I thought to read some of the Psalms in Hebrew, a very slow process for me but with enough online tools, it is possible. I was hoping to gain insight by reading in the original Hebrew. After struggling for a while, I said to my wife "it's not written in iambic pentameter, ya know". She observed that I was showing off by throwing big words around, and asked me what that meant. Busted. I had to look it up. Wikipedia(Iambic_Pentameter)  to the rescue.
A standard line of iambic pentameter is five iambic feet in a row:
da DUM da DUM da DUM da DUM da DUM
A straightforward example of this rhythm can be heard in the opening line of William Shakespeare's Sonnet 12:
When I do count the clock that tells the time
I have been unable to find out "why ten syllables?"
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I already knew Hebrew poetry wasn't like English poetry, but I thought there would be some rhythm and some rhyme. Not so much.
Hebrew poetry is destitute of meter in the strict sense, and also of rhyme, though this last occurs in some isolated cases. No wonder then that western scholars, missing these marks of the poetry which they knew best, failed for so long to note the poetry which the Old Testament contains. Hebrew Poetry 
According to Introduction to Hebrew Poetry , the Old Testament is one third poetry, and according to Poetry in the Hebrew Bible  by Jeff A. Benner, the Old Testament is 75% poetry. Benner argues that even Genesis 1 is poetic in structure. So it is surprising that until the 18th century, Western scholars did not even recognize the OT was filled with poetry - Hebrew Poetry  - except for the fact there is no rhythm or rhyme...
Oddly, the New Testament contains very little poetry.
Poetry in the New Testament: Very little poetry is found in the New Testament, except poetry quoted from the Old Testament or hymns which were included in the worship services of the early church. The Beatitudes (Matt. 5:3-10; Luke 6:20-26) have a definite poetic form. The Gospel of Luke contains several long poems: Zacharias' prophecy, known as the Benedictus (Luke 1:68-79); the song of Mary, known as the Magnificat (Luke 1:46-55); the song of the heavenly host, known as the Gloria in Excelsis (Luke 2:14); and the blessing of Simeon, known as the Nunc Dimittis (Luke 2:29-32). Poetry In The Bible 
Well, what is Hebrew poetry then, and what kind of poetry is it?
Rhythm, rhyme, repetitive sounds and wordplays are not easily reproduced in a translation. However, the key to appreciating biblical poetry, and indeed most of the ancient Near Eastern poetry, is none of these. It is parallelism. The Key to Biblical Poetry I guess that's good news for people reading the Bible in a translation, which is most of us, because those poetic elements we expect (rhythm, rhyme, repetitive sounds and wordplays) for the most part aren't there anyway, so in a sense, less is lost in translation. Those things do exist in Bible poetry, but aren't as important or as common as parallelism.
Parallelism is the expression of one idea in two or more different ways. "The content of one line is repeated, contrasted, or advanced by the content of the next - a type of sense rhythm characterized by thought arrangement rather than by word arrangement or rhyme" - Poetry Of The Bible . It is also called rhyming thoughts. Let's consider an example.
Psalm 8:4 What is man, that thou art mindful of him?
and the son of man, that thou visitest him?
Basically, it says the same thing twice. I picked this verse because the Hebrew actually does some wordplay here, and I need to justify all my efforts. David uses two different words for "man" (enosh and adam), and two similar words for "mindful" (tizkrenu) and "visit" (tifqdenu). The symmetry is easy to see in a fixed width font.
Mah enosh ki tizkrenu (to remember or mark)
uven adam ki tifqdenu (to visit, oversee, care for)
This is an example of synonymous parallelism, where line two repeats the thought of line one. In antithetical parallelism, line two states the same idea in opposite terms so no matter which way you read it, you end up with the same thought.
Psalm 1:6: "The LORD knows the way of the righteous,One more type of parallelism that I'll cover here is the chiasm, but parallelism doesn't stop there, you can read more about poetic structures of Hebrew at references 2, 3, 4, or 6.
