Search results for "Pythagorean"
showing 10 items of 13 documents
Fuzzy tuning systems: the mathematics of musicians
2005
We present some mathematical properties which determine tuning methods. We introduce the concept of fuzzy tuning systems and we analyze four of the systems coexisting within the current orchestras: Pythagorean, Just Intonation, Holder's and Equal Temperament systems. We show that the theoretical and practical tuning methods are the same. We introduce the idea of compatibility between tuning systems and we give some sufficient conditions to determine an appropriate number of notes into which the octave must be divided.
On an Inequality for Trigonometric Polynomials In Several Variables
1990
Publisher Summary This chapter presents trigonometric polynomials in n variables. Using the methods of approximation theory, an inequality can be extended to almost periodic functions and to still more general classes of functions as in the case for Bohr's inequality. However, no analogous result exists in the case of two variables. For the solution of problems containing small divisors, the estimate has to be completed by theorems concerning the best approximation of holomorphic functions by trigonometric polynomials in polystrips. The chapter also presents equations to provide an estimate for a differential operator.
Music and Medicine
2020
The relationship between music and medicine involves the notions of affinity between the human body and musical structures, relief, catharsis and therapy. The Homeric poems attest to the use of healing songs (paeans) and spells (epaoidai). The early Pythagoreans used musical catharsis for both the soul and the body. The doctrine of musical ēthos (whose main source is Plato) presupposes a relationship between music and character based on mimēsis, also establishing a link between therapy and ēthos. According to Aristotle, melodies performed in the rites are able to arou-se the emotions and purify from their excesses (the same dynamics appear in Theophrastus). The musical notions first detecta…
Hermann weyl, the reluctant revolutionary
2003
“Brouwer – that is the revolution!” – with these words from his manifesto “On the New Foundations Crisis in Mathematics” (Weyl 1921) Hermann Weyl jumped headlong into ongoing debates concerning the foundations of set theory and analysis. His decision to do so was not taken lightly, knowing that this dramatic gesture was bound to have immense repercussions not only for him, but for many others within the fragile and politically fragmented European mathematical community. Weyl felt sure that modern mathematics was going to undergo massive changes in the near future. By proclaiming a “new” foundations crisis, he implicitly acknowledged that revolutions had transformed mathematics in the past, …
Correcting ēthos and Purifying the Body. Musical Therapy in Iamblichus’ De vita pythagorica
2015
The tradition relating to the Pythagoreans and music therapy is most widely attested in two Neoplatonic works, Porphyry’s The Life of Pythagoras, and Iamblichus’ On the Pythagorean Way of Life. Although the chronological distance from the early Pythagoreans makes their accounts controversial, they offer interesting evidence on the beneficial effects of music. Iamblichus, whose work will be focused on in this paper, describes the effects of music on health through the notion of catharsis, which he often links with musical ēthos. The latter is not in fact attested before Plato, but Iamblichus, presenting Pythagoras in Platonic terms, emphasizes the importance he gives to the improvement of th…
A Characterization of Quintic Helices
2005
A polynomial curve of degree 5, @a, is a helix if and only if both @[email protected]^'@? and @[email protected]^'@[email protected]^''@? are polynomial functions.
An isoperimetric type problem for primitive Pythagorean hodograph curves
2012
An isoperimetric type problem for primitive Pythagorean hodograph curves is studied. We show how to compute, for each possible degree, the Pythagorean hodograph curve of a given perimeter enclosing the greatest area. We also discuss the existence and construction of smooth solutions, obtaining a relationship with an interesting sequence of Appell polynomials.
On the construction, comparison, and exchangeability of tuning systems
2015
The aim of this article is to describe mathematically different tuning systems, to study their mathematical properties, and to propose a construction allowing their comparison. In order to reach these goals, we introduce a concept of similarity between tuning systems and then we provide two sufficient conditions for the particular case in which a tuning system generated by an interval and a circulating temperament are compared. Finally, we show by means of an example that, for two tuning systems to be exchangeable, some well-known results determining the suitable number of notes per octave are not enough.
Musica, catarsi ed eunomía. I Pitagorici in Magna Grecia e l'uso terapeutico del peana.
2011
Accanto allo sviluppo del pensiero filosofico-scientifico e al sorgere di nuovi ordinamenti politici in diverse poleis, la presenza delle comunità pitagoriche in Magna Grecia sembra aver dato impulso anche a livello religioso -ad esempio, con l’affermarsi del culto di Apollo- e musicale. A tale riguardo, si propone una riflessione sulla musicoterapia presso i Pitagorici antichi, e in particolare sull’uso da parte loro del peana, come emerge ad esempio nella Vita Pythagorica di Giamblico (§ 110: la catarsi “primaverile” di gruppo) e nella Vita Pythagorae di Porfirio (§ 32). Il peana, che nella tradizione appare rivolto soprattutto a calmare l’ira (in particolare quella divina, come emerge da…
The Pythagoreans and the Therapeutic Effects of the Paean between Religion, Paideia and Politics
2016
The interest of the Early Pythagoreans in musical speculation appears in literary sources as strictly linked with religion and education. The use of paeans for healing and calming both rage and anger among the Pythagoreans (see for instance Iamblichus, De vita pythagorica 110; Porphyrius, Vita Pythagorae 30) shows that catharsis was meant within such groups as a “purification” from every kind of excess in which religion, medicine and ethics were blended together in order to provide a harmonious order within the individuals. Music and musical education in Pythagorean communities had also a “political” role, since they were intended to foster social order.