External Applications of Math and Music

The applications of math in the real world are endless. Whether one focuses on science or language or history, math plays a role, sometimes big, sometimes small. It is necessary in everyday life: to manage money, do taxes etc. Music in the real world too. While not necessarily having as practical an application as math, there are countless studies documenting the effect of music on people’s mindsets and attitudes.

With a lot of the focus in this project being on patterns such as the Golden Ratio and the Fibonacci sequence, these specific mathematical phenomenons can be found in places other than music. They are often found in nature as well, in the way many plants and animals develop. This has been most notably demonstrated in the nautilus shell.

Overall throughout the course of this research project I have learned a lot about music as well as the mathematics associated with it. Finding all of these patterns and such within songs has opened my eyes to a new way of viewing music.

The Golden Ratio and Musical Aesthetics and Acoustics

Since it has been previously mentioned that the Fibonacci sequence and the Golden Ratio are related, it’s no surprise that the Golden ratio can be found in many aspects of music as well – namely the design of instruments and how they sound.

Let’s look at the guitar first. In many cases of guitars if you take the measurements of certain dimensions you will see a pattern. One specific design, the Kertsopoulos model, was specifically designed to have such patterns incorporated. Some say this affects the way the guitar sounds, but everyone has different opinions.

The same applies to certain violins.

And in certain recording rooms to purportedly “improve acoustics”.

Whether the sound improves due to these designs a question of differing opinions for different people, but it’s undoubtedly true that the Golden Ratio shows up in many places related to music.

Fibonacci and Frequencies

There are also multiple relationships between the Fibonacci sequence and the pitches of notes.

Firstly, keep in mind that the lowest frequency that a human ear can hear pitch clearly is around 40 Hz; the lowest Fibonacci number above 40 is 55. A normal midrange A is 440 Hz, and translating that pitch up or down an octave only requires multiplying or dividing the frequency by 2. Since 440/8 is 55, we can see that 55 Hz is an A 3 octaves below the one at 440. The next pitch corresponding with a Fibonacci number comes 3 Fibonacci numbers later, at 233 (Hz). This corresponds to Bb. The pattern continues as ever 3 Fibonacci numbers corresponds with a certain pitch. And if we list these pitches in order, we get a pattern of A, Bb, B, E, Db, D, Eb, E, F, Gb, G, Ab, A, and so on. Notice the notes A and E in the sequence. The distance from the and A or E to the next A or E goes 3, 4, 5…. and E is the 5th note above A, 8 half steps above. Also worth noting is that it takes 55 Fibonacci numbers to get all the pitches from Ab to G.

Second, all the pitches’ frequencies are fractions of the 440 Hz A, using numerators and denominators of numbers in the Fibonacci sequence.Refer to the below chart for reference.

Fibonacci ratios and frequencies http://www.goldennumber.net/music/

Fibonacci ratios and frequencies http://www.goldennumber.net/music/

As you can see, when multiplying the 440 Hz frequency by 2/1, you get the A an octave up, 3/2 is an E, 5/3 a C, and much more.

The Fibonacci Sequence Within Musical Scales

As stated previously within this blog,

“The Fibonacci Sequence is a sequence of numbers beginning with 1,1,2,3,5… where each proceeding number in the sequence is the sum of the two numbers before it. Thus an extended version of the sequence shows

1,1,2,3,5,8,13,21,34,55,89,144,233,377… and so on.”

Looking at the first few numbers in the sequence, up until 13 we can see that many of these numbers can be applied to a musical scale. Within one octave of the scale, there are 13 notes total. Any scale, consists of eight notes, and when played in order, the 5th, 3rd, and 1st notes, make up the basic triad, and the difference of two between each of these notes. Going from one note to the next in a scale consists of whole or half steps, where going up one note is 1 half step, and one whole step is 2 half steps. In the most basic major scale, the one without any accidentals (flats or sharps in the key signature), C major, on a piano, the scale consists of 8 white keys, skipping the 5 black keys in between. The 5th note of the scale is the 8th note when counting all keys.

While these sorts of patterns may not influence the way music sounds, it may give us more insight into the way music works and how instruments as well as musical writing was designed

Music and Math as Related Subjects

When looking at music and math separately, from an outside perspective, there may not seem to be many similarities. While math is fairly concrete, with its formulas and rationality, music is often expressive and subjective, able to be interpreted in many ways. So how can these two be so interrelated?

Statistically, and historically, numerous studies have shown that people who studied or learned music also had a higher ability to reason, and that the percentage of musical students to take math majors in college was well above average. This isn’t just a correlation; there are reasons for such a relationship. While interpretations of mathematics and music may be different, they still do share common characteristics. Both subject fields require a somewhat creative mind; the feelings that one gets when solving a math problem may be similar to those when playing a song. And yet, mathematical thinking, too, contains a hint of “musicality”, as people can work together to find, identify, and interpret patterns and results within data.

