An historical note that I didn't think about before: the Ancient Greek modes are NOT exactly equivalent to the Medieval and Renaissance modes, but they are similar and related. Confusingly, regarding the Phrygian and Dorian specifically, the names were swapped around by the Medieval musicians, so that what the Greeks called Phrygian became Dorian to the Medievals, and what the Greeks called Dorian became Phrygian to the Medievals.

If you want to get some sense of the sort of music Plato would have approved of, listen to Medieval and Renaissance music written in Dorian and Phrygian modes. It goes without saying that the texts of Medieval and Renaissance music were overwhelmingly Christian in nature. It also goes without saying that music theory had progressed since Plato's time (particularly in terms of Renaissance polyphony). Thus, while the style and text of some of the music would not have been especially like the music Plato imagined, it is likely that the "spirit" of it would have been.

Of course, you could also listen to actual ancient music, but there simply isn't that much of it and what we do have is all based on reconstruction. Mesomedes' Hymn to the Sun is written in Ancient Greek Dorian (the warrior's mode, according to Plato).

Thomas Tallis' masterpiece Spem in Alium is written in what the ancient Greeks would have called Phrygian (the prayerful mode, according to Plato):
https://www.youtube.com/watch?v=7Cn7ZW8ts3Y

It might be worth adding that I do not claim expertise on any of this and could be completely wrong. Use my research as a helpful guide only.

A characteristic chord progression of (what I think would have been) Plato’s Phrygian mode would be D minor, A minor, G major.

Here is a modern translation and commentary on the Handbook of Epictetus that would be very useful for daily practice.

We know that the Handbook is one of the texts used by the ancients to prepare themselves morally.

Our purpose is to become God-like.

I cannot stress enough the importance of daily practice.

As Plotinus states (Enneads, I.2.6), we are not to focus on purging ourselves of vice, but rather our focus is to be on becoming like God. This distinction is subtle but important.

The responsibility is yours. You will not be saved. Put in the work daily.

Proper music is a divine gift that mirrors the harmony of the cosmos and the divine realm and that aids our soul in its ascent. Music is a tool for the philosopher that, like everything else in our modern societies, has largely degenerated into a hedonistic pursuit. As Plato writes in Timaeus 47 b-d:

"God invented and gave us vision in order that we might observe the circuits of intelligence in the heaven and profit by them for the revolutions of our own thought, which are akin to them, though ours be troubled and they are unperturbed; and that, by learning to know them and acquiring the power to compute them rightly according to nature, we might
imitate the perfectly unerring revolutions of the god and reduce to settled order the wandering motions in ourselves.

Of sound and hearing once more the same account may be given: they are a gift from heaven for the same intent and purpose. […] That part of music that is serviceable with respect to the hearing of sound is given for the sake of harmony; and harmony, whose motions are akin to the revolutions of the soul within us, has been given by the Muses to him whose commerce with them is guided by intelligence, not for the sake of irrational pleasure (which is now thought to be its utility), but as an ally against the inward discord that has come into the revolution of the soul, to bring it into order and consonance with itself."

We can understand the will of nature from those things in which we do not differ from one another. For example, when our neighbor's slave has broken a cup, we are immediately ready to say, 'Well, such things happen.' Understand, then, that when your own cup gets broken you should react in just the same way as when someone else's cup gets broken. Apply the same principle to matters of greater importance. Has someone else's child or wife died? There is no one who would not say, 'Such is the way of things.' But when someone's own child dies they immediately cry, 'Woe is me! How wretched I am!' But we should remember how we feel when we hear of the same thing happening to other people.

Epictetus, The Handbook, 26

"Nevertheless I long - I pine, all my days -
to travel home and see the dawn of my return.
And if a god will wreck me yet again on the wine-dark sea,
I can bear that too, with a spirit tempered to endure.
Much have I suffered, labored long and hard by now
In the waves and wars. Add this to the total -
Bring the trial on!"

Odyssey, 5.221-27

Concerning all the calamities that men suffer by divine fortune,
support your lot with patience - it is what it may be - and never complain about it.
Do what you can to remedy it, and say to yourself as follows:
Fate does not send the greatest portion of these misfortunes to good men.

The Golden Verses

Keep this one truth in mind, that a good man cannot be harmed either in life or in death, and that his affairs are not neglected by the gods.

Plato, Apology 41c-d

Remember, daily (or nightly) self-examination is an ancient practice that was used by the Pythagoreans, the Stoics, and the Neoplatonists. And we are told by the famous Delphic maxim, and by Socrates, to know ourselves.

"What have I done wrong? What have I accomplished? What duty have I neglected?"

