A Multidimensional Polymetric Analysis of Excerpts from...

A MULTIDIMENSIONAL POLYMETRIC ANAYLSIS OF EXCERPTS FROM THE

WIND BAND MUSIC OF DAN WELCHER AND YO GOTŌ

David DeWitt Robinson, Jr., B.M.E., M.M.

Dissertation Prepared for the Degree of

DOCTOR OF MUSICAL ARTS

UNIVERSITY OF NORTH TEXAS

December 2016

APPROVED:

Eugene Migliaro Corporon, Major Professor

Diego Cubero, Committee Member

Dennis Fisher, Committee Member

Benjamin Brand, Director of Graduate Studies

in the College of Music

John Richmond, Dean of the College of Music

Victor Prybutok, Vice Provost of the Toulouse Graduate School

Robinson, David DeWitt, Jr. A Multidimensional Polymetric Analysis of Excerpts from

the Wind Band Music of Dan Welcher and Yo Gotō. Doctor of Musical Arts (Performance),

December 2016, 91 pp., 4 tables, 101 figures, references, 47 titles.

Polymetric writing is an integral technique in contemporary compositional practice. Dan

Welcher and Yo Gotō are principal employers of this practice in the wind band medium. Their

methods endure even the results of modern scholarship showing limited human perception of

polyrhythmic events. This dissertation provides a comprehensive metric analysis of excerpts

from the music of Welcher and Gotō. Five examples are explored from major band works of

each of the two composers. The analytical process in the study utilizes the metrical concept set

forth by Maury Yeston, so that a comparison can be made between the rhythmic components of

the competing meters. The results of the study show that both Welcher and Gotō, in all ten

excerpts, create polymetric sections containing elements that surpass the aural limits proposed by

modern scholarship. Additionally, through identification of the misaligned metric layers causing

each polymeter, pedagogical considerations are offered to aid performance of each identified

excerpt.

ii

Copyright 2016

by

David DeWitt Robinson , Jr

iii

ACKNOWLEDGEMENTS

I would like to thank the faculty of the Wind Studies area, Eugene Migliaro Corporon,

Dennis Fisher, and Nicholas Williams, for their constant support and guidance. Thanks, too, to

Daniel Arthurs and Diego Cubero, who also helped me through the proposal and dissertation-

writing process. I am indebted to my colleagues and students at McMurry University for their

patience as I completed this endeavor. Finally, I am also grateful for the unwavering

encouragement from my family, without whom a project such as this would not be possible.

Appreciation is extended to Theodore Presser Company (the works of Dan Welcher) and

Bravo Music, Inc. (the works of Yo Gotō). All musical examples in this dissertation, except

those in the public domain, are excerpts from their original respective scores and are used with

their permission.

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TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ........................................................................................................... iii

LIST OF TABLES ......................................................................................................................... vi

LIST OF EXAMPLES .................................................................................................................. vii

Chapters

1. INTRODUCTION ...................................................................................................1

Purpose .........................................................................................................1

Significance..................................................................................................2

2. COMPOSER BIOGRAPHIES.................................................................................4

Dan Welcher ................................................................................................4

Yo Gotō ........................................................................................................6

3. POLYMETER..........................................................................................................8

A Definition of Polymeter ...........................................................................8

The Problem of Polymetric Perception ........................................................9

Method of Analysis ....................................................................................11

4. POLYMETRIC ANALYSES OF WELCHER EXCERPTS .................................15

Zion (1994), mm. 214-221 .........................................................................15

Laboring Songs (1997), mm. 161-172 .......................................................20

Circular Marches (1997), mm. 133-147....................................................25

Circular Marches (1997), mm. 196-223....................................................32

Minstrels of the Kells (2001), mm. 109-128 ..............................................37

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5. POLYMETRIC ANALYSES OF GOTŌ EXCERPTS .........................................43

Lachrymae (2005), mm. 55-69 ..................................................................43

Fantasma Lunare (2008), mm. 72-74 ........................................................47

Fantasma Lunare (2008), mm. 78-86 ........................................................53

Fêtes lointains (2009), mm. 56-65 .............................................................58

Fêtes lointains (2009), mm. 105-117 .........................................................70

6. PEDAGOGICAL COMPARISON ........................................................................80

7. CONCLUSION ......................................................................................................87

BIBLIOGRAPHY ..........................................................................................................................89

vi

LIST OF TABLES

Page

1. Excerpts for Polymetric Analysis ......................................................................................12

2. Misalignment of Excerpts by Metric Level .......................................................................80

3. Polymetric Composition of Excerpts with Misalignment at All Levels ............................81

4. Polymetric Composition of Fêtes lointains, Second and Third Strands, mm. 105-117 ....81

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LIST OF EXAMPLES

Musical examples are used by permission from these copyright holders:

*1998/2006 Theodore Presser Company, Bryn Mawr, PA

†2005/2008/2009 Bravo Music, Inc., Deerfield Beach, FL

Page

1. Sample Metric Layering ....................................................................................................13

2. *“Zion’s Security,” excerpt, Zion, mm. 1-5 .......................................................................15

3. *“Zion’s Walls,” excerpt, Zion, mm. 149-152 ...................................................................15

4. *Full Score, excerpt, Zion, mm. 214-217 ..........................................................................16

5. *Score Reduction, Zion, mm. 214-217 ..............................................................................17

6. *Re-notation of “Zion’s Security” Component, Zion, mm. 214-217 ................................18

7. Metric Layering of “Zion’s Security” Component, Zion, mm. 214-217 ...........................18

8. *“Zion’s Walls” Component, Zion, mm. 214-217 .............................................................19

9. Metric Layering of “Zion’s Walls” Component, Zion, mm. 214-217 ...............................19

10. Polymetric Layering, Zion, mm. 214-217 ..........................................................................20

11. “Followers of the Lamb,” excerpt, traditional, mm. 1-8 ....................................................20

12. *Full Score, excerpt, Laboring Songs, mm. 163-164 ........................................................21

13. *Score Reduction, Laboring Songs, mm. 163-164 ............................................................22

14. *Re-notation of Flute Background Component, Laboring Songs, mm. 163-164 ..............23

15. Metric Layering of Flute Background Component, Laboring Songs, mm. 163-164 .........23

16. *Re-notation of “Followers of the Lamb” Component,

Laboring Songs, mm. 163-164...........................................................................................24

17. Metric Layering of “Followers of the Lamb” Component,

Laboring Songs, mm. 163-164...........................................................................................24

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18. Polymetric Layering, Laboring Songs, mm. 163-164........................................................25

19. *“Twelve-tone Ostinato,” excerpt, Circular Marches, mm. 21-24 ...................................26

20. *Full Score, excerpt, Circular Marches, mm. 133-138 .....................................................27

21. *Score Reduction, Circular Marches, mm. 133-138 .........................................................28

22. *“Band 1” Component, Circular Marches, mm. 133-138.................................................29

23. Metric Layering of “Band 1” Component (Upper Woodwinds & Xylophone),

Circular Marches, mm. 133-138 .......................................................................................29

24. *“Band 1” Component (Upper Woodwinds & Xylophone),


25. Metric Layering of “Band 1” Component (Snare Drum),


26. *“Band 2” Component (Snare Drum), Circular Marches, mm. 133-138 .........................30

27. Metric Layering of “Band 2” Component, Circular Marches, mm. 133-138 ...................30

28. Polymetric Layering, Circular Marches, mm. 133-138 ....................................................31

29. *“Compound Melody,” excerpt, Circular Marches, mm. 51-52 .......................................32

30. *“Come Contentment, Lovely Guest,” excerpt, Circular Marches, mm. 97-98 ...............32

31. *Full Score, excerpt, Circular Marches, mm. 196-199 .....................................................33

32. *Score Reduction, Circular Marches, mm. 196-199 .........................................................34

33. *Re-notation of “Compound Melody,” excerpt, Circular Marches, mm. 196-199 ..........34

34. Metric Layering of “Compound Melody,” excerpt, Circular Marches, mm. 196-199 .....35

35. *Re-notation of “Come Contentment, Lovely Guest,” excerpt,


36. Metric Layering of “Come Contentment, Lovely Guest,” excerpt,


37. Polymetric Layering, Circular Marches, mm. 196-199 ....................................................37

38. “Hardiman the Fiddler,” excerpt, traditional, mm. 1-4 ......................................................37

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39. *Full Score, excerpt, Minstrels of the Kells, mvt. 1, mm. 109-110 ...................................38

40. *Score Reduction, Minstrels of the Kells, mvt. 1, mm. 109-110 .......................................39

41. *Re-notation of “Hardiman the Fiddler” Component,

Minstrels of the Kells, mvt. 1, mm. 109-110 .....................................................................40

42. Metric Layering of “Hardiman the Fiddler” Component,


43. *Background Material Component, Minstrels of the Kells, mvt. 1, mm. 109-110 ............41

44. Metric Layering of Background Material Component,


45. Polymetric Layering, Minstrels of the Kells, mvt. 1, mm. 109-110 ..................................42

46. “Lachrimae Antique,” excerpt, Lachrimae, mm. 4-6 ........................................................43

47. †Full Score, excerpt, Lachrymae, mm. 60-64 ....................................................................44

48. †Score Reduction, Lachrymae, mm. 60-64 .......................................................................45

49. †Re-notation of “Lachrimae Antique” Double Reed Version,

Lachrymae, mm. 60-64 ......................................................................................................46

50. Metric Layering of “Lachrimae Antique” Double Reed Version,

Lachrymae, mm. 60-64 ......................................................................................................46

51. †Re-notation of “Lachrimae Antique” Clarinet/Saxophone Version,

Lachrymae, mm. 60-64 ......................................................................................................46

