From this reference by I. Arvanitis, I have found the specific book: Ol. Strunk, Source Readings in Music History, From Classical Antiquity Through the Romantic Era, New York 1950.
Strunk, Source Readings in Music History
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Unless I overlook something, he basically recognises two types of an interval: a tone and a semitone. Equating a tone with 12 moria and a semitone with 9 moria, (Pseudo)-Odo gloriously vindicates Evangelos Soldatos' theory on the diphonies in all modes 
P.S. Don't take the above (too) seriously
P.S. Don't take the above (too) seriously
No diphonies, no Soldati
, only "toni et semitoni carabinati" As useful as these references are, one should exercise caution in extrapolating or conjecturing from these to the intervals one experiences in Byzantine ecclesiastic chant.
At the time of (pseudo)-Odo, those may have been the intervals (pseudo)-Odo (or any of his contemporaries) were experiencing, but how much of that was indigenous to the chant in Syria, Jerusalem, and C/ple? Can one find references from Syria and Jerusalem and C/ple that support (pseudo)-Odo's observations?
As for equivalent thirds (omoies diphonies), the evidence-especially for the soft chromatic- I believe, is strong-especially in light of work by Mr. Haris Symeonidis.
Whether they are evident in other genera awaits further study with software......
NG.
At the time of (pseudo)-Odo, those may have been the intervals (pseudo)-Odo (or any of his contemporaries) were experiencing, but how much of that was indigenous to the chant in Syria, Jerusalem, and C/ple? Can one find references from Syria and Jerusalem and C/ple that support (pseudo)-Odo's observations?
As for equivalent thirds (omoies diphonies), the evidence-especially for the soft chromatic- I believe, is strong-especially in light of work by Mr. Haris Symeonidis.
Whether they are evident in other genera awaits further study with software......
NG.
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One will be pressed hard to find anything, because Chrysanthos in his Great Theoretikon says that measurement of the intervals of the Byzantine chant "was first accomplished in the times of Manuel Protopsaltes" (quoting from memory). This is not to say that there didn't circulate various theories on intervals (primarily epitomised from ancient Greek sources), but their relevance for practice seems to be minimal (or none?): no Papadike or teaching method seems to include any discussion on the intervals except of practical teaching of them through various melodies and theseis.
Therein lies the root of many problems that plague the manuscript approach alone to unravelling the progression and evolution of oral practice. Furthermore, the use of abstractions in the works of Vryennios for example don't help. Even Chrysanthos presents abstractions that, in some instances, are not compatible (at least in the manner manuscript investigators interpret them) with oral practice.
What we can say with certaintly, I believe, is that the oral practices of the psaltai recorded in the 1920s-1940s are closer to the oral practices of the post-Chrysanthine period (i.e. mid-1800s). With the closest being Naypliotis- primarily because of the strict bounds imposed by the Patriarchal Chruch of Ct-George on its musicians.
From that point, everything becomes murky as one goes back in time. Manuscript translation is often arbitrary and according to the whims of the translator (indeed, there is no consensus among even the Greek scholars on a modern "key" of translation of old neumes into the exegeses of Petros and then into the exegeses of Chourmouzios and his contemporaries). I won't even discuss non-Greek scholars who have their own views on the matter that is incompatible with the Greek view OR the oral tradition.
NG.
What we can say with certaintly, I believe, is that the oral practices of the psaltai recorded in the 1920s-1940s are closer to the oral practices of the post-Chrysanthine period (i.e. mid-1800s). With the closest being Naypliotis- primarily because of the strict bounds imposed by the Patriarchal Chruch of Ct-George on its musicians.
From that point, everything becomes murky as one goes back in time. Manuscript translation is often arbitrary and according to the whims of the translator (indeed, there is no consensus among even the Greek scholars on a modern "key" of translation of old neumes into the exegeses of Petros and then into the exegeses of Chourmouzios and his contemporaries). I won't even discuss non-Greek scholars who have their own views on the matter that is incompatible with the Greek view OR the oral tradition.
NG.
There are two interesting articles on the modal structure of the secular Arabic music around Damascene's times:
Owen Wright. Ibn al-Munajjim and the Early Arabian Modes. The Galpin Society Journal. Vol. 19, (Apr., 1966), pp. 27-48.
