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Phthora nenano (Medieval Greek: φθορὰ νενανῶ, also νενανὼ) is the name of one of the two "extra" modes in the Byzantine οctoechos—an eight mode system, which was proclaimed by a synod of 792. The phthorai nenano and nana were favoured by composers at the Monastery Agios Sabas, near Jerusalem, while hymnographers at the Stoudiou Monastery obviously preferred the diatonic mele.

The phthora nenano as part of the Hagiopolitan octoechos
Today the system of eight diatonic modes and two phthorai ("destroyers") is regarded as the modal system of Byzantine chant, and during the eighth century it became also model for the Latin tonaries—introductions into a proper diatonic eight mode system and its psalmody, created by Frankish cantores during the Carolinigian reform. While φθορά νενανῶ was often called "chromatic", the second phthora was named "nana" (gr. φθορά νανὰ) and called "enharmonic", the names were simply taken from the syllables used for the intonation (enechema). The two phthorai were regarded as two proper modes, but also used as transposition or alteration signs. Within the diatonic modes of the octoechos they cause a change into another (chromatic or enharmonic) genus (metavoli kata genos).

The earliest description of phthora nenano and of the eight mode system (octoechos) can be found in the Hagiopolites treatise which is known in a complete form through a fourteenth-century manuscript. The treatise itself can be dated back to the ninth century, when it introduced the book of tropologion, a collection of troparic and heirmologic hymns which was ordered according to the eight-week cycle of the octoechos. The first paragraph of the treatise maintains, that it was written by John of Damascus. The hymns of the tropologion provided the melodic models of one mode called echos (gr. ἦχος), and models for the phthora nenano appeared in some mele of certain echoi like protos and plagios devteros.

"Ἤχους δὲ [ λέγουσιν ] ἐν τούτῳ ὀκτὼ ψάλλεσθαι. ἔστι δὲ τοῦτο ἀπ [ οβλητέον καὶ ] ψευδές· ὁ γὰρ πλάγιος δευτέρου ὡς ἐπὶ τὸ πλεῖστο [ ν μέσος ] δεύτερος ψάλλεται, ὡς τὸ «Νίκην ἔχων Χριστὲ» [ καὶ ὡς τὸ «Σ ] ὲ τὸν ἐπὶ ὑδάτων» καὶ ἄλλα ὅσα πα [ ρὰ τοῦ κυροῦ Κοσμᾶ ] καὶ τοῦ κυροῦ Ἰωάννου τοῦ Δαμασκ [ ηνοῦ ἀπὸ τῆς μουσικῆς ] ἐξ [ ε ] τέθησαν – ὅσα δὲ [ ἐποιήθυσαν ὑπὸ τοῦ κυρ ] οῦ [ Ἰ ] ωσὴφ [ καὶ ] ἄλλ [ ων ] τ [ ινῶν, ε ] ἰ δο [ κιμάσεις αὐτὰ μετὰ τῆς μουσικῆς ψάλλειν, οὐκ ἰσάζουσι διὰ τὸ μὴ ἐκτεθῆναι ὑπ’ αὐτῆς – ὁμοίως δὲ καὶ ὁ πλάγιος τετάρτου ὡς ἐπὶ τὸ πλεῖστον μέσος τέταρτος ψάλλεται, ὡς ] ἐπὶ τὸ «Σταυρὸν χαράξας Μωσῆς» καὶ ἕτερα οὐκ ὀλίγα. ἔστιν οὖκ ἐκ τούτων γνῶναι, ὅτι οὐκ ὀκτὼ μόνοι ψάλλονται ἀλλὰ δέκα."

"For the songs in this book eight Echoi are said to be necessary. But this is not true and should be rejected. In fact the Plagios of Deuteros is mostly sung as Mesos Deuteros—e.g. «Νίκην ἔχων Χριστὲ», the «Σὲ ἐπὶ τῶν ὑδάτων» and other pieces written by Master Cosmas and Master John of Damascus 'from the Mousike'. (If, however, you try to sing the compositions of Master Joseph and others 'with the Mousike', they will not fit, having not been composed 'according to the Mousike'). Similarly the Plagios of tetartos is mostly sung as Mesos tetartos like over «Σταυρὸν χαράξας Μωσῆς» and many others. For these cases we can see that ten Echoi are used (for the repertory of this book) and not eight, only."

The author of the treatise wrote obviously during or after the time of Joseph and his brother Theodore the Studite, when the use the mesos forms, phthorai nenano and nana were no longer popular. The word "mousike" (μουσική) referred an autochthonous theory during the 8th century used by the generation of John of Damascus and Cosmas of Maiuma at Mar Saba, because it was independent from ancient Greek music. But it seems that it was regarded as inappropriate to use these phthorai for the hymn melodies composed by Joseph and other hymns composed since the ninth century, since they must have preferred the diatonic octoechos based on the kyrios and the plagios instead of the mesoi.

The concept of phthora in the Hagiopolites was less concerned that the nenano and nana were somehow bridges between the modes. As an introduction of the tropologion it had to integrate the mele composed in these phthorai within the octoechos order and its weekly cycles. Since they had their own mele and compositions like the other echoi, they were subordinated to the eight diatonic echoi according to the pitches or degrees of the mode (phthongoi) of their cadences.

"φθοραὶ δὲ ὠνομασθήσαν, ὅτι ἐκ τῶν ἰδίων ἤχων  πᾶρχονται, τελειοῦνται δὲ εἰς ἑτέρων ἤχων φθογγὰς αἱ θέσεις αὐτῶν καὶ τὰ  ποτελέσματα." "They were called Phthorai (i.e. destroyers), because they begin from their own Echoi, but the thesis of their cadences and formulas are on notes (phthongoi) from other Echoi."

They had to be classified according to a certain echos of the eight-week cycle by adding the intonation "nenano" to the intonation of the main diatonic echos (usually abbreviated by a modal signature). For example, the intonation formula of echos plagios devteros (E) could be followed by the intonation of nenano which leads to the echos protos (a), as a kind of "mesos devteros", which lies in between the finalis of the kyrios (b natural) and the one of its plagios (E). Usually the diatonic kyrios protos (a) could end on its plagios (D) in the diatonic genus, but the chromatic phthora nenano makes it end in the plagios devteros (E).

The use of phthora nenano in the psaltic art
In the period of the psaltic art (gr. ψαλτικὴ τέχνη, "the art of chant", 1261–1750) the Late Byzantine Notation used four additional phthorai for each mode, including the eight diatonic echoi, in order to indicate the precise moment of a transposition (metavoli kata tonon). The former system of sixteen echoi (4 kyrioi, 4 plagioi, 4 mesoi, and 4 phthorai) which was still used in the old books of the cathedral rite (asmatikon, kontakarion, etc.), was replaced by the Hagiopolitan octoechos and its two phthorai. The new book akolouthiai which replaced the former book and established a mixed rite in Constantinople, introduced into eight diatonic echoi and two phthorai. In rather soloistic chant genres, the devteros echoi were turned into the chromatic genus by an abundant use of the phthora nenano. Hence, it became necessary to distinguish between the proper echos and its phthora, nenano and nana as "extra modes", and their use for temporary changes within the melos of a certain diatonic echos.

The use of six phthorai for all of the ten Hagiopolitan echoi
In his theoretical treatise about psaltic art and in response to the "wrong ideas" that some singers already had some years after the conquest of Constantinople (1458), the famous Maïstoros Manuel Chrysaphes introduced not only into the two phthorai nenano and nana, but also into four phthorai which bind the melos to the diatonic echoi of protos, devteros, tritos, and tetartos.

All six phthora, two of them belonged to the phthora nenano (the phthora of nenano and the one of plagios devteros), could dissolve the former melos and bind it to the melos of the following echos defined by the next medial signature. The diatonic phthora was no longer the destruction of the diatonic modes and its genus, melos, and its tonal system, it could change each mode and its finalis into another echos by a simple transposition. Hence, the list of phthorai mentioned in each Papadikai, was simply a catalogue of transposition signs, which were written over that neume where the transposition has to be done.

Phthora nenano and the plagal second echos
In that respect phthora nenano, as well as nana, stuck out, because within their own melos they were both directed to certain other echoi.

"ὅταν δὲ τεθῇ καὶ εἰς ἄλλου ἤχου μέλος, ποιεῖ μέλος ἴδιον παρ’ ὅ ποιοῦσιν αἱ ἄλλαι φθοραὶ καὶ ἡ κατάληξις ταύτης οὐ γίνεται εἰς ἄλλον ἦχον ποτέ, εἰ μὴ εἰς τὸν πλάγιον δευτέρου. εἰ δὲ θήσει τις ταύτην τὴν φθορὰν καὶ οὐ καταλήξει εἰς πλάγιον δευτέρου, ἄλλ’ εἰς ἕτερον ἦχον, τοῦτο οὐκ ἔστιν ἔντεχνον· προείπομεν γάρ, ὅτι ἔστιν  πὸ παραλλαγῶν ἡ αὐτὴ πρῶτος ἦχος·"

"Whenever it stands in the melody [μέλος] of another echos, it creates its own melody [μέλος] and cadence [κατάληξις] which the other phthorai cannot do, and its resolution never closes into another mode apart from the plagios devteros. If one uses this phthora and it does not resolve [θήσει] into the echos plagios devteros but into another mode, this is not artistic [ἔντεχνον]; for we said before that this is the echos protos by parallage [παραλλαγή]."

It was the psaltic art itself which moved the phthongos of plagios devteros to the one of plagios protos. It is possible, that the phthora of plagios devteros was needed to change again into diatonic genus. According to the New Method (since 1814) echos plagios devteros was always chromatic and based on the phthongos of echos protos, memorised as πα. This was Chrysanthos' way to understand Manuel Chrysaphes—probably a contemporary way, since the 17th-century Papadike introduced a seventh phthora for plagios protos.

