What is motion geometry?

Introduction of motion geometry:

            
                 The motion geometry initiated the notion of geometry which is used throughout middle-grade and higher-level mathematics. These conceptions are obtained by the learning of transformations, translations, rotations and reflections. The motion geometry is one of center grades can offer an influential context for students to extend their reasoning development. Motion geometry is part of daily life.

Definition:

                The operation or method of changing position or place is called as motion geometry. The movement of the geometry object is also called as motion geometry. The motion geometry contains more terms, the terms are follows below. The motion geometry is used to change the position of the geometry object. The terms are following below.

Terms:

Term 1: Transformation

Term 2: Translation

Term 3: Reflection

Term 4: Rotation.

Explanation:

Transformation:

            In motion geometry the transformation is one of the terms. The transformation is the movement of the geometry object that cannot modify the size or shape of a figure is a rigid transformation.

Example:

           From this diagram the triangle A’B’C’ is the transformation triangle of ABC. In transformation process the triangle’s width, length and height cannot change. It’s just transformed from place to another place.

Translation:

            Translation is the process of move a figure with permanent distance in a given direction. To translate the object without turning or flipping.

Example:

This is the translation diagram of geometry object.

 Reflection:

            It is the processes of the geometric figure can be twisted by flip over a line of reflection to obtain a mirror image. The reflection thread needs the reflect figures, find the lines of reflection, and locate the point of objects before they were reflected. Reflections are executed over a line of reflection.

Example:

This is the reflection of the geometric object.

Rotation:

            Rotation is one of the process of transformation that rotates the figure around one center point is known as rotation. The rotation depends on the geometric degrees.

Example:

This is the example of the rotation of object. The object can be rotated by 180 degrees

Related Article on Motion Geometry

Symmetrical Musical Pieces

Colleen Duffy

University of St. Thomas

2115 Summit Ave, Mail #5258

St. Paul, MN 55105

cmduffy@stthomas.edu

Advisor: Dr. Cheri Shakiban

June-August 2001

 

Abstract

Symmetry in decorative arts has played an important role throughout the ages.  Many cultures have used various symmetric groups to decorate vases, bracelets, borders, etc.  Modern artists such as M.C. Escher have also utilized the beauty of transformations.  These patterns are pleasing to the eye and mind because of their symmetric and infinite properties.  There are seven such frieze groups (borders, strips) and seventeen wallpaper patterns.  The next area to explore within the realm of symmetric groups is their application to a different form of art.  Music is naturally intimate with and enhances math.  This paper will explore the application of frieze patterns in composing music. 

Introduction

In this paper the computer algebra system Mathematica is used to represent musical compositions and then apply mathematical transformations to create the melody.  This was not only done with existing melodies, but also is applied to create new pieces.  The object of this paper is to gain knowledge about how transformations and group theory (the group of symmetries) can be applied to music theory.  Furthermore, it is interesting to determine what types of musical characteristics (consonant, dissonant, chaotic, smooth, etc.) these techniques produce, and whether or not it sounds “good”.  Not only is it important to utilize math to compose music of this fashion, but also the auxiliary part of the project is to ascertain to what extent composers have used similar techniques in their music.  The project’s goal can be summed up as trying to discover and create symmetry in music.

“Music and Symmetry”.  Some people might think this juxtaposition of words sounds intriguing, or at least logical, while others might believe this to be at best an oxymoron.  There are, however, numerous ways in which these two seemingly different areas of study can be tied together.  For instance, composers often employ melodic inversion (reverse ascending and descending movement between notes, i.e. horizontal reflection), retrograde (reversed order of notes, i.e. vertical reflection), and retrograde melodic inversion (reversed and inverted, i.e. 180 degree rotation) in their pieces.  With each of these examples a phrase is transformed and interlaced within numerous other melodies and harmonies to produce music.  Most composers blend in their symmetries.  This paper, however, puts them in the spotlight.  It takes the musical concept of symmetry and adds to it that of symmetric groups and frieze patterns to create a unique methodology for composing music.  Compositions in this paper take melodies and perform transformations to them so that the final piece itself exhibits vertical and horizontal reflectional symmetry. There is an endless supply of variations that can be produced using this method.  Different ways of manipulating and playing the music will be discussed throughout the paper.

