Ratio Of Small Integers Research Articles

The Six-planet Resonant Chain of HD 110067

HD 110067 is the brightest star known to have six transiting planets. Each adjacent pair of planets has a period ratio that is nearly equal to a ratio of small integers, suggesting the planets are in a chain of mean-motion resonances, but the limited time span of the available data has prevented firm conclusions. Here, we show that the requirement of long-term dynamical stability implies that all six planets are very likely to form a resonant chain. Dynamical simulations of nonresonant systems with initial conditions compatible with the available data almost always suffer an instability within 25 Myr (∼0.3% of the system’s age). Assuming the system is in resonance, we place upper limits on the planets’ eccentricities and lower limits on the masses of the planets that have not yet been measured. We also predict the characteristics of transit timing variations and the values of the three-body libration centers.

Resonant chains in triple-planet systems

The mean motion resonance is the most important mechanism that may dominate the dynamics of a planetary system. In a multi-planetary system consisting of $N planets, the planets may form a resonant chain when the ratios of orbital periods of planets can be expressed as the ratios of small integers $T_1: T_2: :T_N=k_1: k_2: k_N$. Due to the high degree of freedom, the motion in such systems could be complex and difficult to depict. In this paper, we investigate the dynamics and possible formation of the resonant chain in a triple-planet system. We defined the appropriate Hamiltonian for a three-planet resonant chain and numerically averaged it over the synodic period. The stable stationary solutions -- apsidal corotational resonances (ACRs) -- of this averaged system, corresponding to the local extrema of the Hamiltonian function, can be searched out numerically. The topology of the Hamiltonian around these ACRs reveals their stabilities. We further constructed the dynamical maps on different representative planes to study the dynamics around the stable ACRs, and we calculated the deviation ($ of the resonant angle in the evolution from the uniformly distributed values, by which we distinguished the behaviour of critical angles. Finally, the formation of the resonant chain via convergent planetary migration was simulated and the stable configurations associated with ACRs were verified. We find that the stable ACR families arising from circular orbits always exist for any resonant chain, and they may extend to a high eccentricity region. Around these ACR solutions, regular motion can be found, typically in two types of resonant configurations. One is characterised by libration of both the two-body resonant angles and the three-body Laplace resonant angle, and the other by libration of only two-body resonant angles. The three-body Laplace resonance does not seem to contribute to the stability of the resonant chain much. The resonant chain can be formed via convergent migration, and the resonant configuration evolves along the ACR families to eccentric orbits once the planets are captured into the chain. Ideally, our methods introduced in this paper can be applied to any resonant chain of any number of planets at any eccentricity.

Commonality and variation in mental representations of music revealed by a cross-cultural comparison of rhythm priors in 15 countries

Music is present in every known society but varies from place to place. What, if anything, is universal to music cognition? We measured a signature of mental representations of rhythm in 39 participant groups in 15 countries, spanning urban societies and Indigenous populations. Listeners reproduced random ‘seed’ rhythms; their reproductions were fed back as the stimulus (as in the game of ‘telephone’), such that their biases (the prior) could be estimated from the distribution of reproductions. Every tested group showed a sparse prior with peaks at integer-ratio rhythms. However, the importance of different integer ratios varied across groups, often reflecting local musical practices. Our results suggest a common feature of music cognition: discrete rhythm ‘categories’ at small-integer ratios. These discrete representations plausibly stabilize musical systems in the face of cultural transmission but interact with culture-specific traditions to yield the diversity that is evident when mental representations are probed across many cultures.

A resonant sextuplet of sub-Neptunes transiting the bright star HD 110067.

Planets with radii between that of the Earth and Neptune (hereafter referred to as 'sub-Neptunes') are found in close-in orbits around more than half of all Sun-like stars1,2. However, their composition, formation and evolution remain poorly understood3. The study of multiplanetary systems offers an opportunity to investigate the outcomes of planet formation and evolution while controlling for initial conditions and environment. Those in resonance (with their orbital periods related by a ratio of small integers) are particularly valuable because they imply a system architecture practically unchanged since its birth. Here we present the observations of six transiting planets around the bright nearby star HD 110067. We find that the planets follow a chain of resonant orbits. A dynamical study of the innermost planet triplet allowed the prediction and later confirmation of the orbits of the rest of the planets in the system. The six planets are found to be sub-Neptunes with radii ranging from 1.94R⊕ to 2.85R⊕. Three of the planets have measured masses, yielding low bulk densities that suggest the presence of large hydrogen-dominated atmospheres.

Understanding Design Features of Music and Language: The Choric/Dialogic Distinction.

