JJOURNAL OF MATHEMATICS AND THE ARTShttps://doi.org/10.1080/17513472.2020.1766339
Quasiperiodic music
Darren C. Ong
Department of Mathematics, Xiamen University Malaysia, Sepang, Malaysia
ABSTRACT
Using the definition of quasiperiodic function as a motivation, weintroduce the idea of quasiperiodic music and detail the compositionprocess of a quasiperiodic music piece,
Raindrops in A minor . We alsodiscuss connections between quasiperiodic music and other worksof music theory and composition that make use of aperiodic orderor periodic order with large periods, such as Lindenmayer systems,Vuza canons, Messiaen’s
Quatuor pour la Fin du Temps , and the phasemusic of Steve Reich.
ARTICLE HISTORY
Received 17 July 2019Accepted 5 May 2020
KEYWORDS
Minimal music;quasiperiodicity; aperiodicorder; phasing; Lindenmayersystems
1. Introduction
In recent decades, there has been a surge of interest in aperiodic order. ‘Aperiodicorder’ is a rather broad term that encompasses both discrete and continuous structures.These different structures are loosely grouped under one umbrella term because they
CONTACT
Darren C. Ong [email protected] data for this article can be accessed here. https://doi.org/10.1080/17513472.2020.1766339 © 2020 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis GroupThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
D. C. ONG are aperiodic, at the same time, each of them is close to periodic in some mathemat-ically precise way (although the sense in which they are ‘close’ may differ). See Baakeand Grimm (2013, 2017), Baake et al. (2016), Kellendonk et al. (2015), Barber (2008) andDal Negro (2013) for recent books detailing the impact of aperiodic structures in puremathematics, mathematical physics, condensed matter physics, and optics.As an example of a discrete aperiodic pattern, we take the Fibonacci word, which isan infinite binary string starting with 0100101001001. This is listed as sequence A003849in the Online Encyclopedia of Integer Sequences (Sloane, 2019). Note that the Fibonacci word is different from the more well-known Fibonacci sequence
1, 1, 2, 3, 5, 8, 13, . . . , inwhich every term in the sequence (after the second term) is the sum of the previous two.To form the Fibonacci word, we start with initial bit strings S : = S : =
01, and thenwe form a sequence of bit strings for which every string in the sequence is obtained byconcatenating the preceding two terms. That is, S n + : = S n + (cid:2) S n for n =
0, 1, 2, . . . . Thefirst few strings in the sequence will be: S = S = S = S = S = . . . , which we call the Fibonacci word.This Fibonacci word contains exactly n + n -bit substrings. We define a com-plexity function on a binary string as follows. Let σ ( n ) be the number of substrings oflength n of that string. We can observe that, for the Fibonacci word, σ ( n ) = n +
1. Fora periodic binary string, it is easy to see that σ ( n ) is bounded. By the Morse-HedlundTheorem (Baake & Grimm, 2013, Proposition 4.11), for any aperiodic infinite string σ ( n ) ≥ n +
1. It is in this sense that we can say that the Fibonacci word is aperiodic butvery close to periodic.Let us now consider an example in the continuous setting, with almost periodic func-tions. These are explained in detail in Appendix 1 of Avron and Simon (1981), that webriefly summarize here. Given a function f , let us define the translated function f t ( x ) = f ( x − t ) . A ( Bochner ) almost-periodic function is a bounded continuous function on R ν such that the set of functions { f t | t ∈ R ν } form a precompact set with respect to the supre-mum norm (a precompact set is a set whose closure is compact). The function f is periodicif and only if { f t | t ∈ R ν } is a compact set itself (Remling, 2019), so we see again that analmost-periodic function is close to a periodic one.Fibonacci words and almost periodic functions are two of the most prominent examplesof aperiodic order, but there are many others. Perhaps, the most visually iconic exampleof aperiodic order would be the Penrose tiling (Figure 1), which is an aperiodic tiling ofthe plane. We say that a tiling is periodic if it is invariant under a translation. Imagine aninfinite floor tiled with 1 × OURNAL OF MATHEMATICS AND THE ARTS 3
Figure 1.
