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. References to names of rhythmic durations are to the halved values. The omission of twenty-three sixteenth note pairs is of little consequence. Most replace an ornamental eighth note while some fill in the interval of a fourth. The in­ tervals altered by this simplification occur thousands of times so the overall contour of the interval analysis is 4 not noticeably altered. The omission of sixteenth notes speeds execution of the analytical procedures and allows more efficient use of the computer memory. The omission of the text affects the purely musical analysis to an extent since phrases not separated by rests are no longer distinguishable. It would be possible to in­ put the text separately and overlay it on the music with some alteration of the present computer programs. The ease of error detection and correction possible when various aspects of the music are kept apart during encoding makes this method attractive. Accidentals could also be overlaid on the music. Palestrina was quite conservative in his use of chroma­ ticism, especially in his Masses, and notated accidentals are infrequent. Fi is not an independent tone for Pale­ strina, capable for instance of serving as the root of a chord. It is rather an alternate type of F which is some­ times preferable for melodic or harmonic reasons. The analyses performed investigate the broader aspects of the style and are not concerned with details of chromatic al­ teration. A Mass is depicted as a finite three-dimensional cross-product of pitch, duration, and voice. This suppres­ sion of surface detail does not obscure the fundamental musical processes at the basis of Palestrina's style. CHAPTER II

MELODIC ANALYSIS

The Proq·ram ANLSl The results of ANLSl for all notes in each Mass are contained in Table 2. The unweighted column records the number of occurrences of each pitch, regardless of length. The next column expresses these values in percent for ease of comparison. The weighted column records the total duration in eighth notes of each pitch, stmilarly followed by the values in percent. The average duration column con­ tains the quotient of the weighted total by the unweighted, and is the average length in eighth notes of each pitch. The same factors are also calculated in terms of pitch­ class. The total unweighted and total weighted figures are the sums of the respective columns, and the total average duration is their quotient and is the average length in eighth notes of the notes in the Mass. The means and standard deviations of the unweighted and weighted columns are also recorded. In all four Masses the broad outlines of the pitch profile are quite stmilar. The unweighted and weighted totals are larger near the middle and smaller at the

5 TABLE 2

WEIGHTED AND UNWEIGHTED TOTAL AND AVERAGE DURATION FOR EACH PITCH; AD FUGAM, MARCELLI, BREVIS, AETERNA

Ad Fugam

Coded Unweighted Percent Weighted Percent Average Pitch Pitch Occurrences Unweighted Occurrences Weighted Duration

1 c 0 0.0 0 0.0 0.00 2 D 0 o.o 0 0.0 o.oo 3 E 0 0.0 0 0.0 o.oo 4 F 0 o.o 0 0.0 0.00 5 G 0 o.o 0 o.o o.oo 6 A 0 0.0 0 0.0 0.00 7 B 0 0.0 0 0.0 o.oo 8 c 14 0.4 44 0.5 3.14 9 d 94 2.6 278 2.9 2.96 10 e 80 2.2 199 2.1 2.49 11 f 98 2.7 224 2.3 2.29 12 g 303 8.4 1099 11.5 3.63 13 a 257 7.1 665 6.9 2.59 14 b 173 4.8 397 4.1 2.29 15 c' 320 8.9 822 8.6 2.57 16 d' 360 10.0 1141 11.9 3.17 17 e' 233 6.5 518 5.4 2.22 18 f' 258 7.2 539 5.6 2.09 19 g' 389 10.8 1172 12.2 3.01 20 a' 300 8.3 709 7.4 2.36 21 b' 212 5.9 464 4.8 2.19 22 c' ' 242 6.7 644 6.7 2.66 23 d. ' 154 4.3 383 4.0 2.49 24 e' ' 72 2.0 172 1.8 2.39 25 f' ' 36 1.0 100 1.0 2.78 26 g' ' 3 0.1 6 0.1 2.00 0'\ TABLE 2--Continued

Ad Fugam

Coded Unweighted Percent Weighted Percent Average Pitch Pitch Occurrences Unweighted Occurrences Weighted Duration

27 a I I 0 o.o 0 0.0 0.00 28 bl I 0 0.0 0 0.0 o.oo

Unweighted mean pitch = 16.9024 Unweighted standard deviation = 3.9106 Weighted mean pitch = 16.6819 Weighted standard deviation - 3. 9387

Unweighted Percent Weighted Percent Average Pitch-Class Occurrences Unweighted Occurrences Weighted Duration

F 392 10.9 863 9.0 2.20 G 695 19.3 2277 23.8 3.28 A 557 15.5 1374 14.3 2.47 B 385 10.7 861 9.0 2.24 c 576 16.0 1510 15.8 2.62 D 608 16.9 1802 18.8 2.96 E 385 10.7 889 9.3 2.31 Total unweighted = 3598 Total weighted = 9576 Total average duration (total weighted/total unweighted) = 2.66

-...J TABLE 2--Continued

Marcelli

Coded Unweighted Percent Weighted Percent Average Pitch Pitch Occurrences Unweighted Occurrences Weighted Duration

1 c 0 o.o 0 o.o 0.00 2 D 0 o.o 0 o.o 0. 00 3 E 0 0.0 0 0.0 o.oo 4 F 0 0.0 0 0.0 o.oo 5 G 0 0.0 0 0.0 0.00 6 A 0 0.0 0 0.0 o.oo 7 B 0 0.0 0 o.o 0.00 8 c 191 2.5 932 3.9 4.88 9 d 94 1.2 384 1.6 4.09 10 e 85 1.1 207 0.9 2.44 11 f 201 2.6 652 2.7 3.24 12 g 514 6.6 1891 7.9 3.68 13 a 479 6.2 1341 5.6 2.80 14 b 395 5.1 845 3.5 2.14 15 c' 884 11.4 2853 12.0 3.23 16 d' 723 9.3 2133 9.0 2.95 17 e' 742 9.6 2071 8.7 2.79 18 fl 536 6.9 1238 5.2 2.31 19 g' 942 12.2 3221 13.5 3.42 20 a' 522 6.7 1667 7.0 3.19 21 b' 311 4.0 757 3.2 2.43 22 C I I 442 5.7 1499 6.3 3.39 23 d I I 316 4.1 927 3.9 2.93 24 e I I 262 3.4 845 3.5 3.23 25 f I I 82 1.1 249 1.0 3.04 26 g I I 31 0.4 96 0.4 3.10

00 TABLE 2--Continued

Marcelli Coded Unweighted Percent Weighted Percent Average Pitch Pitch Occurrences Unweighted Occurrences Weighted Duration

27 a I I 0 o.o 0 o.o 0.00 28 b I I 0 o.o 0 0.0 0 .oo Unweighted mean pitch = 16.9576 Unweighted standard deviation - 3.9049 Weighted mean pitch = 16.8120 Weighted standard deviation - 4.0985

Unweighted Percent Weighted Percent Average Pitch-Class Occurrences Unweighted Occurrences Weighted Duration

c 1517 19.6 5284 22.2 3.48 D 1133 14.6 3444 14.5 3. 04 E 1089 14.0 3123 13.1 2.87 F 819 10.6 2139 9.0 2.61 G 1487 19.2 5208 21.9 3.50 A 1001 12.9 3008 12.6 3.00 B 706 9.1 1602 6.7 2.27 Total unweighted = 7752 Total weighted = 23808 Total average duration (total weighted/total unweighted) = 3.07

\0 TABLE 2--Continued -Brevis Coded Unweighted Percent Weighted Percent Average Pitch Pitch Occurrences Unweighted Occurrences Weighted Duration

1 c 0 0.0 0 0.0 0.00 2 D 0 0.0 0 o.o o.oo 3 E 0 0.0 0 0.0 o.oo 4 F 0 0.0 0 0.0 o.oo 5 G 0 0.0 0 o.·o o.oo 6 A 3 0.1 6 o.o 2.00 7 B 28 0.6 103 0.8 3.68 8 c 73 1.6 297 2.2 4.07 9 d 70 1.5 200 1.5 2.86 10 e 47 1.0 90 0.7 1.91 11 f 282 6.0 1042 7.8 3.70 12 g 226 4.8 615 4.6 2.72 13 a 283 6.1 728 5.5 2.57 14 b 264 5.7 635 4.8 2.41 15 c' 482 10.3 1555 11.7 3.23 16 d' 350 7.5 893 6.7 2.55 17 e' 377 8.1 846 6.4 2.24 18 f' 534 11.4 1706 12.8 3.19 19 gl 397 8.5 1083 8.1 2.73 20 a' 427 9.2 1165 8.8 2.73 21 bl 210 4.5 528 4.0 2.51 22 c' ' 311 6.7 990 7.4 3.18 23 d I I 153 3.3 411 3.1 2.69 24 e I I 95 2.0 216 1.6 2.27 25 f' ' 51 1.1 191 1.4 3.75 26 g I I 3 0.1 14 0.1 4.67

~ 0 TABLE 2--Continued

Brevis Coded Unweighted Percent Weighted Percent Average Pitch Pitch Occurrences Unweighted Occurrences Weighted Duration

27 a I I 0 o.o 0 0.0 o.oo 28 b I I 0 o.o 0 o.o 0.00 Unweighted mean pitch = 16.8078 Unweighted standard deviation = 3.9129 Weighted mean pitch = 16.6932 Weighted standard deviation = 4.0359

Unweighted Percent Weighted Percent Average Pitch-Class Occurrences Unweighted Occurrences Weighted Duration

G 626 13.4 1712 12.9 2.73 A 713 15.3 1899 14.3 2.66 B 502 10.8 1266 9.5 2.52 c 866 18.6 2842 21.3 3.28 D 573 12.3 1504 11.3 2.62 E 519 11.1 1152 8.7 2.22 F 867 18.6 2939 22.1 3. 39 Total unweighted = 4666 Total weighted = 13314 Total average duration (total weighted/total unweighted) = 2.85

1-' 1-' TABLE 2--Continued

Aeterna

Coded Unweighted Percent Weighted Percent Average Pitch Pitch Occurrences Unweighted Occurrences Weighted Duration

1 c 0 o.o 0 o.o o.oo 2 D 0 0.0 0 0.0 o.oo 3 E 0 - 0.0 0 0.0 o.oo 4 F 0 0.0 0 0.0 0. 00 5 G 0 0.0 0 0.0 o.oo 6 A 2 0.1 4 o.o 2.00 7 B 35 0.9 109 1.1 3.11 8 c 65 1.6 213 2.1 3.28 9 d 63 1.6 159 1.5 2.52 10 e 53 1.3 103 1.0 1. 9 4 11 f 257 6.4 799 7.8 3.11 12 g 200 5.0 489 4.8 2.45 13 a 250 6.3 543 5.3 2.17 14 b 263 6.6 622 6.1 2.37 15 cl 448 11.2 1243 12.1 2.77 16 d' 378 9.5 939 9.1 2.48 17 e' 391 9.8 799 7.8 2.04 18 fl 513 12.8 1539 15.0 3.00 19 q' 364 9.1 895 8.7 2.46 20 a' 310 7.8 781 7.6 2.52 21 bl 176 4.4 404 3.9 2.30 22 C I I 178 4.5 485 4.7 2.72 23 d I I 49 1.2 142 1.4 2.90 24 e I' 0 o.o 0 0.0 o.oo 25 f I I 0 o.o 0 0.0 0.00 26 g' ' 0 o.o 0 o.o o.oo

1-' 1'\.) TABLE 2--Continued

Aeterna Coded Unweighted Percent Weighted Percent Average Pitch Pitch Occurrences Unweighted Occurrences Weighted Duration

27 a I I 0 o.o 0 o.o 0.00 28 b I I 0 o.o 0 0.0 o.oo Unweighted mean pitch = 16.1544 Unweighted standard deviation = 3.5150 Weighted mean pitch = 16.1059 Weighted standard deviation = 3.5948

Unweighted Percent Weighted Percent Average Pitch-Class Occurrences Unweighted Occurrences Weighted Duration

G 564 14.1 1384 13.5 2.45 A 562 14.1 1328 12.9 2.36 B 474 11.9 1135 11.1 2. 39 c 691 17.3 1941 18.9 2. 81 D 490 12.3 1240 12.1 2.53 E 444 11.1 902 8.8 2.03 F 770 19.3 2338 22.8 3.04 Total unweighted = 3995 Total weighted = 10268 Total average duration (total weighted/total unweighted) = 2.57

1-' w 14 extremes, with a noticeable tendency to lengthen certain bass notes. This is reflected by the fact that the weighted mean is somewhat lower and the weighted standard deviation somewhat higher than the corresponding unweighted values in each Mass. The standard deviation is the square root of the average squared deviation from the mean and reflects the amount of dispersion of the data from the mean. In a normal distribution about 68% of all cases fall within one and 95% within two standard deviations of the mean. The higher value of the weighted standard deviation in this case re­ sults from the great lengthening of some extreme low notes, a usual trait of writing for the bass voice. The mean values cluster just below one-lined e in the three earlier Masses and drop to just above one-lined d in Aeterna. Variations of this type have been used by Dr. Mendel [9] to make deductions concerning longterm variations in musical pitch. This drop might seem to indicate a rising pitch at Rome near the end of the sixteenth century, how­ ever in this casethereis a simpler explanation. The pitch profile of Aeterna reveals not an overall lowering of notated pitch, but simply the absence of notes above two­ lined d in the soprano. This upper bound on the sixth scale degree is also present in the hymn on which the Mass is based [10]. Other works published at the same time use a normal range up to two-lined g in the soprano, so the dif­ ference in means seems to be the result of the hymn source 15 rather than a general rise in pitch. The pitch-class profile provides information concerning the relationship between the repetition and prolongation of notes and the tonality of each Mass. Table 3 presents the results from the three Masses in the Ionian mode. Since two of the Masses were transposed the table is arranged accord­ ing to scale degree from the tonic to the leading tone in­ stead of according to pitch-class. Scale degree one cor­ responds to pitch-class C in Marcelli and F in ·Brevis and Aeterna, and so forth. The mean, M, and standard deviation, S, are calculated for each category. Table 4 contains the values in Table 3 expressed as percentages for ease of com­ parison. It is apparent from these tables that the first and fifth scale degrees, the tonic and dominant, are emphasized while the fourth and seventh degrees, the subdominant and leading tone, are avoided. The uniformity among the Masses in the percentages is striking. The weighted values ex­ hibit the most variability and the average duration values the least, according to the size of the standard deviation. Since the weighted, unweighted, and average duration factors are so intimately related it would be desirable to combine the figures for a scale degree to form a single numerical estimate of its role in the mode. One way to do this would be to average the corresponding percentages, but this would 16

