Resources for medieval musicology and liturgy
While the synthesis between harmonikai and Carolingian chant theory can be studied between Boethius as a Carolingian source (we have no earlier manuscripts) and Hucbald, there has not really survived an early source which testified about the synthesis between the Byzantine octoechos and the Ancient Greek tropes, I mean how the Dorian mode on E—e became that of D—d. Harold Powers (Grove article "mode") wrote:
From the 6th century to the early 9th, when the repertory of Western plainchant achieved its basic forms, there is no record of descriptive or theoretical sources, and of course no notated music. Towards the end of this period a system of eight modal categories, for which there was no genuine precedent in Hellenistic theory, came to be associated with the rapidly stabilizing repertory of Gregorian chant. This system was proximately of medieval Byzantine origin, as indicated by the non-Hellenistic Greek names of the modes in the earliest Western sources from about 800.
The origins of the Eastern Christian system of eight modes – usually called Oktōēchos – are not entirely clear; but it seems more than probable that it was not delimited purely or even primarily by musical criteria. In any case, the octenary property of the modal system of Latin chant in the West was of non-Latin origin
Then he abruptly switched back to the Western synthesis and the reception of the Hagiopolites in Carolingian tonaries. In my imagination, the synthesis can probably explained, that Dorian as the main trope was defined by the fixed degrees of the tone system, the frame of the tetrachords had been B—E—a—b—e—aa, and it must have changed somewhen to A—D—G—a—d—g—aa. But so far, no source had been found which could offer any evidence. Concerning Carolingian theory, this could simply have been a misreading of Boethius, but please read the very original opinion in Pavlos' article.
Please bear in mind some slight modifications suggested by the author. If Pavlos agrees, I can also upload a pdf with the modifications added as a commentary, or Pavlos might upload an updated version of the text as pdf.
The happy meeting between Epitrope and Aristoxenos
While Chrysanthos linear description of the tetrachord α'—δ' (12:11 x 88:81 x 9:8 = 4:3, 151’ + 143’ + 204' = 498’ cents) was based on corrupted arithmetics (108/96/88/81 = 12 + 8 + 7: 8 + 7 + 12 turned into 9 + 7 + 12, which count the fourth into 28 and therefore into 4 minor tones, but not 5 half tones of 6 units), a patriarchal synode in 1883 decided for an equal tempered representation which counted the fourth in 30 units (10 + 8 + 12, 166' + 133' + 200' = 500' cents). Aristoxenos divided the tetrachord into 60 parts. The exact method of division is still a controversial matter.
A 24 mode system mentioned in the Alchemy compilation
The reconstruction of the hardly readable paraechoi was easily done, because they are mentioned later:
Ὥσπερ δὲ τεσσάρων μουσικῶν γενικωτάτων στοιχείων, αον, βον, γον, δον, γίνονται παρ῾ αὐτῶν τῷ εἴδει διάφορα στοιχεῖα κδ῾, κέντροι καὶ ἶσοι καὶ πλάγιοι, καθαροί τε καὶ ἄηχοι [καὶ παράηχοι]· καὶ ἀδύνατον ἄλλως ὑφανθῦναι τὰς κατὰ μέρος ἀπείρους μελωδίας τῶν ὕμνων ἣ θεραπείων ἣ ἄποκαλύψεων ἣ ἄλλον σκέλους τῆς ἱερᾶς ἐπιστήμης, καὶ οἷον ῥεύσεως ἣ φθορᾶς ἄλλον μουσικῶν παθῶν ἐλευθέρας,
My translation as an alternative to Gombosi's one (1940, 40):
As there are 4 basic elements, there are also four musical ways, the πρῶτος, the δεύτερος, the τρίτος, and the τέταρτος, and by their formulas the same generate 24 different elements: the  κέντροι (central),  ἶσοι (basic), and  πλάγιοι (plagal), the  καθαροί (kathartic),  ἄηχοι (aphonic), and [4 παράηχοι (paraphonic)]. Hence, it is impossible to create something outside those infinite melodies of hymns, treatments, revelations, and of other parts of the Holy Wisdom, which is free from the irregularities and spoilages of other musical passions (πάθη).
