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fractionizer: Allows to approximate the fractional numbers in the range [0.005, 1] with sum of 2 or 3 unit fractions of special types.

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This version is deprecated.

Inspired by the ancient Egyptian mathematics. See the ideas description and formulation in the related draft paper.

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Change log CHANGELOG.md
Dependencies base (>=4.13 && <5) [details]
License MIT
Copyright Oleksandr Zhabenko
Author Oleksandr-Zhabenko
Maintainer oleksandr.zhabenko@yahoo.com
Uploaded by OleksandrZhabenko at 2023-04-22T11:39:26Z
Revised Revision 1 made by OleksandrZhabenko at 2023-04-22T11:41:13Z
Category Math, Music, Sound, Language
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Readme for fractionizer-0.6.1.1

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Mathematical problem solution

Let's consider the problem of decomposition (approximation) of fractions that lie in the range [0.005, 2/3) by the sum of two fractions with 1 in the numerators (so called unit fractions) and, possibly, natural, unequal numbers in the denominators.

We will consider one of the two denominators to be a natural number, smaller than the other number in the denominator, and we will look for such a pair of numbers that, when rounded to natural numbers, will give the minimum absolute error of the resulting approximation. Mathematically, this leads to a problem and its further solution below.

Finally, we can decompose (approximate) the fraction in the range [0.005, 1] by the sum of 2, 3, 4, 5 or more unit fractions, with likely different denominators, just as like in the Egyptian mathematics.

In relation to music, this means that we can create rhythmic patterns using these unit fractions by treating the 1 as the music bar (notes between two consequtive barlines), but this leads to patterns that use irrational (non-dyadic) music meters and together to slightly different (hopefully, not perceptible) from 1 summary duration.

The tests show that there are some numerical instability for the numbers with more digits after the dot. So we use just 3 digits as correct after the dot, but the errors are calculated with respect to the original value with all the digits included.

More is by the link in the package description.

Relation to music rhythm and meter

It can be used to produce music by approximation of the meter. This leads to interesting structures.

It also is directly connected to the irrational time signatures in music.

Acknowledgements

Author would like to support the foundation Gastrostars and its founder Emma Kok. The founder inspired him to conduct such a research. Besides, the author is grateful to the Hackage website for publishing the Haskell code related to the research.

On the 2023-04-19 there is the founder's namesday, the memory of St. Emma of Lesum or Emma of Stiepel (also known as Hemma and Imma). If you would like to share some financial support, please, contact the foundation using the URL:

Contact Foundation GASTROSTARS