Обличай блудню еретическую

Внимание! Этот пост предназначен прежде всего для лиц с математическим образованием. (Замечания для "нематематиков" - специально чуть ниже).

Предлагаю ознакомиться с некоторыми статьями (PDF) Т. Нолля (Thomas Noll), одного из редакторов журнала Journal of Mathematics and Music. вот по этой ссылке:
user.cs.tu-berlin.de/~noll

Внимание! Все ссылки ниже - на PDF файлы.

Особенно интересной показалась статья
''Sturmian Sequences and Morphisms: A Music-Theoretical Application (вторая в списке). Неожиданный (для меня) чисто алгебраический подход (т.е. без какой-либо "акустики" и чисел, хотя в качестве иллюстрации возникают (полу)подходящие дроби. Совершенно неожиданно, помимо прочего, двойственность Кристоффеля как бы объясняет "порождение" "диатонических" ладов, причем с их финалисами и кофиналисами, разделением на плагальные и автентические (и дано алгебраическое объяснение "вырожденности" локрийского, который Глареан исключает как раз ).

Для "нематематиков" вполне понятным будет чтение введения из статьи ''Facts and Counterfacts: Mathematical Contributions to Music-theoretical Knowledge" - для простоты я его копипэйстю в самом конце поста.

И также крайне живо написанный ''Dialogue Concerning the Three Chief Tone Systems: the Diatonic, the Triadic, and the Chromatic" (беседа Математика, Теоретика Музыки, Философа и Simplicio).

Ну и понятен вроде бы должен быть обзор про восприятие и закон Вебера-Фехнера: "Tone Apperception, Relativity, and Weber-Fechner's Law"

Ну и в конце для простоты копипейстю обещанную "программу" Abstract Music Theory:

1 Abstract Music Theory
Many traditional music-theoretical concepts, such as ’diatonic scale’, ’third chain’, ’chromatic alteration’ are seemingly anchored in the realm of musical notation. Graphemic elements, such as staff, relative note head positions, accidentals etc. help to communicate these concepts among musicians and theorists. The double articulation of music in terms of scores and performances is mirrored in the common practice of musical analysis. The study of musical scores on the basis of the above-mentioned type of concepts constitutes the core of music-analytical and music-theoretical knowledge. Even pure theoretic concepts like Rameau’s ’fundamental bass’ cannot be understood without the concepts of diatonic third chains and the diatonic fifth progressions. The methodological difficulties to grasp the music-theoretical meanings of these concepts beyond their graphemic anchors caused serious doubts about their scientific value within systematic musicology. We may distinguish two main attitudes or strategies to face this problem. From a pragmatic point of view one may simply do music theory without considering as a ’hard science’. Many music theorists conceive their work as hermeneutic, philological or pedagogical contributions to a general discourse about music. Accordingly does the enrichment of music-theoretical knowledge not need to be measured in evident facts, but rather in terms of plausible modes of access to music, in terms of philological connections (i.e. knowledge about knowledge, with responsibility only for the former one). The second strategy aims at grasping musical facts along the ontological stratification of musical reality and to formulate them on acoustical, psycho-acoustical, cognitive, phenomenological, social and other levels of description. These attempts rely on the modes of access that other sciences -such as physics, psychology, philosophy, sociology, etc. have developed. The coexistence of the complementary strategies has a remarkable history on its own and includes numerous expressions of skepticism on both sides, inspirations as well as attempts to build bridges between them.

It would be misleading, however, to consider the ontologically sensitive approaches as ’scientific’ ones and to oppose them to a mere pragmatically oriented ’unscientific’ music theory. Instead we propose the term abstract music theory in order to refer to attempts which aim at grasping musictheoretical contents within consistent frameworks of abstract musical concepts. In this article we discuss particular approaches to abstract music theory, where mathematics plays a major role in the reformulation of traditional concepts as well as in the investigation of dependencies between them. Many fruitful contributions from the last three decades indicate, that a successful mathematisation of music-theoretical knowledge is by no means restricted to the domain of acoustics or to the quantitative investigation of experimental data (such as in computational psychology). The strengthening of abstract music theory by means of mathematical conceptualization contributes in a threefold way to an enrichment of the music-theoretical knowledge.

1. The elaboration of a consistent conceptual network and its practical usage in musical analyses adds new insights to the ’traditional’ knowledge. In the context of this article we show this in the case of central ideas of Rameau’s music theory. We argue that Rameau’s original plan to explain various music-theoretical facts as a consequence of choosing the fifth and third as generating intervals can indeed be realized to a remarkable degree.

2. Conceptual bridges between abstract music theory and acoustical, psychological, and other levels of description can be based on innermathematical translations between mathematical models for the corresponding structures and processes. The Rameau Diagram (see Section 5) can be seen as a refinement of Fred Lerdal’s model of tonal pitch space, which was developed as a bridge between Music Theory and Cognitive Psychology. Furthermore we formulate some ideas, how abstract music theory can be a source for investigations into a mathematical theory of the phenomenology of the mind.

3. Mathematical formulations of musical facts can be relativized by constructing musical counterfacts, i.e. by constructing alternative models which are not exemplified by any music. These counterfacts either indicate an insufficiency of the mathematical characterizations and/or they may inspire musicians to test them out in practice, and to eventually turn them into musical facts. Our discussion is inspired by Eytan Agmon’s counterfactual recompositions of Schumann’s ”Am Kamin” from Kinderszenen (c.f. [2]) which intend to test analytical assertions by changing the music in such a way, such that the assertion still holds. It is also inspired by Gerard Balzano’s [3] suggestion to experiment with generalized diatonic systems, which share certain properties with the familar one.

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И да, по опыту некоторых обсуждений на форуме - такое вот имею NB: Если вдруг что, прошу избавить меня от риторических восклицаний на тему "вот же, задумал проверить алгеброй гармонию", или "это не нужно для музыки".