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Curtis Hanson Music and Mathematics - A Divine Relationship Most people know the aesthetic beauty music and art can offer. Many, however, may not be aware of the mathematical principles which exist in music and composition. The aesthetic perception of music is governed by the right half of the brain. Mathematical relationships and spatial reasoning are controlled by the left half. This brief article is based on the premise that music theory reflects the laws of mathematics and nature, and that great composition contains mathematical relationships which enhance the perception of its aesthetic beauty. In other words, knowing music exercises the whole-brain, the whole-person. Composers have long been fascinated by numerology. The ancient Greeks knew of the relationships between numbers and what they considered perfection in architectural design. The Golden Sequence, also known as the Golden Section, the Golden Number, or the Divine Proportion, is one such mathematical relationship, the formula of which often occurs in the natural world. (A preliminary internet search on this phenomenon turned up over thirty million hits!) This proportion can be created by dividing a line into two parts. The point of the division should be in such a place that the square of the longer subdivision is equal to the product of the shorter subdivision times the length of the entire line. Put as a formula, A (shorter subdivision) x C (entire line) = B2 (longer subdivision), or proportionally A/B = B/C. Line B is roughly 1.62 times the length of A (B/A), or conversely, line A is approximately .62 times that of B (A/B). The same relationships hold for lines B and C. To the Greeks, the 1.62 figure is known as Phi and the .62 figure is phi. The ancient Greeks believed that a rectangle whose sides were in this
proportion were the most aesthetically pleasing and based their architectural
principles upon it. The most well-known such building is the Parthenon
in Athens. It is also a fundamental formula used by the ancient Egyptians
and is most notably seen in the pyramids. B / A: 8 / 5 = 1.6 13 / 8 = 1.625 21 / 13 = 1.615 34 / 21 = 1.619 How do these formulae and figures relate to music? A simple analysis of elementary music theory reveals profound occurrences of the Fibonacci series. For example, the scale and the triad are the fundamental building blocks of Western music. There are 13 notes in the complete chromatic scale (C-C). The scales and modes most frequently used contain 8 notes in an octave. Another frequently used scale, the pentatonic, uses 5 notes. The modern keyboard contains black note groupings of 2 and 3, a total of 5 in an octave. The most common triad used in harmony is based on 3rds and contain a root (1), 3rd, and 5th. In a full chromatic scale, the notes which create a full octave major chord (C, E, G, C) are 1, 5, 8, and 13. In the diatonic (major) scale, the notes 1, 3, 5, and 8 create the major chord. The chord is created by intervals of 2 scale steps between each of the 3 lower pitches of the chord and 3 scale steps between the upper 2 pitches. In fact, the pitches found in the chromatic scale are all based upon Fibonacci ratios. (For a more detailed analysis of the relationships between music theory and mathematics, visit http://evolutionoftruth.com/goldensection/goldsect.htm and http://evolutionoftruth.com/goldensection/music.htm.) Just as the architectural aestheticism of ancient civilizations has been traced to the Golden Sequence, later revived by renaissance and classical ideology, so have composers utilized such thought in musical architecture. Many have found that a point approximately 3/5 the way through a composition, correlating to the proportion of two successive numbers in the Fibonacci series (13/21, 21/34, etc.), yields a satisfying musical climax or a point of transition to a different texture or other musical variation, and further subdivide their work according to such a principle. (Space limitations prevent a detailed look into how numerical relationships and music interface in various compositions. Only a superficial mention can suffice here.) Debussy often utilized this architectural design in his works. The beauty of his music (right-brain perception) is balanced by a musical form influenced by a left-brain demand for mathematical and spatial relationships. Several of Debussy’s works display clear evidence of Golden Sequence/Fibonacci relationships, in particular his orchestral work Prelude to the Afternoon of a Faun. (For an in-depth analysis of Debussy’s usage of form, particularly those works incorporating the Golden Sequence, the reader is directed to Debussy in Proportion: A Musical Analysis, an excellent study by Roy Howat, published by Cambridge University Press.) Perhaps even more remarkable, and more understood, is J. S. Bach. Most musicians recognize the beauty and soaring expression of Bach’s compositions, right-brain concepts. Yet many scholars have researched Bach’s music and found distinct relationships between numerology and his creative processes. The technical perfection of his fugues and his predisposition to adhere to and expand upon the musical forms of his time only reflects the strong organizational nature of his left-brain. Offsetting this are his toccati, freely improvised expressions uninhibited by a rigid formal structure; his chorales, simple in structure but extremely moving in his interpretations; and the great vocal/choral works which manifest his faith, particularly the B Minor Mass and the St. Matthew Passion. The meditative beauty and soaring expression of these works are those of a man consumed by passion for his art, for his faith, and for his heritage, balancing his architectural genius with aesthetic virtues. On the other brain….er, hand, his usage of the Golden Sequence and the Fibonacci series is most profound in the St. John Passion. The St. John has 68 individual pieces or movements. The overall structure of the work and the location of those movements, upon analysis, display clear evidence of mathematical form and adherence to the Fibonacci series. For example, a palindrome encompasses a major portion of Part II, pieces twenty-seven through fifty-two. A palindrome is formed with one piece as a pivot, in this case #40. Like arch form, related material can be found an equal number of pieces from the pivot forming a mirror image such as in the word “level.” Golden Sequence/Fibonacci relationships can be observed in the following ways:
