scholarly journals Psychoacoustic Properties of Fibonacci Sequences

10.14311/1027 ◽  
2008 ◽  
Vol 48 (4) ◽  
Author(s):  
J. Sokoll ◽  
S. Fingerhuth
Keyword(s):  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Binary Sequence ◽  
Fibonacci Sequence ◽  
Almost Periodic ◽  
Listening Tests ◽  
Spectral Components ◽  
Self Similar ◽  
Set Up ◽  
Fibonacci Sequences

1202, Fibonacci set up one of the most interesting sequences in number theory. This sequence can be represented by so-called Fibonacci Numbers, and by a binary sequence of zeros and ones. If such a binary Fibonacci Sequence is played back as an audio file, a very dissonant sound results. This is caused by the “almost-periodic”, “self-similar” property of the binary sequence. The ratio of zeros and ones converges to the golden ratio, as do the primary and secondary spectral components intheir frequencies and amplitudes. These Fibonacci Sequences will be characterized using listening tests and psychoacoustic analyses. 

scholarly journals Musica fibonacciana: Aesthetic and practical approach

New Sound ◽  
2017 ◽  
pp. 70-90
Author(s):  
Rima Povilionienè
Keyword(s):  
Music Composition ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Laws Of Nature ◽  
Fibonacci Sequence ◽  
Number Sequences ◽  
Music And Mathematics ◽  
To Come ◽  
Composition Structure ◽  
And Mathematics

In the sphere of musical research, the intersection of two seemingly very different subject areas-music and mathematics is in essence related to one of the trends of music-attributing the theory of music to science, to the sphere of mathematica. It is regarded the longest-lasting interdisciplinary dialogue. The implication of numerical proportions and number sequences in the music composition of different epochs is closely related to this sphere. A significant role in creating music was attributed to the so-called infinite Fibonacci sequence. Perhaps the most important feature of the Fibonacci numbers, which attracted the attention of thinkers and creators of different epochs, is the fact that by means of the ratio between them it is possible to come maximally close to the Golden Ratio formula, which expresses the laws of nature. On a practical plane, often the climax, the most important part of any composition, matches the point of the Golden Ratio; groups of notes, rhythm, choice of tone pitches, a grouping of measures, time signature, as well as proportions between a musical composition's parts may be regulated according to Fibonacci principles. The article presents three analytical cases-Chopin's piano prelude, Bourgeois' composition for organ, and Reich's minimalistic piece, attempting to render music composition structure to the logic of Fibonacci numbers.

scholarly journals Generalized Fibonacci Numbers and Music

2018 ◽  
Vol 14 (1) ◽  
pp. 7564-7579
Author(s):  
Anthony G Shannon ◽  
Irina Klamka ◽  
Robert van Gend
Keyword(s):  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Numbers

Mathematics and music have well documented historical connections. Just as the ordinary Fibonacci numbers have links with the golden ratio, this paper considers generalized Fibonacci numbers developed from generalizations of the golden ratio. It is well known that the Fibonacci sequence of numbers underlie certain musical intervals and compositions but to what extent are these connections accidental or structural, coincidental or natural and do generalized Fibonacci numbers share any of these connections?

Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps

2011 ◽  
Vol 63 (3) ◽  
pp. 648-688 ◽  
Author(s):  
Sze-Man Ngai
Keyword(s):  
Golden Ratio ◽  
Iterated Function Systems ◽  
Spectral Dimension ◽  
Open Set Condition ◽  
Bernoulli Convolution ◽  
One Dimensional ◽  
Open Set ◽  
Iterated Function ◽  
Self Similar ◽  
Set Up

AbstractWe set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition.

Generalization of the Fibonacci Sequence to n Dimensions

1966 ◽  
Vol 18 ◽  
pp. 332-349 ◽  
Author(s):  
George N. Raney
Keyword(s):  
Free Semigroup ◽  
Jacobi Polynomials ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Natural Generalization ◽  
Unimodular Group ◽  
Natural Extensions ◽  
Generalized Fibonacci Sequences ◽  
Fibonacci Sequences

We introduce certain n X n matrices with integral elements that constitute a free semigroup with identity and generate the n-dimensional unimodular group. In terms of these matrices we define a certain sequence of n-dimensional vectors, which we show is the natural generalization to n dimensions of the Fibonacci sequence. Connections between the generalized Fibonacci sequences and certain Jacobi polynomials are found. The various basic identities concerning the Fibonacci numbers are shown to have natural extensions to n dimensions, and in some cases the proofs are rendered quite brief by the use of known theorems on matrices.

scholarly journals Phi-Bonacci Butterfly Stroke Numbers to Assess Self-Similarity in Elite Swimmers

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1545
Author(s):  
Cristiano Maria Verrelli ◽  
Cristian Romagnoli ◽  
Roxanne Jackson ◽  
Ivo Ferretti ◽  
Giuseppe Annino ◽  
...  
Keyword(s):  
Golden Ratio ◽  
National Level ◽  
Fibonacci Sequence ◽  
Long Distance ◽  
Front Crawl ◽  
Generalized Fibonacci Sequence ◽  
Time Harmonic ◽  
Elite Swimmers ◽  
Self Similar ◽  
Butterfly Stroke

A harmonically self-similar temporal partition, which turns out to be subtly exhibited by elite swimmers at middle distance pace, is formally defined for one of the most technically advanced swimming strokes—the butterfly. This partition relies on the generalized Fibonacci sequence and the golden ratio. Quantitative indices, named ϕ-bonacci butterfly stroke numbers, are proposed to assess such an aforementioned hidden time-harmonic and self-similar structure. An experimental validation on seven international-level swimmers and two national-level swimmers was included. The results of this paper accordingly extend the previous findings in the literature regarding human walking and running at a comfortable speed and front crawl swimming strokes at a middle/long distance pace.

