Continued fractions, best measurements, and musical scales and intervals

Douthett, Jack; Krantz, Richard

doi:10.1080/17459730601137799

Cited by 15 publications

(18 citation statements)

References 20 publications

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“…Within Definitions 1 and 2 was considered the quality of the approximation of a real number from [1], and also [2][3][4][5][6][7][8][9].…”

Section: Continued Fractions and Best Approximationsmentioning

confidence: 99%

“…Ya. Khinchin [1], M. Thill [2], A. Cortzen [3], J. Douthett and R. Krantz [4] and M. Schechter [6], as shown in Table 1. On the other hand, terms, which were used for the same set of fractions are:…”

Section: Definition 2 Formentioning

confidence: 99%

“…-best Lagrange approximation by S. Khrushchev [5], -principal convergent by S. Khrushchev [5] and J. Douthett and R. Krantz [4], -convergent by A. Honingh [7] and N. Carey and D. Clampitt [8].…”

Section: Definition 2 Formentioning

confidence: 99%

“…The quality of the approximation of a real number by a fraction can also be observed based on the following definition according to [1], and also [2][3][4][5][6][7][8][9].…”

Section: Intermediate Fractionsmentioning

confidence: 99%

“…Many authors in their books and papers analyse not only the continued fractions, but also analyse the other types of fractions. One of the problems in this theory is nomenclature, as there is no consensus about terminology between different sources [2][3][4][5][6][7][8][9]. Some of the most used terms are best (rational) approximation of the first kind, best (rational) approximation of the second kind, intermediate fraction, semiconvergent, bestconvergent, principal convergent, nonprincipal convergent and intermediate convergent.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Continued Fractions, Intermediate Fractions and Their Relation to the Best Approximations

2020

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The subject of this paper is the current state of art in theory of continued fractions, intermediate fractions and their relation to the best rational approximations of the first and second kind. The paper provides an overview of the some well known and even some new properties of continued fractions, and the various terms associated with them. In addition to intermediate fractions, paper considers the fine intermediate fractions and gave some statements to position these fractions in the continued fraction representation of numbers.

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“…Within Definitions 1 and 2 was considered the quality of the approximation of a real number from [1], and also [2][3][4][5][6][7][8][9].…”

Section: Continued Fractions and Best Approximationsmentioning

confidence: 99%

Section: Definition 2 Formentioning

confidence: 99%

Section: Definition 2 Formentioning

confidence: 99%

“…The quality of the approximation of a real number by a fraction can also be observed based on the following definition according to [1], and also [2][3][4][5][6][7][8][9].…”

Section: Intermediate Fractionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Continued Fractions, Intermediate Fractions and Their Relation to the Best Approximations

2020

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Tempered monoids of real numbers, the golden fractal monoid, and the well-tempered harmonic semigroup

Bras-Amorós

2019

Semigroup Forum

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This paper deals with the algebraic structure of the sequence of harmonics when combined with equal temperaments. Fractals and the golden ratio appear surprisingly on the way.The sequence of physical harmonics is an increasingly enumerable submonoid of (R + , +) whose pairs of consecutive terms get arbitrarily close as they grow. These properties suggest to define a new mathematical object which we denote a tempered monoid. Mapping the elements of the tempered monoid of physical harmonics from R to N may be considered tantamount to defining equal temperaments. The number of equal parts of the octave in an equal temperament corresponds to the multiplicity of the related numerical semigroup.Analyzing the sequence of musical harmonics we derive two important properties that tempered monoids may have: that of being product-compatible and that of being fractal. We demonstrate that, up to normalization, there is only one product-compatible tempered monoid, which is the logarithmic monoid, and there is only one nonbisectional fractal monoid which is generated by the golden ratio.The examle of half-closed cylindrical pipes imposes to the sequence of musical harmonics one third property, the so-called odd-filterability property.We prove that the maximum number of equal divisions of the octave such that the discretizations of the golden fractal monoid and the logarithmic monoid coincide, and such that the discretization is oddfilterable is 12. This is nothing else but the number of equal divisions of the octave in classical Western music.

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On the joint subword complexity of automatic sequences

Moshe¹

2009

Theoretical Computer Science

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Continued fractions, best measurements, and musical scales and intervals

Cited by 15 publications

References 20 publications

Continued Fractions, Intermediate Fractions and Their Relation to the Best Approximations

Continued Fractions, Intermediate Fractions and Their Relation to the Best Approximations

Tempered monoids of real numbers, the golden fractal monoid, and the well-tempered harmonic semigroup

On the joint subword complexity of automatic sequences

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