What do those numbers mean? Divide each number by two and you have 3, 8, 13, 21, and 34, all numbers in the Fibonacci series. Furthermore, dividing the total number of pieces in the St. John (68) with the number of pieces not in the palindrome (42) yields an answer of 1.62, the Greek Phi of the Golden Sequence. The same answer results from dividing the number of pieces not in the palindrome (42) with the number of pieces in the palindrome (26). Did Bach deliberately use such musical architecture and mathematical principles, or were they an unconscious reflection of his genius? We will never know, but evidence suggests that he was very much aware of how music and mathematics work in harmony. Richard Strauss’ Death and Transfiguration is another composition which has evidence of the Fibonacci series. In a subjective analysis of the work according to time, the location of the various themes all relate to figures in Fibonacci:
Are such relationships coincidental? The evidence suggests otherwise. Quarter Notes Music in May Sunday, May 6 – 4:00 p.m. Holy Family Catholic Church Suggested donation: $10.00 adults - $5.00 students & seniors
Saturday, May 19 – 1:00 and 2:30 p.m. St. Ann’s Residence no charge for admission * * * * * * * * * * * * * * * * Saturday – Sunday, June 2-3 – 1:30 and 3:00 p.m. Edgewood Vista Assisted Living (Saturday) Keystone Bluffs (Sunday) no charge for admission Opus - Music in May: Paris in the Springtime This year's Music in May concert brings works of four 19th-century Parisian composers to Holy Family Catholic Church in Duluth. The Conservatory's John Duss Choral Society and the Pilgrim Lutheran Church Senior Choir will perform pieces by César Franck, Gabriel Fauré, Charles Gounod, and Claude Debussy. Organist Todd Sletten and pianist Nick Nyderek will accompany the combined choral groups.
Please join the John Duss Choral Society and Pilgrim Lutheran Church Senior Choir at 4:00 p.m. on Sunday, May 6, 2007 as they fill Holy Family Catholic Church with late-nineteenth-century music of four eminent Parisian composers. The church is located at 2430 W. 3rd St. in Duluth. The suggested donation is $10 ($5 for students and seniors). CENTER STAGE— 2007 Graduates
The John Duss Music Conservatory congratulates each of these high
school seniors and is honored to have been an educational partner
in this rewarding journey. Interlude - O Sister, Where Art Thou? Many students, parents and friends ask about Sister Bernardone Rock, Director of the John Duss Music Conservatory, who is currently on sabbatical. Good news! Read on for an update regarding her latest adventures in the art of taking it easy. In early September, Sister Bernardone traveled to Meriden, Connecticut for a well-deserved change of pace from the non-stop responsibilities of steering this ship of music we fondly call, “The Duss”. In the blink of an eye, her e-mails began asking for bits of music from the Conservatory’s archives. “I’m teaching guitar!” At present, it seems there are now piano and voice students adding to the mix. Meriden, CT is home to the Franciscan Life Center Network (www.fsecommunity.org) where the primary services provided to the community are marriage/family counseling, home health care and hospice care. The city is a bit smaller than Duluth with a population of nearly 60,000 and is situated halfway between Hartford and New Haven. Sister Bernardone describes the nature of the area’s small cities that are strung together along the roadways of Connecticut: “I can be traveling along what I think is a street and all of a sudden I am in Cheshire (a neighboring city), or Wallingford or Southington. That happens on the small ‘back roads’. On the freeway of course, it doesn’t work that way.” An unforgettable experience in January was joining in the Pro-Life March in Washington, DC. Sister Bernardone was profoundly impressed with the number, youth, and broad geographic representation of the participants. She would be happy to see a group from Duluth take part in the march next year.The question folks often ask about Sister Bernardone is, “When is she coming back?” Not for a while! Her latest enterprise has been especially exciting. In April she was asked to temporarily replace Sister Naomi, one of the nuns who serves in Jamaica. Yes, Jamaica! In Sister Naomi’s absence, the Franciscan Community’s administrators were wondering if there might be a nun who could work with adults and children and provide an emphasis on music at their Jamaican location called “Above Rocks”. Sounds like destiny for our Sister Bernardone Rock. Her journey landed her in Kingston and from there she took a breathtaking hairpin-turn ride upward to the mission site. The assignment was a welcome chance to collaborate with her lifelong friend, Sister Marcia. They are both from the small town of Dodgeville, Wisconsin and were classmates from grade school through college. They entered the convent together and now have had the opportunity to once again share in the work of their faith. Sister Bernardone’s travels do not diminish her thoughts of Duluth, her colleagues, students, beloved dog Liebchen, and our magnificent Lake Superior. Though
she enjoys the beauty and diversity of this nation, her home
is in the heartland. But
she
adds, “God’s work is there to do, no matter where
we are”. It seems God’s work is never on sabbatical.
