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31295006117831.Pdf (3.308Mb) A COMPUTER-ASSISTED ANALYSIS OF FOUR SELECTED MASSES BY GIOVANNI~PIERLUIGI DA PALESTRINA by STEVEN RALPH BROCK, B.M. A THESIS IN MATHEMATICS Submitted to the Graduate Faculty of Texas Tech University in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE Approved Accepted December, 1974 AC!_ go'S 13 }Cf7tr No. 143 C:: i) r)· Z --1 ACKNOWLEDGMENTS I wish to express my gratitude to the Graduate School and the Department of Mathematics of Texas Tech University for their financial assistance, and to the members of my committee, Dr. Judson Maynard, Dr. Stanley Liberty, Dr. John White, and Dr. Thomas Newman, chairman, for their ad- vice and assistance. .· ii TABLE OF CONTENTS ACKNOWLEDGMENTS. • • • • • • • • • • • • • • • • • • • ii LIST OF TABLES • • • • • • • • • • • • • • • • • • • • iv I. INTRODUCTION. • • • • • • • • • • • • • • • • 1 Source Materials • • • • • • • • • • • • • 2 Encoding Procedure • • • • • • • • • • • • 3 II. MELODIC ANALYSIS. • • • • • • • • • • • • • • 5 The Program ANLSl. • • • • • • • • • • • • 5 The Program ANLS3. • • • • • • • • • • • • 30 The Program ANLS2. • • • • • • • • • • • • 40 The Program ANLS4. • • • • • • • • • • • • 43 The Program ANLS6. • • • • • • • • • • • • 47 The Program ANLSS. • • • • • • • • • • • • 68 The Program ANLS7. • • • • • • • • • • • • 76 III. HARMONIC ANALYSIS • • • • • • • • • • • • • • 82 IV. INFORMATION CONTENT ANALYSIS. • • • • • • • • 92 v. CONCLUSION •• • • • • • • • • • • • • • • • • 104 REFERENCES • • • • • • • • • • • • • • • • • • • • • • 106 LIST OF REFERENCES • • • • • • • • • • • • • • • • • • 108 iii LIST OF TABLES Table Page 1. GENERAL INFORMATION; AD FUGAM, MARCELLI 1 BREVIS, AETERNA ••• ;-••••••••• • • • • 2 2. tffiiGHTED AND UNWEIGHTED TOTAL AND AVERAGE DURATION FOR EACH PITCH; AD FUGAM, MARCELLI, .BREVIS, AETERNA;-•••••••• • • • 6 3. WEIGHTED AND UNWEIGHTED TOTAL AND AVERAGE DURATION FOR EACH PITCH-CLASS; MARCELLI, BREVIS I AETERNA. • • • • • • • • • • • • • • • • 16 4. THE DATA OF TABLE 3 EXPRESSED AS PERCENTAGES; MARCELLI, BREVIS 1 AETERNA • • • • • 17 5. THE DATA OF TABLE 3 EXPRESSED AS STANDARDIZED SCORES; MARCELLI, BREVIS, AETERNA. • • • • • • • • • • • • • • • • • • • • 19 6. AVERAGES OF CORRESPONDING AVERAGE DURATION, WEIGHTED, AND UNWEIGHTED STANDARDIZED SCORES OF TABLE 5; MARCELLI, BREVIS, AETERNA ••••• • • • • • • • 20 7. AVERAGE STANDARDIZED SCORES; AD FUGAM. • • • • • 22 8. AVERAGE STANDARDIZED SCORES FOR INDIVIDUAL VOICES; AD FUGAM • • • • ••••••••• • • • 23 9. AVERAGE STANDARDIZED SCORES; MISSA DE PLUS ---en PLUS, KYRIE, JOHANNES OCKEGHEM. :-. • • • • • 26 10. AVERAGE STANDARDIZED SCORES; SONATA IN C MAJOR K.545, FIRST mvt., EXPOSITION, W. A • MOZART • • • • • • • • • • • • • • • • • • 26 11. TOTAL OCCURRENCES OF EACH RHYTHM WITH PERCENTAGES AND AVERAGE DURATION; MARCELLI, BREVIS, AETERNA, AD FUGAM •• • • • • • 31 • 1V. 12. TOTAL OCCURRENCES OF EACH RHYTHM AT EACH EIGHTH NOTE OFFSET IN THE MEASURE WITH TOTAL PERCENTAGES FOR EACH OFFSET; MARCELLI, BREVIS, AETERNA, AD FUGAM. • • • • • • 32 13. PERCENTAGES OF RHYTHMS NO LONGER THAN A HALF NOTE AT EACH EIGHTH NOTE OFFSET IN THE MEASURE; MARCELLI, BREVIS, AETERNA • • • • • 