52. Metric Layering of “Lachrimae Antique” Clarinet/Saxophone Version,

Lachrymae, mm. 60-64 ......................................................................................................47

53. †Polymetric Layering, Lachrymae, mm. 60-64 .................................................................47

54. Moonlight Sonata, mvt. 2, excerpt, mm. 1-4 .....................................................................48

55. †Full Score, excerpt, Fantasma Lunare, mm. 72-74 .........................................................49

56. †Score Reduction, Fantasma Lunare, mm. 72-74 .............................................................50

57. †Re-notation of Moonlight Sonata, mvt. 2, Double Reed Version,

Fantasma Lunare, mm. 72-74 ...........................................................................................51

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58. Metric Layering of Moonlight Sonata, mvt. 2, Double Reed Version,


59. †Re-notation of Moonlight Sonata, mvt. 2, Clarinet Version,


60. Metric Layering of Moonlight Sonata, mvt. 2, Clarinet Version,


61. Polymetric Layering, Fantasma Lunare, mm. 72-74 ........................................................53

62. Moonlight Sonata, mvt. 1, excerpt, mm. 5-7 .....................................................................53

63. †Full Score, excerpt, Fantasma Lunare, mm. 81-83 .........................................................54

64. †Score Reduction, Fantasma Lunare, mm. 81-83 .............................................................55

65. †Re-notation of Moonlight Sonata, mvt. 1, Clarinet Version,


66. Metric Layering of Moonlight Sonata, mvt. 1, Clarinet Version,


67. †Re-notation of Moonlight Sonata, mvt. 1, Saxophone Version,


68. Metric Layering of Moonlight Sonata, mvt. 1, Saxophone Version,


69. Polymetric Layering, Fantasma Lunare, mm. 81-83 ........................................................57

70. “Canzon septimi toni,” excerpt, Sacrae symphoniae (1597), mm. 15-21..........................58

71. “Canzon septimi toni,” excerpt, Sacrae symphoniae (1597), mm. 22-26..........................59

72. †Full Score, excerpt, Fêtes lointains, mm. 56-62 ........................................................ 60-61

73. †Score Reduction, Fêtes lointains, mm. 56-62 ............................................................ 62-63

74. †Re-notation of “Canzon septimi toni” Double Reed Version,

Fêtes lointains, mm. 56-62 ................................................................................................64

75. Metric Layering of “Canzon septimi toni” Double Reed Version,


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76. †Re-notation of “Canzon septimi toni” Saxophone Version,


77. †Metric Layering of “Canzon septimi toni” Saxophone Version,


78. Re-notation of “Canzon septimi toni” Low Brass Version,


79. †Metric Layering of “Canzon septimi toni” Low Brass Version


80. Re-notation of “Canzon septimi toni” Trumpet/Trombone Version


81. †Metric Layering of “Canzon septimi toni” Trumpet/Trombone Version,


82. Polymetric Layering, Double Reed Version with Saxophone Version,


83. Polymetric Layering, Saxophone Version with Low Brass Version,


84. Polymetric Layering, Low Brass Version with Trumpet/Trombone Version,


85. Nocturnes, mvt. 2, excerpt, mm. 124-126..........................................................................71

86. Re-notation of Nocturnes, mvt. 2, excerpt, mm. 124-126 .................................................71

87. †Full Score, excerpt, Fêtes lointains, mm. 107-110 .................................................... 72-73

88. †Score Reduction, Fêtes lointains, mm. 107-110 ........................................................ 74-75

89. †Re-notation of Nocturnes, mvt. 2, Horn Version,

Fêtes lointains, mm. 107-110 ............................................................................................76

90. Metric Layering of Nocturnes, mvt. 2, Horn Version,


91. †Re-notation of Nocturnes, mvt. 2, Trombone Version,


92. Metric Layering of Nocturnes, mvt. 2, Trombone Version,


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93. †Re-notation of Nocturnes, mvt. 2, Trumpet Version,


94. Metric Layering of Nocturnes, mvt. 2, Trumpet Version,


95. Polymetric Layering, Fêtes lointains, mm. 107-110 .........................................................79

96. †Saxophone Parts in Written and Re-notated Forms,


97. *Competing Strands in Original Notation, Circular Marches, mm. 196-199 ...................83

98. *Competing Strands in Original Notation, Zion, mm. 214-217 ........................................84

99. *Competing Strands in Original Notation, Laboring Songs, mm. 163-164 ......................85

100. *Competing Strands in Original Notation,


101. †Competing Strands in Original Notation, Fêtes lointains, mm. 107-108 ........................86

CHAPTER 1

INTRODUCTION

Purpose

Polymeter can be defined as “the presence of two (or more) concurrent metric

frameworks.”1 Its use in compositional practice has taken many forms throughout Western music

history: the basic construction of Medieval motets, Classical examples such as the “three

orchestras” scene in Mozart’s Don Giovanni, and Twentieth Century uses by composers such as

Charles Ives, Elliott Carter, and Bela Bartok. Polymeter itself is a more developed use of the

musical effect of polyrhythm, or “any two or more separate rhythmic streams in the musical

texture whose periodicities are non-integer multiples.”2 Thus, under these definitions, every

polymeter contains a polyrhythm but not every polyrhythm represents a polymeter.

Nonetheless, composers continue to write sections of music that use multiple meters

simultaneously. Composers have continued this trend in their wind band works in modern times,

among them Dan Welcher and Yo Gotō. Both are recipients of the prestigious Sousa/Ostwald

Award for their wind band works. These composers have employed polymeter in several

passages in their works. Such a fundamental aspect of their works warrants further scrutiny.

In light of recent scholarship, the purpose of this study is to provide a metric analysis of

polymetric sections in the wind band works of Dan Welcher and Yo Gotō. This dissertation

shows how the competing meters are developed through rhythmic layering beyond the pulse

level that creates a polyrhythm. The intent of this information is to provide a resource for these

pieces that displays the metric layers present and provides analysis of how these components

interact.

1 Justin London, Hearing in Time: Psychological Aspects of Musical Meter (New York: Oxford, 2012), 66. 2 Ibid.

1

Significance

The impetus for this study comes from Welcher and Gotō themselves. Both composers

have cited polymeter as a primary device in their compositional languages. Gotō listed polymeter

as one of four major traits of his compositional style.3 Furthermore, he has specifically stated that

he wants the listener to be confronted with multiple meters and the separate pulses that they

suggest.4 Welcher has both used and taught polymetric writing frequently and even instructed his

students on proper ways to write polymetrically.5 Additionally, Welcher is actively interested in

proper performance of these sections. About the performance of a polymetric section in his

Laboring Songs, Welcher stated in a personal interview, “how do these people coordinate? You

have to make it work. You (the conductor) have to understand it enough yourself that you can

perform it physically, or you’ll never be able to teach a band to do it.”6

Other sources do frequently mention the polymetric style of both of these composers but

without metric analysis. These include a 1997 dissertation on Welcher’s Three Places in the

West (a triptych that includes Zion) by Sarah McKoin and several performance guides in the

Teaching Music Through Performance in Band book series.7 No other known sources mention

the use of polymetric writing by these composers. Beyond the aforementioned sources, analytical

writings on the band music of these two important composers is notably absent from current

scholarship. By highlighting the paramount importance of rhythm and polymeter in the works of

both Welcher and Gotō, this study will help fill a gap in existing literature.

3 Yo Gotō, “Voci Lontani for Flute, Trumpet, Percussion, Piano, and String Quartet: Critical Essay and

Score” (MM thesis, University of North Texas, 2004), 12. 4 Ibid. 5 Dan Welcher, personal interview, May 8, 2012. 6 Ibid. 7 Sarah Lynn McKoin, “‘Three Places in the West’ by Dan Welcher: An Analysis and Critical Reference

for Conductors.” (DMA diss., The University of Texas at Austin, 2004), 1997. A complete list of relevant articles

from the Teaching Music series is included in the Bibliography.

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CHAPTER 2

COMPOSER BIOGRAPHIES

Dan Welcher

Dan Welcher (b. 1948) is a native of Rochester, New York. He began his career at age six

by studying piano. Welcher cites the beginning of his compositional career as writing music to

accompany homemade cartoons.8 His musical experiences broadened during his teenage years as

he learned to play the bassoon and in order to perform with his high school band and orchestra.

Welcher, who cites an early interest in both creative writing and music, began his

undergraduate studies at the State University of New York at Potsdam, in order to study both

music and journalism. After two years, he transferred to the Eastman School of Music to study

music exclusively. During his Eastman years, he served as principal bassoon for both the

Eastman Wind Ensemble and Eastman Philharmonia. He later continued his musical studies at

the Manhattan School of Music and completed a master’s degree in composition.

Welcher has balanced the roles of bassoonist, composer, and conductor throughout his

professional career. His first post-graduate job was principal bassoonist in the Louisville

Orchestra. During his years in Louisville, Welcher was appointed to the artist faculty of the

Aspen Music Faculty, where he taught bassoon and composition for fourteen years.

In 1978, Welcher became the Lee Hage Jamail Regents Professor of Composition at the

University of Texas at Austin. There, in addition to conducting the New Music Ensemble, he

now teaches courses in composition and orchestration. His composition teachers have included

Robert Washburn, Samuel Adler, and Warren Benson. Welcher has received wide acclaim for

8 Ysabel Sarte, “Dan Welcher,” in A Composer’s Insight, Vol. 4: Thoughts, Analysis and Commentary on Contemporary Masterpieces for Wind Band, ed. Timothy Salzman (Galesville, MD: Meredith Music, 2009), 240.

