I wrote a while back of how it wasn't until livredor pointed out the use of parallelism to me that I was aware of any structure in Biblical poetry at all; and how R. Chaim Weiner later deepened my insight.
R. Weiner recently gave another דרשה on in Assif on the same theme, on the subject of אשרי (Psalm 145). [The handout gave the whole of אשרי, with the prologue and epilogue verses, but as R. Weiner's talk didn't cover those, I will omit them here.] As this was given in shul on Shabbos, and I didn't type up these notes until a week and a half later, what you get here is what I remember. [There may have been a bit about its importance in the liturgy too.]
As is well known, the psalm forms an alphabetic acrostic with the letter נ missing; it also demonstrates well the device of parallelism, the second half of each verse restating in different words the theme of the first half. Yet there is deeper structure to it too.
The Praise of David:א 1 I extol You, my God, the king; and bless Your name for ever and ever.
ב 2 Every day I bless You and praise Your name for ever and ever [saying]:
Stanza Iג 3 Great is the LORD, and greatly to be praised; and his greatness is unsearchable.
ד 4 One generation shall praise Your works to another, and shall declare Your mighty acts.
ה 5 I will speak of the glorious honour of Your majesty, and of Your wondrous works.
ו 6 And men shall speak of the might of Your terrible acts: and I will declare Your greatness.
Stanza IIז 7 They shall abundantly utter the memory of Your great goodness, and shall sing of Your righteousness.
ח 8 The LORD is gracious, and full of compassion; slow to anger, and of great mercy.
ט 9 The LORD is good to all: and his tender mercies are over all his works.
Interludeי 10 All Your creatures shall praise You, O LORD; and Your faithful shall bless You.
Stanza IIIכ 11 They shall speak of the glory of Your kingship, and talk of Your power;
ל 12 To make known to the sons of men his mighty acts, and the glorious majesty of his kingdom.
מ 13 Your kingship is an everlasting kingship, and Your dominion endures throughout all generations.
Stanza IVס 14 The LORD supports all that fall, and straightens all who are bent.
ע 15 The eyes of all look to You expectantly; and You give them their food on time.
פ 16 You open Your hand, and satisfy the desire of all living.
צ 17 The LORD is righteous in all his ways, and holy in all his works.
ק 18 The LORD is nigh unto all them that call upon him, to all that call upon him in truth.
ר 19 He will fulfil the desire of them that fear him: he also will hear their cry, and will save them.
ש 20 The LORD preserves all them that love him: but all the wicked will he destroy.
Postludeת 21 My mouth shall speak the praise of the LORD: and all flesh shall bless his holy name for ever and ever.
As can be seen, the psalm constitutes a prologue, two stanzas, interlude, two more stanzas and an epilogue. The theme of the prologue, interlude and epilogue is that of the praise of G-d. The theme of each of the stanzas is referred to at both its start and end. For stanza I this is the greatness of G-d, for stanza II G-d's goodness, stanza III G-d's kingship, and stanza IV G-d as one Who supports.
The psalmist has also gone to considerable length to arrange that the three verses on kingship fall on the letters כ, ל and מ, which spell out the word מלך "king".
There are four verses in the first stanza, and three in the second. Together these total seven, a number of great mystical significance: the seven days of Creation, seven Noachide commandments, seven times seven days in the Omer, seven times seven sets of שמיטה years before the Jubilee, etc. Stanza IV also totals seven verses.
Stanzas III and IV together total ten verses, and the first two stanzas are followed by the Interlude on י, symbolising ten. This is also a number of significance: the Ten Commandments, the Ten Days of Repentance, etc.
Stanzas I and III have the common theme of extolling the greatness of G-d; these too come to seven verses; and Stanzas II and IV, describing the benevolence of G-d to man, also come to ten verses. Note how the verses describing G-d's benevolence to men outnumber, and are therefore more important, than those that simply describe G-d's greatness.
There are various explanations for the missing letter נ. Some say it is because the verse that would have been there would have referred to bad things. Others say there was a verse, and it is lost. (There are MSS with a נ verse, but they are mediaeval and inauthentic.) R. Weiner argues rather that it is because it would have spoiled the pattern of the verses given above. [Though it occurs to me the pattern could still have been accommodated by having the postlude two verses, as the prelude.]