39 14. COMBINED TOTAL OCCURRENCES OF EACH RHYTHM ~ AT EACH EIGHTH NOTE OFFSET IN THE MEASURE WITH PERCENTAGES; MARCELLI, BREVIS, AETERNA. • • 40 15. ACTUAL AND EXPECTED OCCURRENCES OF VARIOUS PAIRS OF RHYTHMS AND THEIR RATIO; AETERNA. • • • 42 16. AVERAGE POSITIVE AND NEGATIVE MELODIC VELOCITIES, OVERALL AND FOR EACH INDIVIDUAL VOICE; MARCELLI, BREVIS, AD FUGAM, AETERNA • • • 45 17. TOTAL SIGNED AND ABSOLUTE MELODIC DIS­ PLACEMENT IN UNITS OF SCALE DEGREES; MARCELLI, BREVIS, AD- FUGAM, AETERNA •• • • • • • 46 18. AVERAGE VELOCITY CALCULATED IN AN ALTERNATIVE MANNER, IN UNITS OF SCALE DEGREES; MARCELLI , BREVIS, AD FUGAM, AETERNA • • 48 19. COMBINED TOTAL OCCURRENCES OF EACH MELODIC INTERVAL WITH PERCENTAGES; MARCELLI, BREVIS, AD FUGAM, AETERNA • • • • • • • • • • • • • • • • so 20. RATIOS FROM TABLE 19 OF THE NUMBER OF DESCENDING TO ASCENDING SECONDS, THIRDS, AND FIFTHS, AND ASCENDING TO DESCENDING TO ASCENDING FOURTHS AND OCTAVES, OVERALL AND FOR THE SOPRANO AND BASS VOICES SEPARATELY; MARCELLI, BREVIS, AD- FUGAM, AETERNA. • • • • • • • • • • . • • • • • • • • • 51 21. THE FACTORS OF TABLE 19 AND TABLE 20 CALCULATED FOR THE SOPRANO VOICE OF THE SANCTUS OF BREVIS. • • • • • • • • • • • • • • • 52 22. COMBINED TOTAL OF ADJACENT OCCURRENCES OF EACH PAIR OF MELODIC INTERVALS IN THE SOPRANO, ALTO, AND TENOR VOICES; MARCELLI, BREVIS, AETERNA, AD FUGAM ••••••••• • • • 57 v 23. COMBINED TOTAL OCCURRENCES OF EACH MELODIC INTERVAL FOR THE SOPRANO, ALTO, AND TENOR VOICES WITH PERCENTAGES7 MARCELLI, BREVIS, AETERNA, AD- FUGAM. • • • • • • • • • • • • • • • 58 24. RATIO OF ACTUAL TO EXPECTED OCCURRENCES FOR EACH MELODIC INTERVAL BASED ON TABLE 22 AND TABLE 23; MARCELLI, BREVIS, AETERNA, AD- FUGAM. • 60 25. COMBINED TOTAL OCCURRENCES OF EACH PAIR OF INTERVAL TYPES, LEAP, STEP, OR REPEATED NOTE, IN THE SOPRANO, ALTO, AND TENOR VOICES7 MARCELLI BREVIS, AETERNA, AD FUGAM. • • • • • • 64 1 - 26. DATA OF TABLE 25 EXPRESSED AS ROW PERCENTAGES. • 64 27. PATTERNS OF RHYTHMIC SUCCESSION IN TERMS OF THE INCLUDED MELODIC INTERVAL AND THE DEGREE OF MACRORHYTHMIC STRESS FOR ALL VOICES COM- BINED; AETERNA • • • • • • • • • • • • • • • • • 70 28. PATTERNS OF RHYTHMIC SUCCESSION IN TERMS OF THE INCLUDED MELODIC INTERVAL AND THE DEGREE OF MACRORHYTHMIC STRESS FOR ALL VOICES COM- BINED; BREVIS. • • • • • • • • • • • • • • • • • 71 29. COMBINATION AND CONDENSATION OF TABLE 27 AND TABLE 28 IN TERMS OF ASCENDING MOTION AND ASCENDING LEAPS, EXPRESSED AS ROW PERCENTAGES; AETERNA, BREVIS. • • • • • • • • • • • • • • • • 73 30. MELODIC INTERVALS BETWEEN ADJACENT MELODIC HIGH POINTS WITH PERCENTAGES; MARCELLI, BREVIS, AETERNA, AD FUGAM ••••••••• • • • 80 31. DISTRIBUTION OF COMPLETE AND INCOMPLETE CONSONANCE AND SUSPENSION AND PASSING DISSONANCE WITH RESPECT TO THE STRONG AND WEAK BEATS OF THE MEASURE AND THE SEVEN BASIC CHORDS7 MARCELLI • • • • • • • • • • • • • 83 32. PERCENTAGES OF VARIOUS VERTICAL STRUCTURES ON STRONG AND WEAK BEATS; MARCELLI • • • • • • • 86 33. COMMON SUCCEEDING HARMONIES FOR EACH BASIC HARMONY; MARCELLI ••••••••••••• • • • 88 34. PERCENTAGES OF VARIOUS VERTICAL STRUCTURES ON STRONG AND WEAK BEATS; AETERNA ••••• • • • 89 vi 35. RESULTS OF VARIOUS INFORMATION CONTENT CALCULATIONSJ MARCELLI •••••••• • • • • • 96 vii CHAPTER I INTRODUCTION Giovanni Pierluigi was born in 1524 or 1525 at Pale- strina, a small town not far from Rome, and later added his birthplace to his name according to the custom of the time. From his youth as a choirboy at Santa Maria Maggiore to his death in 1594 as maestro di capella of the Julian choir at St. Peter's, Palestrina devoted himself to the creation of music for the Roman Catholic Church. Twentieth century scholars have substantially reassessed his life and work. The nineteenth century view of Palestrina as a trag- ic, flawless artist struggling to save sacred music from secular corruption has given way to calm historical com- parison, a modern edition of his works has been published, and the artificial and anachronistic theories of J. J. Fux's Gradus ad Parnassum have been replaced by the de­ tailed scholarship of Knud Jeppesen's The Style of Pale­ strina --and the Dissonance. This paper will reexamine the Palestrina style with the aid of a computer and suggest various applications of elementary mathematics to the analysis of music. The re­ sults obtained differ in many ways from the opinions of 1 2 previous writers on the subject and indicate the importance of computer assistance in the analysis of musical style. Source Materials It is commonly conceded that the finest examples of Palestrina's work are found among his 105 surviving set- tings of the Ordinary of the Mass. The Missa ad Fugam, Missa Papae Marcelli, Missa Brevis, and Missa Aeterna Christi Munera in the Roma edition [1] were selected for analysis. These works will be referred to as ~ Fugam, Marcelli, Brevis, and Aeterna for the sake of conciseness throughout. General information on each is contained in Table 1. TABLE 1 GENERAL INFORMATION; AD FUGAM, MARCELLI, BREVIS, AETERNA Mass Ad Fugam Marcelli Brevis Aeterna First Pub- lication 1567 1567 1570 1590 Voices 4 6 4 4 Type canonic free free paraphrase Source original original original hymn Mode Mixolydian Ionian transposed transposed Ionian Ionian Period early middle middle late Source R,iv,p.74 R,iv,p.l67 R,vi,p.62 R,xv,p.l Analysis [2] [4] [7] [5] [3] [6] [8] 3 Encoding Procedure The Masses were encoded in a highly simplified form. Each note was represented by three numbers representing pitch-class, octave, and duration. Seven pitch-classes, C through B, were numbered one through seven with rests as­ signed the value zero. Chromatic alterations were ignored. The octaves were numbered sequentially from zero, repre­ senting the great octave, to three, representing the two­ lined octave, rests again coded zero. The total available pitches from C to two-lined b are numbered from one to twenty-eight. Pitch is recovered by multiplying the octave code by seven and adding the pitch-class. Duration is ex­ pressed as an integral number of eight notes. Occasional sixteenth note pairs were replaced by an appropriate eighth note. Rests of length zero coded 0 0 0 were used to sepa­ rate internal subdivisions at double bars and 0 1 0 was used as the termination symbol for a major section of a Mass. The Roma edition halves all original rhythmic values in translating them into modern notation.
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