The Argument of the Exposition of the Psalms by Hippolytus, (Bishop) of Rome.
1. The book of Psalms contains new doctrine after the law which was given by Moses; and thus it is the second book of doctrine after the Scripture of Moses. After the death, then of Moses and Joshua, and after the judges, David arose, one deemed worthy to be called the father of the Saviour, and he was the first to give the Hebrews a new style of psalmody, by which he did away with the ordinances established by Moses with respect to sacrifice, and introduced a new mode of the worship of God by hymns and acclamations; and many other things also beyond the law of Moses he taught through his whole ministry. And this is the sacredness of the book, and its utility. And the account to be given of its inscription is this: (for) as most of the brethren who believe in Christ think that this book is David's, and inscribe it "Psalms of David," we must state what has reached us with respect to it. The Hebrews give the book the title "Sephra Thelim,"  and in the "Acts of the Apostles" it is called the "Book of Psalms" (the words are these, "as it is written in the Book of Psalms"), but the name (of the author) in the inscription of the book is not found there. And the reason of that is, that the words written there are not the words of one man, but those of several together; Esdra, as tradition says, having collected in one volume, after the captivity, the psalms of several, or rather their words, as they are not all psalms. Thus the name of David is prefixed in the case of some, and that of Solomon in others, and that of Asaph in others. There are some also that belong to Idithum (Jeduthun); and besides these there are others that belong to the sons of Core (Korah), and even to Moses. As they are therefore the words of so many thus collected together, they could not be said by any one who understands the matter to be by David alone.
2. As regards those which have no inscription, we must also inquire to whom we ought to ascribe them. For why is it that even the simplest inscription is wanting in them -- such as the one which runs thus, "A psalm of David," or "Of David," without any addition? Now, my idea is, that wherever this inscription occurs alone, what is written is neither a psalm nor a song, but some sort of utterance under guidance of the Holy Spirit, recorded for the behoof of him who is able to understand it. But the opinion of a certain Hebrew on these last matters has reached me, who held that, when there were many without any inscription, but preceded by one with the inscription "Of David," all these should be reckoned also to be by David. And if this be the case, it follows that those without any inscription are by those (writers) who are rightly reckoned, according to the titles, to be the authors of the psalms preceding these. This book of Psalms before us has also been called by the prophet the "Psalter," because, as they say, the psaltery alone among musical instruments gives back the sound from above when the brass is struck, and not from beneath, after the manner of others. In order, therefore, that those who understand it may be zealous to carry out the analogy of such an appellation, and may also look above, from which direction its melody comes -- for this reason he has styled it the Psalter. For it is entirely the voice and utterance of the most Holy Spirit.
3. Let us inquire, further, why there are one hundred and fifty psalms. That the number fifty is sacred, is manifest from the days of the celebrated festival of Pentecost, which indicates release from labours, and (the possession of) joy. For which reason neither fasting nor bending the knee is decreed for those days.  For this is a symbol of the great assembly that is reserved for future times. Of which times there was a shadow in the land of Israel in the year called among the Hebrews "Jobel" (Jubilee), which is the fiftieth year in number, and brings with it liberty for the slave, and release from debt, and the like. And the holy Gospel knows also the remission of the number fifty, and of that number which is cognate with it, and stands by it, viz., five hundred;  for it is not without a purpose that we have given us there the remission of fifty pence and of five hundred. Thus, then, it was also meet that the hymns to God on account of the destruction of enemies, and in thanksgiving for the goodness of God, should contain not simply one set of fifty, but three such, for the name of Father, and Son, and Holy Spirit.
4. The number fifty, moreover, contains seven sevens, or a Sabbath of Sabbaths; and also over and above these full Sabbaths, a new beginning, in the eight, of a really new rest that remains above the Sabbaths. And let any one who is able, observe this (as it is carried out) in the Psalms with more, indeed, than human accuracy, so as to find out the reasons in each case, as we shall set them forth. Thus, for instance, it is not without a purpose that the eighth psalm has the inscription, "On the wine-presses," as it comprehends the perfection of fruits in the eight; for the time for the enjoyment of the fruits of the true vine could not be before the eight. And again, the second psalm inscribed "On the wine-presses," is the eightieth, containing another eighth number, viz., in the tenth multiple. The eighty-third, again, is made up by the union of two holy numbers, viz., the eight in the tenth multiple, and the three in the first multiple. And the fiftieth psalm is a prayer for the remission of sins, and a confession. For as, according to the Gospel, the fiftieth obtained remission, confirming thereby that understanding of the jubilee, so he who offers up such petitions in full confession hopes to gain remission in no other number than the fiftieth. And again, there are also certain others which are called "Songs of degrees," in number fifteen, as was also the number of the steps of the temple, and which show thereby, perhaps, that the "steps" (or "degrees") are comprehended within the number seven and the number eight. And these songs of degrees begin after the one hundred and twentieth psalm, which is called simply "a psalm," as the more accurate copies give it. And this is the number  of the perfection of the life of man. And the hundredth  psalm, which begins thus, "I will sing of mercy and judgment, O Lord," embraces the life of the saint in fellowship with God. And the one hundred and fiftieth ends with these words, "Let every thing that hath breath praise the Lord."