3

his works, to include five Pulitzer Prize nominations, a fellowship from the Guggenheim

Foundation, and the 1996 American Bandmasters Association Ostwald Award for his Zion.9

Shortly after his appointment at the University of Texas at Austin, he served as Assistant

Conductor of the Austin Symphony Orchestra for ten years. He was the founding conductor of

the New Music Ensemble at UT, and conducted over thirty concerts during an appointment as

Composer-in-Residence with the Honolulu Symphony.10 Welcher guest conducts his pieces in

clinics and receives guest conducting invitations regularly.

Rhythm is one of the defining characteristics of Welcher’s compositions. He has

admitted, “I never fake rhythm. I’m never vague about rhythm. What I write is exactly what I

want.”11 Welcher freely uses advanced rhythmic and metrical techniques in his music, to include

metric modulations, mixed meter, polyrhythm, and polymeter.

Welcher’s use of polymeter arises from several influences. One of his primary teachers,

Arthur Weisberg, compiled his teachings of contemporary performance practice into a 1993 text:

Performing Twentieth-Century Music: A Handbook for Conductors and Instrumentalists.

Welcher, in fact, penned a review for this book, in which he endorsed Weisburg’s “least common

denominator” approach to performing polymetric passages.12 Welcher also lauded composer

Charles Ives about his use of polymeter. About his Circular Marches, he referred using the

“Charles Ives technique” to compose a section of music.13 Many other examples of polymeter

appear in Welcher’s compositional output.

9 The Ostwald Award was renamed the Sousa/Ostwald Award in 2011. 10 Scott Carter. “Songs Without Words,” in Teaching Music Through Performance in Band, Vol. 4, ed.

Richard Miles (Chicago: GIA Publications, 2002), 555. 11 Sarte, “Dan Welcher,” 242. 12 Dan Welcher, review of Performing Twentieth-Century Music: A Handbook for Conductors and

Instrumentalists, Notes 52 (Sept. 1995): 127. 13 Sarte, “Dan Welcher”, 247.

4

Yo Gotō

Yo Gotō (b. 1958) is a native of Akita, Japan. Gotō’s educational career encompasses

studies both in education and composition. He holds a Bachelor of Music Education degree from

Yamagata University in Yamagata Prefecture, Japan, and a Performer’s Certificate in

composition from the Tokyo College of Music. In 2001, Gotō moved to the United States to

attend the University of North Texas, where he earned both the Masters of Music Education

degree and the Master of Music degree in composition. His primary composition teachers have

been Shin-ichiro Ikebe in Japan and Cindy McTee in the United States.

Gotō’s professional career includes work in both the educational and compositional

realms. He has given numerous clinics in both Japan and the United States on an array of

educational topics, including band literature selection and pedagogical philosophy. Gotō has

served as an advisor, board member, and president of the Japan Band Clinic. He has guest

conducted regularly, to include appearances at the World Association of Symphonic Bands and

Ensembles Convention and the Midwest Band and Orchestra Clinic.

After completing studies in Texas, Gotō returned to Japan to work as a free-lance

composer. His compositional output for band contains works from young to professional bands,

including more than thirty compositions or arrangements in publication. Gotō has received

abundant acclaim for these works, including the 2000 Academy Award from the Academic

Society of Japan for Winds and the 2011 Sousa/American Bandmasters Association/Ostwald

Award for his Songs for Wind Ensemble. In 2016, Gotō was appointed Professor of Wind Studies

and Composition at Showa Academia Musicae, where he serves both as band conductor and

graduate composition teacher.

5

One of the most important aspects of Gotō’s music is the listener’s experience of time. In

his master’s compositional thesis, he suggested that “composers should give new consideration

to the modern experience of time.”14 Gotō attempts to avoid a “strictly linear experience of time”

in his music.15 His compositional techniques, therefore, include rhythmic gestures that aim to

avoid traditional metric structures.

Similar to Welcher, Gotō cites Charles Ives as an influence in his use of polymetric

writing. In his thesis, he mentioned Ives’s The Unanswered Question as an influential example of

“simultaneous juxtaposition of different musics.”16 Gotō described this technique as the creation

of two different tempos through divergent rhythmic groupings. He admittedly attempted to

recreate this effect in his master’s thesis composition Voci Lontani, and this technique can be

found in many of Gotō’s other works.17

14 Gotō, “Voci Lontani”, 3. 15 Ibid., 6. 16 Ibid., 4. 17 Ibid., 12.

6

CHAPTER 3

POLYMETER

A Definition of Polymeter

A specific definition of the term “polymeter” is difficult to establish. A basic

understanding of the concept, of course, implies two or more meters in use simultaneously.

However, a proper understanding of this idea must consider the finer points of the meaning of

“meter” itself.

First, many modern theorists have abandoned the idea that meter and time signature are

synonymous. As Christopher Hasty suggested in his Rhythm as Meter, meter is primarily a

product of a rhythmic process free from the time signature and barlines—one that does not

“presuppose an invariant procession of equal beats.”18 In other words, while a time signature

may suggest a metric structure, the written music may or may not reflect that meter. Therefore,

while simultaneous use of different time signatures might signal the use of polymeter, it is not

necessary for different time signatures to be used.

On what, then, should a definition of meter be based? The answer must arise from the

music itself. Several approaches have been proposed by experts on the topic. For example,

Grosvenor Cooper and Leonard Meyer, defined meter as “a grouping of accented and unaccented

pulses.”19 Fred Lerdahl and Ray Jackendoff proposed a theory equating musical meter with

poetic meter.20 Theories such as these are valid and useful in understanding meter in various

ways.

18 Christopher Hasty, Rhythm as Meter (Oxford: Oxford University Press, 1997), 149. 19 Grosvernor Cooper and Leonard B. Meyer, The Rhythmic Structure of Music (Chicago: University of

Chicago Press, 1960), 4. 20 Fred Lerdahl and Ray Jackendoff, A Generative Theory of Tonal Music (Cambridge: Massachusetts

Institute of Technology Press, 1983), 25.

7

Perhaps the most relevant definition to a discussion in polymeter is one set forth by

Maury Yeston. He proposes an idea of meter as a product of different layers of rhythmic activity

in the music. For Yeston, music produces regularly recurring pulses on different rhythmic levels,

which he defines as a pulse layer, a micropulse layer (that divide the pulses), and an interpretive

layer (that group the pulses).21 How these layers interact with each other in music, then, creates

the metric structure.

Yeston’s definition allows for a more thorough understanding of polymeter. Harald Krebs

does exactly this by discussing a similar concept: metric grouping dissonance. Krebs believes

that this is created by a conflict between two different metric structures as exemplified by a

conflict at one or more rhythmic layers (as defined by Yeston).22 This description allows for a

way to qualify polymeter, especially that which is not explicit from the use of simultaneous time

signatures.

The Problem of Polymetric Perception

An expert on rhythm in music, Justin London exposed an issue with polymetric music in

his Hearing in Time. Citing several psychological studies, he emphasized that the perception of

polyrhythms is greatly affected by the human tendency to avoid the interpretation of multiple

stimuli. London remarked that when confronted with a polyrhythm, listeners “either extract a

composite pattern of all of the rhythmic streams present, and then match it to a suitable metric

framework; or focus on one rhythmic stream and entrain to its meter while treating the other

21 Maury Yeston, The Stratification of Musical Rhythm, 2nd ed. (New Haven: Yale University Press, 1976),

27. нн Harald Krebs, Fantasy Pieces: Metric Dissonance in the Music of Robert Schumann (New York: Oxford

University Press, 1999), 23.

8

rhythmic stream(s) as ‘noise’.”23 No fewer than four published psychological studies support

London’s claim.24 Though London here is discussing polyrhythm, there is an impact on how we

understand polymeter.

For example, one of these studies, published in 1981 by Stephen Handel and James S.

Oshinsky, consisted of a series of experiments to test the ability of subjects to tap simple

polyrhythmic pulses. Participants were simultaneously played the two different electronically

created pulses from different speakers and asked to “tap along with the perceived beat.”25

Variables included the polyrhythmic ratio (i.e, 2:3, 3:4, etc.), overall tempo, pitch variance

between streams, and level of subjects’ musical training.26 Handel and Oshinsky found that the

vast majority of responses preferred tapping one of the pulse trains while largely ignoring the

other.27

Other studies since, each with differing sets of variables, have all produced similar

results. It is reasonable to assume, then, that a listener does not hear two rhythmic strands

simultaneously. However, this result only pertains to the perception of a simple polyrhythm.

These experiments only tested two simple pulse trains without any variance of rhythms within

each pulse.

So, while this conclusion does not necessarily apply to polymetric perception, it does rule

out the idea that one senses a polymeter by the conflict of pulses alone. While such a conflict is

necessary, the development of the metric layering of the two different meters is also required.

23 Ibid., 48. 24 These four studies are: “The Meter of Syncopated Auditory Patterns” (1981) by Stephen Handel and

James S. Oshinsky; “Using Polyrhythms to Study Rhythm” (1984) by Stephen Handel; “Test of Attentional

Flexibility in Listening to Polyrhythmic Patterns” (1995) by Mari Riess, Jones, Richard J. Jagacinski, and William

Yee; and “Detecting Perturbations in Polyrhythms: Effects of Complexity and Attentional Strategies” (2013) by

Brian C. Fideli, Éve Poudrier, and Bruno H. Repp. 25 Stephen Handel and James S. Oshinsky, “The Meter of Syncopated Auditory Patterns,” Perception and

Psychophysics 30, No. 1 (January 1981): 3. 26 Ibid. 27 Ibid.

