Note: This piece is the first in a series of academic/scholarly essays about "low-brow" or seldom respected art forms. As such, it's a bit more technical than most of my work. If you'd like to read something a little closer to what I usually write, please enjoy some of the older posts on this blog first, then get back to this one.
Introduction
The irrational number Phi is represented in integers by ~1.618 and is the only working integer of dividing a straight line into two segments, in which the ratio of the line to its larger segment is equal to the ratio of the larger segment to the smaller, and can thus be infinitely squared to smaller segments.
Phi’s relevance is in its persistent appearance throughout seemingly unrelated dynamic systems, be they in nature, animal biology, art, human physiology or otherwise. Phi, its reciprocal phi and their geometric equivalents have been found throughout popular works like The Last Supper and Da Vinci’s Vitruvian Man as well as patterns in sunflower seeds and cephalopod shells. This paper will specifically focus on understanding Phi and its use within the song “Lateralus” by the Los Angeles-based heavy metal musical act Tool.
Understanding Phi and phi in Integer Sequencing and Geometry
To find Phi in Euclidean geometry, a straight line (a) must be divided into two segments: one larger (b) and one smaller (c). The ratio of the length of the entire straight line (a) to the larger segment of the line (b) is the same ratio as the length of the larger segment (b) to the smaller (c) (see Fig. A). Mathematically, a:b::b:c. This is only possible when the first number in the ratio is ~1.618 times larger than the second.
Conversely, Phi’s reciprocal integer, phi, is Phi minus one. To find the length of b in the above example, one must multiply a by ~0.618. Then c is equal to b times 0.618, etc., squared down into the infinitesimal. This brings us to the much simpler analytical tool of 61.8% to examine phi.
Keeping in mind the Phi ratio 1.618:1, which is referred to as The Golden Ratio or The Divine Ratio, is essential to its applications throughout the known universe. It was first discovered by Leonardo da Pisa, alias Fibonacci, b. 1175 A.D.. Fibonacci was presented with a problem at a popular math competition – “Beginning with a pair of rabbits, if each productive pair bears a new pair, which becomes productive when they are one month old, how many rabbits will there be after n months?” (Knott, Quinney) Fibonacci’s resulting equation was that each number in the sequence (Fn), which would have to be a whole number, would be equal to the addition of the previous two numbers in the sequence (represented as Fn-1 + Fn-2), starting with 0 and 1. It follows that 0 + 1 = 1, giving the first three numbers 0, 1 and 1; of course 1 + 1 = 2. The next step is that 1 + 2 = 3. To explain the equation Fn = Fn-1 + Fn-2, we would simply plug in the numbers 3 = 2 + 1. The integer sequence developed through this equation’s whole integer application is known as the Fibonacci Sequence. It proceeds as follows.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…
As the Sequence progresses, the ratio of the newest number (Fn) to the previous number (Fn-1) rapidly approaches The Golden Ratio and only continues to do so. By the time any mathematician adds 34 and 55 to equal 89, the relationship between Fibonacci and Phi is clear: 89:55::~1.618:1.
The final two pieces of the puzzle lie in geometry. The Golden Rectangle is drawn using either the Golden Ratio or the Fibonacci Sequence. Utilizing our first explanation of the Ratio with line segments, the wide rectangle’s x-axis, or long horizontal lines, is our full line (a). It is divided into two segments: a longer (b) and a shorter (c). The wide rectangle’s y-axis, or vertical lines, are equal in length to b, so that the rectangle’s full length to height are a:b. This results in the Golden Rectangle. If we draw a vertical line through The Golden Rectangle where b meets c on the x-axis a, the resulting larger portion of the rectangle is a perfect square, in equal length and height of b, and c is a perfectly proportioned new Golden Rectangle, whose length and height can then be calculated and segmented the same way, as can its smaller rectangle and so on.
To understand The Golden Rectangle with The Fibonacci Sequence, imagine a rectangle as wide in number as a Fibonacci number and as tall as the previous number in the sequence. For this example, let’s use 21 wide by 13 tall. The width of the rectangle, 21, is arrived by adding 8 to 13, so imagine it’s now cut into two pieces: on the left, a square 13 x 13; on the right, a vertical rectangle 13 tall by 8 wide (see Fig. B).
This new rightmost rectangle contains essentially the same ratio or proportions in height to width (8 x 13) that was just used for the initial rectangle’s width to height (21 x 13), but turned 90 degrees. So we can start over and cut it into a square at its bottom and a rectangle at its top, which can then be cut, etc.
When we draw an arc from one corner of the Golden Rectangle to the opposite corner (where we “cut” the rectangle) and repeat for each new Golden Rectangle, we’re drawing the Golden Spiral (see Fig. C).
