9 1 9 a 2 2 2 4 4 4 A 4 A* I 0 4 2 A A ’ 1 2 0 2 kr 0 4 4 ■ - A « - A b* CC ! | 2 | 9 1 j j 1 1 2 4 4 A A 4 kt 2 2 • 2 2 9 j • e 9 * 2 <- ♦ 4 - 2 - - 0*7 2 • 9 < t< j 1 AA - 2 4 4 4 4 kk < 2 2 o t - 2 | * 9 1 ^2 j 1 A A 4 « 2 2 : i « 2 : “> 2 4 cA | 4 AA A t kt 4 A A A 4 I * A A 4 A A A . H I * J AA 4 - 4 4 . 4 4 4 iiC 2 a ! * a i 0 2 - " 4 2 i A^S k s * 2 A •> A 4 A 2 ? 4 4 - . 4 A-5 i, a 2 t A a 4 0 0 O * - <-} 4 2 A "> 4 2 4 4 2 * Ait 1 AA 4 ■2 * c5 4 4 ■) . - • A 4 ? 4 A A 2 2 4 CZ 4 2 2 4 ? 2 A A * 4 2 4 1 ■) * £ *\ 1 2 - ■ 4 4 * 4 4 A 4 4 \£\ 4 4 .. 4 . > 2 2 2 f C/; 4 A A 4 A * 4 - a * 4 - '' ? 2 - i : ' i ' The column of numbers on the left side of each block identify a particular transformation. The 0, 1, and 2 in the five boxes to the right of each transformation number designate the conditions of each of the five variables in that transformation. 94 between the prototypes that they might hear. Spatial references in the form of musical notation or any other graphic representation and labels of interval identification were avoided in the descriptions. It is possible that the subjects used what they knew about musical notation and labels for interval identification from their previous experience while listening to the musical examples in the test. However, labels and notation were not included in the descriptions of the prototypes in order to avoid directing the subjects' attention to how to designate what they heard rather than to what they heard. The three transformations that were used in the practice sets were chosen so that each subject had practice with one transformation from the fifty-four transformations that he/she would hear in the test and two that were among the transformations that other subjects would hear in the test. If the identity of the transformation in the practice set were chosen incorrectly, the correct answer was given and the same practice set was presented again. If the identity of the transformation were chosen correctly, the correct answer was given and the next practice set was presented. The experiment was divided into two sessions so that each subject heard 27 transformations per session. The purpose of the two sessions was to minimize the possibility of subject fatigue. As a result of discussions about fatigue with the subjects who participated in the pilot study a 90-second rest period occurred every nine transformations in each session. During this rest period the subjects were encouraged to remove their earphones and not to think about the test. These rest periods were timed by the computer. Each testing sessions was 95 approximately thirty-one minutes long, including the two 9-second rest periods. (This time varied slightly depending upon the speed with which the subjects read the instructions and responded to the presentation of the material in the test.) PLATO, a computer-based education system was used to conduct the experiment. The prototypes and transformations were presented by a four-voice digital synthesizer known as the Gooch MUSIC BOX. The prototypes and transformations were produced with a reference pitch of a^ = 440 c.p.s. and were presented at a rate of an eighth-note durational unit per one-half second. This educational system was used for several reasons: 1) the production of sound by the digital synthesizer permitted the control of timbre and performance articulations which eliminated them as variables in the presentation of the musical examples; 2) the use of the PLATO terminals allowed the students to participate individually in the experiment and thus eliminate the possible distractions of group participation; 3) the computer recorded, times, and stored the responses, thus guaranteeing accuracy in the collection of the data. The music majors who participated in the experiment were familiar . with the PLATO system and the synthesizer sound because they used the PLATO system for their eartraining classes. The subjects heard the prototypes and transformations through earphones. The two prototypes and the transformations were presented to the subjects in an ABX triad arrangement. In this arrangement, the two prototypes were presented followed by a transformation ("X”) of either Prototype A or B. The order of Prototypes A and B 96 alternated with each presentation of the triad resulting in an ABX or BAX order. Following the presentation of each ABX triad, the subjects were asked to indicate if they were able to recognize the prototype in the transformation by choosing one of the following response categories: a. "The variation is more like melody A." b. "The variation is more like melody B." The terms "variation" instead of "transformation" and "melody" instead of "prototype" were used in the response statements because "variation" and "melody" are terms that are used in musical discourse and thus were more familiar to the students. The instructions were presented on the graphics screen of the PLATO terminal. No musical notation accompanied any of the musical examples. Since the order of the prototypes alternated with each presentation of the triads the subjects were not able to rely on the order of the prototypes for help in remembering which one was labeled "A" and which was labeled "B." Therefore, the order of the prototypes in the triad appeared on the screen of the terminal as each triad was presented. At the same time that the order of the prototypes was presented the two possible answers for identifying "X" in the triad appeared on the screen. The subjects responded by touching the screen to indicate a choice between the two possible answers. In the triads there was a four-second interval between the prototypes and a four-second interval between the last prototype and the transformation. The subjects were given time to respond during a twelve-second interval between each triad. A message to alert the subjects for the 97 next triad appeared on the screen one second before the beginning of the next triad. The subjects were allowed to proceed to the next ABX triad sooner than the twelve-second interval by touching a designated place on the graphics screen. Collection and Analysis of Data The effects of each of the variables were determined by the calculation of the mean percentages of correct ABX triad recogni tions. Each ABX triad had 22 responses since each of the 66 subjects responded to one-third of the total number of ABX triads. Increasing difficulty in recognition was indicated by a decrease in the mean percentage of correct recognitions and conversely, decreasing difficulty was indicated by an increase in the mean. The data were analyzed to determine whether the number of altered conditions of the variables (e.g., change in meter, octave transposi tion) that was present at one time in a transformation had an effect on the difficulty of recognition regardless of which variables occurred in altered conditions. The significance of the effects of the five variables were determined by an Analysis of Variance. Also, significant interactions from the combinations of the five variables were determined by an Analysis of Variance. The significance of the effects of eartraining experience and practice on the accuracy of recognitions was determined by an Analysis of Variance. Also, an Analysis of Variance determined the significance of the effects of eartraining experience and practice on the speed of recognitions. CHAPTER IV RESULTS The results of the following analyses of the data are presented in this chapter: analysis of the percent of correct recognitions according to the number of alterations present in the transformations; Analysis of Variance for the effects of the five variables; Analysis of Variance for the effects of eartraining experience and practice on the percent of correct recognitions; and Analysis of Variance for the effects of eartraining experience and practice on the response time. There was no statistically significant increase or decrease in the difficulty of recognition when octave transposition of the first part of the tone series, octave transposition of the last part of the tone series, change in the order of temporal intervals, and change in meter each occurred alone. The statistically significant two-way interactions were: the interaction between octave transposition of the first part of the tone series and changing the order of temporal intervals was significant at the observed .04 level; and the interaction between meter change and the interpolation of tones was significant at the observed . 0A level. The statistically significant three-way interactions were: the interaction among octave transposition of the last part of the tone series, meter change, and the interpolation of tones was significant 98 99 at the observed .02 level; and the interaction among octave transposi tion of the first part of the tone series, the octave transposition of the last part of the tone series, and the interpolation of tones was significant at the observed .04 level. The interaction between changing the order of the temporal intervals and the interpolation of tones resulted in an F-ratio with a significance level of .08. The significance level of .08 is approaching chance occurrence. However, when it is compared to the significance levels of the remaining interactions which are at or above .2 0, it appears to be positioned at a cut-off point that aligns it with the statistically significant interactions rather than with the interactions having nonstatistically significant F-ratios. For this reason, the interaction between changing the order of temporal intervals and the interpolation of tones is discussed along with the statistically significant interactions. The descriptions of these interactions and their effects are presented in the next chapter to accompany the analyses of the variables that were involved in the significant interactions. Effects of Number of Alterations The 162 transformations used in the experiment contained different numbers of altered conditions of the variables (alterations). The data were analyzed to determine whether the percentages of correct recognitions were affected by the number of altered conditions of the variables present in the transformations regardless of which variables were in altered conditions. 100 There were different numbers of transformations that contained different numbers of altered conditions of the variables. There was 1 transformation that contained the unaltered condition of all five variables; 9 transformations contained one altered condition among the five variables; 32 transformations contained two altered conditions; 56 transformations contained three altered conditions; 48 transforma tions contained four altered conditions; and 16 transformations contained five altered conditions. The analysis involved a "weighting1’ process so that the percentages of correct recognitions for different numbers of alterations (altered conditions of the variables) could be compared despite the different number of transformations. The weighting process involved the following steps: 1. The transformations were grouped according to the number of altered conditions of the variables that they contained. 2. The percentages of correct recognitions for the 162 transformations were tallied according to these groups of transformations. 3. The percentages of the transformations containing a given number of altered conditions of the variables were figured for each percent of correct recognitions. 4. Distributions of the percentages of transformationswith different percentages of correct recognitions were produced for groups of transformations that contained a given number of altered conditions of the variables. 101 The analysis revealed a definite direction to the changes in the percentages of correct recognitions which corresponded to changes in the number of alterations present at one time in the tranformations. As a result of this correspondence, the percent of correct recognitions decreased as the number of altered conditions of the variables present in the transformations increased (Figure 29), There were two indicators for this correspondence: 1 ) the decrease in the percent of correct recognitions for the modes of each of the groups of transforma tions; 2) skewed distributions of the percentages of transformations containing a given number of altered conditions of the variables toward the lower percentages of correct recognitions as the number of altered conditions of the variables present in the transformations increased. The largest percent (mode = 100 percent) of the transformations in which the unaltered conditions of the five variables occurred had 90 percent correct recognitions. The largest percent (mode = 45 percent) of the transformations in which one altered condition of the five variables occurred had 90 percent correct recognitions. The largest percent (mode = 30 percent) of the transformations in which two altered conditions of the five variables occurred had 80 percent correct recognitions. The largest percent (mode = 30 percent) of the transformations in which three altered conditions of the five variables occurred also had 80 percent recognitions. The two largest percentages (modes = 20 percent) of the transformations in which four altered conditions of the five variables occurred had 80 percent and 70 percent correct recognitions. The largest percent (mode - 40) of PLEASE NOTE: In all cases this material has been filmed In the best possible way from the available copy. Problems encountered with this document have been identified here with a check mark w*' . 1. Glossy photographs 2. Colored Illustrations _ _ _ _ 3. Photographs with dark background __ _ _ 4. Illustrations are poor copy ______5. Print shows through as there 1s text on both sides of page 6. Indistinct, broken or small print on several pages throughout ___ 7. Tightly bound copy with print lost 1n spine ______8. Computer printout pages with Indistinct print __ _ _ _ 9. Page(s) lacking when material received, and net available from school or author ______10. Page(s) _ _ _ _ _ seem to be missing 1n numbering only as text follows __ _ _ _ 11. Poor carbon copy ______12. Not original copy, several pages with blurred type __ _ _ _ 13. Appendix pages are poor copy ______14. Original copy with light type ______15. Curling and wrinkled pages 16. Other University Mi cn xi ims International 300 N ZEE5 RD. ANN ARSOR Ml 48106 '3131 761-4700 no one two t hree four f lve altered altered .1 11 e r ed n 11 e r ed al tered altered eond it tons eond it ion eond it ions rond i t ions eond i t ions eond it ions 90 80 k j 70 h j 60 u 50 40 ir 30 m I r 20 . L 10 0 •it• iiii*i fliilMM1! iM'l'i'ri 1*111- i'i# j rm I* Ml'I U P i . lU M ** fi I® l*« * 2 0 job o * } 4 S0 tc So l(b o MO *0 JO 1 ^ * JO 40(0 to IV®I0B o U tofO IB®ID N - 1 N = 9 N = 32 N = 56 N = 48 N = 16 Percentage distributions of transformations within each group (N = no, of transformations) 102 tfi. 29,— Correspondence hetvecn the percent of correct recognitions and the number of altered conditions var iah!es. 103 the transformations in which all five variables occurred in altered conditions had 60 percent correct recognitions. Effects of the Five Variables The initial analysis of the data showed a binomial distribution of subjects' responses that appeared as though pure guessing occurred when tones with durations of sixteenth-note values were interpolated. This condition of interpolation of tones resulted in the greatest number of tones per unit time that occurred in the experiment. It was concluded that when tones were interpolated in this condition the task was so difficult that the responses did not reflect the presence or absence of the altered conditions of the other variables. As a result, further analysis of the data was done excluding the presence of the interpola tion of tones in this condition. A large amount of variability, of which about 14 percent was accounted for, occurred in the experiment. Despite this high variability, one main effect, two two-way interactions, and two three-way interactions were statistically significant. The results of the Analysis of Variance showed that the interpolation of tones was the main effect that was statistically significant. The interpolation of tones resulted in an increase in the difficulty of recognition of melodic contour that was statistically significant at the observed level of .001 (Table 3). When tones were interpolated the mean percent of correct recognitions decreased from 87 percent to 66 percent. TABLE 3 ANALYSIS OF VARIANCE OF THE DIFFERENCES IN THE MEAN PERCENT CORRECT RECOTNITIONS RESULTING FROM THE EFFECTS OF THE FIVE VARIABLES N = 66 Sum of Mean Signif. Source of Variation Squares DF Square F of F Main Effects 14.090 8 1.761 10.226 .001 Meter .510 2 .255 1.480 .228 Order .246 1 .246 1.428 .232 Octave-OF (first part) .061 2 .031 .179 .837 Octave-OL (last part) .081 2 .040 .235 .791 Tones (interpolation) 13.181 1 13.181 76.529 .001 2-way Interactions 4.642 25 .186 1.078 .361 Meter Order .107 2 .053 .310 .734 Meter Octave-OF .255 4 .064 .371 .830 Meter Octave-OL 1.031 4 .258 1.496 .201 *Meter Tones 1.133 2 .567 3.289 .038* *0rder Octave-OF 1.100 2 .550 3.194 .041* Order Octave-OL .355 2 .177 1.030 .357 *Order Tones .530 1 .530 3.078 ,080* Octave'-OF Octave-OL .103 4 .026 .149 .963 Octave--OF Tones ,006 2 .003 .019 .981 Octave--OL Tones .026 2 .013 .075 .928 3-way Interactions 6.924 38 .182 1.058 .376 Meter Order Octave-OF .409 4 .102 .594 . 