1 The Golden Ratio and Fibonacci Sequence in Music A project completed in partial fulfillment of the requirements for the Honors Program by Ryan Blankenship May 8th, 2021 Mathematics Ohio Dominican University Approved by Honors Project Review Committee: James Cottrill, Ph.D., Project Advisor Angela Heck Mueller, D.M.A., Reviewer John Marazita, Ph.D., Reviewer, Honors Committee Accepted by Director, ODU Honors Program 2 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, Ernő 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 3 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: 휑= = with 푎>푏>0. By replacing a with 휑 and b with 1 in this equation and cross multiplying we can find that 휑 = 휑+1. Solving this equation in its quadratic form, 휑 −휑−1=0, gives the exact value of φ which is √ . 4 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 = 휑 −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 5 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 6 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 Note frequencies compared with Fibonacci number ratio frequencies Fibonacci Calculated Natural Note Calculated Natural Frequency Number Ratio Frequency Frequency Frequency when A when A = 432 Hz when A = 440 when A = = 432 Hz Hz 440 Hz 1/1 440 440 A4 432 432 2/1 880 880 A5 864 864 2/3 293.33 293.66 D4 288 288.33 2/5 176 174.62 F3 172.8 171.44 3/2 660 659.26 E5 648 647.27 3/5 264 261.63 C4 259.2 256.87 3/8 165 164.82 E3 162 161.82 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.