The Circle of Fifths Is a Mystery to Many
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Unified Music Theories for General Equal-Temperament Systems
Unified Music Theories for General Equal-Temperament Systems Brandon Tingyeh Wu Research Assistant, Research Center for Information Technology Innovation, Academia Sinica, Taipei, Taiwan ABSTRACT Why are white and black piano keys in an octave arranged as they are today? This article examines the relations between abstract algebra and key signature, scales, degrees, and keyboard configurations in general equal-temperament systems. Without confining the study to the twelve-tone equal-temperament (12-TET) system, we propose a set of basic axioms based on musical observations. The axioms may lead to scales that are reasonable both mathematically and musically in any equal- temperament system. We reexamine the mathematical understandings and interpretations of ideas in classical music theory, such as the circle of fifths, enharmonic equivalent, degrees such as the dominant and the subdominant, and the leading tone, and endow them with meaning outside of the 12-TET system. In the process of deriving scales, we create various kinds of sequences to describe facts in music theory, and we name these sequences systematically and unambiguously with the aim to facilitate future research. - 1 - 1. INTRODUCTION Keyboard configuration and combinatorics The concept of key signatures is based on keyboard-like instruments, such as the piano. If all twelve keys in an octave were white, accidentals and key signatures would be meaningless. Therefore, the arrangement of black and white keys is of crucial importance, and keyboard configuration directly affects scales, degrees, key signatures, and even music theory. To debate the key configuration of the twelve- tone equal-temperament (12-TET) system is of little value because the piano keyboard arrangement is considered the foundation of almost all classical music theories. -
An Exploration of the Relationship Between Mathematics and Music
An Exploration of the Relationship between Mathematics and Music Shah, Saloni 2010 MIMS EPrint: 2010.103 Manchester Institute for Mathematical Sciences School of Mathematics The University of Manchester Reports available from: http://eprints.maths.manchester.ac.uk/ And by contacting: The MIMS Secretary School of Mathematics The University of Manchester Manchester, M13 9PL, UK ISSN 1749-9097 An Exploration of ! Relation"ip Between Ma#ematics and Music MATH30000, 3rd Year Project Saloni Shah, ID 7177223 University of Manchester May 2010 Project Supervisor: Professor Roger Plymen ! 1 TABLE OF CONTENTS Preface! 3 1.0 Music and Mathematics: An Introduction to their Relationship! 6 2.0 Historical Connections Between Mathematics and Music! 9 2.1 Music Theorists and Mathematicians: Are they one in the same?! 9 2.2 Why are mathematicians so fascinated by music theory?! 15 3.0 The Mathematics of Music! 19 3.1 Pythagoras and the Theory of Music Intervals! 19 3.2 The Move Away From Pythagorean Scales! 29 3.3 Rameau Adds to the Discovery of Pythagoras! 32 3.4 Music and Fibonacci! 36 3.5 Circle of Fifths! 42 4.0 Messiaen: The Mathematics of his Musical Language! 45 4.1 Modes of Limited Transposition! 51 4.2 Non-retrogradable Rhythms! 58 5.0 Religious Symbolism and Mathematics in Music! 64 5.1 Numbers are God"s Tools! 65 5.2 Religious Symbolism and Numbers in Bach"s Music! 67 5.3 Messiaen"s Use of Mathematical Ideas to Convey Religious Ones! 73 6.0 Musical Mathematics: The Artistic Aspect of Mathematics! 76 6.1 Mathematics as Art! 78 6.2 Mathematical Periods! 81 6.3 Mathematics Periods vs. -
The Worlds of Rigoletto: Verdi╎s Development of the Title Role in Rigoletto
Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2008 The Worlds of Rigoletto Verdi's Development of the Title Role in Rigoletto Mark D. Walters Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF MUSIC THE WORLDS OF RIGOLETTO VERDI’S DEVELOPMENT OF THE TITLE ROLE IN RIGOLETTO By MARK D. WALTERS A Treatise submitted to the College of Music in partial fulfillment of the requirements for the degree of Doctor of Music Degree Awarded: Spring Semester, 2008 The members of the Committee approve the Treatise of Mark D. Walters defended on September 25, 2007. Douglas Fisher Professor Directing Treatise Svetla Slaveva-Griffin Outside Committee Member Stanford Olsen Committee Member The Office of Graduate Studies has verified and approved the above named committee members. ii I would like to dedicate this treatise to my parents, Dennis and Ruth Ann Walters, who have continually supported me throughout my academic and performing careers. iii ACKNOWLEDGEMENTS I would like to express my gratitude to Professor Douglas Fisher, who guided me through the development of this treatise. As I was working on this project, I found that I needed to raise my levels of score analysis and analytical thinking. Without Professor Fisher’s patience and guidance this would have been very difficult. I would like to convey my appreciation to Professor Stanford Olsen, whose intuitive understanding of musical style at the highest levels and ability to communicate that understanding has been a major factor in elevating my own abilities as a teacher and as a performer. -
