According to aristotle, why did the greek's ancestors make music a part of education?

Because the Greeks believed that, by nature, they wanted a good work ethic but also but also wanted to teach the children a civilized pursuit of leisure.

An excerpt from Aristotle’s Views on Education reads:

Roughly four things are generally taught to children, (a) reading and writing, (b) physical training, (c) music, and (d) not always included, drawing. Reading and writing and drawing are included as useful in daily life in a variety of ways, gymnastics as promoting courage. But about music there could be an immediate doubt. Most men nowadays take part in music for the sake of the pleasure it gives; but originally it was included in education on the ground that our own nature itself, as has often been said, wants to be able not merely to work properly but also to be at leisure in the right way. And leisure is the single fundamental principle of the whole business....

If we need both work and leisure, but the latter is preferable to the former and is its end, we must ask ourselves what are the proper activities of leisure. Obviously not play; for that would inevitably be to make play our end in life, which is impossible. Play has its uses, but they belong rather to the sphere of work; for he who toils needs rest, and play is a way of resting....we must therefore admit play, but keeping it to its proper uses and occasions, and prescribing it as a cure; such movement of the soul is a relaxation, and, because we enjoy it, rest. But leisure seems in itself to contain pleasure, happiness and the blessed life. This is a state attained not by those at work but by those at leisure, because he that is working is working for some hitherto unattained end, and happiness is an end, happiness which is universally regarded as concomitant not with pain but with pleasure.

Admittedly men do not agree as to what that pleasure is; each man decided for himself following his own disposition....Thus it becomes clear that, in order to spend leisure in civilized pursuits, we do require a certain amount of learning and education, and that these branches of education and these subjects studied must have their own intrinsic purpose, as distinct from those necessary occupational subjects which are studied for reasons beyond themselves.

Hence, in the past, men laid down music as part of education, not as being necessary,...nor yet as being useful in the way that a knowledge of reading and writing is useful for business or household administration, for study, and for many of the activities of a citizen, nor as a knowledge of drawing seems useful for the better judging of the products of a skilled worker, nor again as gymnastics is useful for health and vigour—neither of which do we see gained as a result of music. There remains one purpose—for civilized pursuits during leisure; and that is clearly the reason why they do introduce it, for they give it a place in what they regard as the civilized pursuits of free men.

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