Quadrivium

Description "History of Liberal Arts Education" is a survey seminar course in which students take up and discuss short selections of texts about liberal arts education from antiquity to the present. Co-lead by Professors Peterson & Sosa, each two-hour class session will consist of a short introduction to the reading selections by one or both professors, identifying questions which he or they propose for discussion for the rest of the class. They will then enter into conversation, inviting students to participate. As this summer's iteration of the course will be online, students must have a viable Internet connection and a computer with a webcam and microphone. This is a very reading-heavy courseabout 40 pages per classthat relies on student preparation and participation. Because of the condensed summer schedule, preparation before the course and over the weekends is essential. Students should come prepared to each class with some thoughts and questions about the readings. Questions of the Course What is the model for liberal arts education-philosophy, the highest art of which is dialectic-or rhetoric? What is the difference between these? What makes arts "liberal"? What is the "orator" and what is his relation to society? What is the role of faith, culture, and tradition in education? How has liberal arts education changed, and what were the reasons for those changes? Has Christianity changed our understanding of the liberal arts, and if so, how? What is the proper course of study? Does that curriculum change based on regime or circumstance? What has Renaissance humanism meant for our understanding of the liberal arts? How have the liberal arts come to be understood today? What challenges has liberal arts education faced throughout history? What are the perennial obstacles to liberal arts education? What is the relation between the liberal arts and civic education, character education, and advanced study?

En la Real Biblioteca del Monasterio de San Lorenzo de El Escorial
se conserva un texto manuscrito que recoge una traducción al castellano del De institutione musica de Boecio. Tanto el tipo de letra empleado, como cuestiones codicológicas, indican que este texto fue copiado en la segunda mitad del siglo xv. El tratado de Boecio es, sin ninguna duda, el referente teórico más importante para la música especulativa en el marco del quadrivium medieval y renacentista. En él, el autor tardorromano recoge las principales teorías helenísticas sobre ciencia armónica, siguiendo la estela de Nicómaco y Ptolomeo, y las sistematiza dentro de la disciplina música del quadrivium. La música para Boecio es una disciplina matemática que utiliza los procedimientos de la aritmética para demostrar «verdades» armónico-musicales. Por otra parte, esta traducción es la primera conservada a una lengua romance de este texto; las siguientes conocidas son dos traducciones al italiano realizadas ya en el último cuarto del siglo xvi. El manuscrito de El Escorial, por tanto, constituye un caso singular en la historia del texto boeciano. Presentamos un estudio preliminar del texto centrado en algunas cuestiones como: datación y posible origen de la traducción, diferentes manos intervinientes en la elaboración física del documento, traducción y transliteración de términos técnicos musicales.

PALABRAS CLAVE: Boecio, teoría musical, traducción, Biblioteca de El Escorial, quadrivium.

In the Real Biblioteca del Monasterio de San Lorenzo de El Escorial is preserved a Spanish translation of Boethius’ De institutione musica. The handwriting and other codicological issues suggest that the text was copied in the second half of the 15th century. Boethius’ treatise on music is the most important theoretical reference for speculative music of the medieval and renaissance quadrivium. In this text, the late-Roman author, following Nicomachus and Ptolemy, compiles the fundamental Hellenistic theories on harmonic science and systematizes them into the musica disciplina of the quadrivium. Music, according to Boethius, is a mathematical discipline which uses arithmetical procedures to prove musical-harmonic truths. This translation of Boethius’ De musica is the first one to a vernacular language that has come down to us, being the next ones two translations to Italian written in the last quarter of the sixteenth century. The El Escorial manuscript is, then, a singular case in the history of the Boethian text. In this article we present a preliminary study of the text focused on such issues as: dating, possible origin, different participating hands in the document elaboration, translation and transliteration of musical terms.

KEYWORDS: Boethius, music theory, translation, El Escorial Library, quadrivium.

The engraving of the Lab-Oratorium, in the Amphitheatre of Eternal Wisdom of the Paracelsian Heinrich Khunrath of Leipzig (1560-1605) - 'doctor of both medicines and faithful lover of theosophy'- is a well-known image for historians of early modern esotericism, but little has been said about the significance of the musical instruments in the foreground of the image. This article examines various references in Khunrath's writings that concern music and the related theme of harmony in the context of Khunrath's cabalistic and alchemical activities in his Oratory and Laboratory. It examines the influence of pythagorean ideas on Khunrath's theurgical practices, identifies the christian-cabalist source of the polyglot hymn that he links to one of the theosophical images on the table of his Oratory, and offers some reflections about the use of music in this cabalistic dimension of his work. Given that Khunrath is known above all as a practitioner of alchemy, the second part of this article considers several examples of music and song, beginning with a manuscript survival from Khunrath's work and ends with a brief examination of the most famous combination of alchemy and music, namely, the Atalanta fugiens of his admirer, Count Michael Maier (1568-1622).

