A tonary is a liturgical book in the Western Christian Church which lists by incipit various items of Gregorian chant according to the Gregorian mode (tonus) of their melodies within the eight-mode system. Tonaries often include Office antiphons, the mode of which determines the recitation formula for the accompanying text (the psalm tone if the antiphon is sung with a psalm, or canticle tone if the antiphon is sung with a canticle), but a tonary may also or instead list responsories or Mass chants not associated with formulaic recitation. Although some tonaries are stand-alone works, they were frequently used as an appendix to other liturgical books such as antiphonaries, graduals, tropers, and prosers, and are often included in collections of musical treatises. Function and form Tonaries were particularly important as part of the written transmission of plainchant, although they already changed the oral chant transmission of Frankish cantors entirely before musical notation was used systematically in fully notated chant books. Since the Carolingian reform the ordering according to the Octoechos assisted the memorization of chant. The exact order was related to the elements of the "tetrachord of the finales" (D—E—F—G) which were called "Protus, Deuterus, Tritus", and "Tetrardus". Each of them served as the finalis of two toni—the "authentic" (ascending into the higher octave) and the "plagal" one (descending into the lower fourth). The eight tones were ordered in these pairs: "Autentus protus, Plagi Proti, Autentus Deuterus" etc. Since Hucbald of Saint-Amand the eight tones were simply numbered according to this order: Tonus I-VIII. Aquitanian cantors usually used both names for each section. The earliest tonaries, written during the 8th century, were very short and simple without any visible reference to psalmody. Tonaries of the 9th century already ordered a huge repertoire of psalmodic chant into sections of psalmtone endings, even if their melody was not indicated or indicated by later added neumes. Most of the tonaries which have survived until now can be dated back to the 11th and 12th centuries, while some were written during later centuries, especially in Germany. The treatise form usually served as a bridge between the Octoechos theory and the daily practice of prayer: memorizing and performing the liturgy as chant and reciting the psalms. This can be studied at a 10th-century treatise called Commemoratio brevis de tonis et psalmis modulandis, which used the Dasia-signs of the Musica enchiriadis treatise (9th century) in order