congruence…by angular functions:

In Euclidean geometry triangles with equal angle-angle-angle does not provide information regarding the size of the two triangles & hence proves only similarity not congruence in Euclidean space.  However, in spherical geometry and hyperbolic geometry (where the sum of the angles of a triangle varies with size) triangles with equal angle-angle-angle is sufficient for congruence on a given curvature of surface.

 

key:

the group of pitches or scale forming the basis of composition in Western music; features a tonic note & corresponding chords (provides a sense of arrival & rest); the tonic has a unique relationship to the other pitches/cords of the same group; pitches/ chords outside the tonic create tension, resolved when the tonic note/chord returns.  In classical music (1650-1900) the key would be major or minor; longer pieces may have sections in contrasting keys.

 

phrasing:

refers to the way a musician shapes a sequence of notes in a passage of music to express an emotion or impression.  Done through interpreting the music (from memory or sheet music), by altering tone, tempo, dynamics, articulation, inflection & other characteristics.  Phrasing can emphasise a concept in the music or message in the lyrics, or it can digress from the composer's intention (e.g. accelerating tempo or prolonging a note adds tension).

 

chromatic:

terms (diatonic & chromatic) in music theory used to characterize scales; also applied to musical instruments, intervals, chords, notes, musical styles & harmony.  Often used as a pair when applied to contrasting features of the music of the period 1600–1900.  Diatonic refers to musical elements derived from the modes & transpositions of the "white note scale (C–D–E–F–G–A–B); may also include all forms of heptatonic scale in common use in Western music (the major & all forms of the minor).  Chromatic refers to structures derived from the 12-note chromatic scale- all semitones.

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convergence in infinite series:

Given an infinite sequence                           the nth partial sum         is the sum of the first n terms of the sequence.

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A series is convergent if the sequence of its partial sums                           tend to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases.

transformation of elliptic & integrals into multiply periodic-functions: * see Endnote 71

Elliptic integrals are generalizations of the inverse trigonometric functions and provide solutions to a wider class of problems. In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse.  While the arc length of a circle is given as a simple function of the parameter, computing the arc length of an ellipse requires an elliptic integral. 

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Abel: * see Endnote 72

(1802–1829) wrote a number of papers, which played important roles in the development of complex geometry and the elliptic function.  His 1827-28 foundation paper, established the theory of elliptic functions & revealing their double periodicity. 

 

Gauss:

Gauss had thought about the issue of the geometric representation of complex number for some decades at the beginning of the 19th century, but didn’t publish anything on the subject until 1831, in which he specifically defined a complex number z of the form x+iy to correspond to the point (x, y) in the Euclidean two-plane R2, and the usual arithmetic (addition and multiplication) of complex numbers.  His main purpose in this paper was extending the results of his number theory (a paper he wrote in 1801) to the setting of complex numbers, and specifically he discusses complex numbers of the form a + ib where a and b are integers, but emphasizing that such numbers were points in a two-dimensional plane.

abstruse:

hard to understand; recondite; esoteric

 

Divina Comedia:

Dante's  epic poem (1308-1321) & masterpiece of Italian literature, full of symbolism & pathos; author’s imaginative journey (first person) through the 3 realms of the dead: Inferno (hell); Purgatorio (Purgatory); and Paradiso (heaven).  It is a mixture of the Bible, Roman Catholicism, mythology & medieval lore and included some elements of Islamic tradition (Hadiths).  It is intended to be literary, not theological.  It does reflect a deep yearning to understand the mysteries of life and death & has generated tremendous interest over the centuries.

 

Alter of Pergamum: * see Endnote 73

monumental construction built during the reign of Greek King Eumenes II, first half of the 2nd century BC in ancient Greek city of Pergamum (Asia Minor); 35.64 m. wide x 33.4 m. deep.