Uncovering GEMS of Mathematics

Gateway to Exploring Mathematical Sciences (GEMS) is an outreach program offered by the six mathematics departments of the Claremont Colleges for eighth, ninth, and tenth graders. In this paper, we describe our program (in terms of format, participants, mathematical activities and topics involved) and share why we are so enthusiastic about it

La melodía de las matemáticas

Music and Mathematics have been considered like complementary areas of studies since the times of Pythagoras. His contributions regarding these two areas are explained in this project with the objective of being able to show the reader the basis of the existing parallelisms. Many composers, between actual and old composers, have used mathematical structures either conscious or unconsciously. Some were even criticized for their strict inclination with the use of this science for their compositions; however, we cannot assure these practices are common. This project pretends to show that, even though it is not very perceptible, the relationship existing between these two areas is deeper than it seems.La Música y las Matemáticas han sido consideradas como áreas complementarias desde los tiempos de Pitágoras. Sus aportes en relación a estas dos áreas son explicados en este trabajo con el objetivo de mostrar al lector las bases de los paralelismos existentes. Varios compositores, entre actuales y antiguos, han hecho uso de estructuras matemáticas consciente o inconscientemente. Algunos fueron incluso criticados por su estricta inclinación hacia el uso de esta ciencia para sus composiciones; sin embargo, no podemos asegurar que este tipo de casos sean comunes. Este trabajo pretende mostrar que, a pesar de que sea poco perceptible, la relación entre estas dos áreas es más profunda de lo que parece

An alternative approach to generalized Pythagorean scales: generation and properties derived in the frequency domain

This is an Accepted Manuscript of an article published by Taylor & Francis Group in Journal of mathematics and music on march 2020, available online at: http://www.tandfonline.com/10.1080/17459737.2020.1726690.Abstract scales are formalized as a cyclic group of classes of projection functions related to iterations of the scale generator. Their representatives in the frequency domain are used to built cyclic sequences of tone iterates satisfying the closure condition. The refinement of cyclic sequences with regard to the best closure provides a constructive algorithm that allows to determine cyclic scales avoiding continued fractions. New proofs of the main properties are obtained as a consequence of the generating procedure. When the scale tones are generated from the two elementary factors associated with the generic widths of the step intervals we get the partition of the octave leading to the fundamental Bézout’s identity relating several characteristic scale indices. This relationship is generalized to prove a new relationship expressing the partition that the frequency ratios associated with the two sizes composing the different step-intervals induce to a specific set of octaves.Peer ReviewedPostprint (author's final draft

Matematičko modeliranje puhačkih instrumenata

U radu je dana klasifikacija puhačkih instrumenata i pripadne matematičke formulacije proizvodnje zvuka koji oni proizvode. Analiziran je rombos, slobodni aerofon i dane su jednadžbe koje opisuju njegovo gibanje prilikom njegovog sviranja. Izvedena je valna jednadžba za prave aerofone cjevastog oblika, te jednadžbe harmonika. Opisana su osnovna svojstva valova; princip superpozicije i stojni valovi. U ovisnosti o građi pravog aerofona, jesu li obje strane cijevi otvorene (u slučaju flaute) ili je jedna strana zatvorena (u slučaju klarineta) dobivamo pripadni skup harmonika. Od pravih aerofona analizirani su skandinavska frulica, okarina i rog.In this paper the classification of wind instruments is given, alongside the mathematical formulation on making of the sound. The analysis of the bullroarer, the free aerophone and the equations that describe his motion while it’s being played is performed. Derivation of the wave equation is done for proper aerophones that are tube-shaped as well as harmonic equations. Basic wave properties are described such as superposition principle and standing waves. Depending on the structure of proper aerophone, weather the both sides of the tube are open (in case of the flute) or one side of the tube is closed (in case of the clarinet) we’re getting harmonics set accordingly. Proper aerophones that have been analysed are willow flute, ocarina and the horn

Sistema de afinación musical de proporciones áureas

En este Trabajo Final de Máster se va a tratar de realizar un sistema de afinación basado en proporciones áureas. Para ello se van a realizar tres aproximaciones una con doble temperado, a la que se le denominará bilineal, una aproximación parabólica y otra exponencial. Para introducir el tema y justificar este TFM se van a explicar los distintos tipos de afinación y se van a comentar algunos tipos. Una vez generado el sistema de afinación se van a realizar una serie de cálculos para comprobar si los sistemas son compatibles con el sistema temperado igual de 12 notas. Para explicar estos cálculos, se va a hacer una introducción a la lógica borrosa.In this Thesis we will to try create a tuning system based on aureus proportions. For this three approaches will adopted: double tempering, which will be called bilinear, a parabolic approximation and an exponential approximation. To introduce the topic and justify this TFM we will attempt to explain the different types of tuning and will elaborate on some types. Once we have created the tuning system we will perform a series of calculations to see if the systems are compatible with the 12-note equal tempered system. To explain these calculations, we will make an introduction to fuzzy logicBarberá Saiz, J. (2012). Sistema de afinación musical de proporciones áureas. Universitat Politècnica de València. http://hdl.handle.net/10251/18491Archivo delegad

Errata and Addenda to Mathematical Constants

We humbly and briefly offer corrections and supplements to Mathematical Constants (2003) and Mathematical Constants II (2019), both published by Cambridge University Press. Comments are always welcome.Comment: 162 page

Sound bullets from nonlinear granular chains

The propagation of ultrasound along chains of granular particles has some interesting characteristics. These have the potential to dramatically improve the performance of HIFU (High Intensity Focussed Ultrasound) for the use in therapeutic ultrasound treatments and medical imaging. This thesis has investigated a novel approach for the creation of ultrasonic focussed energy in chains composed of spheres. Within these highly sensitive chains, non-linear propagation is possible which leads to the formation of highly robust localised pulses known as sound bullets. Subject to the right conditions, the chain of spheres become a dynamically tunable system where slight changes to the nature of the Herzian contact between the spheres produce drastic changes in the propagation velocity of the solitary wave. The nature and resulting characteristics of the system to variations such as input excitation frequency, effect of loading, changes in length and diameter of the chain were studied. It was observed that the system was highly dependent of each of these factors, with each situation altering the behaviour of the chain of spheres