This article deals with the philosophical idea of worldmaking pursued through techné, meaning the fusion of the technical means of artistic creation, theorizing, and analysis, but specifically occurring in a feedback loop involving introspection and computation (cybernetic phenomenology). The discussion ranges over a wide variety of topics, including (1) the history of music theory from Ancient Greece to the present (including a rebuttal to Daniel Chua’s (2001) account of music history); (2) emergent properties in music arising from self-organization (explored through the Ancient Chinese abstract strategy board game go); (3) the ontology of musical qualities (properties) and categories (including their relation to visual, tactile, and olfactory qualities and categories). 
Various repertoires, artists, and philosophies are referenced. The essay analyzes aspects of Ligeti’s Violin Concerto, Carter’s String Quartet No. 5, and Lachenmann’s Kinderspiel.  Connections to the author’s own previous analytical and theoretical work are also discussed in relation to his interactive algorithmic audio-visual works (such as Fluxations and FluxNOISations). The techné of these is discussed in terms of logistics as well as aesthetic influences, including Wagner, Liszt, Stravinsky, Schoenberg, Crawford, Babbitt, Carter, Xenakis, Parker, Reich, Ligeti, Truax, Oliveros, Kandinsky, Miro, Pollock, Pollock, Hofmann, Rothko, Louis, Frankenthaler, Moholy-Nagy, Newman, Richter, and Brakhage.  The essay touches on philosophical ideas of Bergson, Whitehead, Deleuze, Harman, and Bogost, and more fully engages the philosophies of Nelson Goodman and Hannah Arendt in connection with artistic creation as relating to the theorizing and analysis of artistic works.     

Contact the author at jmailman@alumni.uchicago.edu
Download the article through JSTOR: http://www.jstor.org/stable/10.7757/persnewmusi.54.1.0005
or http://tinyurl.com/zdvkllr

• Illo Humphrey | PhD (2004) | HDR (2014) : Université Paris X-Nanterre •
• Mediævalist | Musicologist | Proto-Philologist | Concert-Baritone •
• Trilingual Simultaneous Interpreter (French A • English A • German C) •
• Research Associate | EA 4593 CLARE - LaPRIL | Université Bordeaux Montaigne •

• Boethii De institutione arithmetica libri duo | Complete princeps proto-philological critical edition: Boethian Text, Glosses in clear Latin, Glosses in Latin stenography (tironian notes), notae sententiarum (cross-reference signs), original Carolingian  punctuation, etc. •

INTRODUCTION (French] •

1. Paris, Bibliothèque nationale de France, Fonds latin 14064 (IXe s.), f. 1-84
(Boethii De institutione arithmetica libri duo : texte, gloses, signes de renvoi) •

Le manuscrit Paris, Bibliothèque nationale de France, Fonds latin 14064 (olim Saint-Germain-des-Prés 302, puis N. 781), est datable de la première moitié du IXe siècle. Il s’agit d’un recueil scolaire ordinaire sans décoration qui a servi pour l’enseignement du premier des quatre volets du quadruvium, en l’occurrence ars arithmetica. Il contient uniquement le traité Boethii De institutione arithmetica libri duo, dont les annotations marginales et interlinéaires furent copiées également dans la première moitié du IXe siècle. Au feuillet 84, l’ex libris du XVIe siècle[?] indique sans équivoque sa provenance de l’abbaye Saint-Germain-des-Prés : Liber sancti Germani de Pratis ; son origine précise, en revanche, est incertaine (peut-être Corbie [?]). Le manuscrit a été abondamment glosé en langue latine ; environ 80 % de ces gloses ont été consignées en sténographie latine (notes dites “tironiennes”). L’ensemble des annotations fut écrit, semble-t-il, par deux mains différentes contemporaines, ayant utilisé deux qualités d’encre différentes. En raison de son ancienneté, son importance dans la tradition textuelle de ce traité au IXe siècle, puis de la quasi intégralité de son texte (les 32 Capitula au début du Livre I furent omis), ce manuscrit a été choisi comme source principale pour cette édition proto-philologique intégrale. Les gloses sont accompagnées de nombreux signes de renvoi, lesquels figurent aux mêmes endroits dans l’édition que dans le manuscrit principal •

2. Descriptif et état physique du codex •

La reliure du codex, dont la date précise est incertaine (XVIe-XVIIe s. [?]), a été faite au moyen de trois double nerfs apparents en ficelle ; elle est dotée de tranchefiles simples sur âme de cuir. La couvrure du manuscrit est une simple feuille de parchemin collée sur les deux ais de carton rigide. Les contregardes, en papier, sont collées sur les remplis pliés au carré. L’état d’usure des premiers et derniers feuillets du manuscrit, i.e. feuillets 1-2v° et 83-84, suggère que le manuscrit est resté longtemps sans reliure, peut-être plusieurs siècles, ou bien qu’il était doté d’une « reliure souple », c’est-à-dire une feuille protectrice de parchemin attachée sommairement à l’ouvrage. Par ailleurs, on observe de petits trous dans le parchemin en marge supérieure gauche du feuillet 2, et en marge inférieure du feuillet 17, puis au cahier IV les feuillets 28, 29 et 30 sont mutilés côté gouttière, faisant ainsi disparaître certaines gloses et, de temps à autre, les dernières lettres de certaines lignes du texte de Boèce.