667 Meter Order Octave-OL .471 4 .118 .684 .603 Meter Order Tones .210 2 ,105 .610 .544 Meter Octave-OF Octave-OL 1.191 8 .149 .864 .546 Meter Octave-OF Tones .349 4 .087 .506 .731 *Meter Octave-OL Tones 2.040 4 .510 2.961 .019* Order Octave-OF Octave-OL .167 4 .042 .242 .914 Order Octave-OF Tones .330 2 .165 .957 .384 Table 3 Continued Sum of Mean Signif. Source of Variation Squares DF Square F of F 3-way Interactions (continued) Order Octave-OL Tones .028 2 .014 .080 .923 *Octave-OF Octave-OL Tones 1.731 4 .433 2.512 .040* 4-way Interactions 4.511 28 .151 .935 .563 Meter Order Octave-OF Octave-OL .992 8 .124 .720 .674 Meter Order Octave-OF Tones .243 4 .061 .353 .842 Meter Order Octave-OL Tones .843 4 .211 1.223 .299 Meter Octave-OF Octave-OL Tones 1.772 8 .221 1.286 .247 Order Octave-OF Octave-OL Tones .667 4 .167 .968 .424 5-way Interactions Meter Order Octave-OF Octave-OL Tones 1.176 8 .147 .853 .556 Explained 31.343 107 .293 1.701 ,001 Residual 186.189 1081 .172 TOTAL 217.532 1188 .183 Interactions to be discussed infra. 106 Effects of Eartraining Experience and Practice The four groups of subjects that differed according to the amount of eartraining experience had a small range of differences in the mean percent of correct recognitions across sessions one and two (from 63 percent for the nonmajors group to 69 percent for the sophomore and mixed groups (Table 4). The results of the Analysis of Variance for the mean percent correct recognitions for the groups showed that these differences were not statistically significant (Table 5). Practice on the recognition task resulted in only a 4 percent increase in the mean percent of correct recognitions across all subjects from session one to session two (Table 4). The results of the Analysis of Variance of the mean percent correct recognitions for sessions showed that this increase was not statistically significant (Table 5). The results of the Analysis of Variance showed that differences among the groups in the amounts of increase from session one to session two (interactions between groups and sessions) were not statistically significant. A "ceiling" of a mean of about 70 percent of correct recognitions appeared to limit the amount of increase from session one to session two for all music majors. The average percent of correct recognitions for the sophomore and mixed groups, which had the highest average percent of correct recognitions, with 68 percent and 67 percent in session one, only increased to 7 0 percent and 71 percent respectively in session two. The average percent of correct recognitions for the 107 TABLE 4 DATA FOR THE ANALYSIS OF VARIANCE OF DIFFERENCES IN THE MEAN PERCENT CORRECT RECOGNITIONS AMONG FOUR GROUPS OF SUBJECTS WITH DIFFERENT AMOUNTS OF EARTRAINING EXPERIENCE Mean Percent of Correct Recognitions Session I Session II Mean MAJORS Freshman 60 67 64 Sophomore 68 70 69 Mixed 67 71 69 Mean 65 69 67 NONMAJORS 61 64 63 TOTAL (Majors and nonmajors) 63 67 TABLE 5 ANALYSIS OF VARIANCE OF DIFFERENCES IN THE MEAN PERCENT CORRECT RECOGNITIONS AMONG FOUR GROUPS OF SUBJECTS WITH DIFFERENT AMOUNTS OF EARTRAINING EXPERIENCE Source df SS MS F Groups 3 .0170275 .0056758 .78 Within groups error 7 .0508811 .00727 687 Sessions 1 .0097018 4.29 Group X session interactions 3 .0025642 . 00 08 5 47 .38 Error 7 .015812 .0022589 TOTAL 21 .0959866 108 freshman group that had 60 percent in session one increased the most, but stopped at 67 percent in session two. Three of the four groups of subjects had essentially the same mean response times across sessions (freshman and nonmajors groups, both had 2,49 seconds mean response time; the sophomore group had 2,47 second) (Table 6). The mixed group had a faster mean response time (about one-half second) across sessions than the mean response times of the other groups. The results of the Analysis of Variance of the mean response times for the groups showed that there was no statistically significant difference among the groups in mean response time (Table 7). Practice on the recognition task resulted in a large decrease of about one-half second in average response times across all subjects from session one to session two. The results of the Analysis of Variance of response times for sessions showed that the decrease from session one to session two was statistically significant at the observed level of .01. The results of the Analysis of Variance showed that these differences among the groups in the amounts of decrease in mean response time from session one to session two (interactions between groups and sessions) were not statistically significant. 109 TABLE 6 DATA FOR THE ANALYSIS OF VARIANCE OF DIFFERENCES IN THE MEAN RESPONSE TIMES AMONG FOUR GROUPS OF SUBJECTS WITH DIFFERENT AMOUNTS OF EARTRAINING EXPERIENCE Mean Response Times (seconds) Session I Session I Mean MAJORS Freshman 2.64 2.33 2.49 Sophomore 2.79 2.16 2.47 Mixed 2.21 1.73 1.97 Mean 2.35 2.07 2.31 NONMAJORS 2.76 2.21 2.49 TOTAL (Majors and nonmajors) 2.66 2.14 TABLE 7 ANALYSIS OF VARIANCE OF DIFFERENCES IN THE MEAN RESPONSE TIMES AMONG FOUR GROUPS OF SUBJECTS WITH DIFFERENT AMOUNTS OF EARTRAINING EXPERIENCE Source df SS MS F Groups 3 .44193 .14731 1. 38 Within groups error 7 .74618 .1065971 Sessions 1 1.63636 103.88** Group X sessions interactions 3 .0297 .0099233 <1.00 Error 7 .11027 .0157259 TOTAL 21 2.96451 *p <.01. I CHAPTER V DISCUSSION OF INTERACTIONS The discussion of the significant interactions is presented as a separate chapter because of the amount of detail that is necessary to adequately describe these interactions. Also, the descriptions of the interactions are needed for clarification when the results of the interactions and the interpretation of the results are discussed. For this reason, the descriptions, the results, and the interpretations of the results of the interactions are presented together in order to avoid unnecessary repetition. The significant interactions and the effects of these interac tions which are discussed were not predicted at the beginning of this study (with the exception of the interactions between meter change and the interpolation of tones). This resulted from the fact that all of the significant interactions in this study seem to involve aspects of the variables that affect "organizational characteristics" of the tone series. Very little is known by musical scholars and psychologists about the effects of different musical variables on each other when they are combined. Further, little is known about the effects of changes in "organizational characteristics" in the tone series in a melody on melodic contour recognition. These 110 Ill problems are not discussed In the literature, nor have they been studied under experimental conditions. Development of complete explanations of these interactions requires more information than is provided from the results of this study. Also, since there is no information from other experimental research which might fill in the gaps in the explanations, the following discussions are necessarily speculative. Some observations are made about results of the interactions which show some patterns in their effects on the recognizability of melodic contour. These observations are presented in order that they may be used as indicators of points of departure in future experimental research about melodic contour recognition. An exhaustive discussion of "organizational characteristics" affecting the recognizability of melodic contour requires the formulation of a conceptual framework and operational definitions which is beyond the scope of this study. However, a working definition can be formulated for the discussion of the interactions. This definition is as follows: The organizational characteristics of melodic contour are those that result in perceived groups of tones in a series in a melody. In studies by Ortmann (1926, 1934), some aspects of melodies he described seem to resemble some of the organizational characteristics of the tone series affected by the variables involved in the interactions. Also, the effects of some of these characteristics on the difficulty of remembering and notating melodies described in 112 these studies resemble effects of some of the Interactions. Therefore, even though Ortmann was concerned with the effects of these character istics on the perception of melody, his descriptions seem pertinent to the results of this study. These studies by Ortmann are the only ones in the literature which seem to address questions of how organizational characteristics of melody could affect its perception. His observations were not made under experimental conditions. However, the procedures he followed did lead to orderly observations which lend support to suggestions presented in this study about points of departure for further experi mental research. For this reason, Ortmann*s observations are cited when they are pertinent to the following discussion. Interactions are graphically displayed in the following analyses. A sample graph is shown in Figure 30. These graphs are used as an aid in interpreting the results of the interactions. The horizontal axis is not a scale; rather, it is used to space discrete conditions of one of the variables in the interaction. The original condition of the variable is at the left hand mark on the horizontal axis and the changed condition is at the right hand mark. Both are captioned. The vertical axis is a scale giving the mean percent of correct recognitions of the melodic contour. Point A represents the mean percent of correct recognitions when both variables were in their original (unchanged) condition. Point B represents the mean percent of correct recognitions when only the variable, interpolation of tones, was changed. 113 100 Order of temporal Intervals 90- 80. 70 60 50. 40- 30. 1 1— without with Interpolation of tones Fig. 30.— Sample graph for illustration of interactions. Point C represents the mean percent of correct recognitions when only the variable, order of temporal intervals, was changed. Point D represents the mean percent of correct recognitions when both variables were changed. Point E also represents the mean percent of correct recognitions when both variables were changed. Lines on such a graph are used to represent trends in the change in the mean percent of correct recognitions between two discrete condi tions. They should not be mistaken as a series of data points. 114 The key to interpretation is in determining whether the line representing a changed condition is parallel with the line representing the original condition. If the lines are parallel, there was no measurable interaction. If the lines are not parallel, then an interaction occurred. The following observations can be made about the example in Figure 30: - The solid line (A-B) shows that when the order of temporal intervals was in the original condition but interpolation of tones occurred the mean percent of correct recognitions decreased form 90 percent to 63 percent. - The dashed line (C-D) shows that when the order of temporal intervals was changed and the interpolation of tones occurred the mean percent of correct recognitions decreased from 83 percent to 66 percent. - The dotted line (C-E) is shown in this sample graph only to illustrate the position parallel to A-B where the line C-D would be located if no interaction had taken place. However, an interaction did take place and the result of the interaction on the mean percent of correct recognitions can be measured. - To determine the change in mean percent of correct recognitions due to an interaction, it is necessary to measure the change in mean percent of correct recognitions between points A and C,and B and D. The changes are added if the lines cross. If the lines do not cross, then the change between B and D is subtracted from the change between A and C. 115 - The remaining step is to determine the direction (i.e., increase or decrease in mean percent of correct recognitions) of the effect of the interaction. - If the point C is at or below point A and the lines cross or the difference in mean percent of correct recognitions between B and D is less than that between A and C, the effect is an increase (i.e., an increase in mean percent of correct recognitions is attributable to the interaction). If the lines do not cross and the difference between B and D is greater than the difference between A and C, then the effect is a decrease (i.e., a decrease in the mean percent of correct recognitions is attributable to the interaction). - If the point C is above A and the lines do not cross and the difference between B and D is greater than the difference between A and C, the effect is an increase in the mean percent of correct recognitions. However, if the lines cross or the difference between B and D is less than the difference between A and C, the effect is a reduction in the mean percent of correct recognitions. Discussion of each of the interactions will be presented in three parts. First, there will be a discussion of the results of the interactions. Second, there will be a description of the combinations of the variables that result in the interactions. Finally, there will be an interpretation of the results of the interactions. The tone series in Prototype A is used in all of the notated examples of the variables that accompany the discussions of the 116 interactions. The eight tones of the original statement of the tone series in Prototype A are indicated by numbers directly above the appropriate notes in the figures. Interaction: Between Interpolation of Tones and Change in Order of Temporal Intervals (F, .08) Results of Interaction The interaction between the Interpolation of tones and change in the order of the temporal intervals of the tone series resulted in an increase (8 percent) in the mean percent of correct recognitions from when each of these occurred alone. The change in the order of temporal intervals alone resulted in a decrease (7 percent) in the mean percent of correct recognitions. The interpolation of tones alone resulted in a decrease (25 percent) in the mean percent of correct recognitions. Figure 31 illustrates the interactions. 117 100 Order of temporal intervals 60 ot CG k- l~ c u 50 30 wi t h Interpolation of tones Fig. 31.— Graph of the interaction between interpolation of tones and change in the order of temporal intervals. Description of Combination The total time of the tone series was delineated into four equal segments by the occurrence of longer temporal intervals succeeding tones nos. 1, 2, 6 and 8 in the original statement (Figure 32). The total time of these four segments was delineated into two parts by the repetition of the temporal intervals between tones nos. 1 , 2 and 3 by tones nos. 6, 7 and 8. (The four equal segments are bracketed by dashed lines and the two parts of the total time of the equal segments are bracketed by solid lines in Figures 32-35.) 118 X 8 H 7 f J------II------j Fig. 32.— Delineations of time in the original statement. When only the order of temporal intervals was changed, the four equal segments were delineated by longer temporal intervals succeeding tones nos. 2, 4, 5 and 7 (Figure 33). The total time of the four equal segments was delineated into two parts by the repetition of the temporal intervals between tones nos. 2, 3, and 4 by tones nos. 5, 6 2 and 7. The pitch of tone no. 2 (g ) is repeated by tone no. 5 at the beginning of the repetition of temporal intervals. The pitch interval size between tones nos. 5 and 6 (m6) that accompanied the repetition of these temporal intervals was different from the pitch interval size between tones nos. 2 and 3 (P4) that accompanied the repeated temporal intervals. (These intervals are labeled and marked with an * in Figure 33.) I 3. 3 V f 6 11 :t* tlT"r I I f'r I r f— n -1 -{ e e M *- * p H 6 I------II______I I______J!______I I______II______I Fig. 33.— Delineations of time when the order of temporal inter vals is changed. 119 When only tones were interpolated, a succession of pitch interval directions was repeated at regular intervals (Figure 34). The begin ning of each repetition coincided with the beginning of the last three equal segments in the original statement. The total time of the occurrence of this succession of pitch interval directions was delineated into two parts by a change in the size of the first interval in the repeated succession at the beginning of the repetition that corresponded with the beginning of the third segment. (The size of the first pitch interval of the succession was a second; in the third segment the size of the first pitch interval was a third.) There were no repetitions to delineate the total time into two parts when only tones were interpolated. The change in the size of the pitch interval was the only means to differentiate the first part from the last part of the total time of the four equal segments. (The pitch interval directions in the repeated successions are labeled in Figures 34 and 35. The differences in sizes of the first pitch intervals in the repeated succession are labeled in Figure 34.) Fig. 34.— Delineations of time when tones are interpolated. 