Some articles on mode, authentic modes, authentic, modes, authentics:
... In addition to the final and cofinal, every mode is distinguished by scale degrees called the mediant and the participant ... The mediant is named from its position—in the authentic modes—between the final and cofinal ... In the authentic modes it is the third degree of the scale, unless that note should happen to be B, in which case C substitutes for it ...
... which are lists of chant titles grouped by mode, appear in western sources around the turn of the 9th century ... oktōēchos, that is, with the four principal (authentic) modes first, then the four plagals, whereas the Latin modes were always grouped the other way, with the authentics and ... Later authors created confusion by applying mode as described by Boethius to explain plainchant modes, which were a wholly different system (Palisca 1984, 222) ...
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