Les dimensions du codex sont de 180 mm x 224 mm, soit une proportion absolue de 1 à 1,2444. Les dimensions de la justification des feuillets sont de 115 mm x 155 mm. Le manuscrit renferme 84 feuillets répartis sur onze cahiers, dont les dix premiers sont des quaternions et le dernier est un binion. La réglure des cahiers fut effectuée selon le « nouveau style » : > < > < | > < > < à la pointe sèche pour ce qui concerne les dix quaternions, puis : > < | > < , pour ce qui concerne le binion du cahier 11. Le manuscrit a été écrit par trois mains différentes, désignées dans l’édition comme suit : Texte : [M1], Texte : [M2], Texte [M3]. La première main intervient du feuillet 1 au feuillet 32 v° : ligne 27, la deuxième main démarre au feuillet 33 : ligne 1 et se poursuit jusqu’au feuillet 37 : ligne 24, la troisième main commence au feuillet 37 v° : ligne 1 et s’arrête au feuillet 38 : ligne 24, puis, enfin, la deuxième main redémarre au feuillet 39 : ligne 1 et continue jusqu’à la fin du manuscrit, semble-t-il. Le texte du De arithmetica, ainsi que les gloses qui l’accompagnent, furent écrits en minuscule caroline à l’encre brune ; les inscriptions et les débuts de phrase sont en écriture onciale et capitale rustique. Les 11 cahiers, dotés de piqûres côté gouttière, ont été réglés avec tantôt 26, tantôt 25 longues lignes par feuillet. Les cahiers de ce manuscrit ne portent pas de signatures.

Notre codex est entré à la Bibliothèque nationale de France pendant la Révolution française, semble-t-il, entre 1792 et 1802 ; cette hypothèse est corroborée par la présence de l’estampille dite « révolutionnaire » tamponnée une fois au feuillet 1, puis une deuxième fois au feuillet 83 verso. L’estampille en question est caractérisée par les lettres RF (i.e. République Française) en écriture majuscule cursive entourées de BIBLIOTHEQUE NATIONALE suivi d’un losange formé de quatre points. Enfin, la cote actuelle, latin 14064, lui fut attribuée en 1865, lors du récolement de la troisième série de l’ancien fonds latin (i.e. les manuscrits cotés 11504 à 14231) (12) •

3. À propos de l’édition proto-philologique princeps :
texte, gloses et signes de renvoi du Boethii De institutione arithmetica •

Une édition critique proto-philologique, néologisme inventé ici pour les besoins de ce travail, est une édition qui reproduit le plus fidèlement possible un manuscrit et son contenu. Une édition critique proto-philologique intégrale est une édition complète d’un texte et ses gloses, basée sur un seul manuscrit, puis reproduite,  avec la même mise en page, les mêmes caractéristiques textuelles, les mêmes qualités et les mêmes défauts, favorisant ainsi l'établissement d’un stemma fiable  •

La présente édition respecte donc la disposition du texte ses gloses, tels que se présentent texte et gloses dans le manuscrit Paris, BnF, Fonds latin 14064. Elle tient compte aussi de toutes les particularités textuelles (orthographe • sténographie latine • signes de renvoi • ponctuation • exponctuation • etc.) que l’on rencontre dans le manuscrit •

Le point d’interrogation en italiques entre crochets carrés, à savoir: [?] ou [-?-], ou bien [?] ou [-?-], à la suite d’un mot, à la suite d’une syllabe, ou isolé entre deux mots ou à la fin d’une phrase, signifie, soit que la graphie est très effacée, soit que la lecture de celle-ci est incertaine •

La ponctuation d’époque est respectée, à savoir : [ • ] (subdistinctio et media distinctio : pause brève et pause moyenne), [• | ;] (distinctio : pause longue), puis [?] (interrogatio ou percontatio : pause longue interrogative) ; de même, tous les signes diacritiques (accents aigus et graves : cf. Isidori Etymologiarum I, 18 ; éd. Migne, Patrologiae. Cursus completus series latina, t. 82 (III-IV) : col. 93-94), les corrections, exponctuations, grattages, traits de liaison entre les syllabes et traits de liaison entre les mots se trouvant dans le manuscrit ont été scrupuleusement conservés. À noter que les endroits où l’usure du parchemin rend l’écriture illisible et les endroits où les mots ont été coupés suite au rognage du manuscrit, sont signalés soit par une série de traits obliques entre crochets carrés [////////], soit par une série de traits obliques sans crochets carrés ///////. Enfin, certaines corrections (omissions du copiste) sont indiquées par les accolades { } •

4. Signes de renvoi (paléographie proto-philologique) •

Les signes de renvoi que l’on rencontre dans le manuscrit sont au nombre de 300 environ, dont 52 différents, et se divisent en trois catégories : signes de correction, signes d’explication, signes d’omission. Il faut savoir également qu’un signe peut servir plusieurs fois, et peut être employé indifféremment dans les trois catégories mentionnées •