The text according to Marcellin Berthelot's edition (1888, ii:219):
Ὥσπερ δὲ δ´ ὄντων τῶν μουσικῶν γενικωτάτων στοχῶν, α´, β´, γ´, δ´, γίνονται παρ῾ αὐτοῖς τῷ εἴδει διάφοροι στοχοὶ κδ´, κέντροι καὶ ἶσοι καὶ πλάγιοι καθαροί τε καὶ ἄηχοι · καὶ ἀδύνατον ἄλλως ὑφανθῦναι τὰς κατὰ μέρος ἀπείρους μελῳδίας τῶν ὕμνων, ἣ θεραπειῶν ἣ ἄποκαλύψεων, ἣ ἄλλου σκέλους τῆς ἱερᾶς ἐπιστήμης, καὶ οἷον ῥεύσεως, ἣ φθορᾶς, ἢ ἄλλων μουσικῶν παθῶν ἐλευθέρας ·
Here another version clearly in the context of the book ceremonies which is closer to Gombosi's quotation (1888, iii:434):
Ὥσπερ δὲ τεσσάρων ὄντων μουσικῶν γενικωτάτων στοχῶν, Α Β Γ Δ, γίνονται παρ῾ αὐτῶν τῷ εἴδει διάφοροι στοχοὶ κδ´, κέντροι καὶ ἶσοι καὶ πλάγιοι, καθαροί τε καὶ ἄηχοι <καὶ παράηχοι> · καὶ ἀδύνατον ἄλλως ὑφανθῆναι τὰς κατὰ μέρος ἀπείρους μελῳδίας τῶν ὕμνων ἣ θεραπειῶν, ἤ ἄποκαλύψεων, ἤ ἄλλου σκέλους τῆς ἱερᾶς ἐπιστήμης, καὶ οἷον ῥεύσεως ἤ φθορᾶς ἤ ἄλλων μουσικῶν παθῶν ἐλευθέρας, τοῦτο κἀνταῦθα ἔστιν εύρεῖν τὸν δυνατὸν ἐπὶ τῆς μιᾶς καὶ ἀληθοῦς κυριωτάτης ὕλης, τῆς ὀρνιθογονίας.
translated as follows (1888, ii:212):
De même que les lignes musicales les plus générales étant au nombre de quatre, Α, Β, Γ, Δ, on forme avec elles 24 lignes d'espèces diverses ; et qu'il y a aussi des centres et des lignes obliques, selon qu'il a été dit à propos des sons, et attendu qu'il est impossible de composer autrement les mélodies innombrables des hymnes, pour le service (du culte ?), la révélation, ou quelque autre partie de la science sacré... (Phrase inintelligible.)
Alia musica compilations
Some of the sources have been already published online, see my list of tonaries. There you will also find links to two corresponding editions. There is a third edition by Karl-Werner Gümpel of the "Nova expositio" part as he found it in a manuscript of the Catalonian monastery Ripoll.
there is no need to make any separate PDF of that article; including the corrections. It would be better to let things be at the state they are. In the future I'll return to these topics with improvements and a more detailed exposition.
Instead, for the moment I'd preferred to concentrate on a methodological issue posed at that paper, namely critical textualism and the way editors improve the texts of the treatises they edit. The case I noticed there, wasn't only an "innocent" philological reconstruction of a text and I explained how that the text has to be read without major "additions / emendations," keeping the reading of the Mss. I remember another such a case of "text correction" where Otto Gombosi modified the Medieval Greek musico-alchemical texts (in relation to the texts as edited by the "Ministre de l'Instruction publique et des Beaux-arts" Marcellin Berthelot [and Ch.-É. Ruelle]) and then translated his reconstruction in German.
