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One Categorization of Microtonal Scales

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One Categorization of Microtonal Scales One Categorization of Microtonal Scales Luka Milinkovića, Branko Maleševićb, Dragana Pavlović –Šumaraca, Bojan Banjacb,c, Miomir Mijića a Department of Telecommunication, School of Electrical Engineering, University of Belgrade, Belgrade, Serbia b Department of Applied Mathematics, School of Electrical Engineering, University of Belgrade, Belgrade, Serbia c Department of Fundamentals Sciences, Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia Corresponding author is Branko Malešević ([email protected]) Luka Milinković ([email protected]), Branko Malešević ([email protected]), Dragana Pavlović –Šumarac ([email protected]), Bojan Banjac ([email protected]), Miomir Mijić ([email protected]) One Categorization of Microtonal Scales This study considers rational approximations of musical constant = , 3 which defines perfect fifth. This constant has been the subject of the numerous2 �2� studies, and this paper determines quality of rational approximations in regards to absolute error. We analysed convergents and secondary convergents (some of these are the best Huygens approximations). Especially, we determined quality of the secondary convergents which are not the best Huygens approximations – in this paper we called them non-convergents approximations. Some of the microtonal scales have been positioned and determined by using non-convergents approximation of music constant which defines perfect fifth. Keywords: 17edo, 31edo, categorization, continued fraction, convergents, microtonal scales, music constant, perfect fifth, secondary convergents 2010 Mathematics Subject Classification: 00A65, 30B70 Introduction Continued fractions belong to the field of mathematics that has influence on theory of music. Much about this topic has been written in 20th century. At that time, convergents and secondary convergents were considered as the part of the continued fraction theory by mathematician A. Ya. Khinchin (Khinchin 1964). We have decided to perform further research this field, and to check whether some fractions with similar optimal properties to convergents and secondary convergents of a continued fraction exists, and to consider their applications. One of the many important applications of the convergents and secondary convergents is in music theory. The most important intervals in music are octave, perfect fifth, and major third, which naturally fit to our hearing. They are based on simple relations 2:1, 3:2, and 5:4, respectively. Perfect fifth and major third have been analysed into more details in papers (Benson D. 2006; Kent J. T. 1986; Krushchev S. 2008.), while in the paper (Liern V. 2015) has been noted through the example 5.9, specific array of fractions which approximates music constant, which in turn defines perfect fifth = = 0,5849625007211 … (1) 3 2 �2� Rational approximations of the music constant , which are discussed in the paper (Liern V. 2015) in the example 5.9, are continued fractions. The goal of this study is to show that there are other approximations of the previously mentioned constant. In this paper, we will do the following: (1) Determine rational approximations of the first kind (Khinchin 1964), which are not continued fractions of the constant (2) Consider possibilities of categorizationβ of remaining rational approximations which are not the best rational approximations of the first kind of the constant . β Considerations of the problem 1) and 2) will solve the general problem of categorization of rational approximations for arbitrary real number. Also, the application of categorization will be considered, with the goal to determine microtonal scales. After the analysis of the problems 1) and 2) in the study, pseudo code which determines quality of rational approximation is given. Continued Fractions Continued fraction is expression given in the following form: = + (2) 1 0 1 1+ 1 2+ 1 + 3 ⋱ where , = {0} , ( 1). Throughout the paper we will 0 0 consider ∈onlyℕ cases ∈ ℕwhen ℕα is∪ positive ∈number,ℕ ≥ while whole analysis can be done analogously in case when α is negative number. Continued fractions are written in shorter form [ ; , , … , ]. For convergents 0 1 2 = ; , , … , , (3) �0 1 2 −1 � we introduce secondary convergents = [ ; , , … , , ] ′ (4) ′ ′ 0 1 2 −1 where 0 < < . ′ ′ ∈ ℕ ∧ Properties of Continued Fractions Basic properties of continued fractions are described through the following. Property 1. GCD( , q ) = 1; ( ) ( ) ∞ n ∈ ℕ0 ∧ ↗ ⟶ ∞ Property 2. lim = lim = 0 +1 1 ⟶∞ �+1 − � ⟶∞ ∙+1 Property 3. < < < < ; ( ) 0 2 3 1 0 2 ⋯ ≤ ≤ ⋯ 3 1 ∈ ℝ Property 4. < ; ( ) 1 2 � − � −1 α ∈ ℝ ∧ ∈ ℕ For the fraction value of absolute error in approximation of the