120 When interpolation of tones and changing the order of temporal intervals were combined, the total time of the four equal segments was delineated into two parts by the repetitions of three pitches and two pitch interval sizes and directions (Figure 35). The pitches of tone no. 5 and the next two interpolated tones (g^, a^, g^) and the sizes (2nd, 2nd) and directions (+, -) of the pitch intervals between them were the same as the pitches of tone no. 2 and the next two interpolated tones and the sizes and directions of pitch intervals between them. g ■I- - - H -t------+ + II______I L II______I I------II______I Fig. 35.— Delineations of time when order of temporal intervals is changed and tones are interpolated. Interpretation of Results The increase in the mean percent of correct recognitions that resulted when the interpolation of tones and changing the order of temporal intervals were combined could be related to the separation of the tone series into two large groups by the repetitions of pitch and temporal characteristics just described. In the study by Ortmann (1934), which was cited earlier, he also made some observations about what organizational characteristics of 121 melodies could affect the ease or difficulty in remembering and writing the melodies in dictation exercises. As mentioned earlier, there is a problem of determining which results of his study are pretinent only to perceptual problems and which are pertinent only to notation and writing problems. However, the effects of the interactions confirm his observations about the effect of organizational characteristics of melodies on the students* ability to accurately hear the melodies. Ortmann discussed the advantages for making melodic dictation exercises easier when tones in a melody are grouped into "higher units" by pitch and duration repetition. Problems in melody dictation may be simplifed by using motives that permit easy grouping into higher units. This device is especially useful when we are dealing with increasing the memory-span. Thus an example such asCDEDEFEFG although it contains nine tones, is relatively easy on account of its grouping into three groups of step-wise progression, each one scale step higher than the preceding. As a result we have but two pitch variables: two steps up, one step down. (p. 45) He also said that repetitions could result in tones in a melody being grouped into larger units within phrases. The element of repetition functions for complete phrases as well as for motives. The 'naturalness* of the following phrases, in spite of the measure change results from their duration parallelism: J J L I (p. 83) 122 Perhaps, the repetitions of the pitches, of the sizes and directions of pitch intervals, and of the temporal intervals that occurred when interpolation of tones and change in order of temporal intervals were combined resulted in the tone series being separated into two groups or "higher units" which made them easier to hear than when there were fewer repetitions to separate the tone series into groups. Thus, the recognition of the contour was less difficult when the interpolation of tones and changing the order of temporal intervals were combined from when they occurred alone. Interactions: Between Octave Transposition of the First Part and Change in Order of Temporal Intervals (F, .04) Results of Interactions The interactions between octave transposition of the first part of the tone series and changing the order of temporal intervals produced different results depending upon whether the octave was raised or lowered. When the octave was raised and combined with a change in the order of temporal intervals, the mean percent of correct recognitions decreased (4 percent) from when each of the changes occurred alone. When the octave was lowered and combined with a change in the order of temporal intervals, the mean percent of correct recognitions increased (11 percent) from when each of the changes occurred alone. Raising the octave of the first part of the tone series alone resulted in a small increase (2 percent) in the mean percent in correct recognitions. Lowering the first part alone resulted in a 123 decrease (7 percent) in the mean percent of correct recognitions. The change in the order of temporal intervals alone resulted in a decrease (5 percent) in the mean percent of correct recognitions. Figure 36 illustrates these interactions. 100 — - order of temporal interval order of temporal Interval 90 — 80 - ea o original -r4 70 — bfiQ ou **v u Ih o u ** $ uV aV c 50 V X 40 — 30 untranspoeed octave, flrat part octave, first pert Fig. 36.— Graph of the interaction between octave transposition of the first part and change in the order of temporal intervals. 124 Description of the Combinations The combination of the octave transposition of the first part of the tones series and change in the order of temporal intervals affected the coincidence or lack of coincidence of the divisions of the tone series into two parts. These divisions resulted from the octave transposition and the repetition of the temporal intervals midway in the tone series. In the original statement the temporal intervals between tones nos. 1, 2 and 3 were repeated by tones nos, 6, 7 and 9 (Figure 37). This repetition delineated the total time of the four equal segments into two parts. The second part began with tone no. 6, (In Figures 37-42 the four equal segments are bracketed by dashed lines and the two parts of the total time of the equal segments are bracketed by solid lines.) 1 2 3 V r 6 7 8 — f -n / P M j. * ' * f— z $ = f H I______II______II______1 !------1 Fig. 37.-— Division of the tone series by the repetition of temporal intervals in the original statement. When only the octave of the first part was raised, tones nos. 1 through 3 were transposed (Figure 38). The division of tone series into two parts by the octave transposition occurred between tones nos. 5 and 6. The last part began with tone no. 6 and coincided 125 with the beginning of the repetition of the temporal intervals at tone no. 6, £ 7 F i JL JL J L. JL Fig. 38.— Division of the tone series when the octave of the first part of the tone series is raised. When only the octave of the first part was lowered tones nos. 1 through 4 were transposed (Figure 39). The division of the tone series into two parts by the octave transposition occurred between tones nos. 4 and 5. The last part began with tone no. 5 and did not coincide with the beginning of the repetition of temporal intervals at tone no. 6. I p ------, f - T F------u 7 F = = f = ; ^ — 11 - ■" ■" ■■ T W - --- r-- — I— — ——} ; S v 1 i T , JL JL -IL -JL Fig. 39.— Division of the tone series when the octave of the first part of the tone series is lowered. 126 When the order of temporal intervals was changed, the temporal intervals between tones nos. 2, 3 and 4 were repeated by tones nos. 5, 6 and 7 (Figure 40). The repetition of temporal intervals began with tone no. 5 and therefore, the second part began with tone no. 5 instead of with tone no. 6 as it occurred in the original statement. ill y S' i It I_____ 11 IL JL J J______[J______J Fig, 40,— Division of the tone series by the repetition of temporal intervals when order of temporal intervals is changed. When the octave of the first part was transposed and the order of the temporal intervals was changed, the coincidence or lack of coincidence of the divisions of the tone series occurred with the opposite direction of transposition from when the first part was transposed without the change in the order of temporal intervals. When the octave of the first part was raised and the order of the temporal intervals was changed, the divisions of the tone series did not coincide (Figure 41). The last part began with tone no. 6. The beginning of the last part did not coincide with the repetition of the temporal intervals that began with tone no. 5. i----- . 1 j------ii------it------1 i______11______i Fig. 41.— Division of the tone series when the octave of the first part is raised and order of temporal intervals is changed. When the octave of the first part was lowered and the order of the temporal intervals was changed, the divisions of the tone series did coincide (Figure 42). The beginning of the last part, with tone no. 5, coincided with the repetition of the temporal intervals that began with tone no. 5. I X 3 H S' i 12 f * * r p t f \ S h ” l r ) / f --- 1 — i $ = / — = 7 - ^ r — 1______II----- II------II______I I______11______1 Fig. 42.— Division of the tone series when the octave of the first part is lowered and order of temporal intervals is changed. Interpretation of Results The results of these interactions could be related to the coincidence or lack of coincidence of the ending of the compound pitch interval between the first and last parts of the tone series with the beginning of the repetition of the temporal intervals. 128 Ortmann (1934) described several types of conditions in a melody which resulted in what he called "emphases" of the tones in the melody. Among the conditions that he described were the repetitions of tones, pitch intervals, and durations in a melody and the "... presence of a wide pitch interval preceded by relatively small intervals." (p. 46) Perhaps, in this study, the repetition of the temporal intervals in the second part of the tone series resulted in an "emphasis" of the tone at the beginning of the repetition. And when the octave of the first part was transposed, the compound interval between the first and last parts of the tone series, which was preceded by simple intervals, resulted in an "emphasis" of the second tone of the compound interval. Ortmann did not specify what he meant by "emphasis" but did discuss it in the following statement as a means of making a tone more prominent than others in a melody. Whenever two or more types of emphasis coincide upon a tone, that tone 'stands out' from the rest. When the types of emphasis do not coincide, or, when one type is in conflict with another, that tone is obscurred by its environment.(p. 46) In the previous statement Ortmann talked about how the coincidence or lack of coincidence of different emphases on a tone could affect whether that tone "stood out" from the other tones in its "environment" or was obscurred by its "environment," He also said that the coincidence or lack of coincidence of emphases, which he called "accents," could affect the ease or difficulty of correctly hearing the entire melody. 129 When metrical, pitch and duration accents coincide upon any tone of the melody, that tone is triply emphasized and the coincidence greatly facilitates the correct hearing and correct writing of the melody.... When the three types do not coincide they tend to neutralize each other, and the resulting ambiguity Increases the difficulty of writing from dictation. Melodies containing such conflicts are invariably more difficult to 'comprehend' in their entirety than the melodies in which the accents coincide.(pp. 82 and 83) Perhaps, when the emphasis from the compound interval and the emphasis from the repetition of the temporal intervals coincided on the tone at the beginning of the second part of the tone series that tone was "doubly emphasized." Thus, hearing the melodic contour was facilitated and resulted in an increase in the mean percent of correct recognitions. When the emphases did not coincide on the tone at the beginning of the second part of the tone series that tone was obscurred. Thus, hearing the melodic contour was more difficult and resulted in a decrease in the mean percent of correct recognitions. Apparently, these interactions were specific to the change in the octave register of the first part of the tone series because the coincidence or lack of coincidence of the emphases just described would have occurred if the octave transposition of the last part of the tone series and changing the order of the temporal intervals had been combined. 130 Interactions: Among Octave Transposition of the First Part of the Tone Series, Octave Transposition of the Last Part of the Tone Series, and the Interpolation of Tones (F, .04) Results of Interactions Three-way interactions The results of the three-way interactions among octave transposi tions of the first and last parts of the tone series and the interpolation of tones differed according to whether the octaves of the first and last parts were transposed to the same or different octaves from each other. Both of the three-way interactions in which the first and last parts were transposed to the same octave resulted in a decrease in the mean percent of correct recognitions. The three-way interactions in which both parts were raised resulted in a greater decrease (16 percent) than the decrease (7 percent) when both parts were lowered. In the three-way interactions in which both parts were transposed to octaves different from each other there was an increase (20 percent) in the mean percent of correct recognitions when the octaves of the first part was raised, the octave of the last part was lowered, and tones were interpolated. A very small decrease (1 percent) occurred when the octave of the first part was lowered, the last part was raised, and tones were interpolated. The amount of this decrease was so small that the difference between the results of the two-way and three-way interactions which involved lowering the first part, raising the last part^ and the interpolations of tones was negligible. 131 Two-way interactions The two-way interactions between octave transpositions of the first and last parts to the same octave resulted in increases. The amount of the increase (9 percent) was the same when both parts were raised or both parts were lowered. The two-way interactions between octave transpositions of the first and last parts to different octaves from each other resulted in a decrease (7 percent) when the first part was raised and the last part was lowered and an increase (2 percent) when the first part was lowered and the last part was raised. The two-way interactions between octave transposition of the first part and interpolation of tones resulted in a decrease (1 percent) by raising the octave and an increase (2 percent) by lowering the octave. The two-way interactions between octave transposition of the last part and interpolation of tones resulted in an increase (4 percent) by raising the octave and a decrease (6 percent) by lowering the octave. Figure 43 illustrates the two-way and three-way interactions. 132 THREE-WAY 100 with lnt*rpHal*i lof| of ton** INTERACTIONS octave, laet part W p 40 urE rin ip o a id uni nm poiid octave, flrat pan octane, fltat pari TWO-WAY 100 without Interpolation of tonea INTERACTIONS 90 70 unt r a n a p o a a d 40 10 ul untranaponad octave, flrai pan octave, flrat parr Fig. 43. — Graphs of the interactions involving octave trans- positions of the first and last parts and the interpolation of tones. 133 Description of the Combinations in the Interactions The combinations of all three variables differed from the combinations of any two variables with respect to the size of the interval between the first and last parts of the tone series when the octave transpositions were combined with the interpolation of tones. Two variables combined The transposition of only one part of the tone series resulted in a compound interval midway in the tone series. For example, the simple interval between tones nos. 5 and 6 in the original statement was changed to a compound interval when the octave of the first part was raised (Figure 44). 1 1 2 H ? 7 a? 0 2 t o --- * ... 0 - . - - .. 7 '-I/— i?— M = / i 3 H sr ' 6 7 *1 > _ _ t hr- f-‘— = - t 1 p ■/ -- -- f ---- 4 ----- y Fig. 44.— Change of simple interval in the original statement to a compound interval when the octave of the first part is raised. 134 When both parts were transposed, the compound interval that resulted from octave transposition of only one part was either changed back to a simple interval or expanded to a larger compound interval (Figures 45 and 46). The transposition of both parts to the same octave changed the compound interval between the first and last parts to a simple interval, FIRST PART RAISED, LAST PART LOWERED t X 3 f 6 7 8 £_* = — - 7 T V fj i---- N # 1 7 “ FIRST PART LOWERED, LAST PART RAISED IX 3 i S' 4 7 s' Fig. 45.— Change of the compound interval to a larger compound interval. 135 The transposition of both parts to different octaves from each other changed the compound interval that was larger than one octave between the first and last parts to a compound interval that was larger than two octaves. FIRST PART RAISED, LAST PART LOWERED i z j v r 6 7 s FIRST PART LOWERED, LAST PART RAISED 3 v r 8 JL' 2 e T? Fig. 46.— Change of the compound interval to a larger compound interval. 13 6 The transposition of only the first part with the interpolation of tones changed a simple interval in the tone series to a compound interval midway in the tone series (Figure 47). FIRST PART RAISED FIRST PART LOWERED 1“ g j Fig. 47.— Change of a simple interval to a compound interval with the interpolation of tones. 137 The transposition of only the last part with the interpolation of tones changed a simple interval in the tone series to a compound interval midway in the tone series (Figure 48). LAST PART RAISED LAST PART LOWERED * u J - Fig. 48.—-Change of a simple interval to a compound interval with the interpolation of tones. 133 Three variables combined When all three variables were combined, the octave transposition of the first and last parts to the same octave changed the compound interval midway in the tone series to a simple interval in the presence of the interpolation of tones (Figure 49). FIRST AND LAST PARTS RAISED P i t — * — * ------!