Dans le manuscrit BnF, latin Fonds 14064, on découvre une variété de signes de renvoi, dont environ 43% proviennent directement des notæ sententiarum d’Isidore de Séville, d’autres de la sténographie latine (environ 24%), d’autres encore de divers alphabets, notamment : grec (environ 0,05%) et runique (environ 0,05%), ainsi qu’un certain nombre de signes non identifiés. Pour l’étude critique proto-philologique intégrale des signes de renvoi, voir pages 231-237 • IH | ih • Explicit •

This article discusses a full-page schematic diagram contained in a twelfth-century manuscript of Boethius’ De institutione arithmetica and De institutione musica from Christ Church Cathedral, Canterbury (Cambridge University Library MS Ii.3.12), which has not yet been the subject of any significant musicological study despite its remarkable scope and comprehensiveness. This diagrammatic tree, or arbor, maps the precepts of the first book of De institutione arithmetica into a unified whole, depicting the ways music and arithmetic are interrelated as sub-branches of the quadrivium. I suggest that this schematic diagram served not only as a conceptual and interpretative device for the scribe working through Boethius’ complex theoretical material, but also as a mnemonic guide to assist the medieval pedagogue wishing to instruct students in the mathematics of musica speculativa. The diagram constitutes a fully developed theoretical exercise in its own right, while also demonstrating the roles Boethian philosophy and mathematics played in twelfth- century musical scholarship.

http://www.brepols.net/Pages/ShowProduct.aspx?prod_id=IS-9780888441911-1


http://www.pims.ca/publications/new-and-recent-titles/publication/thierry-of-chartres-the-commentary-on-the-de-arithmetica-of-boethius

Unlike the commentaries on Plato's Timaeus and on Boethius's Consolatio phlosophiae, medieval exegesis of Boethius's De arithmetica has seldom been subjected to comprehensive and systematic enquiry. Inhabiting the shifting boundary between philosophy and history of science, the De arithmetica itself has been neglected by most medievalists. Yet, from the Carolingian renaissance onward, when the scholarly curriculum came to be based on the seven liberal arts, Boethius's work soon became a canonical text for the study of arithmetic. Indeed, the growing interest in it during the twelfth century is attested by the large number of surviving commentaries in manuscript.

The commentary on the De arithmetica preserved in Stuttgart, Württembergische Landesbibliothek, Cod. math. 4° 33 and edited here for the first time can be securely attributed to Thierry of Chartres. It belongs to a phase when the Chartrian master's interests were mainly directed toward the liberal arts. We can also discern in Thierry's commentary on the De arithmetica themes and problems developed in his Tractatus de sex dierum operibus and more elaborately in his commentaries on Boethius's Opuscula sacra. Indeed, the discovery of this commentary on the De arithmetica might legitimately be said to clarify not only the more intractable passages in the theological writings but also to illuminate Thierry's philosophical project as a whole. At the heart of that vision is a developing trend in twelfth-century philosophy that places number and proportion at the heart of the physical cosmos. In this profoundly 'mathematical Platonism,' all things are based on number and follow the rule of number; or, to quote Thierry himself, “creatio numerorum, rerum est creatio.”

"ABSTRACT
From psophos to noise: origins and transformations of the object in the 17th century

The presence — or absence — of terms denoting "noise" in the Greek, Latin and vernacular (essentially French) semantic fields is considered from the point of view of their respective theoretical traditions. While the Greek psophos accounts for the physical materiality of sound, the medieval musica speculativa neglects the matter. The 17th century causal — rational — approach revitalizes the topic: acoustics, a new scientific discipline that emerges during that period (the first chair at the French royal academy was established in 1700), is in charge of mapping a new field, wider than that of music, in which noise occupies a position that needs to be defined and theorized. The analysis of this object comes up against the limits of mathematical formulation and of univocity; subjectivity cannot be avoided and authorizes discourses on taste (personal, national, etc.) and an aesthetic approach of the pleasant or unpleasant nature of sound, of music, of noise.

____________________________________________________

Théodora Psychoyou, Du psophos au bruit : sur les origines et les transformations de l'objet au xviie siècle

Le constat de la présence ou non de termes visant à désigner le bruit dans les champs sémantiques grec et latin, puis vernaculaire, est mis en parallèle avec les traditions théoriques respectives : le psophos grec tient compte de la matérialité physique du son, alors que la musica speculativa médiévale se désintéresse de la question. Une approche causale fait émerger à nouveau la problématique au xviie siècle ; l'acoustique, qui naît comme discipline scientifique à part entière à l'aube du xviiie siècle, se propose de cartographier un espace plus ample que la musique, dans lequel le bruit occupe une place qu'il convient de définir et de codifier. L'étude de cet objet se heurte aux limites de la mathématisation et de l'univocité ; la subjectivité est inévitable et autorise un discours sur le goût et une approche esthétique sur la nature agréable ou désagréable du son, de la musique, du bruit.
"

