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Considering the fact that Pythagoras formulated the properties of Perfect Fifth and Fourth around 500 BC, why did it take so long to include them in Music?. The first instance of using the Fifth in Polyphony was in Organum around 900 AD. Why did it take so long?
Edit 1: In Howard Goodall's, Story of music Episode 1 - He mentions much of the Greek music was lost and mentions that Organum is the first experiment in Harmony (around 12:30 min mark in the episode).
I have to question the assumption.
Our record of early and ancient music is incredibly incomplete, for two reasons:
First, most people (including presumably most musicians) were illiterate. As recently as about 200 years ago it's estimated that seven out of eight people couldn't write. So throughout history there has been music that was only passed along through oral traditions.
The people who could write between about the fall of Rome and the late medieval period were almost exclusively clergy. So we have a record of liturgical music, but not the secular stuff - for all we know there was barbershop quartet style harmony going on in 650AD.
Second, writing survives only if the medium it's written on is durable enough. I've seen the Dead Sea scrolls in person, and they mostly look like scraps. Things like papyrus and parchment degrade over time.
So the earliest stuff we have a record of was written on stone (like the Seikilos epitaph) or clay (like the Hurrian hymns). And at least some of the competing translations of the Hurrian hymns indicate harmony.
If I restrict your question to "why was liturgical music monophonic for so long", it's because of the purpose of the music: it's meant to be meditative and reflective. The busier it gets, the less it can achieve that aim.
In fact, the rise of polyphony in liturgical music was an issue for the church, and a hot topic at the first Council of Trent. Many of the clergy thought the use of polyphony was obscuring the text of the lyrics. In fact, in 1607 composer Agostino Agazzari wrote that the church came very close to banning polyphony, and that it was saved by the clarity of Palestrina's Missa Papae Marcelli.
The primary focus of Goodall's "Story of Music" is the development of Western classical music — specifically, Western-style harmony. In this regard, he attempts to draw a continuum illustrating that development.
What he is not saying, is that perfect fourths and fifths weren't used in music before 900 CE. He's saying that in the history of written music, we're only aware of their development as harmonic intervals (in the Western sense of "harmony") beginning around 900 CE.
And yet, we've absolutely no idea what the music of these ancient societies actually sounded like. Because they couldn't write their music down, it has disappeared completely. (Episode 1: 5:03 – 5:13)
This doesn't say no music used fourths and fifths. It says we don't know.
... Christian plainchant, which dates from at least the third century AD. (Episode 1: 9:10 – 9:14)
In the earliest form of plainchant, musical monks would sing a meandering tune with no accompaniment, no discernible rhythm, and no harmonizing. (Episode 1: 10:02 – 10:11)
This parallel lines technique, which began in around the ninth century, was called organum. (Episode 1: 12:23 – 12:28)
Taken together, these quotes are saying that, within the known, traceable history of (Western) music, the first use of multiple pitches explicitly organized to function together, came around the ninth century.
In other words, as best as is known, the use of multiple simultaneous sounds operating together in a functional, as opposed to incidental, manner, dates to the ninth century. But this does not mean that other music was strictly monophonic or that it didn't use fourths or fifths; it only means that organum is the earliest that can be documented.
First off, "harmony" to the ancient Greeks had a broader (and somewhat more specific) usage than in the question. Being "in harmony" to the Pythagoreans meant explicitly in a relationship of "harmonious number," but it didn't necessarily refer to notes sounding at the same time, certainly not polyphony.
The Greeks found their "harmony" among notes in relationships among the notes of the scale and in melodic relationships. This definition of "harmony," including melodic intervals, appears for thousands of years in music history, including throughout the renaissance.
So, it didn't take a millennium to use a 3:2 or 4:3 interval in "harmony." The Greeks, in their understanding of harmony, were already doing it.
But I know the question is asking about polyphony too. As other comments and answers have noted, it's likely that much homophonic/polyphonic music wasn't written down, even if it happened. It's also possible (perhaps likely) that besides the potential of polyphony, ancient Greeks sometimes made use of drones in their performances, which would highlight intervallic relationships between the drone and notes of the scale. That would likely involve 3:2 and 4:3 intervals among others.
However, I sense in the question a sort of notion that polyphony should be "obvious" or at least a "natural development" in music. There's a long history of Western musicians infusing the leap to polyphony with a notion of "progress" in music history, often also unfortunately viewing world musics that are monophonic or heterophonic as less interesting or even less "evolved."
The fact is that full-blown Western-style polyphony is not an obvious or common development in music around the world. It wasn't in Western culture, either, where it may have been an aberration until the 1500s or so when the printing press made the possibility of spreading polyphonic written music easier. (Prior to that, Western polyphony only tended to occur among elite circles, nobles who could pay for a small choir of trained singers, etc. It's possible, of course, that polyphonic or at least homophonic works were also being performed by common people, but we have very little written evidence one way or the other.) Polyphony only seems pervasive now due to the spread of global pop -- usually based on Western models -- in the past century.
Most traditional music is primarily monophonic or heterophonic in performance (with or without accompanying drones). That does not mean that such music isn't interested in the construction of scales -- to the contrary, many cultures that don't emphasize polyphony tend to pay a lot more attention to the development of a variety of scales, while emphasizing the nuances of tuning.
So -- even if we make the assumption that just because we don't have written polyphony in the Western tradition that it never occurred before the year 900 or so -- there's no good reason why an appreciation of interval relationships should lead to development of polyphony. Again, Western musical culture is very unusual compared to most traditional musics around the world in that regard.
A perfect fifth may refer to vibrations of the waves. This is notated as 3:2 (also known, in early music theory, as a hemiola), meaning that the upper note makes three vibrations in the same amount of time that the lower note makes two.
This may the scientific ideas around sound that the ancient natural philosopher may have been interested in.
Musicians have added certain terms in regards to interval that reflect the quality of the sound and the tonality of the interval, perfect, major, minor are all intervals that have a sound.
I contend whether polyphony originated in AD900. Polyphony refers to how multiple voices interact with each other, but any time a caveman sang a song while another caveman played a drum they where making polyphony music. (Assuming the drums and the singing was not done in unison.)
Most ancient and medieval music was very monophonic, it was only during the renaissance that musicians started playing together in enough numbers that these issues started getting larger inquiries.