According to the rules of psaltic art phthora nenano could connect the phthongoi devteros and protos as well as protos and tetartos, as can be seen from the solfège diagram called the "parallage of John Plousiadenos" (see the first X in the first row around the centre of the enechema of phthora nenano).

Despite this possibility Manuel Chrysaphes insisted, that phthora nenano and its chromatic melos has always to be resolved as plagios devteros, any other echos would be against the rules of psaltic art. The living tradition today still respects this rule, since phthongos of the plagios devteros (πλβ᾽) has become the same like plagios protos (πλα᾽): πα (D).

The early Persian and Latin reception
Already in the thirteenth century, there were interval descriptions in Latin and Arabic treatises which proved that the use of the chromatic phthora was common not only among Greek psaltes.

Quţb al-Dīn al-Shīrīz distinguished two ways of using the chromatic genus in parde hiğāzī, named after a region of the Arabian Peninsula. The exact proportions were used during changes to the diatonic genus. In both diatonic and chromatic divisions the ring finger fret of the oud keyboard was used. It had the proportion 22:21—between middle and ring finger fret—and was called after the Baghdadi oud player Zalzal. These are the proportions, presented as a division of a tetrachord using the proportions of 22:21 and 7:6:

12:11 x 7:6 x 22:21 = 4:3 (approximate intervals in cents:  151, 267, 80 = 498)

This Persian treatise is the earliest source which tried to measure the exact proportions of a chromatic mode, which can be compared with historical descriptions of phthora nenano.

In his voluminous music treatise Jerome of Moravia described that "Gallian cantores" used to mix the diatonic genus with chromatic and the enharmonic, despite the use of the two latter were excluded according to Latin theorists:

"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."

"Especially when they mix the ecclesiastical chant with the organum mode, they like not only to abandon the first mode [simple plainchant in monophonic realization?], but the confusion of both [plainchant and ars organi] includes another [confusion of the diatonic] with the other genera, because they associate the enharmonic diesis and the chromatic trihemitonium with the diatonic genus. They replace the semitonium by the tonus and vice versa, in doing so they differ from the other nations, as far as chant is concerned."

During the seventies of the thirteenth century Jerome met the famous singers in Paris who were well skilled in the artistic performance of ars organi, which is evident by the chant manuscripts of the Abbey Saint-Maur-des-Fossés, of the Abbey Saint-Denis, and of the Notre Dame school. Despite the fact that no other Latin treatise ever mentioned that the singers were allowed to use enharmonic or chromatic intervals, and certainly not the transposition practice which was used sometimes by Greek psaltes, they obviously felt free enough to use both during the improvisation of organum—and probably, they became so familiar with the described enharmonic chromaticism, that they even used it during the monophonic performance of plainchant. Jerome as an educated listener regarded it as a "confusion" between monophonic and polyphonic performance style. Whatever his opinion about the performance style of Parisian cantores, the detailed description fit well to the use of the phthora nenano as an "echos kratema", as it was mentioned in the later Greek treatises after the end of the Byzantine Empire.

The phthora nenano as kyrios echos and echos kratema
According to a Papadike treatise in a sixteenth-century manuscript (Athens, National Library, Ms. 899 [EBE 899], fol.3f), the anonymous author even argues that phthorai nenano and nana are rather independent modes than phthorai, because singers as well as composers can create whole kratemata out of them.

"Εἰσὶ δὲ καὶ φθοραὶ δύο, αἵτινες ψάλλονται σὺν αὐτοῖς, τὸ νανὰ καὶ τὸ νενανὼ. Εἰσί δὲ καὶ ἄλλαι φθοραὶ τῶν ἄλλων ἤχων ἀλλοὐκ εἰσὶ τέλειαι ὡς αὗται. Εκεῖναι γὰρ δεικνύουσιν ἐναλλαγὴν μερικὴν ἀπὸ ἤχου εἰς ἕτερονἶ αὐταὶ δὲ τέλειαι οὖσαι ἔχουσι καὶ κρατήματα ποινθέντα παρὰ τῶν κατὰ καιροὺς ποιντῶν ὡς εἰς κυρίους ἤχους, καὶ εἰκότως ἄν [ τις ] καλέσειεν αὐτὰς τελείους ἤχους καὶ οὐ φθορὰς."

"There are two phthorai which can be sung as those [of the Octoechos]: νανὰ and νενανὼ. There are also the phthorai of the other [ kyrioi ] echoi, but those are not as perfect [ they have no proper melos like nana and nenano? ], because they just cause a temporary transition [ἐναλλαγή μερική] from one to another echos, while the former have been used by composers of various epochs to create kratemata like they were [ independent ] kyrioi echoi. Hence, it is justified to call them rather 'perfect echoi' than just 'phthorai'."

Kratemata were longer sections sung with abstract syllables in a faster tempo. As a disgression used within other forms in papadic or kalophonic chant genres—soloistic like cherubim chant or a sticheron kalophonikon. From a composer's point of view who composed within the mele of the octoechos, a kratema could not only recapitulate the modal structure of its model, but also create a change into another (chromatic or enharmonic) genus. If a composer or protopsaltes realised a traditional model of a cherubikon or koinonikon within the melos of echos protos, the phthora nenano will always end the form of the kratema in echos plagios devteros, only then the singer could find a way back to the main echos. In the later case the kratema was composed so perfectly in the proper melos of phthora nenano, that it could be performed as a separate composition of its own, as they were already separated compositions in the simpler genres like the troparion and the heirmologic odes of the canon since the 9th century.

Gabriel Hieromonachus (mid fifteenth century) already mentioned that the "nenano phone"—the characteristic step (interval) of nenano—seemed to be in some way halved. On folio 5 verso of the quoted treatise (EBE 899), the author gave a similar description of the intervals used with the intonation formula νε–να–νὼ, and it fitted very well to the description that Jerome gave 300 years ago while he was listening to Parisian singers:

"Ἄκουσον γὰρ τὴν φθορὰν, ὅπως λέγεται: Τότε λέγεται φθορὰ, ὅταν τῆς φωνῆς τὸ ἥμιου εἴπης ἐν ταῖς κατιούσαις, (ἢ κατ’ ἀκριφολογίαν τὸ τρίτον, ἐν δὲ ταῖς ἀνιούσαις) μίαν καὶ ἥμισυ, ὥσπερ εἰς τὸ νενανώ. Ἄκουσον γὰρ:"

νε [ ἴσον ] —να [ δικεντήματα ] —να [ ὀλίγον ] —νω [ ὀλίγον με τῇ διπλῇ ] "Αὕτη ἡ φθορὰ εἰς τὰς ἀνιούσας. Ἰδοὺ γὰρ εἶπε τοῦ νω τὴν φωνὴν τὴν ἥμισυ εἰς τὸ να."

"Please note, what is called 'phthora': phthora is called, if you make a half phonic step in descending [direction], (or more precisely a third of it, while [the second interval] has an ascending) one and a half, as in nenano. Listen:"

νε [ ison ] —να [ dikentimata: small tone ] —να [ oligon: one and a half of the great tone ] —νὼ [ oligon with diple: diesis or quarter tone ]

"This is the intonation of phthora which is ascending. Concerning the final phonic step [ which was a third of tonus in a descending melos], half of it is now part of the [ second ] να step [φωνή] and the rest [ interval is sung ] on νω!"

The upper small tone leading to the final note of the protos, has a slightly different intonation with respect to the melodic movement, at least according to the practice among educated psaltes of the Ottoman Empire during the eighteenth century. But Gabriel Hieromonachos described already in the fifteenth century, that the singers tend to stray away from their original intonation while they were singing the melos of phthora nenano:

"Ὁποπαν γὰρ ψάλλομεν νενανὼ μέλος, οὐκ εἰς ἣν ήρξάμεθα καὶ τελευτῶμεν φωνήν, ἀλλὰ σκοπῶν εὑρήσεις ἐπὶ τὸ κάτω μάλλον ἐρχομένους ἡμᾶς. Αἴτιον δὲ ἡ τοῦ νενανὼ φονή· αὕτη γὰρ ἡμίσεια δοκεῖ πως εἶναι, εἰ καὶ ἡμῖν ἀγνοεῖται· ἄλλως θ' ὅτι ἀσθενεῖς ἐκφέρομεν τὰς τοῦ νενανὼ ἀνιούσας φωνάς, ἵνα ἡ τοῦ νενανὼ ἰδέα χρωματισθῇ τὰς δὲ κατιούσας σῴας, καὶ ἐκ τούτου συμβαίνει τὸ μέλος ὑποχαλᾶν."

"Because when we sing a nenano melody, we don’t end on the tone, from which we started, but if you look at it closer, you will find that we come down to a somewhat lower pitch. The reason for this is the nenano interval; for it seems to be in some way halved, even if we are not aware of it; in other words, we perform the nenano intervals weakly in upward direction, in order to give the characteristic colour of nenano, but in downward direction [we perform them] correctly, and this causes the melody to get out of tune."

Actual usage and meaning
Later use of the enechema (initial incantation formula) of nenano as well as the phthora (alteration and transposition sign) of nenano in manuscripts makes it clear that it is associated with the main form of the second plagal mode as it survives in the current practice of Byzantine (Greek Orthodox) chant. Furthermore, the phthora sign of nenano has survived in the nineteenth-century neo-Byzantine notation system which is still used to switch between a diatonic and chromatic intonation of the tetrachord one fourth below.