(Greek or Latin sources could also be considered not as isolated self-referential environments.) As for the various explanations on the relation of the "Western" and "Eastern" modal systems it would be better to wait for the time that all the new editions of Hagiopolítis, Alia Musica, (and the treatise attributed to Regino of Prüm, of course) will be released. The loose willingness for further research on the subject (and, at the same time, indicative for the state of the research, what a pity, of the "Eastern" systems [I do not include, here, the Ottoman times]) is reflected on the selective way that Otto Gombosi' s (to use once again him as an example) work is cited. Thus, when we read texts about the Latin "regime / domain of musicography" of the Medieval Music Theory, Otto Gombosi is hardly cited, but when the same modern author in the same text reasonably realizes the need to refer to (/its relation to) the Greek "regime / domain of musicography" then ... s/he feels contended to make use of him!
Connection between musical formulas and Oktoechos system I showed in my books :
One more methodological remark on the “generalized” approach that “some preexisting” melodies adapted to the “later” Octoechos (of the “West” or the “East”) system/s.
One have at first to understand both the “medieval theory” and the Melodies (via notation) and then to theorise on the adaptation of the one to the other. Who can claim that the state of understanding of both is adequate enough to proceed on such a level of “post-theorization”?
I definitely agree with your last reply.
I will upload your article as pdf, as soon as somebody will ask me to do so, because I realised that the server of CP is not properly working.
I think, nobody so far has tried to fill this gap, because there is a lack of sources. We are forced to base all our conclusions on vague hypotheses, and so are those by Jørgen Raasted (1966, 7). He quoted Oliver Strunk (1942) and he said, that there was no need to repeat his arguments for a use of tetrachordon ditoniaion. Most of us know this essay very well, and there is not such an argument. On page 192 Oliver Strunk drew very careful conclusions:
The precise nature of the steps within this series remains for the present unknown; for all that we can learn from the Papadike, the step α [prôtos] to β [devteros] may be a whole tone, a half tone, or some other larger or smaller interval. … If we may assume, however, that the interval α [prôtos] to δ [tetartos] is a perfect fourth–a reasonable assumption, to say the least, for a tetrachordal system based on any other interval is virtually inconceivable–the interval δ [tetartos] to α [prôtos], as the difference between an octave and two fourths, becomes a whole tone and the remaining intervals fall readily into line.
We all know that the diazevxis has the proportion 9:8, but it tells us nothing about the tetrachord division!
So it is mainly Egon Wellesz who followed the "Western mainstream," and Chrysanthos himself who wrote in his Theoretikon mega that there is a difference between the Ancient Greeks and the Europeans who followed them by the use of a ditonal tetrachord, while Greek and (other) Ottoman musicians did not. I personally do not agree with this point of view, because the ditonal tetrachord is just one possible choice within all these various divisions testified within Ptolemy's Harmonics.
One reason, why Oriental musicians have a more profound understanding of harmonics than Latin cantors (our discussions here about absonia had simply confirmed it), is, that they use three different intervals for the division of the tetrachord. If we just imagine two transpositions, as they could appear in rather complex compositions of the sticherarion, we soon realise that this case helps to clear all these clouds in Otto Gombosi's argumentation:
Concerning the alchemy treatises, I found that his translation is rather crap.
First of all, the names of the echoi as "elements" seem to me rather inspired by treatises of music therapy (since the author referred to pathe and ethe, while phthorein is still used in pejorative way, in the Hagiopolites phthora is indeed a central modal category), so that I am not sure that this passage is really talking about ecclesiastic chant.
Second, the alchemy treatises are compilations of different treatises, parts of it might really belong to earlier authors like Zosimos of Alexandria, but this seems rather unlikely for the date of those parts which are concerned about modal analogies. But in this context, I am not convinced that stocheios was meant as a music theoretical term.
