This introductory essay sketches the origins and recent development of neo riemannian theory, and positions it with respect to several other genera of music theory, as well as to an evolving poststructuralist critical practice. Neo riemannian theory nrt denotes a range of speculative and analytical studies concerned with the relation of the mathematical structure of tonal pitch materials e. Originally, the tonnetz was some sort of diagram introduced by euler to represent tonal space and the relationships between tones, mainly in terms of fifths and thirds. This article outlines the use of neoriemannian operations nros for the analysis of certain poprock.
All structured data from the file and property namespaces is available under the creative commons cc0 license. The basic theory of riemannian surfaces can be found in many books, see for example the literature given here. This is clearest in the fact that neo riemannian theory is fundamentally atonal, whereas most of riemanns theory is based around the functions of tonic, subdominant, and dominant. Riemannian geometry and the general relativity in the 19th century, mathematicians, scientists and philosophers experienced an extraordinary shock wave. A study of riemannian geometry a thesis submitted to indian institute of science education and research pune in partial ful llment of the requirements for the bsms dual degree programme by safeer k m indian institute of science education and research pune dr. Such characteristics are primarily identified with music of wagner, 167 this content downloaded from 128. It has been accepted for inclusion in masters theses by an authorized administrator of trace.
The rst centres around the introduction and analysis the ricci ow as a geometric heattype partial di erential equation. Media in category riemannian theory the following 21 files are in this category, out of 21 total. Extension of neoriemannian theory hindemith, fugue in e conclusion neoriemannian music theory recent work focuses on the neoriemannian operations p,l, and r. Building generalized neoriemannian groups of musical. They studied the kobayashi metric of the domain bounded by an ellipsoid in c2, and their calculations showed that the kobayashi metric on such a domain exhibits many of the nice properties of a riemannian met. But that merely states that the curvature tensor is a 3covariant, 1contravariant tensor. By the emergence of noneuclidean geometry, the old belief that mathematics o ers external and immutable truths was collapse. Loop quantum gravity lqg is an approach that attempts to face this challenge squarely for details, see, e. The oxford handbook of neoriemannian music theories oxford. Aims of morse theory the palaissmale condition, existence of saddle points local analysis limits of trajectories of the gradient flow floer condition, transversality and z2cohomology orientations and zhomology homotopies graph flows orientations 7 harmonic maps between riemannian manifolds 393. The riemann tensor lecture physics 411 classical mechanics ii september 26th 2007 we have, so far, studied classical mechanics in tensor notation via the lagrangian and hamiltonian formulations, and the special relativistic extension of the classical land to a lesser extent h.
Involving the application of the plr operations, the analysis of the hexatonic systems is a primary component of neoriemannian. The prime number theorem, hardys theorem on the zeros of. Neoriemannian theory is only very loosely related to hugo riemanns theory. O2 bho no name 9f1490d75c19759914f775e2ea0727c7 no file 1988, may. These spaces have enough structure so that they support a very rich theory for analysis and di erential equations, and they also. Neoriemannian theory is a loose collection of ideas present in the writings of music theorists such as david lewin, brian hyer, richard cohn, and henry. Building generalized neoriemannian groups of musical transformations as extensions alexandre popo al. In recent years hugo riemanns ideas have thoroughly captured the musictheoretical imagination, both in the united states and abroad. Neoriemannian theory has proven particularly adept at explaining features of chromatic music where other theoretical approaches have failed, and in so doing has established itself as the leading theoretical approach of our time. Roughly speaking, smooth manifolds are the underlying space, and in ge.
Yale university department of music semantic scholar. Neoriemannian theory is a loose collection of ideas present in the writings of music theorists such as david lewin, brian hyer, richard cohn, and henry klumpenhouwer. Neoriemannian theory arose in response to analytical problems posed by chromatic music that is triadic but not altogether tonally uni fied. The riemannian connection our main theorem is to prove theorem 2. Where necessary, references are indicated in the text. Involving the application of the plr operations, the analysis of the hexatonic systems is a primary component of neo riemannian.
The topic may be viewed as an extension of multivariable calculus from the usual setting of euclidean space to more general spaces, namely riemannian manifolds. P,l, and r generate a dihedral group, called the neo riemannian group. His theoretical writings cover many topics, including musical logic, notation, harmony, melody, phraseology, the history of music theory, etc. Manfredo perdigao do carmo riemannian geometry pdf.
General theory and examples is the perfect resource for graduate students and researchers in. Riemannian geometry is the branch of differential geometry that general relativity introduction mathematical formulation resources fundamental concepts special relativity equivalence principle world line riemannian geometry. What binds these ideas is a central commitment to relating harmonies directly to each other, without necessary reference to a tonic. The oxford handbook of neoriemannian music theories. Neoriemannian the ory nrt denotes a range of speculative and analytical studies concerned with the relation of the mathematica l structure of tonal pitch materials e. The exposition is selfcontained, and required a preliminary knowledge of only the ele. I am submitting herewith a thesis written by laura felicity mason entitled essential neoriemannian theory for todays musician. Neo riemannian theory is only very loosely related to hugo riemanns theory. Broadly speaking, the structure of this book falls into three main topics. They are indeed the key to a good understanding of it and will therefore play a major role throughout this work. There is an extension of the notion of vector eld that we shall need later on.