Ask yourself these questions every day and answer them honestly. When you have done well, praise yourself. When you have not done well, reprimand yourself.

This is one of your tools for walking the path of the philosopher. Just being a metaphysics or history nerd isn't enough.

The divine music of the Medieval mystic Hildegard von Bingen is probably a close image of what Pythagorean-Platonic music would have been like. This piece is even written in the "Timaeus scale" described in Plato's dialogue of that name. It will cleanse your soul, like true music should.

For the ancient philosophers, music was a tool to perfect the soul.

"Pythagoras conceived that the first attention that should be given to men should be addressed to the
senses, as when one perceives beautiful figures and forms, or hears beautiful rhythms and melodies.
Consequently he laid down that the first erudition was that which subsists through music’s melodies
and rhythms, and from these he obtained remedies of human manners and passions, and restored the
pristine harmony of the faculties of the soul. Moreover, he devised medicines calculated to repress
and cure the diseases of both bodies and souls. There is also, by heavens something which deserves to
be mentioned above all: namely, that for his disciples he arranged and adjusted what might be called
apparatus and massage, divinely contriving mingling of certain diatonic, chromatic and enharmonic
melodies through which he easily switched and circulated the passions of the soul in a contrary
direction, whenever they had accumulated recently, irrationally or clandestinely such as sorrow, rage,
pity, over-emulation, fear, manifold desires, angers, appetites, pride, collapses, or spasms. Each of
those he corrected by the use of virtue, attempering them through appropriate melodies, as if through
some salutary medicine." From Iamblichus.

King Zeus, whether we pray or not, give us what is good for us; what is bad for us, give us not, however hard we pray for it.

Socrates, Second Alcibiades, 143

The tetractys is a Pythagorean sacred symbol. It has many applications and can be understood in metaphysical terms. But it can also be interpreted musically.

You have 1 at the top, then 2, then 3, then 4. We can interpret these numbers as the following ratios: 2/1, 3/2, 4/3. These correspond to harmonic ratios that can be expressed respectively as the following musical intervals: the octave, the fifth, and the fourth.

Therefore, you can play the tetractys on a keyboard or a guitar or the like. The notes of the tetractys are E, E (octave lower), A, E (another octave lower).

You can then construct a "scale" from this. I will not go into the details of that here. But the important point is that you can play the tetractys, and the resulting "scale" is essentially what is now called the Phrygian mode. Remember, the ancients taught that music is (literally) medicine for the soul.

Below I will post a short improvised example. The "bells" are playing the tetractys.

How angry you would be if someone handed over your body to just any person who happened to meet you! Are you not ashamed, then, when you hand over your mind to just any person you happen to meet, such that when they abuse you, you are upset and troubled?

Epictetus, The Handbook, 28

“Vessels containing water, perirranteria, are set up at the entrances to the sanctuaries, like the fonts of holy water in Roman Catholic churches; everyone who enters dips his hand in the vessel and sprinkles himself with water. ... The purifying power of fire is joined to the power of water when a log is taken from the altar fire, dipped in water, and used to sprinkle the sanctuary, altar, and participants.” Walter Burkert, Greek Religion

In every undertaking, consider what comes first and what comes after, then proceed to the action itself. Otherwise you will begin with a rush of enthusiasm having failed to think through the consequences, only to find that later, when difficulties appear, you will give up in disgrace. ... Different people are naturally suited to different tasks. Do you think that if you do these things you can still eat in the same way, drink in the same way, give way to anger and irritation, just as you do now? You must go without sleep, endure hardship, live away from home, be looked down on by a slave-boy, be laughed at by those whom you meet, and in everything get the worst of it: in honors, in status, in the law courts, and in every little affair. Consider carefully whether you are willing to pay such a price for peace of mind, freedom, and serenity, for if you are not, do not approach philosophy, and do not behave like children, being first a philosopher, next a tax-collector, then an orator, and later a procurator of the Emperor. These things are not compatible. You must be one person, either good or bad. You must cultivate either your ruling principle or external things, seek to improve things inside or things outside. That is, you must play the role either of a philosopher or an uneducated person.

Epictetus, The Handbook, 29

Zeus I will hymn, the greatest and the noblest,
Wide-seer, king, fulfiller, who converses
Closely with Themis as she leans toward him.
Matchless, world-watching Cronian, be gracious.

Homeric Hymn 23

Hence justice in the soul is to energize in a greater degree intellectually. But temperance is an inner conversion to intellect. And fortitude is apathy, according to a similitude of that to which the soul looks, and which is naturally impassive.

Plotinus, Enneads, 1.2.6