Christiaan Huygens

An artistic investigation into intrinsic interval structures in Nicola Vicentino's 2nd Tuning of 1555. The essay first uses a purely musical typography and presents specific characteristics of the tuning system by means of musical symbols, and then applies these findings creatively in five composed studies for Vicentino's Arciorgano. For scholarly purposes, a commentary is included at the end to make the musical essay easier to read and understand.

Baroque Music History Topic:

My topic will uncover the thoughts of Pythagoras and theories of intervallic relationships. I will research different articles on Pythagoras and also other formulas and topics related to his tuning systems and other musical theories.

Galileo’s Parabola describing the projectile motion passed through hands of all scholars of the classical mechanics. Therefore, it seems to be impossible to bring to this topic anything new. In our approach we will observe the Galileo’s Parabola from Pappus’ Directrix, Apollonius’ Pedal Curve (Line), Galileo’s Empty Focus, Newton’s Evolute, Leibniz’s Subtangent and Subnormal, Ptolemy’s Circle (Hodograph), and Dürer-Simon Parabola. For the description of events on this Galileo’s Parabola (this conic section parabola was discovered by Menaechmus) we will employ the interplay of the directrix of parabola discovered by Pappus of Alexandria, the pedal curve with the pedal point in the focus discovered by Apollonius of Perga (The Great Geometer), and the Galileo’s empty focus that plays an important function, too. We will study properties of this MAG Parabola with the aim to extract some hidden parameters behind that visible parabolic orbit in the Aristotelian World. For the visible Galil...

Recent scholarship suggests that the music of Austrian composer Georg Friedrich Haas can be understood as a dramatic confrontation between “clashing harmonic systems” (Hasegawa, 2015). Building on this observation, the present article focuses on Haas’s recent endeavors in the genre of live electronic music, showing how the composer deploys a relatively straightforward technical procedure—the reinjection loop,orthe delayed playback of recorded sound at various speeds—to juxtapose different modes of pitch organization, including twelve-tone equal temperament, ultrachromatic microtonality, and just intonation. After surveying the composer’s relationship to these historical idioms, the article presents detailed analyses of two pieces (Ein Schattenspieland String Quartet No. 4) to illustrate how the reinjection loop operates at multiple registers simultaneously, bringing the performers into contact with their immediate past, while also bringing Haas into dialogue with the shadows of
his own compositional predecessors.

Historically, Microtonality has been an important component of Karnatic Music, with the microtonal sruthi as the basic unit. Different authorities have ascribed different numbers of sruthis to each octave, with each raga taking different sruthis. With the growing popularity of the 12-semitone equitemperamental tuning system-originally meant for fixed pitch instruments of Western Music-there is pressure on performers to conform to these computed pitches instead of the natural pitch of each raga. Further, the absence of a notation system that can capture both microtonality as well as gamakas makes preservation of microtonality challenging. This paper reviews microtonality in Karnatic music, briefly traces the evolution of the 12-semitone octave and the challenges encountered in tuning fixed pitch instruments, and discusses the need to preserve microtonality of Karnatic music.

Neste artigo, apresento uma análise de ilustrações de óptica presentes em materiais de grande repercussão publicados entre o final do século 17 e o final do século 18. Meu objetivo é mostrar que a imagética da óptica fundamentada em um viés geométrico, como ainda encontramos nos dias atuais, já estava presente neste período de intensos estudos sobre a luz e de desenvolvimento de instrumentos ópticos como o microscópio e o telescópio. Espera-se que esta análise possa contribuir com novas abordagens da óptica em contextos de ensino.

relationship. Ives supported Carter’s musical pursuits as a young man and remained a guiding influence throughout his career. As Jan Swafford writes, however, “Carter, whose mature music would owe a great deal to Ives... would pay back his mentor with a baffling mixture of admiration, advocacy, and cold repudiation ” (Swafford 1998: 334). Although he eventually softened his early criticism of Ives and acknowledged his musical debts, Carter was consistently puzzled by the stress in Ives’s music on musical borrowing—i.e., the procedures by which a composer includes material from or refers to pre–existing musical pieces in the context of an original work.1 As Carter put it, Ives’s reliance on quotations accounted for “a disturbing lack of musical and stylistic continuity ” (Perlis 1974: 145). On another occasion, Carter described Ives’s inclusion of popular songs and hymns as “constantly perplexing ” (Edwards 1971: 63).2 In view of these and numerous other statements, it is equally per...