TABLE 3 WEIGHTED.AND UNWEIGHTED TOTAL AND AVERAGE DURATION FOR EACH PITCH-CLASS; MARCELLI, BREVIS,· AETERNA Scale Degree Marcelli Brevis Aeterna

Unweighted

1 1517 867 770 2 1133 626 564 3 1089 713 562 4 819 502 474 5 1487 866 691 6 1001 573 490 7 706 519 444 M = 1107.4 666.6 570.7 s = 285.0 142.1 111.0

Weighted

1 5284 2939 2338 2 3444 1712 1384 3 3123 1899 1328 4 2139 1266 1135 5 5208 2842 1941 6 3008 1504 1240 7 1602 1152 902 M·= 3401.1 1902.0 1466.9 s = 1302.4 667.7 461.4

Average Duration

1 3.48 3.39 3.04 2 3.04 2.73 2.45 3 2.87 2.66 2.36 4 2.61 2.52 2.39 5 3.50 3.28 2.81 6 3.00 2.62 2.53 7 2.27 2.22 2.03 M - 2.967 2.774 2.516 s - 0.4099 0.3865 0. 3028 17 TABLE 4 THE DATA OF TABLE 3 EXPRESSED AS PERCENTAGES;

MARCELLI 1 BREVIS, AETERNA

Scale Degree Marcelli Brevis Aeterna

Unweighted

1 19.6 18.6 19.3 2 14.6 13.4 14.1 3 14.0 15.3 14.1 4 10.6 10.8 11.9 5 19.2 18.6 17.3 6 12.9 12.3 12.3 7 9.1 11.1 11.1 M - 14.3 14.3 14.3 s - 3.7 3.0 2.8

Weighted

1 22.2 22.1 22.8 2 14.5 12.9 13.5 3 13.1 14.3 12.9 4 9.0 9.5 11.1 5 21.9 21.3 18.9 6 12.6 11.3 12.1 7 6.7 8.7 8.8 M = 14.3 14.3 14.3 s = 5.5 5.0 4.5

Average Duration

1 16.8 17.5 17.3 2 14.6 14.1 13.9 3 13.8 13.7 13.4 4 12.6 13.0 13.6 5 16.9 16.9 16.0 6 14.4 13.5 14.4 7 10.9 11.4 11.5 M = 14.3 14.3 14.3 s = 2.0 2.0 1.7 18 allow the wide variation of the weighted figures to domi­ nate the results. This problem is avoided by converting the raw totals to standardized scores. A standardized score, z, expresses the deviation of each value, X, from the mean, proportional to the standard

deviation, i.e. Z = (X - M)/S for each X. The mean of a distribution of standardized scores is always zero and the standard deviation is always one. Thus a standardized score of -1 indicates the original value was one standard deviation less than the mean while a score of 0.5 indicates half a standard deviation above the mean. Table 5 contains the values from Table 3 converted to standardized scores and Table 6 averages the three scores for each scale degree is repeated and/or prolonged more than average, while a negative score means the scale degree is avoided. The scale degrees in Marcelli and Aeterna rank 1, 5, 2, 3, 6, 4, 7 by Table 6. Brevis differs only in inter­ changing 2 and 3. The ordering seems to show not a gradual decline, but rather a grouping into three classes, the emphasized tones, the neutral tones, and the avoided tones. It is no surprise that the tonic and dominant form the emphasized class, with the tonic somewhat stronger. The second, third, and sixth degrees are treated in a more or less neutral manner with a slight tendency to avoid the sixth. The leading tone or seventh degree is clearly 19

TABLE 5

THE DATA OF TABLE 3 EXPRESSED AS STANDARDIZED SCORES; MARCELLI, BREVIS, AETERNA

Scale Degree Marcelli Brevis Aeterna

Unweighted

1 1.44 1.41 1.79 2 0.09 -0.29 -0.04 3 -0.06 0.33 -0.10 4 -1.01 -1.16 -0.88 5 1.33 1.40 1.08 6 -0.37 -0.66 -0.70 7 -1.41 -1.04 -1.15

Weighted

1 1.45 1.55 1.89 2 0.03 -0.28 -0.17 3 -0.21 -0.01 -0.31 4 -o .91 -0.95 -0.73 5 1.39 1.41 1.03 6 -0.30 -0.60 -0.49 7 -1.38 -1.12 -1.22

Average Duration

1 1.21 1.60 1.89 2 0.17 -0.10 -0.26 3 -0.24 -0.28 -0.57 4 -0.86 -0.64 -0.45 5 1.26 1.31 1.06 6 0.07 -0.39 0.04 7 -1.70 -1.42 -1.78 20 TABLE 6 AVERAGES OF CORRESPONDING AVERAGE DURATION, WEIGHTED, AND UNWEIGHTED STANDARDIZED SCORES OF TABLE 5; MARCELLI, BREVIS, AETERNA

Scale Degree Marcelli Brevis Aeterna

1 1.37 1.52 1.86

2 0.10 -0.22 -0.16

3 -0.17 0.01 -0.33

4 -0.95 -0.92 -0.70

5 1.33 1.37 1.06

6 -0.20 -0.55 -0.38

7 -1.50 -1.19 -1.38

avoided, and so, to a lesser extent, is the subdominant. Since these two degrees are frequently chromatically altered, the actual fourth and seventh degrees occur even less than indicated. The other scale degrees are altered much less often in proportion so the result of performing this analysis in a twelve-note scale framework would be a lowering of four and seven relative to the other degrees and several very low values for the chromatic tones. The subdominant is the third most common chord in these Masses and is often considerably prolonged during a plagal close. The fact that it is avoided and shortened overall is therefore somewhat surprising. Perhaps this is 21 an outgrowth of the tendency of composers of that time to ·avoid the tritone, which is the interval between the sub­ dominant and leading tone in the Ionian mode. Jeppesen noted a similar pentatonic inclination in the Gregorian chant [11] and it appears that this aspect of the chant in­ fluences Palestrina's melodies even in Masses not based strictly on Gregorian models, such as Marcelli and Brevis. It is possible to consider this the result of a shortening of leading tones and/or an avoidance of half-step progres­ sions in general as well as a sensitivity to the tritone dissonance. Viewed in this way the results support editors who consistently raise the fourth degree as a leading tone to the dominant. Analysis of works in other modes should shed additional light on the matter, since the tritone will be located differently with respect to the tonic. Ad Fugam is in the Mixolydian mode, but its pitch dis­ tribution is distorted by the strict canonic procedure. Table 7 contains the average standardized scores for Ad Fugam. The ranking in terms of scale degrees is 1, 5, 4, 2, 6, 3, 7. Only the tonic, dominant and leading tone rank as they did in the Ionian Masses. F and B form the tritone in the Mixolydian mode and rank firmly at the bottom. This seems to indicate that the tritone is the primary factor in determining which tones are most suppressed. Analysis of works in the minor modes would supply further evidence on 22 TABLE 7 AVERAGE STANDARDIZED SCORES; AD FUGAM

Pitch-Class Ad Fugam

G 1.71

A 0.03

B -0.99

c 0.24

D 0.88

E -0.91

F -1.00

this question. The canonic procedure is based upon exact duplication of the bass and alto a fourth above in the tenor and so­ prano. Thus repetition or prolongation of one pitch im­ plies the same to the pitch a fourth higher. The problem is how to achieve an acceptable distribution of pitches overall within this restriction. The ingenuity of Pale­ strina's solution of this problem can best be seen in the pitch-class profiles of the individual voices in Table 8. Each voice is allotted three primary tones and avoids the remainder. In the alto and bass the tones are G, A, and D so the soprano and tenor have C, D, and G a fourth above. This results in G and D being emphasized in all four voices, 23

TABLE 8 AVERAGE STANDARDIZED SCORES FOR INDIVIDUAL VOICES; AD FUGAM

Pitch-Class Soprano Alto Tenor Bass

G 1.06 1.23 1.28 1.87

A -0.28 0.24 -0.69 0.64

B -0.37 -0.48 -0.83 -0.94

c 1.21 -1.03 1.08 -0.44

D 0.35 1.13 0.67 0.51

E -o .87 -0.24 -0.96 -0.68

F -1.10 -o.5o -0.55 -0.96

A and C in only two, and E, B, and F being avoided through­ out. This accounts quite precisely for the pattern of Table 7. It is interesting to note that none of the voices re­ sembl~ the overall profile of Table 7 too highly. In the other Masses this lack of resemblance is much greater. The pitch profiles of individual voices vary substantially from section to section as well as from Mass to Mass. In spite of this the overall contour remains remarkably stable. It seems clear that this contour is a basic aspect of tonality in Palestrina's music. Music of other times, however, does not always conform to these relations. Two hundred years prior to Palestrina 24 greatly prolonged leading tones were regularly employed. Machaut's Messe de Nostra Dame is an excellent example of emphasized leading tones used to establish cadences with unmistakable clarity. In Palestrina's style there is no unmistakable cadential chord. The most common closes are, in modern terminology, the authentic, plagal, and half cadences. These same progressions are used constantly throughout each piece and so are not in themselves the least distinctive. The lack of definite harmonic cues re­ quired Palestrina to employ other means to obtain effective closes. Cadences were characterized by consonance and a low level of rhythmic activity. For contrast, they were often preceded by one or more suspension dissonances and a gradual decline from a relatively high level of rhythmic activity. Information content analysis will delineate this relation in greater detail. In the centuries following Palestrina the dominant­ seventh chord became the common method of establishing tonality. Its ability to imply a tonic not yet sounded because of the regularity of its resolution is analogous to the double leading tone chords of the fourteenth century. In this context the prolongation of the tonic would logic­ ally be less necessary. By the time of Bach several al­ ternative tonal strategies were available. There are some pieces by Bach, such as the familiar Prelude 1 in C major 25 from Book I of the Well-Tempered Clavier, which prolong the tonic and dominant in a manner similar to Palestrina, but there are other pieces, such as Prelude 20 in A minor from Book II, which are completely different. Evidently pitch­ class profiles of this period and those following show con­ siderable variety. Rather than forming profiles of entire works, it might be more revealing to form separate profiles for each major key area or transitional section. For the sake of comparison Tables 9 and 10 contain pitch-class profiles of very short sections of music prior and subsequent to Palestrina. Table 9 is based upon data obtained by Curry [12] for the Kyrie of the Missa de Plus en Plus by Johannes Ockeghem. The standardized scores are for the unweighted pitch-classes only. The ranking of the scale degrees is 1, 5, 4, 2, 6, 3, 7. This ranking is identical to the ranking of Ad Fugarn, though the separation between adjacent ranks is somewhat different. Both of these works are in the Mixolydian mode and Palestrina was writing in the older style so this parallel may be more than coincidence. The internal consistency of the results obtained for the Ionian and Mixolydian modes suggests the hypothesis that each mode has its own distinct pitch-class profile. I The traditional distinction between the modes is expressed in terms of cadence points and, to a lesser extent, 26 TABLE 9 AVERAGE STANDARDIZED SCORES; MISSA DE PLUS EN PLUS, HYRIE, JOHANNES OCKEGHEM

I Pitch-Class Miss a --De Plus en Plus, Kyrie G 1.49 A -0.47 B -0.63 c 0.47 D 1.18 E -0.55 F -1.49