Chrysanthos' exegesis of the phthora nenano
In the chapter "About apechemata", Chrysanthos quoted the medieval apechema of the phthora nenano as a chromatic tetrachord between the pitch (phthongos) of plagios devteros and protos.

This intonation formulas avoids the enharmonic step (diesis) which is expected between tetartos (δ') and protos (α').

His exegesis of this short apechema sets the chromatic or enharmonic tetrachord between plagios protos (πλ α') and tetartos (δ'), so that the diesis lies between tritos (γ') and tetartos (δ'):

The common modern enechema places the tetrachord likewise:

The hard chromatic plagios devteros

Chrysanthos of Madytos offered following exegesis of the traditional echema νεανες which was originally diatonic, but it is currently sung with the chromatic nenano intonation (see νεχὲ ανὲς in Chrysanthos' parallage).

Chrysanthos' exegesis employed the concluding cadence formula of the chromatic plagios devteros which was obviously an exegesis based on psaltic rules, as Manual Chrysaphes had once mentioned them.

He described the correct intonation as follows:

"§. 245. Ἡ χρωματικὴ κλίμαξ, πα [βου ὕφεσις] [γα δίεσις] δι κε [ζω ὕφεσις] [νη δίεσις] Πα, σύγκειται ἀπὸ δύο τετράχορδα· ἐν ἑκατέρῳ δὲ τετραχόρδῳ κεῖνται τὰ ἡμίτονα οὕτως, ὥστε τὸ διάστημα πα βου εἶναι ἴσον μὲ τὸ κε ζω· τὸ δὲ βου γα εἶναι ἴσον μὲ τὸ ζω νη· καὶ τὸ γα δι εἶναι ἴσον μὲ τὸ νη Πα· καὶ ὅλον τὸ πα δι τετράχορδον εἶναι ἴσον μὲ τὸ κε Πα τετράχορδον. Εἶναι δὲ τὸ μὲν πα βου διάστημα ἴσον ἐλαχιστῳ τόνῳ· τὸ δὲ βου γα, τριημιτόνιον· καὶ τὸ γα δι, ἡμιτόνιον· ἤγουν ἴσον 3:12."

"The chromatic scale: D πα—[high E flat]—[high F sharp]—G δι—a κε—[high b flat]—[high c sharp]—d πα', consists of two tetrachords. In each tetrachords the hemitones are placed in a way, that the interval D πα—E βου [ὕφεσις] is the same as a κε—b ζω' [ὕφεσις], Ε βου [ὕφεσις]—F sharp γα [δίεσις] is the same as b ζω' [ὕφεσις]—c sharp νη' [δίεσις], and F sharp γα [δίεσις]—G δι is the same as c sharp νη' [δίεσις]—d πα', so that both tetarchords, D πα—G δ and a κε—d πα', are unisono. This means that the interval D πα—E βου [ὕφεσις] is unisono with the small tone (ἐλάχιστος τόνος), Ε βου [ὕφεσις]—F sharp γα [δίεσις] with the trihemitone, and F sharp γα [δίεσις]—G δι with the hemitone: 3:12—a quarter of the great tone (μείζων τόνος) [7+18+3=28]."

Despite of this tradition modern music teachers tried to translate this sophisticated intonation on a modern piano keyboard as "a kind of gipsy-minor."

The soft chromatic kyrios devteros
In a very similar way—like the classical phthora nenano intonation—also the soft chromatic intonation of the echos devteros is represented as a kind of mesos devteros. Here according Chrysanthos of Madytos the exegesis of the traditional devteros intonation can be sung like this:

He explained that the intonation of the modern echos devteros was not based on tetraphonia, but on trichords or diphonia:

"§. 244. Ἡ χρωματικὴ κλίμαξ νη [πα ὕφεσις] βου γα δι [κε ὕφεσις] ζω Νη σχηματίζει ὄχι τετράχορδα, ἀλλὰ τρίχορδα πάντῃ ὅμοια καὶ συνημμένα τοῦτον τὸν τρόπον·"

νη [πα ὕφεσις] βου, βου γα δι, δι [κε ὕφεσις] ζω, ζω νη Πα.

"Αὕτη ἡ κλίμαξ ἀρχομένη ἀπὸ τοῦ δι, εἰμὲν πρόεισιν ἐπὶ τὸ βαρὺ, θέλει τὸ μὲν δι γα διάστημα τόνον μείζονα· τὸ δὲ γα βου τόνον ἐλάχιστον· τὸ δὲ βου πα, τόνον μείζονα· καὶ τὸ πα νη, τόνον ἐλάχιστον. Εἰδὲ πρόεισιν ἐπὶ τὸ ὀξὺ, θέλει τὸ μὲν δι κε διάστημα τόνον ἐλάχιστον· τὸ δὲ κε ζω, τόνον μείζονα· τὸ δὲ ζω νη, τόνον ἐλάχιστον· καὶ τὸ νη Πα, τόνον μείζονα. Ὥστε ταύτης τῆς χρωματικῆς κλίμακος μόνον οἱ βου γα δι φθόγγοι ταὐτίζονται μὲ τοὺς βου γα δι φθόγγους τῆς διατονικῆς κλίμακος· οἱ δὲ λοιποὶ κινοῦνται. Διότι τὸ βου νη διάστημα κατὰ ταύτην μὲν τὴν κλίμακα περιέχει τόνους μείζονα καὶ ἐλάχιστον· κατὰ δὲ τὴν διατονικὴν κλίμακα περιέχει τόνους ἐλάσσονα καὶ μείζονα· ὁμοίως καὶ τὸ δι ζω διάστημα."

"The chromatic scale C νη—[D flat]—E βου—F γα—G δι—[a flat]—b ζω'—c νη' is not made of tetrachords, but of trichords which are absolutely equal and conjunct with each other—in this way:" C νη—[D flat]—E βου, E βου—F γα—G δι, G δι—[a flat]—b ζω', b ζω'—c νη'—d πα' "If the scale starts on G δι, and it moves towards the lower, the step G δι—F γα requests the interval of a great tone (μείζων τόνος) and the step F γα—E βου a small tone (ἐλάχιστος τόνος); likewise the step E βου—[D flat] πα [ὕφεσις] an interval of μείζων τόνος, and the step πα [ὕφεσις] [D flat]—C νη one of ἐλάχιστος τόνος. When the direction is towards the higher, the step G δι—[a flat] κε [ὕφεσις] requests the interval of a small tone and [a flat] κε [ὕφεσις]—b ζω' that of a great tone; likewise the step b ζω'—c νη' an interval of ἐλάχιστος τόνος, and the step c νη'—Cd πα one of μείζων τόνος. Among the phthongoi of this chromatic scale only the phthongoi βου, γα, and δι can be identified with the same phthongoi of the diatonic scale, while the others are moveable degrees of the mode. While this scale extends between E βου and C νη over one great and one small tone [12+7=19], the diatonic scale extends from the middle (ἐλάσσων τόνος) to the great tone (μείζων τόνος) [12+9=21], for the interval between G δι and b ζω' it is the same."

Phthora nenano as an "Ottoman corruption"
Because of its early status as one of the two mysterious extra modes in the system, nenano has been subject of much attention in Byzantine and post-Byzantine music theory. Papadikai like the manuscript EBE 899 and other late Byzantine manuscripts associate nenano and nana with the chromatic and the enharmonic genus, one of the three genera of tuning during Classical antiquity that fell into early misuse because of its complexity. If the phthora nenano was already chromatic during the 9th century, including the use of one enharmonic diesis, is still a controversial issue, but medieval Arabic, Persian and Latin authors like Jerome of Moravia rather hint to the possibility that it was.

Greek music theoreticians such as Simon Karas continue up to the end of the twentieth century to regard the intonation nenano as "exotic," although they do not always agree, whether the echos plagios devteros intonation is hard or soft chromatic. Anonymous authors like the one of the 16th-century Papadike (EBE 899) maintained, that one of the minor tones in the tetrachord of nenano should be either smaller or larger than a tempered semitone, approaching the smallest interval of a third or a quarter of a tone. The banishment of instrumental musical practice and its theory from the tradition of Byzantine chant has made it very difficult to substantiate any such claims experimentally, and traditional singers use different intonations depending on their school. The only possible conclusions can be drawn indirectly and tentatively through comparisons with the tradition of Ottoman instrumental court music, which important church theoreticians such as the Kyrillos Marmarinos, Archbishop of Tinos considered a necessary complement to liturgical chant. However, Ottoman court music and its theory are also complex and diverging versions of modes exist according to different schools, ethnic traditions or theorists. There, one encounters various versions of the "nenano" tetrachord, both with a narrow and with a wider minor second either at the top or at the bottom, depending on the interval structure of the scale beyond the two ends of the tetrachord.

Although the phthora nenano is already known as one of two additional phthorai used within the Hagiopolitan octoechos, its chromaticism was often misunderstood as a late corruption of Byzantine chant during the Ottoman Empire, but recent comparisons with medieval Arabic treatises proved that this exchange can dated back to a much earlier period, when Arab music was created as a synthesis of Persian music and Byzantine chant of Damascus.

title: User:Platonykiss/sandbox/Nenano


text:
User:Platonykiss/sandbox/Nana

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Phthora nana (Medieval Greek φθορά νανὰ) is one of the ten modes of the Hagiopolitan Octoechos consisting of 8 diatonic echoi and two additional phthorai. It is used in different traditions of Orthodox chant until today (→ Neobyzantine Octoechos). The name "nana" is taken from the syllables (written in ligatures "ʅʅ") sung during the intonation which precedes a melody composed in this mode. The name "phthora" derived from the verb φθείρω and means "destroy" or "corrupt". It was usually referred to the diatonic genus of the eight mode system and as a sign used in Byzantine chant notation it indicated a "change to another genus" (μεταβολὴ κατὰ γένος), in the particular case of phthora nana a change to the enharmonic genus. Today the "nana" intonation has become the standard name of the third authentic mode which is called "echos tritos" (ἦχος τρίτος) in Greek and "third glas" (третий Гласъ) in Old Church Slavonic.

