Further Elements of the Harmonic Vocabulary
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A Group-Theoretical Classification of Three-Tone and Four-Tone Harmonic Chords3
A GROUP-THEORETICAL CLASSIFICATION OF THREE-TONE AND FOUR-TONE HARMONIC CHORDS JASON K.C. POLAK Abstract. We classify three-tone and four-tone chords based on subgroups of the symmetric group acting on chords contained within a twelve-tone scale. The actions are inversion, major- minor duality, and augmented-diminished duality. These actions correspond to elements of symmetric groups, and also correspond directly to intuitive concepts in the harmony theory of music. We produce a graph of how these actions relate different seventh chords that suggests a concept of distance in the theory of harmony. Contents 1. Introduction 1 Acknowledgements 2 2. Three-tone harmonic chords 2 3. Four-tone harmonic chords 4 4. The chord graph 6 References 8 References 8 1. Introduction Early on in music theory we learn of the harmonic triads: major, minor, augmented, and diminished. Later on we find out about four-note chords such as seventh chords. We wish to describe a classification of these types of chords using the action of the finite symmetric groups. We represent notes by a number in the set Z/12 = {0, 1, 2,..., 10, 11}. Under this scheme, for example, 0 represents C, 1 represents C♯, 2 represents D, and so on. We consider only pitch classes modulo the octave. arXiv:2007.03134v1 [math.GR] 6 Jul 2020 We describe the sounding of simultaneous notes by an ordered increasing list of integers in Z/12 surrounded by parentheses. For example, a major second interval M2 would be repre- sented by (0, 2), and a major chord would be represented by (0, 4, 7). -
Many of Us Are Familiar with Popular Major Chord Progressions Like I–IV–V–I
Many of us are familiar with popular major chord progressions like I–IV–V–I. Now it’s time to delve into the exciting world of minor chords. Minor scales give flavor and emotion to a song, adding a level of musical depth that can make a mediocre song moving and distinct from others. Because so many of our favorite songs are in major keys, those that are in minor keys1 can stand out, and some musical styles like rock or jazz thrive on complex minor scales and harmonic wizardry. Minor chord progressions generally contain richer harmonic possibilities than the typical major progressions. Minor key songs frequently modulate to major and back to minor. Sometimes the same chord can appear as major and minor in the very same song! But this heady harmonic mix is nothing to be afraid of. By the end of this article, you’ll not only understand how minor chords are made, but you’ll know some common minor chord progressions, how to write them, and how to use them in your own music. With enough listening practice, you’ll be able to recognize minor chord progressions in songs almost instantly! Table of Contents: 1. A Tale of Two Tonalities 2. Major or Minor? 3. Chords in Minor Scales 4. The Top 3 Chords in Minor Progressions 5. Exercises in Minor 6. Writing Your Own Minor Chord Progressions 7. Your Minor Journey 1 https://www.musical-u.com/learn/the-ultimate-guide-to-minor-keys A Tale of Two Tonalities Western music is dominated by two tonalities: major and minor. -
Music in Theory and Practice
CHAPTER 4 Chords Harmony Primary Triads Roman Numerals TOPICS Chord Triad Position Simple Position Triad Root Position Third Inversion Tertian First Inversion Realization Root Second Inversion Macro Analysis Major Triad Seventh Chords Circle Progression Minor Triad Organum Leading-Tone Progression Diminished Triad Figured Bass Lead Sheet or Fake Sheet Augmented Triad IMPORTANT In the previous chapter, pairs of pitches were assigned specifi c names for identifi cation CONCEPTS purposes. The phenomenon of tones sounding simultaneously frequently includes group- ings of three, four, or more pitches. As with intervals, identifi cation names are assigned to larger tone groupings with specifi c symbols. Harmony is the musical result of tones sounding together. Whereas melody implies the Harmony linear or horizontal aspect of music, harmony refers to the vertical dimension of music. A chord is a harmonic unit with at least three different tones sounding simultaneously. Chord The term includes all possible such sonorities. Figure 4.1 #w w w w w bw & w w w bww w ww w w w w w w w‹ Strictly speaking, a triad is any three-tone chord. However, since western European music Triad of the seventeenth through the nineteenth centuries is tertian (chords containing a super- position of harmonic thirds), the term has come to be limited to a three-note chord built in superposed thirds. The term root refers to the note on which a triad is built. “C major triad” refers to a major Triad Root triad whose root is C. The root is the pitch from which a triad is generated. 73 3711_ben01877_Ch04pp73-94.indd 73 4/10/08 3:58:19 PM Four types of triads are in common use. -
Harmonic Resources in 1980S Hard Rock and Heavy Metal Music
HARMONIC RESOURCES IN 1980S HARD ROCK AND HEAVY METAL MUSIC A thesis submitted to the College of the Arts of Kent State University in partial fulfillment of the requirements for the degree of Master of Arts in Music Theory by Erin M. Vaughn December, 2015 Thesis written by Erin M. Vaughn B.M., The University of Akron, 2003 M.A., Kent State University, 2015 Approved by ____________________________________________ Richard O. Devore, Thesis Advisor ____________________________________________ Ralph Lorenz, Director, School of Music _____________________________________________ John R. Crawford-Spinelli, Dean, College of the Arts ii Table of Contents LIST OF FIGURES ............................................................................................................................... v CHAPTER I........................................................................................................................................ 1 INTRODUCTION ........................................................................................................................... 1 GOALS AND METHODS ................................................................................................................ 3 REVIEW OF RELATED LITERATURE............................................................................................... 5 CHAPTER II..................................................................................................................................... 36 ANALYSIS OF “MASTER OF PUPPETS” ...................................................................................... -