9

The existence of the other layers, as revealed in the music, prevents the two competing strands

from being interpreted into a single metric context. Therefore, for a polymeter to be perceived,

both two (or more) concurrent metric frameworks and a conflict on one or more levels is

required, and the more levels that conflict, the more perceptive the polymeter will be.

Thus, to capture the true essence of a polymetric passage of music, a qualitative

analytical approach is necessary. London and the above studies do successfully show that

polymetric perception cannot solely rest on the underlying polyrhythmic ratio involved. As

London stated, two or more conflicting rhythmic streams that can become integrated will be

heard in one metric framework.28 Therefore, the following analysis focuses on the elements that

help the listener perceive the polymetric nature of the passages beyond the polyrhythmic ratio.

Method of Analysis

This document provides a qualitative metric analysis of the polymetric sections of the

wind band works of Dan Welcher and Yo Gotō. This examination shows the metric layering of

each competing meter in order to show the conflict of multiple rhythmic layers in each example.

The excerpts in the works of Welcher and Gotō that contain polymeter are scattered among

several of their pieces. The identification of polymetric sections in these composers’ works can

be found in articles from the Teaching Music Through Performance in Band series. These

examples are the excerpts analyzed in chronological order in this document and are listed in

Table 1.

28 Ibid., 83.

10

Composer Piece Excerpt

Welcher

Zion (1994) mm. 214-221

Laboring Songs (1997) mm. 161-172

Circular Marches (1997) mm. 133-147

mm. 196-223

Minstrels of the Kells, mvt. 1 (2001) mm. 109-128

Gotō

Lachrymae (2005) mm. 55-69

Fantasma Lunare (2008) mm. 72-74

mm. 78-86

Fêtes lointains (2009) mm. 56-65

mm. 105-117

Excerpts for Polymetric Analysis

Table 1

For feasibility, only the measures necessary to establish each polymeter are used for

analysis. In many examples, the established rhythmic patterns repeat throughout the excerpt.

These repetitions are excluded from the analysis. The remaining measures are first illustrated in

concert-pitch full score.

The method of metric analysis for each example include three steps: score reduction,

metric re-notation, and polymetric analysis. First, each full score is compressed into condensed

score format. All parts that share the same rhythm are represented on the same line. Additionally,

all condensed lines that have similar rhythmic patterns are grouped next to each other in the

score and connected by braces. At this point, all sustained sounds and other rhythms that do not

conform to a regular metric structure are removed.

Next, each remaining component is re-notated to display its implied meter. As detailed by

Krebs, the meter of a passages is properly heard in terms of rhythmic groupings, irrespective of

the given time signature and barlines.29 This re-notation displays each of the rhythmic strands in

the “heard” meter. Several factors help establish these new meters: the original metric function

29 Krebs, Fantasy Pieces, 23.

11

of borrowed material, the metric format of the same material earlier in the composer’s work,

pitch placement, note beaming, mid-measure barlines, and articulation markings such as accents

or slurs. For consistency, all re-notated metric structures in this study contain subdivisions of

eighth notes. New time signatures and barlines are assigned as necessary.

Finally, these re-notated excerpts are converted into a skeletal illustration of their metric

structures. The notation for each is shown in the three layers theorized by Yeston: interpretive

layer (measure), pulse layer (beat), and micropulse layer (subdivision, if exists).30 The division

of beats in the pulse and micropulse layers is represented in traditional notation; the grouping of

these beats into interpretive layers are displayed by bracket above. A sample graphic of a passage

heard as two bars in four-four time is presented in Example 1 below. The diagram shows the

division of the measure into four pulses, each with a duple subdivision.

Sample Metric Layering

Example 1

To show the polymetric nature of each excerpt, the metric layering of each component is

then aligned in respect to the composer’s original. This final result shows how each polymeter

operates aurally through all rhythmic layers of metric motion. Layers of misalignment in the

metric layering are discussed, since this misalignment that causes the polymetric effect to be

aurally perceived. Layers of alignment, regardless of what level that occur, cause the aligning

layers to be heard in the same pulse track, making them more likely heard within a single metric

30 Ibid.

12

structure. A discussion of the misaligning layers and their manner of composition closes each

analytical excerpt.

13

CHAPTER 4

POLYMETRIC ANALYSES OF WELCHER EXCERPTS

Zion (1994), mm. 214-221

Welcher’s Zion is ten-minute symphonic tone poem and one of four portions in his Four

Places in the West.31 Its primary melodic content is derived from two folksongs: “Zion’s

Security” and “Zion’s Walls.” These two melodies exist both melodically and motivically

throughout the work, and the two tunes compete for “musical domination” as the work

develops.32 Examples 2 and 3 present the themes as they appear earlier in Welcher’s work.

Dan Welcher, Zion, mm. 1-5 (Alto Saxophone)

“Zion’s Security,” excerpt

Example 2

Dan Welcher, Zion, mm. 149-152 (Horn 1)

“Zion’s Walls,” excerpt

Example 3

These two themes are layered polymetrically beginning in m. 214. In her 1997

dissertation, Sarah McKoin labels this section as the climax of the piece, with the two

superimposed themes accompanied by other motives introduced elsewhere in the piece.33 This

31 Welcher initially formed a trilogy Three Places in the West from the three standalone pieces Arches

(1984), The Yellowstone Fires (1988), and Zion (1994). Welcher later amended the collection to Four Places in the West with the addition of Glacier (2003).

32 Sarah Lynn McKoin, “’Three Places in the West’ by Dan Welcher: An Analysis and Critical Reference

for Conductors.” (DMA diss., University of Texas at Austin, 1997), 81. 33 Ibid., 94.

14

Dan Welcher, Zion, mm. 214-217

Full Score, excerpt

Example 4

15

polymetric section lasts eight bars. The last four bars are a reorchestrated melodic repeat of the

first four, so are not included in the full score excerpt in Example 4. All parts are notated into a

7/4 time signature.

As shown in Example 5, “Zion’s Security” is scored for upper woodwinds and

glockenspiel while “Zion’s Walls” appears in mid-range woodwinds and upper brass. Welcher

also includes an eighth-note staccato version of “Zion’s Walls” in clarinet and piano. Layers of

unpitched percussion provide a metronomic basis to this section, while sustained harmonic

sounds and the timpani part conform to the 7/4 time signature.


Score Reduction

Example 5

“Zion’s Security”, at this point in the score, is re-notated by Welcher from its original

statement shown in Example 1. The re-notation in Example 6 returns it to a quarter-note pulse.

New barlines are included with the new version to reflect the three-beat pulse of the original.

16


Re-notation of “Zion’s Security” Component

Example 6


Metric Layering of “Zion’s Security” Component

Example 7

“Zion’s Walls” is also augmented from its original 7/8 form to the 7/4 meter reflected in

the time signature. This line is supported metrically by the sustained harmonic sounds and

timpani. The clarinet and piano version of “Zion’s Walls” mirrors the more sustained version but

sometimes trails canonically by a half or full beat, but not in a regular enough pattern to establish

its own meter. The unpitched percussion lines, with overlapping accents, are not fundamental to

establishing any meter. Example 8 returns the “Zion’s Walls” line to the eighth-note subdivision,

while Example 9 shows the metric layering of the 7/4 mixed meter.

17


“Zion’s Walls” Component

Example 8


Metric Layering of “Zion’s Walls” Component

Example 9

Metric analysis shows that this excerpt does contain misalignment of layers of metric

activity. As shown in Example 10, while the micropulse layers of the two components align, the

interpretive and pulse layers do not. The absence of the mixed-meter pulse in the first strand

cause the first to “race ahead” of the second. Therefore, the polymetric effect in this example

relies on the competing pulses and measure groupings (interpretive layers) of these two lines of

music.

18


Polymetric Layering

Example 10

Laboring Songs (1997), mm. 161-172

Laboring Songs is the result of a 1997 commission from L. D. Bell, The Colony, and

Duncanville High Schools in Texas. Its composition reflects Welcher’s interest in the spiritual

practices of the Shaker community.34 The piece is approximately ten minutes in length. It

utilizes six Shaker hymn tunes and melodies, including the shuffle tune, “Followers of the

Lamb” (see Example 11). Laboring Songs, though written and published as a stand-alone piece,

is now considered by Welcher as the first of two movements of Symphony No. 3, “Shaker Life.”

“Followers of the Lamb,” excerpt, traditional

Example 11

34 Dan Welcher, Laboring Songs (Bryn Mawr, PA: Theodore Presser, 2006), 1.

19

“Followers of the Lamb” is the last of the six Shaker tunes to appear in Laboring Songs.

Its first use in the piece is in a four-measure polymetric section. Welcher notates part of the

ensemble in 6/4 and another part simultaneously in 12/16, though the relative lengths of notes are

equal between the two groups. Since Welcher writes two 12/16 bars for every one 6/4 bar, mid-

measure barlines are necessary in the 12/16 instruments. Only the first two measures of the

section are included in Example 12, as the following two measures contain the same rhythmic

content.

Dan Welcher, Laboring Songs, mm. 163-164

Full Score, excerpt

Example 12

20

The “Followers of the Lamb” quotation here is found in the low woodwind group and

tom-toms in this section, all containing 12/16 time signatures. The group of instruments written

in 6/4 consists of a four-part flute background chorale figure of varying rhythms and two short

sixteenth-note runs in the vibraphone part. The single tam-tam note has no relevance to the

metric organization of the passage.


Score Reduction

Example 13

The flute background material notated with a 6/4 time signature is grouped by accents

and slur markings that aurally represent a triple meter. The resulting metric structure of this

section clearly reflects three pulses per bar with a duple subdivision of the beat. Examples 14 and

15 show this re-notation and metric structure using the eighth-note subdivision employed in this

study.