The different functions of phthora nana
In the theory and notation of Byzantine and Orthodox chant nana is the name of a special phthora which had been used in different ways according to its historic context:
 * 1) as a phthora which has its proper melos (intonation and cadence formulas), it may denote a special kind of echos (mode) that has been identified with the echos tritos (third mode) since the 16th century, but deviates from the later diatonic echos tritos by the division of its tonal system and its tetrachord. The name "νανὰ" (ʅʅ) was used for its proper enechema, and within the Hagiopolitan Octoechos whole troparia can be found in the tropologion or the chant book Octoechos which are composed in the melos of phthora nana (φθορά νανὰ).
 * 2) within the heirmologic or sticheraric melos, it may denote a temporary change to the enharmonic genus (μεταβολὴ κατὰ γένος) and to the triphonic tone system (μεταβολὴ κατὰ σύστημα) within another mode, like echos tritos, echos plagios tetartos or echos plagios protos (plagal first mode) for instance, to an intervallic structure and tone system that is proper to phthora nana as an own echos.
 * 3) as echos kratema, phthora nana could used as an own mode which had been used in improvised sections like kratemata or teretismoi which had been sung over abstract syllables. Byzantine composers like Manuel Chrysaphes the Lampadarios mentioned, that the phthora nana always leads to the echos plagios tetartos (fourth plagal mode), before the melos can change into another mode.
 * 4) as exoteric "phthora atzem", the sign was used for the transcription of music composed in makam acem, a makam connected to a certain Persian dastgah.
 * 5) as a modulation sign, phthora nana played a crucial role to change between the third to the plagal fourth mode. Hence, there can be found some parallels between the b flat as used by Guido of Arezzo, and the temporary use of phthora nana in psaltic compositions which changed between the modes. As such phthora nana was used as an alteration sign within the heptaphonic solfeggio of Chrysanthos' New Method, which transcribed its triphonic tone system by a frequent use transposition (a certain μεταβολὴ κατὰ τόνον which turns γα into νη similar to the Guidonian mutation memorised as «F fa ut»).

Hagiopolites treatise about phthora nana
It is supposed that the Hagiopolites treatise served during the 9th century as a manual preceding a chant book called tropologion. The book contained a collection of simple hymns troparia as well as heirmoi which served as melodic models (μέλη) for the 10 modes of Octoechos.

Nana holds the status one of the two "special" additional echoi or "mesoi" (medial forms between authentic and plagal echoi) in the system of the Hagiopolitan Octoechos. The other one is called nenano. Already in the Hagiopolites treatise the phthorai nana and nenano have been characterized as both echoi and "not echoi, but phthorai" (→ phthora). This means that they were proper modes with their own models, but they had to be integrated within the Octoechos and its eight-week cycle. Thus, phthora nana was subordinated to the tetartos, and the treatise also referred to it as "Mesos tetartos": as a third medial mode of the tetartos which was neither authentic (kyrios) nor plagal (plagios).

Phthora nana according to the theory and practice of psaltic art
The concrete intervals of the enharmonic genus are less subject than the explanation of relationships between the modes. Often certain paragraphs of the Hagiopolites concerned about the phthora nana have been re-interpreted again and again according to the current tradition of psaltic art.

Its enechema
In contemporary theoretical explanations the phthora nana had not only been regarded as a melodic model for stichera and heirmoi, but also as a transition model as well, which mainly connected tritos with tetartos echoi.

According to theoretical explanations of the Papadike, phthora nana was not only defined by its melos like a proper echos itself, as such it had been subordinated to certain echoi already in the Hagiopolites treatise. Manuel Chrysaphes, Lampadarios at the Byzantine court, emphasised that phthora nana is only used as a phthora of the echos tritos, but within its melos it causes always a change into the echos plagios tetartos. Any other changes have to be made after the transition into the plagios tetartos:

"εἰ δὲ τρίτου μόνον, ἵνα κατέλθῃς καὶ τὸν πλάγιον τετάρτου ἀνεμποδίστως καὶ ποιήσῃς καὶ τὸν ἦχον, καὶ μετὰ ταῦτα ὡς βούλει εἰς τὰ ἔμποσθεν ποιήσεις."

He also mentioned that the phthora had its own solfège (parallage), so that it solved the diatonic echos tritos and binds it to the echos plagios tetartos (δεσμοῦσι καὶ λύουσι "binding and solving").

"αὕτη δὲ ἡ τοῦ νανὰ φθορά, ἥτις ἐστὶν ἀπὸ παραλλαγῶν ἦχος τρίτος, οὑχ οὕτως,  ἀλλὰ σχεδὸν εἰπεῖν ἔστιν ὡς ἦχος κύριος, διότι καὶ στιχηρὰ ἰδιόμελα πολλὰ ἔχει καὶ προσόμοια καὶ εἱρμοὺς καὶ καλοφωνικὰ στιχηρά, ὡς οἱ κύριοι ἦχοι. ἀλλὰ καὶ ἀλληλουϊάρια εἰς τὸν πολυέλεον καὶ εἰς τὸν ἄμωμον τῶν λαϊκῶν. καὶ εἰκότως ἂν καλέσειε ταύτην ἦχον καὶ οὐ φθοράν, καθὼς ἔχει καὶ ἡ τοῦ νενανῶ γλυκυτάτη καὶ λεπτοτάτη φθορά."

The distinction, that it was by the own solfège (apo parallagon) "like a kyrios echos of its own" (os echos kyrios), meant on the surface, that the tritos could be based on the octave on B flat (heptaphonia) with the finalis F, while it had to change into the tetartos octave which was based on C and therefore used b natural as seventh degree. The triphonic solfège could be solved again from the enharmonic into the diatonic genus, but it was in fact not just a change of the genus (μεταβολὴ κατὰ γένος, metavole kata genos), but also a change from the triphonic into the tetraphonic tone system (μεταβολὴ κατὰ σύστημα, metavole kata systema). It was not organised by separated tetrachords (tetraphonia), but by connected ones (triphonia: C—F—b flat). Hence, the "own triphonic parallage" intermediated between both octave species, which were otherwise very far from each other. Already in this transitional function it formed an alternative melos of the tetartos echoi.

Its tone system
The phthora could intermediate, because it was distinct by its triphonic tone system and its enharmonic genus (the minor tone was always on the top of a tetrachord and smaller than a Western half tone—a diesis). Ioannis Plousiadinos invented an own system of parallage in shape of a square made up by four X. It was constructed to represent the triphonic combinations of conjunct tetrachords, but only the last two X on the bottom described the triphonia of the phthora nana—the triphonia based on echos varys (left) and the triphonia based on the plagios tetartos (right). Both possibilities were illustrated at the end of John Koukouzeles' didactic chant Mega Ison.

Its great sign and its dynamis as echos kratema
The Papadikai list between six and ten phthorai which represent the ten modes of the Hagiopolitan Octoechos—the diatonic eight modes (which did not need eight phthorai, since the echoi just represented the four elements of a tetrachord) and the two additional phthorai as their own echoi.

In this very particular sense, the term "phthora" did neither refer to nenano nor nana, but simply meant the use of a transposition sign in order to indicate the precise place of a temporary transposition (μεταβολὴ κατὰ τόνον).

"γνωστών σοι δὲ ἔστω καὶ τοῦτο, ὅτι ἡ φθορὰ αὕτη οὐκ ἔχει καθώς ἐστιν ἡ τοῦ πρώτου ἢ τοῦ δευτέρου, ἢ τοῦ τρίτου, ἢ τοῦ τετάρτου, ἤ τοῦ πλαγίου δευτέρου, διότι αἱ τούτων τῶν ἤχων φθοραὶ δεσμοῦσι καὶ λύουσι ταχίον καὶ ποιοῦσιν ἐναλλαγὴν μερικὴν  πὸ ἦχον εἰς ἄλλον καὶ ἔχουσι τοῦτο καὶ μόνον."

"Note also, that these phthorai [ νενανῶ and νανὰ ] are different from those of ἦχος πρῶτος, ἦχος δεύτερος, ἦχος τρίτος, and ἦχος τέταρτος or ἦχος πλάγιος τοῦ δευτέρου, since the phthorai of the latter [ diatonic ] echoi bind or solve quickly and thus, they only create a temporary change from one echos to another one, this is the only purpose they have."

The list of phthora did not include one to indicate a change into the melos of phthora nana. This change was instead caused by one of the great signs (μεγάλα σημαδία), the xeron klasma (ξηρὸν κλάσμα), like here in the sticherarion of the Biblioteca ambrosiana, where the medial signature of phthora nana (a temporary change into its melos) is prepared by this great sign.

This sticheron could be sung as well by a protopsaltes in the soloistic kalophonic style. If the medial signature of nana was the end of the section chosen for the realisation of a sticheron kalophonikon, this sign could cause, that a whole kratema, a section in abstract syllables like το—το—το for instance, was created in the phthora nana as an echos kratema.

Nana as exoteric phthora atzem
The phthora nana was called in various theories "phthora atzem" which referred to perde acem (the fret on the long necked lute tambur called "acem" which was a common reference for Ottoman musicians) or even to makam acem—the "Persian makam" or "phthora." Panagiotes Keltzanides' edition offered a seyir—a melodic example to illustrate makam acem—on page 81:

It is close to the melodic models of the Persian Dastgāh-e Šur.