5. But since, as we have already said, to do this in the case of each, and to find out the reasons, is very difficult, and too much for human nature to accomplish, we shall content ourselves with these things by way of an outline. Only let us add this, that the psalms which deal with historical matter are not found in regular historical order. And the only reason for this is to be found in the numbers according to which the psalms are arranged. For instance, the history in the fifty-first is antecedent to the history in the fiftieth. For everybody acknowledges that the matter of Doeg the Idumean calumniating David to Saul is antecedent to the sin with the wife of Urias; yet it is not without good reason that the history which should be second is placed first, since, as we have before said, the place regarding remission has an affinity with the number fifty. He, therefore, who is not worthy of remission, passes the number fifty, as Doeg the Idumean. For the fifty-first is the psalm that treats of him. And, moreover, the third is in the same position, since it was written when David fled from the face of Absalom his son; and thus, as all know who read the books of Kings, it should come properly after the fifty-first and the fiftieth.
And if any one desires to give further attention to these and such like matters, he will find more exact explanations of the history for himself, as well as of the inscriptions and the order of the psalms.
6. It is likely, also, that a similar account is to be given of the fact, that David alone of the prophets prophesied with an instrument, called by the Greeks the "psaltery,"  and by the Hebrews the "nabla," which is the only musical instrument that is quite straight, and has no curve. And the sound does not come from the lower parts, as is the case with the lute and certain other instruments, but from the upper. For in the lute and the lyre the brass when struck gives back the sound from beneath. But this psaltery has the source of its musical numbers above, in order that we, too, may practise seeking things above, and not suffer ourselves to be borne down by the pleasure of melody to the passions of the flesh. And I think that this truth, too, was signified deeply and clearly to us in a prophetic way in the construction of the instrument, viz., that those who have souls well ordered and trained, have the way ready to things above. And again, an instrument having the source of its melodious sound in its upper parts, may be taken as like the body of Christ and His saints -- the only instrument that maintains rectitude; "for He did no sin, neither was guile found in his mouth."  This is indeed an instrument, harmonious, melodious, well-ordered, that took in no human discord, and did nothing out of measure, but maintained in all things, as it were, harmony towards the Father; for, as He says: "He that is of the earth is earthly, and speaketh of the earth: He that cometh from heaven, testifies of what He has seen and heard." 
7. As there are "psalms," and "songs," and "psalms of song," and "songs of psalmody,"  it remains that we discuss the difference between these. We think, then, that the "psalms" are those which are simply played to an instrument, without the accompaniment of the voice, and (which are composed) for the musical melody of the instrument; and that those are called "songs" which are rendered by the voice in concert with the music; and that they are called "psalms of song" when the voice takes the lead, while the appropriate sound is also made to accompany it, rendered harmoniously by the instruments; and "songs of psalmody," when the instrument takes the lead, while the voice has the second place, and accompanies the music of the strings. And thus much as to the letter of what is signified by these terms. But as to the mystical interpretation, it would be a "psalm" when, by smiting the instrument, viz., the body, with good deeds we succeed in good action though not wholly proficient in speculation; and a "song," when, by revolving the mysteries of the truth, apart from the practical, and assenting fully to them, we have the noblest thoughts of God and His oracles, while knowledge enlightens us, and wisdom shines brightly in our souls; and a "song of psalmody," when, while good action takes the lead, according to the word, "If thou desire wisdom, keep the commandments, and the Lord shall give her unto thee,"  we understand wisdom at the same time, and are deemed worthy by God to know the truth of things, till now kept hid from us; and a "psalm of song," when, by revolving with the light of wisdom some of the more abstruse questions pertaining to morals, we first become prudent in action, and then also able to tell what, and when, and how action is to be taken. And perhaps this is the reason why the first inscriptions nowhere contain the word "songs," but only "psalm" or "psalms;" for the saint does not begin with speculation; but when he has become in a simple way a believer, according to orthodoxy, he devotes himself to the actions that are to be done. For this reason, also, are there many "songs" at the end; and wherever there is the word "degrees," there we do not find the word "psalm," whether by itself alone or with any addition, but only "songs." For in the "degrees" (or "ascents"), the saints will be engaged in nothing but in speculation alone. And let the account which we have offered, following the indications given in the interpretation of the Seventy, suffice for this subject in general.