Alberti’s circa 1442-3 Musca — inspired by Guarino’s translation of Lucian’s Encomium muscæ — was not merely a spirited humanist exercise and satire of pompous epideictic rhetoric, but an indication of his complex relationship to the humble and workaday. To bring the heights of virtue and virtuosity down to earth, and catapult a small, common creature up among the stars, Alberti employed a rhetoric that not only extolled the fly’s physical, intellectual, spiritual, moral, martial, and artistic excellence, but its affiliation to the mathematical arts and its practitioners. Taking into account Alberti’s love for mathematics, both applied and abstract, this paper explores Alberti’s references to the quadrivium in the context of the Renaissance paradoxical encomium and with respect to the role that mathematics played in Alberti’s world-making.

Musca, l'opera che Alberti ha composto nel 1442-3 circa, ispirata a una traduzione del Guarino dell'Encomium muscæ di Luciano, non era semplicemente un brioso esercizio umanista e una satira della pomposa retorica epidittica, ma un segnale della complessa relazione di Alberti con l'umile e il quotidiano. Per avvicinare l'apice della virtù e del virtuosismo alla realtà materiale, e per catapultare fin tra le stelle una creatura piccola e comune, Alberti impiega una retorica che non solo celebra l'eccellenza fisica, intellettuale, spirituale, morale, marziale e artistica della mosca, ma la sua affiliazione alle arti matematiche e a coloro che le praticano. Prendendo in considerazione l'amore di Alberti per la matematica, sia applicata sia astratta, questo articolo esplora i riferimenti che Alberti fa al quadrivium nel contesto del paradossale encomio del Rinascimento e rispetto al ruolo che la matematica ha svolto nella concezione del mondo di Alberti.

In preindustrial societies, singing and working was and still is conceived as part of the same activity, of the same life experience. However, the great social transformation that will imply the industrialization of much of the Catalan Countries at the end of the 19th century, and the progressive impetus of the capitalist production system, will often conflict with the cultural practices of workers incorporated directly from an agrarian society to an industrial one. Thus, focusing on the configuration of the Minorcan factories sound space, we want to show the processes of resistance and resilience in the transition from a pre-industrial culture to different one modeled by Fordist capitalism. Through the musical and sound fact in the factory, we review the workers&#39; experience of industrialization in Menorca, the community organization of the sound space to the hierarchy and control of this space: the workplace as a framework where the limits of what is socially possible is modula...

La Disputatio in somnio Scipionis de Favonius Eulogius, opuscule arithmologique et musicologique composé à Carthage dans le premier tiers du Ve siècle est un unicum, conservé dans Bruxelles, KBR, 10078-95. Ce manuscrit, datable de la première moitié du XIe siècle, provient de l’abbaye des Saints-Pierre-et-Exupère de Gembloux. Outre Favonius, il contient de nombreux textes relevant des disciplines du quadrivium. Cet article étudie donc la composition et les sources de ce manuscrit. Cette
étude permet de formuler des hypothèses nouvelles sur la circulation de Favonius en amont de son unique témoin: il aurait existé un manuscrit (aujourd’hui perdu) en Lotharingie à la fin du Xe siècle, probablement à Liège.

On the move from his birthplace in Cremona, a wandering scholar by the name of Gerard settled down in recently reconquered Toledo in or about 1140 and translated from Arabic into Latin, first, the Almagest and then, over four decades, seventy more works of Greek or ‘Islamic’ provenance. Until then, nature-knowledge in medieval Europe had been confined to texts stemming from the effort at simplified translation of selected portions of the Greek corpus that had been undertaken from Cicero to Boethius (p. 31). These texts survived in the few centers of learning that managed to hold their own in Western Europe once this partly Romanized, thinly populated, raiders-battered region was cut off by the early Islamic conquests from its center, the Mediterranean, and found itself willy-nilly compelled to set off on a path of its own.

Britain's Early Philosophers – A two day workshop at Durham University.

Who were Britain’s earliest philosophers? What were Alcuin of York’s contributions to philosophy? To what extent can we consider thinkers such as Hild, Bede, Cuthbert, Gildas, and Cædmon philosophers? How did philosophy reach Britain? Who was reading it, who was writing it, who was teaching it, who was learning it? In this seminal exploratory workshop, we will be considering these questions as well as other questions such as: What counts as philosophy in the early medieval British period? What are the boundary/ies between philosophy and theology? Is there a specifically/uniquely early British philosophical tradition? Just who was reading Alfred’s translation of Boethius?

Resumo: O tratado Speculum Musicae escrito pelo Magister Jacobus (c. 1260-c. 1340) apresenta uma perspectiva da música conforme a qual ela estaria epistemologicamente posicionada em todas as áreas do saber, sendo ao mesmo tempo ciência especulativa e prática, arte liberal e servil. Em nosso artigo temos o objetivo de mostrar como Jacobus elaborou essa concepção ao desenvolver e expandir o arcabouço teórico musical (herdado da antiguidade clássica e dos tratadistas musicais medievais anteriores a ele) sob as bases do aristotelismo latino e escolástico do século XIII.

