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?

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 Curious Generalized Fibonacci Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2588
Author(s):  
Jose L. Herrera ◽  
Jhon J. Bravo ◽  
Carlos A. Gómez
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Numbers

A generalization of the well-known Fibonacci sequence is the k−Fibonacci sequence whose first k terms are 0,…,0,1 and each term afterwards is the sum of the preceding k terms. In this paper, we find all k-Fibonacci numbers that are curious numbers (i.e., numbers whose base ten representation have the form a⋯ab⋯ba⋯a). This work continues and extends the prior result of Trojovský, who found all Fibonacci numbers with a prescribed block of digits, and the result of Alahmadi et al., who searched for k-Fibonacci numbers, which are concatenation of two repdigits.

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 Some Properties of Generalized Fibonacci Numbers: Identities, Recurrence Properties and Closed Forms of the Sum Formulas ∑nk=0 xkWmk+j

2021 ◽  
pp. 11-38
Author(s):  
Yüksel Soykan
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Generalized Fibonacci Sequence ◽  
Summation Formulas ◽  
Special Cases

In this paper, closed forms of the summation formulas ∑nk=0 xkWmk+j for generalized Fibonacci numbers are presented. As special cases, we give summation formulas of Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas numbers. We present the proofs to indicate how these formulas, in general, were discovered. Of course, all the listed formulas may be proved by induction, but that method of proof gives no clue about their discovery. Moreover, we give some identities and recurrence properties of generalized Fibonacci sequence.

Generalization of an Identity Involving the Generalized Fibonacci Numbers and Its Applications

Integers ◽  
2009 ◽  
Vol 9 (5) ◽  
Author(s):  
D. G. Mohammad Farrokhi
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Sufficient Condition ◽  
Generalized Fibonacci Numbers ◽  
Generalized Fibonacci Sequence

AbstractWe first generalize an identity involving the generalized Fibonacci numbers and then apply it to establish some general identities concerning special sums. We also give a sufficient condition on a generalized Fibonacci sequence {

scholarly journals On the problem of Pillai with k-generalized Fibonacci numbers and powers of 3

2020 ◽  
Vol 16 (07) ◽  
pp. 1643-1666
Author(s):  
Mahadi Ddamulira ◽  
Florian Luca
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Numbers ◽  
Generalized Fibonacci Sequence ◽  
Tribonacci Numbers

For an integer [Formula: see text], let [Formula: see text] be the [Formula: see text]-generalized Fibonacci sequence which starts with [Formula: see text] (a total of [Formula: see text] terms) and for which each term afterwards is the sum of the [Formula: see text] preceding terms. In this paper, we find all integers [Formula: see text] with at least two representations as a difference between a [Formula: see text]-generalized Fibonacci number and a power of [Formula: see text]. This paper continues the previous work of the first author for the Fibonacci numbers, and for the Tribonacci numbers.

scholarly journals Motzkin’s Maximal Density and Related Chromatic Numbers

2018 ◽  
Vol 13 (1) ◽  
pp. 27-45
Author(s):  
Anshika Srivastava ◽  
Ram Krishna Pandey ◽  
Om Prakash
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Chromatic Numbers ◽  
Distance Graphs ◽  
Maximal Density ◽  
Generalized Fibonacci Numbers ◽  
Generalized Fibonacci Sequence ◽  
Odd Order ◽  
Upper Density ◽  
Positive Integers

Abstract This paper concerns the problem of determining or estimating the maximal upper density of the sets of nonnegative integers S whose elements do not differ by an element of a given set M of positive integers. We find some exact values and some bounds for the maximal density when the elements of M are generalized Fibonacci numbers of odd order. The generalized Fibonacci sequence of order r is a generalization of the well known Fibonacci sequence, where instead of starting with two predetermined terms, we start with r predetermined terms and each term afterwards is the sum of r preceding terms. We also derive some new properties of the generalized Fibonacci sequence of order r. Furthermore, we discuss some related coloring parameters of distance graphs generated by the set M.

scholarly journals Generalized Fibonacci Numbers, Cosmological Analogies, and an Invariant

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 200
Author(s):  
Valerio Faraoni ◽  
Farah Atieh
Keyword(s):  
General Relativity ◽  
Fibonacci Numbers ◽  
Friedmann Equation ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Numbers ◽  
Isotropic Cosmology ◽  
Spatially Homogeneous

Continuous generalizations of the Fibonacci sequence satisfy ODEs that are formal analogues of the Friedmann equation describing a spatially homogeneous and isotropic cosmology in general relativity. These analogies are presented together with their Lagrangian and Hamiltonian formulations and with an invariant of the Fibonacci sequence.

SOME ALGEBRAICAL, COMBINATORIAL AND ANALYTICAL PROPERTIES OF PARAGRASSMANN VARIABLES

2005 ◽  
Vol 20 (20n21) ◽  
pp. 4797-4819 ◽  
Author(s):  
MATTHIAS SCHORK
Keyword(s):  
Differential Equations ◽  
Fibonacci Numbers ◽  
Delta Function ◽  
Geometrical Interpretation ◽  
Generalized Fibonacci Numbers ◽  
Analytical Properties ◽  
Witten Index ◽  
Grassmann Variables

Some algebraical, combinatorial and analytical aspects of paragrassmann variables are discussed. In particular, the similarity of the combinatorics involved with those of generalized exclusion statistics (Gentile's intermediate statistics) is stressed. It is shown that the dimensions of the algebras of generalized grassmann variables are related to generalized Fibonacci numbers. On the analytical side, some of the simplest differential equations are discussed and a suitably generalized Berezin integral as well as an associated delta-function are considered. Some remarks concerning a geometrical interpretation of recent results about fractional superconformal transformations involving generalized grassmann variables are given. Finally, a quantity related to the Witten index is discussed.

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