Presence in Everyday Life
Phi and phi are not only useful tools in algebra and geometry, but maintain a frequent presence in art and popular culture – usually by accident. In the hit British horror film 28 Days Later, the plot takes a sharp turn from a small group of survivors’ attempts at escape from a zombie-infested England to a mysterious martial law drama reminiscent of Lord of the Flies. If we equate the film’s total length in minutes and seconds as the first line a in the earlier examples of the Golden Ratio and Phi, and divide it into the same proportions of two segments (b and c), the change of focus in plot from zombies to lawlessness occurs within seconds of where b meets c, with b constituting about 62% of the film’s length and c the final 38%. The aforementioned 62% of the film’s duration is nearly exactly phi, 61.8%.
In the 1975 film Jaws, the titular shark is not seen on-screen until ~61.8% of the film’s duration has passed, as well. The John Wayne Western Stagecoach features Wayne’s decision to return to a group of wagon-traveling pioneers at this moment as well – again, nearly down to the second – performing the denouement leading to the film’s final conflict. Similar examples follow suit, with climaxes or major movements changing, in Beethoven’s Fifth and Ninth symphonies, The Last Supper, Vertigo and countless others.
In human physiology, the Golden Ratio is regularly present. Dividing the finger’s length over the length from the second knuckle to the fingertip is generally equal to The Golden Ratio, as is dividing the length of the arm, from shoulder to the fingertip, over the length of the forearm, from the elbow to the fingertip. Dozens of others are found throughout the human body.
Linking The Golden Ratio’s presence in the human body to that in art and media has led many to believe in the Ratio beyond coincidence, that artwork depicting it is more naturally appealing to humans. This in turn has generated several artists to utilize the Golden Ratio or the Fibonacci Sequence either as a means to an end of fame or fortune, or simply to express the artist’s interest in the subject.
Evidence in Tool’s “Lateralus”
One thorough example of Phi in recent musical history is the song “Lateralus” by Tool.
Tool is a progressive metal quartet from Los Angeles. Progressive metal, a sub-genre of heavy metal, is partly a successor to progressive rock, which included bands like Led Zeppelin, Yes and King Crimson. Progressive rock’s main characteristic of time signatures changing regularly within the same song distinguished it from other rock music at the time. Following the advent of heavy metal music, progressive rock translated with metal’s increased usage of loud guitars with overdrive and distortion, steadier bass guitar rhythms, faster drums and more abrasive vocals and became progressive metal.
Tool, at the forefront of the progressive metal scene, released their fourth album, Lateralus, in 2001. Lateralus, and specifically its title track, are in perfect keeping with Tool’s signature sound. Tool’s songs are written regularly in 5/4 time, usually eschewing pop music’s near constant 4/4 time signature. The song “Lateralus” itself does contain several sections in 5/4 and others in 4/4, but its primary refrain is a repeat of three separate signatures – 9/8, 8/8 and 7/8. These constitute a total of 24 even beats before starting again – 9, 8, and 7.
24 is not a “Fibonacci number,” but 987 is. The band’s drummer Danny Carey stated in an interview that this was a coincidence, but singer Maynard James Keenan’s vocal patterns are anything but that. In either of the song’s verses, the singer’s stanzas of lyrics are organized in groups or lines whose syllables strictly adhere to The Fibonacci Sequence, usually increasing or decreasing along the sequence itself. For example, his first lines are “Black / Then / White are / All I see / In my infancy / Red and yellow then came to be / Reaching out to me / Lets me see,” and are sung with deliberate pauses between lines (Keenan). This means the syllables sung proceed as 1, 1, 2, 3, 5, 8 before descending to 5 and 3 again; the first seven digits of the Fibonacci Sequence are 0, 1, 1, 2, 3, 5 and 8.
The following line begins at 13 syllables and works its way back down the sequence to three: “As below so above and beyond I imagine (13) / Drawn beyond the lines of reason (8) / Push the envelope (5) / Watch it bend (3).”
In addition, the second verse of the song plays out along similar lines as the beginning of the first, except that when Keenan arrives at the eight-syllable line “Red and yellow then came to be,” he descends the Fibonacci Sequence one step at a time back to one syllable per line before climbing back up to 13 and down again. “Red and yellow then came to be (8) / Reaching out to me (5) / Lets me see (3) / There is (2) / So (1) / Much (1) / More and (2) / Beckons me (3) / To look through to these (5) / Infinite possibilities (8) / As below so above and beyond I imagine (13) / Drawn outside the lines of reason (8) / Push the envelope (5) / Watch it bend (3).”