The enharmonic echoi of the current Octoechos
The reform of the Byzantine neume notation in the early 19th century redefined the mele according to each genre (troparic, heirmologic, sticheraric, papadic), it also transcribed for the first time the rhythm which was so far part of an oral tradition or method, how to do the thesis of the melos. In these transcriptions the diatonic tritos echoi had little relevance. Chrysanthos of Madytos, together with Chourmouzios the Archivist and Gregorios the Protopsaltes one of the three "great teachers" that undertook the reform, published a treatise explaining the principles of the new system, entitled "Theoretikon Mega tes Mousikes".

While there was once a diatonic form of echos tritos and echos varys, it has no longer any relevance for the current tradition of Orthodox chant, even if it based on Byzantine monodic chant. Orthodox chanters know nowadays only the intonation "nana", when they would like to perform melodies composed in echos tritos. Already Papadikai of the 16th century identified "nana" signature in their lists with the diatonic intonation "aneanes." But its long formula was already replaced by the short of νανὰ in Chrysanthos explanation of the former papadic practice of solfège using the enechemata:

This simple form is used until today, but Chrysanthos also developed its whole melos as a kind of exegesis ("interpretation") of the simple traditional echema of phthora nana:

Chrysanthos' exegesis of the real nana intonation uses the enharmonic intervals:

Διὰ τοῦτο ὅταν τὸ μέλος τοῦ ἐναρμονίου γένους ἄρχηται ἀπὸ τοῦ γα, θέλει νὰ συμφωνῇ μὲ τὸν γα ἡ ζω ὕφεσις, καὶ ὄχι ὁ νη φθόγγος. Καὶ ἐκεῖνο ὅπερ εἰς τὴν διατονικὴν καὶ χρωματικὴν κλίμακα ἐγίνετο διὰ τῆς τετραφωνίας, ἐδῶ γίνεται διὰ τῆς τριφωνίας

 * νη πα [βου δίεσις] γα, γα δι κε [ζω ὕφεσις−6], [ζω ὕφεσις−6] νη πα [βου ὕφεσις−6].

Ὥστε συγκροτοῦνται καὶ ἐδῶ τετράχορδα συνημμένα ὅμοια, διότι ἔχουσι τὰ ἐν μέσῳ διαστήματα ἴσα· τὸ μὲν νη πα ἴσον τῷ γα δι· τὸ δὲ πα [βου δίεσις], ἴσον τῷ δι κε· τὸ δὲ [βου δίεσις] γα, ἴσον τῷ κε [ζω ὕφεσις−6]· καὶ τὰ λοιπά.

"Hence, if the melos of the enharmonic genus starts on F γα, F γα and b flat [ζω' ὕφεσις−6] should be symphonous, and not the phthongos c νη'. And like the diatonic and chromatic scales are made of tetraphonia, here they are made of triphonia:"
 * C νη—D πα—E sharp [βου δίεσις]—F γα, F γα—G δι—a κε—b flat [ζω' ὕφεσις−6], b flat [ζω' ὕφεσις−6]—c νη'—d πα'—e flat [βου' ὕφεσις−6].

"Thus, also conjunct similar tetrachords are constructed by the same intervals in the middle [12+13+3=28]: C νη—D πα is equal to F γα—G δι, D πα—E sharp [βου δίεσις] to G δι—a κε, E sharp [βου δίεσις]—F γα is equal to a κε—b flat [ζω' ὕφεσις−6], etc."

He applied this intonation to the traditional varys enechema:

There is also a diatonic form of echos varys, but according to Chrysanthos' adaption to the Ottoman tone system it was no longer based on a pentachord between kyrios tritos and varys, but on a tritonus on a low b natural.

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title: User:Platonykiss/sandbox/Nana


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User:Platonykiss/sandbox/Hagiopolite Oktoechos

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Oktōēchos (here transcribed "Octoechos"; Greek: ὁ Ὀκτώηχος pronounced in koine: [okˈtoixos]; from ὀκτώ "eight" and ἦχος "sound, mode" called echos; Slavonic: Осмогласие, Osmoglasie from о́смь "eight" and гласъ "voice, sound") is the name of the eight mode system used for the composition of religious chant in most Christian churches during the Middle Ages. In a modified form the octoechos is still regarded as the foundation of the tradition of monodic Orthodox chant today (Neobyzantine Octoechos).

The Octoechos as a liturgical concept which established an organization of the calendar into eight-week cycles, was the invention of monastic hymnographers at Mar Saba in Palestine and in Constantinople. It was formally accepted in the Quinisext Council of 692, which also aimed to replace the exegetic poetry of the kontakion and other homiletic poetry, as it was sung during the morning service (Orthros) of the cathedrals.

A similar eight-mode system was established in Western Europe during the Carolingian reform, and particularly at the Second Council of Nicaea in 787 AD. Quite possibly this was an attempt to follow the example of the Eastern Church.

The evidence for this is an abbreviated chant book called a "tonary", which is a list of incipits of chants ordered according to the intonation formula of each tone in its psalmody. Later on, fully notated and theoretical tonaries were also written. The Byzantine book Octoechos  (9th century) was one of the first hymn books with musical notation and its earliest surviving copies date from the 10th century.

Origins
Students of Orthodox chant today often study the history of Byzantine chant in three periods, identified by the names John of Damascus (675/676-749) as the "beginning", John Koukouzeles (c. 1280–1360) as the "flower" (Papadic Octoechos), and Chrysanthos of Madytos (c. 1770-c. 1840) as the master of the living tradition today (Neobyzantine Octoechos). The latter has the reputation that he once connected in his time the current tradition with the past of Byzantine chant, which was in fact the work of at least four generations of teachers at the New Music School of the Patriarchate.

This division of the history into three periods begins quite late with the 8th century, despite the fact that the octoechos reform was already accepted some decades earlier, before John and Cosmas entered the monastery Mar Saba in Palestine. The earliest sources which gave evidence of the octoechos' use in Byzantine chant, can be dated back to the 6th century.

Jerusalem, Alexandria, or Constantinople
The common schedule and the focus on the circle around John of Damascus is confirmed by a ninth-century treatise called "Hagiopolites" (from hagios polis, "Holy City", referring to Jerusalem) which only survived in a complete form as a late copy. The Hagiopolites treatise served presumably as an introduction of a book called tropologion – a 9th-century chant book which had been replaced soon by the book octoechos, as part of the sticherarion one of the first chant books fully provided with musical notation. The Hagiopolitan emphasis on John of Damascus was obviously the late result of a 9th-century redaction around the Second Council of Nicaea in 787, so it was part of the later Studites reform between Jerusalem and Constantinople and it was motivated theologically, not only because of his contributions to the tropologion, but also because of the keyrole which John of Damascus' polemic against the iconoclasts had during this council.

Nevertheless, the theological and liturgical concept of an eight-week cycle can be traced back to the cathedral rite of Jerusalem during the 5th century, and originally it was the Christian justification of Sunday as the eighth day after Sabbat. Peter Jeffery assumed a first phase during which the concept existed independently in various places, and a second phase during which Palestine became the leading centre of a monastic hymn reform. It established reform models which were also used later by the generation of John of Damascus. Despite that the first paragraph of the "Hagiopolites" ascribes the treatise to John of Damascus, it was probably written about 100 years after his death and went through several redactions during the following centuries.

There is no doubt that the octoechos reform itself had already taken place by 692, because certain passages of the Hagiopolites paraphrase certain law texts (the canons of the synodal decree). Eric Werner assumed that the eight-mode system developed in Jerusalem since the late fifth century and that the reform by the hymnographers of Mar Saba were already a synthesis with the Ancient Greek names used for the tropes, applied to a model of Syrian origin already used in the Byzantine tradition of Jerusalem. During the eighth century, long before Ancient Greek treatises were translated into Arabic and Persian dialects between the ninth and the tenth centuries, there was already a great interest among theorists like Abū Yūsuf al-Kindī, whose Arabic terms were obviously translated from the Greek. He adored the universality of the Greek octoechos:

"Sämtliche Stile aller Völker aber haben Teil an den acht byzantinischen Modi (hiya min al-alhān at-tamāniya ar-rūmīya), die wir erwähnt haben, denn es gibt nichts unter allem, was man hören kann, das nicht zu einem von ihnen gehörte, sei es die Stimme eines Menschen oder eines anderen Lebewesens, wie das Wiehern eines Pferdes oder das Schreien eines Esels oder das Krähen des Hahns. Alles, was an Formen des Schreis einem jeden Lebewesen/Tier eigen ist, ist danach bekannt, zu welchem Modus der acht es gehört, und es ist nicht möglich, daß es sich außerhalb eines von ihnen [bewegt]."

"Every style of any tribe takes part of the Byzantine eight tones (hiya min al-alhān at-tamāniya ar-rūmīya) which I mentioned here. Everything which can be heard, be it the human or be it the animal voice – like the neighing of a horse, the braying of a donkey, or the carking of a cock, can be classified according to one of the eight modes, and it is impossible to find anything outside of the eight mode system."

Al-Kindi demonstrated the intervals on the keyboard of a simple four-stringed oud, starting from the third string as well seven steps in ascending as in descending direction.

According to Eckhard Neubauer, there is another Persian system of seven advār ("cycles"), outside the Arabic reception of the Byzantine octoechos, which was possibly a cultural transfer from Sanskrit treatises. Persian and Ancient Greek sources had been the main reference for the transfer of knowledge in Arabian-Islamic science.

Monastic reform of Mar Saba
According to the Hagiopolites the eight echoi ("modes") were divided in four "kyrioi" (authentic) echoi and their four respective plagioi (enriched, developed) echoi, which were all in the diatonic genus.