TABLE 10 AVERAGE STANDARDIZED SCORES; SONATA IN C MAJOR K.545, FIRST mvt., EXPOSITION, W. A. MOZART

Sonata in C First mvt., Major K.545, Exposition Pitch-Class rmn. 1-12 mm. 13-28

c 1.04 0.90 Ci -0.75 -0.74 D 0.29 1.17 Di -1.18 -0.81 E 0.58 0.26 F 0.65 -1.26 Fi -0.42 -0.15 G 1.36 1.12 Gi -1.18 -0.90 A 0.63 0.78 Ai -1.18 -1.26 B 0.15 0.91 27 idiomatic melodic figures. Since these cadence points and melodic figures comprise only a small fraction of a work, a pitch-class profile of the work as a whole is in a sense a more satisfying distinction. The pitch-class profile im­ plies differences in the preferences for pitch-class and duration which influence the individual voices throughout the work. The most significant difference between the Mixolydian and Ionian modes, in terms of pitch-class, is the favoring of the subdominant in the former and its avoidance in the latter. This agrees well with the usual observation that the subdominant is a common cadence point in the Mixolydian, but not in the Ionian mode. The fact that the interval between the seventh scale degree and the tonic is a whole tone in the first case and half in the second seems less important since the linear treatment of the seventh degree is the same in both modes. The data upon which Table 10 is based were obtained by Hiller and Bean [13] for the exposition of the Mozart Sonata in C major, K. 545. The standardized scores given are the average of the weighted, unweighted, and average duration scores. The exposition has been separated into two sections, the first, rnrn. 1-12, essentially in C major, the second, mrn. 13-28, in G major. All the chromatically altered tones rank far below the natural scale tones in both sections. In the first section the natural scale 28 degrees rank 5, 1, 4, 6, 3, 2, 7 while in the second sec­ tion the order is 5, 1, 3, 4, 2, 6, 7. It seems the tonic and dominant head the ranking with the leading tone last just as in Palestrina. Considerable freedom is evident in the remaining tones. It is interesting to note that the dominant outranks the tonic in both sections. Both the Mozart and Ockeghem samples are far too small to yield any real evidence. They are included to suggest that profiles of this sort are a reasonable tool for the investigation of the nature and evolution of tonality. The large amount of data which must be processed to obtain meaningful profiles implies the necessity of computer as­ sistance in the calculations. There are many ways in which the profiles might be refined. Occurrences of tones could be further weighted according to their position with respect to macrorhythmic stress, vocal range, and textual stress. Accidentals and ligatures could be considered. - Weighting for duration could be varied according to the context rather than being rigidly evaluated in multiples of eighth notes. The standardized scores could be weighted in various ways be­ fore being averaged. Assignment of numerical values to quantities such as these is somewhat arbitrary and care must be taken in in­ terpreting the resulting figures. There is no reason to 29 believe, for instance, that a dotted quarter has exactly three times the weight as an eighth note of the same pitch. However, three is certainly more reasonable than thirty or three-tenths. The pitch-class profile states an exact mathematical relationship between the tones of a mode. The extent to which the figures obtained correspond to audible musical factors is proportional to the accuracy of the fol­ lowing assumption: other things being equal, the subjec­ tive perception of stability and rest increases with dur­ ation and probability of pitch-class. Tonality is surely associated with a feeling of stability and rest so the assumption seems reasonable in terms of the relation be­ tween tonality and repetition and prolongation in the music of Palestrina. Twentieth century attempts to maintain tonality without use of the dominant-seventh chord provide further examples of the general validity of the assumption. The frequent use of ostinatos and pedal points is related to this factor. The music of Bartok in particular is full of instances where tonality is established within a com­ pletely chromatic context simply by repeating and sustain­ ing the tonal center. The average standardized scores should be viewed as an index of the extent of prolongation and repetition of each tone, in comparison with the remainder. In view of the consistency of the results obtained and the general 30 plausibility of the above assumption, the methods described should be useful in the investigation of tonality, particu­ larly in music relatively free from harmonic cadential cues.

The Program ANLS3 ANLS3 counts the occurrences of each rhythmic value and records the distribution of each value among the vari­ ous locations in the measure. A measure is eight eighth or four quarter notes in length and the first measure of each section is assumed to begin with the first note. Al­ most all of the four Masses fit comfortably into these measures although barlines were not present in the original choirbooks. Eleven bars of the Credo of Aeterna were omitted from this calculation since they were clearly written in triple time. Although the Hosanna of Ad Fugam is notated in triple time, all suspension, dissonances and cadences remain correctly positioned if the usual measure is superimposed. This combined with the fact that the canon continues at a distance of a half note indicates that the original mensuration sign was unnecessary or perhaps mistaken, so the change of mensuration was ignored. Tables 11 and 12 contain the results of ANLS3 for the four Masses. The eighth, quarter, dotted quarter, and half notes constitute the bulk of the music. The quarter note is consistently the most common, followed by the half, TABLE 11 TOTAL OCCURRENCES OF EACH RHYTHM WITH PERCENTAGES AND AVERAGE DURATION; MARCELLI, BREVIS, AETERNA, AD FUGAM

Marcelli Brevis Aeterna Ad Fug:am Rhythm Total Percent - Percent Total Percent Total Percent Total t 1353 17.5 755 16.2 688 17.2 688 18.6 J 2904 37.5 2069 44.3 2005 50.2 1583 44.0 J. 475 6.1 289 6.2 246 6.2 248 6.9 d 237.9 30.7 1275 27.3 893 22.4 1037 28.8 d. 169 2.2 81 1.7 60 1.5 3 0.1 0 375 4.8 153 3.3 93 2.3 21 0.6 o. 32 0.4 6 0.1 5 0.1 3 0.1

~ 48 0.6 34 0.7 4 0.1 32 0.9 ~~ 8 0.1 0 -- 0 -- 3 0.1 ,---..... 3 4 0.1 1 o.o 0 '-=' ~ o.o ~~ 0 0 ~ Fl 0 5 0.1 0 -- -- ,.--.... r--.... 1 o.o 0 0 0 t=1 t=l ~ -- -- Average duration in eighth notes 3.07 2.85 2.57 2.66 w ~ 32

TABLE 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

Location 1 2 3 4 5 6 7 8 Rhythm Marcelli

l 101 196 162 308 94 191 82 219 J 57-7 0 775 0 592 4 956 0 ~. 146 0 101 0 133 0 95 0 d 889 0 332 0 865 0 293 0 d. 92 0 0 0 77 0 0 0 0 297 0 0 0 78 0 0 0 o. 12 0 0 0 20 0 0 0 H 48 0 0 0 0 0 0 0 ~ t=t 0 7 0 0 0 1 0 0 0 ~ F=t H 3 0 0 0 0 0 0 0 /--.....~ ~ t=4 0 5 0 0 0 0 0 0 0 ~r--.. F=f t=t Fi 1 0 0 0 0 0 0 0 Percent 28.1 2.5 17.7 4.0 24.0 2.5 18.4 2.8 33

TABLE 12--Continued

Location 1 2 3 4 5 6 7 8 Rhythm Brevis

t 3·3 113 85 163 44 104 71 142 ~ 503 0 546 0 437 0 583 0 ~. 78 0 60 0 71 0 80 0 d 475 0 185 0 493 0 122 0 d. 38 0 0 0 43 0 0 0 0 117 0 0 0 36 0 0 0

o. 2 0 0 0 4 0 0 0 t=4 34 0 0 0 0 0 0 0 ,.---.., 0 0 0 0 0 0 0 0 0 '=='r----.. 4 0 0 0 0 0 0 0 \9~~ ~ ~ 0 0 0 0 0 0 0 0 0 ,...---. ~ 0 0 0 0 0 0 0 0 F\ t=4 ~ Percent 27.5 2.4 18.3 3.5 24.2 2.2 18.3 3.0 34

TABLE 12--Continued

Location 1 2 3 4 5 6 7 8 Rhythm Aeterna I 4·7 74 54 145 52 102 68 146 ~ 465 0 542 0 435 0 563 0 ~. 91 0 so 0 78 0 27 0 d 321 0 115 0 342 0 115 0 d. 37 0 0 0 23 0 0 0 0 77 0 0 0 16 0 0 0 o. 1 0 0 0 4 0 0 0 Ft 4 0 0 0 0 0 0 0 Ho 0 0 0 0 0 0 0 0 ~~ 1 0 0 0 0 0 0 0 ~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ~F4 • Percent 26.1 1.9 19.0 3.6 23.8 2.6 19.3 3.7 35

TABLE 12--Continued

Location 1 2 3 4 5 6 7 8 Rhythm Ad Fugam

I' 46 127 61 109 48 122 55 100 ~ 348 0 442 0 355 0 438 0 J. 48 0 74 0 45 0 81 0 d 425 0 86 0 435 0 91 0 d. 3 0 0 0 0 0 0 0 0 10 0 0 0 11 0 0 0 o. 0 0 0 0 3 0 0 0 t=9 20 0 0 0 12 0 0 0 \::9~0 2 0 0 0 1 0 0 0 ~ 0 0 0 0 0 0 0 0 ~ f=j ~0 0 0 0 0 0 0 0 0 t=9~-~ 0 0 0 0 0 0 0 0 Percent 25.1 3.5 18.4 3.0 25.3 3.4 18.5 2.8 36 eighth, and dotted quarter. The longer note values are less common and their distribution varies considerably. Generally the whole note is most frequent, followed by the dotted half. Values longer than the breve occur mainly as a sort of pedal point at the end of sections. The eighth notes and dotted quarter notes consistently occur about 17% and 6.5%, respectively, despite the larger variations among the other values according to the context. Since a dotted quarter is always followed by an eighth note this indicates a constant proportion of eighths, the fastest rhythm consistently employed. These notes are quite prominent against the slower moving background, and the amount of this activity desired by Palestrina in each voice seems about the same in all four Masses. Generally the higher, more agile voices contain a greater proportion of eighths than the lower voices, especially the bass. There is a noticeable tendency in the more polyphonic sec- tions for eighth notes to occur in clusters at somewhat equally spaced intervals, producing a constant alteration • of tension and relaxation in the texture. Another analysis is planned to explore the alteration in detail by calcu- lating the average duration from measure to measure and comparing the fluctuations with the total average duration. A similar pattern of regular fluctuation will be noticed in the information content of the music. 37 The highest total average duration for an entire Mass is 3.07 for Marcelli, which has six voices. Palestrina left more voices inactive on the longer rhythms and used fewer quarters to keep the texture uncluttered. Brevis, with four voices, has a somewhat lower average of 2.85. The still lower values for Ad Fugam and Aeterna seem mainly due to the special procedures involved in their composition. The low average duration of Aeterna is caused by an un­ usually high proportion of quarter notes, which follows from the largely undifferentiated rhythms of the Gregorian hymn on which the Mass is based [14]. In Ad Fugam long note values cause a sustained interval of a fourth because of the canonic method. This is practical only at cadences so the longer rhythms are restricted exclusively to ca­ dences and the total average duration is low. The distribution according to location clearly re­ veals the hierarchy of the beats within the measure and demonstrates the validity of the measure as a real aspect of the style. Locations 1, 3, 5, and 7 correspond to beats 1, 2, 3, and 4 in ! time while the even locations are the offbeats. All rhythmic values occur on beat 1 while no duration greater than a dotted whole note occurs on beat 3. The half note is the longest rhythm beginning on beats 2 and 4 and nothing but eighth notes fall on the offbeats. Ad Fugam, of course, cannot distinguish between beats 1 and 38 3 due to the canon at a half note offset. The other Masses represent Palestrina's usual style, and the remainder of the discussion refers only to them. Of the 16, 413 notes in the three other Masses there are only two exceptions to these rules. One is an archaic diminution of a consonant fourth suspension figure repeated in imitation four times during the Gloria of Marcelli. The other is a duration of three whole notes placed correctly relative to the end rather than the beginning of the Bene­ dictus of Marcelli. That Benedictus is the only section in all three Masses which does not divide as expected into measures. This distribution has a practical result. More notes are attacked on the first beat than on the third, and on the third than on the second and fourth. The offbeats are lowest of all. The difference in number of attacks between the first and third beats is primarily due to the placement of the long note values. There are 491 whole notes on beat one and 130 on beat three, 86 breves on one and none on three. The hierarchy is still present when no rhythms greater than half notes are considered. Table 13 gives the percentages of these shorter values according to location. The rules are almost invariably apparent, even in totals of a single voice of a single section, often fewer than a hundred notes. 39 TABLE 13 PERCENTAGES OF RHYTHMS NO LONGER THAN A HALF NOTE AT EACH EIGHTH NOTE OFFSET IN THE MEASURE;

lo!ARCELLI 1 BREVIS 1 AETERNA

Location 1 2 3 4 5 6 7 8

Marcelli 24.3 2.8 19.4 4.4 23.9 2.7 20.2 3.1 Brevis 24.8 2.6 20.0 3.7 23.8 2.4 19.5 3.2 Aeterna 24 •. 1 1.9 19.9 3.8 23.7 2.7 20.2 3.8

Jeppesen and Reese have argued that the restriction of suspension dissonances to beats one and three and of pass- ing dissonances to beats two and four implied a greater de- gree of accentuation on the odd beats [15] [16]. Similarly the greater number of attacks recorded in Table 12 and the careful placement of the longer notes implies a somewhat greater accentuation on the first beat than the third, just as was common in later centuries. Those editions which bar each voice individually or mix measures of : and ~ are thus less preferable than the straightforward barring of the Rorna edition, which clearly represents the macrorhythrn against which the microrhythms of the individual voices are contrasted. The nature and even existence of regular accents in the music of the Renaissance has been consider- ably disputed in the past. The correct answer is readily apparent when complete and exact data are available for 40 consideration. Table 14 contains the combined totals and percentages for the three Masses.

TABLE 14 COMBINED TOTAL OCCURRENCES OF EACH RHYTHM AT EACH EIGHTH NOTE OFFSET IN THE MEASURE WITH PERCENTAGES~ l-1ARCELLI, BREVIS, AETERNA

Location 1 2 3 4 5 6 7 8

Total 4506 383 3007 616 3938 401 3055 507 Percent 27.5 2.3 18.3 3.8 24.0 2.4 18.6 3.1

The distribution of melodic high points in the measure is explored in ANLS7. The distribution of the syllables of the text and large harmonic blocks would provide additional information on the question.

The Program ANLS2 Second-order rhythmic events were totaled by ANLS2. Rhythms were classed as normal or syncopated according to their location in the measure. Eighth notes were classed as normal on the beat and syncopated off the beat. Quar- ters, dotted quarters, and halfs were classed normal on beats one and three and syncopated elsewhere. Longer notes were normal if attacked on beat one and syncopated other- wise. Both preceding and succeeding rhythms of all notes were totaled since some differences result from classing 41 as normal or syncopated the first or the second note of each pair.

