The Golden Ratio and Fibonacci Sequence in Music

Microsoft Word - Ryan_Blankenship_Honors_Thesis.docxThe Golden Ratio and Fibonacci Sequence in Music
A project completed in partial fulfillment of the requirements for the Honors Program
by
Committee:
James Cottrill, Ph.D., Project Advisor Angela Heck Mueller, D.M.A., Reviewer John Marazita, Ph.D., Reviewer, Honors Committee
Accepted by
The Golden Ratio and Fibonacci Sequence in Music
The Fibonacci sequence and golden ratio are fascinating ideas to look at on their
own. They are analysed a lot in physical art such as paintings and nature, however, not
much is researched in their connection to music. You have probably heard a friend or two
talk about this connection, but you would not know how factual their statement is because
of this lack of prominent research. This led to my attempt to go in depth and present those
connections. The golden ratio comes from the Fibonacci sequence, so it is important to
incorporate both ideas together. Their connection to music is outlined in detail to help
understand what connection lies between the two and music. Then, historical information
is provided to show what prominent figures used the golden ratio and Fibonacci sequence
in their music. Lastly, I formulated an analysis of the top 25 of the Billboard top 100
decade-end songs of the 2010s to gauge whether those same mathematical and historical
ideas are used in popular music today. I wanted to know, is the golden ratio apparent in
popular music today? Does this ratio appear at the climax of the song or something
different? Beyond that I wanted to see if that relationship had some sort of significance or
if it just was an idea with trivial connections.
Keywords: Fibonacci sequence, golden ratio, music, mathematics
Introduction:
Popular musical theorist, Ern Lendvai, was one of the first authors to discuss how the
golden ratio and Fibonacci sequence were used in the composer Béla Bartók’s music. In
a journal article by Johnathan Kramer, he referenced Lendvai’s book, Bela Bartok: An
Analysis of His Music, and that he talked about how, “there are 55 bars before the main
climax” in one of Bartok’s pieces (Kramer, 1973, p. 120). These 55 bars were out of 89
bars in the piece, making the climax where the golden ratio is in the piece. This inspired
the idea to look for the connection of the golden ratio and Fibonacci sequence to other
pieces of music and modern music. Thus, I began creating a mathematical and musical
foundation for an analysis to test where the golden ratio is in a song and to see if that
point is significant. I wanted to answer two questions. Is the golden ratio a factor in
popular music today? Does this ratio appear at the climax of the song or something
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different?
Mathematical Basics:
Before looking at the history and modern examples of the golden ratio and Fibonacci
sequence in music it is important to understand the basics of each of these ideas and
their connections to music. The golden ratio is an irrational number that uses the Greek
letter φ as a symbol and relates to the geometry of similar rectangles. Like with
triangles, if all the ratios of the sides of two rectangles are equal then the rectangles are
similar. Working from a smaller rectangle, if you add a square, whose side lengths are
the length of the longer side of the rectangle it is possible for the larger rectangle to be
similar to the small one. Below is a diagram to help visualize these similar rectangles.
When the rectangle with sides a and b is similar to the rectangle with the sides a + b and
a then you get a golden rectangle.
Figure 1:
Note. From Golden Rectangle by anonymous, 2021.
This is where you can see the mathematical equation for the golden ratio coming to
fruition: = = with > > 0. By replacing a with and b with 1 in this
equation and cross multiplying we can find that = + 1. Solving this equation in
its quadratic form, − − 1 = 0, gives the exact value of φ which is √ .
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The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144… Each
successive number in this sequence can be found by adding the previous two terms
together. The golden ratio in decimal form is approximately 1.618033988… When
dividing a Fibonacci term by its previous term in the sequence the golden ratio can be
seen in the answer. The farther into the Fibonacci sequence the terms are the more
accurate the value is. For instance: 13/8 = 1.625; 21/13 = 1.61538; 34/21 = 1.61905;
55/34 = 1.61764; 89/55 = 1.61818; 144/89 = 1.61798; and so on, converging to φ. An
important note that will be later incorporated when looking at the historical connections
and modern music analysis is how = − 1 = 0.618033988… This number is where
the separation between the square and rectangle happen above in figure 1 and will be
used to find where the “golden bar” is in a musical piece and song.
