The Canon Diagram and the ‘Lydian’ Foundations
of the Imperial Harmonic System
The centrality of the Imperial Hypolydian key discussed by Ptolemy and Porphyry, and evidenced by the Imperial musical documents, is confirmed by a beautiful diagram that is featured in the earliest codex which preserves ancient Greek music theory (Heidelb. Pal. Gr. 281—1040 AD).
Lynch 2024a, Figure 10: The Canon diagram and the ‘Division of the Musician’s Canon’ handed down by Thrasyllus (Theon Math. Plat. 87.4–93.9). Heidelbergensis Palatinus gr. 281, foll. 173 v–174r.
This diagram represents an ancient monochord (kanṓn) and comes immediately before a passage from Theon of Smyrna’s Mathematics Useful for Reading Plato that describes ‘the division of the Canon’ (ἡ δὲ τοῦ κανόνος κατατομή) handed down by Thrasyllus. This text appears under the heading ‘division of the musician’s Canon’ (μουσικοῦ κανόνος κατατομή) on the facing page of the manuscript, and describes precisely how to slide the bridge of a monochord in order to set up the diatonic and chromatic notes included in the Canon diagram itself (Figure 10).
As shown in Figure 11, the Canon diagram illustrates the Unmodulating Perfect System—the overarching theoretical structure of ancient Greek harmonics—by means of the corresponding Hypolydian notes. This diagram in fact starts from the lowest note of the Hypolydian system—Hypolydian proslambanómenos E2, aptly labelled as ‘the origin’ (archḗ) in keeping with Theon’s usage—and then adds the Hypolydian notation signs that are needed to define the tetrachords that belong to the Greater Perfect System as well as the Lesser Perfect System.
This diagram therefore provides a visual illustration of the correspondence detailed by Porphyry and Ptolemy, showing that the Hypolydian notation tónos replaced the Classical Dorian as the underlying point of reference of the Imperial harmonic system.
Lynch 2024a, Figure 11: The Canon’ (Heidelbergensis Palatinus gr. 281, fol. 173 v) and the Hypolydian Unchanging Perfect System.
The explanatory text that accompanies the Canon diagram likewise notes that a reader who proceeds to reproduce these notes on a monochord ‘shall find the hormasía of the Hypolydian, as anticipated’.
ὁ κανὼν, οὗτος, τῆς ὁρμασίας ἐστὶ τοῦ ἐκεῖθεν
φύλλου· ἀλλὰ κατὰ τῶν τριῶν
γενῶν· τουτέστι διατονικοῦ·
χρωματικοῦ· ἁρμονικοῦ· ἔχει
δὲ τὴν τάξιν τοῦ κανόνος καὶ τὰς
κατατομάς· ὅ περ ἐὰν βούλῃ
ποιῆσαι, ξύλινον ποίησον ὑ-
πόκουφον· τουτέστιν ὑπο-
τύμπανον· ἔχοντα καὶ μίαν
χορδὴν ἐπιτείνουσαν· καὶ ἔχου-
σαν τὸ καβάλιν· καὶ κατὰ γραμμήν
ὑπόσυρον τὸ καβάλιν· καὶ εὑ-
ρήσεις τὴν ὁρμασίαν ὑπολυδί-
ου ὡς προείρηται: ~
This is the Canon of the tuning (hormasías) that is on the previous
leaf; but it is set down according to three genera:
that is to say, diatonic;
chromatic; [en]harmonic. It displays
the organisation of the Canon and its
divisions. If you ever wished to
build it, build a wooden
box as a foundation, that is to say a
‘drum’ [i.e. a sound box]; it must also have a single
string, stretched along [the box]; and the string must have
a bridge (kabálin). Follow the diagram as you
slide the bridge, and you shall find
the tuning (hormasían) of the hypolydian,
as mentioned earlier.
Lynch 2024a, Appendix 1.
This comment links the Canon diagram with the tunings and scales that are detailed in a table called ‘Common Tuning’ (Koinḕ Hormasía) that is provided on the previous page of the codex.
The interpretation of a number of details of this table is problematic, and I will discuss the interpretation offered in the diagram below in a forthcoming book.
But one point is beyond doubt–namely the fact that the tuning recorded in the Hormasía is rooted in the Hypolydian tónos that is reproduced in its right column, precisely as we are told in the text that accompanies the Canon diagram and in full accordance with the Hypolydian system represented in the Canon diagram itself.
Lynch 2024a, Appendix 2A–B: The ‘Common Tuning’ (Koinḕ Hormasía) – Heidelb. Pal. gr. 281 Fol 173r.
For the moment, however, let us keep our focus on the Canon diagram and its significance for Imperial music.
As mentioned above, the Canon diagram comes immediately before a passage that describes ‘the division of the Canon’ (ἡ δὲ τοῦ κανόνος κατατομή) handed down by Thrasyllus. In addition to providing detailed instruction on how to set up the diatonic and chromatic notes reproduced in the Canon diagram, this text sheds light on the abbreviations employed at the bottom of the diagram (Figure 12).
Lynch 2024a, Figure 12: Thrasyllus’ division of the Canon and the abbreviations featured in the Canon diagram
In keeping with the interpretation suggested by Vincent (1847, 257), the abbreviations Φ and Χ correspond to semitones that are not missing from the Hypolydian Canon (Φ for Φαῦλον, ‘missing’ or ‘failing’) and semitones that produce a chromatic division of the relative tetrachords (Χ for Χρωματικόν, ‘chromatic’). As I will show in greater detail in the book, the abbreviation AI likewise indicates a ‘missing’ semitone that corresponds to the note D#4 (Figure 11) and in principle could have belonged to the structure of the Aeolian key that is partly included in the Iasti-aeolian tuning described by Ptolemy, but is in fact missing from Thrasyllus’ division as well as the Canon diagram.
The theoretical picture sketched by the Canon diagram and Thrasyllus’ division is also fully consistent with the evidence of the Imperial musical documents. The notes that correspond to the ‘missing’ semitones marked as Φ in the Canon diagram are either unattested or extremely rare in the Imperial documents (for the relative data, see Lynch 2024a, note 65).
The same is true for the ‘missing’ Iasti-aiolian semitone markedas AI in the Canon diagram, as neither Ø Ø nor × × D#4 are attested in the Imperial record. In contrast, the chromatic notes marked as X in the Canon diagram appear regularly in the Imperial documents, especially in their vocal variety, with the only exception of the lower chromatic note G#2—a note that is interestingly missing from Thrasyllus’ division as well.
The overall picture outlined by the Imperial musical documents also confirms the foundational role of the Hypolydian key illustrated by the Canon diagram: 99.1% of the notes attested in these documents fall within the range of the Hypolydian tónos E2–E4 (Lynch 2024a, Figure 4) and the handful of notes that fall outside this range are clustered in four documents.
In keeping with this, 84.1% of these notes fall within the Hypolydian central octave B2–B3. This figure interestingly rises to 91.7% if we include hyperhypátē A2, in accordance with Thrasyllus, as well as the note
A C#4 that is required to produce the defining tritone of the Higher Mixolydian tuning included in the Imperial harmonic system (Lynch 2024a, Figure 9).
Conversely not a single Dorian mésē P P F3 occurs in the Imperial musical documents, showing that this note lost the central role it enjoyed in the Classical system. As shown in Lynch 2024a, 25–28, many other details of the Canon division handed down by Thrasyllus reflect the metamorphosis of the Imperial harmonic system, including the growing importance of fifths and the parallel weakening of the importance of paramésē.
But how did this transformation happen? The evolution of the Greek notation system can help us bridge the gap.