Generalized Tonnetze and Zeitnetz, and the Topology of Music Concepts
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Chapter 1 Geometric Generalizations of the Tonnetz and Their Relation To
November 22, 2017 12:45 ws-rv9x6 Book Title yustTonnetzSub page 1 Chapter 1 Geometric Generalizations of the Tonnetz and their Relation to Fourier Phases Spaces Jason Yust Boston University School of Music, 855 Comm. Ave., Boston, MA, 02215 [email protected] Some recent work on generalized Tonnetze has examined the topologies resulting from Richard Cohn's common-tone based formulation, while Tymoczko has reformulated the Tonnetz as a network of voice-leading re- lationships and investigated the resulting geometries. This paper adopts the original common-tone based formulation and takes a geometrical ap- proach, showing that Tonnetze can always be realized in toroidal spaces, and that the resulting spaces always correspond to one of the possible Fourier phase spaces. We can therefore use the DFT to optimize the given Tonnetz to the space (or vice-versa). I interpret two-dimensional Tonnetze as triangulations of the 2-torus into regions associated with the representatives of a single trichord type. The natural generalization to three dimensions is therefore a triangulation of the 3-torus. This means that a three-dimensional Tonnetze is, in the general case, a network of three tetrachord-types related by shared trichordal subsets. Other Ton- netze that have been proposed with bounded or otherwise non-toroidal topologies, including Tymoczko's voice-leading Tonnetze, can be under- stood as the embedding of the toroidal Tonnetze in other spaces, or as foldings of toroidal Tonnetze with duplicated interval types. 1. Formulations of the Tonnetz The Tonnetz originated in nineteenth-century German harmonic theory as a network of pitch classes connected by consonant intervals. -
Chord Progressions
Chord Progressions Part I 2 Chord Progressions The Best Free Chord Progression Lessons on the Web "The recipe for music is part melody, lyric, rhythm, and harmony (chord progressions). The term chord progression refers to a succession of tones or chords played in a particular order for a specified duration that harmonizes with the melody. Except for styles such as rap and free jazz, chord progressions are an essential building block of contemporary western music establishing the basic framework of a song. If you take a look at a large number of popular songs, you will find that certain combinations of chords are used repeatedly because the individual chords just simply sound good together. I call these popular chord progressions the money chords. These money chord patterns vary in length from one- or two-chord progressions to sequences lasting for the whole song such as the twelve-bar blues and thirty-two bar rhythm changes." (Excerpt from Chord Progressions For Songwriters © 2003 by Richard J. Scott) Every guitarist should have a working knowledge of how these chord progressions are created and used in popular music. Click below for the best in free chord progressions lessons available on the web. Ascending Augmented (I-I+-I6-I7) - - 4 Ascending Bass Lines - - 5 Basic Progressions (I-IV) - - 10 Basie Blues Changes - - 8 Blues Progressions (I-IV-I-V-I) - - 15 Blues With A Bridge - - 36 Bridge Progressions - - 37 Cadences - - 50 Canons - - 44 Circle Progressions -- 53 Classic Rock Progressions (I-bVII-IV) -- 74 Coltrane Changes -- 67 Combination -
Surviving Set Theory: a Pedagogical Game and Cooperative Learning Approach to Undergraduate Post-Tonal Music Theory
Surviving Set Theory: A Pedagogical Game and Cooperative Learning Approach to Undergraduate Post-Tonal Music Theory DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Angela N. Ripley, M.M. Graduate Program in Music The Ohio State University 2015 Dissertation Committee: David Clampitt, Advisor Anna Gawboy Johanna Devaney Copyright by Angela N. Ripley 2015 Abstract Undergraduate music students often experience a high learning curve when they first encounter pitch-class set theory, an analytical system very different from those they have studied previously. Students sometimes find the abstractions of integer notation and the mathematical orientation of set theory foreign or even frightening (Kleppinger 2010), and the dissonance of the atonal repertoire studied often engenders their resistance (Root 2010). Pedagogical games can help mitigate student resistance and trepidation. Table games like Bingo (Gillespie 2000) and