El Salon Mexico Copland - Free Essay Example

El Salon Mexico by Aaron Copland: A Study and Comparison of the Orchestral Score and Two Transcriptions for Band D. M. A. Document Presented in Partial Fulfillment of the Requirements for the Degree of Doctor of Musical Arts in the Graduate School of The Ohio State University By Erika Kirsten Svanoe, M. M. Graduate Program in Music The Ohio State University 2009 DMA Document Committee: Russel Mikkelson, Advisor Hilary Apfelstadt Richard Blatti Daryl Kinney Copyright by Erika Kirsten Svanoe 2009 Abstract Aaron Copland completed the orchestral score to El Salon Mexico in 1936 marking a turning point in his career. The piece received more performances in the year following its completion than any of his previous orchestral works. It was well received by both critics and audiences due to his focus on melody and shift in thinking towards using the â€Å"simplest possible means† to make the music more accessible to the listener. Mark Hindsley completed a band arrangement of El Sa lon Mexico in 1966 that included numerous changes to the meter and rhythmic notation found in Copland’s orchestral score. The author conducted a comparative analysis of Copland’s published orchestral score, the El Salon Mexico manuscript materials, Bernstein’s arrangements for piano, and Hindsley’s transcription for band. This investigation sought to determine why Hindsley chose to include metric alterations that differ from the orchestral score, and how he decided what meters would be appropriate. The study of Copland’s manuscript materials of El Salon Mexico revealed that Copland simplified the meter and rhythmic notation after the composition was complete. These rhythmic alterations were completed during the orchestration process in an effort to make the piece more performable. Much of Copland’s original conception of rhythmic notation, that appears in his manuscript sketches, also appear in Bernstein’s piano arrangements. In addition, many of the alterations Hindsley utilized were similar to the ii metric and rhythmic notation in Bernstein’s arrangements. In some sections of the music, Bernstein’s and Hindsley’s notation more closely match Copland’s original conception of rhythmic notation than the orchestral score. The comparative analysis also revealed Hindsley’s scoring techniques, including heavy doubling, unnecessary changing of wind instrument timbres and numerous changes to meter and beaming. The author created a new arrangement for band that restores all the orchestral meters and modernizes the instrumentation and orchestration. The intent was to provide today’s conductors the option of using a transcription more closely related to the published orchestral score. iii Dedication Dedicated to my husband and closest friend Erik Evensen. v Acknowledgements I would like to thank my teachers at the Ohio State University for their help and guidance in compl eting this project, including my committee members Hilary Apfelstadt, Daryl Kinney, Richard Blatti, and especially my advisor Russel Mikkelson who proposed the idea for project and guided the work throughout the process. I would also like to thank him and the Ohio State University Wind Symphony for their preparation of my arrangement that resulted in a wonderful performance. Thank you to Philip McCarthy from Boosey Hawkes, and James Kendrick and Jessica Rauch from the Aaron Copland Fund for Music for assisting with the various permissions needed to complete this project. I would also like to thank the librarians at the Library of Congress for their assistance, particularly Loras Schissel for mailing a copy of a necessary manuscript. Finally, I’d like to thank my family for all of their support the past three years. I need to give a special thank you to my husband Erik Evensen, who has been my greatest supporter. v Vita January 26, 1976. Born, Janesville, WI, USA 1999.. B. M. E. University of Wisconsin – Eau Claire 1999-2001.. Music Educator, Mukwonago, WI 2001-2003.. Graduate Assistant, Oklahoma State University 2003.. M. M. Wind Conducting, Oklahoma State University 2003-2006.. Lecturer, University of New Hampshire 2006-2009.. Doctoral Conducting Associate, Ohio State University 2009-present . Director of Bands, Bemidji State University Field of Study Major Field: Music vi Table of Contents Abstract . ii Dedication iv Acknowledgements v Vita . vi List of Figures.. viii List of Photos .. xii List of Tables xiii Chapter 1: Introduction and Procedures .. 1 Chapter 2: El Salon Mexico for Orchestra .. 7 Chapter 3: The Mark Hindsley Arrangement for Band 45 Chapter 4: The Erika Svanoe Arrangement for Band. 82 Chapter 5: Conclusions and Suggestions for further Research .. 92 Bibliography. 100 Appendix A: Copland’s â€Å"Suggested revisions on band arrangement† 103 Appendix B: El Salon Mexico for Band arranged by Erika Svano e. 05 vii List of Figures Figure 2. 1: Folksong material used in El Salon Mexico 23 Figure 2. 2: Copland, El Salon Mexico, melodic material, mm. 8-13 .. 23 Figure 2. 3: Copland, El Salon Mexico, mm. 23-26, Trumpet . 24 Figure 2. 4: Copland, El Salon Mexico, mm. 39-44, Bassoon 1 . 24 Figure 2. 5: Copland, El Salon Mexico, mm. 61-64, Violin 1, 2, Viola (compressed).. 25 Figure 2. 6: Copland, El Salon Mexico, mm. 6-81, Violin 1 25 Figure 2. 7: Copland, El Salon Mexico, mm. 106-110, Clarinet 1 .. 26 Figure 2. 8: Copland, El Salon Mexico, mm. 135-139, Violin 1. 26 Figure 2. 9: Copland, El Salon Mexico, mm. 185-190, Clarinet 1 . 27 Figure 2. 10: Copland, El Salon Mexico, mm. 256-260, English Horn 27 Figure 2. 11: Copland, El Salon Mexico, mm. 211-214, Clarinet 1 28 Figure 2. 2: Copland, ARCO 28-A (Piano Sketch), Rhythmic notation, mm. 1-3. 32 Figure 2. 13: Copland/Bernstein, El Salon Mexico for Two Pianos, Rhythmic notation, mm. 1-3 .. 32 Figure 2. 14: Copland, El Salon Mexico, Rhythmic notation, mm. 1-5 32 Figure 2. 15: Copland, El Salon Mexico, mm. 124-128, Violin 1.. 33 Figure 2. 16: Copland, ARCO 28-A (Piano Sketch), Corresponding music to Figure 2. 5 33 viii Figure 2. 17: Copland, El Salon Mexico, mm. 160-168 (compressed) 34 Figure 2. 18: Copland, ARCO 28-A (Piano Sketch), Corresponding music to Figure 2. 17 34 Figure 2. 19: Copland, El Salon Mexico, mm. 130-135, Strings.. 35 Figure 2. 0: Example of revised rhythmic notation in Symphonic Ode. 37 Figure 3. 1: Hindsley Orchestration Chart 51 Figure 3. 2: Copland/Hindsley, El Salon Mexico for Band, mm. 40-44. Bassoons, Contrabassoon, Alto and Tenor Saxophones . 53 Figure 3. 3: Copland, El Salon Mexico, mm. 247-251, Strings 54 Figure 3. 4: Copland/Hindsley, El Salon Mexico for Band, mm. 247-251, Clarinets 54 Figure 3. 5: Copland, El Salon Mexico, mm. 1-64, Strings 55 Figure 3. 6: Copland/Hindsley, El Salon Mexico for Band, mm. 61-64, Clarinets, Cornets 55 Figure 3. 7: Copland, El Salon Mexico, mm. 73-80, S trings 56 Figure 3. 8: Copland/Hindsley, El Salon Mexico for Band, mm. 73-80, Flutes, Clarinets, Cornets 6 Figure 3. 9: Copland, El Salon Mexico, mm. 1-4, Trumpets 57 Figure 3. 10: Copland/Hindsley, El Salon Mexico for Band, mm. 1-4, Cornets, Trumpets 57 Figure 3. 11: Copland, El Salon Mexico, mm. 20-25, Bass Clarinet, Bassoons, Trombone 1 .. 59 ix Figure 3. 12: Copland/Hindsley, El Salon Mexico for Band, mm. 0-25, Bassoons, Saxophones 59 Figure 3. 13: Copland, El Salon Mexico, mm. 221-226 . 61 Figure 3. 14: Copland/Hindsley, El Salon Mexico for Band, mm. 221-226 .. 62 Figure 3. 15: Copland/Hindsley, El Salon Mexico for Band, 1st Version Manuscript (ARCO 28-D) mm. 73-80 .. 63 Figure 3. 16: Copland, El Salon Mexico, mm. 3-80 . 65 Figure 3. 17: Copland/Hindsley, El Salon Mexico for Band, mm. 73-80 .. 66 Figure 3. 18: Copland, El Salon Mexico, mm. 372-375, Viola 69 Figure 3. 19: Copland/Hindsley, El Salon Mexico for Band, mm. 373-377, Clarinet 2, 3 .. 69 Figure 3. 20: Copland, El S alon Mexico, mm. 377-379, Violin 2, Viola .. 70 Figure 3. 1: Copland/Hindsley, El Salon Mexico for Band, mm. 380-382, Clarinet 2, 3 .. 70 Figure 3. 22: Copland, El Salon Mexico, mm. 14-15, Trumpets. 71 Figure 3. 23: Copland/Hindsley, El Salon Mexico for Band, mm. 14-15, Cornets, Trumpets 71 Figure 3. 24: Copland, El Salon Mexico, mm. 316-319, Trumpet 1, 2. 72 Figure 3. 5: Copland/Hindsley, El Salon Mexico for Band, mm. 316-319, Trumpet 1.. 72 Figure 3. 26: Copland, El Salon Mexico, mm. 324-326, Horns .. 73 Figure 3. 27: Copland/Hindsley, El Salon Mexico for Band, mm. 324-326, Horns 73 Figure 3. 28: Copland, El Salon Mexico, mm. 252-256, Violins .. 75 x Figure 3. 29: Copland/Hindsley, El Salon Mexico for Band, mm. 252-256, Clarinets 1, 2. 75 Figure 3. 30: Copland, El Salon Mexico, mm. 57-258, Viola 76 Figure 3. 31: Copland/Hindsley, El Salon Mexico for Band, mm. 257-258, Clarinet 1 76 Figure 3. 32: Copland, El Salon Mexico, mm. 10, melodic material. 77 Figure 3. 33: Copland/Hindsley , El Salon Mexico for Band, m. 10, melodic material . 77 Figure 3. 34: Comparison of rhythmic notation for music corresponding with orchestral mm. 156-172 in the Piano Sketch (ARCO 28-A), Bernstein arrangement for solo piano, and Hindsley’s arrangement for band 8 Figure 3. 35: Copland, El Salon Mexico, mm. 59, Bassoon 80 Figure 3. 36: Copland/Hindsley, El Salon Mexico for Band, m. 59, Bassoon .. 80 Figure 3. 37: Copland, El Salon Mexico, m. 313, Trumpet 1 in C.. 80 Figure 3. 38: Copland/Hindsley, El Salon Mexico for Band, m. 313, Trumpet (displayed in concert pitch) 80 Figure 3. 39: Copland, El Salon Mexico, mm. 60-161, melodic material in Trombone 1, 2, Violin 1, Viola .. 81 Figure 3. 40: Copland/Hindsley, El Salon Mexico for Band, mm. 160-161, melodic material in Horn 2, 4, Trombone 1, 2 81 xi List of Photos Photo 2. 1: ARCO 28. 1, Music corresponding to published orchestral score m. 321 40 Photo 2. 2: ARCO 28-A, Music corresponding to published orchestral score m. 321.. 41 Photo 2. 3: ARCO 28, Music corresponding to published orchestral score m. 378 2 Photo 2. 4: ARCO 28a copy 2, mm. 24-25, Bass Clarinet, Bassoons, Trumpet in C, Trombone .. 43 Photo 2. 5: ARCO 28-A, Music corresponding to published orchestral score mm. 24-25 . 44 xii List of Tables Table 2. 1: Summary of published scores and manuscripts .. 31 Table 3. 1: Comparison of Instrumentation between Copland Orchestral Score and Hindsley Band Score. 49 Table 4. 1: Comparison of Instrumentation between Copland Orchestral Score, Hindsley Band Score, and Svanoe Band Score. 85 xiii Chapter 1: Introduction and Procedures Background Transcriptions and arrangements of works from other mediums hold an important place in the literature of the wind band. For much of the band’s history, a large part of the available literature included orchestral transcriptions. While there has been an enormous increase in the percentage of original compositions for band in the past several decades, qual ity transcriptions of significant works from other mediums continue to add depth and variety to the literature as a whole. When a conductor is faced with the task of performing an arrangement or transcription, it is important to refer to the original version during score study and preparation. If the arranger of the new version has made changes that may affect the performance of the piece, it is vital to know what these alterations are, and if they are appropriate. In some cases, changes in an arrangement may not accurately reflect the original composer’s intentions, while other changes are appropriate due to the difference in medium. One such band transcription that deserves a thorough comparative analysis and evaluation is Mark Hindsley’s arrangement of Aaron Copland’s El Salon Mexico. While it is considered to be one of Copland’s lighter orchestral works, it is an important piece because of its place in his compositional output as a whole. It was one of the first works that represented a conscious shift in Copland’s compositional style towards using what he 1 called the â€Å"simplest possible terms. † Copland’s perception that the majority of concert audiences were apathetic towards any music but the established classics was responsible for this shift in thinking. As the audience for new music continued to decrease, Copland experimented with music he thought would appeal to a wider audience. 1 El Salon Mexico was the first successful piece in this new style and helped Copland gain widespread popularity. Hindsley’s band arrangement is significant not only because of Copland’s status as one of America’s premiere composers, but because of its widespread use by bands. It has appeared on several state high school contest lists including Texas, Florida, Arkansas, and Virginia and was recorded by both the University of Illinois and the Cincinnati Conservatory. 3 It also appears regularly on collegiate band programs. 4 However, Hindsley made several editorial decisions, particularly regarding meter, which differ significantly from the score of the orchestral work. One purpose of this study is to compare Aaron Copland’s El Salon Mexico for orchestra with Mark Hindsley’s transcription for band and to evaluate the editorial changes made in the band version. Finally, a new arrangement of El Salon Mexico for band was created in which all of Copland’s orchestral meters were restored. Aaron Copland Aaron Copland (1900-1990) is one of the most significant American composers of the 20th century. He won the Pulitzer Prize and the New York Music Critics’ Circle 1 2 Copland, â€Å"Composer from Brooklyn,† xxvi. Berger, Aaron Copland, 30. Shattingermusic. com 4 CBDNA Report 2 Award for Appalachian Spring and his film scores earned him four Academy Award nominations and one win for The Heiress in 1949. He was elected to the American Academy of Arts and Letters in 1954, received the Academy’s Gold Medal in 1956, and served as the Academy’s president in 1971. Other awards included a MacDowell Medal in 1961, a Presidential Medal of Freedom in 1964, a Kennedy Center Honor in 1979, a Medal of the Arts in 1986, and a Congressional Gold Medal in 1986, as well as several honorary doctorates. While Copland’s music has been acknowledged by prestigious awards, it is also recognized by much of the American populous because of its infiltration into popular culture. His music has been used to promote the Olympics, the armed forces and the United States beef industry, because â€Å"when it comes to music that summons up images of America in the minds of American listeners, Copland is unique†¦ in each case the promoters have wanted to tap into deep-seated feelings that, somehow, this music evokes like almost nothing else. 6 The youngest of five children, Aaron Copland was born on November 14, 1900 in Brooklyn, New York to his parents Harris and Sarah Copland. Throughout his youth Copland studied piano, theory, and composition with various teachers and supplemented his education by attending recitals and concerts. In 1921 he traveled to Paris where he studied composition with Nadia Boulanger, his most influential teacher. While studying with Boulanger Copland produced his first orchestral score, Grohg, which he completed upon his return to the United States in 1924. In addition, Boulanger arranged for two 6 Howard Pollack. Copland, Aaron. In Grove Music Online. Burr, â€Å"Copland, the West and American Identity,† 22. 