Addition Theorems. / the Theory of Sound in Organ Pipes
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James Clerk Maxwell
James Clerk Maxwell JAMES CLERK MAXWELL Perspectives on his Life and Work Edited by raymond flood mark mccartney and andrew whitaker 3 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries c Oxford University Press 2014 The moral rights of the authors have been asserted First Edition published in 2014 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2013942195 ISBN 978–0–19–966437–5 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. -
Electrophonic Musical Instruments
G10H CPC COOPERATIVE PATENT CLASSIFICATION G PHYSICS (NOTES omitted) INSTRUMENTS G10 MUSICAL INSTRUMENTS; ACOUSTICS (NOTES omitted) G10H ELECTROPHONIC MUSICAL INSTRUMENTS (electronic circuits in general H03) NOTE This subclass covers musical instruments in which individual notes are constituted as electric oscillations under the control of a performer and the oscillations are converted to sound-vibrations by a loud-speaker or equivalent instrument. WARNING In this subclass non-limiting references (in the sense of paragraph 39 of the Guide to the IPC) may still be displayed in the scheme. 1/00 Details of electrophonic musical instruments 1/053 . during execution only {(voice controlled (keyboards applicable also to other musical instruments G10H 5/005)} instruments G10B, G10C; arrangements for producing 1/0535 . {by switches incorporating a mechanical a reverberation or echo sound G10K 15/08) vibrator, the envelope of the mechanical 1/0008 . {Associated control or indicating means (teaching vibration being used as modulating signal} of music per se G09B 15/00)} 1/055 . by switches with variable impedance 1/0016 . {Means for indicating which keys, frets or strings elements are to be actuated, e.g. using lights or leds} 1/0551 . {using variable capacitors} 1/0025 . {Automatic or semi-automatic music 1/0553 . {using optical or light-responsive means} composition, e.g. producing random music, 1/0555 . {using magnetic or electromagnetic applying rules from music theory or modifying a means} musical piece (automatically producing a series of 1/0556 . {using piezo-electric means} tones G10H 1/26)} 1/0558 . {using variable resistors} 1/0033 . {Recording/reproducing or transmission of 1/057 . by envelope-forming circuits music for electrophonic musical instruments (of 1/0575 . -
Analytic Vortex Solutions on Compact Hyperbolic Surfaces
Analytic vortex solutions on compact hyperbolic surfaces Rafael Maldonado∗ and Nicholas S. Mantony Department of Applied Mathematics and Theoretical Physics, Wilberforce Road, Cambridge CB3 0WA, U.K. August 27, 2018 Abstract We construct, for the first time, Abelian-Higgs vortices on certain compact surfaces of constant negative curvature. Such surfaces are represented by a tessellation of the hyperbolic plane by regular polygons. The Higgs field is given implicitly in terms of Schwarz triangle functions and analytic solutions are available for certain highly symmetric configurations. 1 Introduction Consider the Abelian-Higgs field theory on a background surface M with metric ds2 = Ω(x; y)(dx2+dy2). At critical coupling the static energy functional satisfies a Bogomolny bound 2 1 Z B Ω 2 E = + jD Φj2 + 1 − jΦj2 dxdy ≥ πN; (1) 2 2Ω i 2 where the topological invariant N (the `vortex number') is the number of zeros of Φ counted with multiplicity [1]. In the notation of [2] we have taken e2 = τ = 1. Equality in (1) is attained when the fields satisfy the Bogomolny vortex equations, which are obtained by completing the square in (1). In complex coordinates z = x + iy these are 2 Dz¯Φ = 0;B = Ω (1 − jΦj ): (2) This set of equations has smooth vortex solutions. As we explain in section 2, analytical results are most readily obtained when M is hyperbolic, having constant negative cur- vature K = −1, a case which is of interest in its own right due to the relation between arXiv:1502.01990v1 [hep-th] 6 Feb 2015 hyperbolic vortices and SO(3)-invariant instantons, [3]. -
Orientalism As Represented in the Selected Piano Works by Claude Debussy