Music Theory Through the Lens of Film
Journal of Film Music 5.1-2 (2012) 177-196 ISSN (print) 1087-7142 doi:10.1558/jfm.v5i1-2.177 ISSN (online) 1758-860X ARTICLE Music Theory through the Lens of Film FranK LEhman Tufts University [email protected] Abstract: The encounter of a musical repertoire with a theoretical system benefits the latter even as it serves the former. A robustly applied theoretic apparatus hones our appreciation of a given corpus, especially one such as film music, for which comparatively little analytical attention has been devoted. Just as true, if less frequently offered as a motivator for analysis, is the way in which the chosen music theoretical system stands to see its underlying assumptions clarified and its practical resources enhanced by such contact. The innate programmaticism and aesthetic immediacy of film music makes it especially suited to enrich a number of theoretical practices. A habit particularly ripe for this exposure is tonal hermeneutics: the process of interpreting music through its harmonic relationships. Interpreting cinema through harmony not only sharpens our understanding of various film music idioms, but considerably refines the critical machinery behind its analysis. The theoretical approach focused on here is transformation theory, a system devised for analysis of art music (particularly from the nineteenth century) but nevertheless eminently suited for film music. By attending to the perceptually salient changes rather than static objects of musical discourse, transformation theory avoids some of the bugbears of conventional tonal hermeneutics for film (such as the tyranny of the “15 second rule”) while remaining exceptionally well calibrated towards musical structure and detail. -
Lecture Notes
Notes on Chromatic Harmony © 2010 John Paul Ito Chromatic Modulation Enharmonic modulation Pivot-chord modulation in which the pivot chord is spelled differently in the two keys. Ger +6 (enharmonically equivalent to a dominant seventh) and diminished sevenths are the most popular options. See A/S 534-536 Common-tone modulation Usually a type of phrase modulation. The two keys are distantly related, but the tonic triads share a common tone. (E.g. direct modulation from C major to A-flat major.) In some cases, the common tone may be spelled using an enharmonic equivalent. See A/S 596-599. Common-tone modulations often involve… Chromatic mediant relationships Chromatic mediants were extremely popular with nineteenth-century composers, being used with increasing frequency and freedom as the century progressed. A chromatic mediant relationship is one in which the triadic roots (or tonal centers) are a third apart, and in which at least one of the potential common tones between the triads (or tonic triads of the keys) is not a common tone because of chromatic alterations. In the example above of C major going to A-flat major, the diatonic third relationship with C major would be A minor, which has two common tones (C and E) with C major. Because of the substitution of A-flat major (bVI relative to C) for A minor, there is only one common tone, C, as the C-major triad includes E and the A-flat-major triad contains E-flat. The contrast is even more jarring if A-flat minor is substituted; now there are no common tones. -
When the Leading Tone Doesn't Lead: Musical Qualia in Context
When the Leading Tone Doesn't Lead: Musical Qualia in Context Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Claire Arthur, B.Mus., M.A. Graduate Program in Music The Ohio State University 2016 Dissertation Committee: David Huron, Advisor David Clampitt Anna Gawboy c Copyright by Claire Arthur 2016 Abstract An empirical investigation is made of musical qualia in context. Specifically, scale-degree qualia are evaluated in relation to a local harmonic context, and rhythm qualia are evaluated in relation to a metrical context. After reviewing some of the philosophical background on qualia, and briefly reviewing some theories of musical qualia, three studies are presented. The first builds on Huron's (2006) theory of statistical or implicit learning and melodic probability as significant contributors to musical qualia. Prior statistical models of melodic expectation have focused on the distribution of pitches in melodies, or on their first-order likelihoods as predictors of melodic continuation. Since most Western music is non-monophonic, this first study investigates whether melodic probabilities are altered when the underlying harmonic accompaniment is taken into consideration. This project was carried out by building and analyzing a corpus of classical music containing harmonic analyses. Analysis of the data found that harmony was a significant predictor of scale-degree continuation. In addition, two experiments were carried out to test the perceptual effects of context on musical qualia. In the first experiment participants rated the perceived qualia of individual scale-degrees following various common four-chord progressions that each ended with a different harmony. -
Musical Chord Preference: Cultural Or Universal?