A small section of the Penrose tiling of the P3 Rhombus type. This image was created byUser:Inductiveload from Wikipedia and released under the public domain. perfectly invariant in the same way. Curiously, we can also see the Penrose tiling as closeto periodic in the following way: any finite portion of the Penrose tiling appears infinitelymany times. These patterns are very compelling visually, and indeed they have appearedin the visual arts and architecture dating back almost a millennium ago.Scientists have realized that these patterns are also of great significance in physics andchemistry, especially after the 1984 discovery of the quasicrystal, which is a material withan aperiodically-ordered atomic structure. Since then, there have been many efforts inphysics and mathematics to model quasicrystals and to understand aperiodically-orderedmathematical structures. One of the most important ways to model quasicrystalline struc-tures exploits a quasiperiodic function . We will define quasiperiodic functions preciselylater in Definition 3.1; for now, we can informally say that these functions are generatedby combining periodic functions of different frequencies that never match up precisely.The quasiperiodic functions are a special case of almost periodic functions, as explainedin Theorem A.1.2 of of Avron and Simon (1981).Aperiodically ordered patterns have become very prominent in the visual arts, in sci-ence, and in mathematics, so it is unsurprising that they have also inspired musiciansand music theorists. In particular, musical rhythm has long been understood to have arich mathematical structure (Sethares, 2014). Discrete aperiodic order has been applied torhythmic structures in the form of Lindenmayer systems, for instance.In this article, we creatively explore quasiperiodic music, and we propose, as an appli-cation, the original composition
Raindrops in A minor (Ong, 2020), which is an attempt toexpress musically the structure of a quasiperiodic function.
D. C. ONG
This paper is structured as follows. Section 2 discusses briefly the history of aperiodicorder in the arts and sciences. Section 3 provides a formal mathematical definition ofquasiperiodic functions. This is the only part of the paper that requires any knowledgeof technical mathematics. A reader without a mathematical background could skip thissection and still grasp the main points of the paper. In fact, the main purpose of this sectionis to motivate some choices made in composing
Raindrops in A minor . Section 4 details thecomposition of
Raindrops in A minor , and gives a brief technical analysis of its features.Finally, Section 5 places quasiperiodic music in the context of contemporary music schol-arship. We discuss connections between our work and other studies of music theory andcomposition that are inspired by or related to aperiodic order.
2. A brief history of aperiodic order in the arts and sciences
Aperiodic order has become incredibly important in music, visual arts, physics, chemistry,and mathematics. To fully appreciate quasiperiodic music, one should maybe be familiarwith the artistic and scientific trends that are underpinned by these patterns.Aperiodically ordered structures invoke some very striking images. They have a longhistory in Islamic art and architecture going back to the 12th century. Examples includethe Darb-i Imam shrine and the Friday Mosque in Isfahan, Iran; the walls of the courtyardof the Madrasa al-Attarin in Fez, Morocco; the Seljuk Mama Hatun Mausoleum in Tercan,Turkey; and the external walls of the Gunbad-I Kabud tomb tower in Maragha, Iran (AlAjlouni, 2012, 2013; Lu & Steinhardt,2007). Western interest in these patterns emergedmuch later. In Europe, the earliest examples of aperiodic order are probably the rhombuspatterns in a 1524 doodling by Albrecht Dürer (Kemp, 2005).In the late 20th century, the importance of quasiperiodic structures in mathematics andscience became apparent. Roger Penrose discovered the tilings while investigating pla-nar tilings. The Penrose tilings were the first aperiodic tiling of the plane using just twotiles. Dan Shechtman discovered the quasicrystal in the 1980s, an achievement that wouldlater let him win the Nobel prize in chemistry. Following this discovery, there has beena surge of interest in mathematical models of quasicrystalline structures. As an exam-ple, Artur Avila won the Fields medal in 2014 partially thanks to his investigations onthe Almost Mathieu operator, which is a Schrödinger operator whose associated poten-tial function is quasiperiodic. The Almost Mathieu operator has an important connectionto the Hofstadter butterfly, another iconic mathematical image. This connection betweenquasiperiodicity and the butterfly is thoroughly explored in Chapter 4 of Satija (2016).The scientific importance of aperiodic order is not restricted to mathematical physics.Aperiodic order also appears in consdensed matter physics (Barber, 2008), optics (DalNegro, 2013), and crystallography (Baake & Grimm, 2017).
3. Mathematical background
There are several different ways to define a quasiperiodic function in mathematics. Here isone common definition, which is a restatement of (A.1.2) of Avron and Simon (1981).