------4= — ,------— ------1------■ f — 1r 7 ^ | T ’“ ■} T T ! | ' ' ■ : - J M d — U ' i - J i - J FIRST AND LAST PARTS LOWERED I i 3 i r 6 7 * r n n - ■ . & n ^ r h ^ “ Fig. 49.— Change of a compound interval to a simple interval with the interpolation of tones. 139 When all three variables were combined, the octave transposition of the first and last parts to different octaves from each other changed the compound interval midway in the tone series that was larger than one octave to an interval that was larger than two octaves in the presence of the interpolation of tones (Figure 50). FIRST PART RAISED, LAST PART LOWERED FIRST PART LOWERED, LAST PART RAISED V- t -t- E it e + * f 4. -h_ t-. 1 - 4 - 1 I — ± —:-f— t v L-J Fig. 50.— Change of a compound interval to a larger compound interval with the interpolation of tones. 140 Interpretation of Results The results of the three-way interactions could be related to whether or not the size of the interval midway in the tone series was large enough in comparison to the surrounding intervals to separate the series into two groups. Ortraann (1926) conducted another study in which he was interested in how a change in the pitch of a single tone in a melody could affect how the other tones in the melody were heard. He constructed a test of what he called "melodic memory." In this test the subjcts were asked to compare two presentations of a series of tones in which the pitch of one of the tones was changed in the second presentation. He then analyzed the different pairs of tone series to determine what characteristics of the tone series might have influenced the subjects' ability to hear which tone in the series was altered. Among the characteristics he discussed were factors that could segregate tones in a series into groups. One factor he described was the difference between the interval sizes within a group of tones from the size of the interval between the groups of tones. He said, The clearness of definition of these groups depends upon the ratios existing between the pitch differences of any group and the pitch difference between the groups. If, for example, the tones within a group are all separated by half-tone steps, and the next group begins at the interval of a tenth, the demarkation would be very pronounced. (p. 12) When tones were interpolated there was a greater number of small- size pitch intervals (seconds and thirds) in the tone series from when there were only the original eight tones of the tone series. (In the original tone series a third was the smallest pitch interval and two of the seven intervals were thirds.) Therefore, when a compound interval occurred midway in the tone series, as a result of the octave transposition of one or both parts of the series, the ratio of difference between the size of the compound interval and the sizes of the surrounding intervals would be greater when tones were interpolated than when there were only the original eight tones present. Thus, the "clarity of definition" of the groups of tones before and after the "demarkation" [sic] by the compound interval would be more "pronounced" when the octave transpositions produced a compound interval in the presence of the interpolation of tones than when there were only the original eight tones. Presumably, the larger the compound interval the greater would be the ratio of difference and, therefore, the more "pronounced" the "demarkation" [sic] of the two groups of tones. The pitches of the first tones (tones nos. 1 and 2), middle tone (tone no. 5), and last tone (tone no. 8) were the same for the two tone series. The intervals which differentiated the two tone series occurred within these common pitches. A distinct separation of the series into two groups, that corresponded to the two parts that contained the differentiating intervals, could have helped the listener locate the differentiating intervals. The reduction of the compound interval to a simple interval, when the octave of the first and last parts were transposed to the same octave (both raised or both lowered) with the interpolation of tones, removed the separation of the tone series into two parts. Thus, the differentiating intervals were harder to hear and recognition of the contour was more difficult, 142 resulting in the decrease in the mean percent of correct recognitions. The expansion of the compound interval when the octave of the first part was raised, the last part was lowered, and tones were interpolated produced a distinct separation of the tone series into two parts. Thus, the differentiating intervals were easier to hear and recognition of the contour was less difficult, resulting in the large increase in the mean percent of correct recognitions. Perhaps, the lack of a strong three-way interaction when the octave of the first part was lowered, the last part was raised, and tones were interpolated could be related to the similarity of the arrangement of octave registers of the first and last tones in the series to the arrangement in the original statement of the tone series. In the original statement, the octave of the first tone in the series (g 1 ) was lower than the octave of the last tone (g 9). All of the two- way interactions in which the transposition of one or both of the parts produced this arrangement of octave registers like the original (Figures 45— first and last parts lowered; 46— first part lowered, last part raised; 47— first part lowered) resulted in increases in the mean percent of correct recognitions. All of the two-way interactions in which the transposition of one or both parts produced an arrange ment of octave registers of the first and last tones that was opposite to the original statement (the octave of the first tone in the series higher than the octave of the last tone) (Figures 45— first and last part raised; 46— first part raised, last part lowered; 47— first part raised) resulted in decreases in the mean percent of cor rect recognitions. The similarity in effects, among the two-way 143 Interactions that had in common the arrangement of octave registers of the first and last tones of the series, suggests that the arrangement of octave registers could be a factor that Influenced the interactions. The two-way interactions, in which the arrangement of octave registers was like the original statement, resulted in small changes in the mean percent of correct recognitions. And, the change in the results of the two-way interactions by the three-way interaction in which the arrangement of octave registers was the same as the original was very small. In the two-way and three-way interactions in which the first part was lowered and the last part raised, the expansion of the compound interval midway in the series merely produced an enlarged version (with respect to the number of different octave registers) of the arrangement of the octave registers of the first and last tones in the original statement. Perhaps, the similarity of the arrangement of octave registers over-rode the influence of any other characteristics of the variables, and thus, no strong interactions occurred. Interactions: Between Meter Change and the Interpolation of Tones (F, .038) Results of Interactions The interactions between changes in meter and interpolation of tones resulted in decreases in the mean percent of correct recogni tions from when each variable occurred alone. The amount of the decrease (16 percent) was greater when the meter was changed to three- four than the decrease (7 percent) when the meter was changed to six-eight. IkU The meter changes alone resulted in small increases in the mean percent of correct recognitions. When meter was changed from two- four to three-four the amount of the increase (3 percent) was slightly greater than the increase (1 percent) when meter was changed to six-eight. The interpolation of tones alone resulted in a decrease (13 percent) in the mean percent correct recognitions. Figure 51 illustrates the interactions. ioo — 90 — C£c uo 70 — 4>i_i e r; 4. 50 30 without w i t h Interpolation of tone* Fig. 51.— Graph of the interactions between meter change and the interpolation of tones. 145 Description of Combinations The number of tones per unit time was the same when tones were interpolated with the meter in two-four or when the meter was changed to either six-eight or three-four. However, the difference between the combination of interpolation of tones and the meter changes from two-four to either six-eight or three-four from when each of these changes occurred alone was the greater number of tones, pitch intervals, and changes in direction that occurred in the total time of the tone series. The total time of the tone series in the original statement was sixteen eighth-note durational units long (Figure 52). I a 3 V S 6 7 p * p J w ^ r — A j . 4r- — t— r ~ * 11 1 * \ I------'I______'I______It______I ^ s i i 3 ¥ t i i i 1 2 3 V i 2 3 V / * / i i v 5 < 7 g ? it> H ix n /f /r /t Fig. 52.— Total time in the original statement. 146 When meter was changed from two-four to six-eight or to three- four, the total time of the tone series was extended from sixteen eighth-note durational units to twenty-four eighth-note durational units; but only the eight tones of the original series occurred in this total time (Figure 53). METER CHANGED TO SIX-EIGHT I 1 3 V S 6 7 8 I------!l------1!______II______) / * I 2 3 *1 F i I 1 3 V T 4 I 2 3 Y f 6 /23 Vfi / s < i n ft t r ? id u ntitrttnn m&M METER CHANGED TO THREE-FOUR I X 3 H F 6 7 ? I______I!______II______II______I /= f 2 3 Y rt t x i 'i S~ U 12 3V rt /ll'il'i J* - i x 3 v r* // /i /j /7/f a*aiif Fig. 53.— Extension of total time. 147 When tones were interpolated, one tone occurred every eighth-note durational unit so that sixteen tones occurred in the total time of the tone series (Figure 54). A 3 a m - © + ■H _i t i 3 v i s x 3 H r * 7 i // n tv AT /h Fig. 54.— Number of tones in the total time when tones are interpolated. When meter was changed from two-four to six-eight or three- four and tones were interpolated, twenty-four tones occurred in the total time of the series (Figure 55). There were two differences between the combinations of the interpolation of tones and meter change to six-eight or meter change to three-four. Either or both of these differences could be related to the differences in the amounts of decrease in the mean percent of correct recognitions that result from these interactions. The following discussion of the differences between the combinations of the interpolation of tones and meter change to either six-eight or three-four will refer to Figures 54 and 55. METER CHANGED TO SIX-EIGHT t '— r W J_J - + © - + - + © - + 4- - .1 f * I t J « r * <13 t r t i i t h r t t !■ i « // it a a /r n it /I JC Jl 31 11 *>( METER CHANGED TO THREE-FOUR i 13 i r t 7 * f - ■ 9 * r t v f , , r t r=ff f f - f 4 ------y?— J u i ‘ J Cj J — —© 4- — — 0) t — 4- 4- — — +- i______n______ii______'i------1 /*■ J i J ¥ r * i l 3 ¥ r i il 3 * r t l Z 3 H ft / = I i i 1 f 4 I t 1 >6 i( /I n it /r it n If I* 20 il 11 23 2V Fig. 55.— Number of tones in the total time when tones are interpolated. 149 In these figures, the pitch interval directions in the repeated succession are labeled. The different directions in the repetition of the succession and the one in the original succession from which it differed are circled in the figures which illustrate interpolated tones. The lengths of the four equal segments to which the length of the repeated succession corresponded are bracketed with the dashed lines in all of the figures. When tones were interpolated, a succession of pitch interval directions that occurred with tones nos. 1 and 2 and the interpolated tones between them was repeated at equal intervals (Figure 54). The length of the succession was the same length as the length of each of the equal segments of time In the original statement. A pitch direction that was different from the original succession occurred in the repetition of the succession. The length of time, from the beginning of the succession to the occurrence of this "new" pitch interval direction, was the same as the length of the subdivision of the second of the four equal segments by tone no. 4 in the original statement. One pitch interval occurred in the time of each of the subdivisions of the four segments. When meter was changed from two-four to six-eight and tones were interpolated, the number of subdivisions was the same as when the meter was in two-four but the number of pitch intervals per subdi vision was increased (Figure 55). The length of time, from the beginning of the repeated succession of pitch intervals to the occurrence of the "new" pitch interval directions which occurred halfway in the succession, was extended from two to three eighth-note 150 durational units. The number of pitch intervals that occurred in this time was increased from one pitch interval to two pitch intervals per subdivision. When meter was changed from two-four to three-four and tones were interpolated, the number of subdivisions were increased from two to three, but the number of pitch intervals per subdivision was the same (Figure 55). The "new" pitch interval directions occurred so that the time of the repeated succession was subdivided into thirds. The length of time of these subdivisions was two eighth-note durational units long and one pitch interval occurred per subdivision. Interpretation of Results The decrease in the mean percent of correct recognitions, that resulted from the interactions when interpolation of tones and meter change were combined, could be related to the separation of the tones into groups by the repeated succession of pitch interval directions. As was discussed earlier, Ortmann (1934) thought that repetitions of successions of durations and pitches could result in tones being separated into larger groups which he called "higher units." He thought these repetitions helped the students in his study to hear and write melodies with better accuracy in the dictation exercises. He said that absolute length of a melody was not necessarily a determinant of how difficult it was for the students to remember and write it. Rather, that the presence of repetitions of successions of durations and pitches in a melody could reduce what a listener had to remember by separating the tones into larger groups. 151 Perhaps, the repeated succession of pitch interval directions did not sufficiently separate the tone series into larger groups when meter was changed from two-four to either six-eight or three-four and tones were interpolated. This would have compensated for the increase in number of tones, pitch intervals, and changes in direction when the meter was changed from two-four to six-eight or three-four and combined with the interpolation of tones. Thus, what the listener had to remember was not reduced, which made recognition more difficult, resulting in the decrease in the mean percent of correct recognitions. The greater amount of decrease in the mean percent of correct recognitions when interpolated tones was combined with meter change to three-four than with meter changed to six-eight could be related to the greater variety in the repeated succession of pitch interval directions when the meter was changed to three-fcut. When tones were interpolated with the meter in six-eight, each subdivision contained the same pitch interval directions (except for the third equal segment when the pitch interval directions of the second subdivision were inverted). When tones were interpolated with the meter in three-four, the first two subdivisions contained the same pitch interval directions but the last subdivision contained the opposite direction of the other two subdivisions (except for the third equal segment in which the pitch interval direction of the first subdivision was opposite the last two subdivisions). The result was that three-four, with interpolation of tones, contained more variety within the repeated succession of pitch interval directions than six-eight with interpolation of tones. This was because three-four 152 had the repetition and one nonrepetition while six-eight only had one repetition of pitch interval direction. Ortmann (1934) suggested that repetitions of pitch and duration successions, which resulted in the segregation of tones Into groups or "higher units," could reduce difficulties of correctly hearing a melody which contained a large variety of pitch intervals and durations. He discussed how the segregation of tones into "higher units" by repetitions of durations reduced the amount of variety among the durations in the following example. He said, The higher-unit reaction tends to lessen the pulse-variety error. M 12 involves four changes in time sequence. Reacted to in higher units it involves but two. (p. 23) M 12 J D J J 1 J Perhaps, the greater variety of pitch interval directions within the repeated succession of pitch interval directions, when meter was changed to three-four, might have resulted in the tones being less readily grouped into "higher units" than when there were only repetitions of directions within the successions when the meter was changed to six-eight. Hence, accurate hearing and thus, correct recognitions of melodic contour would be more difficult when meter was changed to three-four and combined with interpolation of tones than when meter was changed to six-eight. Therefore, a greater decrease in mean percent correct recognitions would result when meter was changed to three-four and combined with interpolation of tones than when meter was changed to six-eight. 