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In this article, I try to show how the operation of compounding of ratios as performed with the method of Proposition VIII.4 of Euclid's Elements, has possibly taken shape from experiments on the monochord. This fact, together with the systematic and exclusive use of Proposition VIII.4 in the harmonic field, strengthens the thesis concerning the connection between harmonic theory and Book VIII. A misinterpretation by Zarlino, the most important music theorist of the Renaissance, will help us explain how the musical model may have been partially responsible for the disconnection between compounding and multiplication present in the Elements in spite of the link established by Proposition VIII.5. [ABSTRACT FROM AUTHOR]

The article offers information on the consideration of finding the characterization for a finite Kripke frame in the ambit of modal logic. It says that the modal logic corresponds to the modal algebras such as Boolean algebras. It states the characterization for finite Kripke frame by finding an axiomatization of its theory. Moreover, the finite transitive frame allows a minimal characterization over finite Kripke frames.

The article discusses the inadequate relational semantic for the until operator. It states that the consideration of modal logics with the binary operator Until has shown that the continuum of consistent U-logics without Kripke frames exists and each U-logic whose class of order does not contain the finite frame property. It says that the logics with binary operators Until and Since (US-logics) is complete in terms of its class of US-general frames.

The article offers information on the study to determine the number of non-equivalent modalities in the different modal systems. It mentions that semantic test help in determining the number and the chart of the non-equivalent irreducible modalities. It offers information on Kripke model with formula and propositional variable.

The article discusses a study on the atoms in the equational class of boolean algebras' subvarieties. It analyzes an open problem regarding α-atomicity's sufficient and necessary conditions for the extension of least normal modal logic. It also investigates the α-atomicity of equational class of boolean algebras' extensions, which do not have finite model property.