3 performances of Copland’s organ concerto to be performed by both the New York and Boston Symphony Orchestras with herself as soloist. The performance of the Organ Symphony under the baton of Sergey Koussevitzky initiated an important relationship for Copland. Koussevitzky became one of Copland’s greatest collaborators and champions. 7 Upon Coplandâ₠¬â„¢s return the United States, he felt the need to compose modern music that was identifiably American. He began to incorporate jazz into his symphonic works such as Music for Theatre (1925) and the Piano Concerto (1926). 8 While Copland had the support of Koussevitzky and several other musicians, critics, and artists, much of the press regarded his music with skepticism. His Piano Concerto, performed by the Boston Symphony Orchestra with himself as pianist, had a particularly unfavorable reception. 9 Olin Downs of the New York Times wrote â€Å"It progresses by fits and starts†¦confirming [the listener’s] suspicions that Mr. Copland needs a firmer hold of principles of musical structure†¦ Here is a young man who can surely not remain content with the praise of partisans or knowledge of his own artistic shortcomings. †10 Public audiences had similar reactions. At a performance of the Piano Concerto in Mexico there were so many hisses from the audience during the performance that Copland looked to the conductor, Carlos Chavez, for a sign of whether to continue the performance. 11 Copland’s compositional activity decreased in the late 1920s. He entered a selfreflective period in which he considered his own compositional path, as well as the path of American music. In his 1939 essay â€Å"Composer from Brooklyn† he stated: 7 8 Howard Pollack. Copland, Aaron. In Grove Music Online. Crist, Music for the Common Man, 42. 9 Pollack, â€Å"Copland, Aaron. In Grove Music Online. 10 Berger, Aaron Copland, 24. 11 Copland and Perlis, Copland, 216. 4 I began to feel an increasing dissatisfaction with the relations of the music-loving public and the living composer. The old â€Å"special† public of the modern-music concerts had fallen away, and the conventional concert public continued apathetic of indifferent to anything but the established classics. It seemed to me that we composers were in danger of working in a vacuum. Moreover, an entirely new public for music had grown up around the radio and phonograph. It made no sense to ignore them and to continue writing as if they did not exist. I felt that it was worth the effort to see if I couldn’t say what I had to say in the simplest possible terms. 12 It was El Salon Mexico that â€Å"developed and heralded his new style. 13 It embodied Copland’s new tendency toward â€Å"imposed simplicity. †14 For the untrained listener the use of folksongs and programmatic title helped bridge the gap from absolute music. It was accessible for audiences who did not have musical training or the ability to perceive formal structures. The piece was immediately popular receiving more performances than any of his other orchestral works and brought Copland’s new compositional style to the attention of the public. El Salon Mexico established Copland as a â€Å"successful† composer and was directly responsible for his publishing contract with Boosey Hawkes. Impressive honors would soon follow, including a commission from the Columbia Broadcasting System for Music for Radio (1937), election to the National Institue of Arts and Letters in 1942, and the 1945 Pulitzer Prize in music for Appalachian Spring. 15 In the decade that followed 1935, Copland did not entirely abandon writing in his more abstract style, though most of his efforts had some element of functionality, such as An Outdoor Overture (1939), composed for students, or included external matter that 12 13 Copland â€Å"Composer from Brooklyn,† xxvi. Berger, Aaron Copland, 30. 4 Copland, â€Å"Composer from Brooklyn†, xxvi. 15 Berger, Aaron Copland, 30-31. 5 gave the piece an element of being programmatic, such as Appalachian Spring (1944). Many of Copland’s most popular and well-known works are from this time span, including Fanfare for the Common Man (1942) and Rodeo (1942). Pieces such as his Piano Sonata (1941) and Sonat a for Violin and Piano (1943) are representative of his more abstract style during this time period. However, it was not until he composed his Third Symphony (1946) that he composed for the orchestra without programmatic elements. 6 Copland commented on what was perceived as an abandonment of his more complex music in his 1967 addition to â€Å"Composer from Brooklyn. † The assertion that I wished â€Å"to see if I couldn’t say what I had to say in the simplest possible terms† and the mention of â€Å"an imposed simplicity† were taken to mean that I had renounced my more complex and â€Å"difficult† music†¦ these remarks of mine emphasized a point of view which, although appropriate at the time of writing†¦seems to me to constitute an oversimplification of my aims and intentions, especially when applied to a consideration of my subsequent work and of my work as a whole. 7 While the ten years that followed El Salon Mexico seemed to focus o n Copland’s new accessible style, he later felt that there was no disparity between his two compositional styles, the simple and the complex. Rather, he adapted his technique to the materials with which he chose to work. [There is] a continuing discussion concerning the apparent dichotomy between my â€Å"serious† and my â€Å"popular† works. I can only say that those commentators who would like to split me down the middle into two because I take into account with each new piece the purpose for which it is intended and the nature of the musical materials with which I begin to work. Musical ideas engender pieces, and the ideas by their character dictate the nature of the composition to be written. 18 16 17 Berger, Aaron Copland, 32. Copland, â€Å"Composer from Brooklyn,† xxvi-xxvii. 18 Ibid. , xxxii. 6 By the late 1940s Copland was widely regarded as the leading American composer of his time. While he had lived in Manhattan for many years, he moved to Ossining, New York in 1952. Through the 1950s he continued to lecture, teach and write and in 1958 began a 20-year international conducting career, presenting both this own works and the music of over 80 other composers. In 1961 Copland moved into a larger home located near Peekskill, New York where he lived until his death. He did not compose much after 1972 and began to suffer short-term memory lapses in the mid1970s. After being diagnosed with Alzheimer’s he was under medical supervision by the mid-1980s. He died of respiratory failure on December 2, 1990. 19 El Salon Mexico During a visit to Mexico in the autumn of 1932, Copland conceived of writing a piece based on Mexican themes. From the beginning, he connected the piece with a popular dance hall in Mexico City called Salon Mexico. He realized he did not want to attempt to reflect the profound, historical side of Mexico since he felt he did not truly know the country. Instead he wanted to reflect this tourist â⠂¬Å"hot spot† where he also felt a close connection with the Mexican people. While the work references several Mexican folk songs, Copland transforms the melodies into his own musical language. He said â€Å"It wasn’t the music that I heard, but the spirit that I felt there, which attracted me. Something of that spirit is what I hope to have put into my music. The work was completed in 1936 and premiered by Carlos Chavez and the Orquesta Sinfonica de Mexico the summer of the following year. 19 Pollack, â€Å"Copland, Aaron. † In Grove Music Online. 7 El Salon Mexico is a significant work in Copland’s compositional output for several reasons. First, this piece came at a time when Copland was beginning a shift in thinking toward trying to say things in the â€Å"simplest possible terms. † The works that followed the Symphonic Ode (1929), the composition that marks the culmination of Copland’s austere writing up to that point, gradually evolved towards a generally more accessible aesthetic. Copland’s focus on melody and the use of Mexican folksongs in El Salon Mexico were important elements that helped Copland write in a more accessible style, as well as attain wider audience appeal of his music. Second, this was Copland’s first work to utilize borrowed folk tunes, a tool he would use throughout his career. While some of the material he borrowed is a direct quotation, his treatment of folksongs more often employs a transformation of the materials, making them a part of Copland’s compositional language, deftly retaining the spirit and character of the tunes. Copland’s motive for turning to these melodies was â€Å"an aspect of his campaign to achieve a simple style and a content that would engage the interest of a wider audience. †20 Third, due to greater interest in his music, Copland became enormously successful. El Salon Mexico introduced the composer to a larger audience and ear ned him popular acclaim. Compared to Statements and the Short Symphony, which received hardly any performances, by 1938 El Salon Mexico had been performed by 21 orchestras. Also, it was at the 1938 European premier of this work that Copland met Benjamin Britten, who in turn introduced Copland to his publisher Boosey Hawkes. El Salon 20 Berger, Aaron Copland, 57. 8 Mexico, along with Music for Radio, were the first works to be published by the London firm. El Salon Mexico clearly marks a compositional turning point for Copland in several ways. His conscious efforts to appeal to a wider audience, use of folksong materials, and the success that followed the premiere all contribute to the fact that this work was an important milestone in Copland’s compositional output. Several arrangements of the piece were created. Leonard Bernstein arranged both the solo piano and two piano versions in 1941 and 1943 respectively. A truncated version titled â€Å"Fantasia Mexicana† dated 1952 was adapted and orchestrated by Johnny Green for the MGM motion picture Fiesta. Arturo Toscanini wrote an unpublished arrangement for piano, possibly for his own study of the orchestral score. 21 Mark Hindsley completed his arrangement for concert band in 1966. While Copland created several band arrangements of his own works, as well as composing Emblems (1964) for band, he did not create a band version of El Salon Mexico. His first transcription for band was An Outdoor Overture (1938), originally composed for high school orchestra. The band transcription was completed at the request of Edwin Franko Goldman and was premiered by the Goldman Band in 1942. The other transcriptions Copland completed for band include Variations on a Shaker Melody (1956), Preamble for a Solemn Occasion (1949), Inaugural Fanfare (1977), and Red Pony Suite (1948). 22 Since the Toscanini is an unpublished reduction of the published orchestral score and the Green is a truncated version of the o rchestral score, these two versions did not factor into this study. 2 Briskey, â€Å"The Symphonic Band Repertoire of Aaron Copland,† 38-42. 9 21 Mark Hindsley (1905-1999) created the band arrangement of El Salon Mexico in 1966. Hindlsey was the Director of Bands at the University of Illinois from 1948 until he retired in 1970. He completed dozens of arrangements for band, most of which are selfpublished and still available from his son, Roger Hindsley, who currently distributes his music. 23 The only band arrangement of Hindsley’s that is not currently self-published is El Salon Mexico, published by Boosey Hawkes. According to a 1982 survey by Earle Gregory, it was (and likely still is) one of Hindsley’s most often played transcriptions. The distribution capabilities of Boosey Hawkes certainly contribute to its availability and popularity. 24 Review of Related Research The majority of literature regarding El Salon Mexico relates to the background story of the composition or how the work fits into Copland’s compositional output as a whole. There is no in-depth analysis of the work to date. This lack of analytical research is also noted by Leo Philip Fishman in his recent dissertation â€Å"Theoretical Issues and Presumptions in the Early Music of Aaron Copland† (2007). Fishman states that â€Å"there has been a dearth of useful research concentrating on theoretical aspects of his music while there has been a great deal of work on contextualizing Copland as a way to explain his oeuvre. 25 Fishman’s study concentrates on four early works of Copland’s, none of which is El Salon Mexico. www. hindsleytranscriptions. com Gregory, â€Å"Mark H. Hindsley: The Illinois Years,† 162-3. 5 Fishman, â€Å"Theoretical Issues and Presumptions in the Music of Aaron Copland,† vi. 10 24 23 The most relevant literature that addresses questions posed by this study is either limited in scope or tangential to th e topic. Only two limited analyses of the orchestral El Salon Mexico were discovered. No literature could be found regarding Hindsley’s band transcription. Topics that were considered tangential but supportive to the study include studies of the piano arrangements of El Salon Mexico by Leonard Bernstein, Mark Hindsley’s other arrangements for band, and Copland’s use of metric and rhythmic notation in other works. The most relevant literature within these topics is summarized here. The best information that could be considered an analysis of El Salon Mexico is by Gerald Abraham. When Boosey Hawkes published the miniature score of El Salon, it was traditional for analytical notes to be included. The four-page insert gives a brief summary of the story of the Mexican dance hall and Copland’s inspiration for writing the piece. Most of the information outlines Copland’s use and alteration of Mexican folksongs and where they appear in various guises t hroughout the piece. Abraham outlines his interpretation of the form, which is debatable but certainly workable version of the formal structure. The notes are of high quality and give an excellent summary of the work, but are very limited in scope. 