Nietzsche, Debussy, and the Shadow of Wagner
NIETZSCHE, DEBUSSY, AND THE SHADOW OF WAGNER A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Tekla B. Babyak May 2014 ©2014 Tekla B. Babyak ii ABSTRACT NIETZSCHE, DEBUSSY, AND THE SHADOW OF WAGNER Tekla B. Babyak, Ph.D. Cornell University 2014 Debussy was an ardent nationalist who sought to purge all German (especially Wagnerian) stylistic features from his music. He claimed that he wanted his music to express his French identity. Much of his music, however, is saturated with markers of exoticism. My dissertation explores the relationship between his interest in musical exoticism and his anti-Wagnerian nationalism. I argue that he used exotic markers as a nationalistic reaction against Wagner. He perceived these markers as symbols of French identity. By the time that he started writing exotic music, in the 1890’s, exoticism was a deeply entrenched tradition in French musical culture. Many 19th-century French composers, including Felicien David, Bizet, Massenet, and Saint-Saëns, founded this tradition of musical exoticism and established a lexicon of exotic markers, such as modality, static harmonies, descending chromatic lines and pentatonicism. Through incorporating these markers into his musical style, Debussy gives his music a French nationalistic stamp. I argue that the German philosopher Nietzsche shaped Debussy’s nationalistic attitude toward musical exoticism. In 1888, Nietzsche asserted that Bizet’s musical exoticism was an effective antidote to Wagner. Nietzsche wrote that music should be “Mediterranized,” a dictum that became extremely famous in fin-de-siècle France. -
CIRCLE of FIFTHS WORKSHOP a Useful Concept for When You Want to Jam with Other Instruments Or Write a Song!
CIRCLE OF FIFTHS WORKSHOP A useful concept for when you want to jam with other instruments or write a song! The Circle of Fifths shows the relationships among the twelve tones of the Chromatic Scale, their corresponding key signatures and the associated Major and Minor keys. The word "chromatic" comes from the Greek word chroma meaning "color." The chromatic scale consists of 12 notes each a half step or “semi-tone” apart. It is from the chromatic scale that every other scale or chord in most Western music is derived. Here’s the C chromatic scale going up as an example: C C# D D# E F F# G G# A A# B C In everyday terms: The Circle of Fifths is useful if you’re playing in an open jam with other instruments, e.g., guitars or fiddles around a campfire, or when you’re trying to figure out the chords of a song from scratch, or write your own song. The Circle of Fifths is a music theory picture. Each “segment” on the circle represents a note, a chord, and a key. Why is it called the Circle of Fifths? When going clockwise, each key is a fifth above the last. When going counter-clockwise, each key is a fourth above the last, so you may also hear it called the “circle of fourths”. The perfect fifth interval is said to be consonant, meaning it is a typical “pleasant sound” and sounds stable within music (the groundwork for this concept was set by Pythagoras and based on the sound frequency ratios). -
Finding Alternative Musical Scales
Finding Alternative Musical Scales John Hooker Carnegie Mellon University October 2017 1 Advantages of Classical Scales • Pitch frequencies have simple ratios. – Rich and intelligible harmonies • Multiple keys based on underlying chromatic scale with tempered tuning. – Can play all keys on instrument with fixed tuning. – Complex musical structure. • Can we find new scales with these same properties? – Constraint programming is well suited to solve the problem. 2 Simple Ratios • Acoustic instruments produce multiple harmonic partials. – Frequency of partial = integral multiple of frequency of fundamental. – Coincidence of partials makes chords with simple ratios easy to recognize. Perfect fifth C:G = 2:3 3 Simple Ratios • Acoustic instruments produce multiple harmonic partials. – Frequency of partial = integral multiple of frequency of fundamental. – Coincidence of partials makes chords with simple ratios easy to recognize. Octave C:C = 1:2 4 Simple Ratios • Acoustic instruments produce multiple harmonic partials. – Frequency of partial = integral multiple of frequency of fundamental. – Coincidence of partials makes chords with simple ratios easy to recognize. Major triad C:E:G = 4:5:6 5 Multiple Keys • A classical scale can start from any pitch in a chromatic with 12 semitone intervals. – Resulting in 12 keys. – An instrument with 12 pitches (modulo octaves) can play 12 different keys. – Can move to a different key by changing only a few notes of the scale. 6 Multiple Keys Let C major be the tonic key C 1 D#E 6 A C major b 0 notes F#G -
Modulation: Pivot Chords
