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Terirem and types of chant (monody/poliphony)

Bizzlebin

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Nobody is arguing about the definition of an octave. Everybody agrees that an octave (or more precisely, a perfect octave) is the interval obtained by halving the length of string on a monochord or by doubling the fundamental frequency of a sound. I question your assertion that one can universally generalize octave equivalence. You claim that the modes of ancient Greek music are octave-aware (though I'm not sure what that means), but the systema teleion of classical Greek music encompassed two octaves so that all the modes could fit within it. If the Ancient Greeks had the concept of octave equivalence as we understand it today, the systema teleion would have been an unnecessary construction.
The systema teleion is evidence of their compository technique and probably philosophy, but the inclusion of the diazeuxis shows that even they realized there were issues with harmony if their system was used as-is. In any case, what we'd need is not just evidence that they used the systema teleion generally, but that it was used for harmony, that the harmonies between octaves and other notes was considered equivalent, and that this usage was accepted ecclesially. Is there evidence of any of those 3 links in the logical chain?

The "Byzantine names of the notes" you mention are the parallage according to the new method, created in 1814. The original diatonic system of parallage had but four notes, each with a name corresponding to an ascent to that note and another name corresponding to descending to that note (as I described in my previous post). Furthermore, even though the parallage of the new method is constructed on an octave basis, the underlying tuning does not work on an octave basis. For example, in plagal first mode, the interval between vou (the second scale degree) and high vou' (the 9th scale degree) is typically larger than an octave, because the interval between pa-vou is a lesser tone (ἐλάσσων τόνος), but the interval between high pa and high vou is usually rendered as a greater tone (μείζων τόνος) in order to preserve the perfect fifth between ke and high vou. The chromatic genus (which is probably a late-medieval introduction into Byzantine Music), takes this to the extreme so that no notes are a perfect octave apart (all are either a diminished or augmented octave apart), but all notes have a note which is a perfect fifth above and a perfect fifth below.
Thanks for the info about the date of the solmization. It is not surprising that the change to chromatic is a late one, meaning that clear octaves were present in the earlier music.
 

Cavaradossi

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The systema teleion is evidence of their compository technique and probably philosophy, but the inclusion of the diazeuxis shows that even they realized there were issues with harmony if their system was used as-is. In any case, what we'd need is not just evidence that they used the systema teleion generally, but that it was used for harmony, that the harmonies between octaves and other notes was considered equivalent, and that this usage was accepted ecclesially. Is there evidence of any of those 3 links in the logical chain?
I am not arguing that the suitability of polyphony for worship can be demonstrated by the music of the Ancient Greeks. The original point of contention here was that you have assumed octave equivalence is a universal feature of music (so that you can except men and women singing together at the octave from being polyphony sensu lato), and I do not accept this premise. The systema teleion was being presented as evidence against the universality octave equivalence, for the Ancient Greeks did not have repeating note names at the octave.

Thanks for the info about the date of the solmization. It is not surprising that the change to chromatic is a late one, meaning that clear octaves were present in the earlier music.
That does not at all follow. The first example I gave of non-equivalence at the octave was not in the chromatic genus but in the diatonic genus, which has been present throughout the entire history of ecclesiastical music. If one takes just intonation as the basis for the tuning of the diatonic scale (as is described in some theoretical treatises) the intervals are the greater tone (μείζων τόνος), which is identical in size with the Pythagorean τόνος, being a ratio of 8:9, the lesser tone (ἐλάσσων τόνος), being defined as a ratio of 9:10, and the least tone (ἐλάχιστος τόνος) which is defined as a ratio of 15:16. The the interval between ananes and neanes is a lesser tone, between neanes and nana a least tone, between nana and agia a greater tone, and between agia and ananes a greater tone. Thus when we go from neanes up seven notes, we ascend least, greater, greater, lesser, least, greater, greater, ending on ananes. Expressed mathematically, that is (15/16)*(8/9)*(8/9)*(9/10)*(15/16)*(8/9)*(8/9) = 40/81, which is wider than the octave by one syntonic comma (a ratio of 80:81). From nana, we ascend greater, greater, lesser, least, greater, greater, lesser, ending on neanes, which yields a ratio of 64:135, which exceeds the octave by a greater tone less a least tone, an interval which is about the size of an equal tempered semitone (being just 8 cents narrower).
 
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