26 â€Å"A Comparison and Analysis of Aaron Coplands El Salon Mexico for Orchestra, Piano Solo, and Two Piano Four Hands† by Richard Glazier gives a brief analysis of the orchestral version and documents some of the differences between it and the piano arrangements by Leonard Bernstein. Most of the analysis is drawn from Gerald Abraham’s notes published in the miniature score, though Glazier additionally illustrates 26 Abraham, â€Å"Aaron Copland: El Salon Mexico. † 11 Copland’s use of polyrhythm and polytonality. In comparing the rhythmic notation of the orchestral and piano versions and reading Copland’s essay â€Å"On the notation of rhythm,† Glazier recognizes that Copland simplified th e meter in the orchestral version. Some of the other rhythmic notation that appears in the piano versions he credits to Bernstein, but in some cases this notation originally appears in Copland’s hand in the manuscript materials. It is clear that Copland’s manuscripts of El Salon Mexico were not examined as part of this study. Glazier’s purpose is to document some of the differences in the piano versions and defend them as artistic additions to piano literature. 27 Research regarding Copland’s practice of rewriting rhythmic notation in his orchestral works includes â€Å"The Compositional History of Aaron Copland’s Symphonic Ode† by Elizabeth Bergman Crist. She constructs the history of the composition through existing manuscript materials and correspondence. Crist demonstrates the process and circumstances that led to Copland’s rebarring of the Symphonic Ode, and substantiates that Koussevitzky was largely responsible for initia ting these changes. The article also provides evidence that Copland preferred his original rhythmic notation. This was discovered through his restoration of the original notation in the revised 1955 edition. 28 Research regarding Mark Hindsley’s band transcriptions was done by Earle Suydam Gregory and documented in his dissertation â€Å"Mark H. Hindsley: The Illinois Years. It documents the professional activities of Hindsley with emphasis on his research into the construction of instruments, his contributions to the University of Illinois band building, and his contributions to band literature through transcriptions. 27 28 Glazier, â€Å"A Comparison and Analysis of Aaron Coplands El Salon Mexico. † Crist, â€Å"The Compositional History of Aaron Copland’s Symphonic Ode. † 12 This includes a study of the scoring practices of Hindsley by examining a sample of his orchestral transcriptions for band and an analysis of how each nstrument was utilized. The research confirms and expands much of Hindsley’s own writings in Hindsley On Bands. 29 Procedures and Purpose of the Study I first became familiar with El Salon Mexico when I was playing clarinet on Suite from Billy the Kid with a youth orchestra. It was at this time I became interested in Copland’s other works for orchestra. I did not formally study the piece until I was asked to conduct Mark Hindsley’s band transcription for an audition at the Ohio State University. During the course of my study, I learned that there were several places in Hindsley’s score where the meters published in the band arrangement differ from the meters in Copland’s original orchestral score. These discrepancies led me to several questions. What is happening in the original orchestral score that may have initiated the meter changes in the band score? What are all of the meter changes that Hindsley utilized? Why did the arranger choose to use different meters and how w ere the meters chosen? Do other arrangements of El Salon Mexico also alter the meter? Did Copland approve of the changes Hindsley made? Is a new band arrangement of El Salon Mexico with the orchestral meters restored warranted? These are the questions that led me to develop this study in the manner that follows. At the start of this process, I determined that a new band arrangement of El Salon Mexico was warranted and could be an important addition to band literature. The changing of meters and beaming in arrangements, such as the Hindsley arrangement, can 29 Gregory, â€Å"Mark H. Hindsley: The Illinois Years. † 13 pose difficulties for musicians. In the case of El Salon Mexico, many college wind players learn the piece in the band arrangement, but then must relearn the different meters when called upon to play the original orchestral version. Regardless of whether changing the original meters for an arrangement is an improvement or not, the relearning of music in diff erent meters is a difficult task for both conductors and players. Since it is highly unlikely that a different edition of the orchestral version of El Salon Mexico will be available, I felt it was important that band directors have an arrangement in which the meters of the orchestral version were maintained. In addition, Hindsley’s arrangement was conceived for a large band with over 90 musicians. College bands of this size were much more common in the 1960s than they are today. Today’s collegiate bands use a smaller number of musicians, more in line with a wind ensemble instrumentation. This trend gained acceptance at many colleges and universities in the 1970s. 30 Even the majority of today’s large concert bands are significantly smaller than Hindsley’s model. The new band arrangement of El Salon Mexico was created both to restore the orchestral meters and modernize the instrumentation. I felt many of the questions posed could be answered by doing a comparative analysis between Copland’s orchestral score and Hindsley’s band arrangement. This was completed in two stages. First, I completed an overall study of the orchestral work, with special consideration for rhythm and meter. Second, I compared several aspects of the Hindsley band arrangement to the orchestral score. These aspects included instrumentation, meter, beaming, key signatures, and overall scoring. 30 Battisti, Winds of Change, 68. 14 Since a dialogue with either the composer or arranger was impossible, I decided to review all of the relevant sketches, manuscripts, and correspondence between Copland and Hindsley. To accomplish this I visited the Library of Congress and examined all of the materials relating to El Salon Mexico. It was my premise that an examination of Copland’s sketches and original manuscripts would lead to a deeper understanding of the work and its compositional process, as well as confirm possible errata found during the analysis. To discover any possible contact between Hindsley and Copland, I also searched for and examined correspondence relating to the band arrangement. Chapter Two examines Copland’s orchestral score of El Salon Mexico. This includes a background of its creation and how it fits into a shift in thinking at this point in Copland’s compositional output. It includes a brief formal analysis noting thematic development and use of folksong. The use of meter, beaming, and rhythm is also examined in depth. Chapter Three discusses Hindsley’s arrangement of El Salon Mexico for band. This will include changes made by Hindsley in the band arrangement pertaining to instrumentation, meter, beaming, key signatures, and overall scoring. All metric alterations in the Hindsley are documented and cataloged according to the type of alteration made, and the origins of these alterations are explored. The publication history, relevant correspondence, revisions to the original band manuscript and errata are also be examined. Chapter Four compares the new arrangement for band, created by myself, to the Mark Hindsley band arrangement and Copland’s original orchestral score. It includes a discussion of the decision making process regarding instrumentation, use of key 15 signatures, overall scoring, meters, and beaming. It also documents changes made due to errata found in both the Hindsley band arrangement, as well as the original orchestral score. Chapter Five presents a summary of the findings of the study and provides suggestions for further research. Description of Appendices Appendix A: Copland, â€Å"Suggested Revisions on band arrangement of El Salon Mexico† Appendix B: Full score of El Salon Mexico for band arranged by Erika Svanoe 16 CHAPTER 2: El Salon Mexico for Orchestra In 1935 Copland organized a series of â€Å"one-man concerts† featuring the works of one living composer on each program. In observing the audiences at this series Copland stated â€Å"As I looked around at the all-too-familiar small group at these concerts, I knew that I wanted to see a larger and more varied audience for contemporary music. †31 At this time Copland was finishing El Salon Mexico in which he said he was experimenting with a different style of writing. He was not rejecting one kind of music for another, but felt it was time to try something new. 2 Copland considered the first version of the Symphonic Ode from 1929 to be the piece that marked the end of his most austere and complex compositions. The move toward a simpler style was a gradual transition in the works that followed. In retrospect it seems to me that the Ode marks the end of a certain period in my development as a composer. The works that follow it are no longer so grandly conceived. The Piano Variations (1930), the Short Symphony (1933), the Statements for Orchestra (1935) are more spare in sonority, more lean in texture. They are still compar atively difficult to perform and difficult for an audience to comprehend. 33 El Salon Mexico was first conceived while Copland was simultaneously working on two other works in the fall of 1932 in Mexico: Short Symphony and Statements. â€Å"These three works and their combined compositional histories document Copland’s 31 32 Copland and Perlis, Copland, 244. Copland and Perlis, Copland, 245. 33 Copland, â€Å"Composer in Brooklyn,† xxvi. 17 refinement of a simplified musical idiom that emphasizes aural accessibility and draws on the melodic resources of traditional tunes. 34 It is El Salon Mexico and its position in Copland’s compositional output as one of the first of several works to simplify his musical language that makes it particularly significant. Short Symphony is composed in a similar vein as Copland’s earlier works, such as the Piano Variations, which focuses on structural unity and uses a dissonant avant-guard style. Statements still utilizes this type of style, but focuses less on formal structure and is more episodic. Copland also hoped that the suggestive movement titles, such as â€Å"Militant† and â€Å"Cryptic,† would make the piece more palatable to the listening audience. El Salon Mexico also uses an episodic form, but focuses more on melody than these other works. 35 The reason for this shift toward melody comes primarily from the materials Copland chose to work with, which were inspired by the music he heard during his trip to Mexico in 1932. For several years prior to his trip, Copland had promised Carlos Chavez that he would visit Mexico. When Chavez promised him an all-Copland program by the Conservatorio Nacional de Mexico, he felt the time had come. He left New York on August 24, accompanied by Victor Kraft, and arrived in Laredo September 2, the morning of the concert. Copland remained in Mexico for five months. During Copland’s visit, Chavez took him to a dance hall in Mexico C ity called El Salon Mexico, known to the locals as â€Å"El Marro† or the policeman’s nightstick. It was a popular place for tourists who wanted a taste of how the local lower class sought 34 35 Crist, Music for the Common Man, 43. Crist, Music for the Common Man, 43-4. 18 entertainment. 36 The atmosphere of the place made an impression on Copland and he came away with the idea to create El Salon Mexico. El Salon Mexico had been ‘in the works’ since my first trip to Mexico in 1932 when I came away from that colorful dance hall in Mexico City with Chavez. I had read about the hall for the first time in a guidebook about tourist entertainment: ‘Harlem type night-club for the peepul, grand Cuban orchestra, Salon Mexico. Three halls: one for people dressed in your way, one for people dressed in overalls but shod, and one for the barefoot. ’ A sign on a wall of the dance hall read: ‘Please don’t throw lighted cigarette butts on the floor so the ladies don’t burn their feet. A guard, stationed at the bottom of the steps leading to the three halls, would nonchalantly frisk you as you started up the stairs to be sure you had checked all your ‘artillery’ at the door and to collect the 1 peso charged for admittance to any of the three halls. When the dance hall closed at 5:00 A. M. , it hardly seemed worthwhile to some of the overalled patrons to travel all the way home, so they curled themselves up on the chair around the walls for a quick two-hour snooze before going to a seven o’clock job in the morning. 37 Copland did not want to try to translate the profound side of Mexico into a musical work. He felt he did not know the country well enough to attempt this. Rather he wanted to reflect the spirit of the dance hall and his experiences he had there with the Mexican people as a tourist. The â€Å"people† were reflected in Copland’s use of traditional folksongs. Copland s tated â€Å"I began (as I often did) by collecting musical themes or tunes out of which a composition might eventually emerge. It seemed natural to use popular Mexican melodies for thematic material†¦My purpose was not merely to quote literally but to heighten without in any way falsifying the natural simplicity of Mexican tunes. 38 Having the piece sound â€Å"Mexican† was a concern of Copland’s. He wrote to Chavez expressing his concern â€Å"I am terribly afraid of what you will say of the ‘Salon Mexico’-perhaps it is not Mexican at all, and I would feel so foolish. But in America del 36 37 Crist, Music for the Common Man, 51. Copland and Perlis, Copland, 245. 38 Ibid. , 245. 19 Norte it may sound Mexican! †39 He wrote again to Chavez in 1935: â€Å"What it would sound like in Mexico I can’t imagine, but everyone here for whom I have played it seems to think it is very gay and amusing. 