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Béla Bartók's Axis System
BÉLA BARTÓK’S AXIS SYSTEM APPLICATIONS OF THE HARMONIC WHEEL INTRODUCTION 2 Béla Bartók’s axis system was first published by Ernö Lendvai, one of his disciples, after performing an exhaustive analysis of his work. In short, it says that, if we are in the C Major key, the chords having the Tonic harmonic function are the following: • C and Cm • Their relative chords: Am and E , and also A and E m b b • The relatives of these last chords: F m and G (or F ) # b # © 2009 www.harmonicwheel.com TONIC AXES IN C MAJOR 3 We can represent these 8 chords in a cycle of fifths: © 2009 www.harmonicwheel.com THE OTHER AXES IN C MAJOR 4 The same reasoning can be applied to the chords with Dominant function, which will be: • G and Gm • Their relative chords: Em and B , and also E and B m b b • The relatives of these last chords: C m and D (or C ) # b # Similarly, the chords with Subdominant function will be: • F and Fm • Their relative chords: Dm and A , and also D and A m b b • The relatives of these last chords: Bm and C (or B) © 2009 www.harmonicwheel.com b DOMINANT AXES IN C MAJOR 5 The 8 Dominant chords in a cycle of fifths: © 2009 www.harmonicwheel.com SUBDOMINANT AXES IN C MAJOR 6 The 8 Subdominant chords in a cycle of fifths: © 2009 www.harmonicwheel.com HARMONIC FUNCTIONS 7 Therefore, in each key we can clasify the 24 Major and minor chords into 3 groups of 8 chords: • 8 chords with Tonic function (Group T) • 8 chords with Dominant function (Group D) • 8 chords with Subdominant function (Group S) Thus, we have a sequence of S – T – D functions that repeats itself in a cyclic way, as can be seen in the next figure. -
Scale Essentials for Bass Guitar Course Breakdown
Scale Essentials For Bass Guitar Course Breakdown Volume 1: Scale Fundamentals Lesson 1-0 Introduction In this lesson we look ahead to the Scale Essentials course and the topics we’ll be covering in volume 1 Lesson 1-1 Scale Basics What scales are and how we use them in our music. This includes a look at everything from musical keys to melodies, bass lines and chords. Lesson 1-2 The Major Scale Let’s look at the most common scale in general use: The Major Scale. We’ll look at its construction and how to play it on the bass. Lesson 1-3 Scale Degrees & Intervals Intervals are the building blocks of music and the key to understanding everything that follows in this course Lesson 1-4 Abbreviated Scale Notation Describing scales in terms of complete intervals can be a little long-winded. In this lesson we look at a method for abbreviating the intervals within a scale. Lesson 1-5 Cycle Of Fourths The cycle of fourths is not only essential to understanding key signatures but is also a great system for practicing scales in every key Lesson 1-6 Technique Scales are a common resource for working on your bass technique. In this lesson we deep dive into both left and right hand technique Lesson 1-7 Keys & Tonality Let’s look at the theory behind keys, the tonal system and how scales fit into all of this Lesson 1-8 Triads Chords and chord tones will feature throughout this course so let’s take a look at the chord we use as our basic foundation: The Triad Lesson 1-9 Seventh Chords After working on triads, we can add another note into the mix and create a set of Seventh Chords Lesson 1-10 Chords Of The Major Key Now we understand the basics of chord construction we can generate a set of chords from our Major Scale. -
Stravinsky and the Octatonic: a Reconsideration
Stravinsky and the Octatonic: A Reconsideration Dmitri Tymoczko Recent and not-so-recent studies by Richard Taruskin, Pieter lary, nor that he made explicit, conscious use of the scale in many van den Toorn, and Arthur Berger have called attention to the im- of his compositions. I will, however, argue that the octatonic scale portance of the octatonic scale in Stravinsky’s music.1 What began is less central to Stravinsky’s work than it has been made out to as a trickle has become a torrent, as claims made for the scale be. In particular, I will suggest that many instances of purported have grown more and more sweeping: Berger’s initial 1963 article octatonicism actually result from two other compositional tech- described a few salient octatonic passages in Stravinsky’s music; niques: modal use of non-diatonic minor scales, and superimposi- van den Toorn’s massive 1983 tome attempted to account for a tion of elements belonging to different scales. In Part I, I show vast swath of the composer’s work in terms of the octatonic and that the rst of these techniques links Stravinsky directly to the diatonic scales; while Taruskin’s even more massive two-volume language of French Impressionism: the young Stravinsky, like 1996 opus echoed van den Toorn’s conclusions amid an astonish- Debussy and Ravel, made frequent use of a variety of collections, ing wealth of musicological detail. These efforts aim at nothing including whole-tone, octatonic, and the melodic and harmonic less than a total reevaluation of our image of Stravinsky: the com- minor scales. -
