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The Tn-Type Thesaurus Viewer (2017)

The “Tn-Type Thesaurus” is a harmonic encyclopedia of Tn (transpositional) set types generated by a computational library in C++, which includes a visualizer GUI application in javascript available through the internet. This software is one of the results from the research project “ The formalization and abstraction of paradigmatic psychoacoustical properties of agglomerates of sounds of definite pitch as theoretical grounds for a new discipline of Harmony”, which is being conducted since 2014 at the State University of Maringá, Paraná, Brazil, with funding from a two-year fellowship grant (2014-2016) given by the Araucaria Foundation for the Scientific and Technological Development of the State of Paraná.
This research project aims to construct the theoretical grounds for a new discipline of Harmony and it tries to investigate and define paradigmatic psychoacoustical concepts for agglomerates of sounds of definite pitch, their qualification and mathematical quantification, as well as the enumeration and a qualitative and taxonomical study of a Harmonic Universe defined by these studies. The research includes also the computational implementation in C++ of the quantitative mathematical modellings developed, creating Computer-Assisted Composition software components that aim at the fast taxonomical identification of a harmonic combination and the survey of its properties and comparative interrelationships with other combinations. This project for a Tn-Type Thesaurus is the concluding formalization of that research.

The Structure of the Thesaurus

The Tn-Type Thesaurus is designed to show, in a compact yet comprehensive way, a descriptive study of all the 351 transpositional set types, or Tn-Types (FORTE 1973; RAHN 1980; COSTÉRE 1954). The thesaurus dedicates one complete page for every Tn-Type, in which a detailed description of the type's constitution and its harmonic properties is shown. This description includes the following information:

  • the Tn-Type's label both in Set Theory format and Costère's representative number (FORTE 1973; RAHN 1980; COSTÉRE 1954);
  • the Tn-Type's interval vector (FORTE 1973; RAHN 1980);
  • its classification according to the criterion of symmetry (COSTÉRE 1954);
  • its inverse Tn-Type (COSTÉRE 1954);
  • its isomer Tn-Types (or Z-relations) (HANSON 1960 e FORTE 1973);
  • its cardinality and labelling according to the Forte listing (FORTE 1973);
  • its complement to the chromatic aggregate (FORTE 1973; COSTÉRE 1954);
  • its tonicity and roots, phonicity and vertexes, its azimuth and generatrixes (BITTENCOURT 2011);
  • its Tn-Type estimation of roughness (BITTENCOURT 2011);
  • its table of pertinence of all chromas as root or vertex (revised from PARNCUTT 1988, and BITTENCOURT 2011 and 2014);
  • its cardinal, tonal (triadic) major and minor, and transpositional and inversional gravitational flow tables (COSTÉRE 1954; BITTENCOURT 2011);
  • its invariance table after operations of transposition and inversion, also known as vicinality relationships (RAHN 1980; COSTÉRE 1954);
  • its commonality table relative to its transpositions and inversions and relative to all 24 perfect triads (BITTENCOURT 2011).
  • its classification according to the criteria of cardinal, tonal (triadic), and transpositional gravitational flow (revised from COSTÉRE 1954);
  • the listing of its cardinal poles and subpoles, tonal (triadic) centers and subcenters (COSTÉRE 1954);
  • its properties of autocomplementarity (revised from COSTÉRE 1954);
  • its properties of multimodalization relative to its subsets (revised from COSTÉRE 1954);
  • the auto-imitative properties of its subsets (revised from COSTÉRE 1954);
  • the musical notations for all its transpositions and inversions, including markings for the invariance relative to the set's 0-index transposition;
  • enumeration and description of subdivision and partitioning of the set's 0-index transposition in subsets, including description of multimodalization and auto-imitation properties;
  • enumeration and description of the inclusion of the set's 0-index transposition in supersets.

All these properties and diagnostics are calculated computationally by means of a software library programmed in C++ by the researcher. This code library automatically generates all 351 pages of the Tn-Type Thesaurus in a high-quality vector graphics format file of the EPS type (encapsulated postscript).

The Tn-Type Viewer Online Application

At this time, the Tn-Type Thesaurus is only available experimentally in the form of an online javascript Viewer application available through the internet AT THIS LINK. With this Viewer application, it is possible to browse through all the pages of the Tn-Type Thesaurus and it also has a field in which one can input a pitch or pitch-class in the usual Set Theory format. When a set is input, the application figures out which page contains the Tn-Type the set belongs to and it shows that page in the Viewer. As this is only an experimental public preview of the current state of the Tn-Type Thesaurus (which is still undergoing further development), the quality of the page images in this online javascript version was designed only for a good online viewing and not for printing.

Bibliographic References

BITTENCOURT, Marcus Alessi. Sketches for the foundations of a contemporary experimental treatise on Harmony. In: Proceedings of the II Encontro Internacional de Teoria e Análise Musical. São Paulo: UNESP-USP-UNICAMP, 2011.
BITTENCOURT, M.A. Treatise on Harmony, demonstrated in geometrical order, Book One (Tratado de Harmonia, demonstrado à maneira dos geômetras, Livro Um). Representative research work presented and successfully defended as requirement for the promotion to the rank of Associate Professor at the State University of Maringá, Paraná, Brazil, 2014.
COSTÈRE, Edmond. Lois et Styles des Harmonies Musicales. Paris: Presses Universitaires de France, 1954.
FORTE, Allen. The Structure of Atonal Music. New Haven: Yale University Press, 1973.
HANSON, Howard. Harmonic Materials of Modern Music: Resources of the Tempered Scale. New York: Appleton-Century-Crofts Inc., 1960.
PARNCUTT. R.. Revision of Terhardt's Psychoacoustical Model of the Root(s) of a Musical Chord. Music Perception. Los Angeles: University of California Press, Vol. 6, No. 1 (Fall), pp. 65-93, 1988.
RAHN, J.. Basic Atonal Theory. New York: Schirmer Books, 1980.

This research received funding support from the Araucária Foundation for the Scientific and Technological Development of the State of Paraná, Brazil.


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