But the way of the ungodly shall perish"
The Menorah Psalm
One way the Bible organizes both prose and poetry is by means of parallelism called the chiasm. This means a sequence of ideas is presented, then repeated in reverse order, like a mirror image - What Is A Chiasm? . It doesn't mean the verses are repeated word for word, but the ideas or simply words are repeated. A chiasm draws one's attention to the middle verse. It is the way a Menorah is made, one side balances the other. Psalm 67 is an example of this, and is called the Menorah Psalm. The psalm has an introduction plus seven verses. Verses 2 and 8 are related, verses 3 and 7 too, verses 4 and 6 are identical, and verse 5 forms the center of the chiasm like the central stalk of a Menorah. Verse 5 is a mini chiasm of three phrases, the first and third referring to the nations, which narrows our attention to the very middle phrase (in bold). Note the highlighted words.
2. God be gracious to us, and bless us; and let His face shine upon us. Selah.Psalm 67:2-8
3. That your way may be known on earth, your salvation among all nations.
4. Let the peoples praise you, O God; let all the peoples praise you.
5. O let the nations be glad and sing for joy;
for you shall judge the peoples righteously,
and govern the nations on earth. Selah.
6. Let the peoples praise you, O God; let all the peoples praise you.
8. God shall bless us; let all the ends of the earth fear him.
"The Menorah Psalm"
Not counting the intro, Psalm 67 has 49 words (in Hebrew), arranged in a symmetrical pattern, looking at verses 2-8, taken in order, that's 7, 6, 6, 11, 6, 6, 7. This is structure you can only see in the Hebrew. Because it has 49 words, "it has often been used in conjunction with the counting of the 49 days between Passover and Shavuot" - The Menorah Psalm . In this article, Christopher P. Benton makes the argument that the Priestly Benediction of Numbers 6:24-26 and Psalm 67 are also parallel passages.
Dr. Benton writes:
There are clearly some connections between the language of the Priestly Benediction and the Menorah Psalm. For instance, one cannot fail to notice the similarity between the first and second verses of the Priestly Benediction where it says, “The Lord bless you and keep you; The Lord make His face shine upon you,” and verse 2 of Psalm 67 where we read, “God be gracious to us, and bless us; and let His face shine upon us.” Clearly Psalm 67 continues a theme begun in Numbers 6.
Thus, a picture is starting to emerge of the Priestly Benediction and the Menorah Psalm as complementary parts of a single whole with the Priestly Benediction invoking God’s mercy and the Menorah Psalm addressing God’s justice. - The Menorah Psalm 
Here is the text of the Priestly Benediction, also called the Aaronic Benediction.
24 The LORD bless you, and keep you;
25 The LORD make His face shine on you,
And be gracious to you;
26 The LORD lift up His countenance on you,
And give you peace.
But the Menorah Psalm stands on its own. Like I said above, one's attention is drawn to the middle of a chiasm. The outer verses all speak of bounty and blessings and praise but the middle verse upon which all that depends speaks of righteous judgment and governing the nations. Like the branches of a menorah depend on the central stalk, blessings depend on justice.
Why A Menorah?
|The Menorah in the Tabernacle|
burned oil, not candles
Indeed, why a menorah, a chiasm? One answer is that God creates things in balance, and that balance is pictured in the menorah (which was the means of light in the Holy Place of the Tabernacle). God often reveals understanding in His word by juxtaposing ideas in a chiasm, giving light to the reader. Dr. Denton says that balance is mentioned in Eccl 7:14 "In the day of prosperity be happy, But in the day of adversity consider-- God has made the one as well as the other, So that man will not discover anything that will be after him". A word for word translation of the middle of this verse (highlighted portion) from Hebrew says "this against this made God". The rest of the verse talks of balancing prosperity against adversity, that is "this against this", but the principle is broader, it's how the world was made.