Chapter 4 ORIENTALISM AS REPRESENTED IN THE SELECTED PIANO WORKS BY CLAUDE DEBUSSY A prominent English scholar of French music, Roy Howat, claimed that, out of the many composers who were attracted by the Orient as subject matter, “Debussy is the one who made much of it his own language, even identity.”55 Debussy and Hahn, despite being in the same social circle, never pursued an amicable relationship.56 Even while keeping their distance, both composers were somewhat aware of the other’s career. Hahn, in a public statement from 1890, praised highly Debussy’s musical artistry in L'Apres- midi d'un faune.57 Debussy’s Exposure to Oriental Cultures Debussy’s first exposure to oriental art and philosophy began at Mallarmé’s Symbolist gatherings he frequented in 1887 upon his return to Paris from Rome.58 At the Universal Exposition of 1889, he had his first experience in the theater of Annam (Vietnam) and the Javanese Gamelan orchestra (Indonesia), which is said to be a catalyst 55Roy Howat, The Art of French Piano Music: Debussy, Ravel, Fauré, Chabrier (New Haven, Conn.: Yale University Press, 2009), 110 56Gavoty, 142. 57Ibid., 146. 58François Lesure and Roy Howat. "Debussy, Claude." In Grove Music Online. Oxford Music Online, http://www.oxfordmusiconline.com/subscriber/article/grove/music/07353 (accessed April 4, 2011). 33 34 in his artistic direction. 59 In 1890, Debussy was acquainted with Edmond Bailly, esoteric and oriental scholar, who took part in publishing and selling some of Debussy’s music at his bookstore L’Art Indépendeant. 60 In 1902, Debussy met Louis Laloy, an ethnomusicologist and music critic who eventually became Debussy’s most trusted friend and encouraged his use of Oriental themes.61 After the Universal Exposition in 1889, Debussy had another opportunity to listen to a Gamelan orchestra 11 years later in 1900. -
Technical Analysis on HW Ernst's Six Etudes for Solo Violin in Multiple
Technical Analysis on Heinrich Wilhelm Ernst’s Six Etudes for Solo Violin in Multiple Voices In partial fulfillment of the requirements for the degree of DOCTOR OF MUSICAL ARTS in the Performance Studies Division of the College-Conservatory of Music Violin by Shang Jung Lin M.M. The Boston Conservatory November 2019 Committee Chair: Won-Bin Yim, D.M.A. Abstract Heinrich Wilhelm Ernst was a Moravian violinist and composer who lived between 1814-1865. He was a friend of Brahms, collaborator with Mendelssohn, and was admired by Berlioz and Joachim. He was known as a violin virtuoso and composed many virtuosic works including an arrangement of Schubert’s Erlkönig for solo violin. The focus of this document will be on his Six Etudes for Solo Violin in Multiple Voices (also known as the Six Polyphonic Etudes). These pieces were published without opus number around 1862-1864. The etudes combine many different technical challenges with musical sensitivity. They were so difficult that the composer never gave a public performance of them. No. 6 is the most famous of the set, and has been performed by soloists in recent years. Ernst takes the difficulty level to the extreme and combines different layers of techniques within one hand. For example, the second etude has a passage that combines chords and left-hand pizzicato, and the sixth etude has a passage that combines harmonics with double stops. Etudes from other composers might contain these techniques but not simultaneously. The polyphonic nature allows for this layering of difficulties in Ernst’s Six Polyphonic Etudes. -
Applications of Elliptic Functions in Classical and Algebraic Geometry
APPLICATIONS OF ELLIPTIC FUNCTIONS IN CLASSICAL AND ALGEBRAIC GEOMETRY Jamie Snape Collingwood College, University of Durham Dissertation submitted for the degree of Master of Mathematics at the University of Durham It strikes me that mathematical writing is similar to using a language. To be understood you have to follow some grammatical rules. However, in our case, nobody has taken the trouble of writing down the grammar; we get it as a baby does from parents, by imitation of others. Some mathe- maticians have a good ear; some not... That’s life. JEAN-PIERRE SERRE Jean-Pierre Serre (1926–). Quote taken from Serre (1991). i Contents I Background 1 1 Elliptic Functions 2 1.1 Motivation ............................... 2 1.2 Definition of an elliptic function ................... 2 1.3 Properties of an elliptic function ................... 3 2 Jacobi Elliptic Functions 6 2.1 Motivation ............................... 6 2.2 Definitions of the Jacobi elliptic functions .............. 7 2.3 Properties of the Jacobi elliptic functions ............... 8 2.4 The addition formulæ for the Jacobi elliptic functions . 10 2.5 The constants K and K 0 ........................ 11 2.6 Periodicity of the Jacobi elliptic functions . 11 2.7 Poles and zeroes of the Jacobi elliptic functions . 13 2.8 The theta functions .......................... 13 3 Weierstrass Elliptic Functions 16 3.1 Motivation ............................... 16 3.2 Definition of the Weierstrass elliptic function . 17 3.3 Periodicity and other properties of the Weierstrass elliptic function . 18 3.4 A differential equation satisfied by the Weierstrass elliptic function . 20 3.5 The addition formula for the Weierstrass elliptic function . 21 3.6 The constants e1, e2 and e3 ..................... -
Interval Cycles and the Emergence of Major-Minor Tonality
