Academic literature on the topic 'Pentachoron'

We describe an algorithm to compute trisections of orientable four-manifolds using arbitrary triangulations as input. This results in explicit complexity bounds for the trisection genus of a 4-manifold in terms of the number of pentachora (4-simplices) in a triangulation.

Based on my revision of the types of the Neotropical genera Nyctelia Latreille, 1825 (Nycteliini), Praocis Eschscholtz, 1829 (Praocini), Scotobius Germar, 1824, and Emmallodera Blanchard, 1842 (Scotobiini), nine NEW SYNONYMIES are proposed and illustrated: Nyctelia roigi Marcuzzi, 1977 with N. quadricarinata Fairmaire, 1905; Praocis freyi Marcuzzi, 1977 with P. uretai Kulzer, 1958; Praocis larraini Marcuzzi, 2001 with P. pentachorda Burmeister, 1875; Praocis sivestrii Marcuzzi, 2001 with P. striolicollis Fairmaire, 1883; Praocis baloghi Marcuzzi, 2001 with P. montana Kulzer, 1958; Scotobius freudei Marcuzzi, 1977 with S. akidioides bicostatus Kulzer, 1955; Scotobius kulzeri Marcuzzi, 1977 and S. contrerasi Marcuzzi, 1977 with S. alaticollis Kulzer, 1955; and Emmallodera ceii Marcuzzi, 1977 with E. obesa costata Kulzer, 1955. A statement on the differences between Nyctelia blapoides Fairmaire, 1905 and N. kulzeri Marcuzzi, 1977 is presented, as well as a redescription of N. blapoides.

If there has been one dominant, quasi-official theory for Turkish classicalmusic in the second half of the twentieth century, it is that particularlyassociated with Ezgi and Arel. Their notational conventions have becomestandard, supplanting earlier norms, and the framework they developed is theone still employed in recent general accounts of the modal system, whether thesimplified introductory survey of Yilmaz (1983), for example, or the moredetailed and comprehensive coverage of Ozkan (1984). Both of these follow theanalytical models provided by their predecessors, and begin with an expositionofintervals and the various species of tetrachord and pentachord formed from them before moving on to describe the structure of the makams themselves. Thesequence interval, scale, mode, nevertheless forms a conceptual continuum: theintervals denned are restricted to those deemed to occur in Turkish classicalmusic, and the nomenclature of the various species, for all that they appear asabstract assemblages of intervals, identifies them with characteristic segments ofwell-known and important makams.

This study examined factors guiding the participants' ratings of pentachords representing different pentad classes. Two tests were carried out where the participants (non-musicians, professional music students, and music theorists and composers) heard sequences of five chords. Four of the chords represented one set-class, and these chords formed the context; one chord represented another set-class, and it was the deviant. The participants were asked which chord did not belong to the same group as the others. When the data were analysed, the main interest was in the difference in the consonances of the context chords and the deviant chord. The difference varied greatly from case to case. It was found that the participants selected the deviant chord when there was a clear difference in the consonances of the set-classes. However, the difference in the consonances was not the only factor guiding the participants' ratings. The context was found to be important for both musically trained and untrained participants, but in different ways. Generally, the non-musicians used the set-class identity as a guide in the items where the context was more consonant than the deviant, while in a dissonant context they used chordal characteristics, especially the transpositional level. The composers and theorists, on the other hand, made their ratings according to the set-class identity when the context was more dissonant than the deviant and according to the chordal characteristics when the context was more consonant than the deviant.

