Behold, you delight in truth in the inward being, and you teach me wisdom in the secret heart.   Purge me with hyssop, and I shall be clean; wash me, and I shall be whiter than snow.  Ps 51:6-7 (ESV)

hyssop

Hyssop Plant in Israel

As David is coming to the point as Isaiah did in Isaiah 6, he realizes that purging comes from God’s altar.  He begins by acknowledging that God desires truth beginning deep down.  In the context of David’s heinous sins,  He has revealed the truth in his confession.  Some never come to this point.  They blame others or their circumstances or their upbringing.  I don’t think, based on what is being said here, that purging can begin until truth finds residence deep down in one’s heart.  Only then can one expect to be purged (purified).  Hyssop was used in the OT in connection with purification.  “Purge me with hyssop” = “Purify me.”