I'll do my very best to give you the short answer because the long answer tends to put people to sleep.
I should also say that the argument for exclusive psalmody is one of three expressions of the Regulative Principle of Worship (hereafter RPW) found in the Westminster Confession (21:1). There is the exclusive position; the combined psalm-hymn position, and the hymn only position. I have many friends who believe in the RPW who do not worship via exclusive psalmody (hereafter EP). In the Churches I have preached in that are not EP, each of them have selected only Psalms that Lord's Day for my conscience's sake. This is the charity we should expect in the family of God (No, I do not return the favour when they come to our pulpit. But I do let them preach). ;-)
I take it, by your post, that you are a believer in the RPW? Forgive me if I am mistaken but I will assume that you agree with this principle and forgo establishing the biblical foundation of that doctrine and move on to the fulcrum of your question- the biblical mandate to sing psalms only.
The most obvious reason for singing the Psalms is the Book of Psalms. It has rightly been called the Hymnbook of the Church, and since its delivery (piece by piece) to the Church, she has sang it. However this alone will not satisfy the serious inquirer who wants a more explicit command than simple inference.
There are direct commands to sing the Psalms in the Word,'Oh come, let us sing to the Lord!... Let us come before His presence with thanksgiving, Let us shout joyfully to Him with psalms.(Ps. 95:1-2; cf. 81:2; 98:5; 100:2; 105:2). Add to this the prefatory dedication to the individual psalms to be played or sung (To the chief musician, etc.) and you have the building of the foundation (out of the RPW for ES).
The Old testament is full of examples of the use of Psalms in corperate (cf. 1 Chr. 16; 2 Chr. 5:13; 20:21; 29:30; Ezra 3:11;Ex. 15:1; 2 Sam. 1:18; 2 Chr. 23:13; Ps. 30:4; 137:1 ff). I would imagine that you would have no problem with the use of the Psalms in the Old Testament as the outworking of the RPW, so I will jump to the New Testament.
Keeping in mind the RPW, that the only way of acceptable worship of God is 'instituted by himself, and so limited by his own revealed will, that he may not be worshiped according to the imaginations and devices of men, or any other way not prescribed in the Holy Scripture' (WCF 21:1), let me show you our argument for EP in the New Testament.
To me, as I sift through the vast mountain of information on EP (some good, some bad), I have found that the Lord's own example a powerful illustration of EP. Aside from the normal and established synagogical practice of ES, we find the only place in the NT where Christ is said to be worshiping by song, singing a Psalm.
Matthew 26:30 'And when they had sung an hymn, they went out into the mount of Olives.'
Here the word 'hymn' as Easton's Bible Dictionary says was, 'the latter part of the Hallel, comprehending (Ps 113-118) It was thus a name given to a number of psalms taken together and forming a devotional exercise.' This seems to be a compelling argument for EP in my opinion. The word hymn is used in other places in the New Testament and we should take a moment to look at its use.
Ephesians 5:19 and Colossians 3:16
'Speaking to one another in psalms and hymns and spiritual songs, singing and making melody in your heart to the Lord'(Eph. 5:18-19). 'Let the word of Christ dwell in you richly in all wisdom, teaching and admonishing one another in psalms and hymns and spiritual songs, singing with grace in your hearts to the Lord'(Col. 3:16).
Before we look at the complete text let's remember that Scripture interprets Scripture. We cannot import a definition onto a word that the bible has already established. The difference between the word 'hymn' in Matthew 26 and the word 'hymns' here in these two texts is the plural use 'humnos'in Eph, and Col. It is the same word as Matthew 26:30. This means if we let the Bible interpret itself, that in at least two of the 3 kinds of songs sung in Ephesians 5:19 and Colossians 3:16 are Psalms. However one would like to import a modern definition of hymn (Say, 'The Old Rugged Cross') is importing a non-biblical definition into the text. How do we in 2004 know what Paul meant by hymn in the 1st century? We don't. We do however know that the Word says hymn meant. It was a Psalm.
Now, regarding psalms, hymns, and spiritual songs.
Paul was writing to Greek thinking Christians and so it makes sense that in his usage of the Scriptures he would quote from the Septuagint (The Greek translation of the Hebrew OT). In fact, when we read quotes from Paul from the OT we often find that the quote does not exactly match the TO text in our Bibles. This is because he was using the Septuagint text. This was the usual text that even out Lord used.
Now, I'll try and make this short.
As we look at the Septuagint, we find that the expressions psalm (psalmos), hymn (humnos), and song (odee) Paul used clearly refers to the Old Testament book of Psalms. 'Psalmos'occurs around 87 times in the Septuagint. 'Humnos'occurs 17 times in the Septuagint. In 2 Samuel, 1 & 2 Chronicles and Nehemiah there over 15 examples in which the Psalms are called 'hymns'(humnoi) or 'songs'(odai) and the singing of them is called 'hymning'(humneo, humnodeo, humnesis) is used 80 times in the Septuagint, 45 of which are in the Psalms, 36 in the Psalm titles. The point being, all there terms used in Ephesians and Colossians are biblical terms for psalms in the book of psalms itself from the LXX. I think we should understand how Paul's audience would understand his words (without explanation or qualification) and use that definition in our own usage.
Believe it or not this is the short answer to your question of the Biblical reasons for EP. I realize that much of what I have written is inferential and consequential, but as the WCF says regarding the truth of God's Word it, 'is either expressly set down in Scripture, or by good and necessary consequence may be deduced from Scripture'(WCF 1:6). I believe EP is both, but I respect the right to disagree. But this is where I stand.
Regarding the historical use of the Psalms, both the early Church and the Reformers (not Lutherans) used the Psalms exclusively up until the 1700's with Keach & Watts introducing paraphrased hymns.