Palavras-chave: Speculum Musicae, artes liberais, quadrivium, musica speculativa, música medieval.

Resumen: El tratado Speculum Musicae escrito por Magister Jacobus (c. 1260-c. 1340) presenta una perspectiva de la música según la cual ella se posicionaría epistemológicamente en todas las áreas del conocimiento, siendo tanto ciencia especulativa como práctica, arte liberal y servil. En nuestro artículo, nuestro objetivo es mostrar cómo Jacobus elaboró esta concepción desarrollando y expandiendo el marco musical teórico (heredado de la antigüedad clásica y los tratados musicales medievales anteriores) bajo los cimientos del aristotelismo latino y escolástico del siglo XIII.

Palabras claves: Speculum Musicae, artes liberales, quadrivium, musica speculativa, música medieval.

Abstract: The treatise Speculum Musicae written by Magister Jacobus (c. 1260-c. 1340) brings forward a perspective of music according to which it would be epistemologically positioned in all areas of knowledge, being both a speculative, a practical science and a liberal and servile art. In our article we aim to show how Jacobus elaborated this conception by developing and expanding the theoretical musical framework (inherited from classical antiquity and medieval musical treatises prior to it) under the foundations of 13th century Latin and Scholastic Aristotelianism.

Keywords: Speculum Musicae, liberal arts, quadrivium, speculative music, medieval music.

This article brings together the evidence concerning the subjects of the quadri- vium in Varro’s Disciplines and provides a description of the book’s content, composition, and sources, while at the same time discussing the level of post- school mathematical education in Varro’s time. Polarizing views exist on the matter of post-school mathematical education, with some scholars placing it as early as the beginning of the Hellenistic period, and others linking its emergence to Neoplatonic circles in the fourth century CE. I argue that it is possible to attest to the existence of post-school mathematical education in the first century BCE, even though it was pretty basic in nature and did not go beyond the fundamentals of the subjects of the quadrivium, as the contents of Varro’s book suggest. The first section of the article covers Varro’s unconventional views on the origin of geometry: Varro rejects the Egyptian origin of geometry and traces its invention back to the dawn of human civilization. The second section deals with Varro’s geometrical definitions and their relation to the Euclidean tradition, showing that among his sources, there definitely were some Hellenistic introductions to μαθήματα. The final section focuses on Varro’s conception of optics and canonics; here, his approach to canonics is identified as mostly mathematical with some empirical features.

Tradução do capítulo 5 do livro de Basil Bernstein, "Pedagogia, controle simbólico e identidade". Esta tradução foi feita com apoio do programa CAPES-PIBID e destina-se apenas a usos não-comerciais, em grupos de estudo e aulas. (Ronai Rocha)

CUBAN writer, Severo Sarduy, author of numerous essays, novels, books of poetry, and radio plays, remains even among Latinamericanists a relatively marginal writer; and this, I believe, is due to the inherent difficulty of his neobaroque, post-modern, fragmentary style. And yet there is an order to everything he wrote, and an aesthetic and theoretical project that becomes evident when one puts all the pieces of the puzzle together. For instance much of what Sarduy wrote about art and culture can be found in an understandable--albeit cryptic manner--in his 1974 book, Barroco. And what prima facie seems to be a nonsensical stew of disparate ideas, for Sarduy the ideas and concepts he explored were all in some ways logically connected to each other--even if transversally. For instance, that Sarduy would consider Rothko, Kline, and Larry Bell, as a continuation of the Baroque--art styles that seemed so removed in their &quot;abstraction&quot; and minimalism from the classical Baroque-only makes sense when one considers that it was precisely this group of artists who rediscovered Mannerism in the 1950s. For as Arnold Hauser argued in Mannerism: The Crisis and the Origin of Modern Art, modern artists, like their predecessors, were also concerned with the materiality of art. And it was precisely this general aesthetic that they shared with the Mannerists. If Sarduy dared to put Larry Bell and Robert Morris in the same line as Velazquez and Cervantes it is only because they all shared the belief that, as Sarduy was to put it &quot;the work was in the work,&quot; and nowhere else. And further, if Sarduy could see a relation between Kepler&#39;s theory of elliptical movement and Borromini&#39;s architectural designs, and between the Borrominian ellipse and the hidden code of the anamorphic image, it is only because for Sarduy the structuralist theorist, language, no less than the pictorial image, called not so much for a reading, as a deciphering. Hence, in the pages that follow I will briefly examine how Sarduy puts seemingly unrelated ideas together to give us an anamorphic picture of painterly and writerly language, be it scientific or poetic. And thus begin with his treatment of Galileo. As is well known to the history of science, for the Aristotelian Galileo the natural orbits of celestial bodies had to be downward and centripetal. No other kind of motion would do. And interestingly this Aristotelian metaphysical position also influenced his lesser known theory of aesthetics--something to which Sarduy alludes, and Erwin Panofsky notes in his article, &quot;Galileo as a Critic of the Arts.&quot; According to Panofsky, Galileo seems to have rejected Kepler&#39;s theory of elliptical orbits, primarily because it was incompatible with the very principles which dominated his thought as well as his imagination ... Where we would consider the circle as a special case of the ellipse, Galileo would not but feel that the ellipse is a distorted circle: a form which was, so to speak, unworthy of celestial bodies ... and, which, we may add, was as emphatically rejected by the High Renaissance as it was cherished in Mannerism. (10, 11-12, my emphasis) (1) For Galileo the figure of centrism and circularity represented a moral imperative, founded on the notion of &quot;the natural.&quot; The Italian scientist, says Sarduy, confused the notion of the &quot;natural&quot; with that of the &quot;rational&quot; (OCII/Barroco 1216). Sarduy writes: &quot;[E]l barroco sera extravagancia y artificio, perversion de un orden natural y equilibrado: moral&quot; (OC-II/&#39;Barroco 1216). In other words, whatever was un-natural (e.g., the elongated figures of Parmigianino, the figural distortions of Giuseppe Arcimboldo (c. 1527-1593), the anamorphic, off-centered image of Hans Holbein&#39;s The Ambassadors, etc.) all constituted a species of perversion for Galileo. And of Galileo&#39;s rejection of anamorphic painting vis-a-vis his more specific denunciation of the Mannerist overturning of linear perspective, Sarduy argues: El imperio de lo natural obliga a Galileo a proclamarse agente del terror, como se ha llamado a esa tradicion, moral, denotativa, que asimila las figuras retoricas a una perversion de la naturalidad del relato . …

