Karan Takar Senior project for Jazz Performance
I have decided to share some of my students work in my blog. The goal is to generate discussion and curiosity about diverse topics in music. Also you will realize the brilliant minds that surround me on a daily basis...
Have fun reading about this musical explorations.
Music through the eyes of the Fibonacci Sequence
The Fibonacci sequence is one of the most famous sets of numbers ever discovered and explored in mathematics. The sequence was discovered by the Italian mathematician Leonardo of Pisa, who proposed the idea as an answer to a question about the growth of a rabbit family in his book Liber Abaci. The rule determining the series is that any number is equal to the sum of the previous two numbers, with the first numbers being 1 and 1 (the first few terms are 1, 1, 2, 3, 5, 8, 13, 21, and 34). The most important facet of the Fibonacci sequence for the purpose of this investigation, however, is the Golden Ratio. The ratio, also known by its Greek name φ, equals 1.618 ((1 + sqrt(5))/2) and is the value that the ratio of consecutive terms of the Fibonacci sequence approach as the sequence reaches very large numbers (34/21 = 1.619, 55/34 = 1.6176, 89/55 = 1.6181, etc;). Now enough of the introduction and on to the music!
Our first stop in our mathematical journey will be the marvelous musical miniature known as Frederic Chopin’s Prelude, specifically Prelude No. 1 in C major. For the purposes of this piece, we will focus only on the melodic line in the right hand (it’s the top voice in the following picture. Note that the piece lasts exactly 34 measures: this is our first hint that the Fibonacci sequence may play a role here. The first 28 measures of the piece contains two notes separated by a second (major or minor, with the whole note harmonizing with the left hand). Of specific importance to us is the position of the climaxes in the melody line. The melody begins with the note sequence G – A, rises to a temporary climax at E – D in measure 5 (a Fibonacci number), and then eventually hitting the biggest climax on the D – C notes in measure 21. Not only is 21 a number belonging to the Fibonacci sequence, the ratio 34/21 = 1.619 is extremely close to the golden ratio! In fact, a large number of Chopin’s preludes and more contemporary pieces by composers such as James Tenney make use of this pleasing divide between the build up of a piece and the winding down stage.
Another interesting aspect of music involving the Golden Ratio is the structure of the form of a piece. The most common form structure in music is a binary form (A – B), so let’s take a look at these pieces in particular. There are two general classes of binary form: equal binary form, in which the A and B sections are roughly equal in size, and unequal binary form, where there is a large difference in the length of the A and B sections (equal binary form is the preferred form construction in most jazz music using a binary form). The question we ask is how does a composer assign a length to the two sections of an unequal binary form? The answer, as we will discover shortly, is that composers tend to split the two sections proportional to the Golden Ratio. One of the best examples of this Golden construction can be found in the first movements of the Piano Sonatas of none other than Mozart himself. Mozart’s sonatas were developed in the sonata-allegro form, an invention of the late 1700s. The sonata-allegro form consists of two basic sections, each of which is repeated during the movement: the first part is the exposition, where the musical elements of the piece are first presented (it is repeated so that the listener can fully grasp what is being presented in the piece). The second part of the form contains the development, where the elements of the exposition are distorted and glued together in different ways to increase tension towards the climax, and the recapitulation, where the piece returns to the head, but with a number of subtle changes. Thus the form of the first movement of a Mozart sonata looks like this (this is called rounded binary form because the second part contains a repeat of the first):
||: A :||: B A :||
Out of the 18 piano sonatas composed by Mozart, 17 use the sonata-allegro form (the other one is composed of variations on a theme). A study of the measure length of the respective sections of the sonatas shows that the ratio of second section length to first section length is exactly equal to the Golden Ratio in six of the sonatas (35%), and another 8 (47%) are so close that they are considered to round to the Golden Ratio.
Why does the division of the form into Golden Ratio sections sound so appealing to the listener, and why does jazz not use this form?
Most music students in America struggle with compound additive rhythms such as 3 + 2 + 3/9 (3 beats stressed added to 2 unstressed and then 3 more stressed), but in countries such as Bulgaria these meters are common in folk dance songs and newer popular work. Why does the Western world focus on divisive (breaking the measure down into beats) rhythms while Eastern Europe has developed additive rhythms?
Have fun reading about this musical explorations.