40 Once Chavez heard Copland perform the piano version, he agreed to conduct it once the orchestration was complete. The premiere took place on August 27, 1937 in Mexico City with Orquesta Sinfonica de Mexico at the Palacio de Bellas Artes. The music was well received by the musicians and public with newspapers stating the piece could be taken as Mexican music. The piece was immediately popular. Twenty-one orchestras had performed the piece by 1938. 41 The first American performance was conducted by Koussevitzky with the Boston Symphony Orchestra on October 14, 1938. Another significant performance was at the 1938 International Society for Contemporary Music concert in London where Copland met Benjamin Britten, who was responsible for introducing Copland to the publishing firm Boosey Hawkes. In a letter to Ralph Hawkes Britten wrote â€Å"I’m fearfully anxious for you to cash in on Aaron Copland – the American composer – now without a publisher since Cos Cob Press gave up. His El Salon Mexico was the br ightest thing of the festival†¦. I feel he’s a winner somehow. †42 Ralph Hawkes wrote to Copland August 12, 1938 expressing an interest in publishing El Salon Mexico. 3 Negotiations by written correspondence ensued and Copland eventually convinced the firm to publish Music for Radio as well. It was Ibid. , 246. Ibid. , 246. 41 Crist, Music for the Common Man, 43-4. 42 Mitchell and Reed, eds. , Letters from a Life, 566. 43 Ralph Hawkes to Aaron Copland, 12 August 1938, Aaron Copland Collection, Library of Congress. 20 40 39 Hawkes who suggested that more performances of El Salon Mexico would be possible if the instrumentation was slightly reduced and suggested that having a second version of the piece available with smaller instrumentation. 4 Instrumentation: The full instrumentation for El Salon Mexico is as follows: Piccolo 2 Flutes 2 Oboes English horn Clarinet in E-flat 2 Clarinets in B-flat Bass Clarinet in B-flat 2 Bassoons Contrabassoon 4 Horns in F 3 Trumpet s in C 3 Trombones Tuba Timpani Percussion Piano Strings (Violins, Violas, Cellos, Contrabasses) The percussion part calls for multiple instruments: xylophone, suspended cymbal, gourd, temple blocks, wood block, bass drum, snare drum, and tambour de Provence, which Copland describes as a long drum with a dull sound. The gourd was the only Mexican percussion instrument that he included in the piece. This may have been for the best, since several orchestras of the time had a difficulty acquiring a proper gourd for performances. Some of the wind instruments were marked in the score as â€Å"not essential to performance. † At the request of Ralph Hawkes, and likely seeing the opportunity for 44 Ralph Hawkes to Aaron Copland, 20 September 1938, Copland Collection. 21 more performances with a reduced instrumentation, Copland created an alternate scoring to accommodate this request, eliminating the need for the English Horn, Clarinet in Eflat, Bass Clarinet, Contrabassoon, and Tr umpet 3. Folksong Materials and Form Copland used several Mexican folk songs, found in published collections, as the basis for many of the melodies in El Salon Mexico. Two of the songs, â€Å"El Palo Verde† and â€Å"La Jesuita† were found in Cancionero Mexicano edited by Frances Toor. â€Å"El Mosco† and â€Å"La Malacate,† an indigenous dance tune, were found in El Folklore y la Musica Mexicana by Ruben M. Campos. These melodies are not usually used in their original form, but rather Copland derived new melodic material from them. 45 An excellent summary of Copland’s use of these folk songs comes from musicologist Gerald Abraham. When Boosey Hawkes published the miniature score, it was customary to provide analytical notes about the music. The publisher asked Abraham to write the notes for El Salon Mexico. 46 The four-page insert includes excerpts of the original folksong material and documents Copland’s alteration of these melodies into the thematic material used in the piece. Abraham notes the three most utilized melodies as â€Å"El Palo Verde,† La Jesusita,† and â€Å"El Mosco. † He describes the form of the piece as a â€Å"subtilised and elaborated ternary from, with a long introduction. †47 Abraham illustrates Copland’s alteration of each of the folksongs, as well as documents where material derived from each folksong appears in the piece. 45 Lee, Masterworks of 20th -Century Music, 119. 46 Dickenson, â€Å"Copland’s Earlier British Connections,† 169-70. 47 Abraham, â€Å"Aaron Copland: El Salon Mexico. † 22 Figure 2. 1: Folksong material used in El Salon Mexico48 EL SALON MEXICO  © Copyright 1939 The Aaron Copland Fund for Music, Inc. Copyright Renewed. Boosey Hawkes Inc, sole agent. Reprinted by permission. The melodic material from the opening is derived from the first strain of â€Å"El Palo Verde† Figure 2. : Copland, El Salon Mexi co, melodic material, mm. 8-13 EL SALON MEXICO  © Copyright 1939 The Aaron Copland Fund for Music, Inc. Copyright Renewed. Boosey Hawkes Inc, sole agent. Reprinted by permission. 48 Abraham, â€Å"Aaron Copland: El Salon Mexico. † 23 Beginning in measure 23, the trumpet solo is based on â€Å"La Jesusita† Figure 2. 3: Copland, El Salon Mexico, mm. 23-26, Trumpet EL SALON MEXICO  © Copyright 1939 The Aaron Copland Fund for Music, Inc. Copyright Renewed. Boosey Hawkes Inc, sole agent. Reprinted by permission. The duet between bassoon and bass clarinet starting at measure 39 is a rhythmically altered version of â€Å"El Mosco. Figure 2. 4: Copland, El Salon Mexico, mm. 39-44, Bassoon 1 EL SALON MEXICO  © Copyright 1939 The Aaron Copland Fund for Music, Inc. Copyright Renewed. Boosey Hawkes Inc, sole agent. Reprinted by permission. 24 At measure 61, the strings have material modified from the second strain of â€Å"El Palo Verde† This same material also ap pears twice more during the piece beginning at measures 145 and 353. Figure 2. 5: Copland, El Salon Mexico, mm. 61-64, Violin 1, 2, Viola (compressed) EL SALON MEXICO  © Copyright 1939 The Aaron Copland Fund for Music, Inc. Copyright Renewed. Boosey Hawkes Inc, sole agent. Reprinted by permission. The melody in the strings beginning at measure 76 is derived partly from the second strain of â€Å"El Mosco. † This theme concludes the section that Abraham labels as the introduction. Figure 2. 6: Copland, El Salon Mexico, mm. 76-81, Violin 1 EL SALON MEXICO  © Copyright 1939 The Aaron Copland Fund for Music, Inc. Copyright Renewed. Boosey Hawkes Inc, sole agent. Reprinted by permission. The Allegro Vivace begins at measure 103 and marks the start of what Abraham calls the first section of the ternary form. It begins with material related to the opening measures illustrated above in Figure 2. 2. This is followed by a theme derived from both 25 he altered material from â⠂¬Å"El Palo Verde† at the beginning of the piece, and the second strain of â€Å"El Mosco. † Figure 2. 7: Copland, El Salon Mexico, mm. 106-110, Clarinet 1 EL SALON MEXICO  © Copyright 1939 The Aaron Copland Fund for Music, Inc. Copyright Renewed. Boosey Hawkes Inc, sole agent. Reprinted by permission. New material is added to the beginning of the previous theme at measure 135 but is very similar. Figure 2. 8: Copland, El Salon Mexico, mm. 135-139, Violin 1 EL SALON MEXICO  © Copyright 1939 The Aaron Copland Fund for Music, Inc. Copyright Renewed. Boosey Hawkes Inc, sole agent. Reprinted by permission. 6 The material from Figure 2. 5 returns at measure 145 and is followed by a variant of Figure 2. 2 from measures 175-182. This concludes the Allegro Vivace section and what Abraham labels the first large section of a ternary form. The second section of this form begins at measure 183 at the tempo change marked â€Å"Moderato molto (rubato). † The clarinet sol o that follows (Fig. 2. 9) is a version of Figure 2. 7. Abraham states this version â€Å"recurs several times in the section as a species of refrain, holding it together. † One such reiteration is a rhythmic variant in the English horn at measure 256. (Fig. 2. 10) Figure 2. 9: Copland, El Salon Mexico, mm. 185-190, Clarinet 1 Figure 2. 10: Copland, El Salon Mexico, mm. 256-260, English horn EL SALON MEXICO  © Copyright 1939 The Aaron Copland Fund for Music, Inc. Copyright Renewed. Boosey Hawkes Inc, sole agent. Reprinted by permission. 27 At measure 211, melodic material in the solo clarinet is derived from â€Å"La Jesusita† and is then restated in the strings. Figure 2. 11: Copland, El Salon Mexico, mm. 211-214, Clarinet 1 EL SALON MEXICO  © Copyright 1939 The Aaron Copland Fund for Music, Inc. Copyright Renewed. Boosey Hawkes Inc, sole agent. Reprinted by permission. Abraham describes the remainder of the middle section consisting of rhythmic developmen t of this tune and material from the first section. Copland writes of the section â€Å"before the final climax I present the folk tunes simultaneously in their original keys and rhythms. The result is a kind of polytonality that achieves the frenetic whirl I had in mind before the end, when all is resolved with a plain unadorned triad. †49 The return of the main section occurs at measure 324. The material is very similar to what had been heard previously but with slightly altered keys. The piece ends with the fanfare-like material from the opening measures. 0 Manuscript materials and changes in rhythmic notation Copland himself noted that orchestras had a difficult time performing El Salon Mexico. â€Å"El Salon was not easy to perform; it presented rhythmic intricacies for the 49 50 Copland and Perlis, Copland, 246. Abraham, â€Å"Aaron Copland: El Salon Mexico. † 28 conductor and the players. †51 There are many places in the orchestral version where eighth no tes are beamed over a barline, dotted barlines and brackets are used to indicate an alteration of normal rhythmic stress, and more than one time signature is often indicated or simply implied by the placement of accents. The need for this kind of rhythmic notation can be observed upon examination of the manuscript materials in the Aaron Copland Collection cataloged at the Library of Congress. Description of the manuscript materials The initial â€Å"Sketches† of El Salon Mexico are cataloged at the Library of Congress in the Aaron Copland Collection as ARCO 28. 2. It appears that Copland began his initial sketches of in March of 1933, which is the earliest date noted on the manuscript. These sketches are in pencil, do not always follow the progression of the piece, and have large sections crossed out. They illustrate a working out of melodic material and experimentation with different meters. The â€Å"Piano Sketch† (ARCO 28-A) is dated 1934 on the front cover. T he date at the end of the manuscript notes it was completed in July, 1934. The Piano Sketch is a 21page manuscript that uses between two and four staves and progresses through the piece from beginning to end. Revisions were still being made as some sections are crossed out, but then continue on sequentially with revised material. Copland wrote most of the Piano Sketch initially in pen and his original, more complex concept of the meter and beaming appear in pen. Many of the meters and beaming that appear in Leonard Bernstein’s piano arrangements appear in the Piano Sketch. There are also marks in lead and blue pencil throughout this manuscript. Many of the pencil markings indicate instruments for 51 Copland and Perlis, Copland, 247. 29 orchestration or possible changes to the choice in meter. While the pencil indications of meter do not match the meters of the final published orchestral version exactly, they are certainly less complicated than the original metric indicatio ns that appear in pen and are a step closer to what was published in the orchestral score. There are three orchestral manuscript scores cataloged at the Library of Congress. One is cataloged as the â€Å"Rough Orchestral Score† (ARCO 28. 3) and notes â€Å"began Aug. 9, 1934. † The manuscript is in pencil, but is incomplete and only four pages in length. The second â€Å"Full Score (Draft)† (ARCO 28. 1) is a complete draft in pencil, but has no date indicated. The final â€Å"Full Score† (ARCO 28) is complete and mostly completed in ink with some red and blue pencil marks indicating rehearsal numbers and time signatures respectively. The date on the front cover is 1936. 2 There are several arrangements and other works that are significant to the study of changes in rhythmic notation in El Salon Mexico beyond the manuscript materials. These include Bernstein’s and Hindsley’s various arrangements of the work, as well as Copland’s 192 9 and 1955 versions of the Symphonic Ode for orchestra, which is discussed later in this chapter. The table below summarizes all relevant documents in chronological order. 52 El Salon Mexico manuscripts. Aaron Copland Collection. Library of Congress. 30 Year 1929 1933 1934 1934 c. 1934 1936 1939 1941 1943 1955 1966 1972 Document name Symphonic Ode (for orchestra) Sketches El Salon Mexico Piano Sketch El Salon Mexico Rough Orchestral Score El Salon Mexico Full Score (Draft) El Salon Mexico Full Score El Salon Mexico El Salon Mexico (for orchestra) El Salon Mexico (for piano solo) El Salon Mexico (for two pianos) Revised Symphonic Ode (for orchestra) 1st Version manusciript El Salon Mexico (for band) El Salon Mexico (for band) Composer/Arranger Other information Copland withdrawn, revised in 1955 Copland ARCO 28. 2 manuscript Copland ARCO 28-A manuscript Copland ARCO 28. manuscript Copland ARCO 28. 1 manuscript Copland ARCO 28 manuscript Copland published Copland/Bernstein publishe d Copland/Bernstein published Copland published Copland/Hindsley ARCO 28-D manuscript Copland/Hindsley published Table 2. 1: Summary of published scores and manuscripts Changes in Rhythmic Notation One of the reasons for the â€Å"rhythmic intricacies† that Copland mentioned is that he originally conceived of groupings of eighth notes in groups of twos and threes that would call for shifting irregular time signatures. 53 In the Piano Sketch these groupings occur starting on the beat. While time signatures do not appear in ink on the Piano Sketch manuscript for the measures in the figure below, the groupings and barlines appear to be the same as Leonard Bernstein’s arrangement for two pianos with the appropriate time signatures added. The one change in rhythm between the two examples is the eighth rest that appears at the end of the third measure. However, this rest does appear in the orchestral version. â€Å"Irregular† time signatures are defined for the pur poses of this study as meters that have uneven groupings of eighth notes. For example, a time signature of 7/8 could have an eighth note grouping of 2+2+3. This would be defined as â€Å"irregular. † Other meters that would be included in this definition would be 5/8, 8/8, and 10/8. These irregular meters are considered â€Å"shifting† when the time signatures rapidly change from measure to measure. If a regular meter (3/4, 4/4) appears during a string of irregular meters, the term â€Å"shifting irregular meters† still applies because the effect as a whole is a succession of changes between groups of 2 and 3 eighth notes. 1 53 Figure 2. 12: Copland, ARCO 28-A (Piano Sketch), Rhythmic notation, mm. 1-3 Figure 2. 13: Copland/Bernstein, El Salon Mexico for Two Pianos, Rhythmic notation, mm. 1-3 Figure 2. 14: Copland, El Salon Mexico, Rhythmic notation, mm. 1-5 EL SALON MEXICO  © Copyright 1939 The Aaron Copland Fund for Music, Inc. Copyright Renewed. Boosey Hawkes Inc, sole agent. Reprinted by permission. As the previous examples illustrate, there were drastic changes in meter between the Piano Sketch and the final orchestral version. These kinds of alterations to meter occur throughout the orchestral version and are particularly prevalent during sections with faster tempi, which tend to be more rhythmic. In most cases, the meters are changed from shifting irregular meters to a more constant regular meter of 3/4 or 4/4. This forces rhythms that originally occurred on the beat to be played as syncopations. 32 Figure 2. 15: Copland, El Salon Mexico, mm. 124-128, Violin 1 Figure 2. 16: Copland, ARCO 28-A (Piano Sketch), Corresponding music

