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I scanned and uploaded Milos Velimirovic's article "The Byzantine Heirmos and Heirmologion" that appeared in Gattungen der Musik in Einzeldarstellungen, Gedenkschrift Leo Schrade, Munchen 1973, pp. 192-244.
Although somewhat dated, it is still a useful source of information. Some interesting highlights:
1) The well-known excerpt on the kanon composition by Theodosios the Grammarian is possibly not his, but of Choiroboskos, who, Velimirovic (following Beck) admits, possibly lived in the 7th c. (see p. 193 and footnote 17 on p. 197). Later research however showed that Choiroboskos lived in the 8th-9th c. (see here and here).
2) The 7th c. papyrus Rylands 466 contains two heirmoi (or two pieces that later did reach the Heirmologia) and several other unidentified stanzas (pp. 199-200). The high quality scanned images of this papyrus are available online here and here.
3) The Byzantine Heirmologia typologically can be classified in two groups: KaO and OdO groups. In both groups the heirmoi are arranged in the modal order (first the heirmoi of Mode 1, then the heirmoi of Mode 2 and so on), but while in the mss of the KaO group within each Mode the heirmoi are grouped according to the kanons (e.g. first the heirmoi of, say, the Nativity kanon by Damascene, then the heirmoi of some other kanon in Mode 1, then of the third one in Mode 1 and so on), in the OdO group the heirmoi within each Mode are grouped according to the Odes (i.e. first all the heirmoi of Ode 1 of Mode 1, then all the heirmoi of Ode 2 of Mode 1 and so on). The overwhelming majority of the old Greek Heirmologia mss belong to the KaO group, while the OdO group represents to Velimirovic (and other researchers of the epoch) some type of a mystery. The oldest ms of the OdO group known to Velimirovic was Paris suppl. gr. 1284, which he dates not later than the early 12th c. (see p. 224 and p. 234). The ms is a fragment consisting of two folia only. It has been described by A. Gastoué (see here) and I attach both the description and the photo of 1 folio. When Velimirovic was writing his article, it was known that the old Slavonic Heirmologia all followed the OdO order and they in fact probable reflect the tradition of the mid-11th c. Velimirovic readily admits the Byzantine origin of the OdO type in the Greek mss, but in absence of early examples of the Greek mss with OdO type he finds it puzzling to explain what was the origin of the OdO in the early Slavonic mss (see p. 228). Later research demonstrated that the OdO is norm in the 10th Georgian Heirmologia, which ultimately depend on the Palestinian sources, see the article by Metreveli and Outtier, "Contribution à l'histoire de l'Hirmologion: anciens hirmologia géorgiens", Le Muséon 88, 1975, pp. 331- 359. This is a French translation of the extensive Russian summary of Metreveli's monograph with additional remarks by Outtier. Although most Georgian Heirmologia of the 10th c. are of the OdO type, Metreveli identified three collections which she attributes to the KaO group. None of these collections is Heirmologion in a proper sense of the word: Sin. 26 contains only an index of the heirmoi in a manner described in the article (see pp. 334-335). The scribe of that part of the ms Iovane Zosime gives the reason for compiling the index, the essence of which can be summarised as follows: these particular heirmoi are indicated because their texts are given in full in the ms and whatever feast there is of whatever Mode one can use these heirmoi because of the need (the need of course here means that the user of the ms might simply not know the appropriate heirmoi of the kanon of the feast off heart and if he still wants to say the heirmoi, he can use some other, known ones). A small heirmoi collection in Sin. 34 consists of the heirmoi of 8 Resurrection kanons (1 kanon for each Mode) only and follows the part with Sunday hymnography. The Sunday office was of paramount importance and hence this little collection was primarily of practical importance (the heirmoi in the Resurrection kanons themselves are indicated with incipits only, the heirmoi collection tries to rectify this). The last ms that contains heirmoi arranged according to KaO is Sin. 20, but this is again a small collection with material for Modes 4, Pl. 1 and Pl. 4 only. The reason why only these heirmoi are included is that the ms contains material for weekdays of Modes 4, Pl. 1 and Pl. 4 and these heirmoi are needed for the respective kanons. Thus