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2017 | Buch

Common Parameter Spaces Kapitel lesen Erstes Kapitel lesen

The Topos of Music IV: Roots

Appendices

verfasst von: Prof. Dr. Guerino Mazzola

Verlag: Springer International Publishing

Buchreihe : Computational Music Science

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This is the fourth volume of the second edition of the now classic book “The Topos of Music”. The author presents appendices with background material on sound and auditory physiology; mathematical basics such as sets, relations, transformations, algebraic geometry, and categories; complements in physics, including a discussion on string theory; and tables with chord classes and modulation steps.

Inhaltsverzeichnis

Frontmatter

Appendix: Sound

Frontmatter
A. Common Parameter Spaces
Abstract
This appendix chapter is an overview, not an exhaustive treatise of spaces which parametrize sound objects. These spaces where sounds are positioned always define an aspect, never the totality of music thinking, and every attempt to define a preferred space will narrow the music thinking, not the music. The best that can occur is that we offer an encompassing or at least a representative ensemble of parameter spaces which are interrelated by a precise relation. To this end, it is recommended to distinguish topographic positions, above all in their realities and communicative perspectives. This will also entail the corresponding mathematics.
Guerino Mazzola
B. Auditory Physiology and Psychology
Abstract
“Music listening” is a metonymy of understanding music: For all participants, the ear functions as an interface for perceiving music between physical, psychological, and mathematical reality. But a metonymy is not the matter as such. This is what deaf Beethoven teaches us impressively: his innermost ear was an organ of imagination that was uncoupled from the material ear. And the physiology of the hearing process teaches us that the neural coupling of the ear to the respective cortical regions is extremely complex and still hardly understood. Section B.1 is written to give an overview on auditory physiology.
Guerino Mazzola

Appendix: Mathematical Basics

Frontmatter
C. Sets, Relations, Monoids, Groups
Abstract
The language of sets describes mathematical facts in a classical way. An alternative foundation to sets is the language of categories, see Appendix G.
Guerino Mazzola
D. Rings and Algebras
Abstract
A (unitary) ring is a triple (R, α, μ) where (R, α) is an abelian group whose operation α is written additively (α(r, s) = r + s) with neutral element 0 R , and (R, μ) is monoid, written multiplicatively \( ({\mu}(r, s) = {r}\cdot{s})\) with multiplicative neutral element 1 R such that these operations are coupled by distributivity, i.e., \( (r + s) \cdot {t} = {r} \cdot {t}+{s} \cdot {t}, {t} \cdot (r + s) = {t} \cdot {r} + {t} \cdot {s}\) for all \( {r, s, t} \ {\in} \ {R}\).
Guerino Mazzola
E. Modules, Linear, and Affine Transformations
Abstract
Let R be a ring, then a (left) R-module is a triple (R, M, μ : R × M → M) where M is an additively written abelian group and μ is the scalar multiplication, usually written as μ(r, m) = r.m if μ is clear (R is also called the ring of scalars, and M the group of vectors), with the properties:
Guerino Mazzola
F. Algebraic Geometry
Abstract
Given a topological space X, its system of open sets Open X is viewed as a category with inclusions as morphisms.
Guerino Mazzola
G. Categories, Topoi, and Logic
Summary
For a comprehensive introduction to category theory, see [637]. For topos theory and sheaves see [639], for topos theory and logic, see [376].
Guerino Mazzola
H. Complements on General and Algebraic Topology
Summary
Refer to [527] for general topology, and to [993] for algebraic topology.
Guerino Mazzola
I. Complements on Calculus
Abstract
Abstract on Calculus
Guerino Mazzola
J. More Complements on Mathematics
Summary
This appendix is not self-contained, it completes the mathematical Appendix Part XXI and is built upon those topics.
Guerino Mazzola

Appendix: Complements in Physics

Frontmatter
K. Complements on Physics
Summary
The appendix recalls some special topics in theoretical physics that are referred to in the main text. They are neither self-contained nor complete.
Guerino Mazzola

Appendix: Tables

Frontmatter
L. Euler’s Gradus Function
Abstract
This table lists the rational numbers x/y with Euler’s gradus suavitatis \({\it \Gamma (x/y)}\leq\;10, \mathrm{see \; also}\; [161]\).
Guerino Mazzola
M. Just and Well-Tempered Tuning
Summary
This table lists the just coordinates of the just tuning intervals (with respect to c, second tone in first column) according to Vogel [1089], see Section 7.2.1.4, together with the value in Cents, and the deviation in % from the tempered tuning with 100, 200, 300, etc. Cents.
Guerino Mazzola
N. Chord and Third Chain Classes
Summary
This section contains the list of all isomorphism classes of zero-addressed chords in PiMod12. The meanings of the column items are explained in Section 11.3.7.
Guerino Mazzola
O. Two, Three, and Four Tone Motif Classes
Summary
The order of these representatives is a historical one.
Guerino Mazzola
P. Well-Tempered and Just Modulation Steps
Summary
In the following table, the exclamation sign (!) in column 6 means that quantization is not possible for every translation quantity p in the notation of Theorem 30.
Guerino Mazzola
Q. Counterpoint Steps
Summary
All the following tables relate to representatives of strong dichotomies (X/Y) which are indicated in the table after the counterpoint theorem 33 in Section 31.3.3.
Guerino Mazzola
Backmatter
Metadaten
Titel
The Topos of Music IV: Roots
verfasst von
Prof. Dr. Guerino Mazzola
Copyright-Jahr
2017
Electronic ISBN
978-3-319-64495-0
Print ISBN
978-3-319-64494-3
DOI
https://doi.org/10.1007/978-3-319-64495-0

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