Poker (Gingerich 1991) have been adapted to suit college-level classes in music theory. Familiar television shows provide another source of pedagogical games; for example, Berry (2008; 2015) adapts the show Survivor to frame a unit on theory fundamentals. However, none of these pedagogical games engage pitch- class set theory during a multi-week unit of study. In my dissertation, I adapt the show Survivor to frame a four-week unit on pitch- class set theory (introducing topics ranging from pitch-class sets to twelve-tone rows) during a sophomore-level theory course. As on the show, students of different achievement levels work together in small groups, or “tribes,” to complete worksheets called “challenges”; however, in an important modification to the structure of the show, no students are voted out of their tribes. -
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. -
Chromatic Sequences
CHAPTER 30 Melodic and Harmonic Symmetry Combine: Chromatic Sequences Distinctions Between Diatonic and Chromatic Sequences Sequences are paradoxical musical processes. At the surface level they provide rapid harmonic rhythm, yet at a deeper level they function to suspend tonal motion and prolong an underlying harmony. Tonal sequences move up and down the diatonic scale using scale degrees as stepping-stones. In this chapter, we will explore the consequences of transferring the sequential motions you learned in Chapter 17 from the asymmetry of the diatonic scale to the symmetrical tonal patterns of the nineteenth century. You will likely notice many similarities of behavior between these new chromatic sequences and the old diatonic ones, but there are differences as well. For instance, the stepping-stones for chromatic sequences are no longer the major and minor scales. Furthermore, the chord qualities of each individual harmony inside a chromatic sequence tend to exhibit more homogeneity. Whereas in the past you may have expected major, minor, and diminished chords to alternate inside a diatonic sequence, in the chromatic realm it is not uncommon for all the chords in a sequence to manifest the same quality. EXAMPLE 30.1 Comparison of Diatonic and Chromatic Forms of the D3 ("Pachelbel") Sequence A. 624 CHAPTER 30 MELODIC AND HARMONIC SYMMETRY COMBINE 625 B. Consider Example 30.1A, which contains the D3 ( -4/ +2)-or "descending 5-6"-sequence. The sequence is strongly goal directed (progressing to ii) and diatonic (its harmonies are diatonic to G major). Chord qualities and distances are not consistent, since they conform to the asymmetry of G major. -
Fourier Phase and Pitch-Class Sum
Fourier Phase and Pitch-Class Sum Dmitri Tymoczko1 and Jason Yust2(B) Author Proof 1 Princeton University, Princeton, NJ 08544, USA [email protected] 2 Boston University, Boston, MA 02215, USA [email protected] AQ1 Abstract. Music theorists have proposed two very different geometric models of musical objects, one based on voice leading and the other based on the Fourier transform. On the surface these models are completely different, but they converge in special cases, including many geometries that are of particular analytical interest. Keywords: Voice leading Fourier transform Tonal harmony Musical scales Chord geometry· · · · 1Introduction Early twenty-first century music theory explored a two-pronged generalization of traditional set theory. One prong situated sets and set-classes in continuous, non-Euclidean spaces whose paths represented voice leadings, or ways of mov- ing notes from one chord to another [4,13,16]. This endowed set theory with a contrapuntal aspect it had previously lacked, embedding its discrete entities in arobustlygeometricalcontext.AnotherpronginvolvedtheFouriertransform as applied to pitch-class distributions: this provided alternative coordinates for describing chords and set classes, coordinates that made manifest their harmonic content [1,3,8,10,19–21]. Harmonies could now be described in terms of their resemblance to various equal divisions of the octave, paradigmatic objects such as the augmented triad or diminished seventh chord. These coordinates also had a geometrical aspect, similar to yet distinct from voice-leading geometry. In this paper, we describe a new convergence between these two approaches. Specifically, we show that there exists a class of simple circular voice-leading spaces corresponding, in the case of n-note nearly even chords, to the nth Fourier “phase spaces.” An isomorphism of points exists for all chords regardless of struc- ture; when chords divide the octave evenly, we can extend the isomorphism to paths, which can then be interpreted as voice leadings. -