again the collection has a practical goal and is restricted to the needs of one ms only. My conclusion is that one cannot suppose existence of a KaO type Greek Heirmologion that served as a model for the Georgian KaO type Heirmologia. On the other hand there is no doubt that the Georgian OdO Heirmologia were inspired by the Greek OdO Heirmologia (there are three layers of the Georgian Heirmologia: the most ancient in this or that way probably depended on some Greek original, the later two redactions are obtained by gradual expansion of the ancient version by the newly translated Greek material or Georgian original hymnography). Finally, it should be mentioned that there exists the 8th c. palimpsest Garrett 24 kept in Princeton (the upper layer is Georgian, the lower layer of the text is Greek. As an additional curiosity, the Greek heirmoi are notated with some type of primitive notation; see Raasted's paper "The Princeton Heirmologion Palimpsest" in CIMAGL 62) that is the oldest surviving example of the OdO type Heirmologion (and it predates all the known Greek examples of the KaO type Heirmologion). There seems to be no reason to believe that the idea of the OdO type is in any way later than that of KaO. Of course this does not mean that the surviving Greek OdO type Heirmologia follow some ancient model (and are not for instance just excerpted from the KaO type Heirmologia), but the idea itself is old and as the Georgian sources would indicate, is Palestinian.
4) Velimirovic is careful not to claim that the KaO and OdO ordering did correspond to two different melodic traditions as well, although he seems to admit that some changes did take place in the melos of the Greek Heirmologion somewhere at the turn of the 13-14th c. The possibility of two different melodic traditions is worth investigation (and in particular according to the grouping into KaO and OdO type Heirmologia), although it will be hampered by the fact that the oldest notated Greek witness of the OdO type Heirmologion, Paris suppl. gr. 1284, consists of two folia only and is in paleobyzantine notation. Who knows, such studies might shed some extra light on emergence of the Heirmologion connected with the name of Ioannes Koukouzeles (cf. a relevant topic in the Greek part of the forum).
5) Velimirovic makes some interesting observations on the melodic structure of the Byzantine heirmoi (pp. 234-243).
Although somewhat dated, it is still a useful source of information. Some interesting highlights:
1) The well-known excerpt on the kanon composition by Theodosios the Grammarian is possibly not his, but of Choiroboskos, who, Velimirovic (following Beck) admits, possibly lived in the 7th c. (see p. 193 and footnote 17 on p. 197). Later research however showed that Choiroboskos lived in the 8th-9th c. (see here and here).
2) The 7th c. papyrus Rylands 466 contains two heirmoi (or two pieces that later did reach the Heirmologia) and several other unidentified stanzas (pp. 199-200). The high quality scanned images of this papyrus are available online here and here.
3) The Byzantine Heirmologia typologically can be classified in two groups: KaO and OdO groups. In both groups the heirmoi are arranged in the modal order (first the heirmoi of Mode 1, then the heirmoi of Mode 2 and so on), but while in the mss of the KaO group within each Mode the heirmoi are grouped according to the kanons (e.g. first the heirmoi of, say, the Nativity kanon by Damascene, then the heirmoi of some other kanon in Mode 1, then of the third one in Mode 1 and so on), in the OdO group the heirmoi within each Mode are grouped according to the Odes (i.e. first all the heirmoi of Ode 1 of Mode 1, then all the heirmoi of Ode 2 of Mode 1 and so on). The overwhelming majority of the old Greek Heirmologia mss belong to the KaO group, while the OdO group represents to Velimirovic (and other researchers of the epoch) some type of a mystery. The oldest ms of the OdO group known to Velimirovic was Paris suppl. gr. 1284, which he dates not later than the early 12th c. (see p. 224 and p. 234). The ms is a fragment consisting of two folia only. It has been described by A. Gastoué (see here) and I attach both the description and the photo of 1 folio. When Velimirovic was writing his article, it was known that the old Slavonic Heirmologia all followed the OdO order and they in fact probable reflect the tradition of the mid-11th c. Velimirovic readily admits the Byzantine origin of the OdO type in the Greek mss, but in absence of early examples of the