Third, Otto Gombosi's arguments concerning μεταβολή κατὰ τόνον testify a profound confusion that music theorists suffer, whenever they try to study a Greek harmonics treatise. We should be grateful, that he risked so much. After all my studies, I have only one explanation for it, it is the heritage of the Carolingian renaissance. Maybe if Europeans were less destructive, as they were during the Golf wars, some Orientalists would have had a chance to retranslate some unknown Greek harmonics treatises from some Arabic and Persian translations. Here we realise, how deep this particular feeling of Western inferiority, an abyss indeed, goes, despite all their deliria they used to have of Babylonia. Concerning Egon Wellesz you already quoted Alexander Lingas' article, who wrote explicitely about the Western background of some scholars. Some Greeks, and they find themselves in the same boat with the whole population of the Balkans, think that they are not sure that they are Europeans, but this kind of ignorance should be absolutely not their problem.
But you are one of those few Greek scholars who study Latin treatises as sources for Byzantine music as well. When I wrote my paper about the intonation AIANEOEANE I had no access to the sources, so I could not yet reconstruct this intonation from adiastematic neumes. But now we have and I will not wait for the next edition, since I never trust any edition, as long as I can study the original. And after all, the old editions are not so bad. Chailley did not avoid the trouble to edit all the glosses of the St Emmeram collection (though his datation to the 9th century, on the base that some Hucbald can be found in it, seems to be rather lunatic), while Martin Gerbert already made a good edition of another compilation which you can find in Pa 7211, but this source has no notated echemata.
On the other hand, I advice to distrust the so-called evidence that musicians simply do as theorists write. Some readers might be surprised, when they read in Jerome of Moravia's that Parisian cantors used to sing a kind of nenano tetrachord (Cserba):
Gaudent insuper, cum modum organicum notis ecclesiasticis admiscent, quod etiam non abjicit primus modus, necnon et de admixtione modorum duorum generum relictorum. Nam diesim enharmonicam et trihemitonium chromaticum generi diatonico associant. Semitonium loco toni et e converso commutant, in quo quidem a cunctis nationibus in cantu discordant.
Now, after this introduction, I would like to ask you that I understand your article better. What are exactly the proportions behind that tetrachord which you described as 12—9—9 or 9—9—12 with respect to the Ancient Greek tetrachords?
Thank you for your reply but It would be better not to upload my paper at a separate “state,” even in the case, as you referred, the server of CP is not working properly. Anyone who is interested could find it his/herself.
What I stressed here about Otto Gombosi’s set of articles wasn’t, of course his German translation or the date of the Achemical Corpus, but mainly his “reconstruction” of the Greek text and the “selective” way his is cited by modern musicographers.
New editions of treatises are always welcome.
As for your question, the intervals, as proportion, are the ones of the omalón diátonon, but as I said before I prefer here to remain on methodological issues and not to proceed on further documentation.
In that case, they will have to adress you personally, I just wanted you to know about these problems.
Concerning the alchemy sources (you were quite clear about the fact that Otto Gombosi adapted the Greek text to his interpretation), I hope that they will be available soon, the last time I looked for them, there were still no reproductions available.
Concerning the Persian and the Arabic system, I recommend the often quoted article by Eckhard Neubaur, because the old Persian term concerning melodies is cycle parde (pl. advār) and they were seven, I assume that in Ottoman Turkish it was also used as name for fret, while the contemporary Persian term was dastgah. The concept of naġme was soon after Al-Kindi associated with a poetic concept of rhythm (īqā‘). This association was called "path" (ţariqā) and became soon a problem for creative musicians like Ziryab, the student of Ishaq al-Mawsili. Concerning al-Kindi's translation of "oktoechos", Neubauer was so careful to write that his odd terminology was obviously an early effort to translate Greek terms into Arabic (he could not rely on the translations in Baghdad like al-Farabi, because they were made between the 9th and the 10th century). It does not mean that you are wrong, since στοχεία—the "elements"—was an alchemistic explanation for the 4 elements of the tetrachord.