153 Interactions: Among Meter Change, Interpolation of Tones, and Octave Transposition of the Last Part of the Tone Series (F, .019) Results of Interactions Three-way interactions The results of the three-way interactions involving octave transposition of the last part of the tone series, interpolation of tones, and meter change differed according to whether the meter was changed from two-four to six-eight or to three-four. When the octave transposition and interpolation of tones were combined with a meter change from two-four to six-eight, the mean percent of correct recognitions increased from when any two of these changes were combined. The amount of the increase (8 percent) was the same regardless of whether the octave was raised or lowered. When octave transposition and interpolation of tones were combined with a meter change from two-four to three-four, the mean percent of correct recognitions decreased from when any two of these changes were combined. The amount of the decrease (35 percent) was greater when the octave was raised than the decrease (5 percent) when the octave was lowered. Two-way interactions The two-way interactions, in which interpolation of tones and octave transposition were combined, resulted in an increase (7 percent) when the octave was raised and a decrease (3 percent) when the octave was lowered. 154 The Interactions between interpolation of tones and meter change resulted in decreases. A greater decrease (12 percent) resulted when the meter was changed to six-eight than the decrease (1 percent) when the meter was changed to three-four. The interactions between meter change to six-eight and octave transposition resulted in decreases, with a greater decrease (13 percent) when the octave was lowered than the decrease (6 percent) when the octave was raised. The interactions between meter change to three-four and octave transposition resulted in an Increase (6 percent) when the octave was lowered and a decrease (2 percent) when the octave was raised. Figure 56 illustratesthe two-way and three-way interactions. Description of the Combinations One difference resulted between combinations of three variables from the combinations of two variables. It was the combination of the compound interval, produced by the octave transposition, and the increased number of pitch intervals in the repeated succession that resulted when meter was changed and combined with the interpolation of tones. Two variables combined When tones were interpolated, the repeated succession of pitch interval directions contained three pitch intervals. The directions were arranged so that the time of the repeated succession was subdivided in half. The compound interval produced by the octave 100 ** with Ifiierpolaljan oj tone* THREE-WAY INTERACTIONS § wi &O Mt o 40 untransposed railed untrsnapcewl 1 ro e red octave, last pare octave, Lsst part 100 without tni*rpo-J« Lon of tones TWO-WAY INTERACTIONS 90 s w wi fca uo I i a I 40 wotrsnspossd Mtirt, last part ectsvt, last part Fig. 56. -Graph of the interactions involving meter change, interpolation of tones, and octave transposition of the last part. 156 transposition occurred between tones nos. k and 5 when the octave was raised and between tones nos. 5 and 6 when the octave was lowered (Figures 57 and 58). The directions of the intervals in the repeated succession are labeled. j a a i s 4 7 t n $ r - V + - [ - | Fig. 57.— Simple intervals in the original statement. OCTAVE ON THE LAST PART RAISED f f I £ ffijzzb | ^ ■■■= I : =3 J i| L J ___ 1 *--4 L— J - © +- - © +- -© 4 * - © OCTAVE OF THE LAST PART LOWERED s _ V . r- . . r a . % — © +■ © +■ — © 4- Fig. 58.— Increase of Intervals with a compound interval when tones are interpolated. 157 When meter was changed, the equal segments of time were extended from four eighth-note durational units to six eighth-note durational units. The intervening tones in these equal segments subdivided the time in halves when the meter was in six-eight and in thirds when the meter was in three-four. The compound interval produced by the octave transposition occurred between tones nos. 4 and 5 when the octave was raised and between tones nos. 5 and 6 when the octave was lowered (Figures 59 and 60). OCTAVE OF THE LAST PART RAISED 3 V JT ~f-L H*--;— i £ OCTAVE OF THE LAST PART LOWERED 2 r 0 * Fig. 59.— Change to a compound interval when the meter is changed to six-eight. OCTAVE OF THE LAST PART RAISED I I 3 H T 6 7 8 OCTAVE OF THE LAST PART LOWERED - p # — p — —j------j] J\ 9— z— f ---- 4^ 0 P f ----- ^ ------_ '# b H "d Fig. 60.— Change to a compound interval when the meter is changed to three-four. 159 When meter was changed and tones were Interpolated, the number of pitch intervals In the repeated succession was increased from three to five pitch intervals. The directions were arranged so that the repeated succession was subdivided in halves when the meter was six- eight and in thirds when the meter was three-four (Figures 57 and 61). The labels for the pitch interval directions that occur between the subdivisions are circled in the figures which illustrate meter changes and the interpolation of tones. METER CHANGED TO SIX-EIGHT X 3 H t t 7 8 €------*------j*— i------w— =-- 1— *— * ~f~ f— /— --- i rV , * ■-T', , J t -( ■ -I-- ;-j &—*>—j—Y- -— — — i H 1—--l— i- .4 j— —T r—.— * -— r r * r - * - -■ ■■■ ^ —i---- -I 1— '— , t— i— \— \ '-- —t = £I i- ‘Lbhhir11 I i i t 1V u " d L i i —.. ii i © 4' - + © - + - + - © - + - + © -h - - + ©-+- I » 1------1 1 I 1______I I 1 I______1 I ! I I METER CHANGED TO THREE-FOUR 9 ■* A m 3 3 = $ 'u u ^ t f u i r - © - © +- — © “ © J_ —• © -*- ©>+- Fig. 61.— Increase of pitch intervals when tones are interpolated. 160 Three variables combined When meter was changed to six-eight, tones were interpolated, and the octave was raised, the compound interval occurred between tone no. 5 and the preceding interpolated tone. The compound interval occurred between tones nos. 5 and 6 when the octave was lowered (Figure 62). OCTAVE OF THE LAST PART RAISED X 3 I I___ L - - — -1- © — OCTAVE OF THE LAST PART LOWERED ( X 3 1 J i 7 I 1 ,! J.._ \ -P- ...., = £ = i t ~ M ~ ' - | P f ± = - — *— i— i— * - \ * W — W — jy m ■' W - j|----- I © — ■+■ — + 0 -- H — 4- © — +- J ------1 i------\ I______I :______i i______i i______i i______I Fig. 62,— Change to a compound interval when the meter is changed to six-eight. 161 When meter was changed to three-four, tones were interpolated, and the octave was transposed, the compound interval occurred between tones nos. 4 and 5 when the octave was raised. The compound interval occurred between tones nos. 5 and 6 when the octave was lowered (Figure 63). OCTAVE OF THE LAST PART RAISED OCTAVE OF THE LAST PART LOWERED Fig. 63.— Change to a compound interval when the meter is changed to three-four. 162 Interpretation of Results The increases in the mean percent of correct recognitions from the interactions among octave transposition of the last part of the tone series, interpolation of tones, and meter change to six-eight could be related to the combination of two different factors that resulted in the division of the tones in the series into groups. The possibility of the division of the tone series into two groups resulting from the difference between the interval sizes of the compound interval midway in the tone series and the surrounding intervals, has already been discussed (supra, pp. 140 and 141). Also the segregation of the tones into groups by the repetitions of pitch interval directions within the repeated succession when meter was changed to six-eight was discussed earlier (supra, pp. 150 and 151). The repetitions are bracketed below the labels for the directions in Figures 62 and 63. There were no repetitions of pitch interval directions within the repeated succession when the meter was in two-four and tones were interpolated. The pitch interval direction in the second half of the succession was opposite that in the first half of the succession (Figure 58). Ortmann implies throughout both of his studies (1926, 1934) that accuracy in hearing and notating melodies was facilitated when tones in melodies were divided into groups because the groups reduced what the listener had to remember. Perhaps the repetitions within the repeated succession of pitch interval directions, when meter was changed to six-eight and tones 163 were Interpolated, made the separation of the tone series into two groups by the compound Interval easier to hear than was the case when there were no repetitions within the repeated succession when the meter was in two-four. The segregation of tones into small groups by the repetitions of pitch intervals and into two large groups by the compound interval midway in the tone series should have reduced what the listener had to remember. Thus, the result was an increase in the mean percent of correct recognitions. The fact that the same amount of increase resulted for both directions of octave transposition in these interactions could be because the benefits of the repetitions in the repeated succession would occur regardless of the directions of the octave transposition when they were combined with the separation of the tone series by the compound interval. The decrease in the mean percent of correct recognitions resulting from the interactions among the octave transposition of the last part of the tone series, interpolation of tones, and meter change to three-four, could be related to the combination of the large compound interval with the increase in changes in pitch interval directions. Ortmann (1934), found in his melodic dictation exercises, that melodies containing both changes in pitch interval directions and large intervals were the most difficult to hear and notate correctly. Further, he said that difficulty in correctly notating melodic intervals was directly related to the sizes of the intervals, with 164 the number of errors increasing as the sizes of the intervals increased. When the meter was changed from two-four to three-four and tones were interpolated, the number of pitch intervals in the repeated succession was increased from three to five, and the number of subdivisions of the repeated succession was increased from two to three. If the number of changes in direction between the subdivisions of the repeated succession were counted, there was one change in direction between the subdivisions when the meter was in two-four and two changes in direction when the meter was changed to three-four. Perhaps, the tone series was more difficult to hear when the increased number of changes in pitch interval directions in the repeated succession resulting from the meter change to three-four and the interpolation of tones was combined with the occurrence of a large compound interval midway in the series from when there were fewer changes in direction when the meter was in two-four. The difference in the amount of decrease in the mean percent of correct recognitions between raising or lowering the octave could be related to the concidence or lack of coincidence of the beginning of the second group of the tones, that resulted from the compound interval, with the beginning of the third occurrence of the repeated succession of pitch interval directions. (The beginning of the second group of tones that resulted from the division by the compound interval is marked by a i n Figure 63. And the beginning of the third occurrence of the repeated succession is marked by a ® in Figure 63.) 165 The effect of the coincidence or lack of coincidence of emphasis of tones at the beginnings of groups of tones on correctly hearing a melody has already been discussed (supra, pp. 127-129). The greater decrease, when the octave was raised, could be related to the lack of coincidence of the beginning of the second group of tones at tone no. 5 with the beginning of the third occurrence of the repeated succession. The lesser decrease, when the octave was lowered, could be related to the coincidence of the beginning of the second group of tones at tone no. 6 with the beginning of the third occurrence of the repeated succession. Apparently, the effect of the octave transposition in all of the three-way interactions with the interpolation of tones and meter change was specific to the transposition of the last part of the tone series since the only difference between octave transposition of the first part or the last part was which part of the tone series was changed. CHAPTER VI SUMMARY AND CONCLUSIONS Problem This study Investigated the problem of the recognizabllity of melodic contour under conditions of variation in the classical theme and variation form. As part of this problem, this investigation sought to establish experimentally the physical conditions for variations of five musical elements and to compare the effects of these variations singly and in combination on the recognizabllity of melodic contour. Sixteen Beethoven compositions in the theme and variation form were analyzed. Based on these analyses, four kinds of melodic variation were selected to be used in the experimental portion of the study. The four kinds of melodic variation were: 1) octave trans position of the pitches in the melody; 2) change in the order of the temporal intervals of the tones in the melody; 3) change in the meter and 4) interpolation of new tones between the tones of the melody. These four kinds of melodic variation based on variation procedures derived from the analyses of the Beethoven compositions were used in the experimental portion of the study. The objective of this study was to determine experimentally the effects of the transformations that were derived from Beethoven's 167 theme and variations on the listener's ability to recognize the contour of a series of tones. To pursue this objective, the following fourteen hypotheses were stated: Hypotheses 1. When there is an increase in the number of alterations present at one time in a variation there will be a corresponding increase in the difficulty of recognition of contour. 2. When the octave register of the first part of a tone series is transposed there will be an increase in the difficulty of recognition of the contour. 3. When the octave register of the last part of a tone series is transposed there will be an increase in the difficulty of recognition of the contour. A. When the octave register of the first part of a tone series is transposed the increase in difficulty of recognition of the contour will be greater than when the octave register of the last part of a tone series is transposed. 5. When the octave register of both the first and last parts of a tone series are changed and the pitch interval size between the beginning and ending of the tone series is changed the increase in difficulty of recognition of the contour will be greater than when only the first part of the tone series is transposed. 6. When the octave register of part of a tone series is transposed there will be no difference between the directions of the transposition in the increase in the difficulty of recognition of the contour, 7. When the octave register of both the first and last parts of a tone series are transposed there will be an interaction between the directions of the octave transpositions. a. When the direction of the octave transpositions of the first and last parts of a tone series results in a change of the pitch interval size between the beginning and ending of the tone series there will be an increase in the difficulty of recognition of the contour that is greater than the increase that would result from the summation of the effects of transposition of the octave of each part alone. 168 b. When the direction of the octave transpositions of the first and last parts of a tone series does not result in a change of the pitch interval size beteen the beginning and ending of the tone series there will be a decrease in the difficulty of recognition of the contour from when only the first or last part is transposed, 8. When the order of the temporal intervals that are delineated by the onsets of consecutive tones in a series is changed there will be an increase in the difficulty of recognition of the contour. 9. When the meter of a tone series is changed there will be no increase in difficulty of recognition of the contour. 10. When both the order of consecutive temporal intervals and the meter of the tone series are changed there will be an interaction between these changes that will result in an increase in difficulty of recognition of the contour that is greater than the increase in difficulty in recognition that would result when only the order of the temporal intervals or only meter was changed. 11. When both the order of the temporal intervals and octave register of parts of a series of tones are changed there will be an increase in the difficulty of recognition of the contour that is greater than when the order of the temporal intervals and the meter of the tone series are changed. 12. When tones are interpolated in a tone series there will be an increase in the difficulty of recognition of the contour. 