The Musical Geometry of Genes Generating Rhythms from DNA

Abstract

DNA encodes all sorts of information that makes us human, but, aside from encoding genes, could DNA also encode for a mapping of musical rhythms in a very abstract way? This project sought to generate rhythms out of DNA and compose a musical piece out of a gene's rhythmic sequence. Computational rules inspired by geometric analyses of rhythms guided the mapping of DNA's molecular structure into rhythmic timelines and melodic scales; these basic structures were then used to compose a song according to the sickle cell gene DNA sequence. The rhythms generated by this "genetic analysis" alternate pleasantly between even and odd time signatures.

geometr ic rhyt hms and DNA

The discovery of deoxyribonucleic acids (DNA) and their role in encoding the information that makes us human has inspired scholars from numerous corners of academia. In musicology, DNA's unique geometric properties have typically been explored with sonification studies that primarily generate melodies out of gene sequences [1–5]. To my knowledge, however, no academic studies have attempted to generate geometric rhythms out of DNA's geometric properties, as I could not find other published descriptions of DNA-derived rhythms in literature. This is surprising given the almost natural opportunity to relate rhythms and DNA by looking at their geometry. Rhythms can be ascribed geometric properties like shapes in mathematics if understood as timelines of k sounded onsets and n total pulses that repeat over time in a circle diagram, also known as polygon notation (see Fig. 1 for an example) [6]. DNA is famous for its attractive and highly conserved molecular geometry; seeing such an opportunity, I therefore used geometry as an abstract means to generate rhythms out of DNA molecules, and I used these rhythms to compose a whole musical piece out of a DNA sequence: the sickle cell gene [7].

Fig. 1.

The clave son in polygon notation (k = 5, n = 16). Geometric properties arise from the timeline (a symmetric pentagon; thick contour) and the time signature (4/4, a hexadecagon; thin contour). (© Godfried Toussaint [19])

Turning DNA Molecules into Rhythms

DNA is a long molecule made out of four basic subunits [8]. These precursors are adenosine monophosphate (AMP), thymidine monophosphate (TMP), guanosine monophosphate (GMP) and cytidine monophosphate (CMP; Fig. 2) [9]. These four basic units bond with each other in specific orders to form long sequences called genes. Molecularly, AMP, TMP, GMP and CMP have regular structures. These consist of three chemical groups: one central ribose sugar (central highlight in each panel of Fig. 2), one phosphate group (left-most highlight in each panel of Fig. 2) and one nitrogenous base (rightmost highlight in each panel of Fig. 2). In AMP, the nitrogenous base is adenine (A); in TMP, thymine (T); in GMP, guanine (G); and in CMP, cytosine (C). The important property to note here is neither the names nor the chemistry of these subunits but rather that two out of these three chemical groups are cyclic rings—namely, the central ribose sugar and the nitrogenous base. Given that rhythms can be represented using polygon notation (i.e. essentially, a circle with enclosed polygons; see Fig. 1), the polygons formed by the central ribose sugar and the nitrogenous bases were a for the mapping of DNA molecules into rhythms.

I created four rules to map these rhythms. These rules came about by noting that (a) chemical characteristics within the cyclic rings could be associated to (b) rhythmic outputs into a circle diagram. Namely, the size of the ring, the atoms within it and the functional groups attached to it could be associated with a circle diagram's n total pulses and k sounded onsets. For illustration purposes, all pulses in this work were written as quarter notes; however, this was a stylistic decision that could be changed. The four rules (summarized in Table 1) were:

Rule 1: The number of cyclic rings in the DNA subunit will determine the number of circle diagrams to be drawn. Application: Because AMP and GMP have three cyclic rings, they yield three circle diagrams (TMP and CMP yield two).

Rule 2: The number of atoms in each cyclic ring will determine n, the number of total pulses of its respective circle diagram. Application: Because AMP has a five-membered ribose ring, a six-membered first ring in the base and a five-membered second ring in the base, AMP's three circle diagrams have n = 5, n = 6 and n = 5 (respectively).

Rule 3: Noncarbon atoms within each cyclic ring will determine the bass drum k polygon, i.e. the timeline of a drum kit's bass drum. Application: Because AMP's ribose ring only has one noncarbon atom (i.e. the oxygen atom, O [10]), AMP's ribose ring's circle diagram has a bass drum timeline of [ x . . . . ]. Similarly, AMP's base's diagrams have bass drum timelines of [ . x . x . . . ] and [ x . x . . ] [11].

Rule 4: Functional groups attached to each cyclic ring will determine the snare drum k polygon, i.e. the timeline of a drum kit's snare drum. Application: Because AMP's ribose ring has two –OH groups attached, AMP's ribose ring's circle diagram has a snare drum timeline of [ . . x x . ]. Similarly, AMP's base's diagrams have snare drum timelines of [x . . . . . ] and [ . . . . . ].

