Music, along with its attached rhythm, has been with man for centuries, developing and evolving along with him. Its influence on human behavior and mood can reach levels whose limits are still unknown, especially in everything related to perception, where the whole nervous system is involved. Thus, physiology and psychology become strongly connected areas, while technology, through, for example, the production of music by electronic means, appears as a new unexpected ingredient that traditional composers and musicians of older times could not imagine. Obviously, bioengineering and its multiple branches are not absent either [1]–[4]. The literature is enormous with several specialized journals. When one looks back in time at the evolution of this complex area, the appearance of some kind of sudden jump (as a step function), which took place within a relatively recent short interval, is evident: music is now much more than what it used to be, and rhythm has made a step forward as if resurrecting and renewing the ancient Indian or African drums.
This column deals only with rhythm, setting two objectives:
- first, to explore whether there are, in the current modern music, rhythms other than the common two-, three-, or four-beat tempos
- second, to suggest a scaffold of possible rhythms based on a binary system with fractional exponents.
Odd Rhythms Based on the Traditional Integer Binary Musical System
Occidental modern music sets its rhythmic modalities on patterns marked by the regular succession of strong and weak elements, i.e., there is recurrence in time, where sound durations and silences are perfectly adjusted and inserted within pre-established compartments called measures, organized by time signature and tempo indications. Thus, music (as dance and oral poetry do) lays and maintains what is known as a metric, a basic unit of time that may be audible or implied; it is the pulse, often simply called the beat, a repeating series of identical yet distinct periodic short-duration stimuli acting as points in time. The beat pulse of the rhythm is perceived as basic, with a tempo to which listeners entrain (synchronize) as they tap their foot or dance to a piece of music.
Franco de Colonia (1215?–1270?), a German musician, in 1260, in his Ars Cantus Mensurabilis, was perhaps the first to describe the relationships between the terms maxima, longa, and breve. He also better explained how to use what is known as the semibreves. In turn, Petrus de Cruce (1270?–1300?), a French composer, in around 1300, introduced subdivisions for the breve in even shorter notes. The most relevant theoretician in music was Philippe de Vitry (1291–1361), who, in France in 1322, systematized the time length scheme in his opera Ars Nova. Along the 17th century, the evolution of mensural scriptures and proportions continued until the modern notation, using bars, measures, ties, and other symbols currently in use, was established [5], [6]. Thus, the inception of our current musical system goes as far back as the 13th century.
Musical notes, when played by a given instrument within the context of a melodic line and rhythm, have a relative duration (or time length) that spreads from the longest [or whole note Redonda (Re)] to the shortest [or 64th note Semifusa (SFu)]. Arbitrarily, it is stated that Re stands for a duration Dr = 1 and that the relative time length that follows, represented by Blanca (Bl), is equivalent to Re/2. The full system, with seven possible durations for any of the sounds of the modern temperate scale, belongs, from a mathematical viewpoint, to a perfect binary system, i.e., it follows the scheme shown in Figure 1. Each sound duration is obtained as the value of a whole note (which is 1) divided by a power of 2 with an integer exponent that covers the 0–6 interval and which obviously is in the hands of the performer. Nothing would prevent increasing those possible time lengths to shorter values, as with 2^7 or with 2^8, which would call for one or two new names (say, Half-SFu and Quarter-SFu). In the latter two cases, Re would be equivalent in time to 128 Half-SFu and 256 Quarter-SFu, respectively. When one computes the arithmetic relationships between consecutive durations, i.e., Re/Blanca, Bl/Negra (Ne), Ne/Corchea (Co), Co/Semicorchea (SCo), SCo/Fu, Fu/SFu, SFu/Half-SFu, and Half-SFu/Quarter-SFu, it is easily seen that all are constant and equal to 2, the system’s base. The current music theory also recognizes the double whole note, or 2Re, which the British call Breve, keeping the same ratio 2Re/Re = 2. In other words, on one extreme 2Bl = Re, and, on the other end 64SFu = Re. The noninitiated musical reader should observe that, in all this structure, time is relative as it is not measured in absolute units like the astronomical time of our daily life; thus, any musical piece can be played slowly or quickly without losing its melodic line. Unfortunately, performers often tend to show virtuosity by playing a given composition too fast and, not uncommonly, destroying part of its beauty.