13. When tones are interpolated in a tone series the increase in difficulty in recognition of the contour will be greater for the variation in which there are more interpolated tones per durational unit. 14. When the interpolation of tones in the tone series is combined with each of the other alterations in the temporal and pitch characteristics of the tone series there will be an interaction between these changes that will result in an increase in the difficulty of recognition of the contour that is greater than the increase that would result from the summation of the effects of the interpolation of tones and each of the alterations in the temporal and pitch characteristics. 169 Procedures An experiment was designed to produce comparisons among the effects of five variables— two kinds of octave transposition, two kinds of changes in temporal intervals, and the interpolation of tones, singly and in all combinations, on the subjects* ability to recognize the contours of two differing tone series (designated as Prototypes A and B). Prototypes A and B consisted of a series of eight tones that encompassed the octave g-1-1 to g 7 . The prototypes differed from one another only in the order of the sizes and directions of the pitch intervals between consecutive tones (i.e., the contours). The five variables that altered the pitch and temporal characteristics of Prototypes A and B were: 1) octave transposition of the first part of the tone series (VOF); 2) octave transposition of the last part of the tone series (VOL); 3) change in the order of temporal intervals (VOR) ; 4) change in the length and subdivisions of equal segments of time (referred to as "meter" in the problem statement) (VLS); and 5) interpolation of tones (VI). Sixty-six students at the University of Delaware were subjects in the experiment. They were grouped according to whether they were music majors or nonmusic majors. The music major group was subdivided into three groups according to the amount of formal eartraining that they had had. These groups were: freshman, sophomore, and mixed. A random numbers table was used to choose 162 of the 324 possible transformations and assign each transformation to either Prototype A or Prototype B in the experiment. 170 A 3^ factorial design was constructed In which two degrees of freedom for the four-way interaction are confounded. In this design each subject heard 54 of the 162 transformations. To minimize subject fatigue the experiment was divided into two sessions so that each subject heard 27 transformations per session. PLATO, a computer-based education system, was used to conduct the v experiment. The prototypes and transformations were produced by a four-voice digital synthesizer known as the Gooch MUSIC BOX. The prototypes and transformations were produced with a reference pitch of al3440 c.p. s. and were presented at a rate of an eighth-note durational unit per one-half second (tempo, The subjects heard the prototypes and transformations through earphones. The instructions were presented on the graphics screen of the PLATO terminal. The prototypes and transformations were presented to the subjects in an ABX triad arrangement. Following the presentation of each ABX triad the subjects were asked to indicate if they were able to recognize the prototype in the transformation. The subjects touched the screen to indicate their answer. In the triads there was a four-second interval between the prototypes and between the last prototype and the transformation. Response time was a twelve-second interval between each triad. An alert appeared on the screen one second before the beginning of each triad. The subjects were allowed to proceed to the next ABX triad sooner than the twelve-second interval. All responses and response-times were recorded by the computer. 171 The effects of each of the variables were indicated by the mean percent correct recognitions for the ABX triads for each of the conditions of each variable. Increasing difficulty in recognition of melodic contour was indicated by a decrease in the mean percent of correction recognitions and decreasing difficulty was indicated by an increase in the mean percent of correct recognitions. The data were analyzed to determine whether the number of altered conditions of the variables present at one time had an effect on the difficulty of recognition regardless of which variables occurred in altered conditions. The significance of the effects of the five variables and their interactions was determined by an Analysis of Variance. The significance of the effects of eartraining experience and practice of the accuracy and speed of recognition was determined by an Analysis of Variance. Results This study yielded information which supported hypotheses 1, 9, 12, 13, and a portion of 14, which are discussed below. Hypothesis 1: When there is an increase in the number of alterations present at one time in a variation there will be a corresponding increase in the difficulty of recognition of contour. The results of the analysis of the data revealed a correspondence between the percentages of correct recognitions and the number of alterations present at one time in the transformations. This correspondence resulted in a decrease in the percent of correct 172 recognitions when the number of alterations present at one time in the transformations increased. Thus, hypothesis 1 was supported. Hypothesis 9: When the meter of a tone series is changed there will be no increase in difficulty of recognition of the contour. The results of the Analysis of Variance for the effects of the five variables (supra, p. 104) on mean percent correct recognitions showed that there was no statistically significant increase or decrease in the difficulty of contour recognition when the meter was changed from two-four to six-eight or three-four (F, .228). There fore, hypothesis 9 was supported. Hypothesis 12: When tones are interpolated in a tone series there will be an increase in the difficulty of recognition of the contour. The results of the Analysis of Variance for the effects of the five variables showed that when tones were interpolated there was an increase in the difficulty of contour recognition that was statisti cally significant at the observed level of .001. Therefore, hypothesis 12 was supported. Hypothesis 13: When tones are interpolated in a tone series the increase in difficulty in recognition of the contour will be greater for the condition in which there are more inter polated tones per durational unit. The initial analysis of data showed a binomial distribution of subjects' responses that appeared as though subjects only guessed at the answers when there was the greatest number of tones per unit time that occurred in the experiment. It is presumed that the guessing of the subjects was because of the difficulty of the recognition task 173 when the interpolation of tones occurred in this condition. Therefore, hypothesis 13 was supported. Hypothesis 14: When the interpolation of tones in the tone series is combined with each of the other alterations in the temporal and pitch characteristics of the tone series there will be an interaction between these changes that will result in an increase in the difficulty of recognition of the contour that is greater than the increase that would result from the summation of the effects of the interpolation of tones and each of the alterations in the temporal and pitch characteristics. The two-way interactions between interpolation of tones and meter change were significant at the observed level of .04. Both of the interactions, when meter was changed from two-four to six-eight or three-four, resulted in decreases in the mean percent of correct recognitions from when each of the meter changes and interpolation of tones occurred alone. Therefore, hypothesis 14 was supported for the combination -of the interpolation of tones and meter change. This study yielded information which did not support hypotheses 2, 3, 5, 7, 8, 10, and a portion of 14, which are discussed below. Hypothesis 2: When the octave register of the first part of a tone series is transposed there will be an increase in the difficulty of recognition of the contour. The results of the Analysis of Variance for the effects of the five variables showed that there was no statistically significant increase or decrease in the difficulty of contour recognition when the octave register of the first part was transposed (F, .837). There fore, hypothesis 2 was not supported. Hypothesis 3: When the octave register of the last part of a tone series is transposed there will be an increase in the difficulty of recognition of the contour. 174 The results of the Analysis of Variance for the effects of the five variables showed that there was no statistically significant increase or decrease in the difficulty of contour recognition when the octave register of the last part was transposed {F, .791). Therefore, hypothesis 3 was not supported. Hypothesis 5: When the octave register of both the first and last parts of a tone series are changed and the pitch interval size between the beginning and ending of the tone series is changed the increase in difficulty of recognition of the contour will be greater than when only the first part of the tone series is transposed. Hypothesis 7: When the octave register of both the first and last parts of a tone series are transposed there will be an interaction between the directions of the octave transposition. a. When the direction of the octave transpositions of the first and last parts of a tone series results in a change of the pitch interval size between the beginning and ending of the tone series there will be an increase in the difficulty of recognition of the contour that is greater than the increase that would result from the summation of the effects of transposition of the octaves of each part alone. b. When the direction of the octave transposition of the first and last parts of a tone series does not result in a change of the pitch interval size between the beginning and ending of the tone series there will be a decrease in the difficulty of recognition of the contour from when only the first or last part is transposed. The results of the Analysis of Variance for the effectsof the five variables showed that there was no statistically significant interaction between the octave transpositions of the first and last parts of the tone series (F, .963). Therefore, the lack of a statistically significant increase or decrease in difficulty greater than changes when only one part was transposed did not support hypotheses 3 and 7. 175 Hypothesis 8: When the order of the temporal intervals that are delineated by the onsets of consecutive tones In a series is changed there will be an increase in the difficulty of recognition of the contour. The results of the Analysis of Variance of the effects of the five variables showed that there was no statistically significant increase or decrease in the difficulty of contour recognition when the order of the temporal intervals was changed (F, .232). Therefore, hypothesis 8 was not supported. Hypothesis 9: When both the order of consecutive temporal intervals and the meter of the tone series are changed there will be an interaction between these changes that will result in an increase in difficulty of recognition of the contour that is greater than the increase in difficulty in recogni tion that would result from when only the order of the temporal intervals or only meter was changed. The results of the Analysis of Variance of the effects of the five variables showed that there was no statistically significant interaction between changes in the order of temporal intervals and meter change (F, .734). Therefore, the lack of a statistically significant increase or decrease in difficulty of contour recognition greater than changes in difficulty when only meter or order was changed did not support hypothesis 9. Hypothesis 14: When the interpolation of tones in the tone series is combined with each of the other alterations in the temporal and pitch characteristics of the tone series there will be an interaction between these changes that will result in an increase in the difficulty of recognition of the contour that is greater than the increase that would result from the summation of the effects of the interpolation of tones and each of the alterations in the temporal and pitch characteristics. The results of the Analysis of Variance of the effects of the five variables showed that there was no statistically significant interaction 176 between the interpolation of tones and octave transposition of the first part of the tone series (F, .981), nor between the interpolation of tones and octave transposition of the last part of the tone series (F, .928), Therefore, hypothesis 14 was not supported for the combinations of octave transpositions and the interpolation of tones. The interaction between the interpolation of tones and changes in the order of temporal intervals resulted in an F-ratio with a signifi cance level of .08. Although this significance level is approaching chance occurrence, it appears to be aligned with the statistically significant interactions rather than with the interactions that had nonstatistIcr.Tly significant F-ratios. This interaction between the interpolation of tones and changes in the order of temporal intervals resulted in a decrease in the difficulty of contour recognition that was greater than the change in difficulty when interpolation of tones and change in the order of temporal intervals occurred alone. Therefore, hypothesis 14 was not supported for the combination of change in the order of temporal intervals and the interpolation of tones. The support or lack of support of hypotheses 4, 6 , and 11 could not be determined because of the lack of statistical significance for the effects of the variables pertinent to these hypotheses. These will be discussed below. Hypothesis 4: When the octave register of the first part of a tone series is transposed the increase in difficulty of recognition of the contour will be greater than when the octave register of the last part of a tone series is transposed. 177 Hypothesis 6 : When the octave register of part of a tone series is transposed there will be no difference between the directions of the transposition in the increase in the difficulty of recognition of the contour. The results of the Analysis of Variance of the effects of the five variables showed that there was no statistically significant increase or decrease in the difficulty of contour recognition when only the octave of the first part was transposed, or when only the octave of the last part was transposed. Therefore, the support or lack of support for hypotheses 4 and 6 could not be determined. Hypothesis 11: When both the order of the temporal intervals and octave register of parts of a series of tones are changed there will be an increase in the difficulty of recognition of the contour that is greater than when the order of the temporal intervals and the meter of the tone series is changed. The results of the Analysis of Variance of the effects of the five variables showed that there was no statistically significant interaction between changes in the order of the temporal intervals and meter change. Therefore, the support or lack of support for hypothesis 11 could not be determined. There were four statistically significant interactions. One interaction, having a significance level that approached chance occurrence (F, .08), was aligned with the statistically significant interactions rather than with the nonstatistically significant inter actions. Only one of these five interactions was predicted. That interaction was between the interpolation of tones and meter change. These five interactions are discussed below: 1. The two-way interactions between interpolation of tones and meter change were significant at the observed level of .04. Both of the interactions, when meter was changed from two-four to six-eight or three-four, resulted in decreases in the mean percent of correct recognitions from when each of the meter changes and interpolation of tones occurred alone. The two-way interaction between interpolation of tones and changing the order of temporal intervals was significant at the observed level of .08. This interaction resulted in an increase in the mean percent of correct recognitions from when each variable occurred alone. The two-way interactions between octave transposition of the first part of the tcne series and changing the order of temporal intervals were significant at the observed level of .04. The results of the interactions differed according to the direction of transposition. Raising the octave resulted in a decrease in the mean percent of correct recognitions; lowering the octave resulted in an increase in the mean percent of the correct recognitions from when each variable occurred alone. The three-way interactions among octave transposition of the first part of the tone series, octave transposition of the last part of the tone series, and the interpolation of tones was significant at the observed level of .04. The interactions in which both parts were transposed to the same octave resulted in decreases in the mean percent of correct recog nitions from the results when only two of the variables were combined. The interaction in which the first part was raised 179 and the last part was lowered resulted In an increase. The result of the interaction in which the first part was lowered and the last part was raised was negligible. 