OURNAL OF MATHEMATICS AND THE ARTS 5
Definition 3.1 (Quasiperiodic function):
Let S be the unit circle parameterized as[0, 2 π ) . A function f on R is quasiperiodic if for some positive integer n there exists acontinuous function Q on S n and a frequency vector ω = (ω , ω , . . . , ω n ) such that f ( x ) = Q ( x ω , x ω , . . . , x ω n ) .Technically, under Definition 3.1 all continuous periodic functions are quasiperiodic(although in the rest of this article, when we refer to a function of a piece of music asquasiperiodic, we implicitly mean that it is also aperiodic). To get an aperiodic functionfrom the above definition, we require n to be at least 2 and that the entries of the frequencyvector should be rationally independent: Definition 3.2 (Rational Independence):
We say that a set of real numbers ω , . . . , ω n are rationally independent if for rational numbers y i the only solution of the equation y ω + y ω + . . . + y n ω n = y = y = . . . = y n = f ( x ) = sin ( x ) + sin ( π x ) (1)is a quasiperiodic function that is not periodic. Using Definition 3.1, here n = Q ( v , v ) = sin ( v ) + sin ( v ) and ω = (
1, 2 π ) . Notice that, since 2 π is an irrational num-ber, the only rational solution to 1 · y + π · y = y = y =
0, and thus the set {
1, 2 π } is rationally independent.Indeed, one simple way to obtain a quasiperiodic function is to add two periodic func-tions, one with a rational period and one with an irrational period. We will do somethingsimilar to create our quasiperiodic music composition in Section 4. Even though the quasiperiodic functions we consider here are aperiodic, they can becomevery close to being periodic. This phenomenon is connected to the idea of rational approx-imants of irrational numbers. Definition 3.3 is given in Khinchin (1997) as a best rationalapproximant ‘of the second kind.’ We will just call it a best rational approximant becausein this paper we do not consider the first kind.
Definition 3.3 (Best rational approximant):
A fraction a / b is a best rational approximant of a number y if, for every fraction c / d with 1 ≤ d ≤ b and a / b (cid:6)= c / d , we have | dy − c | > | by − a | .Intuitively, a best rational approximant is a rational number that approximates a (usu-ally irrational) number y better than any fraction with smaller or equal denominator. Forexample, 22/7 is a best rational approximant of π , because it is closer to π than any otherfraction with denominator 1, 2, 3, 4, 5, 6 or 7. D. C. ONG
Example 3.4:
The best rational approximants of π are 3/1, 13/4, 16/5, 19/6, 22/7, 179/57,201 /
64, 223/71, 245/78, 267/85, 289/92, 311/99, 333/106, 355/113, . . . . Example 3.5:
The best rational approximants of ϕ , the golden ratio are fractions ofconsecutive Fibonacci numbers, that is, 2 /
1, 3 /
2, 5 /
3, 8 / / / . . . .These best rational approximants of a number y are derived from the continued fractionexpansion of y . More precisely, the best rational approximants are the ‘convergents’ and‘semi-convergents’ of the continued fraction expression of y . Since it is not necessary forthis paper we will not explain the details; but the interested reader can check the thoroughexplanation in Khinchin (1997).The best rational approximants are associated to points where a quasiperiodic func-tion is close to being periodic. For the function in (1), ω /ω = π . The best rationalapproximants of 2 π then correspond to points where the two components of f ( x ) , thatis, sin ( x ) and sin ( π x ) , are close to repeating at the same time. For example, we can use2 π ≈ /
3. This leads to ( )( ) ≈ ( π )( ) . In other words, 19 repetitions of sin ( π x ) are close to 3 repetitions of sin ( x ) . A period of sin ( x ) begins at x =
0, 2 π , 4 π , 6 π and aperiod of sin ( π x ) begins at x =
0, 1, 2, 3, . . . , 18, 19. The closest pair of points where theperiod begins from this list is 6 π ≈ x ∈ (
0, 19 ) that is closer than this pair. This is a direct consequence of thefact that 19/3 is a better approximation to 2 π than any fraction with denominator 3 orless (where we define ‘better’ in terms of Definition 3.3). We sometimes say that 19 is a quasiperiod of f ( x ) .