8 diatonic echoi of the Hagiopolitan Octoechos
Despite the late copies of the Greek Hagiopolites treatise, the earliest Latin description of the Greek system of eight echoi is an eleventh-century treatise compilation called "alia musica". "Echos" was translated as "sonus" by the anonymous compilator, who commented with a comparison of the Byzantine octoechos:

"Quorum videlicet troporum, sive etiam sonorum, primus graeca lingua dicitur Protus; secundus Deuterus; tertius Tritus; quartus Tetrardus: qui singuli a suis finalibus deorsum pentachordo, quod est diapente, differunt. Superius vero tetrachordum, quod est diatessaron, requirunt, ut unusquisque suam speciem diapason teneat, per quam evagando, sursum ac deorsum libere currat. Cui scilicet diapason plerumque exterius additur, qui emmelis, id est, aptus melo vocatur."

"Sciendum quoque, quod Dorius maxime proto regitur, similiter Phrygius deutero, Lydius trito, mixolydius tetrardo. Quos sonos in quibusdam cantilenis suae plagae quodammodo tangendo libant, ut plaga proti tangat protum, deuteri deuterum, triti tritum, tetrardi tetrardum. Et id fas est experiri in gradalibus antiphonis." "It is known about the tropes, as to say: the ἦχοι, that the Greek language call the First πρῶτος, the Second δεύτερος, the Third τρίτος, the Fourth τέταρτος. Their Finales were separated by a pentachord, that is: a falling fifth (gr. diapente) [between kyrios and plagios]. And above [the pentachord] they require a tetrachord, that is: a fourth (gr. diatessaron), so that each of them has its octave species, in which it can move freely, rambling down and up. For the full octave (gr. diapason) another tone might be added, which is called ὁ ἐμμελής: 'according to the melos'." "It has to be known that the 'Dorian' [ octave species ] is usually ruling in the πρῶτος, as the 'Phrygian' in the δεύτερος, the 'Lydian' in the τρίτος, or the 'Mixolydian' in the τέταρτος. Their πλάγιοι are derived by these ἦχοι in that way, that the formula touch them [going down a fifth]. So the πλάγιος τοῦ πρώτου touch the πρῶτος, the plagal Second [τοῦ δευτέρου] the δεύτερος, the plagal Third [βαρύς] the τρίτος, the plagal Fourth [πλάγιος τοῦ τετάρτου] the τέταρτος. And this should be proved by the melodies of the antiphonal graduals as a divine law."

This Latin description about the octoechos used by Greek singers (psaltes) is very precise, when it says that each kyrios and plagios pair used the same octave, divided into a fifth (pentachord) and a fourth (tetrachord): D—a—d in protos, E—b—e in devteros, F—c—f in tritos, and C—G—c in tetartos. While the kyrioi had the finalis (final, and usually also base note) on the top, the plagioi had the finalis on bottom of the pentachord.

The intonation formulas, called enechema (gr. ἐνήχημα), for the authentic modes or kyrioi echoi, usually descend within the pentachord and turn back to the finalis at the end, while the plagal modes or plagioi echoi just move to the upper third. The later dialogue treatises (Gr. ἐρωταποκρίσεις, erotapokriseis) refer to the Hagiopolitan diatonic eight modes, when they use the kyrioi intonations to find those of the plagioi:

Περὶ πλαγίων

Ἀπο τοῦ πλαγίου πρώτου ἤχου πάλιν καταβαίνεις τέσσαρας φωνάς, καὶ εὑρίσκεται πάλιν πλάγιος πρώτου· ὅυτως δὲ / ἄνανε ἄνες ἀνὲ ἄνες· Ὁμοίως καὶ ὁ β' ἤχος καταβαίνων φωνάς δ', εὑρίσκεις τὸν πλάγιον αὐτοῦ, ἤγουν τὸν πλάγιον τοῦ δευτέρου. πλ Β οὕτως δέ. Ὁμοίως πάλιν ὁ τρίτος καταβαίνεις φωνὰς τέσσαρας, καὶ εὑρίσκεται ὁ πλάγιος αὐτοῦ, ἤγουν ὁ βαρύς, οὕτως·

Ὁμοίως καὶ ἀπὸ τὸν τέταρτον καταβαίνων φωνὰς τέσσαρας, εὑρίσκεις τὸν πλάγιον αὐτοῦ, ὡς ἐστὶ ὁ πλ δ'οὕτως·

Phthorai and mesoi of the Hagia Sophia
The Hagiopolites as "earliest" theoretical treatise said, that two additional phthorai ("destroyers") were like proper modes which did not fit into the diatonic octoechos system, so the Hagiopolitan octoechos was in fact a system of 10 modes. But the chronology of definitions concerned about two phthorai regarded them first as modes of their own because of their proper melos and that their models had to be sung during the eight-week cycle. These mesoi of tetartos and protos, with a finalis and base between kyrios and plagios, were obviously favoured by composer like John of Damascus and his step-brother Kosmas, while the concept of a transition between echoi was established later. It seems that the construction of the eight diatonic echoi was established later by the generation of Theodore the Studite and his brother Joseph.

The later Papadikai mentions that changes between the echos tritos and the echos plagios tetartos were bridged by the enharmonic phthora nana, and changes between the echos protos and the echos plagios devteros by the chromatic phthora nenano.

Nevertheless, the terminology of the Hagiopolites somehow suggested that nenano and nana as phthorai "destroy" one or two diatonic degrees used within one tetarchord of a certain echos, so that the chromatic and enharmonic genera were somehow subordinated and excluded from the diatonic octoechos. This raises the question, when the music in the near eastern Middle Ages became entirely diatonic, since certain melodies were coloured by the other enharmonic and chromatic gene according to the school of Damascus. This is the question about the difference between the Hagiopolitan reform of 692 and in as much it was opposed to the Constantinopolitan tradition and its own modal system.

The author of the Hagiopolites mentioned an alternative system of 16 echoi "sung in the Asma," with 4 phthorai and 4 mesoi beyond the kyrioi and plagioi of the diatonic Octoechos:

"Οἱ μὲν οὖν τέσσαρρεις πρῶτοι οὐκ ἐξ ἄλλων τινων ἀλλ'ἐξ αὐτῶν γινονται. οἱ δὲ τέσσαρεις δεύτεροι, ἤγουν οἱ πλάγιοι, ὁ μὲν πλάγιος πρῶτος ἐκ τῆς ὑπορροῆς τοῦ πρώτου γέγονε. καὶ ἀπὸ τῆς ὑπορροῆς τοῦ πληρώματος τοῦ δευτέρου γέγονεν ὁ πλάγιος δευτέρου· ὡς ἐπὶ τὸ πλεῖστον δὲ καὶ τὰ πληρώματα τοῦ δευτέρου [εἰς τὸν πλάγιον δευτέρου] τελειοῖ. ὁ βαρὺς ὁμοίως καὶ ἀπὸ τοῦ τρίτοῦ· καὶ γὰρ εἰς τὸ ἆσμα ἡ ὑποβολὴ τοῦ βαρέως τρίτος ψάλλεται ἅμα τοῦ τέλους αὐτοῦ. καὶ ἀπὸ τοῦ τετάρτου γέγονεν ὁ πλάγιος τέταρτος. καὶ ἀπὸ τῶν τεσσάρων πλαγίων ἐγεννήθησαν τέσσαρεις μέσοι· καὶ ἀπ'αὐτῶν αἱ τέσσαρες φθοραί. καί ἀνεβιβάσθησαν ἦχοι ις', οἵτινες ψάλλονται εἰς τὸ ἆσμα, οἱ δὲ δέκα ὡς προείπομεν εἰς τὸν Ἁγιοπολίτην."

"The 4 Echoi which come first are generated from themselves, not from others. As to the four which come next, i.e. the Plagal ones, Plagios Prōtos is derived from Prōtos, and Plagios Deuteros from Deuteros – normally Deuteros melodies end in Plagios Deuteros. Similarly, Barys from Tritos – 'for in the Asma Hypobole of Barys is sung as Tritos together with its ending'. From the 4 Plagioi originate the 4 Mesoi, and from these the 4 Phthorai. This makes up the 16 Echoi which are sung in the Asma – as already mentioned, there are sung only 10 in the Hagiopolites."

These "echoi of the Asma" did probably point at the rite at the Patriarchal church or even at the cathedral rite of Constantinople which was also known as "choral" or "sung rite" (ἀκολουθία ᾀσματική). The Constantinopolitan chant books were called asmatikon ("book of the choir"), psaltikon ("book of the soloist called 'monophonaris'"), and kontakarion (the name of the psaltikon, if it included the huge collection of kontakia, sung during the morning service).

Unfortunately no early Constantinopolitan chant manual survived, there is only this short paragraph of the Hagiopolites which says, that the singers of the choir followed in their chant books an own modal system, which was distinct from the Hagiopolitan octoechos. A distinction from Constantinople is not the only possible explanation, because Jerusalem had also its own local cathedral rite. Since the 14th century at latest, the monastic rite was not opposed to the cathedral rite, even monks celebrated it on festival occasions, whenever they expected guests.

The earliest sources are those of the Slavic reception of Constantinople which can be dated back not earlier than to the 12th century, and they used a system of 12 modes. The earliest treatises which mention a modal system, is not a chant manual, but a corpus of alchemic treatises, which testifies a modal system of 24 "elements" (στοιχεῖα) or "aims" (στοχοὶ):

"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,"

"As there are 4 basic elements/targets [ earth, water, air, and fire ] which created their music, the πρῶτος, the δεύτερος, the τρίτος, and the τέταρτος, and by their formulas the same generate 24 different elements: the [4] κέντροι (central), [4] ἶσοι (basic), and [4] πλάγιοι (plagal), the [4] καθαροί (kathartic), [4] 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 corruptions of other musical emotions (πάθη)."

