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Why were ancient philosophers and mathematicians so intrigued by music? Why did philosophers like Plato, Arthur Schopenhaur, Neitsche and etc perceive it differently than other forms of art?

P. S it would be great if you could also give me some sources to read up further on this topic!

Why restrict yourself to philosophers and mathematicians?

Lots of people are interested in music, many more than are actually interested in either of the disciplines mentioned above. And most are moved more by music than by poetry and the literary arts; and likewise, the visual or dramatic arts. Music has been of perennial interest in mankind.

Philosophers and mathematicians being part of mankind are then as likely to be interested in music but unlike most - are able to philosophise about it (or mathematise about it).

It's because the ratios between the tone and the length of strings were in Greece and Inda, at a very early date, worked out. It demonstrates the connection between the mathematical things and the world of appearances. "Music is mathematics in time," rather than in space as with geometry: but the strings and their lengths are in space. Of course, on a side track: "Architecture is frozen music." is an observation that brings this into the visual metaphysics of Goethe.

There is something about this which is likely historical or empirically accurate, in Plato's Laws, concerning the method of teaching or civilizing (with music) the youth in the time prior to Plato, and in his own time. Aristotle says: Plato is a Pythagorean. Any good book on pre-Socratic philosophy or the origins of Western thought should have something on this.

I think this is a rather deep topic, worthy of more than just an answer. However, I think the musings of Alan Watts provide an excellent answer: music is intrinsically close in nature to the act of life. He also suggests another art which is in a similar class: dance.

The existence, the physical universe is basically playful. There is no necessity for it whatsoever. It isn’t going anywhere. That is to say, it doesn’t have some destination that it ought to arrive at.

But that it is best understood by the analogy with music. Because music, as an art form is essentially playful. We say, “You play the piano” You don’t work the piano.

Why? Music differs from say, travel. When you travel you are trying to get somewhere. In music, though, one doesn’t make the end of the composition. The point of the composition. If that were so, the best conductors would be those who played fastest. And there would be composers who only wrote finales. People would go to a concert just to hear one crackling chord… Because that’s the end!

Same way with dancing. You don’t aim at a particular spot in the room because that’s where you will arrive. The whole point of the dancing is the dance.

Here's a video with the reference quote, and a transcript of the video. The video is actually a splicing together of several of his lectures.

The link may be astronomy.

The planets in our solar system are said to be organized in a similar manner to a musical chord. There are also claims that rotating planets produce sounds. In fact, when I was in the Navy, we were told that a certain ultra-low frequency sound we monitored was thought to be Earth's "heartbeat."

I couldn't find a single article that really pulled it all together, and much of the literature is found on websites that delve into astrology, mysticism, etc. But if you search for information on the relationship between planets and musical notes or octaves, you can dig up a lot of information. (This might be of interest: Musica universalis)

Whether these alleged musical-astronomical connections were known to people like Plato or Neitsche, I don't know. (Then again, see the above link.)

Music is also notable for its symbolism and for its extraordinary influence on human emotions and behavior. However, this probably isn't of much interest to mathematicians.

One of music's fundamentals is the way in music, like an octopus, people convincingly imitate other things... then quickly doff the masquerade and swim off.

In the example below, Duane Shinn uses a piano to imitate bells and chimes (i.e. physical bells and chimes, like in a bell tower).

https://www.youtube.com/watch?v=0iWU1St32vo

This is more than just a trick: everything that everybody hears in nature is a composite waveform of some combination of frequencies; and the notes of a major chord represent the five strongest frequencies for simple objects and quiet places.

With more time, I would write an entire theory explaining which thing in nature is being imitated by the different musical expressions. Thinking of chords, this example comes to mind:

Why do minor chords, and to a greater degree diminished chords, produce tension?

In signal interpretation, the most important question is whether the recipient is receiving enough signal to understand the message over the noise. This metric is called the signal-to-noise ratio, and when it is low, the processor must work harder to understand the message because it comes with errors.

If you are speaking with a person in a very loud place, you will not hear the lower frequencies of their voice but the higher frequencies. Since the major chord is only found clearly in the first five overtones, the composite waveform of the person's voice that you hear will be less like a major chord and more like a minor chord or diminished chord... or like the sound of banging adjacent keys on a piano: dissonant.

As a listener, your mind must work harder to understand language when there is more background noise because it knows that there may be more errors to correct once the message is received. That the voice is characterized less by major tones and more by dissonant tones (and here is my theoretical assertion:) indicates to your mind that there will be more work required to get the message, resulting in a (somewhat irrational) association between extra signal-processing work in your mind and dissonant tones.

Musicians who use dissonance, diminished chords, and even minor chords (I assert) make use of this irrational association in people's minds to produce the tension that makes music interesting and gives it power to tell stories-in-abstraction. This technique has been available to musicians since the discovery of music in antiquity.

Here's a silly example of this analogy playing out in a chord progression.

Oh no, a vii_dim7! What's going to happen???

iii? What does that mean?

vi? That's strange.

II- that's nice, but how did that get there?

V7-- Oh, I know what that is....

Wait for it...

I. Relief.

If you liked my little theory, read more about it here.