There also exist more detailed aspects of music that relate with math on an even more specific level; we will get to that later.

Learning Math Through Music

Other than patterns in music, there also exist many other connections between music and mathematics. Another of these relations can be found in the way people learn and how their brains associate things. A study in 2005 suggested that math and music are related in the brain from a very early age. Because people of all ages have a psychological reaction to music, young children can be introduced to and more easily learn mathematics by just unconsciously taking in the patterns, sequences, and counting within music. Other studies have found that children become more engaged when listening to musical beats and that the exposure to these beats, as well as early education and identification of these patterns can allow younger children to recognize patterns both visually and audibly, which in turn hones their cognitive abilities.

While the purpose of musical education may not be to teach mathematics, it nonetheless aids in many ways. When babies are sung lullabies, or songs are sung at parties, young children are able to pick up on these patterns within their daily lives. This builds up the understandings of various mathematical aspects that require the recognition of patterns by making the children make relationships between two different things within their heads. As a song progresses, too, the repetition and development of certain progressions or patterns can aid children in understanding repeating patterns and growing patterns when the time comes to learn math.

Case Study: Pop Music

Ever wonder why a song is so catchy? Did an artist get lucky in thinking of a good song? Why are some songs so much more popular than others?

It turns out that Pop music, not unlike the music from earlier eras like the Baroque and Classical Eras, contains patterns as well. These patterns may not be as complex as the ornaments and counterpoint from the Baroque era, or as clear as the Alberti bass of the Classical era, but they exist nonetheless.

The biggest, and most important pattern in Pop music today (disregarding unmelodic styles of music such as hip-hop), is arguably, the chord progression. Although each song is unique in itself, with different lyrics, different beats, et cetera, but the chord progressions, the progressions of keys that sections of songs go through, actually demonstrate similarities across a wide range of songs. For example, if you replaced the accompaniments or backing tracks to pop songs with just chords played on a piano, you would realize that many songs use the same four chords in the same progression, and other songs perhaps a slightly different progression, but yet the same basic chords. This is demonstrated very well by musical comedy group Axis of Awesome in their song, “Four Chords” (caution: there is a vulgarity near the end of the song).

The Alberti Bass

Back in the 18th century, when piano music was popular and creative, composers needed to develop new patterns and ways to convey music in order to captivate audiences. As the Baroque period transitioned into the Classical era, the style of music changed in a way that reflected social norms and events occurring at the time. While still utilizing the musical devices such as the contrapuntalism of the earlier eras, Classical music become much simpler and lighter. To appeal to the public with this simple music, the Alberti bass was largely utilized, especially among keyboard music pieces during the Classical era. Although it was not developed by Domenico Alberti himself, it was named after him because he was the first to extensively use the device. Mozart too was a notable user of the Alberti Bass.

The Alberti Bass is an accompaniment that is often played by the left hand on a keyboard or lower instruments in an ensemble. The accompaniment consists of a broken chord pattern, where the 3 notes of a triad or inversion (1st, 3rd, and 5th), are played in the order 1, 5, 3, 5 (or 1, 5, 3, 5, 3, 5 in triple meter time signatures). The pattern itself not only allowed for a new form of music but also created a smooth and sustained accompaniment that sounded nice on a wide variety of instruments.

The Alberti bass as can be seen in the left hand (bass clef) http://www.music.vt.edu/MUSICDICTIONARY/texta/AlbertiBass.html

Case Study: Johann Sebastian Bach

Inventions and Sinfonias
Bach’s Inventions and Sinfonias made heavy use of the counterpoint in his pieces, with many repeated melodies that can be heard in both the right and left hands of the pieces. He wrote 15 of each, each invention and sinfonia is in a different key and thus are at levels of varying difficulty.

Bach’s Invention No. 1, http://solomonsmusic.net/bachin1a.gif

Crab Canon
The Crab Canon is a remarkable composition by Bach that not only uses a form of the counterpoint, but uses it in a unique way, such that the piece can be played forwards and backwards. But not only that, the Crab Canon can be played forwards and backwards at the same time to create a unique harmony and impress many people. This is demonstrated in the video below.

https://www.youtube.com/watch?v=xUHQ2ybTejU

Effects of Patterns in Baroque Music

Contrapuntalism
When we look at patterns in music, one of the necessary questions we must ask ourselves is: What effect does this have on the piece? While many people value chords and harmony in music, contrapuntalism does not necessarily rely on that. Instead it just flows with the juxtaposed melodies, with chords sounding where notes incidentally happen to be in the right place. The contrupantalism often gives the piece the ornamented yet somewhat theatrical feel that characterizes many Baroque pieces. Examples of forms that use contrapuntalism are the canon and fugue.

Ornamentation
While not officially a pattern in music, ornamentation was the defining characteristic of Baroque music. Different from other ornaments in music, these ornaments were not completely necessary in the piece and often happened on the beats of the piece, which contributed to the grand theatricality that characterized many Baroque pieces.