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3 books match the query 'Arithmetic'

cover of Criteria for Divisibility

Criteria for Divisibility

N. N. Vorob'ev

University of Chicago Press, 1980

N. N. Vorob'ev's Criteria for Divisibility introduces the high school or early college student to a specific number-theoretic topic and explains the general mathematical structures which underlie the particular concepts discussed. Vorob'ev discusses the ideas of well-ordered sets, partial and linear orderings, equivalence relations, equivalence classes, algorithms, and the relationship between the determinability of algorithms defined on the integers and the well-ordering principle.

All this is done comprehensively with the help of a unique plan for study which encourages the student to skip large sections of the book on first reading and return to them later. The more general and conceptually challenging material appears in small print, so that the student must have a good grasp of the number-theoretic concepts on which the generalizations are based before making the step to generalization. The booklet provides both specific knowledge in a particular field of mathematical investigation and a fine basis on which to continue studies in mathematics.
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cover of The Arithmetic of Listening: Tuning Theory and History for the Impractical Musician

The Arithmetic of Listening

Tuning Theory and History for the Impractical Musician

Kyle Gann

University of Illinois Press, 2019

"Tuning is the secret lens through which the history of music falls into focus," says Kyle Gann. Yet in Western circles, no other musical issue is so ignored, so taken for granted, so shoved into the corners of musical discourse.

A classroom essential and an invaluable reference, The Arithmetic of Listening offers beginners the grounding in music theory necessary to find their own way into microtonality and the places it may take them. Moving from ancient Greece to the present, Kyle Gann delves into the infinite tunings available to any musician who feels straitjacketed by obedience to standardized Western European tuning. He introduces the concept of the harmonic series and demonstrates its relationship to equal-tempered and well-tempered tuning. He also explores recent experimental tuning models that exploit smaller intervals between pitches to create new sounds and harmonies.

Systematic and accessible, The Arithmetic of Listening provides a much-needed primer for the wide range of tuning systems that have informed Western music.

Audio examples demonstrating the musical ideas in The Arithmetic of Listening can be found at: https://www.kylegann.com/Arithmetic.html

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cover of The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number

The Foundations of Arithmetic

A Logico-Mathematical Enquiry into the Concept of Number

Gottlob Frege

Northwestern University Press, 1980

The Foundations of Arithmetic is undoubtedly the best introduction to Frege's thought; it is here that Frege expounds the central notions of his philosophy, subjecting the views of his predecessors and contemporaries to devastating analysis. The book represents the first philosophically sound discussion of the concept of number in Western civilization. It profoundly influenced developments in the philosophy of mathematics and in general ontology.
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