5. The three-way interactions among meter change, octave transposition of the last part, and the interpolation of tones were significant at the observed level of .02. The results of the interactions differed according to whether the meter was changed from two-four to six-eight or three-four. Changing meter to six-eight resulted in increases in the mean percent of correct recognitions from when only two of the variables were combined. Changing the meter to three-four resulted in decreases in the mean percent of correct recognitions. The Analysis of Variance of the effects of the five variables revealed that a large amount of variability, of which about 14 percent was accounted for, occurred in the experiment. This study yielded information about the effects of practice and amount of subjects’ eartraining experience on the accuracy and speed of melodic contour recognition. The results of the Analysis of Variance (supra,p.107) showed that there was no statistically significant increase or decrease in the mean percent correct recognitions between sessions one and two. The Analysis of Variance (supra,p.109)showed that there was a decrease in the mean response times from session one to session two that was statistically significant at the observed .01 levels These results indicate that subjects became considerably faster 180 at responding without improving greatly In accuracy of recognition from session one to session two. None of the differences among the four groups of subjects with respect to the mean percent of correct recognitions and mean response times* nor in differences between sessions one and two in the mean percent correct recognitions and mean response times were statistically significant. Discussion and Conclusions The variables and their range of variation which were used in this experiment were based on theme and variations by Beethoven which had been designated by musical scholars as compositions in which the melodic contour of the theme was a characteristic that the variations and theme had in common. If these scholars were correct, the majority of kinds and ranges of alterations chosen by Beethoven were those which maintained the identity of the melodic contour of the theme. It could therefore be assumed that in experiments which tested the effects of kinds and ranges of alterations based on Beethoven's compositions, no significant increase in difficulty of melodic contour recognition would occur as a result of the majority of alterations. The meter changes, change in the order of temporal intervals, and octave transpositions of the prototype tone series were alterations used in the experiment that had no significant effects on the recognizabllity of the contour of the tone series. Since these variables were derived from the Beethoven compositions, these particular alterations and their range of variation, like Beethoven's, 181 did not affect the recognizabllity of the contours. The difficulty of contour recognition was significantly Increased when tones were interpolated In the prototype tone series. Therefore, it is concluded that the difficulty of recognition of the theme's melodic contour in Beethoven theme and variations is increased significantly in the variations which contain interpolated tones such as Variations one and three, Op. 34. Further, the transformations containing the greatest number of interpolated tones resulted in the greatest increase in the difficulty of recognition. Therefore, in Op. 34, Variation one would result in a greater increase in difficulty of melodic contour recognition than Variation three, because Variation one contains more interpolated tones than Variation three. A common assumption among musicians is that the more differences between the theme statement and the variations, the more difficult it will be for the listener to recognize the theme in that variation. This assumption seems to exist regardless of what elements of the theme have been varied. In the experiment, the recognizabllity of contour differed depending upon the number of altered conditions of the variables present in a transformation at one time, with a corresponding increase in difficulty for an Increase in the number of altered conditions. These results suggest that the assumption mentioned above is correct. The variations in Beethoven's Op, 34 can be ranked from most to least difficult with regard to the recognizability of melodic contour of the theme. 182 1. Variation two, with three alterations 2. Variation one and Variation four, each with two alterations 3. Variation three, with one alteration The analyses of the combinations of the variable in the interac tions showed that the variables affected the recognizabllity of the contour through their effect on the organizational characteristics of the tone series. This information should provide points of departure for further experimental research about changes in those aspects of the variables which might result in significant changes In the recognizabllity of melodic contour. The effects of the differences in the amount of eartraining experience among the groups of subjects may not have appeared because of the diversity in college rank of the subjects in the mixed majors group, and the diversity of academic disciplines and college ranks of the subjects in the nonmajors group. A perusal of the raw data revealed a wide range of differences in the mean percent of correct recognitions among some of the subjects within the nonmajors group. These subjects represented a diverse range of academic disciplines (e.g., computer science, chemistry, elementary education) and all college ranks (freshmen through seniors). This heterogeneity, particularly with respect to the diverse academic disciplines, may have had a greater effect on the responses of the subjects than the differences among the groups in the amount of eartraining experience. On the other hand, it is possible that the recognition task in the experiment was a "leveler 11 among the groups of subjects that had 183 different amounts of eartraining experience. Dowling (1971) found that correlation coefficients were very low between the number of years of music training of the subjects and their ability to recognize melodic contour in his experiment. He concluded that "... precision of pitch and interval judgment encouraged by musical training is of little help in recognizing melodic contour" (p. 528). Perhaps, the listening skills used in the present experiment were different from those learned in the eartraining courses in which the music majors were enrolled. Therefore, the amount of formal eartraining or lack of it was not a deciding factor in how well the subjects were able to recognize melodic contour in the experiment. The lack of statistically significant improvement in recognition from session one to session two could be because of the kind of similarity between the two tone series used as prototypes in the experiment. The first part of one tone series contained pitches which were the same as those occurring in the last part of the other tone series. The similarity of pitches occurring in different parts of the two tone series could have caused a high degree of confusion in the subjects. This confusion would not necessarily be reduced by more practice. The subjects became significantly faster at responding without improving significantly in recognition accuracy. This could be because the recognition task resulted in boredom and the subjects became more prone to respond quickly as the test progressed in order to finish it more quickly. A primary cause of the large amount of error variability in the experiment may have been due to subjects* boredom with the experimental task. The boredom could have been influenced by the many repetitions of the two prototype tone series, or the kind of similarities between the two tone series (the first part of one being like the last part of the other), or the number of ABX triads which the subjects listened to at one time, or any combination of these factors. Post-experimental reports of the subjects indicated that for some portions of the test they had confused the two tone series and identified the opposite one from what they had intended. Apparently, the presentation of the two tone series with every variation did not alleviate this confusion (except at the point where they discovered their mistake), and only added to the boredom of the subjects. Many of the subjects said that they tired of hearing the two tone series so much, and thus paid attention only when the variation was presented. Some subjects said they felt that the number of triads per session (27) was good for them because they felt they improved with practice. Other subjects said that they felt taxed by the number of triads per session even with the three one-and-a-half minute rest periods that were interspersed in the test. However, the individual scores were not consistent with either one of these subject reactions. Another factor that may have contributed to the variability of responses was that the subjects may have attended to different parts of the tone series in different variations. Many of the subjects said that they were able to hear the intervals in the first part of the tone series that differentiated the two tone series, but were not 185 confident of the Identity of the variation until after hearing the differentiating intervals in the last part of the tone series. Some of the subjects said they particularly attended to whether the last interval that they heard was the fourth in Prototype A or the sixth in Prototype B. Perhaps, some variations enabled the subjects to hear the differentiating intervals in one part better than in the other part of the tone series. If the subjects had chosen the part to attend to that was more difficult to hear in a particular variation than it was in a previous variation, the response strategy of the subject could be disrupted and thus contribute to the variability of the responses. Suggestions for Further Research Based on the results of this study, two areas of investigation seem particularly worthwhile: 1) the extension of ranges of alteration of variables that did not significantly affect recognizabllity of contour in this experiment; 2) the alterations of amount of time occupied by different pitch intervals in a tone series and their effect on the recognizabllity of contour of the tone series. The extension of the ranges of alteration of the variables that showed no significant effects in the experiment would provide information about whether or not under different conditions these variables affect the recognizabtlity of melodic contour and, if so, what the limits are for the recognizabllity of melodic contour. The information about these variables and their ranges of alteration could be used as a basis for measurment of the recognizability of melodic 186 contour under conditions of variation in other compositions by other composers. It was suggested earlier (supra, p. 182) that greater changes in those aspects of the variables that affected the organizational characteristics of the tone series might result in significant changes In the recognizability of melodic contour. The results of the analyses of the interactions indicate some ways to change the vari ables that could result in greater changes in the organizational characteristics of the tone series. These indications are amplified by observations made by Ortmann (1934) about the effects of increasing and decreasing the amount of changes in organizational characteristics of melodies on the task of melodic dictation. The analyses in this study showed that the coincidence or lack of coincidence on the same tone of the characteristics that segregated the tone series into groups varied for different combinations of the variables (for example, when the octave transposition of the first part and the repetition of temporal intervals were combined). Ortmann (1934) said that the difficulty in correctly hearing and writing melodies could be Increased or decreased depending upon whether or not factors that segregated the tones of a melody into groups coincided on the same tone. He also said that the difficulty in correctly hearing and writing melodies could be increased or decreased depending upon the number of grouping factors that did or did not coincide on the same tone. An experiment in which the recognizability of melodic contour was tested under conditions with different numbers of characteristics that 187 segregate tones into groups to coincide or not coincide would show whether greater change in these organizational characteristics of a tone series affected the recognizability of melodic contour. The analyses also showed that the amount of repetition in temporal and pitch characteristics of the tone series varied for different combinations of the variables (for example, when the interpolation of tones and changing the order of temporal intervals were combined). Ortmann (1934) found that difficulty in correctly hearing and writing melodies increased or decreased according to the amount of repetition of pitches and durations that was present in a melody. An experiment in which the recognizability of melodic contour was tested under conditions with different amounts of repetition in temporal and pitch characteristics would show whether greater change in these organizational characteristics of a tone series affected the recognizability of melodic contour. The further study of alterations of amount of time occupied by different pitch intervals in a tone series could provide information about the effects of such alterations on the recognizability of melodic contour. In this experiment the variable, changes in the order of temporal intervals, altered the amount of time occupied by different pitch intervals. This variable did not significantly affect the recogniza bility of the contour of the prototypes in the experiment. This result might have been influenced by the kind of prototypes used. Some of the subjects reported after completing the experiment that they listened to the last pitch interval in the tone series as 188 the basis for their decision about the identity of the tone series in the variation. It is assumed that temporal intervals seem to affect the recognizability of contour by influencing the importance of single pitch intervals relative to the whole succession of pitch intervals In distinguishing one contour from another. If this is the case, proto types should be developed which would not allow subjects to use only one differentiating interval as the basis for recognizing the contour. In an earlier discussion of the effects of temporal intervals on melodic contour (supra, pp. 24 and 25), it was stated that the concept of contour, as it is generally accepted among musicians, does not consider all pitch intervals in a series of equal importance when differentiating one melodic contour from another. It was asserted that the temporal intervals in the succession of pitch intervals affected which pitch intervals were important to differentiating one melodic contour from another. The following examples presented during that discussion are presented here (Examples 1-3, Figure 64) to illustrate kinds of alternative prototypes which might be used in future studies of the effect of alterations of temporal intervals in melodic contour recognition. Tone Series A and B (Example 1, Figure 64) are the same with respect to their highest and lowest pitches, but are different in the order of consecutive pitches and pitch intervals. The directions of the consecutive pitch intervals In Tone Series A and B are labeled directly below the staff in Examples 1-3. The range of high and low pitches is the same In both series and these pitches are indicated with a ’I' above the appropriate notes in the examples. Both tone 189 Example 1 i m i 4 _ ♦ + ♦ 4- Example 2 JVI X3 A. S l.3 H € U n 9 4 ♦ Example 3 If I A^ 4 m ------^ji--- r t ^ H HH b n = i y t L 3 H 7 1 1 A . x .3 i" t T t A..: ♦ 4- ♦ 4 + 4- 4- 4- Fig. 64.— The effects of alterations of amount of time occupied by different pitch intervals in two different tone series on the identity of their contours. 190 series consist of the same overall pitch interval directions if only the pitch Interval directions between the high and low pitches in the succession are considered. The directions of the pitch intervals of the high and low pitches in the succession are labeled below the labels for the consecutive intervals in Examples 1-3. The intervening tone (d^) between the high pitch (g^) and low pitch (b^) in Tone Series A becomes part of an interval that has the same direction as the one between the g 2 and b 1 . In Tone Series B, the intervening tone (d^) between the low pitch (b^) and high pitch (g^) becomes part of an interval that has the same direction as the one between b 1 and g 2 . In Example 2, Figure 64, the temporal intervals between pitches i that define the range in the succession (g to g 2 ; g^2 to b 1 ; b-1-1 to g 2) are the same. This makes the presence of the intervening pitch interval before the b^ in Tone Series A (g^ to d^) and after in Tone Series B (d 2 to g2 ) of little importance to differentiating the two tone series. Thus, the two tones series may sound alike. In Example 3, Figure 64, the temporal Intervals between the pitches that define the range in the succession are different from those occurring in Example 2, Figure 64. Also, the intervening pitch intervals before and after the b^ in the respective tone series occur through a larger proportion of the total time of the tone series in Example 3, Figure 64 than in Example 2, Figure 64. It is asserted that the intervening pitch intervals would seem to be more important In differentiating the contours of the tone series in Example 3, Figure 64 than in Example 2, Figure 64. 