Most of the totals are explainable in terms of the first-order structure and the location in the measure as recorded in ANLS3. To separate the second-order effects, the data from ANLS3 are used to calculate the number of expected occurrences of pairs of rhythms if they succeeded each other in a random manner. Actual values greater than those expected indicate a favored pair while lesser values indicate a succession that was avoided to an extent. Table 15 records the actual and expected values and their ratios for the four most common rhythms for Aeterna. N and S mean normal and syncopated and rhythmic durations are given in multiples of eighth notes. Rhythmic succes­ sions are notated as ordered pairs, so (Sl, 2) means an eighth note off the beat followed by a quarter note, (N4, 3) means a normal half followed by a dotted quarter, and so forth. As an example, the expected number of occur­ rences of (N2, 2) in Aeterna from ANLS3 is calculated (465 + 435) (542 + 563)/1534 = 648.3 while ANLS2 records 701 actual occurrences. Three successions, (Nl, 1), (N3, 1), and (S3, 1), are fixed by first order considerations and were omitted from the table. Most of the common pairs do not deviate greatly from random, though certain moderate likes and dislikes are 42

TABLE 15

ACTUAL AND EXPECTED OCCURRENCES OF VARIOUS PAIRS OF RHYTHMS AND THEIR RATIO; AETERNA

Ratio Rhythms Actual Expected (Actual/Expected)

(N2, 1) 32 72 0.45 (N2, 2) 701 648 1.08 (N2, 3) 57 45 1.26 (N2, 4) 1"10 135 0.81 (N4, 1) 10 33 0.30 (N4, 2) 182 299 o. 61 (N4, 3) 38 56 0.68 (N4, 4) 207 220 0.94 (Sl, 1) 118 29 4.07 (Sl, 2) 216 234 0.92 (Sl, 3) 20 33 0.61 (Sl, N4) 46 155 0.30 (Sl, S4) 52 70 0.74 (S2, 1) 26 55 0. 4 7 (S2, 2) 558 499 1.12 (S2, 3) 109 94 1.16 (S2, 4) 344 367 0.94 (S4, 1) 26 18 1.42 (S4, 2) 188 166 1.13 (S4, 3) 5 12 0.43 (S4, 4) 11 34 0.32 43 evident. The highest ratio is 4.07 for (Sl, 1). This im­ plies a strong tendency for eighth notes to occur in groups of three or more. The other undotted rhythms also tend to group, to a lesser extent. The influence of the macro­ rhythm is clear in the preference for (S4, 1) and (S4, 2) and against (S4, 3) and (S4, 4). A concern for variety is noticeable, since the two most common rhythms, halfs and quarters, are not paired as often as expected. Considerable variation from section to section and Mass to Mass is possible in ANLS2. Overall it would be best to say that rhythmic succession in Palestrina is flexible, for most second-order effects are not strong. In order to understand Palestrina's rhythmic procedure melodic factors must be taken into consideration. ANLSS will attack the problem from that point of view.

The Program ANLS4 A quantity analogous to velocity can be computed from a melodic line by dividing each melodic interval by the length of the first note·of the interval. If a melody changed continuously and linearly through the time avail­ able rather than leaping the interval at the last moment its velocity would be computed in this manner. Since this is not actually the case the quantity computed in not a real velocity but simply an indicator of a relationship between rhythm and melodic intervals. The unit employed 44 to measure the velocity is scale degrees per eighth note. For each successive pair of pitches the first is subtracted from the second to obtain the interval. Thus the interval of an ascending second in musical terminology is expressed as +1, a descending third as -2, an ascending octave as +7, and so forth. Repeated notes form intervals of zero and do not contribute to the velocity. The interval is divided by the duration in eighth notes of the first pitch to obtain the velocity. To explore possible differences in treatment of posi­ tive and negative intervals the average positive and nega­ tive velocities were computed separately. Table 16 pre­ sents the results. The average positive velocity exceeds the average negative velocity by about 15%. In general the magnitude of both the average positive and negative veloci­ ties increases gradually from the soprano to the bass. The ordering is still present in well over ninety percent of small samples, a single voice of a single section, which is remarkably consistent considering the relatively small overall percentage difference. To interpret the relationship of these figures to the melodies consider the two factors which could cause the magnitude of the velocity to increase. One would be an overall tendency for larger positive intervals to be used. This would be reflected by a cumulative positive displace- 45 TABLE 16 AVERAGE POSITIVE AND NEGATIVE MELODIC VELOCITIES, OVERALL AND FOR EACH INDIVIDUAL VOICE; MARCELLI, BREVIS, AD- FUGAM,· AET:ERNA Average Positive Average Negative Velocity Velocity

Marcelli 0.814 -0.6 79 soprano 0.728 -0.618 alto 0.802 -0.623 tenor 0.846 -0.713 bass 0.853 -0.719 Brevis 0.731 -0.650 soprano 0.716 -0.585 alto 0. 708 -0.676 tenor 0.741 -0.657 bass 0.767 -0.698 Ad Fugam 0. 743 -0.656 - soprano, 0.720 -0.627 alto 0.719 -0.633 tenor 0.771 -0.682 bass 0.765 -0.689 Aeterna o. 717 -0.668 soprano 0.671 -0.621 alto 0.702 -0.636 tenor 0.749 -0.676 bass 0.745 -0.761 46 ment in the interval totals. Borrowing data from ANLS6, Table 17 records the signed and absolute displacement of each Mass in units of scale steps. The signed displacement is obtained by multiplying the signed size of each interval by its total number of occurrences and adding. The abso­ lute displacement is computed in the same manner except that the absolute value of the interval is used.

TABLE 17 TOTAL SIGNED AND ABSOLUTE MELODIC DISPLACEMENT IN UNITS OF SCALE DEGREES; MARCELLI, BREVIS, AD FUGAM, AETERNA

Signed Displacement Absolute Displacement

Marcelli -780 9634 Brevis -415 5557

Ad Fugam -308 4092

Aeterna - 40 4462

The signed displacement is instead consistently nega- tive. The signed value is. about eight percent of the abso- lute in the earlier Masses, but only one percent for Aeterna. This variation is evidently caused by model Gregorian hymn rather than a general change in style, and will be consid­ ered further in the interval analysis. The different posi- tive and negative velocities thus must reflect a strong tendency to precede positive intervals with the shorter 47 rhythmic values to overcome the negative displacement. ANLSS will discuss the basic relation between rhythm and interval which accounts for the difference in velocity. Another method of calculating velocity has been em­ ployed by Maurita and Ronald Brender [17]. Instead of separating positive and negative velocities, all were averaged together. This method yields less useful results since it is influenced by the ratio of positive to negative intervals as well as the two factors just discussed. In their analysis of thirteenth century motets they found consistently negative velocities, with the largest magni­ tude in the upper voice, less in the middle, and least of all in the lowest voice. These characteristics are all reversed in Palestrina, so it should be interesting to trace the role of velocity through the intervening cen­ turies. The Brender's results may only reflect an excess of negative intervals rather than more precipitous downward motion, however. Table 18 contains the average velocity of each Mass computed in the Brender's fashion. The results are all quite close to zero as the greater number of nega­ tive intervals tends to cancel out the greater positive velocity.

The Program ANLS6 Palestrina's melodies show an obvious concern for balance between rising and falling motion, and the treatment 48 TABLE 18 AVERAGE VELOCITY CALCULATED IN AN ALTERNATIVE MANNER, IN UNITS OF SCALE DEGREES; MARCELLI, BREVIS, AD---- FUGAM, AETERNA Marcelli Brevis Ad Fugam Aeterna

Average Velocity 0.0016 -0.0198 -0.0222 0.0248

of melodic intervals is a key factor in this process. Knud

Jeppesen discussed his findings in detail in ~ Style of

Palestrina ~ the Dissonance, and subsequent writers such as Andrews, Soderland, and Reese accepted his opinions with little or no change. Exact counts of all intervals and interval pairs in the four Masses by ANLS6 indicate ~hat Palestrina's style is considerably more complicated and sophisticated than Jeppesen allowed. In interpreting the results of ANLS6 it is necessary to keep in mind the encoding procedure. Major, minor, per- feet, augmented, and diminished intervals are not distin- guished. Also, the omission of the text made it impossible to differentiate some of the so-called "dead" intervals which occur between the final note of a phrase of text and the first note of the next phrase. Those dead intervals which are separated by a rest are detected and omitted, but if no rest intervenes they are counted as real melodic in- tervals. It seems preferable to divide intervals across 49 textual separation in two classes, dead and elided, the dead intervals being separated by a rest and the elided intervals not separated. Dead intervals as defined are evidently subject to no restrictions except vocal range, for normally forbidden sevenths and ninths seem about as frequent as the normally usable octave, though all three are fairly rare because of limited vocal ranges. Elided intervals, however, expand the normally usable intervals only to include the ascending major sixth and descending sixth in the four Masses examined. Also, elided intervals can freely ignore second-order interval probabilities. At any rate, ANLS6 includes elided but not dead intervals and it will occasionally be necessary to point out some small totals which were perhaps effected by their inclusion. For instance, in Table 19, which records the total occurrences of all intervals in the four Masses, there is one descending sixth. This sixth is an elided interval and can be disregarded, so the allowable intervals in the style are exactly as stated by Jeppesen and others. Little has been said about the relative probabilities of the various intervals except that stepwise motion predominates and sixths and octaves are relatively rare. Table 19 confirms these obvious relations and reveals considerable differen­ tiation of the middle intervals as well. For each size in­ terval there is a definite favoring of either the ascending ; \

50

TABLE 19

COMBINED TOTAL OCCURRENCES OF EACH MELODIC INTERVAL WITH PERCENTAGES; MARCELLI, BREVIS, ----AD FUGAM, AETERNA Interval Total Percent

-8 43 0.2 -7 0 -6 1 o.o -5 480 2.6 -4 312 1.7 -3 1731 9.3 -2 6020 32.4 0 2528 13.6 2 5750 31.0 3 596 3.2 4 729 3.9 5 212 1.1 6 12 0.1 7 0 8 152 0.8

or the descending type. Descending seconds, thirds, and fifths and ascending fourths, sixths, and octaves are pre- ferred. These preferences are consistently apparent even in samples of a hundred or fewer intervals. The natural question is whether these relations owe their origins to melodic or harmonic factors or both.

Table 20 records the ratios of the more favored to the less favored of each interval for all voices combined and 51 TABLE 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, ~ETERNA

Interval ratio -2:2 -3:3 4:-4 -5:5 8:-8 Total 1.05 2.90 2.34 2.26 3.53 Soprano 1.14 2.43 6.79 2.63 7.50 Bass 1.02 3.66 1.69 2.17 2.85

for the soprano and bass separately. Since the preference for ascending minor sixths is total, that ratio is omitted. The preferences are stronger than average in the soprano and less than average in the bass in every case except -3:3. Since the soprano is the most melodically polished of the voices while the bass often performs as much a har- monic as melodic function, these preferences are aspects of the best melodic writing. For further evidence of this fact consider the soprano of the Sanctus of Brevis which Andrews cited as "an example of an almost perfect curvi- linear line" [18]. Table 21 shows the intervals present in this line and their ratios •. The relations are even stronger here. The prominent placement of the favored intervals demonstrates Palestrina's awareness of the 52 relation.

TABLE 21 THE FACTORS OF TABLE 19 AND TABLE 20 CALCULATED FOR THE SOPRANO VOICE OF THE SANCTUS OF BREVIS

Interval Total Percent

-8 0 -7 0 -6 0 -5 0 -4 0 -3 9 7.2 -2 66 52.8 0 3 2.4 2 38 30.4 3 2 1.6 4 5 4.0 5 0 6 0 7 0 8 2 1.6 Interval ratios -2:2 -3:3 4:-4 -5:5 8:-8

Total 1.74 4.50 00 00

A natural place to look for an explanation of these differences is in the Gregorian chant. Unfortunately the chant does not seem very helpful. The lack of rhythmic variety and the frequent repetition of notes prevent it from balancing its melodic curves in the same manner as Palestrina, and its melodic leaps often lack the careful 53 preparation and compensation characteristic of Palestrina. Since the explanation is evidently not purely melodic, it must relate to the polyphonic character of the music. Many writers have mentioned the natural ease and grace with which the music of Palestrina may be sung. It seems rea­ sonable to consider under what conditions intervals are most easily and effectively sung in a polyphonic setting. The two most important factors are clarity of the individual voice lines and ease of hearing and singing the intervals. The octave, for instance, is always easy to hear so the important considerations are vocal range and clarity. To fit within the range of one voice the upper note must be near the top and the lower near the bottom of the range. Since the higher notes are naturally stronger the ascending octave will be more prominent and clear in the texture while the descending octave will seem to disappear among the lower voices. These difficulties are minimized in the lowest voice so one would expect to find most descending octaves in the bass. The interval counts for the bass alone confirm that this is the case. In the smaller intervals the limitations of vocal range are less a factor. Clearly an interval will be easier to sing if its second note is already sounding in another voice, and when a triad is sounding the easiest and most stable note to sing is the root. Now a voice singing 54 the third or fifth of a triad and leaping to the root can span the following four intervals: a descending third, descending fifth, ascending fourth, and ascending sixth, precisely the preferred direction for each type of inter­ val. Considerable variety can be achieved within this framework by changing the harmony as the second note of the interval is sung, since this will not affect the ease of singing the interval at all. Examination of each occurrence of the rare ascending minor sixth provides striking confirmation of this idea. Of the twelve counted six were found to be elided intervals and disregarded. The initial note of each of the remaining six was found to be the third of a complete major triad in root position, though several other harmonizations are theoretically possible. The leap was upward to the root of the triad. The interval was evidently considered so prominent that consistent careful preparation was necessary. The exclusion of the descending minor sixth from the style can be explained by the impossibility of preparing it in this manner. An ascending major sixth could be prepared in this way only by a minor triad, however, the minor triad was felt to be less stable than the major triad and was often corrected by chromatic alteration to a major triad, especially at cadences and the beginnings and endings of sections. The less stable character of the minor triad may 55 account for the exclusion of the ascending major sixth from the style.

