Baroque
Harvard defines baroque as "the period of Western music history extending from the end of the 16th century to ca. 1750; also the musical styles of that period." (It does not deal with other aspects of the same name, specifically architecture.)
The article deals first with the history and usage of the term itself, suggesting its first uses (some time in the 18th century) and two possible derivations that may have been in mind when it was use: one from the Portuguese barroco, or irregular pearl (the generally accepted derivation); the other from the Italian baroco, "a logical term referring to an extraordinary type of syllogism." The term baroque began being used to describe the period in the 18th and 19th centuries "with pejorative intent."
Heinrich Wölfflin, in his Principles of Art History (1915), attempts to distinguish the Renaissance from the Baroque with the "plastic arts through five neat stylistic antitheses." Scholars eventually turned to identifying pieces by their "contemporaneity" to other art forms of the day and their "spiritual and artistic unity" with the post-Renaissance period. Claude Palisca, however, "singles out a static, rationalistic view of human affections and a conviction that music should arouse these affections as two decisive cultural features reflected in all Baroque music." The "spiritual and artistic" qualities we label as "baroque" are present as far back as the late 15th century, so there cannot really be said to be any "clear point of demarcations between the Renaissance and Baroque periods." We have established dating methods "by scholarly convention."
With regard to the style of Baroque music, we see a "homophonic texture in which the uppermost part carries the melody over a bass line with strong harmonic implications," which leads to the development of the basso continuo. Performing forces that differ in function but similar in timbre are united, and specific works organize more or less into three categories: church, chamber, and theater.
"The Baroque period is usually divided into three subperiods." The establishment of monody marks the early Baroque (ca. 1590-1640), and "the pitch hierarchies of modern tonality begin to emerge from the freer tonal practices of the polyphonic madrigalists." The middle Baroque (1640-1690) "is a period of consolidation," where "dissonance is more strictly controlled" and techniques used in opera are applied to other forms. The late Baroque (1690-mid 18th century) is indicated by the presence of large formal patterns, and the music becomes characterized by "insistent, motoristic rhythms that emphasize motives designed to project a single, static affection."
Hemiola
Hemiola, the ratio 3:2, is defined by Harvard as "the lengths of two strings that together sound a perfect fifth" when applied to pitch, and "the use of three notes of equal value in the time normally occupied by two notes of equal value" when applied to rhythm. Its presence in mensural notation is indicated by "red notes if the prevailing notation is black, or black notes if the prevailing notation is white." Hemiola was a common compositional technique of the 15th-century composers, and was used in French Baroque courantes, Viennese waltzes, and the music of Schumann and Brahms.
Mannheim School
The Mannheim School was "a group of composers and performers active at the court of Mannheim in southwestern Germany during the 18th century." Mannheim's musical and cultural significance began in 1720, but "its principal fame...was achieved during the reign of Elector Carl Theodor (1742-78)," who assembled the Mannheim orchestra, "widely regarded as the fines in Europe."
Mannheim is generally considered to be "the residence of a talented, up-to-date, though not pre-eminent group of composers" by scholars today. The symphonic style at Mannheim "unquestionably originated in Italian opera," as did the "'Mannheim sigh,' or melodic appoggiatura, the ÔMannheim rocket' (an arpeggio theme rising through several octaves), and numerous other melodic figures or Manieren associated...with Mannheim." It can be safely said that Mannheim adapted and extended the Italian style to the concert symphony, and provided "the earliest consistent use of four movements in the symphony...and the idiomatic, frequently virtuosic treatment of the orchestra."
Whether there was actually a "Mannheim school" per se is debatable. Early references are about "either...violinists at the electoral court or, more generally, to every aspect of music there, including performance, composition, and taste." The style of the first Mannheim composers refute the premise of a "compositional school," though "such second-generation" composers as Anton Fils, Christian Cannabich, and Carl Joseph all shared a signature compositional quality that can be said to be a "'school' of composition."