Music and spoken language share certain characteristics: both consist of sequences of acoustic elements that are combinatorically combined, and these elements partition the same continuous acoustic dimensions (frequency, formant space and duration). However, the resulting categories differ sharply: scale tones and note durations of small integer ratios appear in music, while speech uses phonemes, lexical tone, and non-isochronous durations. Why did music and language diverge into the two systems we have today, differing in these specific features? We propose a framework based on information theory and a reverse-engineering perspective, suggesting that design features of music and language are a response to their differential deployment along three different continuous dimensions. These include the familiar propositional-aesthetic (‘goal’) and repetitive-novel (‘novelty’) dimensions, and a dialogic-choric (‘interactivity’) dimension that is our focus here. Specifically, we hypothesize that music exhibits specializations enhancing coherent production by several individuals concurrently—the ‘choric’ context. In contrast, language is specialized for exchange in tightly coordinated turn-taking—‘dialogic’ contexts. We examine the evidence for our framework, both from humans and non-human animals, and conclude that many proposed design features of music and language follow naturally from their use in distinct dialogic and choric communicative contexts. Furthermore, the hybrid nature of intermediate systems like poetry, chant, or solo lament follows from their deployment in the less typical interactive context.

Categorical rhythms in a singing primate

What are the origins of musical rhythm? One approach to the biology and evolution of music consists in finding common musical traits across species. These similarities allow biomusicologists to infer when and how musical traits appeared in our species1. A parallel approach to the biology and evolution of music focuses on finding statistical universals in human music2. These include rhythmic features that appear above chance across musical cultures. One such universal is the production of categorical rhythms3, defined as those where temporal intervals between note onsets are distributed categorically rather than uniformly2,4,5. Prominent rhythm categories include those with intervals related by small integer ratios, such as 1:1 (isochrony) and 1:2, which translates as some notes being twice as long as their adjacent ones. In humans, universals are often defined in relation to the beat, a top-down cognitive process of inferring a temporal regularity from a complex musical scene1. Without assuming the presence of the beat in other animals, one can still investigate its downstream products, namely rhythmic categories with small integer ratios detected in recorded signals. Here we combine the comparative and statistical universals approaches, testing the hypothesis that rhythmic categories and small integer ratios should appear in species showing coordinated group singing3. We find that a lemur species displays, in its coordinated songs, the isochronous and 1:2 rhythm categories seen in human music, showing that such categories are not, among mammals, unique to humans3.

A super-Earth and two sub-Neptunes transiting the nearby and quiet M dwarf TOI-270

We report the Transiting Exoplanet Survey Satellite discovery of three small planets transiting one of the nearest and brightest M dwarf hosts to date, TOI-270 (TIC 259377017; K-mag 8.3; 22.5 parsec). The M3V-type star is transited by the super-Earth-sized TOI-270 b (1.247+0.089-0.083 R_earth) and the sub-Neptune-sized exoplanets TOI-270 c (2.42+-0.13 R_earth) and TOI-270 d (2.13+-0.12 R_earth). The planets orbit close to a mean-motion resonant chain, with periods (3.36, 5.66, and 11.38 days) near ratios of small integers (5:3 and 2:1). TOI-270 is a prime target for future studies since: 1) its near-resonance allows detecting transit timing variations for precise mass measurements and dynamical studies; 2) its brightness enables independent radial velocity mass measurements; 3) the outer planets are ideal for atmospheric characterisation via transmission spectroscopy; and 4) the quiet star enables future searches for habitable zone planets. Altogether, very few systems with small, temperate exoplanets are as suitable for such complementary and detailed characterisation as TOI-270.

Why Do Durations in Musical Rhythms Conform to Small Integer Ratios?

One curious aspect of human timing is the organization of rhythmic patterns in small integer ratios. Behavioral and neural research has shown that adjacent time intervals in rhythms tend to be perceived and reproduced as approximate fractions of small numbers (e.g., 3/2). Recent work on iterated learning and reproduction further supports this: given a randomly timed drum pattern to reproduce, participants subconsciously transform it toward small integer ratios. The mechanisms accounting for this “attractor” phenomenon are little understood, but might be explained by combining two theoretical frameworks from psychophysics. The scalar expectancy theory describes time interval perception and reproduction in terms of Weber's law: just detectable durational differences equal a constant fraction of the reference duration. The notion of categorical perception emphasizes the tendency to perceive time intervals in categories, i.e., “short” vs. “long.” In this piece, we put forward the hypothesis that the integer-ratio bias in rhythm perception and production might arise from the interaction of the scalar property of timing with the categorical perception of time intervals, and that neurally it can plausibly be related to oscillatory activity. We support our integrative approach with mathematical derivations to formalize assumptions and provide testable predictions. We present equations to calculate durational ratios by: (i) parameterizing the relationship between durational categories, (ii) assuming a scalar timing constant, and (iii) specifying one (of K) category of ratios. Our derivations provide the basis for future computational, behavioral, and neurophysiological work to test our model.