It is important to define here what symmetry means in the context of this paper.  Unlike most other forms of art, there are only two known ways in which musical notation can be symmetrical.  Reflectional and rotational symmetry can either be done visually or aurally.  Visual symmetry is a graphical application of the motions and simply means that the objects (the notes and the key signature) are translated, reflected, and rotated according to their graphical coordinates.  So, when looking at a piece of sheet music it is symmetric in appearance - sharps and flats are reflected with the notes.  An example of this is a low C# becomes a high A#.  A musical symmetry group, then, means that the music is aurally changed.  The notes are horizontally reflected and rotated in order to maintain the same number of half steps from the middle staff line, middle B in treble clef, but in the opposite direction (inversion).  For instance, low C# is eleven half steps below B.  Its horizontal reflection and its rotation, therefore, would be eleven half steps above, a high A.  Also, the key signature is unchanged in this case because the inversion takes it into account.  This latter form of symmetry often sounds more “pleasing” or “natural” to the ear while the first produces key signatures and intervals that are unusual in more classical types of music.  Samples of both types are found in this paper.

           

 

Method and Analysis

Part 1: Composing Music in Mathematica

Before the transformation and composition of melodies could begin, functions and commands had to be written in Mathematica to represent the notes and then transform them.  Musical notation is comprised of many components.  There are whole notes, quarter notes, eighth notes, rests, bar lines, sharps, flats, naturals, etc. that each need to be represented in the program.  For simplicity, each type of note was broken into two components: note and stem.  That way, the commands only needed two types of notes, open and filled, and four stems, up and down quarters and up and down eighths (sixteenths were also needed later).  Mathematical functions for these, as well as the other pieces mentioned above, were created.  The next step was to write commands to transform lines of music.  There are four basic types of transformations: translation, horizontal reflection, vertical reflection, and a half turn; all of the frieze patterns (discussed later) can be formed using these.  Once the program proved to work properly the project took the next step, creating the frieze patterns.

There is a lot of math, namely group theory, which is represented in this music.  A group is a set with a closed operation such that it has the properties of associativity, existence of an identity element, and existence of inverse elements.  The group of symmetries contains the motions (isometries) that leave a figure invariant, which means that the distances between the points are preserved.  Translations, rotations, and glide-reflections are the three basic types of motion that can be used in the two-dimensional plane (see figure 1).  Translation is a mapping that sends all of the points of the figure the same distance in the same direction.  Rotation is turning a figure about a point.  Reflection is a mapping that sends each point across a line, like a plane mirror.  Finally, glide-reflection is a reflection with a translation; think of footprints.  These types of motions can be broken down into two categories – proper and improper motions.  A motion is “proper” if it preserves orientation; that is if the clockwise direction is the same before and after a motion is applied.  Rotations and translations are proper while reflections are not [1].  Symmetry groups play an important role in numerous areas of science such as crystallography and quantum mechanics.

           

      Translation                         Rotation                           Reflection                Glide reflection

                                                                                               

Figure 1

Frieze patterns are two-dimensional symmetry groups that are an infinite extension of a repeating pattern along a line.  There are only seven different types since by definition they must be discrete (finite number of symmetric motions), infinite in length, and leave a line invariant (wallpaper does not because it extends up and down as well as left and right).  Combining the different types of plane motions makes up these seven types.  The simplest border, frieze pattern, is the one that only exhibits translation.  However, every pattern contains translation.  The musical equivalent of a translation is a repeat.  From there the patterns move up in complexity.

The first melody used to create frieze patterns is a passage from Swan Lake by Tchaikovsky.  This melody was chosen because it is a relatively short phrase and easy to work with.  Like all of the variations in this paper, the phrase was not chosen because of its symmetrical content; one of the objects of the project was putting symmetry in music.  The first step was to enter in the data for the melody and next proceed by representing each of the seven frieze patterns with this melody.  The seven types of patterns, along with their names and description, are shown below in figure 2.  The characteristics of each pattern are represented by the following notation: a “1” in the first position means there is no vertical reflection and an “m” means there is.  In the second position a “1” means there is no horizontal reflection, “m” that there is, “2” means a 180-degree rotation, and “g” that there is glide reflection.