Greek mss with OdO type he finds it puzzling to explain what was the origin of the OdO in the early Slavonic mss (see p. 228). Later research demonstrated that the OdO is norm in the 10th Georgian Heirmologia, which ultimately depend on the Palestinian sources, see the article by Metreveli and Outtier, "Contribution à l'histoire de l'Hirmologion: anciens hirmologia géorgiens", Le Muséon 88, 1975, pp. 331- 359. This is a French translation of the extensive Russian summary of Metreveli's monograph with additional remarks by Outtier. Although most Georgian Heirmologia of the 10th c. are of the OdO type, Metreveli identified three collections which she attributes to the KaO group. None of these collections is Heirmologion in a proper sense of the word: Sin. 26 contains only an index of the heirmoi in a manner described in the article (see pp. 334-335). The scribe of that part of the ms Iovane Zosime gives the reason for compiling the index, the essence of which can be summarised as follows: these particular heirmoi are indicated because their texts are given in full in the ms and whatever feast there is of whatever Mode one can use these heirmoi because of the need (the need of course here means that the user of the ms might simply not know the appropriate heirmoi of the kanon of the feast off heart and if he still wants to say the heirmoi, he can use some other, known ones). A small heirmoi collection in Sin. 34 consists of the heirmoi of 8 Resurrection kanons (1 kanon for each Mode) only and follows the part with Sunday hymnography. The Sunday office was of paramount importance and hence this little collection was primarily of practical importance (the heirmoi in the Resurrection kanons themselves are indicated with incipits only, the heirmoi collection tries to rectify this). The last ms that contains heirmoi arranged according to KaO is Sin. 20, but this is again a small collection with material for Modes 4, Pl. 1 and Pl. 4 only. The reason why only these heirmoi are included is that the ms contains material for weekdays of Modes 4, Pl. 1 and Pl. 4 and these heirmoi are needed for the respective kanons. Thus again the collection has a practical goal and is restricted to the needs of one ms only. My conclusion is that one cannot suppose existence of a KaO type Greek Heirmologion that served as a model for the Georgian KaO type Heirmologia. On the other hand there is no doubt that the Georgian OdO Heirmologia were inspired by the Greek OdO Heirmologia (there are three layers of the Georgian Heirmologia: the most ancient in this or that way probably depended on some Greek original, the later two redactions are obtained by gradual expansion of the ancient version by the newly translated Greek material or Georgian original hymnography). Finally, it should be mentioned that there exists the 8th c. palimpsest Garrett 24 kept in Princeton (the upper layer is Georgian, the lower layer of the text is Greek. As an additional curiosity, the Greek heirmoi are notated with some type of primitive notation; see Raasted's paper "The Princeton Heirmologion Palimpsest" in CIMAGL 62) that is the oldest surviving example of the OdO type Heirmologion (and it predates all the known Greek examples of the KaO type Heirmologion). There seems to be no reason to believe that the idea of the OdO type is in any way later than that of KaO. Of course this does not mean that the surviving Greek OdO type Heirmologia follow some ancient model (and are not for instance just excerpted from the KaO type Heirmologia), but the idea itself is old and as the Georgian sources would indicate, is Palestinian.
4) Velimirovic is careful not to claim that the KaO and OdO ordering did correspond to two different melodic traditions as well, although he seems to admit that some changes did take place in the melos of the Greek Heirmologion somewhere at the turn of the 13-14th c. The possibility of two different melodic traditions is worth investigation (and in particular according to the grouping into KaO and OdO type Heirmologia), although it will be hampered by the fact that the oldest notated Greek witness of the OdO type Heirmologion, Paris suppl. gr. 1284, consists of two folia only and is in paleobyzantine notation. Who knows, such studies might shed some extra light on emergence of the Heirmologion connected with the name of Ioannes Koukouzeles (cf. a relevant topic in the Greek part of the forum).
5) Velimirovic makes some interesting observations on the melodic structure of the Byzantine heirmoi (pp. 234-243).