Computational Approach to Musical Consonance and Dissonance.

In sixth century BC, Pythagoras discovered the mathematical foundation of musical consonance and dissonance. When auditory frequencies in small-integer ratios are combined, the result is a harmonious perception. In contrast, most frequency combinations result in audible, off-centered by-products labeled “beating” or “roughness;” these are reported by most listeners to sound dissonant. In this paper, we consider second-order beats, a kind of beating recognized as a product of neural processing, and demonstrate that the data-driven approach of Recurrence Quantification Analysis (RQA) allows for the reconstruction of the order in which interval ratios are ranked in music theory and harmony. We take advantage of computer-generated sounds containing all intervals over the span of an octave. To visualize second-order beats, we use a glissando from the unison to the octave. This procedure produces a profile of recurrence values that correspond to subsequent epochs along the original signal. We find that the higher recurrence peaks exactly match the epochs corresponding to just intonation frequency ratios. This result indicates a link between consonance and the dynamical features of the signal. Our findings integrate a new element into the existing theoretical models of consonance, thus providing a computational account of consonance in terms of dynamical systems theory. Finally, as it considers general features of acoustic signals, the present approach demonstrates a universal aspect of consonance and dissonance perception and provides a simple mathematical tool that could serve as a common framework for further neuro-psychological and music theory research.

Rhythmic Prototypes Across Cultures

It has long been assumed that rhythm cognition builds on perceptual categories tied to prototypes defined by small-integer ratios, such as 1:1 and 2:1. This study aims to evaluate the relative contributions of both generic constraints and selected cultural particularities in shaping rhythmic prototypes. We experimentally tested musicians’ synchronization (finger tapping) with simple periodic rhythms at two different tempi with participants in Mali, Bulgaria, and Germany. We found support both for the classic assumption that 1:1 and 2:1 prototypes are widespread across cultures and for culture-dependent prototypes characterized by more complex ratios such as 3:2 and 4:3. Our findings suggest that music-cultural environments specify links between music performance patterns and perceptual prototypes.

Cavity length matching and optical resonances in a Raman laser with the multimode pump source.

We report on the observation of local reduction in generation threshold and enhancement in the output energy of a nanosecond Raman laser when the round-trip time of its cavity is matched with that of a longitudinally multimode pump laser or related to the latter as a ratio of small integers. We refer both observations to the synchronous pump effect originating from the periodicity in the small-scale intensity structure of the pump pulse.

Seven temperate terrestrial planets around the nearby ultracool dwarf star TRAPPIST-1

One focus of modern astronomy is to detect temperate terrestrial exoplanets well-suited for atmospheric characterisation. A milestone was recently achieved with the detection of three Earth-sized planets transiting (i.e. passing in front of) a star just 8% the mass of the Sun 12 parsecs away1. Indeed, the transiting configuration of these planets combined with the Jupiter-like size of their host star - named TRAPPIST-1 - makes possible in-depth studies of their atmospheric properties with current and future astronomical facilities1,2,3. Here we report the results of an intensive photometric monitoring campaign of that star from the ground and with the Spitzer Space Telescope. Our observations reveal that at least seven planets with sizes and masses similar to the Earth revolve around TRAPPIST-1. The six inner planets form a near-resonant chain such that their orbital periods (1.51, 2.42, 4.04, 6.06, 9.21, 12.35 days) are near ratios of small integers. This architecture suggests that the planets formed farther from the star and migrated inward4,5. The seven planets have equilibrium temperatures low enough to make possible liquid water on their surfaces6,7,8.

Virtual Pitch and Pitch Shifts in Church Bells

It is well established that musical sounds comprising multiple partials with frequencies approximately in the ratio of small integers give rise to a strong sensation of pitch even if the lowest or fundamental partial is missing—the so-called virtual pitch effect. Experiments on thirty test subjects demonstrate that this virtual pitch is shifted significantly by changes in the spacing of the constituent partials. The experiments measured pitch by comparison of sounds of similar timbre and were automated so that they could be performed remotely across the Internet. Analysis of the test sounds used shows that the pitch shifts are not predicted by Terhardt’s classic model of virtual pitch. The test sounds used were modelled on the sounds of church bells, but a further experiment on seventeen test subjects showed that changes in partial amplitude only had a minor effect on the pitch shifts observed, and that a pitch shift was still observed when two of the lowest frequency partials were removed, so that the effects reported are of general interest.