The four rules generated 10 rhythmic timelines from 10 cyclic rings in all four DNA subunits; these are shown in Fig. 3. To have one timeline per DNA subunit, I combined each subunit's timelines into one timeline [12]. This yielded four rhythms of unusual time signatures (Fig. 4a; online supplemental video S1): AMP's and GMP's timelines are in 15/4 (or 5/4, 6/4 and 5/4), and TMP's and CMP's are in 11/4 (or 5/4 and 6/4). These time signatures were a consequence of AMP and GMP both having a ribose and a double-membered base ring and of TMP and CMP both having a ribose and a singlemembered base ring. The timelines generated are not too different from one another: AMP, TMP and GMP progress similarly, but AMP differs from TMP and GMP in a silent eighth onset, while TMP differs from GMP due to a shorter n. CMP is actually equal to TMP, which is a consequence of TMP and CMP having equal distributions of carbon and noncarbon atoms and functional groups in the base.

Fig. 2.

Chemical structures of the four basic subunits of DNA. Central, leftmost and rightmost highlights in each panel indicate the central ribose sugar, the phosphate group and the nitrogenous base of each subunit (A in AMP, T in TMP, G in GMP and C in CMP). (© Alvaro Yanez) natural source of inspiration

Table 1.

Summary of the geometric rules for the rhythmic timelines.

The rhythmic timeline of the first four letters of the sickle cell gene are illustrated in Fig. 4b. The time signature oddity reminded me of Dave Brubeck's Take Five [13], written in the unusual 5/4 (i.e. k = 5); the oddity in Take Five's time signature has been described to both challenge the listener's ability to keep track of the beat and "keep the brain more active while listening and performing" [14]. These "genetic" rhythms are not only oddly timed, however: They also alternate with even time signatures. This reminds me of the work of Dream Theater, a progressive rock band famous for virtuosic alternation between time signatures. In The Count of Tuscany [15], for instance, the time signature starts in 3/4 (0:00–1:02) and begins alternating in 1:03–2:36 with a repeated sequence of 6/8, 9/8, 6/8 and 12/8 to form the core of the piece. A second instance of this unusual alternation can be found 9:33–10:12, where the piece follows an ABAB structure in which A = 7/8, 15/16, 7/8 and 4/4 and B = 7/8, 15/16, 7/8 and 19/16. Here, the rhythms generated by DNA resemble Dream Theater's style more than that of Dave Brubeck because we obtained oddly timed rhythms whose time signatures alternate. These mixed time signatures could also be written in an arbitrary number of different ways; nonetheless, regardless of how you choose to write them, the underlying rhythmic timelines will never be regular when compared to each other. An additional illustration of how DNA sequences can be mapped into rhythmic timelines using this method is shown in Fig. 4c, where the "rhythm" of the first four letters of the MAOA gene are depicted.

Fig. 3.

Chemical structures and circle diagrams of the resulting rhythmic timelines. Chemical structures (left) and resulting rhythmic timelines (right) of AMP (a), TMP (b), GMP (c) and CMP (d). (k_bass indicated in black shading; k_snare indicated in red shading [gray in print edition].) (© Alvaro Yanez)

Fig. 4.

(a) Full rhythmic timeline of each DNA subunit. Time signature for each full rhythmic timeline indicated by box length (k_bass indicated in black shading; k_snare indicated in red shading [gray in print edition]). (b) Rhythmic timeline of the first four letters of the sickle cell gene: ACAC [20]. (c) Rhythmic timeline of the first four letters of the MAOA gene: CATA [21]. (© Alvaro Yanez)

turning DNA molecules into melodic scales

Having generated four rhythms from DNA, I then generated melodic scales that would help me constrain the compositional process to a narrower range of notes from which to pick. I took advantage of the geometric isomorphism between rhythms and scales described by Toussaint in The Geometry of Musical Rhythm [16]. In short, this isomorphism converts rhythms into melodic scales (and vice versa) by using a rhythm's circle diagram polygon. If we overlay a timeline's polygon into an n = 12 circle diagram, where each "onset" represents a note of the chromatic scale (with C as the zeroth onset), then the k timeline resulting from this overlay indicates which notes should be played. For example, AMP's ribose timeline (Fig. 3a; a triangle with corners at onsets 0, 2 and 3) transposes into the scale C-F-G (Fig. 5a; a triangle with corners at onsets 0, 5 and 7). Using this methodology, I transposed each of the rhythmic timelines generated in Fig. 3 into their respective melodic scales (Fig. 5; online supplemental video S2). (When a polygon's corner landed between onsets rather than on an onset, the nearest clockwise onset was chosen to be sounded.) Having generated both rhythmic timelines and melodic scales out of DNA, I then proceeded to compose a piece out of a well-studied gene sequence: the sickle cell gene.

