From the time Saint Dominic established the devotion to the holy Rosary up to the time when Blessed Alan de la Roche re-established it in 1460, it has always been called the Psalter of Jesus and Mary. This is because it has the same number of Hail Marys as there are psalms in the Book of the Psalms of David. Since simple and uneducated people are not able to say the Psalms of David, the Rosary is held to be just as fruitful for them as David's Psalter is for others.
But the Rosary can be considered to be even more valuable than the latter for three reasons:
1 Firstly, because the Angelic Psalter bears a nobler fruit, that of the Word incarnate, whereas David's Psalter only prophesies his coming;
2 Just as the real thing is more important than its prefiguration and the body surpasses the shadow, so the Psalter of our Lady is greater than David's Psalter, which did no more than prefigure it;
3 Because our Lady's Psalter or the Rosary made up of the Our Father and Hail Mary is the direct work of the Blessed Trinity.
Here is what the learned Carthagena says about it:
The scholarly writer of Aix-la-Chapelle says in his book, The Rose Crown, dedicated to the Emperor Maximilian: "It cannot be maintained that Salutation of Mary is a recent innovation. It spread almost with the Church itself. For at the very beginnings of the Church the more educated members of the faithful celebrated the praises of God in the 150 psalms of David. The ordinary people, who encountered more difficulty in divine service, thus conceived a holy emulation of them. They considered, which is indeed true, that the heavenly praises of the Rosary contained all the divine secrets of the psalms, for, if the psalms sing of the one who is to come, the Rosary proclaims him as having come.
"That is how they began to call their prayer of 150 Salutations 'The Psalter of Mary,' and to precede each decade with an Our Father, as was done by those who recited the psalms."
The Psalter or Rosary of our Lady is divided into three chaplets of five decades each, for the following reasons:
1) to honour the three persons of the Blessed Trinity;
2) to honour the life, death and glory of Jesus Christ;
3) to imitate the Church triumphant, to help the members of the Church militant, and to bring relief to the Church suffering;
4) to imitate the three groups into which the psalms are divided, the first being for the purgative life, the second for the illuminative life, and the third for the unitive life;
5) to give us graces in abundance during life, peace at death, and glory in eternity.
- from The Secret of the Rosary, bySaint Louis-Marie Grignion de Montfort