Musical Frieze Patterns

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Musical Frieze Patterns

• Colleen Duffy
• University of St. Thomas
• Advisor Dr. Cheri Shakiban
• Summer 2001

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Contents

• Transformations
• Frieze patterns
• The seven types
• Using a musical melody to produce a song
• How composers use symmetry

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Symmetry

• Group of Symmetries - those motions (isometries)
  that leave a figure invariant (distance
  preserving)
• Motions/isometries - set of permutations that
  preserve distance between points
• 3 types of plane symmetry
• translation
• reflection
• rotation

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Symmetry con.

• glide reflection
• (combination of horizontal
• reflection and translation)
• Proper motion - one that preserves orientation
• Theorem A motion of the plane is either a
  rotation, translation, or glide-reflection.
  Either all or half of a symmetry groups motions
  are proper.

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Frieze Groups

• Frieze group - symmetry groups that are discrete,
  infinite, contain translations, and leave a line
  invariant ex borders.
• There are seven different groups. There are only
  seven possible ways of combining the motions of
  the plane to satisfy the definition.

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The Seven Types

• 11 - translation
• 12 - 180-degree rotation
• 1m - horizontal reflection
• m1 - vertical reflection

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• 1g - glide reflection
• mg - vertical, glide reflection (180-degree
  rotation)
• mm - horizontal, vertical reflection (180-degree
  rotation, glide reflection)

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Music

• You can combine the various patterns to make
  music, even if it is a little repetitive )

Frieze Swan Lake
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Mt Kings Hall of Echoes

• Here the notes are reflected musically i.e. the
  distance in half steps between notes is
  preserved.

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Mt Kings Hall of Mirrors

• In this version of the song the melody is
  reflected graphically.

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Rotator Solos

• Here is a duet I composed. The first player
  plays the melody and horizontal reflection, the
  second plays the vertical reflection and 180
  degree rotation they read it from opposite
  sides of the table.

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A Crabby Variation

• This is a variation on the previous duet. In
  this duet, one player plays the song forward and
  the other plays it backwards. This is called a
  crab canon.

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How composers use symmetry

• Composers use symmetry in many different ways in
  their music intentionally and unintentionally.
• Some examples are
• Translation repeat
• Horizontal reflection inversion
• Vertical reflection retrograde
• Half-turn retrograde inversion
• These often occur in small parts of the music.
• We looked at symmetry over a whole piece, so it
  is more interesting to look at if composers used
  some of the symmetrical properties on the piece
  as a whole or not.

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Bach

• Bach is known to have studied math, so I looked
  at some of his works for symmetry.
• He wrote numerous canons, many of which have the
  properties we have been discussing.
• Canon - the strictest form of counterpoint in
  which one voice is bound to imitate the rhythm
  and interval content of another voice ex. a
  round
• Different types of canons
• Retrograde Canon/Cancrizans (crab canon) - the
  melody is played forward and backward at the same
  time, i.e. a type m1 frieze pattern. An example
  is Cancrizans from the Musical Offering.
• Canon in Contrary Motion - the second player
  plays the same intervals as the first, but in the
  opposite direction (inversion). This is similar
  to performing a aural horizontal reflection (and
  possibly other transformations), like Mt. Kings
  Hall of Echoes. The slight difference is that
  the interval qualities can be written to stay in
  the key. An example is Musical Offering No. 3.
• Mirror Canon - similar to contrary motion except
  the precise qualities of the intervals are
  mimicked, so there are a lot of accidentals.
  This is closer to aural horizontal reflection
  there are still some differences from how we did
  it. Examples Canon perpetuus which exhibits a
  vertical mirror image, Canon a 2 Querendo
  invenietis where the second player starts later
  and reads upside down.
• These are just a few examples of the application
  of group theory to music.

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Future Directions

• Finding a better way to enter the data into
  Mathematica
• Explore other variations
• Finding more examples of these concepts in
  existing music

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Questions?