Delitzsch on the te-amim

His opening salvo:

The accents are only musical, and indirectly interpunctional, signs for the chanting pronunciation of the synagogue. And moreover we no longer possess the key to the accents of the three metrical (i. e. consisting of symmetrical stichs and strophes) books as musical signs. For the so-called Sarka-tables (which give the value of the accents as notes, beginning with Zarka, zrqa), e. g. at the end of the second edition of Nagelsbach's Gramm., relate only to the reading of the pentateuchal and prophetic pericope, — consequently to the system of prose accents.

 Before we get to his notes, here are the accents and the names I use. How many zarqa-tables would have different names - I cannot answer this. The lack of a consistent transcription of square text to Latin text impedes understanding of this basic question. 

Haïk-Vantoura clearly manages both the prose and the poetic accents with her two related deciphering keys. Delitzsch is clear in his rejection of punctuation. The shape of the phrase comes from the music. 

The Accents of the Hebrew Bible relating to the Music. Updated table here.
Below the text			Above the text
Recitation		Accent name		Ornament	Full name
c	֧	darga	֝֗	ger-rev	revia-mugrash
d	֢ ֛	galgal (prose), tevir (poetry)	֨	pas	pashta
e	ֽ	silluq	֜	ger	geresh
f	֥	merkha	֞	tar	tarsin
g	֭	tifha, (d’khi)	֡	paz	pazer
A	֑	atnah	֔	z-q	zaqef-qatan
B	֣	munah	֕	z-g	zaqef-gadol
C	֚	mahpakh, (yetiv)	֙	qad	qadma
dm	֦	double merkha, kefulah	֒	seg	segol
			֘	zar	zarqa, tsinnor
			֩	t-q	telisha qetana
			֠	t-g	telisha gedolah
			֟	qar	qarne farah
			֓	shl	shalshalet
			֬	ill	illuy
			֫	ole	ole
			֗	rev	revia

So Delitzsch writes:

Pazer and Shalsheleth have a like intonation, which rises with a trill; though Shalsheleth is more prolonged, about a third than that of the prose books.

 Well, they are both ornaments and so do not move the reciting note. The problem with ornaments is that they can all be similar to a trill - SHV has pazer as moving up a third then returning by step to the current reciting note, shalshelet as moving down a minor third and returning to the reciting note by semi-tone. Shalshelet is rare in any case - 7 uses in the 21 books, 39 in the poetry.

Legarme (in form Mahpach or Azla followed by Psik) has a clear high pitch, before Zinnor, however, a deeper and more broken tone; soft tone tending to repose.

Legarmeh is an adjective applied to other accents. It has no pitch in the SHV schema. Here is what the translator of SHV noted: "The attentive reader, familiar with synagogue chant or the grammatical rules used by Hebraists, will notice that the list of names given by Haik-Vantoura for the signs (cf. pp. 97-100) lacks several names found in every modern list of names given by Hebraists or cantors: azla , legarmeh, yetiv, debir, gaya or metheg, and so on. This includes the list given by Mayer Lambert in his treatise (cited by Haik-Vantoura as her source). Those graphical signs which have more than one name in the modern lists are precisely those which are found in more than one grammatical placement relative to the verbal text. In effect, modern specialists have chosen certain names (among the 80 or so found in the ancient treatises) which are found in all the early sources; their lists assign one of these names to each "grammatical accent" used in the text. Haik-Vantoura’s key assigns (from the same group of names) one name for each graphical form used; the names she rejects can be accounted for as later, secondary names added to distinguish between the different grammatical placements of certain signs. Since (as is acknowledged) the grammatical rules of the Masoretes are an arbitrary imposition upon the syntax of the verbal text, only a decipherment of the te‘amim independent of the names given to various signs could clearly demonstrate which names are ancestral and which were added by the Masoretes and later scribes." For the full book see here.

By Silluk the tone then diminishes.

 Silluk is the tonic, assumed if not specified and returned to at the end. So yes, the tone (volume) will likely diminish.

The tone of Mercha is according to its name, andante and sinking into the depths; the tone of Tarcha corresponds to adagio.

 Tone and tempo should not be confused with each other.

Further hints cannot be traced: though we may infer with respect to Ole we-jored (Mercha mahpachatum) and Athnach, that their intonation ought to form a cadence, as that Rebia parvum and Zinnor (Zarqa) had an intonation hurrying on to the following distinctive accent.

 Yes these are the two major cadences, the Atnah in all books, and the ole-veyored only in the poetry books.

Further, if we place Dechi (Tiphcha initiale) and Rebia gereshatum beside the remaining six servi among the notes, we may indeed produce a sarka-table of the metrical accentuation, although we cannot guarantee its exact agreement with the original manner of singing.

 There is no guarantee associated with any zarka-table.

Following Gerbert (De musica sacra) and Martini (Storia della musica), the view is at present very general that in the eight Gregorian tones together with the extra tone (tonus peregrinus)* used only for Ps. cxiii (= cxiv-cxv in the Hebrew numeration), we have a remnant of the ancient Temple song; and this in itself is by no means improbable in connection with the Jewish nationality of the primitive church and its gradual severance at the first from the Temple and synagogue. 

That Dr. D mentions tonus peregrinus is significant. This is the only 'modern' mode with a variable reciting note. Why would the variation in reciting note have been dropped?  It is very expressive of tone of voice. The 'tune' for Psalms 114 as derived from the deciphering key of Haïk-Vantoura is variations on tonus peregrinus. She herself did not notice this. Her key provides a crucial test case for any other deciphering of the music of the accents. (For more on modes see these posts.)

Here is the score for the psalm 114 showing its similarity to tonus peregrinus.