“There is geometry in the humming of strings, there is music in the spacing of the spheres” -Pythagorous When composing a piece of music, whether its classical, rock, rap, country, alternative or any other type of music out there, geometry actually plays a bigger role than you might think. There are a few key concepts that are crucial to understand for anyone wanting to write music of any kind and geometry plays a big part in many of them. For example, the circle of fifths, a musical tool that shows the relationship of twelve tones on the chromatic scale, the associated major and minor keys, and their corresponding key signatures, gives us a visual representation of how different chords are made from triangles and quadrilaterals. In the pictures above, you see two different chords shown on the circle of fifths one is a major chord, and the other a dominant 7th. The photograph farthest to the left shows a C Major chord which is composed of the notes C, G, and E. C Major and all other Major chords form a scalene triangle when shown on the circle of fifths.
The photograph on the right shows an F# Dominant 7th chord which is composed of Bb, Db, Gb, and E. This and all other Dominant 7th chords create quadrilaterals. Another way geometry is involved in music is with diminished chords. In a diminished chord, you have a 1st, or tonic (base note of the chord), flat 3rd, and flat 5th.
In a diminished chord, the flat 3rd is the midpoint of the chord. It is three half steps from both the tonic and the 5th. This type of chord would create a isosceles triangle on the circle of fifths. These concepts could be important to know when writing and composing music because it gives you a visual representation of how chords work. Especially because chords are the basis of almost all music. Not to mention counting and rhythm.This relationship between music and Geometry was originally discovered and documented in a treaties written by Nikolai Diletskii in the late 1670’s.
The idea was later rediscovered and remastered by german baroque composer Johann David Heinichen in 1728 who created the modern day circle of Fifths that heavily relies on math. He brought forth the idea that each scale on the circle relies on the 4th scale degree of the previous scale, which is what allows us to make the triangles that I talked about on the previous paragraph. He decided to do further research on this subject to strengthen his understanding of music and composing, as well as its relationship with math The relationship between music and math can also be found in science. Science tends to heavily rely on math, and when you talk about frequency, music can be added to that connection.
The formula used to find the frequency of a note is represented as f = v or, Frequency =velocitywavelength. This formula will tell you the frequency of a note being played. Frequency is measured in Hertz, The Standard International unit of frequency measurement.
The name “hertz” was derived from the name Heinrich Rudolf Hertz, the person who discovered it in the 19th century. You can calculate the smallest pitches, like the resonant frequency (the point where the sides of the glass vibrate the most easily) of a wine glass is seen as 20,000Hz. You can also measure much larger things. For example, the resonant frequency of the Earth is 432Hz, or the note C#/Db.
The frequency of the Earth is so much lower because as the pitch decreases, so do the number of waves.When teaching these concepts to a class, there are a few ways you could approach the lesson, because it is such a broad subject. One way the relationship between math and music can be taught to a geometry class is by starting with the circle of fifths. The best approach would be to draw the circle and start with a few chords.
Introduce the class to the topic by starting with a few major chords. You could draw the circle of fifths on the board and have different students come up and attempt to draw the chord. Once the students are comfortable with placing chords on the circle of fifths, start giving them more complex chords, maybe even combinations of notes that do not work. Have them do few on their own and have the students write down whether or not the combination of notes work as a chord and if so, what type of chord it is.