The Louisiana Purchase, The Oregon Treaty, And The...

There were many important events that helped to achieve the goal of Manifest Destiny. For example, the Louisiana Purchase, the Oregon Treaty, and the California Gold Rush all helped achieve this goal. All of these events had either increased the amount of land in the United States, or increased the population of people living in Western United States. One event that occurred during the time of Manifest Destiny was the Louisiana Purchase. In the early 1800s, President Thomas Jefferson wanted to get control over the Mississippi River, where many farmers traded. He sent negotiators to France to offer to buy New Orleans. At that time, New Orleans was the main trading port on the Mississippi, whoever controlled New Orleans controlled the Mississippi and the amount of trade on the river. Jefferson told the negotiators to offer France $2 million but he said to go as high as $10 million. 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The native people were so easy to take

Vedic Mathematics Multiplication

Sample details Pages: 30 Words: 9097 Downloads: 4 Date added: 2017/06/26 Category Statistics Essay Did you like this example? Abstract Vedic Mathematics has been the rage in American schools. The clear difference between Asian Indians and average American students approach to solving math problems had been evident for many years, finally prompting concerted research efforts into the subject. Many students have conventionally found the processes of algebraic manipulation, especially factorisation, difficult to learn. Don’t waste time! Our writers will create an original "Vedic Mathematics Multiplication | Mathematics Dissertations" essay for you Create order Research studies have investigated the value of introducing students to a Vedic method of multiplication of numbers that is very visual in its application. The question was whether applying the method to quadratic expressions would improve student understanding, not only of the processes but also the concepts of expansion and factorisation. It was established that there was some evidence that this was the case, and that some students also preferred to use the new method. Introduction Is Vedic mathematics a kind of magic? American students certainly thought so, in seeing the clear edge it gave to their Asian counterparts in public and private schools. Vedic schools and even tuition centers are advertised on the Web. Clearly it has taken the world by storm, and for valid reasons. The results are evident in math scores for every test administered. Vedic mathematics is based on some ancient, but superb logic. And the truth is that it works. Small wonder that it hails from India, purported to be the land that gave us the Zero or cipher. This one digit is the basis for counting or carrying over beyond nine- and is in fact the basis of our whole number system. It is the Arabs and the Indians that we should be indebted to for this favour to the West. The other thing about Vedic mathematics is that it also allows one to counter check whether his or her answer is correct. Thus one is doubly assured of the results. Sometimes this can be done by the Indian student in a shorter time span than it can using the traditional counting and formulas we have developed through Western and European mathematicians. That makes it seem all the more marvellous. If that doesnt sound magical enough, its interesting to note that the word Vedic means coming from Vedas a Sanskrit word meaning divinely revealed. The Hindus believe that these basic truths were revealed to holy men directly once they had achieved a certain position on the path to spirituality. Also certain incantations such as Om are said to have been revealed by the Heavens themselves. According to popular beliefs, Vedic Mathematics is the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to him, all Mathematics is based on sixteen Sutras or word-formulas. Based on Vedic logic, these formulas solve the problem in the way the mind naturally works and are therefore a great help to the student of logic. Perhaps the most outstanding feature of the Vedic system is its coherence. The whole system is beautifully consistent and unified- the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots. Added to that, these are all simply understood. This unifying quality is very satisfying, as it makes learning mathematics easy and enjoyable. The Vedic system also provides for the solution of difficult problems in parts; they can then be combined to solve the whole problem by the Vedic method. These magical yet logical methods are but a part of the whole system of Vedic mathematics which is far more systematic than the modern Western system. In fact it is safe to say that Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, straight and easy. The ease of Vedic Mathematics means that calculations can be carried out mentally-though the methods can also be written down. There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one accurate method. This leads to more creative, fascinated and intelligent pupils. Interest in the Vedic system is increasing in education where mathematics teachers are looking for something better. Finding the Vedic system is the answer. Research is being carried out in many areas as well as the effects of learning Vedic Maths on children; developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, computing etc. But the real beauty and success of Vedic Mathematics cannot be fully appreciated without actually practising the system. One can then see that it is perhaps the most sophisticated and efficient mathematical system possible. Now having known that even the 16 sutras are the Jagadguru Sankaracharyas invention we mention the name of the sutras and the sub sutras or corollaries in this paper. The First Sutra: Ekdhikena Prvena The relevant Sutra reads Ekdhikena Prvena which rendered into English simply says By one more than the previous one. Its application and modus operandi are as follows. (1) The last digit of the denominator in this case being 1 and the previous one being 1 one more than the previous one evidently means 2. Further the proposition by (in the sutra) indicates that the arithmetical operation prescribed is either multiplication or division. Let us first deal with the case of a fraction say 1/19. 1/19 where denominator ends in 9. By the Vedic one line mental method. A. First method B. Second Method This is the whole working. And the modus operandi is explained below. Modus operandi chart is as follows: (i) We put down 1 as the right-hand most digit 1 (ii) We multiply that last digit 1 by 2 and put the 2 down as the immediately preceding digit. (iii) We multiply that 2 by 2 and put 4 down as the next previous digit. (iv) We multiply that 4 by 2 and put it down thus 8 4 2 1 (v) We multiply that 8 by 2 and get 16 as the product. But this has two digits. We therefore put the product. But this has two digits we therefore put the 6 down immediately to the left of the 8 and keep the 1 on hand to be carried over to the left at the next step (as we always do in all multiplication e.g. of 69 2 = 138 and so on). (vi) We now multiply 6 by 2 get 12 as product, add thereto the 1 (kept to be carried over from the right at the last step), get 13 as the consolidated product, put the 3 down and keep the 1 on hand for carrying over to the left at the next step. (vii) We then multiply 3 by 2 add the one carried over from the right one, get 7 as the consolidated product. But as this is a single digit number with nothing to carry over to the left, we put it down as our next multiplicand. (viii) and xviii) we follow this procedure continually until we reach the 18th digit counting leftwards from the right, when we find that the whole decimal has begun to repeat itself. We therefore put up the usual recurring marks (dots) on the first and the last digit of the answer (from betokening that the whole of it is a Recurring Decimal) and stop the multiplication there. Our chart now reads as follows: The Second Sutra: Nikhilam Navatacaramam Daatah Now we proceed on to the next sutra Nikhilam sutra The sutra reads Nikhilam Navatacaramam Daatah, which literally translated means: all from 9 and the last from 10. We shall and applications of this cryptical-sounding formula and then give details about the three corollaries. He has given a very simple multiplication. Suppose we have to multiply 9 by 7. 1. We should take, as base for our calculations that power of 10 which is nearest to the numbers to be multiplied. In this case 10 itself is that power. Put the numbers 9 and 7 above and below on the left hand side (as shown in the working alongside here on the right hand side margin); 3. Subtract each of them from the base (10) and write down the remainders (1 and 3) on the right hand side with a connecting minus sign () between them, to show that the numbers to be multiplied are both of them less than 10. 4. The product will have two parts, one on the left side and one on the right. A vertical dividing line may be drawn for the purpose of demarcation of the two parts. 5. Now, Subtract the base 10 from the sum of the given numbers (9 and 7 i.e. 16). And put (16 10) i.e. 6 as the left hand part of the answer 9 + 7 10 = 6 The First Corollary The first corollary naturally arising out of the Nikhilam Sutra reads in English whatever the extent of its deficiency lessen it still further to that very extent, and also set up the square of that deficiency. This evidently deals with the squaring of the numbers. A few elementary examples will suffice to make its meaning and application clear: Suppose one wants to square 9, the following are the successive stages in our mental working. (i) We would take up the nearest power of 10, i.e. 10 itself as our base. (ii) As 9 is 1 less than 10 we should decrease it still further by 1 and set 8 down as our left side portion of the answer 8/ (iii) And on the right hand we put down the square of that deficiency 12 (iv) Thus 92 = 81 The Second Corollary The second corollary in applicable only to a special case under the first corollary i.e. the squaring of numbers ending in 5 and other cognate numbers. Its wording is exactly the same as that of the sutra which we used at the outset for the conversion of vulgar fractions into their recurring decimal equivalents. The sutra now takes a totally different meaning and in fact relates to a wholly different setup and context. Its literal meaning is the same as before (i.e. by one more than the previous one) but it now relates to the squaring of numbers ending in 5. For example we want to multiply 15. Here the last digit is 5 and the previous one is 1. So one more than that is 2. Now sutra in this context tells us to multiply the previous digit by one more than itself i.e. by 2. So the left hand side digit is 1 2 and the right hand side is the vertical multiplication product i.e. 25 as usual. Thus 152 = 1 2 / 25 = 2 / 25. Now we proceed on to give the third corollary. The Third Corollary Then comes the third corollary to the Nikhilam sutra which relates to a very special type of multiplication and which is not frequently in requisition elsewhere but is often required in mathematical astronomy etc. It relates to and provides for multiplications where the multiplier digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows: i) Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product; or take the Ekanyuna and subtract therefrom the previous i.e. the excess portion on the left; and ii) Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product. The following example will make it clear: The Third Sutra: rdhva Tiryagbhym rdhva Tiryagbhym sutra which is the General Formula applicable to all cases of multiplication and will also be found very useful later on in the division of a large number by another large number. The formula itself is very short and terse, consisting of only one compound word and means vertically and cross-wise. The applications of this brief and terse sutra are manifold. A simple example will suffice to clarify the modus operandi thereof. Suppose we have to multiply 12 by 13. (i) We multiply the left hand most digit 1 of the multiplicand vertically by the left hand most digit 1 of the multiplier get their product 1 and set down as the left hand most part of the answer; (ii) We then multiply 1 and 3 and 1 and 2 crosswise add the two get 5 as the sum and set it down as the middle part of the answer; and (iii) We multiply 2 and 3 vertically get 6 as their product and put it down as the last the right hand most part of the answer. Thus 12 13 = 156. The Fourth Sutra: Parvartya Yojayet The term Parvartya Yojayet which means Transpose and Apply. Here he claims that the Vedic system gave a number is applications one of which is discussed here. The very acceptance of the existence of polynomials and