21


Re-notation of Flute Background Component

Example 14


Metric Layering of Flute Background Component

Example 15

The “Followers of the Lamb” component is properly heard in 6/8, as it appears in the

traditional folksong. This re-notation creates twice as many bars as Welcher’s version with each

bar having two beats, each with a triple subdivision. The remaining pianissimo vibraphone part,

though notated originally in 6/4, does not fit metrically into the 3/2 groupings of the flute

background. It does fit rather comfortably into the 6/8 pattern established by the “Followers”

component, as shown in Example 16.

22


Re-notation of “Followers of the Lamb” Component

Example 16


Metric Layering of “Followers of the Lamb” Component

Example 17

The implied metric structures in this excerpt do have a degree of alignment, as shown in

Example 18. Four interpretive groupings of the second strand fit into one of the first, and four

micropulses into four of the same. The only level that does not align is the pulse level. In each

bar of the original, three beats of the flute background pass for every eight beats of the

“Followers of the Lamb” segment. Other layers interlock (four fit into one, sixteenth equals

23

sixteenth). So, the polymetric effect in this excerpt is created only by the conflict at the pulse

level.


Polymetric Layering

Example 18

Circular Marches (1997), mm. 133-147

Commissioned by the American Bandmasters Association, Circular Marches is the

second movement of Symphony No. 3 “Shaker Life.” It was premiered in 1998 by the United

States Air Force Band. It, too, quotes Shaker tunes, but also utilizes material composed originally

by Welcher. The most prevalent of these is a ground-bass-type ostinato that contains all twelve

chromatic pitches in each of its four-measure cycles. It first occurs early in the work as it appears

in Example 19.

24

Dan Welcher, Circular Marches, mm. 21-24 (Bassoon 1)

“Twelve-tone Ostinato,” excerpt

Example 19

This “twelve-tone ostinato” line returns later in the work and bleeds into a polymetric

section at m. 133. Reminiscent of Charles Ives’s “two-band effect” in Three Places in New

England, this section is scored by Welcher into “Band 1” and “Band 2.”35 The instruments of

each of the two “bands” are grouped together, disregarding the standard instrumental order at

this point in the score. Each of the two groupings also contain its own set of unaligned barlines,

meaning that the instruments of each of the two “bands” are necessarily listed out of the usual

order at this point in the score. Of the fifteen bars of this section, three bars of “Band 1” pass for

every two bars of “Band 2” (thus three beats of “Band 1” for every two of “Band 2”). Due to

significant repetition, only the first six bars (four for “Band 2”) are necessary for metric analysis.

A reduction of the score shows that the melodic construction within each of the two

“bands” is relatively simple. “Band 1” is represented by an articulate, fast-paced melody in upper

woodwinds and xylophone. A metronomic line in the snare drum serves to reinforce the metric

nature of this group against some syncopations in the melodic line. The “Band 2” reduction also

shows that the low woodwind, low brass, piano, and double bass line compress into the “twelve-

tone ostinato” present elsewhere in the piece. The first tom part serves to articulate the rhythm of

the ostinato, while the second serves to reinforce the nature of this line against syncopations.

35 One section of “Putnam’s Camp” depicts two marching bands marching toward each other at different

tempos. Ives’s original version represents these two bands by assigning two different simultaneous tempos in the

piece.

25

Dan Welcher, Circular Marches, mm. 133-138

Full Score, excerpt

Example 20

26


Score Reduction

Example 21

Since Welcher has already grouped the two “bands” together in the score, a regrouping of

this excerpt is not necessary. The “Band 1” component already exists as its own unit with its own

cut-time time signature. Welcher does, however, call for the conductor to beat the quarter-note

pulse of this group instead of the half-note pulse implied by the time signature.36 This and the

absence of half-note figures suggest that the woodwind melody is more properly heard with four

beats to the bar in duple subdivision, as shown in Examples 22 and 23.

The snare line in “Band 1,” though it shares the same pulse as the woodwind line,

contains a repeating rhythmic pattern of seven beats instead of four. Confirmed by an accent that

signals each new recurrence, this component is illustrated in Examples 24 and 25.

36 Dan Welcher, Circular Marches (Bryn Mawr, PA: Theodore Presser, 2006), 19.

27


“Band 1” Component (Upper Woodwinds & Xylophone)

Example 22


“Band 1” Component (Upper Woodwinds & Xylophone)

Example 23


“Band 1” Component (Snare Drum)

Example 24

28


“Band 1” Component (Snare Drum)

Example 25

The “Band 2” grouping is also clearly heard in 4/4 time. The metric layering of this also

component yields four beats per bar at a duple subdivision, as shown in Example 26 and 27.


“Band 2” Component

Example 26


Metric Layering of “Band 2” Component

Example 27

A high level of polymetric activity exists among the three metric structures. As shown in

Example 28, the two components of “Band 1” share the same pulse and micropulse, but conflict

29

on the interpretive layer. The metric structures between the “Band 1” upper woodwind and

xylophone component and “Band 2” component are identical but pass at a three-for-two rate on

all levels, respectively. None of the layers align between these two components. The conflict

between the “Band 1” snare component and “Band 2” component share the same three-to-two

rate on the pulse and micropulse levels. The interpretive layers, however, conflict at a seven-to-

six rate.


Polymetric Layering

Example 28

30

Circular Marches (1997), mm. 196-223

A second example of polymeter occurs later in Circular Marches. This passage utilizes

two other melodic components from earlier in the piece. The first is a quick, lively dance tune in

alternating bars of 6/8 and 9/8 (see Example 29). The other is another Shaker tune, “Come

Contentment, Lovely Guest,” which is first stated in a simple 4/4 pattern as shown in Example

30.

Dan Welcher, Circular Marches, mm. 51-54 (Flute 1)

“Compound Melody,” excerpt

Example 29

Dan Welcher, Circular Marches, mm. 97-98 (Flute 1)

“Come Contentment, Lovely Guest,” excerpt

Example 30

As in the polymetric excerpt from Zion, Welcher superimposes these two melodies

toward the end of the piece in polymetric fashion. Over a 28-bar section, the two themes compete

in the same rhythmic fashion through a variety of different orchestrational assignments. The

passage, as written by Welcher, is notated in its entirety in 3/4. Only the first four bars of the

section are necessary for metric analysis as shown in Example 31.

31


Full Score, excerpt

Example 31

Discounting the variety of sustained sounds present, the melodic activity in this passage

is complete in three strands. The first, containing the “compound melody” is found in the upper

woodwinds. A separate bongo part reinforces the triplet nature of this melody with sporadic

triplets across the four bars. A second strand, consisting of all dotted rhythms, collapses into a

chorale-like version of “Come Contentment, Lovely Guest.”

32


Score Reduction

Example 32

Since the “compound melody” was heard earlier in the work, the listener is already

aurally familiar with the metric feel of the line. Example 33 shows this component re-notated

into its initial form, with triple subdivision throughout and alternation between three beats per

bar and two beats per bar. Since this exchange between three and two does not fit perfectly into

the three-beat pattern of the original, this leaves an incomplete bar at the end of the re-notated

excerpt.


Re-notation of “Compound Melody” Component

Example 33

33


Metric Layering of “Compound Melody” Component

Example 34

By this point in the piece, the listener is also familiar with the “Come Contentment,

Lovely Guest” tune. In the polymetric excerpt, to fit the 3/4 time signature, Welcher divided each

bar in half and stretched each of the two beats into three by dotting every note. The re-notation in

Example 35 removes these adaptations. The result reflects the initial four beats with duple

subdivision as shown in Example 36.


Re-notation of “Come Contentment, Lovely Guest” Component

Example 35

34


Metric Layering of “Come Contentment, Lovely Guest” Component

Example 36

The resulting polymetric structure has a high degree of misalignment. Three pulses of the

“compound melody” component compete with every two pulses of the “Come Contentment,

Lovely Guest” portion. The micropulse level contains a nine-to-four mismatch. A misalignment

also exists at the interpretive layer, further complicated by the alternation of beat lengths in the

first strand. Alignment does exist at every third pulse of the “compound melody” segment at both

the pulse layer and micropulse layer. The three-to-two and nine-to-four cycles both align at this

point (the originally notated barlines of the excerpt). Thus, the polymetric activity in this excerpt

is mostly contained within the barlines given by Welcher.

35


Polymetric Layering

Example 37

Minstrels of the Kells, mvt. 1 (2001), mm. 109-128

The Big Twelve Band Directors Association commissioned Minstrels of the Kells to

honor former Texas Tech University Director of Bands James Sudduth. Welcher chose to base

the composition on Irish folksongs to honor Suddeth’s curiosity for “all things Celtic”.37 The

work is complete in two movements, “Airs in the Mist” and “Reelin’ and Jiggin’.” Minstrels of

the Kells quotes no fewer than nine Irish folk songs, including the traditional tune “Hardiman the

Fiddler.”

“Hardiman the Fiddler,” excerpt, traditional (mm. 1-4)

Example 38

37 Albert Nguyen. “Minstrels of the Kells,” in Teaching Music Through Performance in Band, Vol. 7, ed.

Richard Miles (Chicago: GIA Publications, 2009), 715.

36

The “Hardiman the Fiddler” folksong is used extensively in the last section of the first

movement of Minstrels of the Kells. Over a period of twenty measures, a version of the tune

appears five different times, each time over a period of roughly two measures apiece. Each

appearance has the same essential rhythmic construction, so only the first two measures of the

section are necessary for metric analysis.