Jeppesen explained Palestrina's idiosyncratic treat­

ment of sixths in purely melodic terms. He states that the

ascending minor sixth was heard as an "energetic expansion"

of the perfect fifth and regularly resolved into the fifth

while descending leaps were not treated with such regular­

ity and could not be heard in this fashion. Of the six

instances in the four Masses, however, two resolve downward

by a third and three others essentially resolve down by a

third with the fifth interpolated as an eighth note passing

tone. Only one of the six examples is strongly followed

by a descending minor second as Jeppesen's explanation re­

quired. In other works there are examples of the relation­

ship he suggested, but it does not seem as essential as he

thought. It would be desirable to modify the harmonic

analysis to make more complete data available on the

preparation of leaps.

Since the bass voice tends to sound the root of each

chord the theory is less satisfactory for it than for the

upper voices. Harmonic factors may be present as well. It

is interesting to recall Schoenberg's ranking of the rela­

tive strengths of chord progressions [20]. Progressions in which the root moves down a fifth (or up a fourth) and down

a third are classed as strong, while progressions down a 56 fourth (or up a fifth) and up a third are classed as weaker. Root progressions of a second are called super-strong. The parallel with Palestrina's use of intervals is quite pre­ cise, though possible reasons for this, if any, are not clear except for the bass voice as it moves from root to root.

It is pleasing to think that a single consideration, ease of singing,· seems responsible for all the first-order interval probabilities from the most obvious, general step­ wise movement, to the most subtle, the ascending minor sixth. Lacking exact counts of intervals in large samples of music, traditional scholarship missed all but the most plain first-order relations, and without a thorough under­ standing of these their attempts to understand the second­ order relations of interval succession was doomed to failure. The succession of intervals is an important factor in the linear technique of Palestrina. The bass often assumes a harmonic role which entails a greater number of leaps and several idiomatic melodic patterns not common in the upper voices. In order to form the clearest possible picture of the best linear practice of Palestrina the upper voices will be considered separately. Table 22 contains the total number of occurrences of each pair in the soprano, alto, and tenor voices in the TABLE 22 COMBINED TOTAL OF ADJACENT OCCURRENCES OF EACH PAIR OF MELODIC INTERVALS IN THE SOPRANO, ALTO, AND TENOR VOICES; MARCELLI, BREVIS, AETERNA, AD FUGAM ==- 13394 Interval Pairs -a -6 -5 -4 -3 -2 0 2 3 4 5 6 8

-a 0 0 0 0 0 0 0 12 2 2 1 0 4 -6 0 0 0 0 0 0 0 1 0 0 0 0 0 -5 0 0 1 1 1 4 17 63 18 24 16 1 34

-4 0 0 1 0 1 1 15 63 18 18 7 0 3

-3 0 0 0 1 51 138 111 689 57 135 32 3 24

-2 2 0 34 10 282 2406 420 993 163 151 20 4 28

0 2 0 49 19 186 516 516 547 49 64 6 0 0

2 17 1 120 106 586 1002 406 1826 126 30 14 1 1

3 1 0 2 9 7 274 80 96 3 2 0 0 0

4 0 0 13 1 69 274 61 28 3 0 0 0 0

5 0 0 5 3 13 25 22 24 2 1 0 0 0

6 0 0 0 0 2 8 2 0 0 0 0 0 0 8 0 0 5 1 14 62 10 1 0 0 0 0 0

~------U1 --.] 58 four Masses. The values of Table 22 reflect both first and second-order probabilities so the first-order probabilities were used to project expected values of each pair and the ratio of actual to expected occurrences was formed for each pair in Table 24. Table 23 contains the first-order totals for the upper voices on which Table 24 was based.

TABLE 23

COMBINED TOTAL OCCURRENCES OF EACH MELODIC INTERVAL FOR THE SOPRANO, ALTO, AND TENOR VOICES WITH PERCENTAGES; MARCELLI, BREVIS, AETERNA, AD FUGAM

14447 Total Intervals Interval Total Percent

-8 23 0.2 -7 0 -6 1 o.o -5 248 1.7 -4 155 1.1 -3 1306 9.0 -2 4890 33.8 0 2029 14.0 2 4640 32.1 3 480 3.3 4 463 3.2 5 105 0.7 6 12 0.1 7 0 8 95 0.7 59 As an example the entry for (3, 2) in Table 24 is cal­ culated from Tables 22 and 23 as follows: Predicted occurrences: (480/14447) (4640/14447)13394

= 142.9 Actual occurrences: 96

Ratio (actual/predicted): 96/142.9 = 0.7. The ratio is less than one, which means that ascending thirds are followed by ascending seconds noticeably less than the random expectation. Ratios near one indicate little second-order effect. Favored pairs will have ratios substantially greater than one. Note that 0.5 and 2.0 or 0.2 and 5.0 are analogous magnitudes of the ratios. As the number of occurrences of a pair increases the ratio becomes quite stable, but when the pair is infrequent only very large or small ratios are sure indicators. The technique of Palestrina is so consistent, however, that the pattern of the results even among pairs which occur less than ten times is remarkably consistent. As an example of the power of this method of viewing the data consider Jeppesen's opinion that (-3, -2) is "an ordinary phrase in Palestrina style" while (3, 2) is "not found quite so often" [21]. Examination of Table 20 seems to confirm this for (-3, -2) occurs 138 times while (3, 2) occurs only 96 times. Table 24, however, shows that (-3, -2) occurs only 0.3 as often as expected while, as we saw TABLE 24 RATIO OF ACTUAL TO EXPECTED OCCURRENCES FOR EACH MELODIC INTERVAL BASED ON TABLE 22 AND TABLE 23; MARCELLI, BREVIS, AETERNA, AD FUGAM

-a -6 -5 -4 -3 -2 0 2 3 4 5 6 8

-a - - - - o.o ·o. o 0.3 1.7 2.8 2.9 6.5 - 28.6 -6 ------3.4 -5 - - 0.3 0.4 o.o 0.1 0.5 0.9 2."4 3.3 9.6 5.2 22.5 -4 - - 0.4 o.o 0.1 o.o 0.7 1.4 3.8 3.9 6.7 - 3.2 -3 o.o - o.o 0.1 0.5 0.3 0.7 1.8 1.4 3.5 3.6 3.0 3.0 -2 0.3 - 0.4 0.2 0.7 1.6 0.7 0.7 1.1 1.0 0.6 1.1 0.9 0 0.7 - 1.5 0.9 1.1 0.8 2.0 0.9 0.8 1.1 0.4 o.o 0.0 2 2.5 3.4 1.6 2.3 1.5 0.7 0.7 1.3 0.9 0.2 0.4 0.3 o.o 3 1.4 - 0.3 1.9 0.2 1.8 1.3 0.7 0.2 0.1 o.o - o.o 4 o.o - 1.8 0.2 1.8 1.9 1.0 0.2 0.2 o.o o.o - 0.0 5 - - 3.0 2.9 1.5 0.8 1.6 0.8 0.6 0.3 0.0 - o.o 6 - - - - 2.0 2.1 0.6 0.0 - - - - 8 -- 3.3 1.1 1.8 2.1 0.8 0.0 o.o o.o o.o - o.o

NOTE: - indicates the interval pair occurred zero times and was expected to occur zero times, i.e. the number of expected occurrences was less than one-half.

0\ 0 61 above, (3, 2) occurs about 0.7 as often as expected. The disparity in the raw number of occurrences is thus the re- sult of the preference for descending thirds over ascending , thirds demonstrated in Table 20, while Palestrina's prefer- ence between the two pairs is in fact the opposite of what Jeppesen thought.

Errors such as this permeate the traditional under- standing of Palestrina's style. Another instance is a theory of Palestrina's treatment of consecutive leaps in the same direction which receives considerable attention in all the various books on Palestrina's technique. Andrews states it as follows: Sometimes two leaps in the same di­ rection are found. In ascending passages of this kind the larger leap is generally taken first, followed by the smaller; • • • in downward progressions the shorter leap generally comes first. • • • The same principle applies to leaps followed or preceded by steps in the same direc­ tion [22]. Inspection of Table 22 shows that (-3, -3), which occurs 51 times, is the only pair of leaps in the same direction that occurs with any frequency. Its inversion, (3, 3), occurs only three times, each time to the words et ascendit indi­ cating Palestrina's sensitivity to the pair. Since these leaps are about the same size they contribute little to confirming the theory. The remaining larger interval pairs where the tendency should be most evident group as eight 62 confirming and six opposing instances, counting (-5, -5) as opposing. It seemed possible that most of the opposing in­ stances were elided intervals which should not technically be counted, so each instance was examined individually. With elided intervals eliminated there remained the follow­ ing pairs: (-3, -4), (-5, -5), (-5, -4), (-4, -3), (3, 4), and three (4, 3), four confirming and four opposed. One would have expected a stronger showing in view of the space authors have devoted to the subject. This is the sort of rule which can be self-perpetuating since editors of the music who believe it to be true can place the text in such a way that many exceptions are elided. Turning to the alternation of stepwise motion with leaps in the same direction, there is support for the theory in, the descending direction but not in ascending motion. The pairs (-2, -8), (-2, -5), (-2, -4), and (-2, -3) are tolerated more than the reverse pairs, although all are avoided consistently. In ascending motion, however, (2, 3), (2, 4), and (2, 6) are preferred over (3, 2), (4, 2), and (6, 2). Only (5, 2) exceeds (2, 5), while (8, 2) and (2, 8) each occurrence. There are many elided inter­ vals among these pairs, but I see no reason to expect that removing them would greatly affect the proportions since they did not in the other case. Measures twelve through fifteen of the Sanctus of Marcelli, for instance, contain 63 several prominent exceptions which could easily have been avoided had Palestrina wished. The bass voice seems to be distributed in about the same way except for a liking of (-4, -5). It seems more reasonable to ascribe this to tonic-dominant harmonic considerations rather than melodic factors.

These questions concern the fringes of the style. Table 25 gives an overall perspective of Palestrina's second-order interval technique. The intervals are classed as leaps (L), steps (S), and repeated tones (R), with plus and minus signs to indicate direction. Table 26 expresses these values as percent with each row totaling one-hundred percent. It is immediately noticeable that no type has a very automatic succession. There are no percentages over sixty. Clearly stepwise motion tends to continue in the same di­ rection, most strongly in descending motion and somewhat less so ascending. The tendency to reverse stepwise motion or repeat notes is about the same in both directions, but the treatment of leaps varies. Descending steps are left by leap about equally in both directions while ascending stepwise motion is left by downward leap almost five times as often as by upward leap. Repeated notes are followed by further repeats or stepwise motion in about equal pro­ portions, while leaps are somewhat less common, especially 64 TABLE 25

COMBINED TOTAL OCCURRENCES OF EACH PAIR OF INTERVAL TYPES, LEAP, STEP, OR REPEATED NOTE, IN THE SOPRANO, ALTO, AND TENOR VOICES; MARCELLI, BREVIS, AETERNA, AD FUGAM

-L -s R s L

-L 57 143 144 828 399

-s 328 2406 420 993 366

R 256 516 516 547 119

s 830 1002 406 1826 172

L 144 643 175 149 11

TABLE 26

DATA OF TABLE 25 EXPRESSED AS ROW PERCENTAGES

-L -s R s L

-L 3.6 9.1 9.2 52.7 25.4 -s 7.3 53.3 9.3 22.0 8.1 R 13.1 26.4 26.4 28.0 6.1

s 19.6 23.7 9.6 43.1 4.1

L 12.8 57.3 15.6 13.3 1.0 65 upward leaps. Leaps are followed by stepwise motion in the opposite direction slightly more than half the time. Re­ peated notes, leaps in the opposite direction, and con­ tinued motion upward by step are about equally common fol­ lowing ascending leaps, while descending leaps greatly prefer the leaps in the opposite direction. Consecutive leaps in the same direction are uncommon except for (-3, -3).

Jeppesen characterizes Palestrina's general treatment of intervals in the following way: Leaps upward are succeeded more regu­ larly by conjunct movement in the contrary direction than leaps downward • • • excep­ tions to this procedure are much rarer than when the leaps descend. • • • With leaps still larger, such as sixths and octaves, the difference in treatment of ascending and descending intervals is distinctly greater [23]. The value in Table 26 for (L, -S) is 57.3% while (-L, S) is 52.7%. The tendency is only slight. Exceptions are not "much rarer." Nor does Table 24 provide support for the idea that the larger intervals were more and more carefully compensated in ascending motion. Jeppesen maintained that the ascending minor sixth was allowed, though the same de­ scending interval was not, because of the great regularity of its resolution a step downward. By Table 24 (6, -2) has ratio 2.1 while (6, -3) has ratio 2.0. Thus the resolution by step and by leap are favored about equally. Nor is 66 (6, -2) noticeably more preferred than ( 3, -2), (4, -2), or (8, -2) with ratios 1.8, 1.9, and 2.1 respectively. Table 24 provides a striking picture of Palestrina's actual second-order preferences. The symmetrical relation of the quadrants is plain. Ignoring pairs involving re­ peated notes the pattern may be succinctly stated. If one of the two intervals is a leap then pairs of the same sign are avoided and pairs of opposite sign are favored. If both intervals are steps then the opposite is the case. There are two idomatic factors which account for the seven exceptions to the rule. First, stepwise motion in conjunc­ tion with ascending or descending fifths is slightly avoided except for (2, -5) . Compensation and preparation by leap is preferred over (-5, 2), (-2, 5), and (5, -2) which ordinarily would have been favored. The second fac­ tor is quite strong. Pairs which would outline portions of a triad first ascending then descending are very rare.

The pairs (4, -8) 1 (3, -5) 1 (4, -4), and (3, -3) occur much less than expected. Jeppesen thought that all leaps upward then downward were avoided because of the exposed position of the high note. This idea does not conform with the solid preference for (5 1 -4) 1 (4, -5) 1 (3, -4), and (4, -3). Perhaps outlining triads in this order was felt to give a very prominent and trivial harmonic character to the melody. Evidently this deficiency was not felt when the second 67 interval was smaller than the first since (8, -5) and (5, -3) were quite acceptable. When the central note was lower rather than higher the outlining of portions of triads was permitted, also. A general concern that the melody be easy to follow in a polyphonic texture is present in these re­ lations.