In the edition of the treatise by Otto Gombosi, the four "elements" (α', β', γ', δ') were associated with certain colours—πρῶτος with black (all colours together), δεύτερος with white (no colour at all), τρίτος with yellow (an elementary coulour), and τέταρτος with purple (a combination of elementary colours). These passages could be easily compiled with Zosimos of Panopolis' treatise about the process of bleaching.

The system favoured 3 four tetrachord sets (either modes by themselves or simply degree of the modes with different functions), called κέντροι, ἷσοι, and πλάγιοι. Kέντρος would be probably an early name for μέσος, if it lied between the ἶσος and πλάγιος, it could be as well used as an early name for κύριος ἦχος, because it is mentioned here first, while ἶσος could mean "equivalent", or just basis notes.

The exact point of reference concerning this 24 mode system was not clarified in the treatise, but it is evident, that there was a canonised wisdom which was connected with an ethical doctrine excluding certain passions (πάθη, pathe) as corruptions. Inside this wisdom, there was a Neoplatonic concept of an ideal and divine existence, which can be found and classified according to a modal scheme based on four elements. The term "element" (στοχείον) was less meant as a technical term or modal category, it was rather an alchemistic interpretation of the 24 musical modes.

In comparison, the Hagiopolitan terminology already included the "corruption" (φθορά) as an acceptable modal category in itself, which was neither excluded in the Hagiopolitan Octoechos nor in the modal system of a certain cathedral rite, which was made of 16 echoi. On the other hand, the described system, whether it meant 24 echoi including 12 pathologic echoi, called "aechoi" and "paraechoi", and associated with 4 "katharoi" or just cadential degrees or other modal functions. It is not clear, whether the latter name was simply meant in a geographical or ethnical way or whether it was here connected with a kind of music therapy which included certain pathe as a kind of antidote. Medical treatises of the Mediterranean had been developed later on by the association of melodic modes with 4 elements and 4 humours.

Latin reception

The introduction of the eight mode system in Western chant traditions was part of the Carolingian reform. Officially, it was motivated by Pope Adrian I's confirmation of an earlier Eastern chant reform during the synode in 787, during which he accepted the reform for the Western traditions as well. Nevertheless, a Carolingian interest for the Byzantine octoechos can already be dated back to a visit some years earlier, when a Byzantine legacy introduced a series of antiphons sung during a procession for Epiphany. These antiphons served as a model for the eight modes according to the Hagiopolitan system.

The contemporary invention of a proper Latin version of the eight mode system was mainly studied from two perspectives:

 * the reception of Ancient Greek music theory since Boethius and the synthesis between music theory as a science and a liberal art of the mathematic Quadrivium on the one hand, and as a medium of chant transmission on the other hand. The eight church tones were called after the names of octave species, which were not connected with modal patterns and plainchant theory in earlier times.
 * the simplification of chant transmission by a Western manuscript type called tonary which allowed the transfer of a huge chant repertoire like the Roman one, but also its deductive modal classification which changed the oral transmission of chant entirely.

Synthesis in Latin music theory
Latin theorists who knew the Hellenic tropes only through Boethius' 6th-century translation of Ptolemy (De institutione musica), made the synthesis of the Ancient Greek music theory with the Octoechos as a system of eight church tones, identified with the tropes. The synthesis had not been done earlier than during the Carolingian reform (usually dated according to Charlemagne's admonitio generalis which was decreed in 789), before music theory as science was strictly separated from chant transmission and the cantor as a profession dedicated to church music.

The terms tropus (transposition octave) and modus (the octave genre defined by the position of the tonus, the whole tone with the proportion of 9:8, and the semitonium, the half tone with the proportion of 256:243) were taken from Boethius' translation. But the Antique names of the seven modi were applied to the eight church tones called toni. The first attempt to connect Ancient Greek music theory (as expressed in Boethius) and the theory of plainchant can be found in the treatise De harmonica institutione by Hucbald of Saint-Amand Abbey, written by the end of the 9th century, in which the author addressed his treatise explicitly to cantors and not to mathematicians, whereas the reduction of 4 "finales" which made up the tetrachord D—E—F—G, was already done in Carolingian times in the treatises Musica and Scolica enchiriadis. Musica enchiriadis is also the only Latin treatise which testifies to the presence of a tetraphonic tone system, represented by 4 Dasia-signs and therefore called "Dasia system", and even the practical use of transposition (metabolē kata tonon) in plainchant, called "absonia". Its name probably derived from "sonus", the Latin term for ἦχος, but in the context of this treatise the use of absonia is reserved to describe a primitive form of polyphony or heterophony, rather than serving as a precise description of transposition in monodic chant, as it was used in certain genres of Byzantine chant.

Hucbald used an idiosyncratic Greek letter system which referred to the double octave system (systēma teleion) and called the four elements known as "finales" according to the Greek system:

"Lycanos ypaton scilicet autentum protum· & plagis eiusdem· id est primum & secundum; Hypate mesonʕ autentum deuterum· & plagis eiusʕ iii & iiii· Parypate mesonʕ autentum tritum & plagis eiusʕ id est v· & vi. Lycanos mesonʕ autentum tetrardum. & plagis eiusʕ id est vii· & viii. Ita ut ad aliquam ipsarum ·quatuor. quamvis ul[tra] citraque variabiliter circumacta necessario omnis quaecumque fuerit redigatur· cantilena· Unde et e[a]edem finales appellatae quod finem in ipsis cuncta quae canuntur accipiant."

"Λιχανὸς ὑπάτων [D] is [the φθόγγος of] the autentus protus and its plagal which are [tonus] I and II, ὑπάτη μέσων [E] of the autentus deuterus and its plagal which are [tonus] III and IV, παρυπάτη μέσων [F] of the autentus tritus and its plagal which are [tonus] V and VI, λιχανὸς μέσων [G] of the autentus tetrardus and its plagal which are [tonus] VII and VIII, for the reason, that these four very present ones surround necessarily each melody, so that they, however they might be, can be reduced to them. These four [φθόγγοι] are called 'finales', since in all those [melodies] which are sung, they are perceived as their end."

According to the Latin synthesis the plagal and authentic tones of protus, deuterus, tritus, and tetrardus did not use the same ambitus as in the Hagiopolitan Octoechos, but authentic and plagal tones used both the finalis of the plagios, so that the finalis of the kyrios, the fifth degree of the mode, was no longer used as finalis, but as "repercussa": the recitation tone used in a simple form of psalmody which was another genuine invention by the Carolingian reformers. The ambitus of the authentic tones was made up the same way as used in the Greek Octoechos, while the plagal tones used a lower ambitus: not the tetrachord above the pentachord, but below it. Hence, the hypodorian octave referred the "tonus secundus" and was constructed A—D—a, and the dorian as "tonus primus" D—a—d, both tones of the protus used D as finalis, the hypophrygian octave was B—E—b and was the ambitus of the "tonus quartus", and the phrygian octave E—b—e was related to the "tonus tertius" and its finalis E belonged to the deuterus, the hypolydian octave C—F—c was connected with the "tonus sixtus", the lydian octave F—c—f with the "tonus quintus" and both shared the finalis F called "tritus", the last was the seventh octave G—d—g called "mixolydian" which referred to the "tonus septimus" and its finalis G.

Tonary

The earliest chant theory connected with the Carolingian octoechos was related to the book tonary. It played a key role in memorising chant and the earliest tonaries referred to the Greek names as elements of a tetrachord: πρῶτος, δεύτερος, τρίτος, and τέταρτος. They were translated into Latin as "protus", "deuterus", "tritus", and "tetrardus", but only the tetrachord D—E—F—G was supposed to contain the final notes ("finales") for the eight tones used in the Latin octoechos. Since the 10th century the eight tones were applied to eight simplified models of psalmody, which soon adopted in their terminations the melodic beginnings of the antiphons, which were sung as refrains during psalm recitation. This practice made the transitions smoother, and in the list of the antiphons which can be found since the earliest tonaries, it was enough to refer to the melodic beginnings or incipits of the text. In the earliest tonaries no models of psalmody had been given and incipits from all chant genres were listed, probably just for a modal classification (see the section for the "Autentus protus" of the Saint Riquier tonary).

According to Michel Huglo, there was a prototype tonary which initiated the Carolingian reform. But in a later study he mentioned an even earlier tonary which was brought as a present by a Byzantine legacy which celebrated procession antiphons for Epiphany in a Latin translation.

Already during the 10th century tonaries became so widespread in different regions, that they do not only allow to study the difference between local schools according to its modal classification, its redaction of modal patterns, and its own way of using Carolingian psalmody. They also showed a fundamental difference between the written transmission of Latin and Greek chant traditions, as it had developed between the 10th and 12th centuries. The main concern of Latin cantors and their tonaries was a precise and unambiguous classification of whatever melody type according to the local perception of the Octoechos system.

Greek psaltes were not interested at all in this question. They knew the models of each modes by certain simple chant genres as the troparion and the heirmoi (the melodic models used to create poetry in the meter of the heirmologic odes), but other genres like sticheron and kontakion could change the echos within their melos, so their main interest was the relationship between the echoi to compose elegant and discrete changes between them.

In contrary, the very particular form and function of the tonary within chant transmission made it evident, that the modal classification of Latin cantors according to the eight tones of the Octoechos had to be done a posteriori, deduced by the modal analysis of the chant and its melodic patterns, while the transmission of the traditional chant itself did not provide any model except of the psalm tones used for the recitation of the psalms and the canticles.