As we take a look at the young generation of composers in Poland, one of the bright lights is Adam Porębski. As he underlines, his music can be described by such words as: theatralisation, interdisciplinary, micromelodies and algorithm. In one of the latest compositions entitled „Iluzje” (Ilusions, 2020) for symphonic orchestra, he used so-called technique of controlled performatic gestures that he invented. It was important for the composer to decide how the musicians would move during the performance, what would be their gestures. One of the other aspects of „Iluzje” is that the music is based on microtonal scales: a 17-pitch-row and 16-pitch-row. These scales are a result of usage of mathematical cycles. Although the composition might sound (and look) like a piece, where orchestra has a lot of freedom, there are a lot of processes that were taken into the consideration before the piece was composed.
In the article the ways Porębski uses microtones in „Iluzje” are shown as well as how they help to create an illusion of freedom and why these pitches were chosen. Both the problem of realization of proposed musical material by the orchestra and the way the composition (and especially microtones) is notated are discussed.

Our workshop aims to present the tangent line with the didactical use of historical sources from Greek mathematics and Galileo Galilei. The discussion starts with an analysis of the definition of the tangent line in Greek mathematics in the case of the circle, from Euclid's Elements. Then a method to draw the tangent line at a point on a parabola is presented. The description is taken from the Fourth Day of Discorsi e dimostrazioni matematiche intorno a due nuove scienze by Galileo Galilei (1638). Galilei's approach comes from Greek mathematics and is different from the algebraic one usually used at school. An application of this geometrical technique to high school exercises is carried out, with examples from textbooks.

In 2009, an unusual seventeenth-century table clock by Parisian clockmaker Isaac Thuret came to light in a French auction room. The unique shape of the case and empty holes on the movement plates strongly suggested that the clock had originally been made to demonstrate the then-new technology of the spiral balance spring. Through study of the clock, its restoration, seventeenth century sources, and recent scholarship, the article presents the dispute over patent rights to the spiral balance spring and opens a discussion on methodologies for dating religieuse clocks of the 1670s.

Abstract
The article deals with the specific interval relations in Lithuanian folk melodies before the 1940s, which may preserve archaic tuning in the vocal and instrumental tradition. A study of archival sound recordings of Lithuanian folk songs and their instrumental versions was carried out in order to determine interval distances and possible tone deviations compared to natural harmonic series and 12-TET. For this purpose, regularly recurring scale degrees corresponding to the particular tones and microtonal overtones of the natural harmonic series were identified. It was found that the interval distances in the scale are very close to the interval ratios of the natural harmonic series (3:2, 4:3, 5:4, 9:8, 21:16, 11:8). By comparing the interval distances of the tones in the scale, the dominant interval distances in Lithuanian folk melodies were determined in order to generate a commonly used interval distance for all vocal and instrumental music. In a separate study, a formula for recurrent/popular tones and intervals was derived. This research revealed the prevalence and use of microtonal interval structures in the tuning of folk music instruments, especially in the multi-part sutartinės tradition and in the performance of monodies, which redefine the interval structure and tuning of the traditional scale. The other aim of this study was to integrate the results into a compositional system to preserve or transform the original sonority of archaic sutartinė scales. Thus, for the composition "L'Astéroïde B-612" for two violins, a 19-tone scale was constructed; the transformation of the archaic microstructure in one piece appears as an integrated four-tone scale of a sutartinė. The composition "No Titre" for shakuhachi and string quartet is made up of 5-tone scales of three sutartinės, a microstructure of 9 tones (4 tempered and 5 microtones) integrated into a 21-tone scale. For the scale of the piece "Avalanche" for orchestra, a model of natural overtones was chosen; a progression of 5 harmonics of the wooden trumpet ensemble, a 24-tone scale was constructed by shifting micro-sonorities on the basis of vocal and instrumental sutartinės. The article presents the strategies for shaping the archaic microtonal sound structures via music composition.
Keywords: microtones, microtonal overtones, interval relations, Lithuanian folk songs, Vytautas Germanavicius, natural harmonic series, interval distances, 12-TET, multi-part chants sutartinė.
Keywords: microtones, microtonal overtones, interval relations, Lithuanian folk songs, Vytautas Germanavicius, natural harmonic series, interval distances, 12-TET, multi-part chants sutartinė.

This paper describes a 12-note musical scale generated from the ratio of consecutive prime numbers that is used as the basis for the author’s electronic composition The Music of the Primes. The scale is a type of extended just intonation, a tuning system whose intervals are derived from the harmonic (overtone) series and uses partials above the first six partials. The ratio of consecutive primes sequence makes a potentially infinite number of prime ratios available to the composer.