the consequent remainder theorem during the Vedic times is a big question so we dont wish to give this application to those polynomials. However the four steps given by them in the polynomial division are given below: Divide x3 + 72 + 6x + 5 by x 2. i. x3 divided by x gives us x2 which is therefore the first term of the quotient x2 2 = 22 but we have 72 in the divident. This means that we have to get 92 more. This must result from the multiplication of x by 9x. Hence the 2nd term of the divisor must be 9x As for the third term we already have 2 9x = 18x. But we have 6x in the dividend. We must therefore get an additional 24x. Thus can only come in by the multiplication of x by 24. This is the third term of the quotient. Q = x2 + 9x + 24 Now the last term of the quotient multiplied by 2 gives us 48. But the absolute term in the dividend is 5. We have therefore to get an additional 53 from some where. But there is no further term left in the dividend. This means that the 53 will remain as the remainder Q = x2 + 9x + 24 and R = 53. The Fifth Sutra: Snyam Samyasamuccaye Samuccaya is a technical term which has several meanings in different contexts which we shall explain one at a time. Samuccaya firstly means a term which occurs as a common factor in all the terms concerned. Samuccaya secondly means the product of independent terms. Samuccaya thirdly means the sum of the denominators of two fractions having same numerical numerator. Fourthly Samuccaya means combination or total. Fifth meaning: With the same meaning i.e. total of the word (Samuccaya) there is a fifth kind of application possible with quadratic equations. Sixth meaning With the same sense (total of the word Samuccaya) but in a different application it comes in handy to solve harder equations equated to zero. Thus one has to imagine how the six shades of meanings have been perceived by the Jagadguru Sankaracharya that too from the Vedas when such types of equations had not even been invented in the world at that point of time. The Sixth Sutra: nurpye nyamanyat As said by Dani [32] we see the 6th sutra happens to be the subsutra of the first sutra. Its mention is made in {pp. 51, 74, 249 and 286 of [51]}. The two small subsutras (i) Anurpyena and (ii) Adayamadyenantyamantyena of the sutras 1 and 3 which mean proportionately and the first by the first and the last by the last. Here the later subsutra acquires a new and beautiful double application and significance. It works out as follows: i. Split the middle coefficient into two such parts so that the ratio of the first coefficient to the first part is the same as the ratio of that second part to the last coefficient. Thus in the quadratic 22 + 5x + 2 the middle term 5 is split into two such parts 4 and 1 so that the ratio of the first coefficient to the first part of the middle coefficient i.e. 2 : 4 and the ratio of the second part to the last coefficient i.e. 1 : 2 are the same. Now this ratio i.e. x + 2 is one factor. ii. And the second factor is obtained by dividing the first coefficient of the quadratic by the first coefficient of the factor already found and the last coefficient of the quadratic by the last coefficient of that factor. In other words the second binomial factor is obtained thus Thus 22 + 5x + 2 = (x + 2) (2x + 1). This sutra has Yavadunam Tavadunam to be its subsutra which the book claims to have been used. The Seventh Sutra: Sankalana Vyavakalanbhym Sankalana Vyavakalan process and the Adyamadya rule together from the seventh sutra. The procedure adopted is one of alternate destruction of the highest and the lowest powers by a suitable multiplication of the coefficients and the addition or subtraction of the multiples. A concrete example will elucidate the process. Suppose we have to find the HCF (Highest Common factor) of (x2 + 7x + 6) and x2 5x 6 x2 + 7x + 6 = (x + 1) (x + 6) and x2 5x 6 = (x + 1) ( x 6) the HCF is x + 1 but where the sutra is deployed is not clear. The Eight Sutra: Puranpuranbhym Puranpuranbhym means by the completion or not completion of the square or the cube or forth power etc. But when the very existence of polynomials, quadratic equations etc. was not defined it is a miracle the Jagadguru could contemplate of the completion of squares (quadratic) cubic and forth degree equation. This has a subsutra Antyayor dasakepi use of which is not mentioned in that section. The Ninth Sutra: Calan kalanbhym The term (Calan kalanbhym) means differential calculus according to Jagadguru Sankaracharya. The Tenth Sutra: Yvadnam Yvadnam Sutra (for cubing) is the tenth sutra. It has a subsutra called Samuccayagunitah. The Eleventh Sutra: Vyastisamastih Sutra Vyastisamastih sutra teaches one how to use the average or exact middle binomial for breaking the biquadratic down into a simple quadratic by the easy device of mutual cancellations of the odd powers. However the modus operandi is missing. The Twelfth Sutra: esnyankena Caramena The sutra esnyankena Caramena means The remainders by the last digit. For instance if one wants to find decimal value of 1/7. The remainders are 3, 2, 6, 4, 5 and 1. Multiplied by 7 these remainders give successively 21, 14, 42, 28, 35 and 7. Ignoring the left hand side digits we simply put down the last digit of each product and we get 1/7 = .14 28 57! Now this 12th sutra has a subsutra Vilokanam. Vilokanam means mere observation He has given a few trivial examples for the same. The Thirteen Sutra: Sopantyadvayamantyam The sutra Sopantyadvayamantyam means the ultimate and twice the penultimate which gives the answer immediately. No mention is made about the immediate subsutra. The illustration given by them. The proof of this is as follows. The General Algebraic Proof is as follows. Let d be the common difference Canceling the factors A (A + d) of the denominators and d of the numerators: It is a pity that all samples given by the book form a special pattern. The Fourteenth Sutra: Ekanynena Prvena The Ekanynena Prvena Sutra sounds as if it were the converse of the Ekadhika Sutra. It actually relates and provides for multiplications where the multiplier the digits consists entirely of nines. The procedure applicable in this case is therefore evidently as follows. For instance 43 9. i. Divide the multiplicand off by a vertical line into a right hand portion consisting of as many digits as the multiplier; and subtract from the multiplicand one more than the whole excess portion on the left. This gives us the left hand side portion of the product or take the Ekanyuna and subtract it from the previous i.e. the excess portion on the left and ii. Subtract the right hand side part of the multiplicand by the Nikhilam rule. This will give you the right hand side of the product The Fifthteen Sutra: Gunitasamuccayah Gunitasamuccayah rule i.e. the principle already explained with regard to the Sc of the product being the same as the product of the Sc of the factors. Let us take a concrete example and see how this method (p. 81) can be made use of. Suppose we have to factorize x3 + 62 + 11x + 6 and by some method, we know (x + 1) to be a factor. We first use the corollary of the 3rd sutra viz. Adayamadyena formula and thus mechanically put down x2 and 6 as the first and the last coefficients in the quotient; i.e. the product of the remaining two binomial factors. But we know already that the Sc of the given expression is 24 and as the Sc of (x + 1) = 2 we therefore know that the Sc of the quotient must be 12. And as the first and the last digits thereof are already known to be 1 and 6, their total is 7. And therefore the middle term must be 12 7 = 5. So, the quotient x2 + 5x + 6. This is a very simple and easy but absolutely certain and effective process. The Sixteen Sutra :Gunakasamuccayah. It means the product of the sum of the coefficients in the factors is equal to the sum of the coefficients in the product. In symbols we may put this principle as follows: Sc of the product = Product of the Sc (in factors). For example (x + 7) (x + 9) = x2 + 16 x + 63 and we observe (1 + 7) (1 + 9) = 1 + 16 + 63 = 80. Similarly in the case of cubics, biquadratics etc. the same rule holds good. For example (x + 1) (x + 2) (x + 3) = x3 + 62 + 11 x + 6 2 3 4 = 1 + 6 + 11 + 6 = 24. Thus if and when some factors are known this rule helps us to fill in the gaps. Literature Research has documented the difficulties students face in algebra and how these can often be traced to their limited understanding of numbers and their operations (Stacey MacGregor, 1997; Warren, 2001). Of growing concern is the artificial separation of algebra and arithmetic, since knowledge of mathematical structure seems essential for a successful transition. In particular, this mathematical structure is concerned with (i) relationships between quantities, (ii) group properties of operations, (iii) relationships between the operations and (iv) Relationships across the quantities (Warren, 2003). Thus it has been suggested by Stacey and MacGregor (1997) that the best preparation for learning algebra is a good understanding of how the arithmetic system works. An understanding of the general properties of numbers and the relationships between them may be crucial, and students need to have thought about the general effects of operations on numbers (MacGregor Stacey, 1999). This study sought to test the hypothesis that arithmetic knowledge can improve algebraic ability by applying a Vedic method of multiplying arithmetic numbers to algebra, based on the similarity of structural presentation. Vedic mathematics has its origins in the ancient Indian texts, the Vedas, an integrated and holistic system of knowledge composed in Sanskrit and transmitted orally from one generation to the next. The first versions of these texts were possibly recorded around 2000 BC, and the works contain the genesis of the modern science of mathematics (number, geometry and algebra) and astronomy in India (Datta Singh, 2001; Joseph, 2000). Sri Tirthaji (1965) has expounded 16 sutras or word formulas and 13 sub-sutras that he claims have been reconstructed from the Vedas. The sutras, or rules as aphorisms, are condensed statements of a very precise nature, written in a poetic style and dealing with different concepts (Joseph, 2000; Shan Bailey, 1991). A sutra, which literally means thread, expresses fundamental principles and may contain a rule, an idea, a mnemonic or a method of working based on fundamental principles that run like threads through diverse mathematical topics, unifying them. As Williams (2002) describes them: We use our mind in certain specific ways: we might extend an idea or reverse it or compare or combine it with another. Each of these types of mental activity is described by one of the Vedic sutras. They describe the ways in which the mind can work and so they tell the student how to go about solving a problem. (Williams, 2002, p. 2). Examples of the sutras are the Vertically and Crosswise sutra, which embodies a method of multiplication with applications to determinants, simultaneous equations, and trigonometric functions, etc. (this is the sutra used in the research reported here see Figure 3), and the All from nine and the last from ten sutra that may be used in subtraction, vincula, multiplication and division. Barnard and Tall (1997, p. 41) have introduced the idea of a cognitive unit, A piece of cognitive structure that can be held in the focus of attention all at one time, and may include other ideas that can be immediately linked to it. This enables compression of ideas, so that a collection of ideas or symbols that is too big for the focus of attention can be compressed into a single unit. It seems as if the sutras nicely fit this description, with the mnemonic or other memory device being used as a peg to hang the collection of ideas on. Thus the theoretical advantage of using the sutras is that they allow encapsulation of a process into a manageable chunk, or cognitive unit, that can then be processed more easily, sometimes using a visual reminder, such as in the Vertically and Crosswise sutra. Here the essential procedure is signified holistically by the symbol 5, unlike the symbol FOIL that signifies in turn four separate procedures. It might be possible for a symbol such as to be used in much the same way for FOIL, but this may appear more visually complex, and it is not usually separated from the accompanying binomials like this. In this way sutras often make use of the power of visualisation, which has been shown to be effective in learning in various areas of mathematics (Booth Thomas, 2000; Presmeg, 1986; van Hiele, 2002). Such visualisation accesses the brains holistic activity (Tall Thomas, 1991) and intuition, and this assists in providing an overview of the mathematical structure. The sutras also aid intuitive thinking (Williams, 2002) and being based on patterns and mnemonics they make recall much easier, reducing the cognitive load on the individual (Morrow, 1998; Sweller, 1994). The sutras were originally envisaged as applying both to arithmetic and algebra, and Joseph (2000) and Bhatanagar (1976) have explained that since polynomials may be perceived as simply arithmetic sequences, the principles apply equally well to them. This research considered a possible role of the vertically and Crosswise sutra for improving facility with, and understanding of, the expansion of algebraic binomials and the factorisation of quadratic expressions. Methodology The research employed a case study methodology, using a single class of Year 10 (age 15 years) students. The school used is a co-educational state secondary school in Auckland, New Zealand and the class contained 19 students, 11 boys and 8 girls. The students, who included 9 recent immigrants, were drawn from several cultural backgrounds, and accordingly have been exposed to different approaches and teaching environments with respect to learning mathematics. This also meant that nine of the students have a first language other than English and these language difficulties tend to hinder their learning (for example, three of the students are on a literacy program at the school). Two anonymous questionnaires (see Figure 1 for some questions from the second) were constructed using concepts we identified as important in developing a structural understanding of binomial expansion and factorisation, such as testing the concept of a factor and the ability to apply a procedure in reverse. Questions