The Consecutive-Semitone Constraint on Scalar Structure: a Link Between Impressionism and Jazz1
The Consecutive-Semitone Constraint on Scalar Structure: A Link Between Impressionism and Jazz1 Dmitri Tymoczko The diatonic scale, considered as a subset of the twelve chromatic pitch classes, possesses some remarkable mathematical properties. It is, for example, a "deep scale," containing each of the six diatonic intervals a unique number of times; it represents a "maximally even" division of the octave into seven nearly-equal parts; it is capable of participating in a "maximally smooth" cycle of transpositions that differ only by the shift of a single pitch by a single semitone; and it has "Myhill's property," in the sense that every distinct two-note diatonic interval (e.g., a third) comes in exactly two distinct chromatic varieties (e.g., major and minor). Many theorists have used these properties to describe and even explain the role of the diatonic scale in traditional tonal music.2 Tonal music, however, is not exclusively diatonic, and the two nondiatonic minor scales possess none of the properties mentioned above. Thus, to the extent that we emphasize the mathematical uniqueness of the diatonic scale, we must downplay the musical significance of the other scales, for example by treating the melodic and harmonic minor scales merely as modifications of the natural minor. The difficulty is compounded when we consider the music of the late-nineteenth and twentieth centuries, in which composers expanded their musical vocabularies to include new scales (for instance, the whole-tone and the octatonic) which again shared few of the diatonic scale's interesting characteristics. This suggests that many of the features *I would like to thank David Lewin, John Thow, and Robert Wason for their assistance in preparing this article. -
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. -
Tonal Frequencies, Consonance, Dissonance: a Math-Bio Intersection Steve Mathew1,2* 1
Tonal Frequencies, Consonance, Dissonance: A Math-Bio Intersection Steve Mathew1,2* 1. Indian Institute of Technology Madras, Chennai, IN 2. Vellore Institute of Technology, Vellore, IN *Corresponding Author Email: [email protected] Abstract To date, calculating the frequencies of musical notes requires one to know the frequency of some reference note. In this study, first-order ordinary differential equations are used to arrive at a mathematical model to determine tonal frequencies using their respective note indices. In the next part of the study, an analysis that is based on the fundamental musical frequencies is conducted to theoretically and neurobiologically explain the consonance and dissonance caused by the different musical notes in the chromatic scale which is based on the fact that systematic patterns of sound invoke pleasure. The reason behind the richness of harmony and the sonic interference and degree of consonance in musical chords are discussed. Since a human mind analyses everything relatively, anything other than the most consonant notes sounds dissonant. In conclusion, the study explains clearly why musical notes and in toto, music sounds the way it does. Keywords: Cognitive Musicology, Consonance, Mathematical Modelling, Music Analysis, Tension. 1. Introduction The frequencies of musical notes follow an exponential increase on the keyboard. It is the basis upon which we choose to use the first-order ordinary differential equations to draw the model. In the next part of the study, we analyze the frequencies using sine waves by plotting them against each other and look for patterns that fundamentally unravel the reason why a few notes of a chromatic scale harmonize with the tonic and some disharmonize. -
Alternative Neo-Riemannian Approaches to Carl Nielsen
ALTERNATIVE NEO-RIEMANNIAN APPROACHES TO CARL NIELSEN By Svend Hvidtfelt Nielsen Riemannian and other Theories Nielsen’s music has shown itself capable of delivering interesting results to a variety of analytical approaches ranging from Schenkerian analysis (Fanning, 1997, Reynolds 2010) to investigations into ‘directed’ (Simpson, 1952) or ‘paired’ tonality (Krebs, 1994, Devoto, 1994). Only to a very small extent has Riemannian analysis been applied (Fjeldsøe, 1999, 153-57), which might seem strange in the light of Nielsen’s prevailing use of tonal harmony1. The reason may be that Riemannian theory has been given only little – if any – attention outside Germany and Scandinavia until recently. To be sure, the neo-Rie- mannian transformation theory (Lewin, 1982, Hyer, 1995 and Cohn, 1996, 1999) has revived some of Riemann’s concepts and it has become an essential part of contem- porary