4. Creating ‘Raindrops in A minor’
In this section, we explain the process behind the creation of the composition
Raindrops inA minor (Ong, 2020) and we discuss its mathematical and musical properties. This piececonsists of two simple elements: a repeating measure in 4/4 time running at 240 beats perminute, and a repeating measure in 6/4 time running at 360 /ϕ beats per minute, where ϕ ≈ /ϕ was intended to make the tempi ofthe two measures approximately equal. That is, 360 /ϕ ≈ ϕ is an irrational number, and thus the length of two measures (in terms of time)are rationally independent. To be precise, the first measure takes one second, while thesecond measure takes exactly ϕ seconds.Let us now directly connect this music with a quasiperiodic function as defined inDefinition 3.1. We interpret the domain R as time. We define the codomain to be R , whichrepresents the amplitude envelopes corresponding to the four musical notes played in thispiece (that is, A4, E5, C5, G5).In the notation of Definition 3.1, we define Q : S → R in the following way. Wedefine Q ( x ω , x ω ) = Q ( x ω ) + Q ( x ω ) , where Q ( x ω ) maps to the A4 and E5 com-ponents, and Q ( x ω ) maps to the C5 and G5 components. Note that the A4 and E5components should be 1-periodic in x , since Q ( x ω ) , the part that plays the A4 and E5notes, repeats every second. Similarly, the C5 and G5 components repeat every ϕ sec-onds, so Q ( x ω ) must be ϕ -periodic in x . However, since Q is defined as a function on OURNAL OF MATHEMATICS AND THE ARTS 7
Figure 2.
Raindrops in A minor is generated by playing these two infinitely repeating measures simulta-neously. © Darren C. Ong.
Figure 3.
After 1,2,3,5,8 or 13 seconds, observe that the two measures in Figure 2 begin at almost exactlythe same time. The larger the Fibonacci number, the closer the two measures are to starting in sync. Notethat the numbers in the bottom of the figure are approximate, since ϕ is an irrational number. This musicis written in treble clef. S parametrized as [0, 2 π ) , to achieve the desired periodicity we must set ω = π and ω = π/ϕ . Given these definitions, f ( x ) will be a quasiperiodic function that indicatesthe amplitudes corresponding to the four notes at a time x (Figure 2 ).This music never repeats exactly even if it could be played for an infinite amount oftime. However, there will be points in time where the music seems very close to repeating.These points are associated with rational approximants of the golden ratio. For example,since ϕ ≈ , this implies 13 · ≈ · ϕ . In other words, thirteen periods of the first mea-sure takes roughly the same time as eight periods of the second measure. Indeed, we cancalculate that the eighth period of the second measure ends after roughly 12.94 seconds,and the thirteenth period of the first measure obviously ends after exactly 13 seconds (seeFigure 3). Following our discussion in Section 3.2, we say that this musical compositionhas a quasiperiod of 13 seconds, since the music almost repeats exactly after that amountof time. Raindrops in A minor has the pleasant property that its quasiperiods are after1, 2, 3, 5, 8, 13, 21, . . . seconds. These are precisely the Fibonacci numbers!We can observe that the larger the quasiperiod, the closer this piece gets to repeatingexactly. For example, using the approximation ϕ ≈ , we find that 34 periods of the sec-ond measure take about 55.01 seconds, whereas 55 periods of the first measure takes exactly55 seconds. Using larger and larger Fibonacci numbers, we can get the two periods to syn-chronize as closely as we wish, but it will never synchronize exactly. This explains why thismusic is non-repeating, even when played for infinite time.We should note however, that the above properties only apply for an ‘idealized’ versionof this music. In practical terms, we created this sound file using a computer, and computersare not capable of dealing with irrational numbers like ϕ , resorting instead to a floating-point approximation. Thus, the full sound file we created will repeat, albeit after a largeamount of time. Even if we could overcome the issues with the floating-point approxi-mation, listeners would not be able to perceive music notes with infinite precision. In a D. C. ONG real-world situation, a human listener would not be able to perceive that even a perfectlyplayed version of this music is not periodic.In terms of the music, we chose simple melodies from an A minor 7th chord. Bellsseemed like a good choice of instrument, since the ringing of bells often produces notesthat are played almost simultaneously. Also, a ‘canon-like’ musical piece played with per-cussion can remind us of gamelan music. In terms of software, we used Musescore to writethe music, and Audacity to perform the sound mixing.
Raindrops in A minor was intended to be a simple example of this quasiperiodic musicthat demonstrates its mathematical properties. Of course, it is possible to consider morecomplex extensions of this idea. One could use irrational numbers other than the goldenratio ϕ . Different irrational numbers have different best rational approximants. This canresult in a different set of quasiperiods, and so different features in the quasiperiodic music.We could also consider writing quasiperiodic music with more parts. To create aquasiperiodic musical piece with n parts for a postive integer n , we first find a set of n rationally indepent numbers ω , . . . , ω n , as in Definition 3.2. We then create musicwhere the first part repeats every ω second, the second part repeats every ω seconds,and so on, so the n th part repeats every ω n seconds. One could even imagine an entireorchestra where each instrument is playing a part with a different rationally independentperiod!