Exercises of petersens riemannian geometry peter petersen zujin zhang january 14, 2010 to my parents, jihe zhang and yulan ouyang contents 1 riemannian metrics 2 2 curvature 7 3 examples 29 4 hypersurfaces 34 5 geodesics and distance 38 6 sectional curvature comparison i 43 1. An important application to number theory is also the question on the spectrum of the laplacian in riemannian geometry. The most basic neoriemannian transformations shift one note and keep two common tones. Original signatures are on file with official student records. The famous nash embedding theorem published in 1956 was aimed for, in the hope that if riemannian manifolds could be regarded as riemannian submanifolds, this would then yield the opportunity to. Introduction to differential geometry and general relativity for pdf file click here. Pdf on jan 1, 2005, edward gollin and others published neoriemannian theory find, read and cite all the research you need on researchgate. Recall that riemannian geometry provides the appropriate. Riemannian approach journal of music theory pedagogy 21 and roig. In the late nineteenth century, composers often used triadic progressions that confound conventional. Neoriemannian theory not only allowed for the mathematical construction and description of these sets, some of which later were called the hexatonic systems, but also a means to explain how these systems were used. Neo riemannian theory not only allowed for the mathematical construction and description of these sets, some of which later were called the hexatonic systems, but also a means to explain how these systems were used. Lecture 1 basic concepts i riemannian geometry july 28, 2009 these lectures are entirely expository and no originality is claimed. To some extent, riemannian geometry is the theory of this ode.
This is one of the few works to combine both the geometric parts of riemannian geometry and the analytic aspects of the theory, while also presenting the most uptodate research. In the oxford handbook of neoriemannian music theories, ed. This introductory essay sketches the origins and recent development of neoriemannian theory, and positions it with respect to several other genera of music theory, as well as to an evolving poststructuralist critical practice. Certain positions of neo riemannian theory can be seen as direct responses to the main points of criticism. As well see, this group is dual to the ti group in the sense of lewin.
May 05, 2015 an introduction into the basics of neo riemannian theory, as developed by david lewin and richard cohn. I have examined the final electronic copy of this thesis for form and content and recommend that it be accepted in partial fulfillment of the requirements for the degree of master of music, with a major in music. The inverse problem is to determine information on g from the dn map g. Neo riemannian theory is a loose collection of ideas present in the writings of music theorists such as david lewin, brian hyer, richard cohn, and henry klumpenhouwer. P,l, and r generate a dihedral group, called the neoriemannian group. Neoriemannian triadic progressions open music theory. A special case in point is the interesting paper 11. Riemannian geometry without the quadratic restriction 2. This is a band new adventure and our past experience with other forces can not serve as a reliable guide. A course in riemannian geometry trinity college dublin. Neo riemannian theory has proven particularly adept at explaining features of chromatic music where other theoretical approaches have failed, and in so doing has established itself as the leading theoretical approach of our time.
Exercises of petersens riemannian geometry peter petersen. Differentiable manifolds, the tangent space, the tangent bundle, riemannian manifolds, the levicivita connection, geodesics, the riemann curvature tensor, curvature and local geometry. Curvature of riemannian manifolds uc davis mathematics. It then introduces the riemann curvature tensor, and quickly moves on to submanifold theory in order to give the curvature tensor a concrete quantitative interpretation. Curvature of riemannian manifolds wikipedia, the free encyclopedia 33110 1. This is clearest in the fact that neoriemannian theory is fundamentally atonal, whereas most of riemanns theory is based around the functions of tonic, subdominant, and dominant. Download fulltext pdf optimal control problems on riemannian manifolds. Free pdf download c documents and settings hp owner local settings temp k 43cc. This is a subject with no lack of interesting examples. Today ill talk about an application of neo riemannian theory, namely the tonnetz. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics,without which one cannot claim to be doing riemannian geometry. This course is an introduction to analysis on manifolds. I am submitting herewith a thesis written by laura felicity mason entitled essential neo riemannian theory for todays musician. This has a lot of number theory involved, see selbergs eigenvalue conjecture.
The last chapter is more advanced in nature and not usually treated in the rstyear di erential geometry course. An introduction to riemannian geometry download book. Curvature of riemannian manifolds from wikipedia, the free encyclopedia in mathematics, specifically differential geometry, the infinitesimal geometry of riemannian manifolds with dimension at least 3 is too complicated to be described by a. This book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and stokes theorem.
Studies in transformational theory western university. Cardinality transformations in diatonic space springerlink. It provides an introduction to the theory of characteristic classes, explaining how these could be generated by looking for. Files are available under licenses specified on their description page. To proceed further, we must discuss a little more machinery. Neoriemannian theory nrt denotes a range of speculative and analytical studies concerned with the relation of the mathematical structure of tonal pitch materials e. Namely, say we have a coordinate transform of the metric. Neoriemannian theory elegantly unites harmonic change and voiceleading efficiency, but deals exclusively with set class 037 in a 12gamut pcset space. Then it is a solution to the pde given above, and furthermore it then must satisfy the integrability conditions. Riemannian geometry from wikipedia, the free encyclopedia elliptic geometry is also sometimes called riemannian geometry. This book will appeal to readers with a knowledge of standard manifold theory, including such topics as. Every neoriemannian transformation toggles between one major and one minor triad. Schenkerian theory, neoriemannian theory and late schubert. Riemannian manifolds might in the future play a role in quantum mechanics similar to the role played by the riemannian manifolds in classical mechanics.1100 193 920 443 175 237 775 196 1555 1179 1543 1191 1024 218 1382 1216 1496 1498 679 1175 930 1039 361 846 331 855 880 154 1416 968 199 585 613 1314 1496 712 349 154 1405 43