11 - Translation

 

12 – 180-degree rotation

 

1m – Horizontal reflection

 

m1 – Vertical reflection

 

1g – Glide reflection

 

mg – Vertical, glide reflection (180-degree rotation)

 

mm – Horizontal, vertical reflection (180-degree rotation, glide reflection)

 

Figure 2

Frieze patterns are infinite in length, so these patterns extend indefinitely in both directions.  The musical notation in all of these pieces deviates from the norm in some cases because of the way the motions are applied.  Stems, sharps, and naturals, etc. sometimes appear on the wrong side of the note or upside down.  This occurs because reflections cause the image to look the opposite of the object.  Also, rotations cause the object to become opposite and upside down.  In order to preserve the nature of the transformations, normal musical notation was sacrificed.

A performance of these patterns would sound repetitive and boring.  Therefore, the next step to be taken was to apply the same technique to longer melodies to create variations based on the group of symmetries.  The same phrase from Swan Lake became subject to experiments once again to determine what types of sounds and new melody lines could be produced.  It would have worked well except for the fact that because the melody was so short, it created a repetitive composition.  This result gave rise to trying a longer melody, “In the Hall of the Mountain King” by Edvard Grieg.  Furthermore, the new trial used only one section of a frieze pattern.  Both of these things helped to make the end result more pleasing.  The creation of a new composition steps up to the next level from an alternation and variation of a melody.  Two appear later in this paper.  The first is based on the Tabletop duet by Mozart, where one player reads the music upside down, and the second is a crab canon, where one player reads the song backwards.  Making these musical frieze patterns resulted in some interesting sounding music, but enjoyable in a different sort of way.  These efforts are discussed and shown below. 

The following two variations employ the frieze patterns of types mg and mm chosen because they utilize the melody and all three transformations, and so contain the most variety.  They utilize sixteen measures of “In the Hall of the Mountain King” by Grieg, the first as a graphical reflection (see figure 3) and the second as reflected musically (see figure 4).  Note: the last half (last eight lines) of each piece belongs to the right of the first half so that it is the vertical reflection of the first half.  Also, the dashed lines represent the lines of reflection.

The Mt. King’s Hall of Mirrors

Arranged by Colleen Duffy

 

Figure 3

The Mt. King’s Hall of Echoes

Arranged by Colleen Duffy

 

Figure 4

The differences between the two can be easily seen and even more easily heard.  The horizontal reflection and rotation change when the manner of reflection changes while the vertical reflections are the same in both.  These are both type mm, but they can be rearranged into type mg by cutting apart each of the four sections and putting them in the order of melody, vertical reflection, horizontal reflection, and 180-degree rotation.  Several options not only exist for how to compose the piece (all seven patterns), but also for how to play it.  Reading it traditionally so the different phrases are alternated or reading each block as a whole  (melody, horizontal reflection, vertical reflection, and rotation) produce different effects.  Another possible variation is to mix up the lines a little bit; it is not necessary to keep all of the phrases of the melody together.  With all of these different ways of rearranging the piece, every person can have a unique piece of music to decorate his or her house and to perform.

            The “Tabletop Duet” by Mozart inspired this next composition.  In his duet one player reads from one side of the table and the other reads it from the other “up-side down”.  For “Rotator Solos” the first player performs the composed melody and its horizontal reflection.  At the same time the other person plays it the opposite direction, and so ends up playing the vertical reflection and rotation.  This one exhibits visual symmetry.

Rotator Solos

Colleen Duffy

Part 2: Composers’ Uses of Symmetry

As stated in the beginning, this section mentions a few ways in which musicians have employed these same techniques.  The most obvious way is one that was noted above - translation acts like a repeat in music.  This occurs frequently in compositions.  Also, composers employ what is termed melodic inversion, retrograde and retrograde inversion, which can be understood as analogous to horizontal and vertical reflection and a half-turn.  What is interesting, however, is whether or not composers have somehow used these transformations to the whole piece as was done above.  Bach is known to have studied math and to have put mathematical concepts in his work, so his works made a good place to start looking.  This led to studying different types of canons, some of which correspond to what has been discussed in this paper.