Music through the eyes of the Fibonacci Sequence
The Fibonacci sequence is one of the most famous sets of numbers ever discovered and explored in mathematics. The sequence was discovered by the Italian mathematician Leonardo of Pisa, who proposed the idea as an answer to a question about the growth of a rabbit family in his book Liber Abaci. The rule determining the series is that any number is equal to the sum of the previous two numbers, with the first numbers being 1 and 1 (the first few terms are 1, 1, 2, 3, 5, 8, 13, 21, and 34). The most important facet of the Fibonacci sequence for the purpose of this investigation, however, is the Golden Ratio. The ratio, also known by its Greek name φ, equals 1.618 ((1 + sqrt(5))/2) and is the value that the ratio of consecutive terms of the Fibonacci sequence approach as the sequence reaches very large numbers (34/21 = 1.619, 55/34 = 1.6176, 89/55 = 1.6181, etc;). Now enough of the introduction and on to the music!
Our first stop in our mathematical journey will be the marvelous musical miniature known as Frederic Chopin’s Prelude, specifically Prelude No. 1 in C major. For the purposes of this piece, we will focus only on the melodic line in the right hand (it’s the top voice in the following picture. Note that the piece lasts exactly 34 measures: this is our first hint that the Fibonacci sequence may play a role here. The first 28 measures of the piece contains two notes separated by a second (major or minor, with the whole note harmonizing with the left hand). Of specific importance to us is the position of the climaxes in the melody line. The melody begins with the note sequence G – A, rises to a temporary climax at E – D in measure 5 (a Fibonacci number), and then eventually hitting the biggest climax on the D – C notes in measure 21. Not only is 21 a number belonging to the Fibonacci sequence, the ratio 34/21 = 1.619 is extremely close to the golden ratio! In fact, a large number of Chopin’s preludes and more contemporary pieces by composers such as James Tenney make use of this pleasing divide between the build up of a piece and the winding down stage.
Another interesting aspect of music involving the Golden Ratio is the structure of the form of a piece. The most common form structure in music is a binary form (A – B), so let’s take a look at these pieces in particular. There are two general classes of binary form: equal binary form, in which the A and B sections are roughly equal in size, and unequal binary form, where there is a large difference in the length of the A and B sections (equal binary form is the preferred form construction in most jazz music using a binary form). The question we ask is how does a composer assign a length to the two sections of an unequal binary form? The answer, as we will discover shortly, is that composers tend to split the two sections proportional to the Golden Ratio. One of the best examples of this Golden construction can be found in the first movements of the Piano Sonatas of none other than Mozart himself. Mozart’s sonatas were developed in the sonata-allegro form, an invention of the late 1700s. The sonata-allegro form consists of two basic sections, each of which is repeated during the movement: the first part is the exposition, where the musical elements of the piece are first presented (it is repeated so that the listener can fully grasp what is being presented in the piece). The second part of the form contains the development, where the elements of the exposition are distorted and glued together in different ways to increase tension towards the climax, and the recapitulation, where the piece returns to the head, but with a number of subtle changes. Thus the form of the first movement of a Mozart sonata looks like this (this is called rounded binary form because the second part contains a repeat of the first):
||: A :||: B A :||
Out of the 18 piano sonatas composed by Mozart, 17 use the sonata-allegro form (the other one is composed of variations on a theme). A study of the measure length of the respective sections of the sonatas shows that the ratio of second section length to first section length is exactly equal to the Golden Ratio in six of the sonatas (35%), and another 8 (47%) are so close that they are considered to round to the Golden Ratio.
Why does the division of the form into Golden Ratio sections sound so appealing to the listener, and why does jazz not use this form?
Most music students in America struggle with compound additive rhythms such as 3 + 2 + 3/9 (3 beats stressed added to 2 unstressed and then 3 more stressed), but in countries such as Bulgaria these meters are common in folk dance songs and newer popular work. Why does the Western world focus on divisive (breaking the measure down into beats) rhythms while Eastern Europe has developed additive rhythms?
posted by Francisco Pais at
6:16 AM
1 Comments:
Mathematics in Jazz and Music
This is really interesting stuff. It is neat how you were able to take something from math and find a way to incorporate that into music. It is kind of weird how perfectly the Fibbonacci Sequence fits into some music that we play all the time we are playing jazz. Do you know if there is math in other songs and if there are any other math theorems in music? Or is there is math in our instruments?
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