Of course it’s possible to look into any facet of the song and find what the seeker is looking for, if one tries hard enough. The second chorus of the song has the lyric “Feed my will to feel this moment, urging me to cross the line.” The first part of that line is eight syllables, and the second part is seven syllables. Though the first is a Fibonacci number, the seven syllables of the second part are sung “Urging – me to – cross the line,” which one could argue is 2+2+3, which are all Fibonacci numbers. However, this level of scrutiny feels unnatural at best, if not far-reaching and a bit obsessive.
Other examinations of “Lateralus” in search of Fibonacci prove futile as well. If we examine the song’s movements in terms of duration, as we did successfully for Jaws, 28 Days Later, Stagecoach and others, we find that 61.8% of the way through the song is about 5 minutes and 48.5 seconds, which is just shy of the halfway mark in the bridge, and not within several seconds of any changes in composition whatsoever. Also, although the song’s climax is mostly contained in eight exact lines of eight syllables each, it is paired with a couplet of 16 syllables (minus one), totaling in 95 syllables – also not a number in the sequence.
Analysis and Interpretation
However, in spite of – or perhaps because of – this obsessive and fruitless search, we find the meaning of the song and its relationship to Phi culture. “Lateralus” is a word invented by the band, although the Latin word “lateralis” means “of the side” (Dictionary.com Unabridged). Nearly all uses of the word “lateral” denote a movement to the side or a spreading out horizontally, including a lateral pass in football. Tool has often written songs and expressed their belief in expanding the human consciousness and gaining new perspectives, by any means necessary – whether thinking about a situation in life from the other person’s point of view, taking perception-altering drugs or, in this case, studying mathematics. Hence, it seems that moving laterally is being used as a metaphor for broadening our minds.
This idea is reinforced by the verses, when Keenan states “Black and white are all I see in my infancy / Red and yellow then came to be / Reaching out to me / Lets me see.” In the first verse, he then says “As below, so above and beyond I imagine / Drawn beyond the lines of reason / Push the envelope, watch it bend.” The metaphor being used here is that newborn babies can only see in black and white at birth, and shades of gray; one week after birth, babies can see in red, yellow, orange and green (Heiting). By this time, all these new possibilities about their world should fascinate babies. Likewise, at first, the singer – or the narrator, if we’re to believe he’s singing in character – had a limited view of the world, which he refers to as “my infancy.” As more colors became visible to him, which could easily mean as he gained new insight into the world, his imagination ran wild and he becomes excited at what could lay outside the frame of his logical mind. As he considers greater and more alien possibilities in the world, which are drawn “beyond the lines of reason,” he realizes just how much he’s “thinking outside the box,” to use a common phrase.
In the second verse, Keenan adds that seeing the colors “Lets me see / There is so much more / And [it] beckons me to look through to these infinite possibilities” before returning to the “As below…” lyric. This further emphasizes his point that his fascination with gaining knowledge and getting a more objective view of the world is like an addiction, that it “beckons me” to “look through to” – or consider – the “infinite possibilities” that are out there.
With regards to the tool he’s using to widen his gaze, it’s clear that Keenan is using Phi and all its aforementioned components at least partially as subject matter. The formulation of lyrics in both verses, using the Fibonacci Sequence, as well as the song’s six individual mentions of spirals, tell us that much. The interest in the Golden Spiral is compounded in the bridge, when Keenan sings “I embrace my desire to […] Swing on the spiral of our divinity and still be a human.” It is very likely that Keenan is referring to the Golden Spiral at least somewhat with the phrase “spiral of our divinity,” as the Golden Ratio is often called The Divine Ratio and the verses’ lyrics are structured with Fibonacci numbers.
The entire bridge’s lyrics are as follows:
“I embrace my desire to
Feel the rhythm, to feel connected
Enough to step aside and weep like a widow
To feel inspired, to fathom the power
To witness the beauty, to bathe in the fountain,
To swing on the spiral of our divinity
And still be a human.”
Since Phi and the Golden Ratio are found in so many unrelated places and systems throughout the known universe, it’s only natural that once one can see that it exists virtually everywhere, one would “feel connected enough to step aside” and simply weep, be it from the epiphany, the beauty of oneness in the universe or even from fear of what implications such unity could have. The inclusion of “like a widow” implies a possible sorrow of the death of something, unless Keenan is simply using consonance to follow the “w” in “weep.” If there is the implication of loss, perhaps the weeping is over his previous perception of the world dying.