8. But again, as we found in the Seventy, and in Theodotion, and in Symmachus, in some psalms, and these not a few, the word diapsalma inserted,  we endeavoured to make out whether those who placed it there meant to mark a change at those places in rhythm or melody, or any alteration in the mode of instruction, or in thought, or in force of language. It is found, however, neither in Aquila nor in the Hebrew; but there, instead of diapsalma (= an intervening musical symphony), we find the word aei (= ever). And further, let not this fact escape thee, O man of learning, that the Hebrews also divided the Psalter into five books, so that it might be another Pentateuch. For from Ps. i. to xl. they reckoned one book; and from xli. to lxxi. they reckoned a second; and from lxxii. to lxxxviii. they counted a third book; and from lxxxix. to cv. a fourth; and from cvi. to cl. they made up the fifth. For they judged that each psalm closing with the words, "Blessed be the Lord, Amen, amen," formed the conclusion of a book. And in them we have "prayer," viz., supplication offered to God for anything requisite; and the "vow," i.e., engagement; and the "hymn," which is the song of blessing to God for benefits enjoyed; and "praise" or "extolling," which is the laudation of the wonders of God. For laudation is nothing else but just the superlative of praise.
9. However it may be with the "time when and the manner" in which this idea of the Psalms has hit upon by the inspired David, he at least seems to have been the first, and indeed the only one, concerned in it, and that, too, at the earliest period, when he taught his fingers to tune the psaltery. For if any other before him showed the use of the psaltery and lute, it was at any rate in a very different way that such an one did it, only putting together some rude and clumsy contrivance, or simply employing the instrument, without singing either to melody or to words, but only amusing himself with a rude sort of pleasure. But after such he was the first to reduce the affair to rhythm, and order, and art, and also to wed the singing of the song with the melody. And, what is of greater importance, this most inspired of men sang to God, or of God, beginning in this wise even at the period when he was among the shepherds and youths in a simpler and humbler style, and afterwards when he became a man and a king, attempting something loftier and of more public interest. And he is said to have made this advance, especially after he had brought back the ark into the city. At that time he often danced before the ark, and often sang songs of thanksgiving and songs to celebrate its recovery. And then by and by, allocating the whole tribe of the Levites to the duty, he appointed four leaders of the choirs, viz., Asaph, Aman (Heman), Ethan, and Idithum (Jeduthun), inasmuch as there are also in all things visible four primal principles. And he then formed choirs of men, selected from the rest. And he fixed their number at seventy-two, having respect, I think, to the number of the tongues that were confused, or rather divided, at the time of the building of the tower. And what was typified by this, but that hereafter all tongues shall again unite in one common confession, when the Word takes possession of the whole world?
 Simon de Magistris, Acta Martyrum Ostiensium, Append., p. 439.  That is an attempt to express in Greek letters the Hebrew title, viz., sphr thls = Book of Praises  [See vol. iii. pp. 94, 103.]  Genesis 6:3.  i.e., in our version the 101st.  [See learned remarks of Pusey, p. 27 of his Lectures on Daniel.]  Isaiah 53:9. [Vol. i. cap. iv. p. 50.]  John 3:31.  The Greek is: onton psalmon, kai ouson odon, kai psalmon odes, kai odon psalmou.  Ecclus. i. 26.  [Our author throws no great light on this vexed word, but the article Selah in Smith's Dict. of the Bible is truly valuable.]
 That is an attempt to express in Greek letters the Hebrew title, viz., sphr thls = Book of Praises
 [See vol. iii. pp. 94, 103.]
 Genesis 6:3.
 i.e., in our version the 101st.
 [See learned remarks of Pusey, p. 27 of his Lectures on Daniel.]
 Isaiah 53:9. [Vol. i. cap. iv. p. 50.]
 John 3:31.
 The Greek is: onton psalmon, kai ouson odon, kai psalmon odes, kai odon psalmou.
 Ecclus. i. 26.
 [Our author throws no great light on this vexed word, but the article Selah in Smith's Dict. of the Bible is truly valuable.]