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Harry Partch’s microtonal tuning and just-intonation concepts have not only affected American microtonalists such as Lou Harrison and Ben Johnston, but also European composers interested in experimenting with microtones, including Georg Friedrich Haas and Wolfgang von Schweinitz. Yet no other European figure has absorbed Partch’s ideas more than German composer Manfred Stahnke, who came to the United States in 1979 to study at the University of Illinois Urbana-Champaign with Partch’s friend and supporter Ben Johnston. This essay approaches Partch’s rebellion not as an isolated paradigm, but in relation to Stahnke’s aesthetic and music—who has also sought to expand the intonational and tuning paradigms of Western art music, whom Partch’s microtonal theories have influenced.
As Partch did, Stahnke has rejected the dominance of twelve-tone equal temperament, grappling with just intonation, as well as non-Western and ancient tuning systems. The analysis of the aesthetic decisions, compositional procedures, and microtonal structures in Partch’s and Stahnke’s works articulates the strong link between both composers. In their works, microtonality goes beyond functioning as a mere formative element; it becomes a means to mediate their cultural discourse on the limited scope of the commercialized Western art music. I demonstrate that Partch’s theory of microtonality—especially his concepts of tonality diamond, otonality, and utonality have influenced Stahnke’s opera Der Untergang des Hauses Usher (1981) based on a short novel by Edgar Allen Poe, as well as his instrumental pieces Diamantenpracht (2005) and Partch Zither (2007).

This work evaluates four violins from three distinct manufacturers, notably Eurostring, Stentor, and Suzuki, using a scientific approach. Eurostring1 and Eurostring2 were the names given to the two Eurostring units. The purpose of this study is to identify elements in various violins that could be used as tools for selecting a pleasantsounding violin by having them classified by a professional violinist. The signal’s time varying frequency was evaluated using a frequency spectrum and a time frequency plane, and the combination of frequency spectrum and time frequency domain is utilised. PicoScope oscilloscopes and Adobe Audition version 3 were used to record the acoustic spectra in terms of time and frequency. The time frequency plane is identified, and time frequency analysis (TFA) is produced by Adobe Audition spectrograms. The sound was processed in order to generate Fast Fourier Transform analysis: Fourier spectra (using PicoScope) and spectrograms (using Adobe Audition). Fourie...

Is light wave? Or particle?
Haven’t we often pondered over how Quantum Physics is rather Philosophy than mathematical derivation?
Is it Philosophy or Physics?
Haven’t you often pondered over why and how Compton effect bears a striking resemblance with Photoelectric Effect, that we find remarkable, while exploring these two phenomena? 
Here comes an elaborate review of Wave-Particle Duality with a Philosophical outlook.
The key periods in the history of Physics research, principal historical turning points, along with the key topics and subtopics have been reviewed and discussed here. Newton’s model versus Huygens’ model, Thomas Young’s double-slit experiment, Blackbody Radiation, Ultraviolet Catastrophe, Photoelectric Effect, Copenhagen Interpretation, wave function collapse, Compton Scattering, concept of photon, de Broglie hypothesis of “Matter Wave” and Niels Bohr’s Complementarity Principle have been reassessed. Link between Einstein’s mass-energy relation and Planck’s equation has been analyzed with a present-day Philosophical standpoint, rather than intricate mathematical approach.

In this paper it is shown the performing of an optical transform to state the scalar diffraction in the formulation of the wavelet transform and the 'wave equations'. From there, a bridge is build between equations of spherical waves presented in 1678 by Huygens and the continuous wavelet transform. For such a purpose, wavelets are introduced that meet the principles of waves and the properties of wavelets. The following equations are applied in solution to show a correspondence between the Huygens-Fresnel diffraction and the wavelet transform.

In this article, we will present how we can create, organize and understand pitches of Just Intonation using 1) the theory of sieves (XENAKIS, 1992) to filter the numbers of the harmonic series, 2) the decomposition by prime-numbers (EXARCHOS et al., 2011) applied to the numbers resulted from the sieve, 3) the concept of harmonic identities by Johnston (2006), and finally, 4) the Combinations-Product Set (CPS) by Erv Wilson (NARUSHIMA, 2018), that allows us to see the relationships between harmonic identities. When we understand that only prime numbers generate new harmonic identities in the interval sense (JOHNSTON, 2006), and odd-numbers create just a new pitch-class from staking prime numbers, we can apply the prime-decomposition for all the numbers of one sieve and claim that: all the non-prime numbers can be understood how intersections between distinct harmonics series. Based on Exarchos' (2011) analysis of sieves, this idea can help us find a way to use the sieves of Xenakis in the Just Intonation context. Complementary with this compositional idea, we will present two OpenMusic libraries developed to help the compositional process.