Retrospective Review of the Development of the Concepts of Consonance and Dissonance: On the Problem of Musical and Aesthetic as well as Physical and Mathematical Interpretation

"The research aimed at a comprehensive retrospective analysis of the development of “consonance and dissonance” as a musical and aesthetic category in physical and mathematical interpretation; defining the notion of consonance as a musical theoretical and aesthetic category, confirming the relevance of its use in physical and mathematical terminology to explain acoustic phenomena. Research methodology is based on the use of the retrospective method (or the retrospection method), which allowed identifying the theories of consonance and dissonance in the historical retrospective. The periodization method was used to find out individual stages in the development of science in order to discover the leading directions of scientific thought, identify new elements relating to various aspects of “consonance” and “dissonance”. The study of the retrospective review of the development of the notions of consonance and dissonance in the physical and mathematical interpretation involved an interdisciplinary method a way of organizing research work, providing for the interaction of music and mathematics in the study of consonance and dissonance. Scientific novelty. This study is the first to reflect the general tendency towards the mathematization of the humanities and the humanitarization of the physical and mathematical areas of modern culture. Some provisions of musical acoustics were clarified in the context of creating a harmonious conceptual structure. The article presents the author’s concept of clarifying the notion of phase in relation to the spectral structure of an audio signal based on the notion “slightly mistuned consonance”. Conclusions. A retrospective review of the development of the notions of consonance and dissonance in the physical and mathematical interpretation was carried out and presented as a comprehensive description and review of the formation of concepts in the temporal sequence of their creation. Retrospectiveness through a review of significant discoveries and achievements in music and natural science allowed tracing the formation of the theories of consonance and dissonance from the standpoint of the interdisciplinarity of modern knowledge in the humanities. The further process of the development of sound musical art technologies requires a significantly higher scientific level of their study, the creation of a coherent conceptual system based on modern physical and mathematical sciences as well as computer science to explain sound acoustic phenomena. Keywords: consonance, dissonance, mistuned consonance, overtone, harmonics, beats, phase, spectral component"

Of the Nature of Math and Math in Nature – a Sequel About Symmetry for Children and Adults

The paper describes activities for developing the notion of symmetries. The activities were performed by a group of children between the ages of 5 and 9 and a parent of two of them. Through play and active work the group explores symmetries in sports, architecture, biology, language, music and mathematics. The activities were carried out outdoors in nature. Ideas for complementary activities with computers are presented. As a result the children demonstrate the ability to recognize symmetries in areas and situations different from the ones set in the activities they performed. This proves the main thesis of the paper: the nature of mathematics can be captured successfully in an informal environment at any age.

Numerological symbols in Schoenberg's creation

Expressionism in music was the movement that brought the contrast to previous eras promoting a tense music, exacerbation of means of expression and destruction of tonal balance. The works that belong to this innovative artistic movement involve diverse interpretations, because they capture the general tense atmosphere, from an aestheticphilosophical perspective, highlighting some text-sonority correspondences and the expressiveness of the elements of musical language. Over time, various connections have been made between music and mathematics, proving that mathematics can give a solid, deep foundation to music, without having a major impact on the rendering of feelings. This research presents some aspects of how numerology influenced and marked the creation of the expressionist composer Arnold Schoenberg. By using the symbolism of numbers, we can better understand the way the composer thought and created his music, strongly marked by the dissolution of tonality, the twelve-tone technique, and the transition from belcanto to "Sprechgesang" and “Sprechmelodie”.

Music and Mathematics

This chapter explores connections between music and mathematics. It includes rationales for integrating music and math, common links between the two disciplines, and a discussion of the Common Core State Standards for Mathematics and the National Core Arts Standards. Tables clearly show the standards that are incorporated throughout the lessons and examples. The chapter contains four detailed, full-length lessons that integrate music and mathematics. These include lessons exploring part/whole relationships in music and math; analyzing structure and pattern, using the song “Wide Open Spaces” as a model; studying the Fibonacci sequence in math and music; and creating an amusement park ride with a theme song. The chapter ends with an inventory of ideas detailing fourteen additional lesson topics, specific teaching strategies, and recommended activities.

An interdisciplinary approach to learning programming through the creation of electronic music in the sonic pi audio programming environment

With the development of computer technologies, there is a rapid development of applied music based on computer technology: music for computer games, for advertising, for voicing household and other equipment, music in movies and animations, TV music, and academic music. This indicates the relevance of such an area of use of information technologies as program (algorithmic) sound synthesis, and the need for schoolchildren to familiarize themselves with this promising IT area. The fact that music and mathematics are interconnected has been known since ancient times. Now we can observe the close connection of music and informatics. And not just the processing of music in synthesis programs, but the programmatic creation of music using code. The article shows the relationship of musical art with the art of programming, considers the concept of the Sonic Pi audio programming environment, demonstrates the main features of this environment, language tools for programming algorithms, data structures and musical compositions in Ruby.

Pythagoras' Legacy

The history of mathematics starts in earnest with one of Pythagoras’ most important proofs, the Pythagorean theorem. This proof was the first link in a chain of ground-breaking ideas, all interconnected with each other, that turned mathematics into an “art of the mind.” The chain continues to be extended today. There would be no computers, science, engineering, or philosophy without Pythagoras’ legacy. This book sketches an outline of that legacy by presenting and discussing ten of the greatest ideas in the mathematical chain. Its aim is to illustrate why mathematics can be designated an intellectual art, a creative enterprise that mirrors any art, from music to painting. Pythagoras actually connected music and mathematics into a theory of the world called the Harmony of the Spheres. The book is intended for a general audience, and especially those who may think that mathematics is uninteresting or boring. Each of its ten chapter ends with five exploratory puzzles that will allow readers to become engaged in some of the ideas treated in the chapter without any technical knowledge. This will allow readers to use this book as well as a collection of fairly easy math problems.

Fractions and Pythagorean Tuning—An Interdisciplinary Study in Secondary Education

Formal education is experiencing a series of reforms that favor the integration of the contents of different areas in the teaching and learning of the different educational stages. The present study examined the use of an interdisciplinary music and mathematics experience in Secondary Education in Galicia (Spain) in the 2016/17 academic year. A descriptive–exploratory design was used, through a Likert questionnaire applied to 197 students with a diagnostic test and a reference test, and a study of multiple cases was carried out in which information was collected through classroom observations. The results show improvements in the understanding of mathematical and musical concepts, and attitudes and procedures so we can argue that the use of interdisciplinary activities have favored the development of teaching–learning opportunities in mathematical and musical training.