MODULATION: PIVOT CHORDS APPLICATIONS OF THE HARMONIC WHEEL CONTENTS 2 The Modulation consists in changing from one Key to another. To do that, it is common to use a Pivot Chord, which is a chord that belongs to both keys, and then the new key is affirmed by a Cadence, which may consist, simply, in the chords V7 I. In this presentation, an explanation is given on how to obtain the pivot chords between two keys, using the Harmonic Wheel. For the sake of simplicity, only consonant pivot chords are considered, that is, Major or minor. © 2009 www.harmonicwheel.com CHORDS OF A GIVEN KEY 3 Every Major or natural minor key contains 6 consonant chords: 3 Major and 3 minor. On the Harmonic Wheel, these 6 chords are inside a curved rectangle and they are placed taking into account their affinity. For example, in the C Major / A minor key these 6 chords appear in the following arrangement: Dm Am Em F C G © 2009 www.harmonicwheel.com CHORDS OF A GIVEN KEY 4 © 2009 www.harmonicwheel.com CHORDS OF A GIVEN KEY 5 Remember that the scale tonic is indicated with and the rest of the notes with . Traditionally, the modulation is studied with the aid of the cycle of fifths, where the keys are sorted according to their key signatures. On its own, the Harmonic Wheel is a much more complete representation than a cycle of fifths, since it gives a panoramic view of chords and keys, as well as the existing relationships among them. © 2009 www.harmonicwheel.com MODUL. -
The Piano Trio, the Duo Sonata, and the Sonatine As Seen by Brahms, Tchaikovsky, and Ravel
ABSTRACT Title of Dissertation: REVISITING OLD FORMS: THE PIANO TRIO, THE DUO SONATA, AND THE SONATINE AS SEEN BY BRAHMS, TCHAIKOVSKY, AND RAVEL Hsiang-Ling Hsiao, Doctor of Musical Arts, 2017 Dissertation directed by: Professor Rita Sloan School of Music This performance dissertation explored three significant piano trios, two major instrumental sonatas and a solo piano sonatine over the course of three recitals. Each recital featured the work of either Brahms, Tchaikovsky or Ravel. Each of these three composers had a special reverence for older musical forms and genres. The piano trio originated from various forms of trio ensemble in the Baroque period, which consisted of a dominating keyboard part, an accompanying violin, and an optional cello. By the time Brahms and Tchaikovsky wrote their landmark trios, the form had taken on symphonic effects and proportions. The Ravel Trio, another high point of the genre, written in the early twentieth century, went even further exploring new ways of using all possibilities of each instrument and combining them. The duo repertoire has come equally far: duos featuring a string instrument with piano grew from a humble Baroque form into a multifaceted, flexible classical form. Starting with Bach and continuing with Mozart and Beethoven, the form traveled into the Romantic era and beyond, taking on many new guises and personalities. In Brahms’ two cello sonatas, even though the cello was treated as a soloist, the piano still maintained its traditional prominence. In Ravel’s jazz-influenced violin sonata, he treated the two instruments with equal importance, but worked with their different natures and created an innovative sound combination. -
Reviving Ballet in the Nineteenth Century: Music, Narrative, and Dance in Delibes's Coppélia by Arthur E. Lafex Submitted To
Reviving Ballet in the Nineteenth Century: Music, Narrative, and Dance in Delibes’s Coppélia By Copyright 2013 Arthur E. Lafex Submitted to the graduate degree program in Music and the Graduate Faculty of the University of Kansas in partial fulfillment of the requirements for the degree of Master of Music. ________________________________ Chairperson Alicia Levin ________________________________ Paul R. Laird ________________________________ David Alan Street Date Defended: April 15, 2013 The Thesis Committee for Author (Arthur E. Lafex) certifies that this is the approved version of the following thesis: Reviving Ballet in the Nineteenth Century: Music, Narrative, and Dance in Delibes’s Coppélia ________________________________ Chairperson Alicia Levin Date approved: April 15, 2013 ii Abstract Léo Delibes (1836-1891) wrote ballet scores that have inspired composers and have entertained generations of ballet lovers. His scores have been cited for their tunefulness, appropriateness for their narrative, and for their danceability. However, Delibes remains an obscure figure in music history, outside the musical canon of the nineteenth century. Likewise, his ballet music, whose harmonic resources are conventional and whose forms are variants of basic structures, has not received much scholarly and theoretical attention. This thesis addresses Delibes’s music by examining his ballet score for Coppélia, its support of narrative and also its support of dance. Chapter 1 begins with a historical view of ballet and ballet music up to the time of Delibes. Following a biographical sketch of the composer, a review of aspects of the score for Giselle by his mentor, Adolphe Adam (1803-1856) establishes a background upon which Delibes’s ballets can be considered. -
