Fibonacci in Music

This is an excerpt from Master Fibonacci: The Man Who Changed Math. All citations are catalogued on the Citations page.

MUSICAL COMPOSITION

(Previous Section: Fibonacci in Art & Architecture)

It is not uncommon for musicians to find mathematics appealing; both disciplines involve precision, organization, and structure. Pythagoras, for example, developed musical theories based on “mathematical harmonics in frequency ratios of whole number intervals” and Galileo’s father Vincenzo, a lutenist, wrote a treatise on string theory (pitch and string tension)” (Hunt). In the 17th century, Gottfried Leibniz wrote that “music is the pleasure the human mind experiences from counting without being aware that it is counting.” One might say math dances with music in the mind. Indeed, Pythagoras “married” math and music when he “heard the sound of hammers on anvils and produced a formula connecting their mass to the sound they made” (Paphides). Nevertheless, while investigating possible relationships between the Golden Section and musical structure, one should not unreservedly determine the aesthetic aims of particular composers without documented testimony or material evidence, for errors are often made in measurement. The numerical value for the Golden Ratio, 5/8, is easy to confuse with the simple proportion of thirds (Fischler 31). Analysis of artwork by the cubist Juan Gris found that he may have used the diagonal of a Golden Rectangle; however, Gris categorically denied in a letter that he used the Golden atio to proportion his paintings (Fischler 31). Similar caution is warranted before drawing conclusions that any particular artist was consciously guided or inspired by the Fibonacci numbers when creating their works.

Whether or not Gris used elements of the Fibonacci sequence, diverse studies suggest composers often incorporate the golden proportion in musical compositions, perhaps due to its power in constructing well-balanced, beautiful and dynamic movements, rhythms and melodies. For example, Claude Debussy’s music “contains intricate proportional systems” based on the Golden Ratio; specifically, the “dramatic climax of Cloches a travers les feuilles occurs when “the ratio of the total number of bars to the climax bar is approximately 1.618” (Van Gend). Fibonacci numbers harmonize naturally and the exponential growth in nature defined by the Fibonacci sequence “is made present in music by using Fibonacci notes” (Sinha). Specifically, when the Golden Section – expressed by the sequence of Fibonacci ratios – is used by a composer, it is “either used to generate rhythmic changes or to develop a melody line” (Beer 4). “The grammar of music – rhythm and pitch – has mathematical foundations. Rhythm depends on arithmetic, harmony draws from basic numerical relationships, and the development of musical themes reflects the world of symmetry and geometry.” Composers rely on symmetry to create progressions in theme and variation; they count on mathematical structure such as “prime numbers to create a sense of unease” and whispers of dissonance by creating unexpected or nontraditional rhythms and meter, such as Messiaen does in his famous Quartet for the End of Time. Conversely, simple ratios suggest harmony. “When we hear two notes an octave apart, the frequencies of the two notes are in an exact 1:2 ratio, so we feel we’re hearing the same note.” They are so similar in sound that “we give them the same name” (Du Sautoy).

Natalie Hoijer investigated compositions of concert music and compared the use of mathematical patterns such as the Fibonacci Series and the Golden Mean to other mathematical symmetries such as palindromes, crab canons, and fractals, and found that “the Fibonacci Series and Golden Mean were the most effective compositional tools and yielded the most aesthetically pleasing results” (Hoijer). This may explain why many composers, from Bartók to Debussy, have found that “the organic sense of growth found in the Fibonacci sequence of numbers is “an appealing framework” for orchestrating unique, memorable, and distinct combinations of melodic courtship (DuSautoy). Emeritus Professor of Music Theory Michael R. Rogers says, “the prevalence (and almost ubiquity) of Golden Sections throughout the so-called common-practice period is well-documented in many studies” (Rogers 249).