Henry George Farmer. The Old Arabian Melodic Modes. The Journal of the Royal Asiatic Society of Great Britain and Ireland. No. 3/4 (Oct., 1965), pp. 99-102.
My library has only hard copies of the second journal, so if anybody uploads the two articles here, I'd be grateful. From the little I've seen and heard about the sources which Wright and Farmer discuss, the Arab theorists talk about eight (!) diatonic modes which are described in Pythagorean terms. Now this does not necessarily mean there's a link with the ecclesiastic music, but still I find the fact very peculiar indeed.
Owen Wright. Ibn al-Munajjim and the Early Arabian Modes. The Galpin Society Journal. Vol. 19, (Apr., 1966), pp. 27-48.
Henry George Farmer. The Old Arabian Melodic Modes. The Journal of the Royal Asiatic Society of Great Britain and Ireland. No. 3/4 (Oct., 1965), pp. 99-102.
My library has only hard copies of the second journal, so if anybody uploads the two articles here, I'd be grateful. From the little I've seen and heard about the sources which Wright and Farmer discuss, the Arab theorists talk about eight (!) diatonic modes which are described in Pythagorean terms. Now this does not necessarily mean there's a link with the ecclesiastic music, but still I find the fact very peculiar indeed.
I have the first from jei stor but I'm afraid about the copyright.
For the second, it is not free. But search about Farmer's book "Historical facts on the arabian influence in the archive.org. There is a chapter "on the old arabian theory" (I have the book but cannot upload it because of my slow connection".
There is also a chapter on the Arabic music by Farmer in The New Oxford History of Music, Vol. I. I read it a long time ago but I remember that he speaks about a kind of Byzantine influence (which, as far as I remember, he identifies with pythagorean intervals). He speaks about the same modes there, too. The old arabic terms used as names come from fingering ("usta" eg. is the middle finger) on the oud. It seems that a collective name for them is "dasatin", which should be an arabic plural form* of the Persian word "dast", meaning "hand", a word found also in the Persian term "dastgah", meaning literally "place of the hand".
* an "arabization" of the plural form exists eg also for the greek word "jins"=genos, arabic plural "ajnas"
For the second, it is not free. But search about Farmer's book "Historical facts on the arabian influence in the archive.org. There is a chapter "on the old arabian theory" (I have the book but cannot upload it because of my slow connection".
There is also a chapter on the Arabic music by Farmer in The New Oxford History of Music, Vol. I. I read it a long time ago but I remember that he speaks about a kind of Byzantine influence (which, as far as I remember, he identifies with pythagorean intervals). He speaks about the same modes there, too. The old arabic terms used as names come from fingering ("usta" eg. is the middle finger) on the oud. It seems that a collective name for them is "dasatin", which should be an arabic plural form* of the Persian word "dast", meaning "hand", a word found also in the Persian term "dastgah", meaning literally "place of the hand".
* an "arabization" of the plural form exists eg also for the greek word "jins"=genos, arabic plural "ajnas"
Thanks for the reference to Farmer's book. It's an interesting reading, although not very concrete as far as technicalities are concerned. From the little I read now he maintains that (I provide a little summary to organise my thoughts and also maybe because it will be interesting to other people).
1) "Arabian culture and civilisation did not originate with nomads nor with Islam" and "Long before Islam we read, here and there, of the musical proclivities of the ancient Arabs" (p. 49).
2) Arabs had indigenous musical system which they were able to reduce to the theory long before the conquest of Iran in the 7th c. (pp. 50-52).
3) Influence of the Persian and Byzantine musical systems and later on of the ancient Greek principles on the Arab music is readily admitted (p. 52) with the converse being also true (pp. 52-53). "The alien influences in Arabian music were for the most part quite superficial, and, at first, had no bearing on theory" (p. 53). What exactly the Arabs borrowed from Persia and Byzantium in terms of theory is unclear (pp. 53-56). Here Farmer seems to clearly distinguish between ancient Greek musical system and the music of Byzantium. "The first information that we have of the definite influence of Persia and Byzantium in Arabian musical theory, also tells us of an indigenous Arabian system" (p. 56). Quoting the chronicler, whatever was found disagreeable, was rejected, "for instance the intervals (nabarat) and modes (nagham)... in the song (ghina) of the Persians and Byzantines, which were alien to the Arabian song" (p. 57). What was not borrowed from Persia or Byzantium is listed on pp. 57-58. What is known on Arabian music prior to the Greek scholiasts is sufficient to assert that the Arabian musical system was different from the Persian and the Greek (p. 59).