In considering the ratios of intervals following re­ peated notes there is an interesting reconfirmation of the first-order distribution of leaps. Note that (0, -5), (0, -3), and (0, 4) are slightly favored while (0, 5), (0, 3), and (0, -4) are slightly avoided. These are the same preferences evident in Table 20. The reason for this is that as repeated notes intervene the distance from the last interval change becomes greater and the need to compensate for it in any way decreases. The second interval is more freely chosen in this situation so first-order preferences reassert themselves somewhat. The ratio of 2.0 for (0, 0) means that repeated notes tend to cluster. This is most familiar in the Gloria and Credo where notes are repeated simply to fit all the syllables of the words in as small a space as possible without complicating the texture. Re­ peated notes tend to occur more frequently than expected after ascending thirds, fourths, and fifths. This is part of a general desire to prolong notes after upward motion which ANLS5 studies in detail. 68 Jeppesen based his theory of Palestrina's technique for creating a balanced melodic line on specific patterns of interval successions. Most of these patterns are weak or nonexistent. Stepwise compensation is not favored over compensation by leap as he supposed, even in ascending motion. Consider, for instance, the ratios of (4, -2), (4, -3), and (4, -5) which are 1.9, 1.8, and 1.8, respec­ tively. The differences in raw totals for these pairs are due to first-order, not second-order, interval preferences. Jeppesen realized that the key to Palestrina's master­ ful melodic balance was an understanding of the treatment I of ascending leap., which naturally arouse heightened at- tention and require careful compensation. He placed the major burden of compensation for ascending leaps on a tendency to stepwise downward resolution. Palestrina's basis for interval succession is more abstract than spe- cific and highly differentiated as he thought. He was closer to the truth when he observed that upward leaps tend to fall on accented beats, fitting the macrorhythm, however he failed to comment on the crucial relation between in- tervals and the microrhythms of the individual voices, which is an essential factor in Palestrina's sensitively balanced melodic lines.

The Program ANLSS In order to investigate the relation between rhythm 69 and interval in Palestrina•s melodic lines the number of occurrences of every possible rhythm-interval-rhythm se­ quence was counted. The succeeding rhythm was classed as either normal or syncopated in the same way as in ANLS2. The important relationships observed depend upon whether the first rhythm is shorter, equal, or longer than the second rhythm and the results are summarized in these cate­ gories. Table 27 summarizes the results of ANLSS for Aeterna, Table 28 for Brevis. Intervals with normal succeeding rhythms affirm the macrorhythm, while intervals with syncopated succeeding rhythms may conflict. Two fundamental methods of creating microrhythmic stress are lengthening a tone relative to its predecessor and approaching it by leap. ANLSS provides in­ formation on the way these two methods are used to affirm and conflict with the macrorhythm. When rhythms move from shorter to longer values the second tone is normal about twice as often as syncopated in both tables, stressing the macrorhythm and still pro­ viding substantial variety in the microrhythm. Long-short sequences have the opposite tendency to a lesser extent, and equal rhythms are fairly evenly distributed. Ascending seconds and several of the larger intervals have a tendency to fall in a normal position, and the remainder are fairly evenly divided. It seems safe to conclude with Jeppesen 70

TABLE 27

PATTERNS OF RHYTHMIC SUCCESSION IN TERMS OF THE INCLUDED MELODIC INTERVAL AND THE DEGREE OF MACRORHYTHMIC STRESS FOR ALL VOICES COMBINED; AETERNA

Interval Total Short-Long Equal Long-Short

Normal Succeeding Rhythm

-8 0 0 0 0 -6 0 0 0 0 -5 38 7 19 12 -4 51 12 21 18 -3 130 42 57 31 -2 626 262 232 132 0 169 44 80 45 2 813 333 403 77 3 98 35 45 18 4 46 19 22 5 5 33 17 12 4 6 2 1 1 0 8 1 1 0 0

Total 2007 773 892 342

Syncopated Succeeding Rhythm

-8 5 0 4 1 -6 1 0 1 0 -5 11 1 5 5 -4 23 3 15 5 -3 153 4 124 25 -2 601 68 236 297 0 306 62 160 84 2 519 115 331 73 3 56 26 30 0 4 52 33 19 0 5 8 3 4 1 6 1 1 0 0 8 13 9 4 0

Total 1749 325 933 491 71

TABLE 28

PATTERNS OF RHYTHMIC SUCCESSION IN TERMS OF THE INCLUDED MELODIC INTERVAL AND THE DEGREE OF MACRORHYTHMIC STRESS FOR ALL VOICES COMBINED; BREVIS

Interval Total Short-Long Equal Long-Short

Normal Succeeding Rhythm

-8 3 0 2 1 -6 0 0 0 0 -5 76 14 36 26 -4 40 7 20 13 -3 272 48 146 78 -2 756 239 280 237 0 176 54 67 55 2 870 431 356 83 3 48 25 18 5 4 111 55 46 10 5 36 19 13 4 6 0 0 0 0 8 2 0 2 0

Total 2390 892 986 512

Syncopated Succeeding Rhythm

-8 2 1 1 0 -6 0 0 0 0 -5 21 4 14 3 -4 10 1 7 2 -3 200 25 117 58 -2 713 109 263 341 0 340 57 194 89 2 512 144 287 81 3 62 35 26 1 4 73 62 8 3 5 14 9 4 1 6 0 0 0 0 8 29 23 6 0

Total 1976 470 927 579 72 that "there is no very intimate correlation of the dynamic and melodic accents in Palestrina, at any rate where the movement progresses in the time units" [24]. The "dynamic" accent referred to is the greater stress placed on the normal rhythms.

Jeppesen discusses in detail the psychological and linguistic bases of melodic and durational accents. The main correlation he finds between these accents is that leaps upward from normal eighth notes are forbidden. In the four Masses there is in fact only one exception to this rule. The situations in which this rule is applied to leaps account for less than two percent of all rhythmic successions. For the remainder Jeppesen can only suggest weakly that there is an analogous tendency in the longer rhythms though exceptions are much more frequent. There is a consistent correlation for all durations present in the tables which involves the way intervals connect rhythms rather than the way intervals follow normal rhythms. As a rule, ascending motion avoids shorter suc­ ceeding rhythms. The percentages over all ascending motion and over ascending leaps are recorded in Table 29. The rule is very strongly observed in ascending leaps to synco­ pated rhythms. Thus when the microrhythm is to conflict with the macrorhythm both methods of creating microrhythmic stress are consistently employed together. When stepwise 73 TABLE 29 COMBINATION AND CONDENSATION OF TABLE 27 AND TABLE 28 IN TERMS OF ASCENDING MOTION AND ASCENDING LEAPS, EXPRESSED AS ROW PERCENTAGES; AETERNA, BREVIS

Short-Long Equal Long-Short

Normal Succeeding Rhythm

Ascending motion 45.4 44.6 10.0 Ascending leaps 45.6 42.2 12.2

Syncopated Succeeding Rhythm

Ascending motion 34.4 53.7 11.9

Ascending leaps 65.3 32.8 1.9

motion is included and when leaps fall on normal rhythms exceptions are more frequent, since the imbalance is either supported by the macrorhythm or is less noticeable. The character of the exceptions indicates a continuing consideration of the rule rather than simple disregard. With note values longer than half notes, for instance, the disparity in length is much less audible. Also, half and dotted quarter notes each leave the following beat silent, so heard in relation to the macrorhythm they are essentially the same length. Palestrina used these rhythms in ascending motion, and though they were counted as exceptions to the rule by the program, they do not seriously violate its spirit. Example 1 contains another type of technical 74 exception which does not audibly violate the rule. Were the two quarter notes replaced by a half note the line would follow the rule perfectly. The resemblance is so great, here even the vowel is the same, that figures of this type are freely used, especially in the Gloria and Credo.

Ex. 1. Cantus, Kyrie, Ad Fugam, nun. 9-11

I~~ - J I j J r. I r .. II CL.-i - 1+c. c. - lc.i - Sol'\

Real exceptions to the rule occur mainly during pas- sages of continuing ascending stepwise motion, as in Ex- ample 2. In each case the note indicated conforms to the rule with respect to its successor rather than its pre- decessor. The fact that the final note is the highest of the group, combined with the stress it receives due to its duration and metric placement, cause the preceding notes to be heard mainly in relation the high note.

Ex. 2. • II

Continued ascending stepwise motion treated according to the rule is the key factor which elides the violation. 75

Example 3 contains two pair of similar melodic figures. A and C violate the rule by moving upward to a shorter rhyth­ mic value. Since they do not continue upward by step to disguise the imbalance, they are quite rare in Palestrina. B and D are inversions of A and c and are favorite melodic figures.

Fig. 3.

JC

rl 0 ~~· J. ~};Q J ~~¢ J.

Jeppesen discusses these and other similar situations

in terms of the placement of melodic high points, a ten- dency to swell on high notes, avoidance of upward leaps from accented beats, special treatment of eighth and six- teenth notes, and patterns of interval succession. It appears that much of this complexity is unnecessary. The principle of avoiding ascending motion to shorter rhythms and favoring ascending motion to longer rhythms accounts for a considerable portion of Palestrina's melodic tech-

nique. Descending motion exhibits the opposite disposition to a mild extent, simply because the motion from longer to shorter durations has to be put somewhere. The bass voice conforms less closely to the principle, perhaps because the 76 greater effectiveness of descending leaps in the lower

range requires different rhythmic reinforcement. Excep­

tions are also more frequent in Marcelli where the extra

voices cover the inner movements. Aeterna follows the rule

most consistently so it may be that Palestrina gradually • became more sensitive to the imbalance created by upward

motion to shorter durations. Certainly composers of the

previous century were less sensitive to this factor. Care-

ful study of the extent and nature of exceptions to this

principle might contribute to the solution of various

chronological problems concerning the music of the Renais- sance.

The Program ANLS7

Many writers have commented that Palestrina exercised

special care with respect to the highest and most exposed

tones in his melodies. For the purpose of this analysis

two different types of exposed points were considered. The

melodic peak refers to the highest pitch of each rest-to-

rest segment of notes. There will be only one melodic peak

pitch for each segment of notes not separated by a rest,

although there may be several different tones of the peak

pitch at various locations in the segment. A high point on the other hand is a particular note which is higher than its immediate surroundings, notwithstanding that it may be exceeded later in the melody. Each peak then is a high 77 point, but each high point need not be a peak. Jeppesen felt that Palestrina avoided repetition of the peak pitch in the phrase [25] and sustained high points [26].

With respect to melodic peaks ANLS7 counts the n~ber of repetitions of the peak pitch of each segment, the num­ ber of times each pitch was a peak value in each different voice, the rhythm of each note of peak pitch and its loca­ tion in the measure, and the intervals by which each peak was approached and left. Segments of seven notes or less were excluded as too short to reveal standard melodic con­ tour. The results of these counts need some interpretation in order to resemble the subjective quality associated with melodic peaks. First, it should be noted that different phrases of text not separated by rests were considered as one line and had only one peak pitch. This alters the totals both for and against Jeppesen's idea so the errors cancel to an ex­ tent. Repetitions of the peak from phrase to phrase remain noticeable and somewhat cloying to the modern ear and should be counted as exceptions. Also, each segment with three repetitions of a peak pitch represents three exceptions to the supposed rule rather than just one. Finally, all cases where the peak is repeated on adjacent notes represent one peak prolonged instead of two separate repetitions of the peak. With these factors modifying the raw totals the 78 results over the four Masses are 854 contrary instances and 586 confirming instances. This count is obviously not at all precise in terms of its purely musical meaning, but it does seem fair to conclude that Palestrina was not as sensi­ tive to the repetition of the peak pitch as Jeppesen thought, particularly in the lower voices. The totals from the soprano alone avoid the repetition somewhat more often. Each voice tends to have only three or four notes around the top of its range which are consistent melodic peaks, so it would be awkward to avoid fairly frequent repetitions. In the untransposed Masses, Marcelli and Ad Fugam, B is almost never a peak note, in either the Ionian or Mixolydian modes. In the transposed mode it is, of course, E which is absent. For the remaining pitch-classes the general distribution over peak pitches is similar to the pitch-class profile of ANLSl, the tonic and dominant favored and the subdominant avoided. The other scale de­ grees are more flexible than in ANLSl and occasionally even overshadow the tonic. Considerations of vocal range in re­ lation to the mode account for these perturbations. The distribution of rhythmic durations over the peaks is as would be expected considering ANLS5. Eighth notes are quite rare and quarters are less common than they are normally. The favored rhythms are dotted quarter, half, and dotted half notes. These rhythms are distributed over 79 the four beats of the measure in a quite irregular fashion, varying widely from section to section and Mass to Mass. The weaker beats contain almost as many peaks as the stronger beats altogether.

The distribution of surrounding intervals shows that repeated notes are less common than usual at the peaks of the four-voice Masses but occur with the expected frequency in Marcelli. The three earlier works leap to and from peaks more often than the second-order probabilities pre­ dict. This combined with the stricter avoidance of as­ cending motion to shorter rhythms result in Aeterna's almost excessively balanced melodic lines. Peaks consti­ tute about ten to thirteen percent of the notes in the Masses. In locating high points the following procedure was employed. Each note was examined individually. First the preceding notes were considered in reverse order moving toward the beginning of the melody. If two notes of lower pitch or a rest occurred before a note of higher pitch was found then the succeeding notes were checked. If this criteria was satisfied in both directions the note was de­ fined to be a high point. This definition is quite arbi­ trary, but it seems to correspond to Palestrina's melodic technique since when counted in this way high points account for just over seventeen percent of the notes in 80 each Mass. Specifically, Marcelli has 17.7%, Brevis 17.3%, Ad Fugam 17.2%, and Aeterna 17.2% highpoints. This simi­ larity seems too great to be coincidental and indicates that Palestrina placed high points at the same rate through­ out all four Masses.