Empirical Musicology Review Vol. 5, No. 3, 2010 Modes on the Move: Interval Cycles and the Emergence of Major-Minor Tonality MATTHEW WOOLHOUSE Centre for Music and Science, Faculty of Music, University of Cambridge, United Kingdom ABSTRACT: The issue of the emergence of major-minor tonality is addressed by recourse to a novel pitch grouping process, referred to as interval cycle proximity (ICP). An interval cycle is the minimum number of (additive) iterations of an interval that are required for octave-related pitches to be re-stated, a property conjectured to be responsible for tonal attraction. It is hypothesised that the actuation of ICP in cognition, possibly in the latter part of the sixteenth century, led to a hierarchy of tonal attraction which favoured certain pitches over others, ostensibly the tonics of the modern major and minor system. An ICP model is described that calculates the level of tonal attraction between adjacent musical elements. The predictions of the model are shown to be consistent with music-theoretic accounts of common practice period tonality, including Piston’s Table of Usual Root Progressions. The development of tonality is illustrated with the historical quotations of commentators from the sixteenth to the eighteenth centuries, and can be characterised as follows. At the beginning of the seventeenth century multiple ‘finals’ were possible, each associated with a different interval configuration (mode). By the end of the seventeenth century, however, only two interval configurations were in regular use: those pertaining to the modern major- minor key system. The implications of this development are discussed with respect interval cycles and their hypothesised effect within music. -
A Natural System of Music
A Natural System of Music based on the Approximation of Nature by Augusto Novaro Mexico, DF 1927 Translated by M. Turner. San Francisco, 2003 Chapter I Music Music is a combination of art and science in which both complement each other in a marvelous fashion. Although it is true that a work of music performed or composed without the benefit of Art comes across as dry and expressionless, it is also true that a work of art cannot be considered as such without also satisfying the principles of science. Leaving behind for a moment the matters of harmony and the art of combining sounds, let us concentrate for the moment only on the central matter of the basis of music: The unity tone and it’s parts in vibrations, which can be divided into three groups: the physical, the mathematical and the physiological. The physical aspect describes for us how the sound is produced, the mathematical it’s numerical and geometrical relationships, the physiological being the study of the impressions that such a tone produces in our emotions.. Musical sounds are infinite. However we can break them down and classify them into seven degrees and denominations: Do Re Mi Fa So La Ti 1° 2° 3° 4° 5° 6° 7° Climbing to ‘Ti’ we reach the duplicate of ‘Do’, that is to say Do², the unity note of the previous series. Ascending in this same manner, we reach Do3, Do4, etc. We shall define the musical scale in seven steps, not because it invariably must be seven; we could also choose to use five or nine divisions, perhaps they would be more practical. -
V. Analytic Continuation and Riemann Sur- Faces
MTH6111 Complex Analysis 2009-10 Lecture Notes c Shaun Bullett 2009 V. Analytic Continuation and Riemann Sur- faces This section will introduce various ideas that are important for more advanced work in complex analysis. 1. Continuation along paths A holomorphic function f defined on some open disc containing z0, has a Taylor series ∞ (n) X f (z0) (z − z )n n! 0 n=0 We can use this series to define f at any points inside the disc D(z0, r0) of convergence of the series where f has not already been defined. Now choosing a new point z1 near the edge of D(z0, r0) we can repeat the same procedure for the Taylor series ∞ (n) X f (z1) (z − z )n n! 1 n=0 and this will enable us to define f in a new disc D(z1, r1), overlapping the first disc. This process is known as analytic continuation of f. An analytic function is one that can be written as the sum of a power series: note that we can only carry out the continuation process because of Taylor’s Theorem, which says that a holomorphic function is an analytic function. The most interesting examples of this process occur when f is one branch of a multi-valued function, and we follow the analytic continuation of this branch along some path, or even round a loop back to where we started. This is best illustrated with an example. √ Let f(z) = z. Around the point z = 1, one branch of this function can be defined by the power series: (1/2) (1/2)(−1/2) (1/2)(−1/2)(−3/2) (1 + (z − 1))1/2 = 1 + (z − 1) + (z − 1)2 + (z − 1)3 + .. -