analytical theory. But when it comes to the central aspect of the relation of chords to an operating tonic, the transformation theory is just as remote from tra- ditional Riemannian theory as is Schenkerian analysis (Schenker 1979, Salzer 1952, Forte, 1982) in its evaluation of harmonic foreground progressions. The neo-Riemannian theories of Lewin, Hyer and Cohn are however not the only attempts to rethink the Riemann legacy. Two Danish scholars and composers, Jörgen Jersild (1970, 1985, 1989) and Jan Maegaard (1981, 1989) have offered new methods of performing harmonic analysis more or less directly based on Riemann’s theories. They thus both offer an alternative, Danish, neo-Riemannian approach to harmony, both dealing mainly with the chromaticism and major/minor fl ux of the music of the romantic era, that is, the music that constitutes the harmonic environ- ment of Nielsen’s upbringing. -
Harmonic Major – Part I, Blackbird by Adam Coulombe
Harmonic Major – Part I, Blackbird By Adam Coulombe The tonality of harmonic there is an absence of seven triad on the flat sixth degree. G major is a fundamental tonality in primary chords and their scales. harmonic major is spelled : music harmony. There are those Usually, only the G A B C D E♭ F♯ G that have never heard of it, and tonalities of major, melodic minor, My belief is that those that have heard of it but do and harmonic minor are taught and eventually everyone will have the not understand that it is a used when analysing the harmony a opportunity to understand tertian completely tertian and diatonic piece of music moves through. This harmony correctly, and completely, tonality. Tertian harmony is the can create an incorrect analysis of that there are four tonalities, term given to chords spelled with a the harmony in a piece when major, harmonic major, melodic major or minor third interval chords from harmonic major are minor, and harmonic minor. I between each chord tone in root used but not recognized as such. believe this should be a standard position, to start with. Diatonic For example, the sheet music to model for all musicians to study as refers to one note being the center Paul McCartney’s Blackbird has a basis to understand all other of the tonality, called the tonic been consistently printed forms of applied harmony. note, and that the harmony created incorrectly due to this lack of I am interested to know in this tonality relates ‘to the tonic.’ knowledge of the tonality of what other institutions and Why this tonality is largely harmonic major. -
Applications of Béla Bartók's Techniques of Pitch Organisation to Jazz Improvisation
Edith Cowan University Research Online Theses: Doctorates and Masters Theses 2021 Applications of Béla Bartók's techniques of pitch organisation to jazz improvisation Niran Jay Dasika Edith Cowan University Follow this and additional works at: https://ro.ecu.edu.au/theses Recommended Citation Dasika, N. J. (2021). Applications of Béla Bartók's techniques of pitch organisation to jazz improvisation. https://ro.ecu.edu.au/theses/2408 This Thesis is posted at Research Online. https://ro.ecu.edu.au/theses/2408 Edith Cowan University Copyright Warning You may print or download ONE copy of this document for the purpose of your own research or study. The University does not authorize you to copy, communicate or otherwise make available electronically to any other person any copyright material contained on this site. You are reminded of the following: Copyright owners are entitled to take legal action against persons who infringe their copyright. A reproduction of material that is protected by copyright may be a copyright infringement. Where the reproduction of such material is done without attribution of authorship, with false attribution of authorship or the authorship is treated in a derogatory manner, this may be a breach of the author’s moral rights contained in Part IX of the Copyright Act 1968 (Cth). Courts have the power to impose a wide range of civil and criminal sanctions for infringement of copyright, infringement of moral rights and other offences under the Copyright Act 1968 (Cth). Higher penalties may apply, and higher damages may be awarded, for offences and infringements involving the conversion of material into digital or electronic form. -
Spatial and Psychoacoustic Factors in Atonal Prolongation