Composing a song out of a gene

The composition of a musical piece out of the sickle cell gene followed a similar methodology to that of eighteenth- and nineteenth-century artists playing composing games. These games challenged artists to compose waltzes on the spot using two dice. The number sequence drawn by the dice decided a priori the meter, scale and chords that the artist should use. Artists would draw on a mental "library" that associated (also a priori) every dice combination to precomposed chord progressions in the meter and scale indicated [17].

Inspired by the concept of using dice's randomness to compose a piece's "backbone," I used the first 52 letters of the sickle cell gene sequence to accomplish the same purpose. These letters are ACACTCGCTT CTGGAACGTC TGAGGTTATC AATAAGCTCC TAGTCCAGAC GC [18], where A stands for AMP, C for CMP, T for TMP and G for GMP. I arrived at the number 52 by experimenting with several values first. I thought 52 letters would be significant enough 48 Yanez, The Musical Geometry of Genes to compose a two-minute long piece with verses, choruses, a bridge and a solo; using all 586 letters in the sequence, on the other hand, would have given a very long piece.

The song's rhythmic "backbone" was generated by substituting each letter with its respective timeline (according to the "rhythmic library" in Fig. 4). The accompanying chord progression was generated by trying out progressions from each time-line's transposed scale (i.e. the "melodic library" in Fig. 5) and by modifying these chords at will to obtain a pleasurable progression, as the scales generated did not inherently sound pleasing. This method gave me both a rhythm and a chord progression from which I could start improvising creatively, like artists playing composing games. Adding a creative component to the composition was important to me because I wanted the final composition to be perceived as a fluid song, rather than the rigid outcome of a geometric "code."

To facilitate the creative process, I subdivided the 52-lettered sequence into smaller sequences that I could compose piece-wise. I decided that these smaller structures would be multiples of four, based on The Count of Tuscany's alternating four-tuple main theme (1:03–2:36) and ABAB structure (9:33–10:12). Subdividing the sequence eased the compositional process because the piece's time signature sequence was as random as the gene sequence itself. Giving me a localized structural context to work with facilitated the creative aspect of this exercise.

Fig. 5.

Chemical structures and circle diagrams of the resulting melodic scales. Chemical structures (left) and resulting melodic scales (right) of AMP (a), TMP (b), GMP (c) and CMP (d). (Sounded onsets indicated in black and red shading [gray in print edition]). (© Alvaro Yanez)

The final product of this project, a two-minute-long piece that I titled E8Q, can be found in the online supplemental files (Audio S1). Like the gene sequence's letters, the rhythm of the piece does not repeat itself, but this lack of repetition does not seem to sacrifice musicality. Astoundingly, the song's rhythm holds a somewhat regular and periodic structure. This is entirely due to the choice of DNA as a source of musical inspiration. Because DNA consists of a sequential combination of four subunits from which two are bigger than the other two (i.e. AMP and GMP versus TMP and CMP), the resulting rhythmic sequence (i.e. the two different 15/4 timelines and the two equal 11/4 timelines) produce a structure that sounds odd and unusual, but not too unusual when permuted. Consequently, the chemical similarity between DNA subunits, and its ability to combine into a sequence, allows DNA to produce unusual rhythms that are musically attractive despite being irregular.

Conclusion

A deep analysis of geometry in the context of rhythms and biochemistry enabled me to map DNA sequences to rhythms and scales. Using rules inspired in DNA's cyclical rings, I generated geometric polygons that could be used both as rhythmic timelines and melodic scales. These timelines and scales, when ordered according to the gene sequence of the sickle cell gene, created a "backbone" from which I composed a two-minute song that alternated randomly but pleasantly between two unusual time signatures: 11/4 and 15/4. I know of no such prior compositional exercise in musicology. Therefore, the main lesson of this project—that chemical structures can be related to rhythmic structures to create a musical "backbone" over which one can improvise and compose—has the potential to serve as an original and abstract compositional technique for artists seeking to relate biology and music even more intimately.

Supplementary Material

• [ Click to download ] Final composition: E8Q by Alvaro Yanez.
Copyright credit: Alvaro Yanez

• [ Click to download ] Rhythmic timelines for AMP, TMP, GMP and CMP.
Copyright credit: Alvaro Yanez

• [ Click to download ] Melodic scales for AMP, TMP, GMP and CMP.
Copyright credit: Alvaro Yanez