Consonance & Dissonance
UIUC Physics 406 Acoustical Physics of Music Consonance & Dissonance: Consonance: A combination of two (or more) tones of different frequencies that results in a musically pleasing sound. Why??? Dissonance: A combination of two (or more) tones of different frequencies that results in a musically displeasing sound. Why??? n.b. Perception of sounds is also wired into (different of) our emotional centers!!! Why???/How did this happen??? The Greek scholar Pythagoras discovered & studied the phenomenon of consonance & dissonance, using an instrument called a monochord (see below) – a simple 1-stringed instrument with a movable bridge, dividing the string of length L into two segments, x and L–x. Thus, the two string segments can have any desired ratio, R x/(L–x). (movable) L x x L When the monochord is played, both string segments vibrate simultaneously. Since the two segments of the string have a common tension, T, and the mass per unit length, = M/L is the same on both sides of the string, then the speed of propagation of waves on each of the two segments of the string is the same, v = T/, and therefore on the x-segment of string, the wavelength (of the fundamental) is x = 2x = v/fx and on the (L–x) segment of the string, we have Lx = 2(L–x) = v/fLx. Thus, the two frequencies associated with the two vibrating string segments x and L – x on either side of the movable bridge are: v f x 2x v f Lx 2L x - 1 - Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2002 - 2017. -
Study of Bartok's Compositions by Focusing on Theories of Lendvai, Karpati and Nettl
Science Arena Publications Specialty Journal of Humanities and Cultural Science ISSN: 2520-3274 Available online at www.sciarena.com 2018, Vol, 3 (4): 9-25 Study of Bartok's Compositions by Focusing on Theories of Lendvai, Karpati and Nettl Amir Alavi Lecturer, Tehran Art University, Tehran, Iran. Abstract: Bela Bartok is one of the most influential composers of the twentieth century whose works have been attended by many theorists. Among the most prominent musical theorists focusing on Bartok's works, Erno Lendvai, Jonas Karpati, Burno Nettl, and Milton Babbitt can be mentioned. These scholars have offered great theories by studying hidden, melodic and harmonic relations, and formal structures in the Bartok’s composition. The breadth of Bartok's works, diversity in using pitch organization, and lack of Bartok's personal views on analyzing his works, encouraged theorists to study his work more. The reason for focusing on Bartok in this research can be attributed to his prominent state as a composer in the twentieth century, as well as the way he employs folk melodies as the main component of composing structure. By focusing on tonal aspects, pitch organization, chord relations, and harmonic structures, this study deals with melodic lines and formal principles, and then dissects the theories and indicators in accordance with the mentioned theorists’ views. Keywords: Bartok, Axis System, Pole and Counter Pole, Overtone. INTRODUCTION Bartok's musical structure was initially influenced by the folk music of Eastern Europe. However, he has not provided any explanation with regard to his music. John Vinton's paper titled “Bartok in His Own Music” can be considered as the main source of reflection of Bartok's thoughts. -
Leonardo of Pisa AKA Fibonacci Was a 12Th Century Mathematician Most Famously Know for Introducing the Fibonacci Sequence
Presented by Angie Schirck-Matthews Broward College Davie Florida [email protected] Leonardo of Pisa • Leonardo of Pisa AKA Fibonacci was a 12th century mathematician most famously know for introducing the Fibonacci Sequence. • The sequence was discovered as an answer to the following riddle: • “How many pairs of rabbits would there be in a year if it was assumed that every month each pair produces a new pair, which begins to bear young two months after its own birth?” Rabbits • Fibonacci's question led to the following sequence known as the Fibonacci Sequence: • 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, … • Each term of the sequence is the sum of the previous two terms. The Fibonacci Sequence The most interesting sequence in the world! What’s so special about this sequence? • I’m glad you asked! • Eerily, many things in nature adhere to this sequence of numbers. • Things that are visually pleasing follow this sequence. • Things that are acoustically pleasing conform to this sequence. • The human body yields to this sequence. Fibonacci in Nature Spirals Fibonacci Spiral Flower petals 1 2 LUCKY CLOVER! 1 2 4 3 Trees and Plants Honey Bee Genealogy 8 5 3 2 1 1 Ocean Life Ocean waves Fibonacci in Art The Golden Ratio • The further you go out in the Fibonacci sequence, the ratio of consecutive numbers approaches the golden ratio (called ) The Golden Ratio 1.618 A Golden Rectangle • A rectangle with length and width consecutive Fibonacci numbers is called a golden rectangle. • Artists and Architects knew this was an aesthetically pleasing shape and it was incorporated in ancient art and architecture.