Dan Welcher, Minstrels of the Kells, mm. 109-110

Full Score, excerpt

Example 39

37

In these two measures, “Hardiman the Fiddler” is scored in piccolo. A countermelodic

line similar in rhythmic structure appears in tambourine. Both parts are written into a 9/16 time

signature, which begins on beat two of the prevailing 4/4 time. Mid-measure barlines are

necessary in both lines to delineate the faster 9/16 bars; each of these passes at one-and-a-half

beats of 4/4 time.

The note lengths between the two groupings are not mathematically equivalent with each

other. The 9/16 notes function essentially as triplets in the other time signature. Also, since the

9/16 structure does not fit evenly into the 4/4 time, incomplete measures are necessary at the

beginning (left as a single beat in 4/4) and end of the line in Welcher’s notation.


Score Reduction

Example 40

The “Hardiman the Fiddler” component already exists in its proper metric context as

notated by Welcher. Re-notation of the melody in Example 41 ignores the quarter rest at the

38

beginning of the excerpt. The line assumes three beats per measure with a triple subdivision, as

shown in Example 42.


Re-notation of “Hardiman the Fiddler” Component

Example 41


Metric Layering of “Hardiman the Fiddler” Component

Example 42

The remaining melodic material functions as background to the folksong at this point. No

other quoted tunes are present at this point, except two pick-up notes on the last beat of the

excerpt in oboe and clarinet. These and sustained tones are not necessary for further

consideration. The remaining material fits comfortably into 4/4 time: the first bassoon part that

marks the subdivision and vibraphone and piano part that helps delineate the major beats of the

meter.

39


Background Material Component

Example 43


Metric Layering of Background Material Component

Example 44

The polymetric effect, here, rises mostly from a misalignment at the interpretive layer.

Since the “Hardiman” measure length represents one-and-a-half beats of the four-beat structure,

if the prevailing pattern continued, eight “Hardiman” measures would pass at the rate of three of

the background measures. While it appears that further misalignment exists at the other layers,

the pulse layer of the “Hardiman” component equals exactly the micropulse layer of the other.

The beat of the 9/16 portion and its micropulses, then, can be heard consonantly within the

subdivision of the original time. Therefore, these do not contribute to the polymetric effect in this

excerpt.

40


Polymetric Layering

Example 45

41

CHAPTER 5

POLYMETRIC ANALYSES OF GOTŌ EXCERPTS

Lachrymae (2005), mm. 55-69

Yo Gotō’s Lachrymae (Latin for “tears”) was commissioned by Japan’s Executive

Committee for Twenty-First-Century Wind Music as a requiem for victims of political and

religious conflicts.38 The eight-and-a-half-minute composition contains an assortment of fanfare

figures and indeterminate motivic gestures. Gotō also incorporates the quotation of the first of

seven pavanes in John Dowland’s 1604 collection Lachrimae. The 24-bar “Lachrimae Antique”

uses a five-voice texture as illustrated in Example 46.

John Dowland, Lachrimae “Lachrimae Antique,” excerpt (mm. 4-6)

Example 46

“Lachrimae Antique” is quoted four times in Gotō’s Lachrymae. The first two statements

overlap each other during a 15-bar section, beginning in m. 55. The second quotation does not

enter until m. 61, interrupting the seventh bar of the first statement. The two overlap for two-and-

38 Erin Bodnar. “Lachrymae,” in Teaching Music Through Performance in Band, Vol. 7, ed. Richard Miles

(Chicago: GIA Publications, 2009), 523-524.

42

a-half measures before the second statement completes the last six measures of the section alone.

The five-measure excerpt in Example 47 exhibits the overlapped segment.

Yo Gotō, Lachrymae, mm. 60-64

Full Score, excerpt

Example 47

The seven parts present in this excerpt divide clearly into two groups: a double reed

group and a single reed group. The trio of double reed parts are highly syncopated but align

rhythmically amongst themselves in many places. The clarinet and trio of saxophones are far less

syncopated but also move together themselves. This demarcation is shown in Example 48.

43


Score Reduction

Example 48

In the first quotation of “Lachrymae Antique” by the double reeds, Gotō compresses the

relative speed by reducing the value of each note by one quarter (i.e. quarters become dotted

eighths). Example 49 shows the re-notation into Dowland’s original values. In the restored

version, the rhythms and intervals of the outer voices are exact, except for the displacement of

the last oboe note of the second bar by one beat. Gotō composed the middle voice selectively

from the three remaining original voices in order to present the most balanced three-part texture.

The resulting metric structure represents a clear 4/4 meter. However, since the re-notation

reveals no subdivision of any beat, the micropulse layer is omitted in the metric structure

rendered in Example 50.

The second statement of “Lachrimae Antique” by the clarinet and saxophones quotes a

portion of the same excerpt (Example 46) as the double reeds. This version is already written in

Dowland’s original meter and enters with material found later in the excerpt than the first.

Similar to the first quotation, the second uses the outer voices exactly (with displacement of the

same note in the top part) and contains freely adapted inner voices from the original. This

44

version (see Examples 51 and 52), too, reflects the same four-beat, duple-subdivision metric

structure.


Re-notation of “Lachrimae Antique” Double Reed Version

Example 49


Metric Layering of “Lachrimae Antique” Double Reed Version

Example 50


“Lachrimae Antique” Clarinet/Saxophone Version

Example 51

45


Metric Layering of “Lachrimae Antique” Clarinet/Saxophone Version

Example 52

Since the two groupings have the same metric structure, the displacement of one over the

other causes misalignment at every level of metric activity. Every four bars of the first strand

pass for every three of the second. Because the metric structures are equivalent, this four-to-three

ratio then applies to all three layers.


Polymetric Layering

Example 53

Fantasma Lunare (2008), mm. 72-74

Fantasma Lunare (Latin for “ghost moon”) was commissioned by the Kanazawa

Municipal Technical High School Symphonic Band for a performance at an all-Beethoven

46

festival. Given this opportunity, Gotō chose to construct the piece from motives of Moonlight

Sonata for piano. The longest fragment quoted in Fantasma Lunare is found at the beginning of

Beethoven’s second movement, “Allegretto.” The first four bars of the segment are illustrated in

Example 54.

Ludwig van Beethoven, Moonlight Sonata, mvt. 2 (mm. 1-4)

Example 54

Gotō introduces this material in two quickly overlapping segments at a transition point in

the piece. The first statement emerges from a loud polyrhythmic texture of repeated notes, and

the second joins only two beats after. The entire excerpt is presented completely in the three bars

shown in Example 55.

As in Lachrymae, the two quotations occur in two separate woodwind choirs. The first

occurs in a double reed quartet, with the bottom bassoon voice not entering until the final three

notes. The second statement is made by four clarinet voices, with the bass clarinet not entering

until the last two notes. The polyrhythmic texture of repeated notes and other sustained sounds

are irrelevant to the layering of the two quotations and not considered in the analysis.

47

Yo Gotō, Fantasma Lunare, mm. 72-74

Full Score, excerpt

Example 55

48


Score Reduction

Example 56

Though the first statement of the Moonlight Sonata, mvt. 2, fragment is proportionally

exact, it is highly syncopated as notated. From the original, Gotō first reduced all note values by

half. He then displaced the material forward in time by one sixteenth of a beat before moving the

barlines to fit it into 4/4 time. Example 57 shows Gotō’s end product re-notated back into the

original three-beat, duple-subdivision format conceived by Beethoven.

49


Re-notation of Moonlight Sonata, mvt. 2, Double Reed Version

Example 57


Metric Layering of Moonlight Sonata, mvt. 2, Double Reed Version

Example 58

The second statement is also exact but again highly syncopated by Gotō. To create this

version, he first shifted the barlines of the original all forward by one beat. He then shortened all

of the note values by one third, essentially making all values triplets. Finally he reassigned

barlines to fit the 4/4 and 3/4 of the composition at that point. Example 59 shows the reversal of

these back into Beethoven’s three-beat, duple-subdivision original.

50


Re-notation of Moonlight Sonata, mvt. 2, Clarinet Version

Example 59


Metric Layering of Moonlight Sonata, mvt. 2, Clarinet Version

Example 60

The overlapping of metric structures in this excerpt is quite complex. No alignment exists

at any level of metric activity (see Example 61). The two strands only align with the original

notated pulse (which is not audibly present in any part at this point). Noting the points of

alignment through the original pulse, four micropulses in the first strand pass for every three in

the other. This proportion applies for the other layers as well, ensuring a high degree of

misalignment in all layers.

51


Polymetric Layering

Example 61

Fantasma Lunare (2008), mm. 78-86

Perhaps the most distinguishable motivic idea from the Moonlight Sonata is the triplet

eighth-note arpeggios from the first movement. This line simply outlines the various harmonies

underneath the sustained melody and above the octave-doubled bass line. The motive keeps the

same rhythmic and melodic form for almost the entire movement, though it expands and

contracts in intervals for feasibility in performance. A sample of this motive is exhibited in

Example 62.

Ludwig van Beethoven, Moonlight Sonata, mvt. 1 (mm. 5-7)

Example 62

52

Gotō uses this arpeggiated motive several times throughout Fantasma Lunare, sometimes

in traditional notation and other times in indeterminate notation. While polymetric possibilities

exist with the indeterminate sections of the piece, their aleatoric nature precludes a definitive

analysis. Therefore, this study will only consider the portions of the piece in traditional notation.

Example 62 displays three measures of one of these sections in traditional notation. Two lines of

indeterminate repeated motives by percussion instruments are excluded from the full score and

the analysis that follows.


