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Paradoxes of Pitch Space

Identifieur interne : 000147 ( Main/Exploration ); précédent : 000146; suivant : 000148

Paradoxes of Pitch Space

Auteurs : Candace Brower

Source :

RBID : ISTEX:65197A076B5FC6036B3A5A30C0F840D8C0959494

English descriptors

Abstract

Parallels between the mathematics of tiling, which describes geometries of visual space, and neo‐Riemannian theory, which describes geometries of musical space, make it possible to show that certain paradoxes featured in the visual artworks of M. C. Escher also appear in the pitch space modelled by the neo‐Riemannian Tonnetz. This article makes these paradoxes visually apparent by constructing an embodied model of triadic pitch space in accordance with principles drawn from the mathematics of tiling, on the one hand, and from cognitive science, on the other – specifically, the notion that our experience of pitch relationships is governed in part by the metaphorical projection of patterns abstracted from embodied experience known as image schemas. These paradoxes are illustrated with reference to passages drawn from four compositions to whose expressive character such paradoxes contribute: the fifteenth‐century motet ‘Absalon fili mi’; the finale of Haydn's String Quartet in G major, Op. 76 No. 1; Brahms's Intermezzo in B minor, Op. 119 No. 1; and Wagner's Parsifal.

Url:
DOI: 10.1111/j.1468-2249.2008.00268.x


Affiliations:


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Le document en format XML

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<term>Absalon</term>
<term>Acoustical society</term>
<term>Alternative pathways</term>
<term>American musicological society</term>
<term>Arnheim</term>
<term>Balance schema</term>
<term>Balance schemas</term>
<term>Basic books</term>
<term>Basic pitch space</term>
<term>Basic shape</term>
<term>Black birds</term>
<term>Blackwell</term>
<term>Blackwell publishing</term>
<term>Bodily basis</term>
<term>Bodily experience</term>
<term>Bodily sensation</term>
<term>Bodily sensations</term>
<term>Brower</term>
<term>Cadence</term>
<term>California press</term>
<term>Candace</term>
<term>Candace brower</term>
<term>Central role</term>
<term>Centre</term>
<term>Certain paradoxes</term>
<term>Chicago press</term>
<term>Chromatic music</term>
<term>Chromatic regions</term>
<term>Chromatic scale</term>
<term>Circular path</term>
<term>Circular pathways</term>
<term>Circular route</term>
<term>Circular staircase</term>
<term>Closure</term>
<term>Cognitive</term>
<term>Cognitive science</term>
<term>Cognitive theory</term>
<term>Compilation</term>
<term>Complex pitch</term>
<term>Congruence transformations</term>
<term>Consonant triad</term>
<term>Container schema</term>
<term>Container schemas</term>
<term>Cycle schema</term>
<term>Cylindrical loop</term>
<term>Diatonic</term>
<term>Diatonic collection</term>
<term>Diatonic space</term>
<term>Diatonic strip</term>
<term>Double chain</term>
<term>Double trajectory</term>
<term>Downwards</term>
<term>Downwards movement</term>
<term>Enharmonic</term>
<term>Enharmonic equivalence</term>
<term>Enharmonic modulation</term>
<term>Enharmonically equivalent keys</term>
<term>Equal temperament</term>
<term>Equilateral triangle</term>
<term>Escher</term>
<term>Everyday diatonicism</term>
<term>Experience movement</term>
<term>Frequency spectrum</term>
<term>Fths</term>
<term>Geometric terms</term>
<term>Grail</term>
<term>Graphic artists</term>
<term>Gure</term>
<term>Gures</term>
<term>Harmonic</term>
<term>Harmonic motion</term>
<term>Hexatonic</term>
<term>Hexatonic collection</term>
<term>Hexatonic space</term>
<term>Hexatonic system</term>
<term>Hexatonic systems</term>
<term>Historical perspective</term>
<term>Hofstadter</term>
<term>Holy grail</term>
<term>Horizontal axes</term>
<term>Horizontal dimension</term>
<term>Howard mayer</term>
<term>Image schemas</term>
<term>Important role</term>
<term>Intermezzo</term>
<term>Inverse relation</term>
<term>Journal compilation</term>
<term>Koninklijke vereniging voor nederlandse muziekgeschiedenis</term>
<term>Lakoff</term>
<term>Lewin</term>
<term>Major keys</term>
<term>Major mode</term>
<term>Major thirds</term>
<term>Major triad</term>
<term>Major triads</term>
<term>Mathematical models</term>
<term>Mathematical systems</term>
<term>Maximally</term>
<term>Melodic motion</term>
<term>Melodic space</term>
<term>Mental image</term>
<term>Metaphorical projection</term>
<term>Minor keys</term>
<term>Minor mode</term>
<term>Minor thirds</term>
<term>Minor triad</term>
<term>Minor triads</term>
<term>Modulation</term>
<term>Motet</term>
<term>Mouch</term>
<term>Music analysis</term>
<term>Music perception</term>
<term>Music theory</term>
<term>Musical forces</term>
<term>Musical object</term>
<term>Musical objects</term>
<term>Musical plane</term>
<term>Musical space</term>
<term>Musicalischen composition</term>
<term>Natural home</term>
<term>Necker cube</term>
<term>Neoriemannian tonnetz</term>
<term>Netherlands motet</term>
<term>Nineteenth centuries</term>
<term>Nineteenth century</term>
<term>Octatonic</term>
<term>Octatonic collections</term>
<term>Octatonic modulation</term>
<term>Octatonic spaces</term>
<term>Octave equivalence</term>
<term>Opening bars</term>
<term>Opposite ends</term>
<term>Overtone</term>
<term>Overtone series</term>
<term>Oxford university press</term>
<term>Paradox</term>
<term>Paradoxical features</term>
<term>Paradoxical nature</term>
<term>Parallelogram</term>
<term>Parsifal</term>
<term>Path schema</term>
<term>Pathway</term>
<term>Period parallelogram</term>
<term>Periodic tiling</term>
<term>Physical objects</term>
<term>Physical space</term>
<term>Physical world</term>
<term>Pitch</term>
<term>Pitch class</term>
<term>Pitch space</term>
<term>Present model</term>
<term>Primary triads</term>
<term>Princeton university press</term>
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<term>Relative stability</term>
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<term>Riemann space</term>
<term>Rochester press</term>
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<term>Same time</term>
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<term>Seamless surface</term>
<term>Sharp keys</term>
<term>Sharp side</term>
<term>Similar paradoxes</term>
<term>Single object</term>
<term>Somatosensory</term>
<term>Somatosensory experience</term>
<term>Somatosensory level</term>
<term>Spatial orientation</term>
<term>Spiral staircase</term>
<term>Steblin</term>
<term>String quartet</term>
<term>Strong consensus</term>
<term>Structural skeleton</term>
<term>Stufen</term>
<term>Stufen space</term>
<term>Subdominant</term>
<term>Such paradoxes</term>
<term>Syntonic comma</term>
<term>Tensing</term>
<term>Tiling</term>
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<term>Tonal pitch space</term>
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<term>Transformational theory</term>
<term>Triad</term>
<term>Triadic</term>
<term>Triadic centre</term>
<term>Triadic harmony</term>
<term>Triadic pitch space</term>
<term>Triadic space</term>
<term>Triangle</term>
<term>Unusual treatment</term>
<term>Upper voices</term>
<term>Vertical arrangement</term>
<term>Vertical axis</term>
<term>Vertical line</term>
<term>Vertical orientation</term>
<term>Vertical path</term>
<term>Vertical path schema</term>
<term>Verticality schema</term>
<term>Virtual pitch</term>
<term>Visual form</term>
<term>Visual images</term>
<term>Visual model</term>
<term>Visual perception</term>
<term>Visual space</term>
<term>Wason</term>
<term>Western music</term>
<term>White birds</term>
<term>Young children</term>
<term>Zigzag chain</term>
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<front>
<div type="abstract" xml:lang="en">Parallels between the mathematics of tiling, which describes geometries of visual space, and neo‐Riemannian theory, which describes geometries of musical space, make it possible to show that certain paradoxes featured in the visual artworks of M. C. Escher also appear in the pitch space modelled by the neo‐Riemannian Tonnetz. This article makes these paradoxes visually apparent by constructing an embodied model of triadic pitch space in accordance with principles drawn from the mathematics of tiling, on the one hand, and from cognitive science, on the other – specifically, the notion that our experience of pitch relationships is governed in part by the metaphorical projection of patterns abstracted from embodied experience known as image schemas. These paradoxes are illustrated with reference to passages drawn from four compositions to whose expressive character such paradoxes contribute: the fifteenth‐century motet ‘Absalon fili mi’; the finale of Haydn's String Quartet in G major, Op. 76 No. 1; Brahms's Intermezzo in B minor, Op. 119 No. 1; and Wagner's Parsifal.</div>
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