included: multiplication of numbers; multiplication of binomial expressions; factorisation of quadratic expressions; word problems on addition and subtraction of like terms; and expansion of expressions in a practical context. Some questions also involved description of procedures and meanings attached to words. In particular, the second questionnaire contained items on the use of the Vedic method applied to binomial expansion and factorisation. The lessons were taught by the first-named author in 2003 in a supportive classroom environment that encouraged student-to-student and teacherstudent interactions. Students were assured that the teacher was genuinely interested in their mathematical thinking and respected their attempts, that it was fine to make mistakes and that understanding how the mistake occurred was a learning opportunity for everyone concerned. Students were encouraged to explain and check the validity of their answers, and positive contributions were praised. The first teaching session comprised work on multiplication of numbers and revision of work on algebra that the students had learned in Year 9 (age 14 years). Substitution, collection of like terms and multiplication of a binomial expression by a single value were revised, using, for example, expressions such as 5(x 4), (p + 2)4, and k(4 + k). Students were also reminded of the meanings of words such as term, expression, factor, expansion, coefficient and simplify. Diagrammatic representations of 3(5 + 6) and k(4 + k) using rectangles were drawn and discussed, and then students drew similar rectangle diagrams representing multiplications such as (3 + 5)(2 + 5) and (k + 2)(k + 4) (see Figure 2). Following a review of factorisation of expressions such as 15p + 10, the FOIL (First, Outside, Inside, Last) method of expanding binomials was taught, where the First terms in each bracket are multiplied together, then the Outside terms, the Inside terms and then the Last term in each bracket, to give four products. Finally factorisation of quadratic expressions, followed by a guess and check method for factorising quadratic expressions was covered. The students did not find these topics easy, especially factorising of quadratic expressions. This took a total of four hours, after which, questionnaire one was administered. Students were then exposed for one hour to the Vedic vertically and crosswise method, where initially they practised multiplying two- and three-digit numbers with this approach. Subsequently, the next three hours were spent expanding binomials and factorising quadratic expressions with the vertically and crosswise method. This method (see Figure 3) involves a sequence of four multiplications, the answers to each of which are placed into a single answer line. The middle two terms are added together mentally to supply the final answer. Results The first question (1a) in each questionnaire was a two-digit multiplication. In the first, it was 37 58, and the second 23 47, and in this second case the question asked that this be done by the vertically and crosswise method. The aim was to check students facility with arithmetic multiplication and to see if the vertically and crosswise sutra was of assistance in this area. In the event 11 of the 18 (61%) students who completed both questionnaires, correctly answered the question in the first test, and 13 (72%) in the second, with only one student not using the sutra in that test. There was no statistical difference between these proportions (c2 = 0.125). When asked to explain what they had done using the Vedic approach, students who were able to write down the final answer were often able to write something like student 4s explanation for 1c), 32 69: 2 times 9 is 18 3 times 9 + 2 times 6 is 39 + carried 1 = 40 3 times 6 + carried 4 = 22 Expansion of binomials A summary of the results in the first of the algebra questions (Q2 see Figure 1 for format), requiring students to multiply together two basic binomials, is given in Table 1. Question 2 was presented in each case with the binomials in the same line. Since the first questionnaire was administered immediately after the FOIL method was taught, most students employed this method, and only 8 (44.4%) correctly performed the expansion in the simplest example, question 2a). In the second questionnaire, where students were asked to use the vertically and Crosswise method, 10 (55.6%) were correct, but there was no statistically significant difference (c2 = 0.11, df = 1, ns). However, when the presentation format was changed from a single line to the grid format of question 3 (see Figure 1), the facility on 3a) (Multiply x + 3 and x + 4) improved to 14 correct answers (77.8%). Thus there was weak evidence (c2 = 2.92, df = 1, p = 0.1) of an improvement in students performance on this basic expansion of binomials in the second questionnaire, using the Vedic method with a grid presentation. However, the improvement was not sustained for question 3b) which had negative signs (see Table 2: c2 = 0.45), or for 2b), which involved 2x and negative signs. It seems that the arithmetic complexity caused problems, with only 3 students answering 2b) correctly on each test. The other results of question 3 on the second questionnaire also support the idea of improved understanding; Table 2 gives the results of the multiplications in this question. In this case question 3c) was the question of a standard corresponding to 2b) in the first test (2x 3 times x + 4), and yet the students did significantly better (50% correct) than they did on that question (c2 = 3.13, df = 1, p 0.05). It is also of interest that on question 2 in the first test all eight students who successfully applied the FOIL method wrote out all four terms and then added the two middle terms. No student wrote the final answer without the intermediate step of giving both middle terms. In contrast, when using the vertically and crosswise approach, all 14 students who were successful were able to write the answers straight down, as seen in the example in Figure 4. Furthermore, 5 students (12, 15, 16, 17, and 18) who were unable to multiply binomials using the standard method achieved success using the Vedic method. Figure 5 shows the corresponding working of student 18 on these questions. In test 1, following the FOIL method she was unable to multiply any terms together and even seemed to confuse the question with factorisation. However, with the Vedic method she makes a good attempt, correctly answering two parts. It might be argued that the improvement above was due to students spending more time learning how to perform such expansions, or that the previous FOIL learning took time to assimilate. However, it should be noted that the questions where the improvement occurred specifically involved use of the Vedic method, which, while it can be related to FOIL by an experienced mathematician, would no doubt look quite different to these students, since it is set out in a grid format rather than being performed in a single line. Furthermore, it was noted during the teaching episode that there was some resistance from the students to learning a second method when they already knew the FOIL method, and this could be expected to have a detrimental effect on performance. Factorisation of quadratic expressions Table 3 contains a summary of the results of question 3 from the first questionnaire and question 5 in the second, these being corresponding single line, traditional format factorisation questions (see Figure 1). Individually the results of these questions did not show any statistical difference between the performance before and after the Vedic method was introduced. For example, between questions 3a) and 5a), c2 = 0.9, which is not significant. However, if the question parts are grouped together and the number of students correct on question 3 compared with those correct on question 5 then we find that there is a significant improvement (Q3 v Q5 c2 = 6.65, p 0.01) on the second questionnaire. For question 4 in each questionnaire the students were asked to supply the missing terms in two binomials that were multiplied together. Boxes were provided for the missing terms, which were in the binomials, the quadratic, or both (see Figure 1). While these are not standard straight factorisation questions they do require students to work backwards from the answer and thus display some conceptual knowledge of how to undo multiplication of binomials. There was no difference in facility on any of these matching questions (see Table 4), or on a comparison of the total number of correct scores between the two questionnaires (c2 = 0.39, ns). The students appeared to be able to do them equally well using either guess or check and decomposition or the Vedic approach. While the discussion above shows that the evidence for a better performance on individual questions following the teaching of the Vedic method was rarely present, a consideration of the students overall scores on the expansion and factorisation algebra questions did show a significantly better performance on the second test, (_x1 = 41.4%,_x2 = 51.5%, t = 2.66, p 0.05). Thus it appears that, overall, the Vedic approach may have contributed to student understanding of the methods, either by cementing in place the previous methods, or by complementing them. It is worth noting too that some students preferred to use the vertically and Crosswise method even when not directed to do so. For example, in question 2 of the second questionnaire students were simply asked to multiply two binomials together, with no method specified. In the event 4 students (4, 5, 16, and 18) chose to use the vertically and Crosswise approach, setting out their work in a grid. In addition, 3 of these students (15, 16 and 18) used it for the factorisation in question 5. An example of their work is shown in Figure 6. While they used the method with varying levels of success, it seems to have benefited both student 5, who answered no algebra questions correctly in test 1 (and 2 in test 2), and student 16 who used it for factorisation and went from 3 correct to 8 correct (see Figure 6). While student 18 preferred the Vedic method to expand and to factorise expressions, she used the traditional method to multiply numbers even when asked to multiply by the vertically and Crosswise method. This seems to suggest that she was comfortable using different methods in algebra from those employed in arithmetic. Discussion India has good reasons to be proud of a rich heritage in science, philosophy and culture in general, coming to us down the ages. In mathematics, which is my own area of specialization, the ancient Indians not only took great strides long before the Greek advent, which is a standard reference point in the Western historical perspective, but also enriched it for a long period making in particular some very fundamental contributions such as the place-value system for writing numbers as we have today, introduction of zero and so on. Further, the sustained development of mathematics in India in the post-Greek period was indirectly instrumental in the revival in Europe after its dark ages. Notwithstanding the enviable background, lack of adequate attention to academic pursuits over a prolonged period, occasioned by several factors, together with about two centuries of Macaulayan educational system, has unfortunately resulted, on the one hand, in a lack of awareness of our historical role in actual terms and, on the other, an empty sense of pride which is more of an emotional reaction to the colonial domination rather than an intellectual challenge. Together they provide a convenient ground for extremist and misguided elements in society to reconstruct history from nonexistent or concocted source material to whip up popular euphoria. That this anti-intellectual endeavour is counter-productive in the long run and, more important, harmful to our image as a mature society, is either not recognized or ignored in favour of short-term considerations. Along with the obvious need to accelerate the process of creating an awareness of our past achievements, on the strength of authentic information, a more urgent need has also arisen to confront and expose such baseless constructs before it is too late. This is not merely a question of setting the record straight. The motivated versions have a way of corrupting the intellectual processes in society and weakening their very foundations in the long run, which needs to be prevented at all costs. The so-called Vedic Mathematics is a case in point. A book by that name written by Jagadguru Swami Shri Bharati Krishna Tirthaji Maharaja (Tirthaji, 1965) is at the centre of this pursuit, which has now acquired wide following; Tirthaji was the Shankaracharya of Govardhan Math, Puri, from 1925 until he passed away in 1960. The book was published posthumously, but he had been carrying out a campaign on the theme for a long time, apparently for several decades, by means of lectures, blackboard demonstrations, classes and so on. It has been known from the beginning that there is no evidence of the contents of the book being of Vedic origin; the Foreword to the book by the General Editor, Dr. A.S.Agrawala, and an account of the genesis of the work written by Manjula Trivedi, a disciple of the swamiji, makes this clear even before one gets to the text of the book. No one has come up with any positive evidence subsequently either. There has, however, been a persistent propaganda that the material is from the Vedas. In the face of a false sense of national pride associated with it and the neglect, on the part of the knowledgeable, in countering the propaganda, even educated and well meaning people have tended to accept it uncritically. The vested interests have also involved politicians in the propaganda process to gain state support. Several leaders have lent support to the Vedic Mathematics over the years, evidently in the belief of its being from ancient scriptures. In the current environment, when a label as ancient seems to carry considerable premium irrespective of its authenticity or merit, the purveyors would have it going easy. Large sums have been spent both by the Government and several private agencies to support this Vedic Mathematics, while authentic Vedic studies continue to be neglected. People, especially children, are encouraged to learn and spread the contents of the book, largely on the