The musical operation of scalar transposition shifts every note in a melody by the same number of scale steps. The musical operation of chromatic transposition shifts every note in a melody by the same distance in pitch class space. In general, for a given scale S, the scalar transpositions of a line L can be grouped into categories, or transpositional set classes, whose members are related by chromatic transposition. In diatonic set theory cardinality equals variety when, for any melodic line L in a particular scale S, the number of these classes is equal to the number of distinct pitch classes in the line L. For example, the melodic line C-D-E has three distinct pitch classes. When transposed diatonically to all scale degrees in the C major scale, we obtain three interval patterns: M2-M2, M2-m2, m2-M2. Melodic lines in the C major scale with n distinct pitch classes always generate n distinct patterns. The property was first described by John Clough and in "Variety and Multiplicity in Diatonic Systems" (1985) (Johnson 2003, p. 68, 151). Cardinality equals variety in the diatonic collection and the pentatonic scale, and, more generally, what Carey and Clampitt (1989) call "nondegenerate well-formed scales." "Nondegenerate well-formed scales" are those that possess Myhill's property. (Wikipedia).
Introduction to the Cardinality of Sets and a Countability Proof
Introduction to Cardinality, Finite Sets, Infinite Sets, Countable Sets, and a Countability Proof - Definition of Cardinality. Two sets A, B have the same cardinality if there is a bijection between them. - Definition of finite and infinite sets. - Definition of a cardinal number. - Discu
From playlist Set Theory
Determine the Cardinality of Sets From a List of Set
This video explains how to determine the cardinality of sets given as lists. It includes union, intersection, and complement of sets. http://mathispower4u.com
From playlist Sets
Determine the Cardinality of Sets: Set Notation, Intersection
This video explains how to determine the cardinality of a set given using set notation.
From playlist Sets (Discrete Math)
BM9.1. Cardinality 1: Finite Sets
Basic Methods: We define cardinality as an equivalence relation on sets using one-one correspondences. In this talk, we consider finite sets and counting rules.
From playlist Math Major Basics
What is the Cardinality of a Set? | Set Theory, Empty Set
What is the cardinality of a set? In this video we go over just that, defining cardinality with examples both easy and hard. To find the cardinality of a set, you need only to count the elements in the set. The cardinality of the empty set is 0, the cardinality of the set A = {0, 1, 2} is
From playlist Set Theory
Cardinality Example with [0,1]
Real Analysis: We show that the sets [0,1], (0,1], and (0,1) have the cardinality by constructing one-one correspondences. Then we expand the method to construct a one-one correspondence between [0,1] and the irrationals in [0,1].
From playlist Real Analysis
The Cardinality of the Union of Three Sets
This video provides an explanation of the formula for the cardinality of the union of three sets.
From playlist Counting (Discrete Math)
Determine How Many Subsets Have More Than a Given Cardinality
This lesson provides examples of how to determine the number of subsets of a given set under various conditions.
From playlist Counting (Discrete Math)
Finding Cardinalities of Sets | Set Theory
Let's find the cardinality of some simple sets in set builder notation! Recall the cardinality of a set is simply the number of elements it contains. We'll write some sets that have been given in set builder notation and identify their cardinalities. We also briefly discuss the cardinality
From playlist Set Theory
Nonlinear algebra, Lecture 13: "Polytopes and Matroids ", by Mateusz Michalek
This is the thirteenth lecture in the IMPRS Ringvorlesung, the advanced graduate course at the Max Planck Institute for Mathematics in the Sciences.
From playlist IMPRS Ringvorlesung - Introduction to Nonlinear Algebra
Local-Global Compatibility and Monodromy - Ana Caraiani
Ana Caraiani Harvard University January 20, 2011 Given a cuspidal automorphic representation of GL(n) which is regular algebraic and conjugate self-dual, one can associate to it a Galois representation. This Galois representation is known in most cases to be compatible with local Langlands
From playlist Mathematics
Valentijn Karemaker, Mass formulae for supersingular abelian varieties
VaNTAGe seminar, Jan 18, 2022 License: CC-BY-NC-SA Links to some of the papers mentioned in this talk: Oort: https://link.springer.com/chapter/10.1007/978-3-0348-8303-0_13 Honda: https://doi.org/10.2969/jmsj/02010083 Tate: https://link.springer.com/article/10.1007/BF01404549 Tate: https
From playlist Curves and abelian varieties over finite fields
Galois theory: Transcendental extensions
This lecture is part of an online graduate course on Galois theory. We describe transcendental extension of fields and transcendence bases. As applications we classify algebraically closed fields and show hw to define the dimension of an algebraic variety.
From playlist Galois theory
[BOURBAKI 2017] 17/06/2017 - 4/4 - Nicolas BERGERON
Variétés en expansion [d'après Gromov, Guth, ...] ---------------------------------- Vous pouvez nous rejoindre sur les réseaux sociaux pour suivre nos actualités. Facebook : https://www.facebook.com/InstitutHenriPoincare/ Twitter : https://twitter.com/InHenriPoincare Instagram : https:/
From playlist BOURBAKI - 2017
Lec 16 | MIT 6.042J Mathematics for Computer Science, Fall 2010
Lecture 16: Counting Rules I Instructor: Marten van Dijk View the complete course: http://ocw.mit.edu/6-042JF10 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu
From playlist MIT 6.042J Mathematics for Computer Science, Fall 2010
D. Loughran - Sieving rational points on algebraic varieties
Sieves are an important tool in analytic number theory. In a typical sieve problem, one is given a list of p-adic conditions for all primes p, and the challenge is to count the number of integers which satisfy all these p-adic conditions. In this talk we present some versions of sieves for
From playlist Ecole d'été 2017 - Géométrie d'Arakelov et applications diophantiennes
R. Dujardin - Some problems of arithmetic origin in complex dynamics and geometry (part3)
Some themes inspired from number theory have been playing an important role in holomorphic and algebraic dynamics (iteration of rational mappings) in the past ten years. In these lectures I would like to present a few recent results in this direction. This should include: the dynamica
From playlist Ecole d'été 2017 - Géométrie d'Arakelov et applications diophantiennes
Local-global compatibility and monodromy - Ana Caraiani
Ana Caraiani Harvard University March 22, 2011 For more videos, visit http://video.ias.edu
From playlist Mathematics
[BOURBAKI 2018] 31/03/2018 - 3/3 - Antoine CHAMBERT-LOIR
Antoine CHAMBERT-LOIR — Relations de Hodge–Riemann et matroïdes Les matroïdes finis sont des structures combinatoires qui expriment la notion d’indépendance linéaire. En 1964, G.-C. Rota conjectura que les coefficients du « polynôme caractéristique » d’un matroïde M, polynôme dont les coe
From playlist BOURBAKI - 2018
BM9.2. Cardinality 2: Infinite Sets
Basic Methods: We continue the study of cardinality with infinite sets. First the class of countably infinite sets is considered, and basic results given. Then we give examples of uncountable sets using Cantor diagonalization arguments.
From playlist Math Major Basics