Connection to Music:
It is important to talk about the difference between Fibonacci numbers and the
Fibonacci sequence. In this section are a few ideas that involve the next 6 numbers after
the first two terms of the Fibonacci sequence, 1, 2, 3, 5, 8, 13. While these are numbers
are present in the sequence they are not in a sequential order. This means that their
prominence should be taken as numbers of the sequence and not as significant as if they
were to come up in the sequential order from each other. This same idea applies
throughout the paper and should be remembered when reading.
When looking at a keyboard the entries in the Fibonacci sequence are present.
There are 13 keys in an octave. A major scale can be created in that octave using 8
notes. Within that major scale the 1st, 3rd, and 5th notes create the most basic major
chord. Major scales also use a pattern of moving up chromatically by intervals of 1 and
2 notes on the keyboard, otherwise known as half and whole steps, respectively. Also,
the ratio of white to black keys within any octave is 8:5. Looking at figures 2-4 below
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there is an interesting property when mapping the 8 notes of the major scale onto the 13
notes of the chromatic scale. In figure 2, the 8 notes of the C major scale are numbered
1-8. In Figure 3, the 13 notes of the C chromatic scale are numbered 1-13. Figures 2 and
3 are then superimposed to create figure 4, subtracting out all the non-Fibonacci
numbers and the number 2 from figure 3. This gives us Fibonacci sequence ratios 1:1,
3:2, 5:3, 8:5, and 13:8.
Figure 2 Figure 3 Figure 4
Note. From Music and the Fibonacci Sequence and Phi by Gary Meisner, 2012.
More, an interesting calculation can also be made when comparing notes, Fibonacci
ratios, and pitches in music. The audio frequency of 440 Hz is the standard for musical
pitches and corresponds to the note A right above middle C on a keyboard. 440 Hz
became this standard in 1975 by means of the International Organization for
Standardization. This standard is reviewed every five years and was last reviewed in
2017. Information regarding this standard can be found on the International
Organization for Standardization under ISO 16:1975. Another situation to note is that
on the internet you can find hundreds of thousands of articles and videos talking about
432 Hz vs. 440 Hz for which should be the standard pitch. Some claim 432 Hz is better
to listen to and the more natural frequency. Others claim there is no real difference and
that there is no significance to the pitch standard. Thus, in recent years there has been a
series of medical studies that claim music tuned to 432 Hz has benefits to physical and
mental health. These studies found results such as, in 33 volunteers, slight decreases in
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blood pressure and heart rate (Calamassi & Pomponi, 2019) and for dentist patients,
lower salivary cortisol levels (Aravena et al., 2020). These studies have popped up in
the last 2 years which indicates a present scholarly interest in this tuning difference.
Considering 432 Hz being the only challenger to the 440 Hz standard pitch it was
included in the calculations below.
Table 1
Fibonacci
5/2 1100 1108.72 C#6 1080 1088.57
5/3 733.33 739.99 F#5 720 726.53
5/8 275 277.18 C#4 270 272.14
8/3 1173.33 1174.64 D6 1152 1153.30
8/5 704 698.46 F5 691.2 685.76
Note. Adapted from Music and the Fibonacci Sequence and Phi by Gary Meisner, 2012.
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Something that sticks out in the table are how in the 3/8 row, the highlighted numbers
on table 1 are approximately 100 times φ. While some numbers that were calculated are
not exactly what their natural frequencies are, the largest difference, F#5, has 0.908%
difference between its calculated and natural frequency. So, there is less than a 1%
difference between these calculated frequencies. Back to the first point made in this
section, these are not exactly Fibonacci ratios since they are not sequential ratios like
13/8 or 21/13 but instead ratios created with Fibonacci numbers.
Another relation to music from the Fibonacci sequence is how musical
instruments are created. φ was used to create Stradivarius violins, the most sought after
and expensive violins in the world, one of which sold for $3.6 million (Stradivarius
Violin Price, 2020). Below in figure 5 are a series of distances between parts of the
violin that are labelled. First, a1 is the length of the neck, pegbox, and scroll and a2 is
the length of the entire body. Next, b1 is the length of the neck and b2 is the length of
the upper bout and waist. Lastly, c1 is the length of the waist and c2 is the length of the
= = =
Figure 5:
Note. From The Fibonacci sequence and the golden ratio in music by Robert Van Gend.