Mansi (Voguls) is one of the Ob-Ugric peoples of Western Siberia. Mansi have a lot of pho-noinstruments (including folk-music instruments): string (zither, lute, harp), shaman’s drum, hunter’s whistles, reindeer herder’s and household’s sounding items, children’s sound toys, etc. The most popu-lar of them is the zither named saŋkvəltap. This instrument is tuned over the diatonic pentachord: c-d-e-f-g or c-d-es-f-g, and there are other variants of tuning. When the traditional lifestyle still remained, the zither was played during rituals (such as a bear-feast, sacrifice ceremony, shamanic rite) and in other situations, accompanying dances and singing. Currently, the zither is played mainly by young musicians during the concerts. The purpose of the article is to describe the principles of the pitch organization of Mansi instru-mental music. The author had made recordings of playing of several old musicians, then the note tran-scription of that melodies has been made. The algorithm for the analysis is shown on the example of a single piece that is the personal melody of the deity in the image of a bear (“The Melody of the Clawed Old Man”). In the collection of analyzed tunes the following types of scales were found: 1) frequent: c-d-e-f-g, c-d-es-f-g, (c)-d-e-f-g, (c)-d-es-f-g; 2) rare: (c)-d-es-e-g, c-d-es-(f)-g, c-(d)-es-e-g. The tone functions can be such as a supporting tone, a semi-supporting tone, a non-supporting tone, an auxiliary tone. The tone functions are determined in each segment by the following signs: 1) accentuation; 2) location on the strong beat; 3) appearance on the boundaries of the composition units; 4) length of sounding; 5) multiple or frequent repetition. Semi-supporting tones are identified by the same criteria as the supporting ones, but they must have less ways of marking than the supporting tones. The auxiliary tones are rare, they don’t enter into the frame of melody and they are used only for variation. As a result of the analysis of the collection of tunes, the following patterns of the pitch-sound or-ganization of the instrumental music of Mansi are defined: 1) both of quarta and quinta modes are frequent; 2) the number of mode supporting tones is two or three, rarely – one or four, the semi-supporting tones are frequent; 3) the main supporting tones are at the highest and the lowest step of the scale. There is no strict correlation between the type of mode and the genre of the tune, but the following trend is noticeable. Melodies of deities and men’s dances in the modes of a quarta volume are performed more frequently. The epic or the song melodies and the melodies of women’s dances mainly sound in the modes of the quinta volume. On the basis of a comparison of several of the same name tunes (“Melody of the Old Man the Owl”), the following conclusion is drawn. The most important parameter of a pitch organization is the correlation of supporting tones and the sequence of appearance of typed intonational moves.

Spherical, Euclidean, and hyperbolic simplices can be characterized by the dihedral angles on their codimension-two faces. These characterizations analyze the Gram matrix, a matrix with entries given by cosines of dihedral angles. Hyperideal hyperbolic simplices are non-compact generalizations of hyperbolic simplices wherein the vertices lie outside hyperbolic space. We extend recent characterization results to include fully general hyperideal simplices. Our analysis utilizes the Gram matrix, however we use inversive distances instead of dihedral angles to accommodate fully general hyperideal simplices.For two-dimensional triangulations, an angle structure is an assignment of three face angles to each triangle. An angle structure permits a globally consistent scaling provided the faces can be simultaneously scaled so that any two contiguous faces assign the same length to their common edge. We show that a class of symmetric Euclidean angle structures permits globally consistent scalings. We develop a notion of virtual scaling to accommodate spherical and hyperbolic triangles of differing curvatures and show that a class of symmetric spherical and hyperbolic angle structures permit globally consistent virtual scalings.The double tetrahedron is a triangulation of the three-sphere obtained by gluing two congruent tetrahedra along their boundaries. The pentachoron is a triangulation of the three-sphere obtained from the boundary of the 4-simplex. As piecewise flat manifolds, the geometries of the double tetrahedron and pentachoron are determined by edge lengths that gives rise to a notion of a metric. We study notions of Einstein metrics on the double tetrahedron and pentachoron. Our analysis utilizes Regge's Einstein-Hilbert functional, a piecewise flat analogue of the Einstein-Hilbert (or total scalar curvature) functional on Riemannian manifolds.A notion of conformal structure on a two dimensional piecewise flat manifold is given by a set of edge constants wherein edge lengths are calculated from the edge constants and vertex based parameters. A conformal variation is a smooth one parameter family of the vertex parameters. The analysis of conformal variations often involves the study of degenerating triangles, where a face angle approaches zero. We show for a conformal variation that remains weighted Delaunay, if the conformal parameters are bounded then no triangle degenerations can occur.