Dubspot’s Rory PQ explores the relationship between the Fibonacci Sequence and the Golden Mean in music.
The Fibonacci Sequence and the Golden Mean reveal an extraordinary phenomenon that occurs throughout nature, art, music, and mathematics. Also known as the Golden Ratio or Golden Section, the Golden Mean is a mathematical ratio that artists, architects, and musicians have used to craft their art form for centuries.
Understanding the Fibonacci Sequence
The Fibonacci Sequence is a series of numbers that exhibits a fascinating numerical pattern that was originally discovered by Leonardo Pisano Bigollo. Famously known as Fibonacci, Leonardo was a 13th-century Italian mathematician that popularized the Hindu–Arabic number system in the Western World and introduced Europe to the sequence of Fibonacci numbers.
The numbers in the Fibonacci Sequence are formed by taking the sum of the previous two numbers in the series to get the next number in the sequence. For example, the sequence starts by adding 0 and 1 together to equal 1. Then add 1 and 1 together to equal 2. Proceed in the same manner, adding the last two numbers of the sequence together to get the next number. The sequence of numbers begins to looks like this:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144…
The relationship between the Fibonacci Sequence and the Golden Ratio occurs when you take any two adjacent numbers and divide the smaller number into the larger number, which results in a number close to 1.618. The further along the sequence you go, the resulting division of the small number into the next larger number gets closer and closer to 1.618. That number is commonly symbolized by the Greek letter “Phi” or otherwise known as the Golden Mean.
You’re probably asking yourself, what the heck does this have to do with music? Well, the ratio 1.618, has been used by architects, artists, and musicians as a way to structure their artistic creations. For example, the Golden Mean is found in the length and height of the Great Pyramid of Giza. Leonardo Da Vinci called the Golden Mean “Divine Proportion” and featured it in many of his paintings. In his famous “Last Supper” painting he used it to define the dimensions of the table to the proportions of the walls and windows in the backgrounds to achieve balance and beauty. In music, the Golden Mean can be found in many compositions mainly because it is a “natural” way of dealing with divisions of time. Legendary composer Bela Bartok used the Golden Mean to structure one of the best-known compositions “Music for Strings, Percussion and Celesta,” so that musically important events occurred in bar numbers represented by the Fibonacci Sequence of numbers.
How the Fibonacci Sequence Influences Music
Now that we have a general understanding about the Fibonacci Sequence let’s explore how it influences music and instrument design.
Numbers in the Fibonacci sequence can be seen on a piano keyboard and in the musical scales. For example, scales along a piano keyboard are composed of thirteen keys in the span of a full octave which consists of eight white keys and five black keys that are arranged in groups of two and three along the keyboard. Fibonacci numbers are also present in the notes that make up musical scales. For example, the Pentatonic scale has five notes, the Diatonic scale has eight notes, and the Chromatic scale has thirteen notes. In addition, the 1st, 3rd, and 5th notes in any scale create the basic foundation of chords. Do you see the pattern? There are even more relationships, but we don’t want to go to deep.
Sylvain Lalonde demonstrates the relationship between music harmony and the Fibonacci Sequence further in this video.
The most pleasing combination of frequencies are those having simple ratios of harmonics that derive musical intervals. Going down the rabbit hole further, we can even associate the Fibonacci numbers and the Golden Mean with note frequencies. Notes in the Chromatic and Diatonic scales are based on natural harmonies that are created by ratios of frequencies. Looking at the chart below, you can see that the first seven numbers of the Fibonacci Sequence (0, 1, 1, 2, 3, 5, 8) are related to key frequency ratios.
Let’s talk about harmonics before going all Fibonacci. Harmonics are a series of frequencies that are multiples of a fundamental frequency. A pure note consisting entirely of one frequency will sound boring. This series of related frequency combinations sound the most pleasing because there are multiple overtones that work together to make a sound more interesting than a pure note consisting entirely of one frequency. For example, let’s use A3 as our fundamental frequency. The note frequency for A3 is 440 Hz. The second harmonic will be twice this frequency. If we double 440 Hz to find the next harmonic frequency, we get 880 Hz, which is the note frequency for A4. The third harmonic will be three times the fundamental frequency. In this example, the third harmonic is 1,320 Hz, which is the note frequency for E5. The third harmonic is a perfect fifth above the second harmonic because the E is seven semitones above the fundamental note and includes the first five notes from the major and natural minor scales. Hope you’re still with me!
The Fibonacci numbers relate to ratios of harmonic frequencies because a root note has a ratio of 1/1, its octave has a ratio of 2/1, a fifth above it has a ratio of 3/2 and so on for the other notes in the scale. These simple ratios of harmonics all contain numbers found in the Fibonacci sequence.