Concerning Ptolemy's ("our") equal diatonic division (I checked in his treatise). It is:
10:9 x 11:10 x 12:11 = 4:3
Hence, there is no 12. But if 9 is supposed to by a rough description of 12:11, I come to this tetrachord:
9:8 x 12:11 x 88:81 = 4:3
The earliest evidence is al-Farabi (10th century) who found it on the ud's keyboard, if you use the ring finger fret named after the famous Baghdadi ud player Zalzal. Chrysanthos used the same tetrachord and his arithmetic description 12 + 9 + 7 was not correct, it must be 12 + 8 + 7, because the middle and the minor tone are very close to each other.
I mean, quite possible that you are right, since Neubauer found this anecdote about a famous singer of Baghdad, who played a Greek song for Ishaq according to the style and ornaments of his school in order to challange him. If it was so far from the Greek song, it could hardly have challenged him. It could have been as well derived from Greek musicians, for reasons which you will find in Neubauer's essay.
I will not comment here nor on this anecdote referred also by (Max Haas and Owen Wright) nor on the other Neubauer’s important articles nor on the scientific method of Ptolemy’s Harmonics. What I want to add here is a comment on a description about me. I understand its positive meaning, and thank you for that, but “But you are one of those few Greek scholars who study Latin treatises as sources for Byzantine music as well.” is not my case. So I remain on methodological issues; such a description gives the impression that someone fills an intellectual box with data from many different and exotic areas which s/he considers as “relative” (“interdisciplinary approach” as they say; the issue what is relative or not is the key issue here), taxonomizes and hierarchizes them into the box the way that fit to his/her agenda, and in the end takes out from that box “the conclusion” like the magician takes out the rabbit from his hat and impress his/her audience.
You can trust me, I think your study is just worth to be discussed, and not just by Byzantinists also by scholars occupied with Latin sources, because you had been very courageous and far beyond the horizon of Harold Powers and this should provoke further questions and discussions. The essays of Owen Wright and Eckhard Neubauer can be read indeed like a very inspired correspondence between the authors.
But in this particular case, Ptolemy translated into Aristoxenian units was not the right hat, unless the rabbit is 11 + 10 + 9 which might have been closer to the homalon proportions.
I just suggested to change to another hat, so that the magic of 12 + 9 + 9 ist still there, which did not fail to make a great impression on me.
As long as we are focussed on the Hagiopolite oktoechos (because of the tropologion papyri), the dominance of a pure diatonic oktoechos turned the phthora into the most important category of monastic chant, even if we do not believe in Vogel's enharmonic hypothesis. This means, that this music was in a radical way different from Ancient Greek music, in fact, it was rather invented to be different from the Constantinopolitan cathedral rite and the kontakia. The only thing we can study in the parpyri are modal signatures, if the protos signature meant a mode on a, b natural or E remains hypothetical.
Thanks to the magic of 12 + 9 + 9 (in descending direction), it is no longer necessary to answer the difficult question, whether the fixed degrees of the tone system (ἑστῶτες φθόγγοι) had changed their position from α'—πλ β' to δ'— πλ α', and you even could integrate the ditonal diatonic 12 + 12 + 6 in α'—πλ β' to argue in favor for the tetrachords in ecclesiastical chant, just in that sense how Chrysanthos wrote about the Ancient Greek and you will even satisfy Raasted!
No doubt that this is an extremely harmonising approach to harmonics!
In its essence, this is very similar to the chromatic trick with the phthora nenano, they can never tell you, whether it should be sung on the phthongos of the diatonic protos or devteros, νε- is always the first element of the tetrachord (α') and it leads the psaltes to the last one (δ') during an initial intonation… and always ends as plagios devteros according to Manuel Chrysaphes (at least this was the rule by the end of the Empire, but it likely already existed for centuries)—with the result that πλ α' finally became the phthongos of plagios devteros, too.
With Ptolemy's διατονὸν ὁμαλὸν this would never work, with Chrysanthos' διατονὸν μαλακὸν it does, if we would like to believe that it already existed in Damascus since the 7th century, when Arab music was still created as it had been later described in the treatises of the autochthonous music theory.