The tonary was the very heart of the mainly oral chant transmission used during the Carolingian reform and as its medium it must have had a strong impact on the melodic memory of the cantors who used it in order to memorize the Roman chant, after a synode confirmed Charlemagne's admonitio generalis. The written transmission by fully notated chant manuscripts, the object of chant studies today, cannot be dated back to an earlier time than nearly 200 years after the admonitio—the last third of the 10th century. And it seems that Roman cantors whose tradition had to be learnt, followed at least 100 years later by the transcription of their chant repertory and no document has survived which can testify the use of tonaries among Roman cantors. Pope Adrian I's confirmation of the Eastern octoechos reform had probably no consequences on the tradition of Roman chant, which might be an explanation for the distinct written transmission, as it can be studied between Roman Frankish and Old Roman chant manuscripts.

The eight sections of the Latin tonary are usually ordered "Tonus primus Autentus Protus", "Tonus secundus Plagi Proti", "Tonus tertius Autentus deuterus" etc. Each section is opened by an intonation formula using the names like "Noannoeane" for the authentic and "Noeagis" for the plagal tones. In his theoretical tonary "Musica disciplina" Aurelian of Réôme asked a Greek about the meaning of the syllables, and reported that they had no meaning, they were rather an expression of joy as used by peasants to communicate with their working animals like horses. There was usually no exact resemblance of the Latin syllables to the names of the Greek intonations or enechemata which were identified with the diatonic kyrioi and plagioi echoi, but Aurelian's question made it obvious that the practice was taken from Greek singers. Unlike the Hagiopolitan octoechos, which used two additional phthorai with the syllables Nana and Nenano for changes into the enharmonic and chromatic genus, the enharmonic and chromatic genus was excluded from the Latin octoechos, at least according to Carolingian theorists.

Since the 10th century tonaries also include the mnemic verses of certain model antiphons which memorise each tone by one verse. The most common among all tonaries was also used by Guido of Arezzo in his treatise Micrologus: "Primum querite regnum dei", "Secundum autem simile est huic" etc. Another characteristic was that melodic melisms called neumae followed the intonation formulas or mnemic verses. Usually they differed more among different tonaries than the preceding intonations or verses, but they all demonstrated the generative and creative aspect within chant transmission.

In comparison with Byzantine psaltes who always used notation in a more or less stenographic way, the exact patterns used during the so-called "thesis of the melos" belonged to the oral tradition of a local school, its own modal system and its genre. But already the question of chant genre was connected with local traditions in medieval times and the point of reference for the psaltes who performed a certain genre: the Hagiopolitan octoechos and its genres (the odes according to the models of the heirmologion, the troparia of the octoechos or tropologion), or the Constantinopolitan cathedral rite (akolouthia asmatike) and its books asmatikon, psaltikon, and kontakarion might serve here as examples.

Question of intervals and their transposition
The exact proportions which divided a tetrachord, had never been a subject of Greek medieval treatises concerned about Byzantine chant. The separation between the mathematical science harmonikai and chant theory gave space to various speculations, even to the assumption that the same division was used as described in Latin music theory, operating with two diatonic intervals like tonus (9:8) and semitonium (256:243). Nevertheless, some treatises referred the tetrachord division into three intervals called the "great tone" (μείζων τόνος) which often corresponded to the prominent position of the whole tone (9:8), the "middle tone" (ἐλάσσων τόνος) between α and β, and the "small tone" (ἐλάχιστος τόνος) between β and γ which was usually a much larger interval than the half tone, and this division was common among most divisions by different ancient Greek theorists that were mentioned by Ptolemy in his Harmonics. Before Chrysanthos' Theoretika (the Eisagoge was simply an extract, while the Mega Theoretikon was published by his student Panagiotes Pelopides), exact proportions were never mentioned in Greek chant theory. His system of 68 commata which is based on a corrupted use of arithmetics, can be traced back to the division of 12:11 x 88:81 x 9:8 = 4:3 between α and δ.

Pitches and their tonal system
Although Chrysanthos did not mention his name, the first who mentioned precisely these proportions starting from the open string of the third or middle chord of the oud, was the Arab theorist Al-Farabi in his Kitab al-Musiqa al-Kabir which was written during the first half of the 10th century. His explicit references to Persian and Ancient Greek music theory were possible, because they had been recently translated into Arabic and Persian dialects in the library of Baghdad. Thanks to them Al-Farabi had also an excellent knowledge of Ancient Greek music theory. The method of demonstrating the intervals by the frets of the oud keyboard was probably taken from Al-Kindi. Here the intervals are not referred to the Byzantine phthongoi, but to the name of the frets. And the fret corresponding to β was called "ring finger fret of Zalzal" (wuṣtā Zalzal), named after the famous Baghdadi oud player Zalzal. It seems that the proportion of the Zalzal fret was a refined one in Bagdad using a large middle tone that came very close to the interval of the small tone, while the Mawsili school used 13:12 instead of 12:11. There is no indication that this division had been of Byzantine origin, so Western scholars felt seduced to ascribe the use of the division called "soft diatonic" (diatonikos malakos) and the chromaticism derived from it as an influence of the Ottoman Empire and to regard their view of the systema teleion also as a norm for the Byzantine tonal system. As Phanariotes (Phanar was the Greek district of Istanbul with the residence of the Patriarchate) who composed as well in the makamlar, the teachers of the New Music School of the Patriarchate around Chrysanthos had certainly exchanges with Sephardic, Armenian, and Sufi musicians, but an intensive exchange between Byzantine, Arab and Persian musicians had already a history of more than 1000 years.

Unlike Latin treatises only a few Greek treatises of chant have survived and their authors wrote nothing about the intervals, about microtonal shifts as part of a certain melos and its echos, or about the practice of ison singing (isokratema). Nevertheless, these practices remained undisputed, because they are still part of the living tradition today, while Western plainchant became rediscovered during the 19th century. Neither musicians nor musicologists were longer familiar with them which explains why various descriptions, as they can be found in certain Latin treatises, were ignored for quite a long time.

Ancient Greek music theory had always been a point of reference in Latin chant treatises, something similar cannot be found in Greek chant treatises before the 14th century, but there were a few Latin treatises of the 11th century which did not only refer to Ancient music theory and the systema teleion together with the Greek names of its elements, they even had parts dedicated to Byzantine chant. The appreciation for Byzantine chant is surprising, because there were very few authors except Boethius who had really studied Greek treatises and who were also capable to translate them.

The systema teleion was present by the Boethian diagram which represented it for the diatonic, the chromatic, and the enharmonic genus. Several tonaries used letters which referred to the positions of this diagram. The most famous example is the letter notation of William of Volpiano which he developed for the Cluniac reforms by the end of the 10th century. In his school a unique tonary was already written, when he was reforming abbot of St. Benignus of Dijon. The tonary shows the Roman-Frankish mass chant written out in neume and pitch notation. The repertory is classified according to the Carolingian tonary and its entirely diatonic octoechos. The use of tyronic letters clearly shows, that the enharmonic diesis was used as a kind of melodic attraction within the diatonic genus, which sharpened the semitonium. Even in Guido of Arezzo's treatise Micrologus, at least in earlier copies, there is still a passage which explains, how the diesis can be found on the monochord. It sharpens the semitonium by replacing the usual whole tone (9:8) between re—mi (D—E, G—a, or a—b) by an even larger one in the proportion of 7:6 which was usually perceived as an attraction towards fa.

But there were as well other practices which could not be explained by the Boethian diagram and its use of tonus and semitonium. The authors of one theoretical tonary of the compilation called alia musica used an alternative intonation with the name AIANEOEANE, the name was obviously taken from a Byzantine enechema ἅγια νεανὲς, a kind of Mesos tetartos with the finalis and basis on a low E, and applied the Byzantine practice to certain pieces of Roman-Frankish chant which were classified as "tonus tertius" or "Autentus deuterus". In the following section "De quarto tono" the author quotes Aristoxenos' description of the enharmonic and chromatic division of the tetrachord, the remark on it in precisely this section had been probably motivated by the Hagiopolitan concept of the phthora nenano which connected the echos protos on a with the plagios devteros on E.

Medieval use of transposition (μεταβολή κατὰ τόνον)
Latin cantors knew about the theoretical concept of the practice of transposition since Boethius' translation of Ptolemy. Very few can be said, if they ever understood the practical use of it. Nevertheless, there was a rudimentary knowledge which can be found in the Carolingian treatises Musica and Scolica enchiriadis. The Musica enchiriadis was also the only Latin treatise which documented a second tone system beside the systema teleion, but it does not explain at all, how these both systems worked together in practice.

The Hagiopolites did neither explain it nor did it mention any tone system nor the metabole kata tonon, but this was probably, because the hymn reform of Jerusalem was mainly concerned with simple models exemplified by heirmoi or troparia. Greek protopsaltes used the transposition only in very few compositions of the sticherarion, for instance the compositions passing through all the modes of the Octoechos, or certain melismatic elaborations of troparia in the psaltic style, the soloistic style of the Constantinopolitan cathedral rite. This might explain that Charles Atkinson discussed Carolingian theory in comparison with the later papadikai, in which all possible transpositions were represented by the Koukouzelian wheel or by the kanônion.

Wheels are also used in Arabic music theory since the 13th century, and Al-Farabi was the first who started a long tradition of science, which did not only find the proportions of the untransposed diatonic system on the oud keyboard, but also those of all possible transpositions. The use of instruments had to adapt to a very complex tradition which had probably been a rather vocal tradition in its origins.

Greek chant treatises

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Latin treatises and tonaries (6t–12th century)

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Latin chant books and notated tonaries

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