Relationships between Fibonacci numbers and the Golden Mean are frequently found when analyzing the structure of compositions and musical patterns dating back several centuries. For example, the climax of many songs or even an important measure where the song changes significantly such as the bridge often occurs near the songs Phi point. As we mentioned earlier, Phi is the basis for the Golden Ratio, Section or Mean. Considerable evidence suggests famous composers such as Mozart, Beethoven, Bella Bartok, and many others have used Fibonacci numbers and the Golden Mean to compose measures of music and structure their compositions.
The Phi point can be determined by taking the number of measures of a song and multiplying them by 0.618. More often than not the result will mark the beginning of a measure where a significant change occurs. I tested this theory on Disclosure’s hit song “Latch.” Using Ableton Live, I determined the song is roughly 129 measures long. I then multiplied 129 by 0.618 and got approximately 80. Sure enough, measure 80 marked the beginning of a tension building bridge that occurs before the final drop. Hear for yourself; measure 80 starts around 2:37.
Composer and teacher Gary Ewer developed a formula that works well to determine where the bridge or highest point in a musical piece often occurs in most modern and historical compositions. This clever musical equation takes the full length of a song and converts it in seconds, which is then multiplied by Phi and converted back to minutes to reveal an approximate time where a significant change may occur in that song. For example, Disclosure song “Latch” is about 4:13 long. To find Phi, we would first convert the minutes into seconds and then add the remaining seconds to get a total amount of seconds for the song. Next, multiply the total amount of seconds by Phi, which is approximately 0.618. Lastly, divide the results by 60 to convert back to minutes, and then multiply the decimals by 60 to find the seconds. Combine the results to determine the time where the climax or bridge of the song may occur. For this example, the formula would like this:
4 x 60 + 13 = 253 (total seconds)
253 x 0.618 = 156.35
156.35 / 60 = 2.60 (2 minutes)
0.60 x 60 = 36 (36 seconds)
Resulting time: 2:36
At 2:36 the same tension building bridge begins that we explored earlier. Very interesting! The following video demonstrates this theory further.
Violins crafted by the master luthier Antonio Stradivari are famously known for their exquisite tonal quality and aesthetic form. Genuine Stradivarius violins are highly sought after and are the most valuable instruments in the string-playing world because of their unmatched sound. However, what is most amazing about his violins was that they were designed and build around the Golden Ratio. The proportions of the violin conform to the ratios of Phi. Even the spiral of a violin scroll reveals how precisely his instruments reflect the Golden Ratio.
Jody Espina, a highly regarded jazz musician and designer of Jody Jazz Saxophone and Clarinet Mouthpieces has applied Phi in the design of his premier saxophone mouthpiece, the JodyJazz DV.
“Every measurement was analyzed with Phi in mind and used when applicable. This included the length of the bore, the width of the shank walls, the beak of the mcp, the depth of the bore at the facing and others. The amount of harmonics in the sound, and therefore the projection of the mouthpiece, is huge. This eliminates the annoying shrillness that is associated with loud, bright mouthpieces.” – Jody Espina
The Recording Institute of Detroit claims they built a “Golden Section Studio” to Phi proportions. They have stated:
“The result of using the Golden Section in studio construction is a remarkable “even” quality with regard to frequencies. Your voice has pretty much the same frequencies present when you talk in any part of the room; the reverb has the same frequency spectrum as the direct sound. Drummers love the way their drums sound and record in this room. There is only approximately 33% of the surface area treated for acoustical absorption making this room quite live. It is a great room to record a distant mic on a lead guitar.” – Recording Institute of Detroit
Speaker Wire Design
The high-end cable manufacturer Cardas Audio applies the Golden Ratio to the design of their speaker wire which they call “Golden Section Stranding.” The individual conductor strands of their cables are arranged in a Phi proportion to the others allowing them to be uniquely musical and pure. According to Cardas:
“An infinitely indivisible progression known as the Fibonacci sequence or Golden Section is the key to resonance control. The ratio of Phi, or 1 to 1.6180339887…to infinity, is the Golden Mean, called Golden Ratio or Golden Proportion.
In Golden Section Stranding, strands are arranged so that every strand is coupled to another, whose note is irrational with its own, to dissipate conductor resonance. This creates a silenced conductor, allowing Cardas cable to produce the purest possible audio signal. No other cable geometry, no other conductor design, can create the listening magic of Golden Section Stranding.” – Cardas
While exploring the relationships between music and the Fibonacci Sequence, I discovered several other fascinating findings linking the numbers to all types of things. If you found this exploration interesting, I encourage you all to investigate this amazing phenomenon further.
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