Full Score, excerpt

Example 63

The eight parts in this excerpt collapse into five strands of rhythmic activity. The two

flute parts, when combined, create an unbroken stream of triplet sixteenth notes. The bottom

53

clarinet parts and alto saxophone parts function similarly, generating streams of sixteenth notes

and eighth-note triplets respectively. The entrances of each strand are staggered with the higher-

pitched strands entering progressively later.


Score Reduction

Example 64

The excerpt above is, as notated, only a five-layer polyrhythmic structure. Since the

rhythms simply repeat at one continuous rate for each line, no multi-layered metric structure

could be perceived in computerized performance. However, since the motives in three of the

lines would have a different weight to them in live performance, a strong-to-weak alternation can

be assumed to create a metric structure within the lines. The combined line, if played through by

only one instrument or group (as in the other two parts), would have no metric potential, and

thus, no polymetric possibilities. Therefore, the piccolo and first clarinet lines must be omitted

from polymetric analysis. Each of the two remaining strands collapses back into the 12/8 feel of

54

Beethoven’s original, implying a four-beat, triple-subdivision metric structure, as shown in

Examples 65 through 68.


Re-notation of Moonlight Sonata, mvt. 1, Clarinet Version

Example 65


Metric Layering of Moonlight Sonata, mvt. 1, Clarinet Version

Example 66


Re-notation of Moonlight Sonata, mvt. 1, Saxophone Version

Example 67

55


Metric Layering of Moonlight Sonata, mvt. 1, Saxophone Version

Example 68

A polymetric layering of these two lines shows that misalignment of metric structures

exists at all levels. Equivalency of the two strands passing at different rates creates a high level

of metric dissonance, the given displacement of beats between the two. Four pulses in the first

group pass for every three of the other, and the four-to-three ratio holds for all layers of activity.


Polymetric Layering

Example 69

56

Fêtes lointains (2009), mm. 56-65

Fêtes lointains (French for “distant celebrations”) was written by Gotō for the Osaka

Municipal Symphonic Band. The piece was commissioned by the band to honor the 120th

anniversary of that city’s municipal founding. Gotō incorporates indeterminate sounds, fanfare

figures, syncopated melodies, and quotations of pre-existing works into Fêtes lointains.

The first borrowed material in this work comes from Giovanni Gabrieli’s collection

Sacrae symphoniae (1597). “Canzon septimi toni No. 2” was composed for two antiphonal

choirs of four brass instruments each. For most of the canzon, the two choirs are scored

independently of each other. The quoted excerpts from each of the choirs are presented in

Examples 70 and 71.

Giovanni Gabrieli, Sacrae symphoniae (1597) “Canzon septimi toni,” excerpt (mm. 15-21)

Example 70

57

Giovanni Gabrieli, Sacrae symphoniae (1597) “Canzon septimi toni,” excerpt (mm. 22-26)

Example 71

“Canzon septimi toni No. 2” is quoted four different times over a seven-measure segment

of Fêtes lointains, beginning at m. 56. Since the rhythmic nature of each strand is different in

each quotation, all seven measures are necessary for polymetric analysis. The full score is

illustrated across two images in Example 72.

The condensed version in Example 73 shows the four quotations more clearly. Each

statement enters at a separate time and is made by a different quartet of instrumentalists: double

reeds, saxophones, euphoniums and tubas, and trumpets and trombones. This material is

surrounded by a cluster chord that descends quasi-chromatically through flutes and clarinets. The

cluster chord, along with the sustained percussion sounds and tutti chord that begin the segment,

will not be necessary for analysis.

58

Yo Gotō, Fêtes lointains, mm. 56-62

Full Score, excerpt

Example 72

59


Full Score, excerpt

Example 72

60


Score Reduction

Example 73

61


Score Reduction

Example 73

62

The first quotation of “Canzon septimi toni” utilizes material from Example 70. Gotō

borrows the material verbatim in all four parts from Gabrieli but notates the segment with halved

note values. Thus, each two measures of the original encompasses the time of one in Gotō’s

work. The re-notation portrays a four-beat, duple-subdivision metric structure.


Re-notation of “Canzon septimi toni” Double Reed Version

Example 74


Metric Layering of “Canzon septimi toni” Double Reed Version

Example 75

The saxophone material that enters two bars begins with the same material as the double

reed version. Instead of shortening each note by half, Gotō contracts each of them by only one-

fourth. Each quarter note then becomes a dotted eighth note; each half note becomes a dotted

63

quarter. Example 76 restores the notation to Gabrieli’s original, and Example 77 confirms the

same four-beat, duple-subdivision metric structure.


Re-notation of “Canzon septimi toni” Saxophone Version

Example 76


Metric Layering of “Canzon septimi toni” Saxophone Version

Example 77

The third quotation in this section employs a different excerpt than the first two. The

material played by euphoniums and tubas (see Example 71) immediately follows the previous

excerpt in the original. Gotō shortens each note length by third in this fragment, essentially

turning every note into a triplet. Since the excerpt begins on an upbeat, this causes the resulting

64

notation to become highly syncopated. Example 78 restores Gabrieli’s notation, again in four

beats and duple subdivision.


Re-notation of “Canzon septimi toni” Low Brass Version

Example 78


Metric Layering of “Canzon septimi toni” Low Brass Version

Example 79

The final statement uses the same material as the fourth, with note lengths halved from

the original. Though this passage enters a full bar after the third, the shorter note lengths cause

the two statements to end simultaneously at the end of the 5/4 bar. Example 80 shows the re-

notation back to the original four-beat, duple subdivision structure.

65


Re-notation of “Canzon septimi toni” Trumpet/Trombone Version

Example 80


Metric Layering of “Canzon septimi toni” Trumpet/Trombone Version

Example 81

The polymetric effect in this passage is a progressive one. Since no more than two of the

quartets are sounding simultaneously, each moment in the music that does overlap has its own

polymetric feel. As the first quotation conflicts with the second, three beats pass at the time of

two, respectively. Since the two metric organizations are identical, misalignment exists at every

metric level (as shown in Example 82).

66


Polymetric Layering, Double Reed Version with Saxophone Version

Example 82

The overlapping of the second quartet with the third creates the most complicated of the

three polymetric structures. Not only do the pulses pass at different rates, but they do not actually

align at any point in the excerpt. Both do, however, align with the original pulse (not aurally

perceptible at this point in the music). As shown in Example 82, the top part passes at four beats

for each three original pulses, and three beats pass for each two of the bottom. Thus, it would

take six original pulses for each part to align, eight for the top for each nine of the bottom.

Misalignment, again, exists between all metric layers.

67


Polymetric Layering, Saxophone Version with Low Brass Version

Example 83

The final point of conflict occurs between the third and fourth quartets. Again, no layers

align, but three bars pass at the rate of four for the other. Since the metric structures are

equivalent but offset, the same proportion applies for the other layers here as well. Therefore, the

polymetric effect is quite strong throughout this entire section, since none of the quartets align

metrically at any level.

68


Polymetric Layering, Low Brass Version with Trumpet/Trombone Version

Example 84

Fêtes lointains (2009), mm. 105-117

The other quotation in Fêtes lointains is taken from the second movement of Claude

Debussy’s three-movement Nocturnes for orchestra. The original excerpt (for trumpets in F)

contains triplets in a 2/4 time signature. Since the triplets imply a different subdivision of the

beat, the excerpt is re-notated in Example 86.

The thirteen-measure section of Fêtes lointains that uses the Nocturnes quote is the most

rhythmically active of the piece. Gotō introduces several layers of rhythmic activity and shapes

the different layers dynamically over this segment. Example 87 illustrates four bars toward the

beginning of this section. The bars that precede and follow contain the same material, so are not

necessary for analytical consideration.

69

Claude Debussy, Nocturnes, mvt. 2 (mm. 124-126)

Example 85

Claude Debussy, Nocturnes, mvt. 2 (mm. 124-126)

Re-notation into a Simplest Metric Presentation

Example 86

There are five distinct layers of rhythmic activity, all similar in motivic structure to

Debussy’s original shown in Examples 85 and 86. Two of the five strands contain the same

rhythmic values as others—the saxophones continues the same sixteenth-note-triplet line as the

horns, and the piccolo and flutes do the same for the trumpets. A variety of other sustained

sounds and isolated motives are scored for a variety of woodwind, low brass and percussive

instruments. These do not contribute to metric overlapping of the other strands and will not be

analyzed. Example 86 illustrates the different layers as grouped by similarity of function.

70


Full Score, excerpt

Example 87

71


Full Score, excerpt

Example 87

72


Score Reduction

Example 88

73


Score Reduction

Example 88

74

The first strand, in the horns, is similar in construction to Debussy’s excerpt. Gotō

assigns it the same sixteenth-note-triplet values as the original. Example 89 shows it re-notated

to the triple-subdivision implied by the triplets. The re-notation assumes the same four-beat

grouping from Example 86.


Re-notation of Nocturnes, mvt. 2, Horn Version

Example 89


Metric Layering of Nocturnes, mvt. 2, Horn Version

Example 90

Gotō, however, modifies the trombone line from Debussy’s initial rhythms. While

keeping the same chordal structure, Gotō repeats occasional notes and compacts the triplets into

sixteenth notes. Examples 91 and 92 show that the result reflects the same four-beat structure as

the first strand, but implies a duple instead of triple subdivision.

75


Re-notation of Nocturnes, mvt. 2, Trombone Version

Example 91


Metric Layering of Nocturnes, mvt. 2, Trombone Version

Example 92

Since there are again no triplets, the third strand is similar in construction to the second.

This line, however, uses thirty-second notes instead of sixteenths. Therefore, the trumpets are

playing as twice as fast as the trombones. Example 93 shows the line re-notated to match the

second, and Example 94 shows that it also contains the four-beat, duple-subdivision metric

structure.