O tratado Speculum Musicae do Magister Jacobus de Ispania (ou Jacobus Leodiensis) não recebeu a devida visibilidade nas instituições de ensino e de investigação intelectual tanto de sua época quanto posteriormente. A partir da segunda metade do século XX o tratado foi gradualmente ganhando maior visibilidade e nos últimos anos tem recebido por parte de diversos estudiosos um maior esforço de investigação. A presente dissertação visa expor o contexto filosófico e teológico desse tratado, investigando os seus principais conceitos norteadores a partir de uma leitura dos primeiros capítulos do seu primeiro livro. A partir da exposição dessas concepções fundamentais, buscamos então esclarecer algumas das questões mais fundamentais a respeito da musica speculativa, a saber, sua função no programa das artes liberais, sua concepção enquanto ciência e sua relação com a teologia sagrada.

Resumo: No tratado Speculum Musicae, Magister Jacobus (c. 1260 – após 1340) designa a cadentia como o movimento de um intervalo musical ou concórdia imperfeita para uma mais perfeita, afirmando ser tal movimento em direção a um melhor ser – sicut ad melius esse (Speculum Musicae IV, L). A relação intervalar entre as notas musicais – remontando ao pensamento pitagórico – era compreendida na tradição do pensamento especulativo musical medieval e da antiguidade como de natureza quantitativa aritmética. A medida de perfeição das relações entre as notas era compreendida como de proporção aritmética, sendo que a imperfeição corresponderia às relações menos proporcionais, e a perfeição à maior unidade na proporção. No entanto, ao afirmar que o movimento da consonância imperfeita para a perfeita é para um melius esse, Magister Jacobus – sem renunciar à constituição aritmética das relações intervalares – atribui aos intervalos uma medida ou grandeza de perfeição não aritmética, mas metafísica, que é aquela compreendida por Santo Tomás de Aquino como quantitas virtualis (Summa Theologiae, I, q.42, a.1, ad 1) ou quantitas intensiva (De Veritate, q.2, a.9). Ao atrelar uma perfeição aritmética enquanto concretizada num objeto da física (número sonoro) a uma perfeição de ordem metafísica, Magister Jacobus nos permite ver como a música era por ele compreendida enquanto subordinada às três ciências especulativas (Speculum Musicae livro I, VIII) e assim, não reduzida ao estatuto epistemológico de scientia media mas, eminentemente compreendida como integrando a metafísica.

Abstract: In the treatise Speculum Musicae, Magister Jacobus (c. 1260 – after 1340) designates the cadentia as the movement of a musical interval or imperfect concord to a more perfect one, claiming to be such a movement towards a better act of being - sicut ad melius esse (Speculum Musicae IV, L). The interval relationship between musical notes – going back to Pythagorean thought – was understood in the tradition of ancient and medieval speculative musical thought as being of an arithmetic quantitative nature. The measure of perfection of the relations between the notes was understood to be of arithmetic proportion, with imperfection corresponding to less proportional relations and perfection to greater unity in proportion. However, when affirming that the movement from imperfect to perfect consonance is towards a melius esse, Magister Jacobus - without renouncing the arithmetic constitution of interval relations – attributes to intervals a measure or type of perfection that is not arithmetical but metaphysical. Saint Thomas Aquinas understands this measure as quantitas virtualis (Summa Theologiae, I, q.42, a.1, ad 1) or quantitas intensiva (De Veritate, q.2, a.9). By linking an arithmetical perfection materialized in a physical object (numerus sonorus) to a metaphysical perfection, Magister Jacobus allows us to see how he understood music as subordinated to the three speculative sciences (Speculum Musicae book I, VIII). Thus music was not reduced to the epistemological status of scientia media but eminently understood as integrating metaphysics.