Working Paper Series MUSICAL CHORD PREFERENCE: CULTURAL OR UNIVERSAL? DATA FROM A NATIVE AMAZONIAN SOCIETY EDUARDO A. UNDURRAGA1,*, NICHOLAS Q. EMLEM2, MAXIMILIEN GUEZE3, DAN T. EISENBERG4, TOMAS HUANCA5, VICTORIA REYES-GARCÍA1,3, 6 VICTORIA RAZUMOVA7, KAREN GODOY8, TAPS BOLIVIA STUDY TEAM9, AND RICARDO GODOY3 1 Heller School, Brandeis University, Waltham, MA 02154, USA 2 Department of Anthropology, University of Michigan, Ann Arbor, MI 48109 3 Institut de Ciència i Tecnologia Ambientals, Universitat Autònoma de Barcelona, 08193 Bellatera, Barcelona, Spain 4 Department of Anthropology, Northwestern University, Evanston, Illinois 60208, USA 5 Centro Boliviano de Investigación y Desarrollo Socio-integral (CBIDSI), Correo Central, San Borja, Beni, Bolivia 6 ICREA, Universitat Autònoma de Barcelona, 08193 Bellatera, Barcelona, Spain 7 147 Sycamore St, Sommerville, MA 02145, USA 8 Pediatric Medical Care Inc., 1000 Broadway, Chelsea, MA 02150. USA 9 Tsimane’ Amazonian Panel Study, Correo Central, San Borja, Beni, Bolivia *To whom correspondence should be addressed. E-mail: [email protected] Word count: 9,372 (excluding abstract and cover page) 2 Abstract (338 words) Purpose. Recent evidence suggests that when listening to Western music, subjects cross- culturally experience similar emotions. However, we do not know whether cross-cultural regularities in affect response to music also emerge when listening to the building blocks of Western music, such as major and minor chords. In Western music, major chords are associated with happiness, a basic pan-human emotion, but the relation has not been tested in a remote non- Western setting. Here we address this question by measuring the relation between (i) listening to major and minor chords in major and minor keys and (ii) self-reported happiness in a remote society of native Amazonian hunters, gatherers, and farmers in Bolivia (Tsimane’). -
Musicianship IV Syllabus
University of Missouri-Kansas City Conservatory of Music and Dance CONS 242: Musicianship IV Spring 2015 Credit hours: 4.0 CRN: 17576 Instructor: Dr. David Thurmaier, Associate Professor of Music Theory Office: 302 Grant Hall Phone: 235-2898 Email: [email protected] Office Hours: M, T, W from 10-10:50 and by appointment Teaching Assistant: Taylor Carmona Office: 304 Grant Hall Email: [email protected] Catalog Description Continuation of CONS 241. Study of late-nineteenth century chromaticism and analytical and compositional methods of twentieth and twenty-first century music, including set theory and twelve-tone theory. Particular attention is given to the development of critical writing skills and the creation of stylistic compositions. Prerequisite: CONS 241 Meeting Time and Location Monday-Friday, 9-9:50 am, Grant Hall 122 Required Materials Kostka, Stefan and Roger Graybill. Anthology of Music for Analysis. Upper Saddle River, NJ: Pearson Prentice Hall, 2004. Laitz, Steven G., The Complete Musician: An Integrated Approach to Tonal Theory, Analysis, and Listening. 3rd Edition. New York: Oxford University Press, 2012. Roig-Francolí, Miguel. Understanding Post-Tonal Music (text and anthology). Boston: McGraw Hill, 2007. Notebook, music paper and pens/pencils In addition, you will be required to use the Finale notation program (or equivalent) for composition assignments. This is available for personal purchase at a substantial student discount http://www.finalemusic.com. I recommend against using such free programs as Notepad, as you are not able to take advantage of the many features of Finale. Continual failure to purchase and/or bring required books will result in deductions on homework or exams. -
Secondary Leading Tone Chords the Most Typical Way to Tonicize a Major
Secondary Leading Tone Chords The most typical way to tonicize a major or minor chord (NOT a diminished chord!) is to approach/precede it with a secondary dominant. Approaching a chord (such as a ii, iii, IV, V or vi) with its own relative dominant emulates the recognizable and strong push that a regular V has as it moves to I. So when a V/V goes to V (as it should), the V chord is arrived at in an emphasized way, and the V/V chord stands out because it has a chromatic note in it–that musical moment stands out in the texture. But similar to V (or V7) moving to I (or a V/V moving to V) is when viio moves to I. Recall that viio can act as a substitute for V(7); viio is also, in fact, a type of dominant-functioning chord. Because viio also functions as a dominant, it can function as a secondary dominant, and tonicize a chord a half-step above it–in the same way that viio resolves to I up a half- step. In that capacity, we call it a secondary leading tone chord. In as much as there can be a V/V, or a V7/V, there can be a viio/V (or viio/anything). (There can even be a viio7/V, or a viiø7/V, but we’ll just focus on the triad version...). Just as a V/V is built off a root a 4th below the V chord, a viio/V is built off a root a half-step below the V chord. -