constant α is ∆ given by = . (5) � − � Definition 1. For , the fraction is approximation of the first kind (the best ∈ ℝ Huygens approximation) if for every fractions stands (0 < ): ≤ < . (6) � − � � − � In the book (Krushchev S. 2008) approximations of the first kind are called the best Huygens approximations. Definition 2. For , the fraction is approximation of the second kind if for every ∈ ℝ fraction stands (0 < ): ≤ | | < | |. (7) − − Property 5 (Khinchin 1964). Every best approximation of the second kind is convergent. Every best approximation of the first kind can be convergent or secondary convergent. Remark. Sequence of approximations of the second kind consists only of convergents. We called them continued approximations (continued fractions) and labeled them as (c). Besides them, array of approximations of the first kind can contain some secondary convergents, and those fractions are called semi-continued approximations (semi- continued fractions) and are labeled with (sc). A. Ya. Khinchin called semi-continued approximations intermediate fractions [1]. Secondary convergents ′ are considered semi-continued approximation if ′ following is true < < ′ . (8) −1 ′ � − � � − � � − −1� Determination of convergents and secondary convergents can be done in one of the ways described in the papers (Cortzen A. 2011; Malešević B. 1988; Malešević B, Milinković L. 2014.). Property 6. If there exist semi-continued approximation[ ; , … , , ], with ′ 0 1 −1 lowest value , then fractions ; , … , , + ], for 0< < , are semi- ′ ′ ′ 0 1 −1 continued approximations too. � − Properties 1-5 are described based on (Khinchin 1964), while Property 6 is based on (Malešević B. 1988.) Example 1. We will use number π to show that between two continued fractions can be found fractions which can be better or worse approximation of number π when compared with existing continued fractions. Continued fraction of the number π is = 3.14159265359 … = [3; 7,15,1,292,1,1,1,2,1,1,3 … ]. (9) Number π and his initial rational approximations are shown on the Figure 1. First continued approximation is (A), and second continued approximation is (G). 3 22 1 7 First semi-continued approximation is (D), second semi-continued approximation is 13 4 (E), and third semi-continued approximation is (F). 16 19 5 6 There are another two rational approximations and (B and C) between two 6 9 2 3 consecutive continued approximations. For denominator 2 numerator is determined as 6 = 2 , while for denominator 3 numerator is determined as 9= 3 . These two fractions⌊ ⌋ are not better approximations of number π then first fraction,⌊ ⌋. They are 3 1 duplicates of . These two fractions in section Defining Categories are called multiples. 3 1 Figure 1 is showing first continued fraction (A) that represents the worst approximation (with the biggest absolute error) of number π. Every next semi-continued or continued approximation (from D to G) has diminishing absolute error while tends to be closer towards real value of π. Please note, on Figure 1 the values of fraction are shown on ordinate and the values of denominators of fractions are shown on abscissa. Figure 1. Rational approximation of the number π In this paper, fractions that are neither continued nor semi-continued are called non-continued approximations (non-continued fractions). Based on previous consideration, non-continued fractions are second convergents which are not the best Huygens approximations. For this example, non-continued fractions are and (H 25 28 8 9 and I), Figure 1. ■ Convergents and secondary convergents are being considered and applied in many different fields, among others in following: • Music (Barbour J. M. 1948; Benson D. 2006; Carey N, Clampitt D. 1989; Douthett J, Krantz R. 2007; Hall R. W, Josi K. 2001; Haluska J. 2003; Kent J. T. 1986; Krantz R. J. 1998; Krantz R. J, Douthett J. 2001; Krushchev S. 2008; Liern V. 2015; Noll T. 2006; Savage M 2014; Schechter M. 1980; Smoyer L. 2005) • Astronomy for approximation of various constants related to universe, such as tropical year, and synodic moon (Malešević B, Milinković L. 2014), • Mathematics for approximation of math constants such as Euler’s number e, then π and others (Malešević B. 1988; Cortzen A. 2011.) • Mechanic at Antikythera Mechanism (Malešević B, Milinković L. 2014; Popkonstantinović P, Obradović R, Jeli Z, Mišić S. 2014;), • Cryptography at RSA algorithms and in generation of random numbers (Malešević B, Milinković L. 2012) etc. Non-continued Fractions and Their Properties In this paper we consider the best rational approximations of the real number α in the following form (10) where , and denominator are defined with = for = 1,2, … , . Now we∈ ℕ form an array ℕ = = 1,2, … (11) � ∶ � where + 0,5 , 2 = = ( ) = (12) , = 2 ⌊ ∙ ⌋ ⌊ ∙ ⌋ ≠ ∧
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