Spatial and Psychoacoustic Factors in Atonal Prolongation By Fred Lerdahl Consider the sequences of letters in example 1 and think of them as strings of objects perceived and parsed in space or time. In case (a), each object is distinct. There may be degrees of similarity among them, meas ured by whatever means, but no member of the string is a point of refer ence for the others. In case (b), Xl repeats literally as X2. One might say that X extends in space or is prolonged in time. In cases (c) and (d), the repetition of X creates a frame or context for Y. The two Xs connect per ceptually and Y is perceived inside that connection. In other words, Y is subordinate within the context XC X2. If moving from one object to the next is experienced as a path, the motion XI~Y represents a departure and Y~X2 represents a return. If for some reason, say relative temporal proximity, Y groups with Xl' as in case (c), Y belongs to Xl in the context XC X2; similarly, in case (d) Y belongs to X2 in the context XC X2. In these instances one can speak of a constituent hierarchy (that is, the subordi nate element is not merely subordinate within its context but is subordi nate to a single superordinate element). Sometimes it is not proximity but patterns of repetition· that cause the internal grouping, as in case (e). But let us leave to one side the issue of subsegments within a string and look at a few other whole patterns. -
AN EXPLORATION of BARTOK's FUGAL STYLE THESIS Presented
li AN EXPLORATION OF BARTOK'S FUGAL STYLE THESIS Presented to the Graduate Council of the North Texas State University in Partial Fulfillment of the Requirments For the Degree of Master of Music By John L. Willett, B.M. Ed. Denton, Texas December, 1984 TABLE OF CONTENTS Page LIST OF EXAMPLES.0.a. ................. *.*........iv Chapter I. INTRODUCTION. .............. .*.&..*.........1 II. THE STRUCTURE OF BARTOK'S MUSIC ..a................7 Basic Tonal Principles...........................10 Formal Principles............................23 Harmony.-..-..-..-..-......................28 III. ANALYSIS OF THE FUGUES........... ............... 46 The Subject.............. ....................47 The Answer.......-..-........................64 The Countersubject...............*............66 Overall Organization of the Fugues...... .... ....... 69 IV. CONCLUSION.. ............................. 94 BIBLIOGRAPHY..... ...--- -- . .0.0. ...... 98 iii LIST OF EXAMPLES Example Page 1. Pentatonic scale. ... ............... 10 2. Circle of fifths.. ... .... .......... 11 3. The tonic, dominant, and subdominant axes . 12 4. Related keys arranged in the axis system (M = major mode, m = minor mode)..........................13 5. C major-minor chord ......... ................... 14 6. The major third and minor seventh as dominants . 15 7. Resolution of the tritone in opposite directions . 15 8. B61a Bartok, Opus 7, Introduzione to the 3rd movement, measures 7-8 and 14-15 of the violoncello part . 19 9. B6la Bart6k, Opus 102, 1st movement, measures 4-6 of the violoncello part (Karpsti's hypothetical form followed by Bart6k's "mistuned" form). ..... 20 10. Linear representation of the GS proportion.... .... 26 11. GS proportions used symmetrically by a) union, and b) intersection.. .. .a..a........27........ 12. Example of GS proportions superposed..... .... 27 13. Lendvai's formal analysis of the Sonata for Two Pianos and Percussion, measures 2-17, with tonal axes..-........-....................28 14. -
Expanded Tonality in Three Early Piano Works of Bela Bartok (1881 -1945)
EXPANDED TONALITY IN THREE EARLY PIANO WORKS OF BELA BARTOK (1881 -1945) by JEFFREY JAMES BRUKMAN submitted in fulfilment of the requirements for the degree of MASTER OF MUSICOLOGY in the DEPARTMENT OF MUSICOLOGY at the UNIVERSITY OF SOUTH AFRICA SUPERVISOR: PROF BS VAN DER LINDE 30 NOVEMBER 1998 (i) TITLE: Expanded tonality in three early piano works of Bela Bart6k (1881 - 1945) ABSTRACT: Bart6k's own expanded tonal ("supradiatonic") pronouncements reveal that his music, notwithstanding tonally camouflaging surface details, clearly had a tonal foundation which in many respects is a reaction to the emerging atonalism of Schonberg. Analysis of three piano works (1908 - 1916) reveal that Bart6k's tonal language embraced intuitively the expanded tonal idiom. The harmonic resources Bart6k employed to obscure tonicisation embrace double-degree constructions, quartal formations, chords of addition and omission and other irregular constructions. Diatonic tonal pillars are evident in pedal points, tonic triads and dominant to tonic root movement. Through an application of the Riemann function theory expanded by Hartmann's supposition of fully-chromaticised scales tonal syntax (especially second phase Strauss cadences or closes) becomes apparent within an expanded tonal product. The analyses conclude that Bart6k's inimitable "sound-world" is a twentieth-century manifestation of traditional tonality's primary tenets. KEY TERMS: Expanded tonality; Fully-chromaticised scales; Bitonal process; Bitonal product; Double-degree chords; Strauss cadences; Pedal points; Key-introductory six-four chord; Chord of addition; Chord of omission; Quartal constructions; Chord streaming. (ii) DEDICATED to the Memory of my late brother Robert Douglas Brukman (1947 - 1995) who was always an inspiring source of encouragement as I furthered my studies.