Beethoven

Rogers describes how Beethoven’s Piano Sonata No. 14 in C# Minor, op. 27, no. 2 serves as an archetype of golden-section form in general and represents a model of tonal clarity, the kind of temporal model with which Chopin was familiar. In the first movement of the Beethoven sonata, there are regularly expanding tonal blocks. “Each new point of arrival develops from the preceding tonal area and simultaneously prepares for the next.” As each new goal cadence relates temporally both to what has gone before and to what is going to follow, “the feeling of gathering strength is inescapable.” While “the arrivals are spaced further and further apart, their durational ratios to one another remain constant [and] the harmonic control is metered out” (Rogers 248).

Haylock believes he discovered a number of occurrences of Golden Ratio in the first movement of Beethoven’s Fifth Symphony. “That’s the one that starts with the famous motto theme: ‘da, da, da, daah’!” He says, “In Beethoven’s original score there are 600 bars before the final statement of the opening motto. But a statement of the opening motto also appears at bar 372. So, we have this structure for the three main statements of the motto: motto starts … 372 bars … motto starts… 228 bars … motto starts”…divided in Golden Ratio proportion (Haylock).

Haylock mentions two other pieces of evidence he offers as proof that Beethoven used the Golden Section to create his most famous masterpiece: the ‘exposition’ in the first movement and the coda that is 129 bars long. “Divide this coda up using the Golden Section and you get 49 bars and 80 bars.” After 49 bars of the coda, Beethoven “introduces a completely new tune that has not appeared in the movement so far. Before Beethoven no-one would ever introduce new material in a coda! So this is a very significant point in the coda.” Haylock admits that we can never know whether Beethoven was “signaling his piece of radical creativity in this very long coda by linking it with the Golden Section” (Haylock).

Mozart

It is well-documented that Mozart had an early fascination with and predisposition for all things mathematical, even to the extent of loving musical gematria, or symbolic number combinations in music. “His sister Nannerl mentioned he was always playing with numbers and even scribbled mathematical equations for probabilities in the margins of some compositions (e.g. Fantasia and Fugue in C Major, K394).” Some of these may have been Fibonacci number calculations. Elements of the Golden Section appear to balance his musical lines (ratios of theme to development or musical exposition to recapitulation) and it seems likely Mozart used the Fibonacci sequence in his Piano Sonata #1 in C major K279 as well (Hunt).

Bach

Loïc Sylvestre and Marco Costa found, through analysis of the mathematical architecture of the printed edition of Bach’s compositions, that he “intentionally manipulated the bar structure of many of his collections so that they could relate to one another at different levels of their construction with simple ratios such as 1:1, 2:1, 1:2, 2:3.” The Art of Fugue (1751), they say, shows that the whole work was conceived on the basis of the Fibonacci series and the Golden Ratio based on bar counts. For example, “distribution of Golden Ratios is evident in the numbers of bars in brackets, and in each of the subdivisions of counterpoints 8-14” (179-180).

Chopin

One of the macro-rhythmic organizational principles underpinning the harmonic and melodic ambiguities in Chopin’s Prelude in A Minor is the Golden Section. (Rogers 245). Rogers asserts, “Golden sections are created on a melodic level within each of the first and second appearances of the minor-seventh descent” (247). Wondering why Chopin would embed one Golden Section (calculated in beats rather than measures) within another within yet another, he decides “this is a tactical timing strategy that works as a series of signals, strategically placed and deliberately paced, which regulate the harmonic ambiguities and help to foreshadow the ultimate establishment of tonal stability – a stability that arrives, in this case, just in the nick of time – at the very end.” These “produce a kind of gradually emerging and increasingly focused view of tonal centricity” (248).

Handel

Examples of deliberate application of the Golden Ratio can be found in Handel’s Messiah, which consists of 94 measures. In measure 34, after 8/13 of the first 57 measures, the entrance of the theme “The kingdom of this world…” marks an essential point (34/57). There is a very important solo trumpet entrance (“King of Kings”) in measures 57 to 58, after about 8/13 of the whole piece (57/94) and another one after 8/13 of the second 37 measures, in measure 79 (“And He shall reign…”) (Beer 5).

