Music: A Mathematical Offering

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The lines drawn represent Equal Tempered Tritones of cents. With the Circle of Fifths you have to swap the direction to get the same interval. Just Intonated intervals are larger or smaller then the Equal Tempered ones. When stacking Just Intervals a spiral appears, not a closed tone-circle such as the Chromatic Circle. So, even though Just Intonated intervals are more consonant, sound more natural Harmonic Series , are more pleasing to the ear, the geometric shapes will be less then perfect.

You can see the Tetractys as a geometric representation of many Just interval ratios when relating a row to one or more above. As you can see, many intervals can be formed by looking at the Tetractys as geometric representation of musical ratios. A scale is a serie of tones at specific distances intervals from each other. There are many different scales, too many to list them all in this article.

Music: A mathematical offering

Most scales including the most common Major and Minor scales are asymmetrical, only a small number of scales are symmetrical. I will only list a few of the most used scales, as well as several less often used but symmetrical scales. Another interesting thing happens when you superimpose both polygons. The polygons of the other Greek Modes can be found by simply rotating the polygon:. The 11 th, 22 nd, 33 rd, and 44 th step in the lines placing each one at a new radian of Only a musician or someone who knows some music theory will see the locations.

The are Harmonic Progression in Music. The 11 th, 22 th and 44 th are Tritones. The most obvious is that of the 1st harmonic fundamental : et cetera. The Circle of Fifths can be used both ways, the interval stays the same only the direction changes, ascending clockwise vs.

How composers from Mozart to Bach made their music add up

In the Bible a 44 day period began on the day Jesus was crucified and ended with his resurrection. These 7-tone scales — when split in the middle — are mirrored on both sides of the center interval:. Just like with scales also chords form geometric shapes. These shapes are the same though for all 12 tonalities. There are of course many more chords variations with additional notes such as for example 4, 6, 9, 11 and more. When you draw a square around the Circle of Fifths and draw lines between the corners of the square, 4 triangles appear.

In each triangle we see the chord progressions that belong to the Major tonality in the center of the outer ring, highlighted by the green triangle.


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Diminished chords are usually not drawn into the tone circle, but to complete the picture I have added it into a 3rd ring in the center. This concept only works with the Circle of Fifths. It does not work this way with the Chromatic Circle.

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The example below is based on C Major, but you could turn the Circle of Fifths in any direction to see the chord progressions for any other Major tonality. In the first image you see the polygon of G 7 and the the tonic of C. In image 3 you see the mirrored polygram of G 7 , that turns out to be a F Minor 6 as noted in the 4 th image. Substituting G 7 with Fm 6 is changing the direction of approach to resolve at C. The tones connected diagonally from left-top to right-bottom are a Minor 3 rd apart, the tones connected diagonally from bottom-left to top-right are a Major 3 rd apart.


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  • With tose 3 lines one of each color triangles can be formed. So far in this article we have looked at the geometric relationships between tone pitch and geometric shapes polygons and polygrams. Music though is more then tones alone. An important aspect of music is rhythm and just as with tones, polygons can be used to visualize rhythm.

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    Well a circle could be divided in any kind of time signatures. On the left you see a division of a circle into 32 parts You could draw various standard geometric shapes like for example squa res , triangles , lines , hexagons , hexagrams , et cetera. In this example on the left you see 3 geometric shapes:. The blue uneven 5-sided polygon visualized the kick drum, the yellow dodecagon visualizes the closed hi-hat and the red line visualizes the snare on the 2 and 4.

    In the example on the left you see 4 geometric shapes:. These were just two very simple examples, but you could make these patterns as complicated as you wish. These principles generalize polyrhythms, additive, and Euclidean rhythms. Furthermore, rhythms can be smoothly morphed between, and irrational rhythms with no regular pulse can also be easily constructed. Each polygon can play an independent sound, and XronoMorph comes with a useful selection of samples to play the rhythms. Watch the following videos to get a better idea about what you can do with this amazing software:.

    If you connect the circled tones a dodecagram is formed. NOTE: There is a lot more that can be said about the Coltrane Circle as well as the geometric relationships between chords and chord progressions in some of his music, in particular the album Giant Steps.

    Music And Measure Theory

    Et cetera …. Of course one could assign the Circle of Fifths to the Torus Knot as well, but that is not what we see in the image provided. If you know a little about astrology, then you must have recognized some of the geometric shapes used in the tone circles earlier in this article, in particular those of the intervals.

    Music: A Mathematical Offering

    The tone circles might have also reminded you of the way the Zodiac Circle is draw. Traditionally the Zodiac Signs are drawn counterclockwise with Aries on the left.


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    Other polygons — not found in the standard tone circles — would also create interesting tones. Of course Hertz and Degrees is not the same thing. But there does seem to be a lot of similarities between the math behind the units, as between many things in the universe. The music system aligns with that model conceptually, though of course the frequency differences of the actual tri-tone notes are not equal.

    The reason to mention these is because of their obvious relation with a few of the shapes found earlier in this article, in particular the Trigon triangle and Quadragon Square. The 3 Squares that represent the Minor Thirds and Major Sixths relationship within the tone circle only forms half a Hexahedron.

    Music: a Mathematical Offering

    You would need to cover 2 octaves to generate 6 Squares to complete the Hexahedron. Advanced Search Search Help. The Mathematical Intelligencer.

    In this issue 21 articles Page is not a valid page number. Please enter a number between 1 and 2 of 2. Letter Letters to the Editor. OriginalPaper Happy birthday! OriginalPaper The mathematical intelligencer. OriginalPaper Refuge from misery and suffering. OriginalPaper Williams college student course survey form: Instructor modified version. OriginalPaper Encounter at far point. Review Classical and quantum orthogonal polynomials in one variable. Review Grothendieck-serre correspondence. Review Mathematical form: John pickering and the architecture of the inversion principle.

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