Musical Techniques
Musical Techniques Musical Techniques Frequencies and Harmony Dominique Paret Serge Sibony First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd John Wiley & Sons, Inc. 27-37 St George’s Road 111 River Street London SW19 4EU Hoboken, NJ 07030 UK USA www.iste.co.uk www.wiley.com © ISTE Ltd 2017 The rights of Dominique Paret and Serge Sibony to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2016960997 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-058-4 Contents Preface ........................................... xiii Introduction ........................................ xv Part 1. Laying the Foundations ............................ 1 Introduction to Part 1 .................................. 3 Chapter 1. Sounds, Creation and Generation of Notes ................................... 5 1.1. Physical and physiological notions of a sound .................. 5 1.1.1. Auditory apparatus ............................... 5 1.1.2. Physical concepts of a sound .......................... 7 1.1.3. -
Tuning: at the Grcssroads
WendyGarloo ?O. Box1024 Tuning:At the New YorkCit, New York 10276USA Grcssroads lntrodrciion planned the construction of instruments that per- formed within the new "tunitrg of choice," and all The arena o{ musical scales and tuning has cer_ publishedpapers or books demonstretingthe supe- tainly not been a quiet place to be for the past thlee dority of their new scales in at least some way over hundred yeals. But it might iust as well have beenif €qual temperament.The tradition has continued we iudge by the results: the same 12V2 equally with Yunik and Swi{t {1980),Blackwood (1982),and temperedscale established then as the best avail- the presentauthor (Milano 1986),and shows no able tuning compromise, by J. S. Bach and many sign of slowing down despite the apparent apathy otheis lHelrnholtz 1954j Apel 1972),remains to with which the musical mainstream has regularly this day essentially the only scale heard in Westem grceted eech new proposal. music. That monopoly crossesall musical styles, of course therc's a perf€ctly reasonable explana- {rom the most contemporary of jazz and av^rf,t' tion lor the mainstream's evident preferetrce to rc- "rut-bound" gardeclassical, and musical masteeieces from the main when by now there are at least past, to the latest technopop rock with fancy s)'n- a dozen clearly better-sounding ways to tune our thesizers,and everwvherein between.Instruments scales,i{ only for at least part of the time: it re' ol the symphonyorchestra a((empr with varyirrg quires a lot of effort ol several kinds. I'm typing this deSreesof successto live up ro lhe 100-centsemi manuscript using a Dvorak keyboard (lor the ffrst tone, even though many would find it inherently far time!), and I assureyou it's not easyto unlearn the easierto do otherwise: the stdngs to "lapse" into QWERTY habits of a lifetime, even though I can Pythagoieen tuning, the brass into several keys of akeady feel the actual superiodty of this unloved lust irtonation lBarbour 1953).And th€se easily but demonstrablv better kevboard. -
Towards a Generative Framework for Understanding Musical Modes
Table of Contents Introduction & Key Terms................................................................................1 Chapter I. Heptatonic Modes.............................................................................3 Section 1.1: The Church Mode Set..............................................................3 Section 1.2: The Melodic Minor Mode Set...................................................10 Section 1.3: The Neapolitan Mode Set........................................................16 Section 1.4: The Harmonic Major and Minor Mode Sets...................................21 Section 1.5: The Harmonic Lydian, Harmonic Phrygian, and Double Harmonic Mode Sets..................................................................26 Chapter II. Pentatonic Modes..........................................................................29 Section 2.1: The Pentatonic Church Mode Set...............................................29 Section 2.2: The Pentatonic Melodic Minor Mode Set......................................34 Chapter III. Rhythmic Modes..........................................................................40 Section 3.1: Rhythmic Modes in a Twelve-Beat Cycle.....................................40 Section 3.2: Rhythmic Modes in a Sixteen-Beat Cycle.....................................41 Applications of the Generative Modal Framework..................................................45 Bibliography.............................................................................................46 O1 O Introduction Western