À propos Hagiopolites. I would like to ask you another question. You wrote:
There is a passage in the Hagiopolites Ms., which describes the intervallic content of the tetrachords in the Aristoxenian manner referred to previously. For Aristoxenians in general, there are only two divisions of the diatonic tetrachord and unlike the five of Ptolemy! For Aristoxenos the two diatonic divisions are the intense diatonic, almost identical to the ditonal diatonic, and the soft diatonic. In Hagiopolites, as well as in its 10th-century predecessor the Anonymous Bellermani, there is a slight difference from Aristoxenos. Bellerman, the former editor of this Anonymous treatise—as well as the latter Najock—felt the need to insert a sentence into the text, disregarding the manuscript tradition; instead of including the medieval intervallic content of (9-9-12) which originally existed, they reconstructed the text according to its Ancient content giving (6-9-15)! Raasted, in his preliminary edition of Hagiopolites however, avoided this reconstruction.
Would you like to quote this passage, even if it might be a hypothetical reconstruction?
What is the manuscript tradition of the Hagiopolites, since we have only this chant manual for centuries? Or do you refer here to traditions of harmonikai with the distinction between Aristoxenos and Ptolemy?
Do you think, that the Hagiopolites did refer to certain proportions by the use of Aristoxenian measures or is this a hypothesis, that there was once a passage which has been lost in both fragments, so that you prefer to wait for a newer critical edition which will also include other sources than the earlier ones by Raasted and Najock?
It would be better not to proceed to comment in full the documentation of that paper in order to avoid confusion between a) documentation, for example as that Proportions are referred to in Bryénnios treatise and Mória in ANONYMUS III / Hagiopolítis and b) other issues “not relative” for my methodological approach like the theoretical work of Chrýsanthos (19th Ottoman century). It is better not to give any exotic flavor to the issue.
But let’s continue the skepticism about methodological issues, established subjective beliefs and about the low level of the research of the Medieval Music Theory as described in the Greek sources. Should we suppose that the intervals of Ecclesiastical music in Greek were the ditonal (i.e. tones and semitones, just to make things easier to understand all the people who read these posts), then what leads to us to suppose that the interval between echos protos and echos deuteros is tone and not semitone (and the one between deuteros and tritos is semitone and not tone) since the notation is not intervallic and in the theoretical treatises found within the papadikai* there is nothing relative? What inspired the belief to suppose that the relation between Echoi and intervals of Ecclesiastical music in Greek is the one commonly believed to?
* Especially for the Greek case (instead of the Latin where the use of Boethius is almost universal) is believed a “generalized” high degree of isolation of the audiences of the treatises of Ecclesiastical and Ancient Greek musical treatises! And Otto Gombosi in the end of the set of his articles he emphasized: mit einem Wort: auf das spezifisch Kirchentonartlische der Kirschentöne, von keinem Einfluss gewesen sein kann.
I will answer all your questions because they go very far, but I also would like to remind you to answer mine, if you please.
I just found in Chrysanthos' suggestion the right hat which fits to your rabbit and it is an easier reference point for the present and its New Method than Otto Gombosi. Nevertheless, proportions were already well-known during the 9th century, just a pity that he avoided to mention al-Farabi which would have been nearly contemporary to the Hagiopolites.
With exotism you have a very good point to let answer history those questions, which 9th-century authors did not offer to us. Then let us leave the Ottoman period of Chrysanthos, when the distinction between esoteric and exoteric music was a pure formality depending on the printed publication, which did not tell us anything about the secrets of the former one.
Jerusalem, Damascus, and the attraction of Hejaz
Let us now have a look on the time span between the 7th and the 10th century (to Bryennios I would like to come later). The whole span is absorbed by a Saint called John of Damascus—at least according to the point of view of most of the Orthodox singers I met so far, who are usually more or less students of Chrysanthos. His Syrian name was Yahyā ibn Manşūr ibn Sarjun, his family was Christian like the majority of the Damascene population of his time (the town had been conquered by Khalid ibn al-Walid in 635), but one of the most privileged ones in the caliphate, because his grandfather, probably already logothetes of Byzantine Damascus, opened the gates for the conquerors in order to get rid of the Byzantines (there must have been a conflict between Syrian and Byzantine Christians)—later during the 9th century, two Patriarchs of Jerusalem were descendants of the Mansur family.