Interval singing links to phenotypic quality in a songbird

Darwin was fascinated by melodic performances of insects, fish, birds, mammals, and men. He considered the ability to produce musical notes without direct use the most mysterious endowment of mankind. Bird song is attributed to sexual selection, but it remains unknown how the expected relationship between melodic performance and phenotypic quality arises. Melodies consist of sequences of notes, and both Pythagoras and music theorists in the Middle Ages found that their tonal frequencies form simple ratios that correspond to small-integer proportions derived from the harmonic series. Harmonics are acoustically predictable, and thus form the basis of the natural, just tuning system in music. Here I analyze the songs of the great tit (Parus major), a bird with a stereotyped song of typically two notes, and test the prediction that the deviations of the intervals from small-integer frequency ratios based on the harmonic series are related to the quality of the singer. I show that the birds with the smallest deviations from small-integer ratios possess the largest melanin-based black ventral tie, a signal that has been demonstrated to indicate social status and dominance, past exposure to parasites, and reproductive potential. The singing of notes with exact frequency relationships requires high levels of motor control and auditory sensory feedback. The finding provides a missing link between melodic precision and phenotypic quality of individuals, which is key for understanding the evolution of vocal melodic expression in animals, and elucidates pathways for the evolution of melodic expression in music.

DYNAMICS AND TRANSIT VARIATIONS OF RESONANT EXOPLANETS

ABSTRACT Transit timing variations (TTVs) are deviations of the measured midtransit times from the exact periodicity. One of the most interesting causes of TTVs is the gravitational interaction between planets. Here we consider a case of two planets in a mean motion resonance (orbital periods in a ratio of small integers). This case is important because the resonant interaction can amplify the TTV effect and allow planets to be detected more easily. We develop an analytic model of the resonant dynamics valid for small orbital eccentricities and use it to derive the principal TTV terms. We find that a resonant system should show TTV terms with two basic periods (and their harmonics). The resonant TTV period is proportional (m/M *)−2/3, where m and M * are the planetary and stellar masses. For m = 10−4 M *, for example, the TTV period exceeds the orbital period by about two orders of magnitude. The amplitude of the resonant TTV terms scales linearly with the libration amplitude. The ratio of the TTV amplitudes of two resonant planets is inversely proportional to the ratio of their masses. These and other relationships discussed in the main text can be used to aid the interpretation of TTV observations.

A resonant chain of four transiting, sub-Neptune planets

Surveys have revealed many multi-planet systems containing super-Earths and Neptunes in orbits of a few days to a few months. There is debate whether in situ assembly or inward migration is the dominant mechanism of the formation of such planetary systems. Simulations suggest that migration creates tightly packed systems with planets whose orbital periods may be expressed as ratios of small integers (resonances), often in a many-planet series (chain). In the hundreds of multi-planet systems of sub-Neptunes, more planet pairs are observed near resonances than would generally be expected, but no individual system has hitherto been identified that must have been formed by migration. Proximity to resonance enables the detection of planets perturbing each other. Here we report transit timing variations of the four planets in the Kepler-223 system, model these variations as resonant-angle librations, and compute the long-term stability of the resonant chain. The architecture of Kepler-223 is too finely tuned to have been formed by scattering, and our numerical simulations demonstrate that its properties are natural outcomes of the migration hypothesis. Similar systems could be destabilized by any of several mechanisms, contributing to the observed orbital-period distribution, where many planets are not in resonances. Planetesimal interactions in particular are thought to be responsible for establishing the current orbits of the four giant planets in the Solar System by disrupting a theoretical initial resonant chain similar to that observed in Kepler-223.

Ratios of Small Integers in Multiplicative Subgroups of Residue Rings

We obtain a new estimate, on average, over primes p in a dyadic interval, on the number of integers u, v of absolute value at most h which fall in a given multiplicative subgroup of the residue ring modulo p. Our result is based on an application of a large sieve inequality. We also present some numerical results comparing the new bound with several previously known bounds.

Principles of parsimony: fMRI correlates of beat-based versus duration-based sensorimotor synchronization.