The intervals between adjacent high points are recorded in Table 30. Clearly their distribution has little in com- mon with that of the melodic intervals. Instead it reflects the tendency of the melodies to remain in the tessitura of the vocal range decline gradually in pitch. The very high percentage of repeated notes implies that Palestrina was not sensitive to the intervals formed and so the contour is due to other factors.

TABLE 30 MELODIC INTERVALS BETWEEN ADJACENT MELODIC HIGH POINTS WITH PERCENTAGES; MARCELLI, BREVIS, AETERNA, AD FUGAM

Interval Total Percent

-8 1 0.1 -7 6 0.3 -6 11 0.5 -s 50 2.4 -4 100 4.9 -3 216 10.5 -2 359 17.5 0 765 37.3 2 299 14.6 3 145 7.1 4 65 3.2 5 29 1.4 6 4 0.2 7 0 8 0 81 The music of Bach seems to show a much greater concern with the exact intervals between high points. It is easy to find lengthy periods of stepwise motion from high point to high point. This was an inherent part of his extensive use of sequences. Palestrina, on the other hand, almost never employs sequential melodies. The modern musician has trained himself to listen for and appreciate relations be­ tween widely separated high points because they are an im­ portant factor in much of his standard repertoire. Pale­ strina does not seem to be greatly concerned with relation­ ships of this type, however, so attempts to describe or evaluate his music in these terms are anachronistic. CHAPTER III

HARMONIC ANALYSIS

The harmonic analysis is performed in terms of the pitch-class and the simple intervals above the lowest sounding pitch •. By a simple interval in meant an interval smaller than an octave, so tenths are indistinguishable from thirds, twelfths from fiths, and so forth. The pro­ gram HRMNY can retrieve the vertical structure in both man­ ners at each eighth note offset in a Mass. HRMNY records both first-order and second-order probabilities with re­ spect to any regular pattern of offsets desired. It will, for instance, total separately all harmonies on the strong beats, the weak beats, the offbeats, and all offsets com­ bined. Second-order calculations were made to record the harmonic progressions from strong beats to weak, weak beats to strong, and offbeats to following beats. The program is easily altered to consider any other patterns desired. The results obtained are not surprising, though exceptions to some rules seem more frequent than most writers admit. Some of the results for Marcelli are summarized in Table 31 in terms of complete triads, incomplete conso­ nances, and suspension dissonances which tend to resolve

82 TABLE 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 CHORDS; MARCELLI

Strong Beats Weak Beats

Consonance Suspension Consonance

Harmony Complete Incomplete Correct Incorrect Complete Incomplete

c 302 41 22 11 320 23

D 115 20 31 5 132 23

E 20 18 7 0 51 12

F 150 20 33 2 146 15

G 215 35 55 6 287 37

A 124 16 10 3 108 14

B 1 5 11 0 15 9 Passing Dissonance 83

(X) w 84 to each harmony. Incorrect suspensions are those in which the resolution of the suspension is already sounding when the suspension dissonance occurs. The tonic and dominant harmonies are most common, while the mediant is avoided. The diminished triad is relegated to the weaker beats. Marcelli is a six-voice work so incomplete consonances are relatively infrequent and incorrect suspensions more common than in the four-voice works. One particularly interesting suspension is shown below. Ex. 4. Brevis-Benedictus, rom. 48-50

j_ I I - - rl rx (lfr, .....J l ..-

A suspension theoretically consists of three parts, prepa- ration, suspension dissonance, and resolution. The prepa­ ration is ideally a consonance between the two voices in- volved on a weak beat. One of the voices then moves to form a dissonance on the following strong beat, and the other voice resolves downward by step to a consonance on the next beat. The indicated note prepares a suspension with a dossonant minor seventh. Patrick's study of suspen- sions in Josquin [27] found that preparations such as this occur occasionally in spite of the usual theory of 85 suspensions which allows only the fourth as a dossonant preparation. Evidently the imitation in the upper voices was considered sufficiently valuable to override the usual suspension procedures. The bass was also engaged in imita­ tion and so was not changed. This is a good example of the primacy of linear elements in spite of the generally triadic harmonic ideal. Table 32 presents the results for Marcelli in terms of the intervals present above the lowest sounding voice. The relative instability of what in modern terminology is a first inversion with respect to a triad in root position is clear. The first inversion occurs mainly on the weak beats and performs a more linear than harmonic function. The second-order totals will show that the first inversion is seldom sustained for more than a single beat. Suspension dissonances are more common than passing dissonances on the weak beats, with a suspension once every three measures on the average, compared to once in seven measures for passing tones. Passing dissonance on the offbeats is evidently about as common as the suspension, though exact totals can­ not be recovered from the analysis in its present state. The strong beats because of the resolution of suspensions. The second-order counts from strong to weak beats in Marcelli record 108 first inversions on the strong beats, of which only 22 are followed by a first inversion on the TABLE 32

PERCENTAGES OF VARIOUS VERTICAL STRUCTURES ON STRONG AND WEAK BEATS 1 MARCELLI

Strong Beats Weak Beats Type of Harmony Percent Type of Harmony Percent

Triad-root position 62.6 Triad-root position 69.6

Triad-first inversion 8.4 Triad-first inversion 13.3

Incomplete consonance 12.1 Incomplete consonance 10.1

Correct Suspension 14.5 Passing dissonance 6.9

Incorrect suspension 2.2

Q) 0\ 87 succeeding beat. Root position triads occur 797 times, followed by 611 root position triads on the weak beats. These totals include possible changes of root. The first inversion then is sustained twenty percent of the time, and the root position seventy-seven percent. From weak beats to strong beats the first inversion is sustained eleven percent and the root position seventy percent of the time. It is clear that the first inversion was not stable enough to be sustained for long even in the interior of the music. Of the triads in root position the major triads are more - consistently sustained than the minor. This is probably a result of the modes of the Masses. Works in the Dorian or Phrygian modes may have entirely different tendencies. The most common chord progressions from each chord are tabulated in Table 33. The amount of diversity of harmonic succession caused by linear movement is striking. An ac- curate harmonic analysis of music of this type would have to consider and remove linear elements by constructing the harmonic background of each work. Works with five or more voices should be easily analyzed in this fashion. Works with fewer voices tend to have a much greater proportion of incomplete consonances which are harmonically ambiguous. A harmonic third or sixth, for instance, sometimes seems to function as the root and third of a triad and at others as the third and fifth. Compare Table 34 for Aeterna with 88 TABLE 33 COMMON SUCCEEDING PARMONIES FOR EACH BASIC HARMONY ; MARCELLI

Harmony Conunon Succeeding Harmonies

c G, F

D G, A

E F, A F G, c, D G c A F, D, E B c

Table 32. The large percentage of incomplete consonances, most of which are thirds and sixths not fifths and unisons, makes the harmonic characteristics of Aeterna play a lesser role in the total effect than in Marcelli. The linear basis of the two works is essentially the same, however. This suggests the possibility that many of the supposed harmonic characteristics may result from linear considera- tions combined with a general desire to prolong the tonic and dominant while avoiding the tritone. One example is the choice of tones to form suspended dissonances. In Marcelli the four most common suspensions each prolong C or G and so shorten the B or F to which they resolve. The development of dominant seventh and augmented sixth chords from what were originally linear considerations is a more TABLE 34 PERCENTAGES OF VARIOUS VERTICAL STRUCTURES ON STRONG AND WEAK BEATS; AETERNA

Strong Beats Weak Beats Type of Harmony Percent Type of Harmony Percent

Triad-root position 39.4 Triad-root position 42.4 Triad-first inversion 9.6 Triad-first inversion 14.1 Incomplete consonance 34.5 Incomplete consonance 39.3 Suspension dissonance 16.5 Passing dissonance 4.1

(X) \0 90 recent example.

Two of the principle differences in the harmonies of Josquin and Palestrina are Palestrina's preference for com­ plete triads and root progressions of fourths and fifths instead of seconds. The difficulty in maintaining complete triads while using stepwise motion and root progressions by seconds is the constant threat of parallel fifths. One of the upper voices must usually leap in order to move to a complete triad. Josquin was less concerned with complete triads than Palestrina, so root movement by seconds was the easiest way to achieve maximum stepwise motion, especially in canonic writing. When roots progress by fourths and fifths all the upper voices can move by step while moving between two complete triads. The necessary leap is in the bass where it is less prominent, and the presence of a common tone aids in the prolongation of the tonic and domi­ nant while root progression by seconds would tend to equal­ ize the different scale degrees. Viewed in this way it is clear why root movement by thirds is the least common of all in many different musical styles. The possibility for stepwise movement in the voices is minimized. It seems plausible to conclude that the change in preference for chord progressions was the inevitable result of the desire for·complete consonances while retaintng predominately stepwise movement, especially in the upper voices. The 91 dominant-tonic chord progression which became such an end in itself in later centuries was originally favored for its ability to combine smooth voice leading with full sonority. In moving to the tonic chord by a fourth or fifth the common tone is the tonic or dominant. The desire to prolong the tonic and dominant at cadences while maxi­ mizing the opportunity for stepwise motion explains Pale­ strina's preference for what came to be known as plagal and authenic cadences. Thus the only definite purely harmonic preference discernable in Palestrina seems to be towards the complete consonance. The cadences and progressions and favored chords are the result of the first-order pitch and interval probabilities. CHAPTER IV

INFORMATION CONTENT ANALYSIS

Throughout the analyses there have been observations that one characteristic was favored and another avoided or one note prolonged and another shortened. Expressed in another way, some events in the music are highly probable, others less so, and some are excluded altogether. In gen­ eral the most probable events are associated with an aural impression of stability, certainty, or repose, while the less probable events create uncertainty, tension, and the expectation of resolution. The resolution must occur, for the less probable events by definition cannot continue for long without becoming more probable in context. The care­ ful introduction and resolution of the less probable events is a basic process in the creation of the middle level of musical structure. The lower levels of musical structure are the proba­ bilities imposed upon the intervals, chords and chord pro­ gressions, notes, and so forth, while the upper levels con­ cern the organization of musical events as a whole. Music theory has traditionally been more successful in describing the upper and lower levels than the middle levels of musical

92 93 structure. Heinrich Schenker began to speak to this prob- lem with his concepts of foreground and middle-ground. His analytical techniques can locate the landmarks on the musical landscape in greater or lesser detail, but the drama of the journey through the landscape is suppressed. The drama is related to the pattern of dispersion of certain less probable events within the musical structure. The preceding analyses provide a good estimate of the prob­ abilities of various fundamental musical quantities in Palestrina's music. One way to estimate how the less prob­ able events are distributed is to combine the various prob­ abilities of all events in some small unit of music. The values for various units can then be compared and the loca­ tion of concentrations of less probable events determined. The difficulty is to express the various probabilities in units which allow meaningful addition and comparison. Information theory provides this capability. The music is viewed as a message and the information content of a mes­ sage is expressed in units of binary digits or bits. The information in a message is the theoretical minimum number of binary digits necessary to transmit the message assuming perfect coding. The basic equation of information theory used in the analysis is due to the work of Claude Shannon

[28]. In an alphabet of N symbols, n1 , n2, •••, nN, with p(ni) the probability of the occurrence of ni' Shannon 94 gives the following expression for HN, the information con­ tent in bits per symbol.

N (1) ~ =- t p(n.)log p(n.) i=l 1 2 1

N t p(ni)log (1/p(ni)) bits/symbol i=l 2

If p(ni) = 1/N for each i, all symbols are equally probable and maximum disorder or information will result.

N t (l/N)log N­ log N bits/symbol ~max- i=l 2 2

On the other hand, if one symbol has probability one the information content is defined to be zero. Thus infor- mation content ranges from zero in a totally certain message to log N for a random message using N symbols. 2 To find the total information, H, in a message simply multiply (1) by the total number of symbols in the message,

T.