John's father had one of the highest offices at the court of Damascus, John grow up together with Muawiya's children, his father was even entrusted with the caliph 'Abd al-Malik who later expelled Christians from offices of caliphate administration, his son Yahya might have already followed him, before he was forced to leave. Yahya was posthumously anathemised under his Syrian name by the iconoclastic council in 754, the next council declared him as a Saint. He called himself Ioannes, when he became a monk at Mar Saba in the Patriarchate of Jerusalem, but it was just the Greek form of his Syrian name. Some believe that he was educated by a Calabrian monk Kosmas, who was kidnapped by Saracene slave traders and sold on the slave market of Damascus, where his father bought him, that he will serve as a teacher for John, when he was already over 20. Ioannes as a hieromonk wrote mainly dogmatic treatises, whose rhetoric polemic style had been imitated in several languages over centuries, but he also contributed as a hymnographer in the monastic genres of the oktoechos reform of 692.
During the time of Arab Sicily, Greek monks in Southern Italy had always been one the main targets, because well educated Greeks had a high value at the slave markets of the whole Mediterranean (but Greek monks in Sicily and Calabria had a very particular and anachoretic tradition within Eastern Orthodoxy). Ioannes was obviously not the author of the Hagiopolites, even if some polemic phrases against certain phthorai and mesoi were taken from the council acts of 692, the author was rather a monk at the Stoudios Monastery of Constantinople short after the generation of Iosef the Hymnographer, and the treatise served as an introduction of a new kind of tropologion which favoured compositions of Ioannes, his foster brother Kosmas of Maiouma, and Andrew of Crete. But they had also new compositions like those by Joseph the Hymnoprapher (ca. 816-886), and it is said that Ioannes avoided the two "exotic" phthorai strictly and composed ἀπὸ τὴν μουσικὴν, while Joseph (a Sicilian with a quite similar destiny like Ioannes' teacher, only that he fled from Italy as a young boy, but when he followed later a papal invitation he ended up to be sold as a slave in Crete, exactly like Kosmas), as a composer of the 9th century, obviously liked them.
Max Haas wrote that the same word μουσική was used by Arabic treatises the 9th and 10th century to make a difference between the science ἁρμονικαὶ and the autochthonous music theory, where two exotic naġme had been avoided by certain composers, by others not. If it is true, what we can read in the foreword of some Arabic divans that Arab music was created by gifted composers of the 7th century who took the best from Byzantine music of Damascus and from Persian music, it seems to me that they also took certain aesthetic preferences from the Greeks.
So what was exotic for them?
Tunes with nenano-like intervals were usually described in Arabic under the name "Hejaz" which was a name of a province around Mecca and Medina along the Red Sea coast on the Arabian peninsula. During the 9th and 10th century the musical idiom was associated with Udhrite mystic love poetry, where the troubadours would later steal most of their ideas. If you try to describe its intervals with the terms of ἁρμονικαὶ, you will do it quite similar like Jerome of Moravia during the 13th century, and you will complain that the musicians mix chromatic minor tones with smaller hemitona or enharmonic dieseis. With regard to the integrative use of the Hagiopolite oktoechos in the papadikai, it looks like the nenano was rather regarded as chromatic and the nana as enharmonic.
What was ἀπὸ τὴν μουσικὴν for them?
I personal found your hypothesis about 12 + 9 + 9 much more plausible than all these 12 + 12 + 6 hypotheses which I find extremely Latinocentric. But this διατονὸν μαλακὸν is definitely not a tetrachord which fits well into the divisions quoted by Ptolemy, and I believe that Byzantine music has changed, despite certain allusions to Hellenic culture, the heirmologic genre with its odes is full of Ancient Greek drama and the epic function of its choir around the scene.