(ProQuest: ... denotes formula omitted.)The temporal processing of successive intervals requires two different mechanisms: beat-based and duration-based timing (Essens & Povel, 1985; Grahn & McAuley, 2009; Keele, Nicoletti, Ivry, & Pokorny, 1989; Pashler, 2001; Teki, Grube, & Griffiths, 2012), which seem to engage different cerebral networks (Sakai et al., 1999; Teki, Grube, Kumar, & Griffiths, 2011). Most studies comparing beat-based and duration-based timing focused on perceptual tasks (Grube, Cooper, Chinnery, & Griffiths, 2010; Grube, Lee, Griffiths, Barker, & Woodruff, 2010; Teki et al., 2011) or memory tasks (Sakai et al., 1999). Despite the fact that timing perception and production involve different cerebral networks- though with some overlap (Coull, Cheng, & Meck, 2011; Wiener, Turkeltaub, & Coslett, 2010)-no studies reported human neuroimaging results of direct comparison between beat-based and duration-based sensorimotor synchronization (SMS). The reason might ensue from the difficulty to create SMS tasks in an irregular context that can be compared to corresponding tasks in a regular context.Duration-based timing requires encoding and retaining the absolute duration of discrete intervals. Beat-based timing involves an internal pulsation emerging from the perceived underlying regularity (or beat) of a sequence. The beat is a psychological construct: listeners can extract a pulsation from an auditory signal without it being physically present (Honing, 2013; Sethares, 2007) and without effort (Grahn & Rowe, 2013). Beat-based timing uses this underlying pulsation as a reference (Teki et al., 2011). Rhythmic patterns based on small-integer ratios, such as the 2:1 pattern, induce internal pulsation more strongly than those based on more complex ratios (Parncutt, 1994).SMS refers to synchronizing actions with external events and implies anticipation (see Repp, 2005 for a review). Many studies have shown the benefits of metrical beat subdivisions (i.e., based on integer ratios) on synchronization accuracy (e.g., Patel, Iversen, Chen, & Repp, 2005; Repp, 2003; Wohlschlager & Koch, 2000). In SMS tasks, error correction responses occurred for phase perturbations of the stimuli introduced at the level of the subdivision, that is, not interfering with the expected intertap intervals (ITI), even for subdivisions not directly preceding the target tone (Repp, 2008). This suggests hierarchical metrical processing: all temporal information available contributes to creating an internal model of the stimulus. Thus, regular patterns of noninteger ratios are likely to engage error-correction mechanisms. Such patterns tend to be adjusted and produced as integer ratio patterns (Essens, 1986; Franek , Radil, & Indra, 1988; Sakai et al., 1999).The cortical and subcortical structures most cited for their implication in both timing perception and production are the cerebellum, the supplementary motor area (SMA), and the basal ganglia (e.g., Coull et al., 2011; Grahn & Brett, 2009; Ivry & Spencer, 2004; Matell & Meck, 2000; Thaut, 2003; Wiener et al., 2010; Wing, 2002). In a review, Coull and collaborators (2011) reported differential activations for perception versus production tasks. Both types of tasks, though more rarely perception tasks, resulted in activation of the cerebellum, especially of lobule VI. At the cortical level, perceptual timing tasks activated almost exclusively the pre-SMA. Motor timing tasks also activated the SMA proper. At the level of the basal ganglia, motor tasks involved mostly the putamen, whereas perceptual tasks also activated the caudate nucleus and the globus pallidus.The basal ganglia and the pre-SMA/SMA have been mainly associated with beat processing (Grahn, 2009; Grahn & Brett, 2007, 2009; Schwartze, Keller, Patel, & Kotz, 2011; Teki et al., 2011). These structures show strong anatomical connections (Johansen-Berg et al. …

Stochastic entrainment of a stochastic oscillator

In this work, we consider a stochastic oscillator described by a discrete-state continuous-time Markov chain, in which the states are arranged in a circle, and there is a constant probability per unit time of jumping from one state to the next in a specified direction around the circle. At each of a sequence of equally spaced times, the oscillator has a specified probability of being reset to a particular state. The focus of this work is the entrainment of the oscillator by this periodic but stochastic stimulus. We consider a distinguished limit, in which (i) the number of states of the oscillator approaches infinity, as does the probability per unit time of jumping from one state to the next, so that the natural mean period of the oscillator remains constant, (ii) the resetting probability approaches zero, and (iii) the period of the resetting signal approaches a multiple, by a ratio of small integers, of the natural mean period of the oscillator. In this distinguished limit, we use analytic and numerical methods to study the extent to which entrainment occurs.

A biological rationale for musical consonance

The basis of musical consonance has been debated for centuries without resolution. Three interpretations have been considered: (i) that consonance derives from the mathematical simplicity of small integer ratios; (ii) that consonance derives from the physical absence of interference between harmonic spectra; and (iii) that consonance derives from the advantages of recognizing biological vocalization and human vocalization in particular. Whereas the mathematical and physical explanations are at odds with the evidence that has now accumulated, biology provides a plausible explanation for this central issue in music and audition.