4) The modes of Arabs, Persians and Byzantines were different in the 9th c. (pp. 60-61; this claim has to be interpreted in a sense that all three modal systems, even though they could have had something in common, still possessed a number of distinctive properties).
5) The influence of the ancient Greek musical theory on the Arab music is dealt with in a separate chapter "The Greek Scholiasts" (p. 63ff). The music became one of the courses of scientific study (p. 64). This was a scholastic exercise and the Arab theorists do not fail to draw a line between this "theory of the ancients (=ancient Greeks)" and the native practical art (p. 65). The borrowings were nomenclature dealing e.g. with intervals (pp. 66-67). The Arabs didn't merely copy the speculative music theory, but further developed it (pp. 67-69) and scrutinised the ancient Greek theory. Among notable Arab developments are "the Figures of the Melody" (melous schemata) and the doctrine of ethos (p. 71).
The book has some very valuable appendices. Of course it has a vigorous and polemical tone and is likely to exaggerate at places, but in this it probably fades away in comparison to some of the bombastic statements and views it is criticising
1) "Arabian culture and civilisation did not originate with nomads nor with Islam" and "Long before Islam we read, here and there, of the musical proclivities of the ancient Arabs" (p. 49).
2) Arabs had indigenous musical system which they were able to reduce to the theory long before the conquest of Iran in the 7th c. (pp. 50-52).
3) Influence of the Persian and Byzantine musical systems and later on of the ancient Greek principles on the Arab music is readily admitted (p. 52) with the converse being also true (pp. 52-53). "The alien influences in Arabian music were for the most part quite superficial, and, at first, had no bearing on theory" (p. 53). What exactly the Arabs borrowed from Persia and Byzantium in terms of theory is unclear (pp. 53-56). Here Farmer seems to clearly distinguish between ancient Greek musical system and the music of Byzantium. "The first information that we have of the definite influence of Persia and Byzantium in Arabian musical theory, also tells us of an indigenous Arabian system" (p. 56). Quoting the chronicler, whatever was found disagreeable, was rejected, "for instance the intervals (nabarat) and modes (nagham)... in the song (ghina) of the Persians and Byzantines, which were alien to the Arabian song" (p. 57). What was not borrowed from Persia or Byzantium is listed on pp. 57-58. What is known on Arabian music prior to the Greek scholiasts is sufficient to assert that the Arabian musical system was different from the Persian and the Greek (p. 59).
4) The modes of Arabs, Persians and Byzantines were different in the 9th c. (pp. 60-61; this claim has to be interpreted in a sense that all three modal systems, even though they could have had something in common, still possessed a number of distinctive properties).
5) The influence of the ancient Greek musical theory on the Arab music is dealt with in a separate chapter "The Greek Scholiasts" (p. 63ff). The music became one of the courses of scientific study (p. 64). This was a scholastic exercise and the Arab theorists do not fail to draw a line between this "theory of the ancients (=ancient Greeks)" and the native practical art (p. 65). The borrowings were nomenclature dealing e.g. with intervals (pp. 66-67). The Arabs didn't merely copy the speculative music theory, but further developed it (pp. 67-69) and scrutinised the ancient Greek theory. Among notable Arab developments are "the Figures of the Melody" (melous schemata) and the doctrine of ethos (p. 71).
The book has some very valuable appendices. Of course it has a vigorous and polemical tone and is likely to exaggerate at places, but in this it probably fades away in comparison to some of the bombastic statements and views it is criticising
Another point here is that one cannot a priori discard the fact that the musical system at use in Constantinople could have been different from the one in oriental provinces (this can be expected in particular for the period prior to the 10th c.). There is no definite evidence, but for instance the distinction between the Hagiopolites system and the Asma made in the Hagiopolites treatise could be a relic of this (unfortunately it is not entirely clear what Asma denotes in the latter treatise). The Syrian/Arabic milieu in which the two oriental Patriarchates, Jerusalem and Antioch, existed must have been different from conditions in Constantinople. A considerable difference between the liturgies of Constantinople and Jerusalem can be another hint of possibility of employment of different musical systems. Of course, at a certain point, with growing influence of Constantinople, the traditions should have converged.