Most frequent are the 2/4, 3/4, 4/4, and 6/8 modes, which mean that, in each measure, respectively, fit 2Ne, 3Ne, 4Ne, and 6Co (Figure 1). Innumerable famous and long-lasting compositions, either as classical music or as folk melodies, have been produced within these rhythmic frameworks; however, always within the strict binary integer structure of Figure 1, other rhythms have been used by composers that are more difficult to play and that produce attractive and different perceptions. One beautiful example that is already famous is the theme song Mission Impossible, by Lalo Schifrin (born in 1932 and currently residing in California), written on the 5/4 signature, which means that, within each measure, there are five quarter notes, Ne, or equivalent silences. It offers a pleasant and catching perceptive effect that tends to persist even after its playing stops (Figure 2).
Several years before, however, other musicians experimented with less frequent rhythms, for example, Dave Brubeck (1920–2012), with pieces such as Pick Up Sticks in 6/4, Unsquare Dance in 7/8, and Blue Rondo à la Turk in 9/8. In addition, Paul Dresmond (his actual name was Paul Emil Breitenfeld, 1924–1977), a long-time close collaborator with Brubeck, came up with his very successful and amazing Take Five in 5/4, written in 1959, which gives the sensation of smoothly riding on a sleigh down a snowy hill. All of these pieces can be found on the Internet and experienced by the interested reader and potential listener or player.
However, these few examples represent only the tip of the iceberg because, digging back into history, one is surprised to learn that Western music includes a long list of pieces based on unusual time signatures, some extremely odd and not easy to figure out. Musical notation allows for more than one written representation of a particular composition. The chosen time signature largely depends upon the musical context and the personal composer’s taste. For example, the Piano Sonata No. 14 in C-sharp minor Quasi una Fantasia, Op. 27, No. 2, by Ludwig van Beethoven (also known as Moonlight Sonata), includes two examples. In the relatively slow first movement, the predominant rhythm is in triplet eighth notes (quavers), whereas, in the second movement, the basic tempo is faster, but the score is notated in quarter notes (crotchets) and half notes (minims), offering a clue to phrasing. The first movement could be written in 12/8 and still convey the same rhythm, phrasing, and tempo, and the second movement could be notated in 3/8 instead of 3/4 without changing its interpretation.
Many other strange rhythms can be discovered, such as in the Scherzo (Second Movement) of the Symphony No. 2, by Alexander Borodin (1833–1887, a Russian composer and chemist), in its Prestissimo, with a 1/1 time signature, except for the trio, which is in Allegretto 6/4; the unbelievable 1/√π/√2/3, in No. 41a of the Study for Player Piano by Conlon Nancarrow (1912–1997), a Mexican composer of U.S. origin, written in a time signature whose numerator is one over the square root of pi and whose denominator is the square root of 2/3; or Mädchentotenlieder (Song of a Dead Girl), by Bo Nilsson (a Swedish composer born in 1937), with Bar 97 in 3/4 time. The number of these strange signatures is very large, and very many are exceedingly odd, so much so that they make one wonder how these pieces are or should be played [7].
Binary System with Fractional Exponents
Looking at the relative durations Dr column in Figure 1, if vertically the integer numbers are sequentially completed, it can be seen that between 2 and 4, only one needs to be included, which, when binarized, corresponds to 2^1.585, obviously requiring an exponent intermediate between 1 and 2. In a similar way, one can see that between 4 and 8, there are 
5=2^2.322, 6 = 2^2.585 and 
7=2^2.808, all also with fractional exponents now lying between 2 and 3, and so proceeding until the sequence is completed. Figure 3 displays the full list up to 64 SFu.