My opinion is, that great Greeks (in the sense of Magna Grecia) should not be irritated by the fact, that the interest and the admiration of Muslims (the militaric enemy of the Byzantine Empire) for Hellenic as well as for Byzantine culture was the motivation for their conquests. Nevertheless, on Byzantine territory they were quite scared of any kind of iconoclasm. Under their influence they never developed the same destructive force as Muslim rulers in India, when they destroyed all the Hindu and Brahman temples and their sculptures.
There must have also been a semitic influence concerning biblical recitation of Greek psaltes, which might have caused a certain break with the heritage of chromatic and enharmonic melodies, but these melodies could always be described by Pythagorean terms and categories. Despite all conflicts between the theory and the practice of ecclesiastical chant, the former also transforms a tradition, as far as it helps to remember and to memorise chant.
I wait now for your answers concerning my questions about the Hagiopolites, before I continue with questions about the psaltic art of the Palaiologan dynasty.
Please take your time.
If you prefer rather an ahistorical structuralist methodology, we can say that the "four elements" (earth, air, fire and water, if you like) of the tetrachord allows any flexibility to choose 3 different intervals—usually called "maior, middle," and "minor tone"—as steps (φωναί) between them. The terminology of protos, devteros, tritos, and tetartos which was also imitated by the Carolingians, had always been more important than the names of the tropes (τρώπη) concerning the Byzantine oktoechos.
Your magic trick with 12 + 9 + 9 works, because the middle and minor tone look like the same thing, if you look at them through "Aristoxenian" glasses, but as proportions they are definitely well distinguished, 9:8 x 88:81 x 12:11 (the descending direction).
Despite the fact, that the idea to choose the ditonus with just two different intervals 12 + 12 + 6 was not an invention of Boethius, but can be traced back to Eratosthenes (some say even back to Mesopotamia), so it might be more Ancient Greek than 12 + 9 + 9, does not say that it ever had the importance within harmonikai as it had for the Latin Middle Ages.
There is an oriental flexibility, which got definitely lost in Latin Europe, when they discovered the chain of ascending fifths and descending fourths (which was before Guido of Arezzo, Odo of Arezzo who was often confused with Odo of Cluny). It was an important condition for Guido's introduction of hexachords.
I would rather avoid to say, that authors of medieval Greek chant manuals were on a low level, simply because they did not mention the proportions of the intervals. I think it was not their problem and not their business, this changed later, when Manuel Bryennios combined harmonics with the discussion of Byzantine music.
The low level exists until today among those who are writing about music theory without any experience to perform more complex Orthodox chant which make use of transposition (μεταβολή κατὰ τόνον)—you need at least 40 minutes to perform them without any interruption. You will get in trouble like Otto Gombosi, when you try to understand it (I use this graphic to demonstrate, what even al-Farabi understood, but probably just a very few readers of Boethius). Since these stichera oktaecha need a profound knowledge of harmonikai, I would rather not conclude, that the author of the Hagiopolites had a very limited knowledge there.
As far as you are talking about the difference between Aristoxenian and Ptolemaic theory, it is usually understood, if a certain author refers to the Great Perfect System (GPS), like Ptolemy and Boethius, or to the Lesser Perfect System (LPS using also names for the tropes like Iastian and Aiolian), like Alypius (but he was not known among Latin scholars), Martianus Capella, and Cassiodorus (who was quoted precisely by Aurelianus of Réôme). The triphonic tone system of LPS is even more flexible, since the chromatic and enharmonic intervals are not limited for the use within a tetrachord (like in GPS and Oliver Strunk referred to it), they could even establish an own tetrachord. It seems that this was no longer understood since Cassiodorus, and it troubled Carolingian readers of Martianus Capella.
I inserted another very fine article by Jacques Chailley which was not quoted in your essay. As far as it discusses the well-known fact that the τρώπη are not a modal category like the church tone (the discussion was probably caused by his review of Henry Potiron), you will realise its year of publication. But since he discussed the reception of Ancient Greek harmonikai in different treatises over the centuries, it is still worth reading.