Fondé en 1257 à Paris par Robert de Sorbon, le collège de Sorbonne constitue le premier modèle de collège séculier. L’institution devait fournir un soutien pour les pauvres étudiants en théologie. Dès ses premières années, la communauté bénéficia des libéralités de nombreux bienfaiteurs, dont le roi Louis IX. Grâce aux dons et aux legs, le collège ne tarda pas à accueillir une importante bibliothèque, qui deviendra l’une des plus riches d’Europe au début du XIVe siècle. Le rayonnement de cette bibliothèque fut considérable, ouverte au reste de l’université, elle reflétait l’ensemble du savoir médiéval. Cette thèse s’intéresse à la collection scientifique du collège de Sorbonne, plus spécifiquement aux disciplines mathématiques (quadrivium). Les sciences quadriviales sont peu documentées en ce qui concerne l’université de Paris, et la bibliothèque de Sorbonne, par son riche corpus de livres mathématiques, constitue un exemple sans précédent dans le milieu universitaire, et contraste avec le silence des statuts universitaires. Les catalogues du XIVe siècle des deux dépôts du collège —Parva libraria et Libraria communis —permettent de reconstituer l’origine d’une collection majoritairement fondée sur des bibliothèques privées attestant des intérêts de leurs anciens possesseurs. L’étude de l’évolution du quadrivium permet dans un premier temps de saisir la portée d’une telle bibliothèque, et son statutau sein d’une discipline renouvelée par les traductions arabo-latines et gréco-latines. Le collège fut certainement l’un des vecteursde l’assimilation de ces œuvres nouvellement traduiteset des adaptations. La reconstitution de la collection réside sur les manuscrits subsistants et les identifications des œuvres. Elle met en valeur les bibliothèques privées de Richard de Fournival, chancelier d’Amiens († 1260), et du magnus astronomus de Sorbonne, Pierre de Limoges (†1306), qui une fois intégrées au collège, ont composé la majeure partie de la collection dédiée aux mathématiques. La question d’unenseignement se pose alors, mais aucun document n’en atteste. C’est là qu’entre en jeu l’étude des usages des manuscrits par les lecteurs, par le biais de leurs annotations. Une étude de cas de Jean des Murs, calculatorparisien du XIVe siècle, et promoteur des tables alphonsines dans les années 1320, permet d’analyser la réception que connue cette collection scientifique. Dans une perspective comparatiste, la bibliothèque du collège de Sorbonne est mise en parallèle avec un foyer de mathématiciens oxoniens du XIVe siècle: Merton College, Oxford. L’étude des sources de la bibliothèque de Merton permettra de souligner les aspects originaux ou traditionnels de la collection quadriviale de Sorbonne. Au XVe siècle, alors que l’université de Paris connaît un déclin certain, en grande partie due à la guerre civile, la maison de Sorbonne sut tirer partie de l’effondrement du collège de Navarre, et accueillir en son sein des lecteurs versés dans les sciences. L’analyse des emprunts des sociétaires et extraneioffrel’ultime représentation de l’usage fait de cette collection.

Статья представляет текст доклада «Об институциональной коэволюции философии и математики в античности», сделанного на Международном круглом столе «Как и почему возникают вопросы соотнесения проблем философии, науки и теологии?» (Reflections on philosophical, scientific, and theological dimensions of interfacing), проведенном Институтом «Высшая религиозно-философская школа» (ВРФШ) 16–18 ноября 2011 г. в Санкт-Петербурге. Кроме некоторых стилистических и технических поправок изменения в текст не вносились. Добавления и комментарии, которые я посчитал нужным сделать к тексту, вынесены в постраничные примечания.
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В статье затрагивается тема генетического родства философии, науки и богословия; и, более узко, институционального единства философии и математики в античности. В статье рассматриваются вопросы (1) совместного становления философии и тех дисциплин, которые впоследствии стали называть «математическими», (2) сближения этих дисциплин в рамках философии в качестве единого исследовательского и учебного блока, (3) роли, которую играли «математические» дисциплины в структуре античной философии и, в частности, в платонизме.
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Shiyan T.A. About institutional co-evolution of philosophy and mathematics in antiquity (On the problem of philosophy, science and theology correlation). ANNOTATION: This article is devoted to the subject of genetic relationship of philosophy, science, and theology; and specifically, to institutional unity of philosophy and mathematics in antiquity. The article covers the following problems: 1) the joint formation of philosophy and those disciplines, which later were named “mathematical”, 2) the approaching between these disciplines within the framework of philosophy and conversion them in united researching and educational complex, 3) the role played by “mathematical” disciplines in the structure of antique philosophy, and especially in platonic tradition.