For medieval and Renaissance musical training, available treatises give some interesting clues, sometimes supplemented by evidence from the careers of noted musicians and composers.
Of course, lots of instrumentalists and minstrels might be very skilled performers without lots of formal training, and the remarks that follow might apply more to composers and performers of composed polyphony, etc.
Many medieval and Renaissance authors draw a distinction between cantus planus or plainsong, learned first, and musica mensurata or measured polyphony. Interestingly, the 16th-century Spanish theorist Tomas de Santa Maria refers to polyphony as canto de organo, a name which reminds me of the medieval term organum for "organized ensemble music" and usually more specifically polyphony.
At any rate, the discipline of plainsong would include the learning of Guido's gamut and of solmization, which could be applied not only to singing, but to the keyboard. Thus Tomas de Santa Maria discussed the "defects" or unsupported accidentals of a keyboard in the prevailing tunings of his times (very likely meantone) in terms of missing notes in hexachords built on various degrees.
The study of discant, counterpoint, or musica mensurata, or canto de organo (all terms used for polyphonic techniques at different times) would typically focus not only on composition but on improvisation to a given melody, for example a liturgical chant. Thomas Morley (1597; 2) gives us a feeling of this tradition when he presents a kind of debate between two brother students, one of whom celebrates the tradition of "discant" (in the sense of clever improvisations to a given melody); Morley pokes fun of the "crude" techniques followed by such traditionalists, and also expresses a view that writing compositions is a worthier goal than performing extemporized discant which lasts no longer than it is heard. Thus a Renaissance author treats the perennial issue of composition and improvisation.
One term which may give some clue as to the tradition of performance is tenorista, which literally translated, might mean "one who sings the tenor part" (typically the basis for the organization of a composition in medieval and early Renaissance times). However, it would seem that actually this position is more analogous to that of conductor: the tenorista would be responsible for coordinating the other performers. Thus the comment of one writer around the 13th or 14th century that a group of singers without their tenor would be like a company of soldiers without their captain.
At least in the early 17th century, as the discussion on basso continuo technique has suggested, traditional counterpoint was the basis for continuo technique: in 1607, Agazzari notes that the student should already be familiar with the rules for proceeding to the nearest consonance and resolving dissonances, and that those who do not know must learn.
Similarly, both Zarlino (1558) and Tomas de Santa Maria (1565; 1) combine traditional two-voice counterpoint with extended discussion of harmony for three or more voices, where the arrangement of the consonances in compound sonorities becomes a vital ingredient of technique and style.
Likewise in the mid-17th century, Christoph Bernard (c. 1655) proceeds through two-voice counterpoint to some guidelines for multi-voice composition and a fascinating discussion of the musical "rhetoric" used by Monteverdi and other recent composers, using Greek terms to classify the idioms or musical "figures of speech" used to good effect in this modern music.
While the treatises are only one source, they can give at least some clues about the musical culture and training of these centuries.
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