The word translated "eight-string lyre" is shem-ee-NEETH, which means "eighth," or "octave." It really isn't clear whether this instruction is to play the psalm on an eight-stringed instrument, or if it denotes a very low singing - the lowest note sung by men's voices, or a certain key to be played in.
Both Psalms 6 and 12 carry instructions for the shem-ee-NEETH, as well as 1Chronicles 15:21.
David obviously knows that something he's done deserves rebuke and chastening. David is an amazing guy. He knew God intimately, yet sometimes he found himself sinning grievously. He committed adultery with Bathsheba, and then arranged to have her husband killed when she got pregnant. When he was brought before Achish, the king of Gath, he pretended he was crazy and let saliva run down his beard and scribbled on the walls. He cut the fringe from King Saul's robe while he was going to the bathroom in a cave, and after God told him not to count the people, he did anyway. But the Bible also calls David a man after God's own heart (1Sam 13:14).
How could such a sinful man be after God's own heart? Here's the key: even though David was a sinner just like everyone else, he was a man of repentance. He acknowledged his sin before men and before God.
Many of us are busy trying to impress others with our maturity in the Lord by hiding our shortcomings. We sin, but we cover it up. We falter in faith, but we keep the mask on. We are defeated, but are too prideful to admit that we're not doing okay. Our marriages aren't as happy as we make them appear. Our prayer lives aren't as extensive as we'd like others to believe. Our lives aren't as spotless as we hope they look.
God's sees maturity in the people who are honest and confess their sin. The man after God's own heart is not the one who appears sinless. He is one who acknowledges that he has fallen short. The one who is continually repenting for his sin.
In Psalm 32, David wrote,
Ps. 32:5 I acknowledged my sin to Thee, and my iniquity I did not hide; I said, "I will confess my transgressions to the LORD"; And Thou didst forgive the guilt of my sin. Selah.
God has reassured us that...
1John 1:9 If we confess our sins, He is faithful and righteous to forgive us our sins and to cleanse us from all unrighteousness.
There is also the benefit of confessing our sins to others.
James 5:16 Therefore, confess your sins to one another, and pray for one another, so that you may be healed....
Sometimes, the cure for our ailments is simply confessing what we've tried to bottle up and keep hidden. David also wrote in Psalm 32,
Ps. 32:3 When I kept silent {about my sin} , my body wasted away
If you're wasting away, stop keeping silent.
David's sin was causing him to waste away, to decline in spirit and health. He prays, "Don't punish me, Lord, for I am already feeling the physical and emotional effects of my sin, and I am repenting." As he repents of his sin, he asks for healing from the effects of his sin, and not having confessed earlier.
He asks for God to be gracious, because he knows he is simply reaping what he's sown. He doesn't deserve to be healed, but he appeals to God's grace.
David knows that his sin has separated him from the intimacy he had with the Lord before.
Isa. 59:2 But your iniquities have made a separation between you and your God, and your sins have hidden His face from you, so that He does not hear.
He begs God to return to that closeness, and to save him from this destruction. Again, he asks for salvation based on God's mercy, not on his own righteousness.
David, fearing that his life is over if God doesn't restore him, says,
Ps. 6:5 For there is no mention of Thee in death; In Sheol who will give Thee thanks?
Now, this is an opportunity to point something out. Not everything in the Bible is true. That is not to say that the Word of God is not totally inspired, but people quoted in Scripture are not always speaking truth. Remember in Joshua, Rahab is quoted as lying to the king's men who were after the two spies. It is a quote, but not an approval for lying. Eliphaz the Temanite, Bildad the Shuhite, and Zophar the Naamathite are quoted extensively in the book of Job, yet most of what they say is inaccurate, misrepresenting God.
David here is wrong in what he says. We know from Jesus' description of the life after death in Luke 16 that David is wrong. After we die, we have consciousness, and the ability to praise the Lord.
When you're under the condemnation of sin, you feel separated from God and on your own. That's why Jude wrote,
Jude 21 keep yourselves in the love of God...
When sin separates us from the love of God, and we're out from under His covering, left on our own, it's a prime time for the enemy to come in and attack us.
The Lord has heard David's cry of repentance. He has received his prayer.
James 4:7-10 "Submit therefore to God. Resist the devil and he will flee from you. Draw near to God and He will draw near to you. Cleanse your hands, you sinners; and purify your hearts, you double-minded. Be miserable and mourn and weep; let your laughter be turned into mourning, and your joy to gloom. Humble yourselves in the presence of the Lord, and He will exalt you."
Notice the formula for us to follow in these verses.
Ps. 6:8-9 ...The LORD has heard the voice of my weeping. The LORD has heard my supplication, the LORD receives my prayer.
There is sorrow over sin - The Lord has heard the voice of my weeping.
There is repentance from sin - The Lord has heard my supplication.
Then there is forgiveness of sin - The Lord receives my prayer.
2Cor. 7:10 For the sorrow that is according to {the will} {of} God produces a repentance without regret, {leading} to salvation; but the sorrow of the world produces death.
Rom. 2:4 ...The kindness of God leads you to repentance
1John 1:9 ...He is faithful and righteous to forgive us our sins...