Figure 4 displays the M/(M-1)
ratio as a function of 
M from 1 to 64, which could be extended further to 128 or 256 or even more. The differences to two of the same 
M sequence, from 
M=14 up, keep essentially constant and almost equal to 1, i.e., they meet the constancy requirement of the pure binary system. Subtracting 14 from 64, one is left with 50 time lengths that could be used to build a new musical rhythmic scheme and taking Dr = 14 as reference (in some way similar to the Re of the pure binary system). In other words, 
Dr = R14 = 2^3.808 = 14, verifying 
Rn/Rn-1 = .1, from a maximum of 
R15/R14 = 1.071 down to a minimum of R64R63 = 1.016.
As a result, the error covers the 7.1–1.6% range, always decreasing as the sound duration decreases. Figure 5 suggests a possible duration scheme based on the available 50 values described before. It makes up a tree with an origin in R14 and three levels, L1, L2, and L3, that meet the indicated temporal relationships. Another scheme based on Figure 3 is that of Figure 6, taking Re as the reference duration.
Discussion
We have briefly scanned a few historical hallmarks of music notation, especially those related to note durations and rhythm, and thereafter explored if other rhythms that are different from the most frequent 2-, 3-, and 4-beat schemes could be located. Surprisingly, a very long list showed up, and some of those rhythms defy the musician, the composer, and the bioengineer, who face a challenging fascinating area of study and experimentation. One example involves trying to understand and resolve the composition by Nancarrow (mentioned earlier in the article) that is written in a time signature whose numerator is one over the square root of pi (3.1416) and whose denominator is the square root of 2/3. Regarding this highly uncommon music creator, we must mention and recommend visiting YouTube to get a feeling of his compositions. The comments are made in German, subtitled in Spanish, by Georgy Ligeti, and the music is played on an old, already museum-piece mechanical piano (pianola) driven by a punched rotating cylinder.
On reviewing the IBMS and after completing the sequence of integer numbers, still other possibilities are foreseen, such as those outlined in Figures 5 and 6. The question stands on its practical implementation; perhaps a computer program could offer a solution. We leave the matter dangling and hope for ideas.
The proposed Theoretical Binary Fractional System (TBFS) could be a useful tool that somehow might simplify or optimize the writing of the handmade rhythms used, for example, in shamanic drumming, making specific sequences easier to write down and referring to the temporal durations of every beat. No less important is the novel way to compose music that this could also bring about, somehow breaking the established scheme and creating possible new rhythmic structures that are not allowed in the traditional system. As to the use of music in medicine as a clinical tool in the treatment of certain pathologies, the literature is full of examples [8], [9].
In conclusion, be it based on the IBMS or the TBFS, the human mind seems to surpass the infinity of the universe, reminding us once again of the seven lamps of bioengineering [10], and music has an ever-increasing future of endless possibilities.
References
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- M. E. Valentinuzzi and N. E. Arias, “Human psychophysiological perception of musical scales and nontraditional music,” IEEE Eng. Med. Biol. Mag., vol. 18, no. 2, pp. 54–60, 1999.
- M. E. Valentinuzzi, D. González, and G. Palloti, “The perception of musical consonance,” Trans. Univ. Coll. Env. Sci. (Radom, Poland), vol. 13, no. 1, pp. 5–22, Mar. 2009.
- D. L. Gonzalez, G. Pallotti, and M. E. Valentinuzzi, “Pitch perception and cochlear biomechanics,” in Proc. 17th Int. Conf. Mechanics in Medicine and Biology (ICMMB-17), Kraków, Poland, 19–24 Sept. 2010.
- (2014, Mar. 9). Notación mensural. [Online].
- Ricci Adams. (2000–2014). Music theory. [Online].
- (2014, Apr. 27). Long list of musical works in unusual time signatures. [Online].
- J. Strong. (1998, Feb.). Rhythmic entrainment intervention: A theoretical perspective. Open Ear J. [Online].
- D. S. Berger and D. J. Schneck, The Music Effect: Music Physiology and Clinical Applications. London, UK: Jessica Kingsley.
- M. E. Valentinuzzi, “The seven lamps of bioengineering,” IEEE Pulse, vol. 1, no. 1, pp. 15–16, July/Aug. 2010.