If the saying “bloom where you’re planted” can
describe Sister Bernardone’s “vacation”,
the next harvest looks promising. Spotlight — The
Conservatory's Advisory Board: In 2004, tthe Conservatory established an Advisory Board. The Board currently consists of twelve community members, an administration representative, a meeting facilitator, a faculty representative, and four ex-officio members from the Franciscan Life Center Network. Its role is to promote the Conservatory in the community and to guide its efforts to remain a successful, vibrant music center. This is the last of three "introductions" to these movers and shakers who give generously of their time and various areas of expertise by serving on the Advisory Board. Board Member: John Hinzmann We regret that Board member Ken Hogg was not available for an interview in time for the release of this issue of The Score. Ken has been with the Board from the start, and his contributions are greatly appreciated. Music affects people from all walks of life in lasting, meaningful, positive ways. The Conservatory is very fortunate to have the voices of a diverse group of advisors to add to its growth and development. Our sincere thanks go out to this group of dedicated volunteers. Standing Ovation On Sunday, March 11, Conservatory faculty and several guest musicians performed in the annual Faculty Recital. This was the fifth year the event was held at Gloria Dei Lutheran Church in their newly renovated sanctuary. Faculty who performed included Mary Lee, Paul Peterson, Carolyn Carver, David Lewis, Curtis Hanson, Sr. Margaret Dahl, Julianna Webb, Doug Maguire, and Barry Schwartz. Guest artists included Paul’s wife Darlene and son Matt, Dave Slattery, Billy Southern, and Lyz Jaakola. Nota bene [ n. b.] - (no'tä be'na) [L.] note well; take particular notice (Webster, 1979, pg. 973)
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 © Copyright 2003-2008 Franciscan Life Center Network, Inc. dba John S. Duss Memorial Music Conservatory |
The Score
César
Franck (1822-1890), although born in Belgium, spent the greater part
of his life in Paris, from 1844 until his death. From 1872-1890
he was professor of organ at the Paris Conservatory. His day jobs as
a teacher and organist eventually paid off as he built a reputation
as a skilled organist. He is considered by many to be the greatest
organ
composer after J.S. Bach. Besides his organ works, he also wrote for
piano, orchestra, and voice. The Music in May concert will feature
his choral setting of Psalm 150, which calls for praising God with
music,
dance, and all manner of instruments. Those students who know of Franck
mainly through his instructional books for beginning pianists will
enjoy hearing this moving, melodic piece. 
Gabriel Fauré (1845-1924)
also lived and composed in Paris during the second half of the 19th
century. Like Franck, he, too, was an educator
and organist. He taught composition at the Paris Conservatory beginning
in 1896, serving as its director from 1905-1920. Most of his compositions
were songs and chamber music pieces, but he also wrote several major
works: two operas and a Requiem. Second on the Music in May program
is the Requiem of 1888. While a Mass for the Dead might sound like
a melancholy
offering for a spring concert, it is a lighter-hearted expression
of mourning; it has even been referred to as a "lullaby of death." Fauré said
of this work, "...it is thus that I see death: as a happy deliverance,
an aspiration towards happiness above, rather than as a painful experience." With
its subtler effects, Fauré's Requiem departs from the other
19th century requiems of Berlioz and Verdi and their heavy, bombastic
approach.
This performance features solos by soprano Barb Bourman and baritone
Tim Ingersoll.
Charles Gounod
(1818-1893) was born in Paris. He, like Franck and Fauré,
was a student of the Paris Conservatory. His early training there
yielded an early cantata, Ferdinand, for which he won the prestigious
Prix de
Rome in 1839. Thereafter he went to Italy to study 16th-century
sacred music. His Messe Solennelle (Solemn Mass, also known as the
St. Cecilia
Mass) of 1851 established his reputation as a noteworthy composer.
His later operas Faust and Romeo et Juliette further strengthened
his position
in French musical circles. Later in life he again returned to religious
composition. His "Sanctus/Benedictus" (from his St. Cecilia
Mass, named for the patron saint of music) will be heard at the
Music in May concert.
Achille-Claude
Debussy (1862-1918), a contemporary of Fauré, was
born in St-Germain-en-Laye, France. When he was 10, his godparents
sent him to study at that venerable French music factory at the
Paris Conservatory.
As an adult, he traveled and performed widely throughout Europe
and Russia, but his home base was Paris. In the upcoming performance,
the choirs,
along with soloists Curtis Hanson and Bob Asleson, will sing
the last portion of Debussy's cantata L'Enfant Prodigue (The Prodigal
Son), one
of his early works. He was only 21 years old and a student at
the
Paris Conservatory when, in 1884, he won the Prix de Rome for
this cantata.
Although Debussy's Conservatory professors did not think he had
much of a future because he refused to follow the traditional
rules of composition
(he said he wrote only for his own pleasure), his mature style
has prompted others to consider him one of the greatest and most
influential French
composers of all time.
Heather Morgan 
Ryan Kephart 
Trey Callahan 
Katie Gableman