5. Related musical work
There are quite a few interesting connections between quasiperiodic music and ideaspresent in modern musical composition and theory. We will briefly discuss some examplesin this section in order to place quasiperiodic music in the context of music scholarship.
Lindenmayer systems (or L-systems) originated in biology as a way to model plant growth,and then inspired also musical composition. As an example, we consider an alphabet withtwo symbols, A and B , with the rules A → AB and B → A . Let us start with a stringconsisting of a single letter A , and then let us apply the rules to each letter in the stringrepeatedly, to get progressively longer strings. The first few iterations will then be A , AB , ABA , ABAAB , ABAABABA . The reader might notice that this is exactly the Fibonacciword we mentioned in the introduction. Indeed, these Lindenmayer systems are associ-ated to substitution sequences, which are one of the most prominent discrete examples ofaperiodic order (see Chapter 4 of Baake & Grimm, 2013).These patterns have caught the interest of musicians. Possibily the first example ofLindenmayer systems being used in music is described in Prusinkiewicz (1986), with analgorithmic musical composition process. We quote from the conclusion of that paper:
This paper presents a technique for generating musical scores, which consists of three steps:(1) A string of symbols is generated by an L-system,(2) This string is interpreted graphically as a sequence of commands controlling a turtle,(3) The resulting figure is interpreted musically, as a sequence of notes with pitch andduration determined by the coordinates of the figure segments.
OURNAL OF MATHEMATICS AND THE ARTS 9
The scores generated using the above method are quite interesting. They are relatively com-plex (in spite of the simplicity of the underlying productions) but they also have a legibleinternal structure (they do not make the impression of sounds accidentally put together).
These ideas have been built upon by composers such as Hanspeter Kyburz, Enno Poppe,Phillipe Manoury, and Alberto Posada: Besada (2019) contains an interesting discussionof the impact of these Lindenmayer systems on contemporary music.These compositions based on Lindenmayer systems are perhaps the closest antecedentsof our work. Substitution sequences (which are equivalent to these Lindenmayer systems)are discrete aperiodically ordered structures, whereas quasiperiodic functions are contin-uous aperiodically ordered structures. The music we introduce thus shares a lot of featureswith the music based on Lindenayer systems. In particular, I think of Prusinkiewicz’scomments on his technique producing music that is ‘relatively complex in spite of simpleunderlying productions’ and with ‘legible internal structure.’ This may also easily apply toquasiperiodic music. In some way, quasiperiodic music can be considered as a continuousanalogue of the music based on Lindenmayer systems.
A very well-studied example of using period lengths to evoke an illusion of a music pieceextending to eternity comes from Olivier Messaien’s
Liturgie de Cristal , which is the firstmovement of
Quatuor pour la Fin du Temps (that is,
Quartet for the End of Time ). Thispiece is for violin, clarinet, cello, and piano. This movement’s use of prime numbers hasbeen well-noted in the literature. For example, on page 25 of Dingle (2016): ‘Liturgie de Cristal’ is undoubtedly the most famous example of a compositional machinein Messiaen’s music. As has been discussed at length the piano part has a cycle of 29 chordscombined with a set of 17 durations. Both being prime numbers, it would take 17 times 29cycles to return to the same starting point. To this should be added the cello, which plays acycle of 5 pitches and a sequence of 15 durations. For this machine to return to its startingpoint would take 4 hours 40 minutes, but Messiaen simply presents a short fragment. Theeffect is of a door being opened to reveal the ongoing workings, and then being closed again.
There are a lot of other works noting Messiaen’s use of mathematics here, for example(Pople, 1998).
Raindrops in A minor differs from Messiaen’s composition because it does not play withdifferent lengths of periods in order to create an illusion of eternity; in some sense, in itsidealized form, the performance of
Raindrops is truly ‘eternal.’