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Using all this information will help as a reference to look at when going into the
musical history of the Fibonacci sequence and golden ratio as well as my analysis of
modern music.
Back to Ern Lendvai’s analysis of Béla Bartók’s music, Lendvai’s book, Bela
Bartok: An Analysis of His Music, has largely been discredited. One such source I found
was written by Johnathan Kramer just 2 years later in 1973 for the Journal of Music
Theory. He alludes to how Lendvai messes with musical notation in a way that made
the Fibonacci sequence look present when it was not. One such example Kramer goes
over on page 120 is:
…the climax occurs after 55 bars (i.e., in bar 56) while the mutes are removed
and the timpani enter in bar 34 (i.e., after 33 bars). For another example, a major
textural change (under- lined by the celeste entrance) occurs at bar 78, but it is
omitted from Lendvai's schematic, perhaps because it does not fit his analysis.
Another author, Roy Howat, talks about how “number of bars begun are sometimes
confused with bars completed” (Howat, 1983, p. 72). This distinction is important
because it shows how a piece can be manipulated by 1 or 2 bars to look like the golden
ratio is a factor. This is exactly what Lendvai does in his analysis and is pointed out by
Kramer in the quote on page 120 above.
These ideas led me to question if the golden ratio is something of significance in
music. Lendvai brought up how it was prominent in a certain classical musician’s
repertoire but that was quickly marginalized. I needed to see if the golden ratio and
Fibonacci sequence were in other classical musician’s pieces and if I could use those
ideas in my analysis. This led me to the works of Chopin, Bartók, and Handel, three
popular classical musicians. These composer’s music and another lesser-known
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composer, Stockhausen, will be looked at in the next section to supplement the lack of
credibility of Lendvai.
Historical Connections to the Fibonacci Sequence and Golden Ratio:
In the musical connections subsection of the introduction was a small piece about how
different authors analysed the composer Béla Bartók. But there are other classical pieces
of music that refer to the Fibonacci sequence in some form, such as in the time
signatures, pitches, or rhythmic structures, to name a few areas. Below, in figure 6, is
sheet music describing points in Chopin’s Prelude No. 1 C major, Op. 28 where the
Fibonacci sequence is present. The annotations on the sheet music were done by
Povilionien, 2017.
Figure 6:
Note. From Musica Fibonacciana: Aesthetic and Practical Approach by Rima Povilionien, 2017.
One important annotation to point out is that the lowest and highest notes played in the
piece are in measures numbered in the Fibonacci sequence, 8 and 21, respectively. In
this prelude by Chopin, the climax happens in the “golden measure” of the piece. This is
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measure 21 out of 34 which when dividing the former by the latter give the number
0.61765 or about φ/1. “Golden measure” is referred to here, but “golden bar” will be
referenced in my analysis later. Bar and measure can be used synonymously. Next, a
popular classical piece the “Hallelujah” chorus, from Handel’s Messiah. This piece of
music lasts for 94 bars and the climax of the piece is when “King of Kings” is sung.
This happens from bars 57 to 58 which is about 8/13 or 0.61538 way through the
“Hallelujah” chorus. Another thing to note about the piece is that 8/13 through the first
57 bars starts the theme, “The kingdom of glory” (Perition, 2013). Looking at Béla
Bartók’s Music for Strings, Percussion, and Celeste the opening xylophone
instrumentation follows the rhythmic pattern, 1, 1, 2, 3, 5, 8, 5, 3, 2, 1, 1 (Roberts,
2018).