scholarly journals On Third-Order Bronze Fibonacci Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2606
Author(s):  
Mücahit Akbiyik ◽  
Jeta Alo
Keyword(s):  
Generating Functions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Second Order ◽  
Matrix Representations ◽  
Third Order ◽  
The Third ◽  
Encryption And Decryption ◽  
Fibonacci Sequences

In this study, we firstly obtain De Moivre-type identities for the second-order Bronze Fibonacci sequences. Next, we construct and define the third-order Bronze Fibonacci, third-order Bronze Lucas and modified third-order Bronze Fibonacci sequences. Then, we define the generalized third-order Bronze Fibonacci sequence and calculate the De Moivre-type identities for these sequences. Moreover, we find the generating functions, Binet’s formulas, Cassini’s identities and matrix representations of these sequences and examine some interesting identities related to the third-order Bronze Fibonacci sequences. Finally, we present an encryption and decryption application that uses our obtained results and we present an illustrative example.

scholarly journals On the Sum of Powers of Two k-Fibonacci Numbers which Belongs to the Sequence of k-Lucas Numbers

2016 ◽  
Vol 67 (1) ◽  
pp. 41-46
Author(s):  
Pavel Trojovský
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Lucas Numbers ◽  
Initial Values ◽  
Lucas Sequence ◽  
Positive Integers

Abstract Let k ≥ 1 and denote (Fk,n)n≥0, the k-Fibonacci sequence whose terms satisfy the recurrence relation Fk,n = kFk,n−1 +Fk,n−2, with initial conditions Fk,0 = 0 and Fk,1 = 1. In the same way, the k-Lucas sequence (Lk,n)n≥0 is defined by satisfying the same recurrence relation with initial values Lk,0 = 2 and Lk,1 = k. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that Fk,n+1 + Fk,n−1 = Lk,n, for all k ≥ 1 and n ≥ 0. In this paper, we shall prove that if k ≥ 1 and $F_{k,n + 1}^s + F_{k,n - 1}^s \in \left( {L_{k,m} } \right)_{m \ge 1} $ for infinitely many positive integers n, then s =1.

scholarly journals Wavelet introscopy of human organism bionets

2021 ◽  
pp. 63-72
Author(s):  
Gennady M. Aldonin ◽  
◽  
Vasily V. Cherepanov ◽  
Keyword(s):  
Functional State ◽  
Human Body ◽  
Dynamic Models ◽  
Golden Ratio ◽  
The State ◽  
The Body ◽  
Human Organism ◽  
Fractal Structures ◽  
Self Similar ◽  
The Creation

In domestic and foreign practice, a great deal of experience has been accumulated in the creation of means for monitoring the functional state of the human body. The existing complexes mainly analyze the electrocardiogram, blood pressure and a number of other physiological parameters. Diagnostics is often based on formal statistical data which are not always correct due to the nonstationarity of bioprocesses and without taking into account their physical nature. An urgent task of monitoring the state of the cardiovascular system is the creation of effective algorithms for computer technologies to process biosignals based on nonlinear dynamic models of body systems since biosystems and bioprocesses have a nonlinear nature and fractal structure. The nervous and muscular systems of the heart, the vascular and bronchial systems of the human body are examples of such structures. The connection of body systems with their organization in the form of self-similar fractal structures with scaling close to the “golden ratio” makes it possible to diagnose them topically. It is possible to obtain detailed information about the state of the human body’s bio-networks for topical diagnostics on the basis of the wavelet analysis of biosignals (the so-called wavelet-introscopy). With the help of wavelet transform, it is possible to reveal the structure of biosystems and bioprocesses, as a picture of the lines of local extrema of wavelet diagrams of biosignals. Mathematical models and software for wavelet introscopy make it possible to extract additional information from biosignals about the state of biosystems. Early detection of latent forms of diseases using wavelet introscopy can shorten the cure time and reduce the consequences of disorders of the functional state of the body (FSO), and reduce the risk of disability. Taking into account the factors of organizing the body’s biosystems in the form of self-similar fractal structures with a scaling close to the “golden ratio” makes it possible to create a technique for topical diagnostics of the most important biosystems of the human body.

scholarly journals On Generalization of Fibonacci, Lucas and Mulatu Numbers

2020 ◽  
Vol 1 (3) ◽  
pp. 112-122
Author(s):  
Agung Prabowo
Keyword(s):  
Continued Fraction ◽  
Golden Ratio ◽  
Fibonacci Numbers ◽  
Lucas Numbers ◽  
Binet Formula

Fibonacci numbers, Lucas numbers and Mulatu numbers are built in the same method. The three numbers differ in the first term, while the second term is entirely the same. The next terms are the sum of two successive terms. In this article, generalizations of Fibonacci, Lucas and Mulatu (GFLM) numbers are built which are generalizations of the three types of numbers. The Binet formula is then built for the GFLM numbers, and determines the golden ratio, silver ratio and Bronze ratio of the GFLM numbers. This article also presents generalizations of these three types of ratios, called Metallic ratios. In the last part we state the Metallic ratio in the form of continued fraction and nested radicals.

scholarly journals On Fibonacci Numbers and its Applications

2019 ◽  
Vol 8 (12) ◽  
pp. 453-455
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Sequence

In this article, we explore the representation of the product of k consecutive Fibonacci numbers as the sum of kth power of Fibonacci numbers. We also present a formula for finding the coefficients of the Fibonacci numbers appearing in this representation. Finally, we extend the idea to the case of generalized Fibonacci sequence and also, we produce another formula for finding the coefficients of Fibonacci numbers appearing in the representation of three consecutive Fibonacci numbers as a particular case. Also, we point out some amazing applications of Fibonacci numbers.

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