            First of all, a canon is “the strictest form of counterpoint in which one voice is bound to imitate the rhythm and interval content of another voice” [6].  The word itself comes from the Greek word for rule or law.  A round (like “Row, Row, Row Your Boat”) is a type of canon.  Basically canon means that the second voice mimics the first voice, but different ways exist for doing this.  This project is only concerned with a few of these ways.  The Retrograde Canon, also known as Cancrizans (“crab” canon), is one of the more intriguing types.  In this one the melody is played forward and backward at the same time.  This translates to taking the melody as one line and reflecting it vertically, type m1 frieze pattern. “Cancrizans” from the Musical Offering by Bach offers an example of this [7*].  Another type of canon is known as a Canon in Contrary Motion.  Here the second player plays the same intervals as the first, but in the opposite direction (inversion).  This provides the mathematical equivalent of writing the melody as one line and performing an aural horizontal reflection (and possibly other transformations) like “Mt. King’s Hall of Echoes”.  The slight difference in the canon is that the interval qualities can be written to stay in the key.  Bach, among others, wrote many of these including “Musical Offering No. 3”.  The last type relevant here and now is called a Mirror Canon.  This is similar to contrary motion except that the precise qualities of the intervals are mimicked, so there are a lot of accidentals.  This is even closer to aural horizontal reflection; however, there are still some differences from the technique used in this research.  Some examples of the Mirror Canon are the “Canon perpetuus” which exhibits a vertical mirror image and the “Canon a 2 Querendo invenietis” where the second player starts later and reads upside down.  Both are from Bach’s Musical Offering [7].  Most canons include some mathematical transformations.  Therefore, known composers use some of the same mathematical concepts that have been laid out here.

           

 

Future Directions

Many accomplishments were made in the course of this research project.  It was found how to represent music in Mathematica and then perform transformations to it.  Also, music was created that sounded chaotic or dissonant at times, but overall was interesting to listen to and study.  Lastly, existing samples of frieze patterns in music were found.  Even with all that was accomplished, a lot can still be done.  It would be helpful to find a more concise way of entering in and transforming music in Mathematica, or a different program.  Also, the different variations (the two types of symmetry, the order in which the parts are played, splitting up the melody, etc.) can be explored in more detail, both in finding other possibilities and in listening to which make the “best” music.  Lastly, it would be nice to find more examples of these concepts in existing music.  This would not only show that group theory has real applications in music, but could also lead to more ideas on the implications of group theory in composition.  There are many variations and possibilities present in this method of creating music that have yet to be discovered.  Hopefully it can now be seen that music and symmetry do indeed belong together.

           

A crab canon closes this composition.

Crabby Variation

Colleen Duffy

Works Consulted

[1]  Durbin, John R.  Modern Algebra: An Introduction.  4^th ed.  New York: John Wiley & Sons,

Inc., 2000.  230-239.

[2]  Mathematica.  Vers. 4.0 and 4.1.

[3]  Music Scores.  Tchaikovsky.  “Swan Lake.” June 2001 <http://www.music-scores.com>.

[4]  Shakiban, Cheri.  “Chapter 5: Geometric Symmetry.”

[5]  Sheet Music Archive.  Edvard Grieg.  “In the Hall of the Mountain King.”  University

Society, Inc., 1918.  June 2001 <http://www.sheetmusicarchive.com>.

[6]  Smith, Timothy A.  “Anatomy of a Canon.” Sojurn 1996.  29 Aug. 2001

<http://jan.ucc.nau.edu/~tas3/canonanatomy.html>.

[7]  Smith, Timothy A.  “Canons of the Musical Offering.” Sojurn 1996.  29 Aug. 2001

<http://jan.ucc.nau.edu/~tas3/musoffcanons.html>.

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* See this source for a more detailed explanation of what is mentioned here and for more examples.