EMBO Conference on Fission Yeast: Pombe 2013 7Th International Fission Yeast Meeting London, United Kingdom, 24 - 29 June 2013
Abstracts of papers presented at the EMBO Conference on Fission Yeast: pombe 2013 7th International Fission Yeast Meeting London, United Kingdom, 24 - 29 June 2013 Meeting Organizers: Jürg Bähler UCL, UK Jacqueline Hayles CRUK-LRI, UK Scientific Programme: Robin Allshire UK Rob Martienssen USA Paco Antequera Spain Hisao Masai Japan Francois Bachand Canada Jonathan Millar UK Jürg Bähler UK Sergio Moreno Spain Pernilla Bjerling Sweden Jo Murray UK Fred Chang USA Toru Nakamura USA Gordon Chua Canada Chris Norbury UK Peter Espenshade USA Kunihiro Ohta Japan Kathy Gould USA Snezhka Oliferenko Singapore Juraj Gregan Austria Janni Petersen UK Edgar Hartsuiker UK Paul Russell USA Jacqueline Hayles UK Geneviève Thon Denmark Elena Hidalgo Spain Iva Tolic-Nørrelykke Germany Charlie Hoffman USA Elizabeth Veal UK Zoi Lygerou Greece Yoshi Watanabe Japan Henry Levin USA Jenny Wu France These abstracts may not be cited in bibliographies. Material contained herein should be treated as personal communication and should be cited as such only with the consent of the authors. Printed by SLS Print, London, UK Page 1 Poster Prize Judges: Coordinated by Sara Mole & Mike Bond UCL, UK Poster Prizes sponsored by UCL, London’s Global University Rosa Aligué Spain Hiroshi Murakami Japan José Ayté Spain Eishi Noguchi USA Hugh Cam USA Martin Převorovský Czech Republic Rafael Daga Spain Luis Rokeach Canada Jacob Dalgaard UK Ken Sawin UK Da-Qiao Ding Japan Melanie Styers USA Tim Humphrey UK Irene Tang USA Norbert Käufer Germany Masaru Ueno Japan Makoto Kawamukai Japan -
MTO 25.1 Examples: Willis, Comprehensibility and Ben Johnston’S String Quartet No
MTO 25.1 Examples: Willis, Comprehensibility and Ben Johnston’s String Quartet No. 9 (Note: audio, video, and other interactive examples are only available online) http://mtosmt.org/issues/mto.19.25.1/mto.19.25.1.willis.html Example 1. Partch’s 5-limit tonality diamond, after Partch (1974, 110) also given with pitch names and cent deviations from twelve-tone equal temperament Example 2. Partch’s 11-limit tonality diamond, after Partch (1974, 159) Example 3. The just-intoned diatonic shown in ratios, cents, and on a Tonnetz Example 4. Johnston’s accidentals, the 5-limit ration they inflect, the target ratio they bring about, their ratio and cent value. As an example of how this table works, take row 11. If we multiply 4:3 by 33:32 it sums to 11:8. This is equivalent to raising a perfect fourth by Johnston’s 11 chroma. Example 5. The overtone and undertone series of C notated using Johnston’s method Example 6. A Tonnetz with the syntonic diatonic highlighted in grey. The solid lines connect the two 5-limit pitches that may be inflected to produce a tonal or tonal seventh against the C. This makes clear why the tonal seventh of C is notated as lowered by a syntonic comma in addition to an inverse 7 sign. It is because the pitch is tuned relative to the D- (10:9), which is a syntonic comma lower than the D that appears in the diatonic gamut. Example 7. Johnston, String Quartet No. 9/I, m. 109. A comma pump progression shown with Roman Numerals and a Tonnetz. -
From Schritte and Wechsel to Coxeter Groups 3
From Schritte and Wechsel to Coxeter Groups Markus Schmidmeier1 Abstract: The PLR-moves of neo-Riemannian theory, when considered as re- flections on the edges of an equilateral triangle, define the Coxeter group S3. The elements are in a natural one-to-one correspondence with the trianglese in the infinite Tonnetz. The left action of S3 on the Tonnetz gives rise to interest- ing chord sequences. We compare the systeme of transformations in S3 with the system of Schritte and Wechsel introduced by Hugo Riemann in 1880e . Finally, we consider the point reflection group as it captures well the transition from Riemann’s infinite Tonnetz to the finite Tonnetz of neo-Riemannian theory. Keywords: Tonnetz, neo-Riemannian theory, Coxeter groups. 1. PLR-moves revisited In neo-Riemannian theory, chord progressions are analyzed in terms of elementary moves in the Tonnetz. For example, the process of going from tonic to dominant (which differs from the tonic chord in two notes) is decomposed as a product of two elementary moves of which each changes only one note. The three elementary moves considered are the PLR-transformations; they map a major or minor triad to the minor or major triad adjacent to one of the edges of the triangle representing the chord in the Tonnetz. See [4] and [5]. C♯ G♯ .. .. .. .. ... ... ... ... ... PR ... ... PL ... A ................E ................ B′ .. .. R.. .. L .. .. ... ... ... ... ... ... ... LR ... ... ∗ ... ... RL ... ( ) ′ F′ ................C ................ G ................ D .. .. P .. .. ... ... ... ... ... LP ... ... RP ... A♭ ................E♭ ................ B♭′ arXiv:1901.05106v3 [math.CO] 4 Apr 2019 Figure 1. Triads in the vicinity of the C-E-G-chord Our paper is motivated by the observation that PLR-moves, while they provide a tool to measure distance between triads, are not continuous as operations on the Tonnetz: Let s be a sequence of PLR-moves. -
A Tonnetz Model for Pentachords
A Tonnetz model for pentachords Luis A. Piovan KEYWORDS. neo-Riemann network, pentachord, contextual group, Tessellation, Poincaré disk, David Lewin, Charles Koechlin, Igor Stravinsky. ABSTRACT. This article deals with the construction of surfaces that are suitable for repre- senting pentachords or 5-pitch segments that are in the same T {I class. It is a generalization of the well known Öttingen-Riemann torus for triads of neo-Riemannian theories. Two pen- tachords are near if they differ by a particular set of contextual inversions and the whole contextual group of inversions produces a Tiling (Tessellation) by pentagons on the surfaces. A description of the surfaces as coverings of a particular Tiling is given in the twelve-tone enharmonic scale case. 1. Introduction The interest in generalizing the Öttingen-Riemann Tonnetz was felt after the careful analysis David Lewin made of Stockhausen’s Klavierstück III [25, Ch. 2], where he basically shows that the whole work is constructed with transformations upon the single pentachord xC,C#, D, D#, F #y. A tiled torus with equal tiles like the usual Tonnetz of Major and Minor triads is not possible by using pentagons (you cannot tile a torus or plane by regular convex pentagons). Therefore one is forced to look at other surfaces and fortunately there is an infinite set of closed surfaces where one can gather regular pentagonal Tilings. These surfaces (called hyperbolic) are distinguished by a single topological invariant: the genus or arXiv:1301.4255v1 [math.HO] 17 Jan 2013 number of holes the surface has (see Figure 8)1. The analysis2 of Schoenberg’s, Opus 23, Number 3, made clear the type of transfor- mations3 to be used. -
1 Original Place of Publication: Murphy, Scott. (2014) “Scoring Loss in Some Recent Popular Film and Television.” Music Theo
Original place of publication: Murphy, Scott. (2014) “Scoring Loss in Some Recent Popular Film and Television.” Music Theory Spectrum, 36(2), 295-314. Original date of publication: 7 Oct 2014 URL: http://mts.oxfordjournals.org/content/early/2014/09/24/mts.mtu014 ABSTRACT A certain tonally- and temporally-oriented progression of two triads, dwelt upon usually through undulation, accompanies scenes depicting the contemplation of a considerable sorrowful loss in many popular films and throughout one television program produced between 1985 and 2012. In lieu of any strong stylistic precedent for this musico-dramatic association, certain structural relationships between the two triads relative to other triadic pairings may account for possible motivations of the association. Keywords: film music, television music, tonality, triadic harmony, triadic transformation, mediant triad, loss, sorrow, homology, James Horner, Cocoon The narrative of Ron Howard’s 1985 movie Cocoon brings two parties into conflict: a peaceable and immortal alien race suffering from the regrettable but necessary abandonment of twenty of their kind on Earth thousands of years ago, and a group of present-day retirement-home residents in Florida suffering from degradation of health and vitality.*To counter these adversities, both parties use a lifeforce generated in a neglected swimming pool by a team of four aliens, led by Walter (Brian Dennehy), sent back to Earth to retrieve their stranded and cocooned comrades. The pool restores the abandoned aliens to full health, a requirement for their interstellar journey home, but it also acts as a “fountain of youth” for a trio of elderly men—Art (Don Ameche), Ben (Wilford Brimley), and Joe (Hume Cronyn)—who surreptitiously swim there; for example, the alien lifeforce miraculously throws Joe’s leukemia into 1 remission.