There are some lesser-known composers that also used the Fibonacci sequence
and golden ratio in their music. One such composer, Karlheinz Stockhausen, used
Fibonacci numbers for time signatures in a 13 stretch of bars in a piece of his called
Klavierstück IX. These time signatures are , , , , , , , , , , ,
, , respectively (Kramer, 1973). The most fascinating use of the Fibonacci sequence
in the piece is during the coda. There are groups of bars in this part of the piece that are
easy to overlook at first glance. This will be expressed next as (group with x number of
bars) → (number of eighth notes). 1 bar group → 1 eighth note, 2 → 3, 3 → 6, 4 → 11,
5 → 19, 6 → 32, 7 → 53, 8 → 87. The eighth notes here are the Fibonacci sequence
numbers minus 2, so what? Well, the difference between each of these eighth notes are
2, 3, 5, 8, 13, 21, 34, the Fibonacci sequence (Kramer, 1973). These ideas are furtherer
analysed by Kramer in his article The Fibonacci Series in Twentieth-Century Music and
show some of the coolest ways you can use the Fibonacci sequence in music. He also
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talks about the Italian composer Luigi Nono to further his analysis of the Fibonacci
sequence in classical music.
Looking more modern, in a song by the band Tool called “Lateralus,” “the
syllables in the first verse count 1, 1, 2, 3, 5, 8, 5, 3, 13, 8, 5, 3... The time signatures of
the chorus change from 9/8 to 8/8 to 7/8” (Periton, 2013, p. 2). The time signatures of
the choruses are interesting to point out because 987 is a number in the Fibonacci
sequence. In Lady Gaga’s song Perfect Illusion she incorporates a key change at about
the φ point in her song. The song lasts 179 seconds and her key change happens at 111
seconds. 179/111 is approximately 1.61261 which is as close to the golden ratio as you
can get with a song length of 179 seconds. While these are interesting instances where
the Fibonacci sequence and golden ratio are present, I want to look at a general body of
musical pieces to see if most of them use these same ideas.
Methods and Analysis:
Before diving into my analysis, I must first establish some parameters. The purpose of
this analysis is to see if the golden ratio is apparent in “good” or popular music. Good
music in this case will be described as the top songs on the Billboard charts. The criteria
for Billboard’s top music were briefly established in a quote on Billboard’s website,
"Generally speaking, our Hot 100 formula targets a ratio of sales (35-45%), airplay (30-
40%) and streaming (20-30%)" (Trust, 2014). Billboard is the standard for music charts,
which is why their top 100 list was used. The songs in the table below are the top 25
songs, in order, of the Billboard top 100 of the 2010-2019 decade. As described in the
quote above, the criteria for these songs to be in top 100 is formulated by Billboard and
in the case of this decade end list, does not fluctuate with any new music purchases,
airplay, or streams. Due to in-access to sheet music since it was behind a hefty paywall,
I had to download the official songs from the internet and then convert them into WAV
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files. These files were then put into a program called Studio One, a digital audio
workstation or DAW, that I use to record my own music in. This allows me to find and
change the tempo for each song to be able to find the total measure count, helping build
the foundation of my analysis.
Golden Bar:
To test where the golden ratio of the song is I will be multiplying the total bars in the
song by 0.618, the φ point of the song. Why this number? Well, as stated above, 0.618
is the reciprocal of φ which means we can see where the golden ratio is within a song
length of 1 whole or 61.803% through the song. Then I will look at the significance of
that point. Will it be the climax or something different? This is what is labelled as the
“golden bar” on the table. When looking at bar ranges in the table the ending number of
the range includes that bar. For instance, in “Uptown Funk” the breakdown of the song
goes from the start of bar 81 to the end of bar 98. I also measure how far away the
“golden bar” is from significant portions of each song to see if the golden ratio
potentially measures something other than the climax. The “negative” bars are when the
point in the song closest to the golden bar precedes that bar.
Climax/Breakdown/Build-up:
In literature, the climax of story is the turning point or resolution of the highest tension.
This same principle applies to music. In a song this high point of tension could be the
build-up of instruments in a song, a scream or high instrumental note, or a pitch change,
to name a few. The release of this tension comes right after the climax, in music’s case,
the bar after the climax. More criteria came from a website talking about climax was
also used to gain a deeper understanding of possible ways the climax of a song can be
found, so that there was as little subjectivity as possible in the analysis.i The climax is a
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lot of times the last bar of the breakdown or build-up due to the tension it builds which
is why…