Vuza canons are another way that aperiodicity appears in music theory. These rhythmicstructures arise from combining aperiodic canons to create another canon having a largeperiod. These canons are closely related to the problem of tiling the integers with ape-riodic tiles (as in the title of Jedrzejewski, 2009). Again, there is an interplay betweenperiodicity and aperiodicity, and complex patterns arising from repetitions of simpler ones.There is a complicated mathematical structure in these canons, including connectionswith group theory and cyclotomic polynomials. It is unsurprising that these canons haveattracted much interest from both music theorists and mathematicians, enough so that an entire special issue of the
Journal of Mathematics and Music is devoted to them (Andreatta& Agon, 2009). See also Andreatta (2011) for a historical survey.
Phasing is a musical technique in minimal music first developed by Steve Reich, and againthere are inescapable thematic connections between phase music and quasiperiodic music.Phasing is associated with minimal music. Similar to our quasiperiodic music compo-sition, a typical phase music composition is formed from two repetitive phrases. However,typically these two phrases are identical, and initially they are played in unison. Gradually,the phrases drift out of sync, with the second phrase lagging behind the first, intially cre-ating an ‘echo’ effect, and eventually a complex combination of sounds. The phrases thendrift back into unison. Note that the two phrases are typically played at approximately thesame tempo . The slowdown to make one phrase lag behind the other is very gradual.Probably the most famous piece of phase music is Steve Reich’s
Piano Phase . Thefollowing description of this music is in page 386 of Schwarz (1980):
In this work two performers begin in unison playing the identical rhythmic/melodic pattern.As the first performer’s pattern remains unvarying, the second pianist increases his tempo veryslightly (this gradual phasing process is indicated in Reich’s scores by dotted lines betweenmeasures) until he is finally one sixteenth note ahead of the unchanged figure of the firstpianist. The phasing process pauses at this point, as the newly shifted rhythmic configura-tion is repeated several times. Soon however, the second pianist again moves slowly forwardof the first, finally ending two sixteenth notes ahead of the original pattern. This sequence ofgradual phase shift and repetition is repeated until the two pianists are back in unison; at thisjuncture the pattern changes and the whole process begins anew.
The structure of Steve Reich’s phase music has received considerable mathematicalanalysis. In Colannino et al. (2009) and Cohn (1992) the authors perform a systematicmathematical analysis on the rhythmic structure of
Piano Phase and its sister piece,
ViolinPhase . While these pieces show an interesting mathematical depth, they are neverthelessfinite.Conversely, quasiperiodic music is infinite and aperiodic. It never perfectly repeats, evenif the pieces could be played forever. Despite these differences, it is easy to see thematicsimilarities between quasiperiodic music and phase music. They are both generated by asmall number of finite repeating measures, which result in complex polyrythmic soundsemerging from the combination. Quasiperiodic music and phase music both feature musi-cal parts drifting in and out of sync. In phase music, this occurs through the gradual slowingof the tempo of one part of the music, and in quasiperiodic music this occurs as the start-ing point of two parts get closer and closer together after each quasiperiod. In some sense,quasiperiodic music might be viewed as an alternative approach to phasing.
6. Conclusion
We used a common mathematical definition of quasiperiodic function, taken from Avronand Simon (1981). We then applied the structure of a quasiperiodic function to music,describing an original music composition (Ong, 2020). Then, we contextualized this cre-ative experiment within recent studies on mathematical music theory. Finally, we made
OURNAL OF MATHEMATICS AND THE ARTS 11 comparisons between quasiperiodic music and related ideas in music composition andtheory, for example, algorithmic music (with, in particular, Lindenmayer systems), andphase music.
Acknowledgments
This work was partially supported by a grant from the Fundamental Research Grant Scheme fromthe Malaysian Ministry of Education (Grant No: FRGS/1/2018/STG06/XMU/02/1) and two Xia-men University Malaysia Research Funds (Grant Numbers: XMUMRF/2018-C1/IMAT/0001 andXMUMRF/2020-C5/IMAT/0011). The author also wishes to thank Jake and Jamie Fillman for manyhelpful discussions, Maria Mannone for creating that delightful design for the graphical abstract, aswell as the anonymous reviewers for many comments that improved the content and presentationof the paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Funding
This work was partially supported by a grant from the Fundamental Research Grant Scheme fromthe Malaysian Ministry of Education (Grant No: FRGS/1/2018/STG06/XMU/02/1) and two Xia-men University Malaysia Research Funds (Grant Numbers: XMUMRF/2018-C1/IMAT/0001 andXMUMRF/2020-C5/IMAT/0011).
ORCID
Darren C. Ong http://orcid.org/0000-0003-4942-4376
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