191 These observations are worthy of further study because of their importance in determining the recognizability of similarities and differences between melodies and melodic contours. The variation techniques in the Beethoven theme and variations, which were used as a basis for the variables in the experiment, are generally considered by musicians to be simple kinds of musical variations. However, these had never been used in an experimental situation. These variables are examples of how complex such musical variables become when they are used as stimuli in an experiment. The variables were complex with respect to the many ways that they could be characterized. The composite of characteristics of the variables when they were combined was different from the characteristics of the variables when they occurred alone. This study demonstrated that it is possible to describe these complex musical variables according to their physical characteristics and to define them independently from each other so that they could be manipulated under controlled conditions in an experiment. Recognition of musical elements— such as melodic contour— under conditions of variation is one of the everyday skills required of musicians. However, training to improve this skill in eartraining courses is generally approached from a round-about way. This research and subsequent research could provide information about methods for directly approaching the improvement of this skill. The experimental determination of how variables used by composers affect the listener serves to identify those compositional techniques that matter to the listener. By identifying these techniques, eartraining procedures can be focused on them. 192 The Idea that no rules could be made about how to decisively determine whether or not a musical element was present under conditions of variations was sometimes expressed in the literature in which the theme and variation form was discussed (e.g., Berry* 1966; Eschmann, 1968). The authors explained this was because the ultimate decision about the invariance of a musical element must be left to the judgments of the individual listener since the amount of perceived invariance was influenced by the musical knowledge and experience of the listener. It is certainly true that the veracity of musical knowledge ultimately must be tested by the human perceiver. It is also true that the growth of musical knowledge is dependent upon collective decisions among musical scholars about individual musical experience. Experimental investigation of the experiences of the music listener is the means by which these experiences may provide an organized set of principles against which assertions about who should be hearing what can be tested. Then there will be a factual basis for the col lective decisions of musical scholars which contribute to the growth of musical knowledge. APPENDIX A List of Beethoven Theme and Variations Used As a Basis for the Transformations 193 Symphony No. 3, fourth movement Symphony No. 5, second movement Symphony No. 9, second movement Op. 14, No. 2, second movement Op. 13, No. 5, third movement Op. 26, first movement Op. 30, No. 1, second movement Op. 34, Six Variations on an original theme. Op. 35, Fifteen Variations on a theme from Eroica Symphony Kighini Variations, Twenty-four Variations in D major. Op. 57, second movement Op. 61 second movement Op, 97, third movement Op. 109, second movement Op. Ill, second movement Op. 127, second movement APPENDIX B Notation of the 162 Transformations Used in the Experiment The number with the symbol "X" at the left of the notation corresponds to the number used to identify the transformation in the statistical design. 195 196 X-l T- - * J * r------~:r-— — » *17 . 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KZt i y G X - ^ J . - i_ j -T... ,r X - i*- £ -tl*- -M- I___ 1 f X - / 5 " 3 JitlV -r# '------nfc------i l> ------h£------= ^ = ------l_;------.------L ------r-r------* ------4 ^ t = T" ~ r ~ X-M_ - ^ 1 t— p f W 7 X-/51 k XZ u j y i £ i f l_L J L T V c\| I CM — tun CM I ' _ ♦ H I ♦ i tl 1 h- s til •H k * 1k __ ,. ■f + * f -- till > <. „ Ilf -- 1IH i- til ■ +l tt L‘-k J I > j — -K k % - 'A.T * ' - t i fftt . _ . .. \ :> t ; : f c -WV I— 11 » ’, i T" "s 1 >1 J- — k. Ob o- e :: *n X -4 +i» h i I APPENDIX C Introductory Material Presented to Subjects 223 224 Make sure That the music box is turned on and that your headphones are plugged in. In this stv.o'y you. will be asked to compare s o m e ’-'=-r i at : cns o f two m e 1 o clies to- the: i r or i :gi na 1 versions. Th e me I oldies will be labeled "Mel'do ft" and Mis loot' 5". The purpose of these compar i sons is to deter rr, ire if M e l o d y ft. and Meic-dy Ed can be reconr: z sc when they have urder-o'ore c h a n g e s i n meter, octav-5 tranepos. 111 or, charges in agognc accents, and the add it ion ■-i" ornamental tones, (Mote: The tempo or the me lo ck as and the variations is. the same in all the pres art at i one- j Ms 1 ody H and Melody £ can be recoin iced based on P r e s s the n e ; S'--1 t O' hear the m s had i as 1 isteri caret u.i how thev. diiian Press NEXT each time you finish reading. 1— Lx/j 'i-1 1 L-L T -_JLlcr*l-_Lr^r i l l f_-* ! 1 i *3* _-' L _'_*-_'L^ ' I . ,■ • J , J t n r i. ■ .■*> ^ i * ■—r-. ; —■—=j ' f-L i Li t. '±2 ^ * L L^ T cLU 'Zr'Lj J- l’-'l-j- Li '^v L, '=:: I T ■-■-r r^iSLioo g .V*I=14 pus y c;.^ t T ^,||j ~j< j ^ — . I - ” "L i Li=r=':'v'[^.«-:-; J.U-='-i-=. J J -p T O A . i O J L L I J O SUJ r—-i-1 * ! L-_J r- ’ ■- * ~ Z.' '— ' — * > [ ^ ' - _ t _ _-<- — ' — - _j_ b 5 L |C '1 1 O O 1 It JO .lOp'-lO O'-J y ■ T isvVem u i a i u o .i—q^o ujouj :;:> es t. p o ; s-uj 3*44 ^ 4 1 sd t ycy.j SZZ 226 The difference in ups and downs between Melody ft and Melody & occurs between tones 3 & 4 and 6 7 . Press NZXT and 1 isten to the melodies once aoair and not ice vtare the intervals in Melody ft arc Melody £ are different. 227 The di f ferences in interva1s between Melody Fi and Melodvt B occur between tones 2 & 3, 4 & 5, 5 & 6, and 7 & 3. 228 The differences in intervals between Melody ft and Melody B occur between tones 2 & 3, 4 & 5 , 5 & 6, and 7 & S . Listen c-nce mere to Melody ft and Melody B, arid be sure you. hear whe'"e these differences bet wee-’ the melo d i s s -:c cur. 229 You. will hear 2 7 different ve.r i at i ons of e i t h e r Melody ft or Melody B. Each of these variations will always be preceded by Melodies ft and B. Lach pairirie; of the meiodie s w i t h o n e of t h e 2 7 d i f fen en:t var i at i o ns i s 03.1 led a ccmroar i son s e t . ftfter the presentation of each comparison set you. I'.ii 1 1 be asked to choose between the fo 1 1 owir::g statements the one that best indicat* the ioientity of the variation that you heard l rr the s e t . r 230 Touch the boy that i rd i cates- how that variation sounded to you. more m o r e 1 I ike 1 ’ T- -= me 1 o m e 1 ody ft B Y o u will be asked to choose one of these statements by touching* the terminal screen in the square which contains the appropriate statement for- your answer. Touch either one now to go on. 231 Touch "the box that indicates how that variation sounded to you. m o r e 1 ike rne 1 oof mel cciv Be sure the arrows show on your screen after you touch the bc-x. The arrows indicate only that, your response is recorded--not whether it is right or wrong. Press NEXT to continue. 232 The no 1 1 -owinor three comparison sets ere foA-U practice. ftf'ter vov. have raspon-ded t-o esc ■comparison set, you. will foe to Id which melody was varied in that set. This i.: for oractice. The identity -of t lie v a n at l ore. will not o;; veri t-o yc-u. in the test. 233 -P ♦ fH -P * ■ct ■ H ■p ■ H -P H ■ti e— r 0 i' > C il in i 0 V 0 II p L i rri u i.O 0 V t e 10 0 c T H i i £ < r-H Q ~ J ■»H X f £ — nI*-* qi H in r V U o t— o V 2 1 - V D- Q -p -p -H ■P • H ' pH w vi ii 1.0 nj r 0 I- o 234 Practi cee compari son set 1: rou are now hear i ng; rne 1 ody R You. are now hearing melody B You. are now hear i ng a practi e varla t i on . Touch the box that indicates how that variation sounded to vou.. more more 1 ike 1 ike m s 1ody rne 1 cob R 235 P i gh t , the y?.r i a t i on i rt practice comparison set 1 is mc-re 1 ike M e l o d y ft. Press NEXT for pract i ce compar i son set 2 . 236 Pract i ce compar i son set 2 : You. are now hearing melody fl. You are now hearing melooly B. You. are now hear 1 ng a practice vaniat ion, Touch the box that indicates :ha.o variation sounded t-o you. mo r e mo r e 1 ike I ike me 1 O'do m e 1ody p B 237 Right, the variation in practice comparison set is more 1 ike Melc-dv R. Press NEXT for practi ce comoari son set 3. ✓ 238 Pr act i ce corripar i so n set: 3 : You a r e now hearirig melody Ft. You are now hearing: melody B. You are now h e a r i ng: a pract i ce vat'- i a t i on , Touch the box that indicates how that va.riat 1 ori sounded to you. m o r e I r r i ' O r e l i1 f c - s i * 1 i k e r e i ' s i i'-.r *! 1 o d v A ! B . . . . i 239 R i ght f the var i at i on i n pr-act i ce compar i son set is more like Fie1odv B . Frees NEk'.T to continue. In comparison set 3, Melody 5 was varied the addition of ornamental tones. Press NEXT to hear how Melodo M sounds 241 You. a r e n o w hearing melody fi. You are now hearing the variation of ft. In comparison set 3. Melody B was varied by the addi t ion o f ornamentsI tones. Press NEXT to hear how Melody ft sounds when it is varied in the same way. 242 The order of melodies ft and B will a l t e r n a t e with each presentation of the comparison sets. You will be shown the order on the screen. Pay careful attention to the order of Melody ft arid B on the screen. Hi is will h e l p you. a v o i d c o n f'u.s 1 rig M e 1 od y ft a n d B . Y o u will have 12 seconds to respond to each cornparison set. If you wish, you may proceed to the next set sooner than this, but you cannot take longer than 12 seconds to respond. Please respond to all presentations. If you find some you are not sure of, r e s p o n d with y o u r b e s t guess. If y o u find some that are difficult, r e m e m b e r that there wi11 be easy ones too! P r e s s NEXT to begin. BIBLIOGRAPHY Apel, W. (ed.), 197 0. Harvard dictionary of music. 2nd ed. revised. Cambridge, Mass.: Belknap Press of Harvard University Press. Berry, W. 1966. Form in music: An examination of traditional tech niques of musical structure and their application in historical and contemporary styles. New Jersey: Prentice-Hall, Inc. Colies, H. C. 1954. "Variations," Grove’s dictionary of music and musicians. 5th ed. E-ic Blom (ed.) 10 vols. New York: St. Martin's Press. Deutsch, D. 1972. "Octave generalization and tune recognition," Perception and Psychophysics. 11, 411-412. Dowling, W. J. 1971. "Recognition of inversions, melodies and melodic contours." Perception and Psychophysics. _9 (3B), 348-349. ______. 1972. "Recognition of melodic transformations: Inversion, retrograde, and retrograde inversion." Perception and Psychophysics. 12 (5), 1972, 417-421. Dowling, W. S. and D. S. Fujitani. 1971. "Contour, interval, and pitch recognition in memory for melodies." Journal of the Acoustical Society of America. 49 (2), 524-531. Eschman, K. 1968. Changing forms in modern music. 2nd ed. Boston, Mass.: E. C. Schirmer Music Co. Godwin, P. M. 1972. A study of concepts of melody, with particular reference to some music of the twentieth century and examples from the compositions of SchBenberg, Webern and Berg. Unpublished Ph.D. dissertation, The Ohio State University. Goetschius, P. 1915. The larger forms of musical compositions: An exhaustive explanation of the variations, rondos, and sonata designs, for the general student of musical analysis, and for the special student of structural composition. New York: G. Schirmer. Goldshine, R. F, 1971. Temporal dimensions of music. Unpublished M,A. thesis, The Ohio State University. 243 244 Hofstetter, F. T, 1977. "Music dream machines: New realities for computer-based musical instruction." Creative Computing. Vol. 2 (2)» 50-54. MacPherson, S. 1915. Form in music: With special reference to the designs of instrumental music. New York: Mills Music, Inc. Nelson, R. U. 1948. The technique of variation. Berkeley; University of California Press. ______. 1962. "Stravinsky's concept of variations." Musical Quarterly. 48 (3), 327-339. Ortmann, 0. 1926. "On the melodic relativity of tones." Psychological Monographs. 35 ( U » Whole No. 162, 1-47. ______. 1934. "Problems in the elements of ear-dictation." Research Studies In Music. No. 2. Baltimore, Md.: Peabody Conservatory of Music. Peabody Institute of the City of Baltimore. Parry, C. H. 1940. "Variations," Vol. 5, pp. 440-445 in Grove * s dictionary of music and musicians. Ed. by H. C. Colles. (4th ed.; 6 vols.) New York: St. Martin's Press. Prout, E. 1895. Applied Forms: A sequel to 'Musical Form.’ London: Augener's Edition No. 9188. Stanford, C. V. 1912. Musical Compositions: A short treatise for students. New York; The Macmillan Co. Stein, L. 1962. Structure and style: The study and analysis of musical forms. Evanston, 111. : Summy-Birehard Co. Tovey, D. F. 1944. Beethoven. London: Oxford Univ. Press. ______. 1935-1939. "Variations," Vol. 2, in Essays in musical analysis. (7 vols.) London: Oxford University Press. White, B. 1960. "Recognition of distorted melodies." The American Journal of Psychology. 73, 100-107. MUSICAL SCORES Beethoven, L. Fifteen Variations on a Theme from Eroica Symphony, Op. 35. Vol. I. New York: G. Schirmer, Inc., 1926. _ . Six Variations on an Original Theme, Op. 34. Vol. I. New York: G. Schirmer, Inc., 1926. _ . Sonata, Op. 14, No. 2, Vol. I. New York: Kalmus, n.d. 245 ______. Sonata, Op. 26. Vol. I. New York: Kalmus, n.d. ______. Sonata, Op. 30, No. 1. New York: Kalmus, No. 756, n.d. ______. Sonata, Op. 57, Vol. I. New York: Kalmus, n.d. ______. Sonata, Op. 109, Vol. II. New York: Peters, n.d. ______. Sonata, Op. Ill, Vol. II. New York: Peters, n.d. ______• String Quartet, Op. 18, No. 5. Vol. I. New York: Kalmus, No. 759, 1968. ______. String Quartet, Op. 127. Vol. III. New York: Kalmus, No. 761, 1968. ______. Symphony No. 3, Op. 55. New York: Bonanza Books, 1935. ______. Symphony No. 5, Op. 67. New York: Bonanza Books, 1935. ______. Symphony No. 9, Op. 125. New York: Bonanza Books, 1935. ______. Trio, Op. 97. New York: Eulenberg, Ltd., n.d. _____ . Twenty-four Variations In D major. Righlni Variations. Vol. I. Hew York: G, Schirmer, Inc., 1926. ______. Violin Concerto, Op. 61. New York: Kalmus, No. 312, n.d. ADDITIONAL SOURCES Backus, J. 1969. The acoustical foundations of music. New York: W. W. Norton and Co., Inc. Blom, E. 1954. "Idee fixe," in Grove’s dictionary of music and musicians. Vol. 4, pp. 438-439. Ed. by E. Blom. (5th ed.; 10 vols.) New York: St. Martin's Press. ______. 1954. "Theme," in Grove's dictionary of music and musicians. Vol. 8 , pp. 409-410. Ed. by E. Blom. (5th ed.; 10 vols.) New York: St. Martinis Press. ' Boring, E. G. 1942. Sensation and perception in the history of experimental psychology. New York: Appleton-Century-Crofts. Colies, H. C. 1954. "Leitmotiv," in Grove's dictionary of music and musicians. Vol. 5, pp. 121-122. Ed. by E. Blom. (5th ed.; 10 vols.) New York: St. Martin's Press. 246 Dallin, L. 1964. Techniques of twentieth century composition. 2nd ed. Dubuque, Iowa: Wm. C. Brown Co., Inc. Dolmetsch, A. 1946. The interpretation of the music of the 17th and 18th centuries. London: Hovello and Co., Ltd. Donlngton, R. 1954. "Ornamentation," in Grove's dictionary of music and musicians. Vol. 6 , pp. 365-384. Ed. by E. Blom (5th ed. ; 10 vols.) New York:St. Martin's Press. Gibson, J. J. 1966. The senses considered as perceptual systems. Boston: Houghton Mifflin Co. Guilford, J. P. and R. A. Hilton. 1933. "Some configurational properties of short musical melodies." Journal of Experimental Psychology. 16, 32-44. Goetschius, P. 1933. The material used in musical composition: A system of harmony designed originally for use in the English harmony classes of the Conservatory of Music at Stllttgart. New York: G. Schirmer, Inc. LaRue, J. 1961. "Significant and coincidental resemblance between classical themes." Journal of American Musicological Society. 14. Summer, No. 2, 224-234. Liberman, A. M. et al. 1957. "The discrimination of speech sounds within and across phoneme boundaries." Journal of Experimental Psychology. 54, No. 5, 358-368. Parry, C. H. 1954. "Transformation," in Grove's dictionary of music and musicians. Vol. 8 , pp. 531-532. Ed. by E. Blom. (5th ed.; 10 vols.) New York: St. Martin's Press. Roederer, J. G. 1975. Introduction to the physics and psychophysics of music. New York: Springer-Verlag. Rosen, C. 1971. The classical style: Hayden, Mozart, Beethoven. London: Faber and Faber. Schoenberg, A. 1967. Fundamentals of musical composition. Gerald Strang, ed. New York: St. Martin's Press. Tovey, D. F. 1957. The forms of music. London: Oxford Univ. Press. Underwood, B. J. 1957. Psychological research. New York: Appleton- Century-Crofts, Inc.