Tese completa

The current version is dedicated with fondness and gratitude to my esteemed colleague Kerry Snyder on the occasion of her eightieth birthday. Many musical scores from the later seventeenth century give the impression that the notation of meter and rhythm had nearly evolved to present-day forms. Their deceptively modern appearance disguises, however, the continued operation of elements of the old mensural system, adapted, not always successfully and certainly without consistency, to the newer musical styles. As with the modal system, it took more than a few decades to eradicate all traces of this ancient heritage from musical theory and practice. One question that interests us especially here is the extent to which composers still drew upon mensural conventions to communicate tempo relationships. The first obstacle to an easy answer is that by the early seventeenth century there was no longer a uniform practice for the notation of meter and tempo; the adaptation of the old mensural system to new musical demands seems often to have been done on an experimental, ad hoc basis. Some composers decided on their own rules, which may have been understood by those in their immediate circles, although not necessarily by the world at large. Writers of pedagogical tracts responded to the resulting chaos either by describing––as well as they could––the wide range of notational options and their possible interpretations, or by coming up with some logical and consistent system in the vain hope that it would meet all needs and enjoy universal adoption. An example of the former is found in a copy by Jan Adam Reincken of Sweelinck's rules of composition, 1 an example of the latter in the Zangh-Bloemzel, by Joan Albert Ban. 2 Before turning to the actual practice of a specific composer, I shall briefly summarize the treatments of the subject by these two authors.

2020

2021

For a video presentation of the conclusions of the work see: https://www.youtube.com/watch?v=fN4fU4laue4 Three times in the history of Western Music - at the end of the 14th century, the end of the 16th century, and the beginning of the 20th century – has there been a flowering in the development of non-dyadic rational rhythmic hierarchies. Only in the last of these occurrences has this development persisted continuously to the present; each time before, rhythmic complexity collapsed into a system dominated by dyadic- and/or triadic-rational rhythmic hierarchies. By the 17th century, even triadic-rational rhythmic hierarchies had totally disappeared from musical discourse to be supplanted by our modern system of dyadic-rational time signatures. Even into the 21st century, dyadic-rational time signatures remain predominant, despite work by composers like Henry Cowell and Conlon Nancarrow, which suggested the possibility of a rhythmic paradigm shift during the early- and mid-20th century. Despite the persistent hold of dyadic-rational time signatures, developments in prescriptive rhythmic complexity during the 20th century have continued to the present, persisting over multiple generations of composers and forming distinct schools of musical discourse popular in contemporary classical music today. Among these composers are not only Charles Ives, Henry Cowell, and Conlon Nancarrow, but also Thomas Ades, Brian Ferneyhough, Michael Gordon, Karen Khachaturian, Milton Babbitt, Elliott Carter, and Jonathan Dawe among others. Within their oeuvre, each of these composers have encountered the need for a broader exploration, development, and notation of rhythmic structure beyond our current dyadic-rational system, allowing in their music pan-rational time signatures, irrational time-signatures, and/or dense and/or indivisible rhythmic hierarchies – all of these levels of rhythmic prescription either not seen since the 16th century or altogether never before seen in Western music. Given the present state of our system of music notation and rhythmic prescription within it, what are we doing and what can we do now in the 21st century with the rhythmic tools developed in the past one hundred years? By thoroughly understanding the history of prescriptive rhythmic experimentation in Western Music, we can possibly better understand why certain systems of rhythmic notation have persisted while others have been forgotten; through such better understanding of the history of rhythmic notation we might fashion a notational system today that overcomes our present limitations in rhythmic prescription better than previous failed models. To this end, I will trace the historical development of systems of rhythmic hierarchy from Medieval to Modern music, focusing on music with exceptional prescriptive, precise, mathematically defined rhythmic structures, excluding aleatoric and spatially based rhythmic notation. In doing this, we will gain a historical contextualization of the rise of pan-rational systems of rhythmic notation. Following this, we will survey a variety of modern compositional methods that expand standard prescriptive rhythmic notation, beginning with Charles Ives and Henry Cowell and ending with living composers like Thomas Ades and Michael Gordon. Last, this dissertation will address my own compositional work in the context of pan-rational systems of rhythmic hierarchies and propose a new addition to the lexicon of rhythmic notation that will emancipate the composer from traditional dyadically-rational rhythmic notation.

zeuxilogy.home.ro

Before being discovered by Francisco Pizzaro, the Incas managed to build at least 23,000 kilometers of roads but missed to invent the wheel. This historical fact made me think that it might be precisely due to their unquestionable value that, sometimes, important, useful discoveries, postpone the quest for other viable alternatives to themselves: "what's the use of a wheel as long as we have good roads?", one may ask.

The best of the best about harmonix.

Fundamentals of Music Theory, 2021

Performance Practice Review, 1989

Revista Música Hodie, Goiânia, 12:2, 258-273, 2012

This paper suggests that rhythmic notational practice seen in complex music adhere to the principle that the number of beams/flags of any note is dependent upon the number of iterations per temporal unit. In the first instance this may seem obvious, since this principle forms the basis of rational rhythmic notation in music. However, with music featuring nested tuplets of two levels or more, an interesting phenomena begins to occur in which the divisions of a unit become decoupled from the expectations of rhythmic ratios, such as three in the space of the last note of a triplet. Therefore, the intent of this paper is to provide a series of proofs to show that the number of iterations per (divisions of) unit take precedence over the expectations of ratios when such mismatches occur.

Music Theory Spectrum, 2008