Bartók

Another of the most famous classical composers believed to have been inspired to use the Fibonacci numbers in musical composition was Hungarian Béla Bartók (1881-1945). Hungarian musicologist Erno Lendvai investigated Bartók’s works “painstakingly” and published books and articles testifying that “from stylistic analyses” he was “able to conclude that the chief feature of his chromatic technique is obedience to the laws of Golden Section in every movement” (Livio 188). For example, the eighty-nine measures of Music for Strings, Percussion and Celesta are “divided into two parts, one with fifty-five measures and the other with thirty-four measures, by the pyramid peak” (in terms of volume) of the movement (Beers 189).

Nevertheless, other musicologists have disputed Lendvai’s conclusions and it is hard to determine Bartók’s true intentions when he himself “said nothing or very little about his own compositions” and “did not leave any sketches to indicate that he derived rhythms or scales numerically” (Livio 190).

Modern Music

In 2009, American Jazz artist Vijay Iyer explained why he and his trio preferred Fibonacci numbers when composing and performing music. He said the sequence’s scaling property is very interesting; “because the ratios get successively closer to the Golden Ratio, the ratio 5:3 is not the same as, but ‘similar’ to the ratio 8:5, which is ‘similar’ to the ratio 13:8, or 144:89, or 6,765:4,181.” His trio enjoyed exploring compositions that are “asymmetrical in a Fibonacci way: a short chord and then a long chord, three beats plus five beats, totaling eight beats.” With a beat that is “standard four-four time,” you could step to the beat, hearing a “chord when you take your first step, and then another chord while your knee is aloft between the second and third steps.” He said, “This is a rhythm that you hear in all kinds of places – like Michael Jackson’s Billie Jean” (Iyer). Working with asymmetry and “move it through Fibonacci-like transformations,” the trio may “perform an asymmetric ‘stretch’ that maintains the same ‘golden’ balance over the entire measure,” but they transform it a bit, trying to “preserve an ‘impression’ of the original – the short-and-long-ness of it – to see if [they]…can achieve that feeling of similarity” (Iyer).

Fibonacci in Instruments: The Violin

>In “Stradivarius: Music of the Golden Ratio,” Feng Wu voices the claim made by many that master luthier Antonio Stradivari (1644-1737) of Cremona, Italy, deliberately used the Golden Ratio to make what most people in the world consider the greatest string instruments ever created. His violins are highly prized for their tone quality and their aesthetic form (Wu). As Wu explains, Stradivari employed the system of design used by the ancient Greeks and the masters of Renaissance architecture, deriving all of his linkage dimensions and placements from a single length, the length of symmetry. “He employed the golden-section ratio and the proportion of the sides of simple triangles as well as the relationship of musical intervals to divide this length” (Bidwell). In addition, Stradivari “took special care to place the ‘eyes’ of the f-holes geometrically, at positions determined by the Golden Ratio” (Livio 184). The Golden Ratio \phi = \frac{1+\sqrt{5}}{2} = 1.61803 can be found throughout the violin, the “Lady Blunt,” by dividing lengths of specific parts of the violin (Wu):

Fibonacci in Instruments: The Piano

Some claim that there is significance in the fact that the intervals between keys on a piano of the same scales are Fibonacci numbers (Sinha). The first few Fibonacci numbers appear to be represented by the arrangement of thirteen keys along the keyboard in groups of two and three black keys between the eight white which comprise a full octave (Rory). However, Livio is a member of the choir which dismisses such notion; he explains in his book, The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number, that “the chromatic scale (from C to B), which is fundamental to Western music, is really composed of twelve, not thirteen, semitones” and, more importantly, the arrangement of the keys on a piano “in two rows, with the sharp and flats being grouped in twos and threes in the upper row, dates back to the early fifteenth century, long before … any serious understanding of Fibonacci’s numbers” (Livio 185).