76


Re-notation of Nocturnes, mvt. 2, Trumpet Version

Example 93


Metric Layering of Nocturnes, mvt. 2, Trumpet Version

Example 94

Polymetric analysis of the three levels shows a mix of alignment and misalignment.

Between the first two strands, the pulse layers align while the micropulse layers conflict between

triples and duples. The interpretive layers are equivalent but displaced, due to the canonic nature

of Gotō’s writing. Since the pulse layer of the third strand matches the micropulse layer of the

second, the bottom two strands sound metrically consonant with one another. However, since the

first two strands do not line up, the first contains the same misalignment with the third.

Therefore, the polymetric effect in this excerpt arises solely from the horn part conflicting with

the other two strands at the micropulse layer with the other two strands.

77


Polymetric Layering

Example 95

78

ул

CHAPTER 6

PEDAGOGICAL COMPARISON

Of the ten excerpts included in this study, all ten include misalignment of metric

structures on at least one level of metric activity. Therefore, all ten examples can potentially be

heard as polymetric in some manner. A comprehensive list of all passages is presented in Table

2, indicating the presence of misalignment in each layer.

Piece Excerpt

Interpretive

Layer

Misalignment

Pulse

Layer

Misalignment

Micropulse

Layer

Misalignment

Zion mm. 214-221 YES YES NO

Laboring Songs mm. 161-172 NO YES NO

Circular Marches mm. 133-147 YES YES YES

mm. 196-223 YES YES YES

Minstrels of the Kells, mvt. 1 mm. 109-128 YES NO NO

Lachrymae mm. 55-69 YES YES ----------

Fantasma Lunare mm. 72-74 YES YES ----------

mm. 78-86 YES YES YES

Fêtes lointains mm. 56-65 YES YES YES

mm. 105-117 NO NO YES

Misalignment of Excerpts by Metric Level

Table 2

Four of the ten passages contain misalignment in all three layers. Another two passages

contain misalignment to interpretive and pulse layers but are missing a micropulse layer. Since in

these two examples both strands contain the same quoted material, misalignment can be assumed

since either choice of subdivision would conflict, if present. Therefore, these six excerpts exhibit

the highest aural effect of polymeter according to the definition of metric dissonance. The

analytical results of these excerpts are displayed in Table 3. Details for all three occurrences of

polymetric layering in the first Fêtes lointains segment are listed separately.

79

Piece Excerpt First

Strand

Second

Strand

Polymetric

Ratio

Circular Marches

mm. 133-1474 pulses

2 micropulses

4 pulses

2 micropulses 3 to 2 (all levels)

mm. 196-2233+2 pulses

3 micropulses

4 pulses

2 micropulses

3 to 2 (pulses)

9 to 4 (micropulses)

Lachrymae mm. 55-694 pulses

----------

4 pulses


Fantasma Lunare

mm. 72-743 pulses

----------

3 pulses

---------- 4 to 3 (all levels)

mm. 78-864 pulses

3 micropulses

4 pulses


Fêtes lointains

mm. 56-65

Dbl Reed/Sax

4 pulses

2 micropulses

4 pulses


mm. 56-65

Sax/Low Br

4 pulses

2 micropulses

4 pulses


mm. 56-65

Low Br/Tpt-Tbn

4 pulses

2 micropulses

4 pulses


Polymetric Composition of Excerpts with Misalignment at All Levels

Table 3

In seven of these eight polymetric locations, the metric composition of the two strands is

identical. The misalignment in these locations is caused by taking the same metric structure and

elongating one of them, causing all of the layers to misalign. However, this technique alone is

not sufficient to ensure complete misalignment. As shown in Table 4, the second and third

strands of the other Fêtes lointains excerpt displace the same material but do not have complete

alignment.

Piece Excerpt Second

Strand

Third

Strand

Polymetric

Ratio

Fêtes lointains mm. 105-1174 pulses

2 micropulses

4 pulses


Polymetric Composition of Fêtes lointains, Second and Third Strands, mm. 105-117

Table 4

80

The reason for this exception is the choice of polymetric ratio. The example from Fêtes

lointains, mm. 105-117, contains a two-to-four ratio, which causes layers that misalign to realign

with other layers. In other words, in this example, the subdivision of one layer is the same as the

pulse of another. Since all layers realign with other layers, no metric dissonance exists between

these two strands. The first strand, containing three micropulses, is necessary to create the

polymetric effect against the other two strands in this excerpt.

The other excerpts do not have this problem because of different choices of polymetric

ratios. The aforementioned example contains metric groupings entirely in groups of twos (noting

that fours can be heard in groups of two). The choice of a ratio that also has groupings of twos

ensures realignment. The fully misaligned excerpts from Table 3 all contain a ratio of duple-to-

triple in some manner (since nine-to-four and eight-to-nine contain factors of twos and threes).

For conductors of these pieces, the excerpts from Table 3 should have the highest degree

of polymetric effect if performed with rhythmic precision. Care should be taken to make certain

that performers (who do not have the benefit of the score) understand the implied metric

structures. For example, in Fêtes lointains, the saxophonists in mm. 58-60 should understand that

their dotted eighth notes equate to the actual pulse of their part (see Example 96). Since Gotō

uses pre-existing music in all of these excerpts, familiarity with the original is also critical.

The second excerpt from Circular Marches (mm. 196-223) contains a similar

performance challenge (see Example 97). Instead of layering the same material at different

speeds, Welcher prefers to superimpose different themes already introduced earlier in the work.

The performance challenge in the excerpt is the same, however. Performers should be reminded

of the previous use of each of the separate melodies and to be encouraged to retain the metric

feel of the original statements.

81


Saxophone Parts in Written and Re-notated Forms

Example 96


Competing Strands in Original Notation

Example 97

The performance challenge in the remaining four excerpts is more problematic, though.

In these examples, the polymetric effect is diminished because of alignment in some metric

levels. It is important, then, for conductors and performers to focus on what does misalign, since

perception of the polymetric effect rests solely on those layers.

In Zion, only the interpretive and pulse layers conflict. It is imperative, then, that the

performers help define the conflicting beat and measure groupings. Attention to Welcher’s

articulation markings, which help define the two different meters, is critical (as shown in

82

Example 98). Again, knowledge of the two folksongs’ original statement can also assist in

placing the proper agogic stress for each line.



Example 98

The other three excerpts rely on only one conflicting rhythmic level each to help the

polymetric effect be heard. Except for one level of rhythmic activity in each, all of their rhythmic

information aligns into one metric framework. These, then, are the most precarious polymetric

examples of this study. Therefore, if attention is not paid to the single conflicting level of

activity, the polymetric effect will be absent in performance.

In Laboring Songs, only the pulse levels misalign. Performers here should be encouraged

to execute Welcher’s accents (see Example 99), which help define the separate pulses. The

conductor should also rehearse the competing groups separately, so that each can perceive its

own pulse independently of the other.

83



Example 99

Misalignment only occurs in the interpretive layer in the Minstrels of the Kells excerpt.

Here, the performance of the accents that define each measure grouping is important for any

polymetric effect to be heard. Additionally, the conductor must explain to the performers how

the two time signatures function with each other. The piccolo and tambourine players should be

aware that their note beamings will not align with the conducting time, as shown in Example

100.

Finally, in Fêtes lointains, only the micropulse layer misaligns. This arises from the

conflict between the competing strands of sixteenth-note triplets, sixteenth notes, and thirty-

second notes (see Example 101). Since the second and third strands (sixteenth notes and thirty-

second notes) are heard in the same metric context, proper performance of the sixteenth-note

triplet is imperative. Performers should avoid rushing these triplets in this passage, so that the

conflict between the figures remains aurally perceptible.

84



Example 100



Example 101

85

CHAPTER 7

CONCLUSION

This document provides an in-depth look into the mechanics of a critical aspect of

Welcher’s and Gotō’s works. Since both composers express similar influences on their

compositional processes, a certain amount of likeness can be expected in comparison of their

polymetric music. However, in review of the analyses present in this study, this does not appear

to be the case.

Of the five excerpts from Welcher in this study, all five have well-defined polymeter.

Welcher mostly uses a combination of folk songs in these polymetric sections; therefore, the

melodic content in these areas tends to be comprised of single melodic lines (sometimes with

new harmonies that support one or both of the lines). The assumed meters in each of these

sections vary greatly, but are often distantly related meters that align at certain points close to

barlines. Welcher typically uses the same or related keys between the different folk songs, but

not exclusively. Additionally, these sections usually appear toward the end of their pieces; each

excerpt, thus, represents a “simultaneous recapitulation” of sorts that uses the respective folk

songs.

The five examples of Gotō’s music all have clear uses of polymeter. Gotō mainly quotes

preexisting material from major instrumental or choral works; therefore, each component of the

polymeter in these places is often itself a three-, four-, or five-part texture. The implied meters in

these sections, for Gotō, are usually the same, since he typically quotes different excerpts from

the same work within the polymeter. The polymetric effect, here, is intensified by the delayed

entries of one or more polymetric components, as well as the use of distant keys between the

different parts. In Gotō’s works, these sections of music usually appear toward the middle of

86

their pieces; these excerpts do not function as “simultaneous recapitulations” since the melodic

content usually does not appear earlier in the work.

Through these analyzed excerpts, Welcher and Gotō provide two unique solutions to the

problem of making polymeter aurally present to the listener. This study demonstrates how the

two composers have each developed a distinctive yet individual expression of polymeter in their

own musical language. These results are a snapshot into two separate approaches to the

polymetric “problem,” and are intended to be of use to the wider audience as polymetric writing

continues to be a viable compositional technique.

87

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