N H- T t p(ni)log2 (1/p(ni)) bits i=l

If t. is the number of occurrences of n. then p(n.) - 1 1 1 t./T. Substituting in the above equation yields 1

N H- t tilog (1/p(ni)) bits i=l 2 95 which indicates that each occurrence of n. contributes 1 log2(1/p(ni)) bits of information to the message. The opposite of information is predictability or re­ dundancy. Redundancy may be expressed in percent as

R = 100 (~ max

Marcelli provides the examples of information calcula­ tions based on an alphabet of seven symbols, the pitch­ classes as in ANLSl, in Table 35. The more probable notes contribute less information with each occurrence, but still account for the bulk of the total information because of their greater numbers. HN is quite near the maximum value so the redundancy is low. If chromatic tones were consid- ered their rarity would increase HN and R considerably max while leaving ~ about the same. Using these facts it is possible to determine the information content of small por­ tions of music with respect to the overall probabilities of the various symbols present in the music. Higher or lower amounts of information in equal portions of music will in- dicate greater or lesser concentrations of the less prob- able events. The measure of length four quarter notes was chosen as the unit of music over which comparisons of information con- tent would be made. Each eighth note in which C sounded added 2.17 bits to the information content of the measure, 96 TABLE 35 RESULTS OF VARIOUS INFORMATION CONTENT CALCULATIONS; MARCELLI

Pitch-Class p (n.) log (1/p(ni)) 1 2 c 0.222 2.17 D 0.145 2.79 E 0.131 2.93 F 0.090 3.48 G 0.219 2.19 A 0.126 2.98 B 0.067 3.89

~max - log2 7 - 2.81 bits/symbol

7 HN =iLl p(ni)log2 (1/p(ni)) - 2.70 bits/symbol

R - 3.9% each eighth note of D added 2.79 bits, and so forth, with rests counted as zero. If C sounded through all six voices all the measure, the information content of that measure would be 6 x 8 x 2.17 = 104.16 bits. The maximum possible would be whole note B's in every voice or 6 x 8 x 3.89 = 186.72 bits, which of course never happens. The analysis calculates the total information in bits first from the first through the eighth eighth note offset, then from the 97 second to the ninth, third to tenth, and so on to the end of a section. The process may be visualized as a window one measure long sliding one eighth note at a time through the Mass. The total information visible in the window is calculated at each eighth note offset. Information content was calculated with respect to four other sets of probabilities in addition to pitch­ class: pitch, pitch and voice, harmony, and interval. The pitch probabilities were obtained from the pitch profile of ANLSl for Marcelli. The pitch and voice probabilities re­ flect the probability of a pitch relative to the voice in which it occurs, i.e. two-lined A is very common in the alto and less so in the tenor, so the same note contributes less information in the alto than the tenor in this calcu­ lation. The harmonic probabilities were computed without regard to doublings and inversions but consider the loca­ tion in the measure. Separate probabilities were used for the intervals in the upper voices and intervals in the bass voice. In order to consider eight possible intervals for each measure the window for the interval calculations was nine eighth notes in length rather than eight as for the others. For this reason Examples 5-8 include the extra pitches on the ninth eighth which are considered only in the interval information content. The values are given in bits. 98

Ex. 5. Marcelli-Kyrie, m. 12 Infonnation (in bits) l Pitch 181.41

.J r I r Pitch-class 116.99 ~ . j\ .i:l lll Pitch/Voice ' 125.10 ' I .,.. . .. Hannony · 21.62 I Interval 20.56

Ex. 6. Marcelli-Gloria, m. 110

L ) Pitch - ' I 137.56 . I . J 'j Pitch-class 97.51 J J Pitch/Voice 104.51 4 .... - ) I 1 Harmony 44.52 - Interval 12.29

Ex. 7. Marcelli-Credo, m. 46

\ J J I Pitch 218.71

I Pitch-class 149.08 ~ ~ J .e) u • Pitch/Voice 169.68 , I J I I J. . J Harmony 39.21 . I Interval 44.26

The information content of the harmonies and intervals seems to fluctuate rather haphazardly. There is no obvious pattern, so it appears that the uncommon harmonies and in- tervals are not distributed through the music in a regular pattern. 99 The three pitch-related calculations, however, show a strong tendency to increase gradually for a few measures, then decrease gradually, then increase again, setting up a fairly regular oscillatory pattern. Sometimes the ascent is very quick and followed by several measures of descent, at others the ascent is slower and the descent rapid, and at others they are more equally spaced. This trait is most apparent in the pitch-class calculations. Cadences are usually prepared by an ascent to a very high information level followed by a slow decline to a low point at the cadence note. The changes between the values in the fol­ lowing example cadence are smooth and uniform, and the eight measures preceding it increase slowly from 100.69 to the high point of 142.12 with which it begins. The numbers for each measure are the pitch-class information values. The first value is computed from the pitch-classes on the first eighth of the measure through the following seven eighths, the second value begins with the second eighth and continues through the first eighth of the following m~asure, and so forth. Example 9 is the Hosanna subsection of the Sanctus. The pitch-class information content is recorded for the initial offset of the measure only, except that when a high point occurs in the center of a measure its value is in­ cluded with an appropriate bracket to indicate the offsets 100 Ex. 8. Marcelli-Credo, mrn. 191-7 I~~ .., J. I l I I I I I I I I hi l

~r p I I I l \ .... -· !. I ·~-.t I l.J J j 1 I J• - 1 .p..· , , I •- - ' I•· 4' - ~.I l - - . m. 191 m. 192 m. 193 m. 194 m. 195 m. 196 m. 197 1 142.12 128.78 116.02 119.14 133.12 140.76 116.61 2 141.38 125.48 115.38 119.98 136.22 137.59 3 140.62 122.18 114.74 120.83 139.32 134.42 4 137.96 121.51 116.00 121.32 139.42 133.33 5 135.30 120.84 117.27 121.81 141.23 130.52 6 132.45 119.71 117.51 124.51 141.72 127.35 7 129.59 118.59 117.75 127.21 142.22 124.18 8 129.19 117.30 118.44 130.16 141.49 120.40 covered. Changes of direction of information flow tend to fall on strong beats like harmonic changes. There is a noticeable tendency for long ascents or descents to be broken by a slight movement in the opposite direction near the center, for instance, in mrn. 63-4, mm. 71-2, and m. 77. The descents from the high points in m. 65 and m. 77 are smooth and uninterrupted. The cadences on c stand out clearly as low points of information content relative to their surroundings. The downturn in m. 64 marks a cadence on G. Throughout the Mass cadences are related to low and/or unfluctuating levels of information, while high information content is associated with harmonic tension and is seldom greatly sustained except immediately prior to important cadences for the sake of 101 Ex. 9. Marce11i-Sanctus, mm. 60-80

~~ I I I

I I ' ~ - --- If.-_...... '" .- ~ !.J r:J "j ..... _J ..d I .J.. . J_ ~ l .J_'

I . 88.5 82.1 92.4 125.4 122.5 I I 133.4 ~~~ --- J. 11 1 I J_ ~,-., 1- ~·1 I I I

I j__.' I I 1 .I I II +.,. _, •I -·1 I _l I 1 J!! I . I I ' I I ,,_ -:-...... r r".- , ~ .,.

I I 144.2 138.1 131.6 112.9 101.7

-- 17ft I J I I I

I I I ' -...... L.!..LfJ1 +I j _V'f 1... 1.~ l· --- - ._ I .... 1 ,---

I • I I v J. -, f I .... I -A ~I '11':-. • .. I ~ ]' .•

141.8 130.9 -138.5 94.0 122.0 I I 145.4

, I J j J. }l~r-, I IM

1 I~~ 4f I '-.t• •• - -"''ll .I I

•• . I I I I ..L . I ' 113.9 118.3 141.6 121.0 119.4 110.6 102 contrast. The smoothness of the changes in information con­ tent indicate a tendency for the harmonic progressions to move between chords of gradually changing pitch-class infor­ mation content. This technique of analysis is capable of considerably greater refinement. The main difficulty seems to be that many less probable events are not musically audible or sig­ nificant. The harmonic calculations for information fail to be of much interest primarily, I suspect, because pass­ ing tones on the offbeats and incomplete consonances are given too much weight and distort the totals. The pitch and voice information values count the rare low notes in the soprano as contributing as highly to the information as the rare high notes. Unfortunately, the low notes are weak and often covered by the other voices, while high notes are prominent and important. If the probabilities were altered to correspond to a greater extent with audible musical events, perhaps the other analyses would reveal some structure. The pitch-classes sounding at any moment are always significant, so it is not surprising that they are carefully controlled. It would be interesting to make the probabilities of the pitch-classes somewhat dependent on the immediate con­ text rather than permanently fixed. This might permit in­ ternal cadences on chords other than the tonic to show 103 greater similarity to the cadences on the tonic, and would reveal the extent to which interior cadences are prepared by pitch-class distributions more characteristic of the mode of the cadence than the overall mode of the work. Hiller and Bean's information analysis of sonata ex­ positions indicated that more recent composers tend to balance their music by sections with respect to informa­ tion [29]. A section of high information would, for in­ stance, be followed by a lower section, then another high one; or the sections might gradually increase or decrease over a long period. It is reasonable that Palestrina's overwhelming concern for balance would cause the compensa­ tion to occur on a measure to measure basis with an im­ balance never being sustained through an entire section. The distribution of the various pitch-classes is thus an important aspect of the middle level phrase structure and the demarcation of sections through cadences in the music of Palestrina. CHAPTER V

CONCLUSION

Donald Grout has stated that "no other composer's technique has been subjected to more minute scrutiny" than Palestrina's [30.]. It should be sobering for the musicolo­ gist to realize the extent of the divergence between tra­ ditional scholarship and the results of these straight­ forward analyses in what was thought to be a very well understood area. The tremendous similarity of all twenti­ eth century writings concerning the technical aspects of Palestrina's style suggests the difficulty. Preconceived opinions of the types of patterns to expect and the methods of analysis to employ prevented scholarship from continuing to progress. The accuracy of an analysis by computer is useful in minimizing the effect of previous misconceptions and revealing new relationships concerning the music. The principle problem in computer analysis is formu­ lating procedures whose results correspond to audible musical events. Certainly many important musical phenomena are highly subjective, however there is no reason to sup­ pose that knowledge of their treatment is more accurate than that of factors which are subject to objective

104 105 verification. After realizing that Jeppesen misunderstood the nature of interval succession, omitted consideration of the linear basis of harmonic progression, and failed to clearly state the essential relation between rhythm and melody in Palestrina, I find myself considerably less will­ ing to accept his conclusions on matters more difficult to investigate. Furthermore, the process of attempting exact characterizations of these subjective events is itself highly educational. The business, governmental, and scientific communi­ ties have been quick to appreciate the benefits of sta­ tistical and computer support in reaching decisions con­ cerning large masses of data, for to them incorrect de­ cisions will be costly. If musicians want to understand the nature of musical styles they will have to attack the problem with the same powerful tools used by a marketing analyst seeking to determine just what factors the American people prefer in a deodorant. REFERENCES

1R. Casimiri et al., ed., Opere Complete (Rome, 1939-).

2J. Roche, Palestrina (London, 1971), pp. 10-12.

3 H. K. Andrews, The· Technique of Palestrina (London, 1958)' pp. 158-61.

4 K. Jeppesen, "Marcellus-ProLleme," Acta Musicologica, Vol. XVI-XVII.

5R. Marshall, "The Paraphrase Technique of Pale­ strina," Journal of the American Musicological Society,

XVI (1963) 1 pp. 350-8, 366-7.

6H. K. Andrews, op. cit., pp. 215-7.

7J. Roche, op. cit., pp. 12-3.

8J. Roche, op. cit., pp. 13-4.

9A. Mendel, Musical Pitch (Armsterdam, 1968).

10R. Marshall, op. cit., p. 350.

11K. Jeppesen, Counterpoint (London, 1946), p. 69.

12J. L. curry, "A Computer-aided Analytical Study of Kyries in Selected Masses by Johannes Ockeghem" (Ph.D. dis­ sertation, University of Iowa, 1969), p. 114.

13L. Hiller and c. Bean, "Information Theory Analyses of Four Sonata Expositions," Journal of Music Theory, Spring 1966, p. 104.

14 R. Marshall, op. cit., pp. 350-4.

. 106 107 15 K. Jeppesen, The Style of Palestrina and the Dis- sonance (London, 194~), pp. 2o-30. 16 G. Reese, Music in the Renaissance (New York, 1959), p. 461.

17 M. and R. Brender, "Computer Transcription and Analysis of Mid-Thirteenth Century Musical Notation," Journal of Music Theory, Winter 1967, p. 206. 18 H. K. Andrews, op cit., pp. 46-7.

19 K. Jeppesen, The Style of Palestrina and the Dis- sonance, p. 78.

20A. Schoenberg,· Theory of Harmony (1948), pp. 70-5.

21K. Jeppesen, The Style of Palestrina and the Dis­ sonance,· p. 77.

22H. K• And rews, op. c1•t ., pp. 48 - 9 •

23K. Jeppesen, The Style of Palestrina and the Dis­ sonance, pp. 77-8.

24Ibid., p. 61.

25K. Jeppesen, Counterpoint, p. 95.

26K. Jeppesen, The Style of Palestrina and the Dis­ sonance, p. 59.

27P. Patrick, "A Computer Study of a Suspension­ Formation in the Masses of Josquin DesPres" (Ph.D. disser­ tation, Princeton University, 1973), pp. 30-37.

28c. E. Shannon and w. Weaver, The Mathematical Theory of communication (Urbana, Illinois, 1949).

29L. Hiller and c. Bean, op. cit., PP• 118-21.

30o. J. Grout, A History of Western Music (New York, 1960) 1 P• 239 • LIST OF REFERENCES

Andrews, H. K. The Technique of Palestrina. London: Novello, 1958. Brender, M. and R. "Computer Transcription and Analysis of Mid-Thirteenth Century Musical Notation." Jour­ nal of Music Theory (Winter 1967), 206. Casimiri, Ret al., ed. Le Opere Complete. Rome: Frateli Scalera, 1939. Curry, J. L. "A Computer-aided Analytical Study of Kyries in Selected Masses by Johannes Ockeghem." Ph.D. dis­ sertation, University of Iowa, 1969. Grout, D. J. A History of Western Music. New York: w. w. Norton, 1960. Hiller, L. and Bean, c. "Information Theory Analyses of Four Sonata Expositions." Journal of Music Theory (Spring 1966), 104. Jeppesen, K. "Marcellus-Probleme." Acta Musicologica, XVI (1932), 1-32. Jeppesen, K. Counterpoint. New York: Prentice Hall, 1939. Jeppesen, K. of Palestrina and the Dissonance. London: Marshall, R. "The Paraphrase Technique of Palestrina." Journal of the American Musicological Society, XVI (1963), 350-8, 366-7. Mendel, A. Musical Pitch. Amsterdam: Frits Knuf, 1968. Patrick, P. "A Computer Study of a Suspension-Formation in the Masses of Josquin DesPres." Ph.D. disser­ tation, Princeton University, 1973. Reese, G. Music in the Renaissance. New York: W. W. Norton, 1959. 108 109 Roche, J. Palestrina. London: Oxford University Press, 1971. Shannon, c. E. and Weaver, w. The Mathematical Theory of Communication. Urbana, Ill.: University of Illinois Press, 1949. Schoenberg, A. Theory of Harmony. New York: Philosophical Library, 1948.