Keenan continues to meditate on this new world view, which – as we learned from the verses – is a great way to expand our horizons and a goal he seems to condone. The extended sentence finds him proclaiming that he is embracing his desire to feel the inspiration, awe and beauty that would come with this scientific enlightenment, which is summarized in the line “To swing on the spiral of our divinity,” or to use mathematic and scientific tools like Phi and the Golden Spiral as tools or toys to come to the same enjoyment of life and the universe that others may find in religion, philosophy or LSD. This section ends with the caveat that he wishes to do all these things “and still be a human.” Interpretations on this final line vary, but it seems as though a level of humility and imperfection – “being human” – would need to stay in mind for Keenan so as not to become self-righteous. A not entirely dissimilar idea is expressed several songs later on the same album. “Reflection” contains the lyric “I must crucify the ego before it’s far too late,” and a variation of it.
As Keenan seems to be expressing the idea of using various tools – in mathematics and science, in this case – to open our minds to fantastic perceptions of the known universe, he also mentions that this process can’t be forced or faked, lest the person risks consequences of disconnecting from the entire endeavor. Nowhere is this clearer than in the first chorus of “Lateralus,” which reads as follows: “Over-thinking, over-analyzing separates the body from the mind / Withering my intuition, missing opportunities and I must / Feed my will to feel my moment, drawing way outside the lines.” In other words, to keep a strong intuition and stay on the same wavelength as this sort of manual evolution, things like working falsely towards the answer one wants is counterproductive – like our earlier examples of seeking to break down lyrical syllables that don’t fit the mold or assign connotation to its lack of a Jaws-like denouement at 62% of its duration.
Conclusion
Any attempt to fully analyze and interpret lyrics from Tool songs will likely be met with a degree of woe. In a 1997 interview in Alternative Press Magazine, Keenan himself had the following to say about fans’ varying takes on their music: “I’m not going to spoon-feed anybody and rob them of their personal experiences. I read the interpretations of the lyrics that people send to [the Tool web] page. They’re way off. But that’s fine” (Pettigrew). Tool have never included lyrics with their albums, only much later releasing differing drafts of the lyrics to a fan’s website.
The same may or may not be said about the song’s popularity. In a recent poll on toolshed.down.net, the internet’s leading Tool website, “Lateralus” was voted Tool’s most popular song in a grand prix-style series of “either/or” votes, beating six other songs and winning its final round by a 2:1 vote against Tool’s next fan favorite, “Third Eye” (Akhtar). It’s impossible to gauge whether the supposed “naturally pleasing” presence of Phi and the Fibonacci Sequence play a part in the popularity of “Lateralus,” or if knowledge of the band’s intentional use of those was admired and respected enough by fans to win it some votes. As is often the case with Tool, fans should probably read into it however much they like, save for a grain of salt.
Phi and its subsequent studies permeate countless facets of nature and have played a subtle role in several wildly popular and successful pieces of artwork. Some of their uses have been intentional by the user, as with “Lateralus,” and some have been unintentional, as with Phi’s permeation throughout our skeletal systems. Whether utilized naturally or artificially, implementing Phi is a universal phenomenon worth studying and can help humankind take one step closer to understanding the very fabric of nature – Tool’s song is just one tribute and a call to arms for those ready and willing to unravel the mystery.
Bibliography
28 Days Later (2002). Dir. Danny Boyle. DNA Films and British Film Council.
Akhtar, Kabir. “Tool Madness.” The Tool Page. Web. 7 July 2011. <http://toolshed.down.net/news/madness/>
Dictionary.com Unabridged. “lateral.” Random House, Inc. 28 January 2012. <http://dictionary.reference.com/browse/lateral>
Heiting, Gary, OD. “Your Infant’s Vision Development.” All About Vision. October 2010. Web. 28 January 2012. <http://www.allaboutvision.com/parents/infants.htm>
Jaws (1975). Dir. Steven Spielberg. Universal Studios.
Kator, Andrew, and Legaz, Jennifer. “Quick Tips in Art and Design: Pattern.” Web. 28 January 2012. <http://articles.katorlegaz.com/quicktipsinartanddesign/pattern/>
Keenan, Maynard James, and Tool. “Lateralus” and “Reflection.” Lateralus. CD, Volcano Entertainment, 2001.
Keenan, Maynard James, and Tool. “Lateralus.” The Tool Page. Web. 7 July 2011 and 28 January 2012. <http://toolshed.down.net/lyrics/lateraluslyrics.php#09>
Knott, Quinney. “The life and numbers of Fibonacci.” Plus Magazine. 31 August 1997. Web. 7 July 2011. <http://plus.maths.org/content/os/issue3/fibonacci/index>
Pettigrew, Jason. “Nobody’s Tool.” Alternative Press. March 1997: 52. Print.
PhiPoint Solutions, LLC. “Phi for Neophites.” Goldennumber.net Web. 7 July 2011. <http://www.goldennumber.net/neophite.htm>