baseless premise of their being from the Vedas. With missionary zeal several devotees of this cause have striven to take the message around the world; not surprisingly, they have even met with some success in the West, not unlike some of the gurus and yogis peddling their own versions of Indian philosophy. Several people are also engaged in research in the new Vedic Mathematics. To top it all, when in the early nineties the Uttar Pradesh Government introduced Vedic Mathematics in school text books, the contents of the swamijis book were treated as if they were genuinely from the Vedas; this also naturally seems to have led them to include a list of the swamijis sutras on one of the opening pages (presumably for the students to learn them by heart and recite!) and to accord the swamiji a place of honour in the brief history of Indian mathematics described in the beginning of the textbook, together with a chart, which curiously has Srinivasa Ramanujans as the only other name from the twentieth century! For all their concern to inculcate a sense of national pride in children, those responsible for this have not cared for the simple fact that modern India has also produced several notable mathematicians and built a worthwhile edifice in mathematics (as also in many other areas). Harish Chandras work is held in great esteem all over the world and several leading seats of learning of our times pride themselves in having members pursuing his ideas; (see, for instance, Langlands, 1993). Even among those based in India, several like Syamdas Mukhopadhyay, Ganesh Prasad, B.N.Prasad, K.Anand Rau, T.Vijayaraghavan, S.S.Pillai, S.Minakshisundaram, Hansraj Gupta, K.G.Ramanathan, B.S.Madhava Rao, V.V.Narlikar, P.L.Bhatnagar and so on and also many living Indian mathematicians have carved a niche for themselves on the international mathematical scene (see Narasimhan, 1991). Ignoring all this while introducing the swamijis name in the brief history would inevitably create a warped perspective in childrens minds, favouring gimmickry rather than professional work. What does the swamijis Vedic Mathematics seek to do and what does it achieve? In his preface of the book, grandly titled A Descriptive Prefatory Note on the astounding Wonders of Ancient Indian Vedic Mathematics, the swamiji tells us that he strove from his childhood to study the Vedas critically to prove to ourselves (and to others) the correctness (or otherwise)of the derivational meaning of Veda that the Vedas should contain within themselves all the knowledge needed by the mankind relating not only to spiritual matters but also those usually described as purely secular, temporal or worldly; in other words, simply because of the meaning of the word Veda, everything that is worth knowing is expected to be contained in the vedas and the swamiji seeks to prove it to be the case! It may be worthwh ile to point out here that there would be room for starting such an enterprise with the word science! He also describes how the contemptuous or at best patronising attitude of Orientalists, Indologists and so on strengthened his determination to unravel the too-long-hidden mysteries of philosophy and science contained in ancient Indias Vedic lore, with the consequence that, after eight years of concentrated contemplation in forest solitude, we were at long last able to recover the long lost keys which alone could unlock the portals thereof. The mindset revealed in this can hardly be said to be suitable in scientific and objective inquiry or pursuit of knowledge, but perhaps one should not grudge it in someone from a totally different milieu, if the outcome is positive. One would have thought that with all the commitment and grit the author would have come up with at least a few new things which can be attributed to the Vedas, with solid evidence. This would have made a worthwhile contribution to our understanding of our heritage. Instead, all said and done there is only the authors certificate that we were agreeably astonished and intensely gratified to find that exceedingly though mathematical problems can be easily and readily solved with the help of these ultra-easy Vedic sutras (or mathematical aphorisms) contained in the Parishishta (the appendix portion) of the Atharva Veda in a few simple steps and by methods which can be conscientiously described as mere mental arithmetic (paragraph 9 in the preface). That passing reference to the Atharva Veda is all that is ever said by way of source material for the contents. The sutras, incidentally, which appeared later, scattered in the book, are short phrases of just about two to four words in Sanskrit, such as Ekadhikena Purvena or Anurupye Shunyam Anyat. (There are 16 of them and in addition there are 13 of what are called sub-sutras, similar in nature to the sutras). The mathematics of today concerns a great variety of objects beyond the high school level, involving various kinds of abstract objects generalising numbers, shapes, geometries, measures and so on and several combinations of such structures, various kinds of operations, often involving infinitely many entities; this is not the case only about the frontiers of mathematics but a whole lot of it, including many topics applied in physics, engineering, medicine, finance and various other subjects. Despite its entire pretentious verbiage page after page, the swamijis book offers nothing worthwhile in advanced mathematics whether concretely or by way of insight. Modern mathematics with its multitude of disciplines (group theory, topology, algebraic geometry, harmonic analysis, ergodic theory, combinatorial mathematics-to name just a few) would be a long way from the level of the swamijis book. There are occasionally reports of some researchers applying the swamijis Vedic Mathematics to advanced problems such as Keplers problem, but such work involves nothing more than tinkering superficially with the topic, in the manner of the swamijis treatment of calculus, and offers nothing of interest to professionals in the area. Even at the western teaching Vedic Mathematics deals only with a small part and, more importantly, there too it concerns itself with only one particular aspect, that of faster computation. One of the main aims of mathematics education even at the western teaching consists of developing familiarity with a variety of concepts and their significance. Not only does the approach of Vedic Mathematics not contributes anything towards this crucial objective, but in fact might work to its detriment, because of the undue emphasis laid on faster computation. The swamijis assertion 8 months (or 12 months) at an average rate of 2 or 3 hours per day should be enough for finishing the whole course of mathematical studies on these Vedic lines instead of 15 or 20 years required according to the existing systems of the Indian and also foreign universities, is patently absurd and hopefully nobody takes it seriously, even among the activists in the area. It would work as a cruel joke if some people choose to make such a substitution in respect of their children. It is often claimed that Vedic Mathematics is well appreciated in other countries, and even taught in some schools in UK etc. In the normal course one would not have the means to examine such claims, especially since few details are generally supplied while making the claims. Recent years have brought about the development of powerful tools for verifying specifications of hardware and software systems. By now, major companies, such as Intel, IBM, and [emailprotected]/* */ Molecules have realized the impact and importance of such tools in their own design and implementation processes as a means of coping with the ever-increasing complexity of chip and computing tools. Protocols, networks, and distributed systems can generally not be described by code of some deterministic programming language. Such systems exhibit concurrent behavior and they are typically reactive in the sense that their behavior depends on what the environment can offer (e.g. Is the printer busy?). Computation tree logic (CTL) is currently one of the popular frameworks used in verifying properties of concurrent systems. We study its syntax and semantics, and use those insights to design an automated verification algorithm which takes a description of a system and specifications of expected behavior as input and checks whether that system meets those expectations. That algorithm is the foundation for a tool, the symbolic model verifier (SMV), which they use to evaluate some basic designs, e.g. simple elevator systems and a mutual exclusion protocol (3-4 weeks). Exposure to a labeling algorithm for finite-state verification illustrates depth-first backwards search in a directed graph; this search is recursive and the recursion is driven by the logical structure of the specified behavior, written as a CTL formula. The description and evaluation of small designs with the tool SMV makes students appreciate how such graphs can be modeled with a modular guarded-command language with non-deterministic assignment. The discussion of program logic contains the linear algorithm for computing minimal-sum sections of integer arrays as a case study. Finally, binary decision diagrams require algorithms that implement the familiar logical operations on such diagrams. Some of these algorithms illustrate dynamic programming at an accessible level. Conclusion In this study students were taught an appropriate Vedic sutra following teaching of the traditional FOIL method of multiplication of binomials, and the decomposition method for factorisation. We found that afterwards the students performed significantly better overall on these types of algebra questions, and specifically on the factorisations, and there was weak evidence of better results on expansion using a grid format. The reasons for the improvement are not easy to pinpoint since they appear in some areas and not in others. This seems to indicate that the value of the method may lie in what it adds to the students overall algebraic conceptions and knowledge of mathematical structure. Thus we have found no evidence that it should be seen as a replacement for the former approaches, but our results suggest it could rather be recommended as a useful adjunct, a complementary method. While this may take longer in terms of teaching time, the results indicate that possession of a range of strategies may have value above and beyond their individual benefit. References Barnard, T. Tall, D. O. (1997). Cognitive units, connections and mathematical proof. In E. Pekhonen (Ed.), Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education, Vol. 2 (pp. 4148). Lahti, Finland. Bhatnagar, P. L. (1976). Some Vedic Sutras for arithmetical operations or a fresh look at arithmetic after learning some algebra. Mathematics Teacher (India) 11A, 85120. Booth, R. D. L. Thomas, M. O. J. (2000). Visualisation in mathematics learning: Arithmetic problem-solving and student difficulties. Journal of Mathematical Behavior, 18(2), 169190. Datta, B. B. Singh, A. N. (2001). History of Hindu Mathematics: A Source Book Parts I and II. New Delhi: Bharatiya Kala Prakashan. Joseph, G. G. (2000). The Crest of the Peacock: Non-European Roots of Mathematics. Harmondsworth: Penguin. MacGregor, M. Stacey, K. (1999). A flying start to algebra. Teaching Children Mathematics, 6(2), 7885. Morrow, L. J. (1998). Wither algorithms? Mathematics educators express their views. In L. J. Morrow M. J. Kenney (Eds), The Teaching and Learning of Algorithms in School Mathematics (pp. 119), National Council of Teachers of Mathematics. Presmeg, N. C. (1986). Visualisation in high school mathematics. For the Learning of Mathematics, 6(3), 4246. Shan, S. J. Bailey, P. (1991). Multiple Factors: Classroom Mathematics for Equality and Justice. Trentham Books. Stacey, K. MacGregor, M. (1997). Building foundations for algebra. Mathematics in the Middle School, 2, 253260. Sweller, J. (1994). Cognitive load theory, learning difficulty, and instructional design. Learning and Instruction, 4, 295312. Tall, D. Thomas, M. (1991). Encouraging versatile thinking in algebra using the computer. Educational Studies in Mathematics, 22, 125147. Tirthaji, B. K. (2001). Vedic Mathematics. New Delhi: Motilal Banarasidass. Van Hiele, P. (2002). Similarities and differences between the theory of learning and teaching of Skemp and the van Hiele levels. In D. O. Tall M. O. J. Thomas (Eds), Intelligence, Learning and Understanding in Mathematics: A Tribute to Richard Skemp (pp. 2748). Flaxton, Queensland: Post Pressed. Warren, E. (2001). Algebraic understanding: The importance of learning in the early years. In H. Chick, K. Stacey, J. Vincent J. Vincent (Eds), The Future of the Teaching and Learning of Algebra, Proceedings of the 12th ICMI Study Conference, Vol. 2 (pp. 633646). Melbourne, Australia. Warren, E. (2003). The role of arithmetic structure in the transition from arithmetic to algebra. Mathematics Education Research Journal, 15(2), 122137. Williams, K. (2002). The cosmic computer course. Obtained from www.vedicmaths.com. Abhyankar, K.D. A rational approach to study ancient literature, Current science, 87 (Aug.2004) 415-416. Adams, E.S., and D.A. Farber. Beyond the Formalism Debate: Expert Reasoning, Fuzzy Logic and Complex Statutes, Vanderbilt Law Review, 52 (1999), 1243-1340. https://law.vanderbilt.edu/lawreview/vol525/adams.pdf Allen, J., S. Bhattacharya and F. Smarandache. Fuzziness and Funds allocation in Portfolio Optimization. https://lanl.arxiv.org/ftp/math/papers/0203/0203073. Anitha, V. Application of Bidirectional Associative Memory Model to Study Female Infanticide, Masters Dissertation, Guide: Dr. W. B. Vasantha Kandasamy, Department of Mathematics, Indian Institute of Technology, Chennai, March 2000. Ashbacher, C. Introduction to Neutrosophic Logic, American Research Press, Rehoboth, 2002. https://www.gallup.unm.edu/~smarandache/IntrodNeutLogic. Axelord, R. (ed.) Structure of Decision: The Cognitive Maps of Political Elites, Princeton Univ. Press, New Jersey, 1976. Bibliography Discover Mathematics, Kenneth Williams Vertically and Crosswise, Kenneth Williams Vedic Mathematics, Bharati Krsna Tirthaji www.hinduism.co.za/vedic.htm https://vedicmathsindia.blogspot.com/search/label/vedic%20math%20research https://vedicmathsindia.blogspot.com/ https://vedicmathsindia.blogspot.com/2007/06/10-questions-to-kenneth-r-williams-on.html