The Perceptual Attraction of Pre-Dominant Chords 1
Running Head: THE PERCEPTUAL ATTRACTION OF PRE-DOMINANT CHORDS 1 The Perceptual Attraction of Pre-Dominant Chords Jenine Brown1, Daphne Tan2, David John Baker3 1Peabody Institute of The Johns Hopkins University 2University of Toronto 3Goldsmiths, University of London [ACCEPTED AT MUSIC PERCEPTION IN APRIL 2021] Author Note Jenine Brown, Department of Music Theory, Peabody Institute of the Johns Hopkins University, Baltimore, MD, USA; Daphne Tan, Faculty of Music, University of Toronto, Toronto, ON, Canada; David John Baker, Department of Computing, Goldsmiths, University of London, London, United Kingdom. Corresponding Author: Jenine Brown, Peabody Institute of The John Hopkins University, 1 E. Mt. Vernon Pl., Baltimore, MD, 21202, [email protected] 1 THE PERCEPTUAL ATTRACTION OF PRE-DOMINANT CHORDS 2 Abstract Among the three primary tonal functions described in modern theory textbooks, the pre-dominant has the highest number of representative chords. We posit that one unifying feature of the pre-dominant function is its attraction to V, and the experiment reported here investigates factors that may contribute to this perception. Participants were junior/senior music majors, freshman music majors, and people from the general population recruited on Prolific.co. In each trial four Shepard-tone sounds in the key of C were presented: 1) the tonic note, 2) one of 31 different chords, 3) the dominant triad, and 4) the tonic note. Participants rated the strength of attraction between the second and third chords. Across all individuals, diatonic and chromatic pre-dominant chords were rated significantly higher than non-pre-dominant chords and bridge chords. Further, music theory training moderated this relationship, with individuals with more theory training rating pre-dominant chords as being more attracted to the dominant. -
Evaluating Prolongation in Extended Tonality
Evaluating Prolongation in Extended Tonality <http://mod7.shorturl.com/prolongation.htm> Robert T. Kelley Florida State University In this paper I shall offer strategies for deciding what is structural in extended-tonal music and provide new theoretical qualifications that allow for a conservative evaluation of prolongational analyses. Straus (1987) provides several criteria for finding post-tonal prolongation, but these can simply be reduced down to one important consideration: Non-tertian music clouds the distinction between harmonic and melodic intervals. Because linear analysis depends upon this distinction, any expansion of the prolongational approach for non-tertian music must find alternative means for defining the ways in which transient tones elaborate upon structural chord tones to foster a sense of prolongation. The theoretical work that Straus criticizes (Salzer 1952, Travis 1959, 1966, 1970, Morgan 1976, et al.) fails to provide a method for discriminating structural tones from transient tones. More recently, Santa (1999) has provided associational models for hierarchical analysis of some post-tonal music. Further, Santa has devised systems for determining salience as a basis for the assertion of structural chords and melodic pitches in a hierarchical analysis. While a true prolongational perspective cannot be extended to address most post-tonal music, it may be possible to salvage a prolongational approach in a restricted body of post-tonal music that retains some features of tonality, such as harmonic function, parsimonious voice leading, or an underlying diatonic collection. Taking into consideration Straus’s theoretical proviso, we can build a model for prolongational analysis of non-tertian music by establishing how non-tertian chords may attain the status of structural harmonies.