Hi everyone, this will be a lengthy post. I wouldn't feel right without warning you all first. A large portion of it will consist of theories and facts brought forth by the community; some of it I have been dwelling on for months but have yet to see anyone mention it. Now I feel the need to open up before the Taken King comes out and we lose focus on the Vault for a few months. I wasn't necessarily withholding information, I just really didn't wanna be the guy to say "Illuminati confirmed." The post will be broken up in three sections for accessibility and subject matter, but in theory they go hand-in-hand.
Freemasonry
Freemasons are part of a fraternal organization that sets moral and spiritual values above all else. For a quick overview about Freemasonry, here's a link. Its members consist of mostly rich, mature adults with a belief in at least one Supreme Being. They are religiously diverse and welcome all religions except for atheism, because they believe that any given religion provides moral guidelines on living one's life and bettering oneself. Religion often manifests itself in the different symbols used in the Masonic rituals and ranks. Freemasonry describes itself as "a beautiful system of morality, veiled in allegory and illustrated by symbols." Indeed, almost the entirety of the Freemasonry system is based around the symbolism in numbers, shapes, and tools of the mason trade. For example, there are many numbers that are sacred to a Mason, but none more sacred than the number 3. Thus, the number 3 is used repeatedly in Freemasonry, notably the traditional 3 ranks of Apprentice, Fellowcraft, and Master Mason. The symbolism of 3 is important because of the spiritual significance it holds throughout the history of all religions, which is the foundation on which Freemasonry is structured.
The symbolism that these numbers and items represent is mirrored heavily in Destiny, from the number of classes available to the designs on our gear and in the tower. The Crest of Alpha Lupi has an image of a double-headed eagle, a symbol of great importance in Freemasonry. Even the Tower crest has a heavy Freemason influence; it makes careful use of the square and compass, or shape-wise, the square and triangle. The square and compass are easily the Freemasons' most recognized symbol. This is appropriate for Bungie, as the sum of the sides of a square and triangle, or the sum of the two perfect numbers, 3 and 4, equals 7. The importance of these two shapes can also be reflected in the symbol for the 47th problem of Euclid, or better known as the Pythagorean Theorem. "The essence of the Pythagorean Theorem (also called the 47th Problem of Euclid) is about the importance of establishing an architecturally true (correct) foundation based on use of the square." (Most recent link) Here is a video that provides more examples (and in a Scottish accent!)*
*I really don't know if that's a Scottish accent; it kind of sounds like it, but I did think it was British at first
The size of the Vault of Glass better portrays the use of symbolism found throughout Destiny because it has less elements to decipher through. There are:
3 sync plates to defend from 4 possible spawn points of Praetorians
3 phases of Conflux
3 phases of legions
3 possible sacrifices to a Conflux before a 4th negates you
3 possible cleanse opportunities at Oracles with 4 total opportunities to succeed
7 total Oracles with 2 waves of 3, 2 waves of 5, 2 waves of 7 and 1 wave of 9, with a total of 7 Oracle waves
3 raid chests without a trigger (if we define raid chest as a chest that drops raid loot and does not disappear when opened as long as the fireteam stays in the area)
3 time periods of past, present, future with corresponding 3 gatekeepers
a total of 7 confluxes in the raid
a total of 10 possible weapon drops, as the sum of the two perfect numbers, 3 and 7.
A strong example would be the 2nd triangular number seen on the scope of the Vex Mythoclast. Freemasons used this symbol as a period in writings, as it symbolized becoming god-like, or enlightened. This is also symbolized in the Triple Tau.
I have written about the Greek references the Vault of Glass contains, which had been somewhat elaborated on by u/sanecoin64902 in this post, where he too mentions Freemasonry briefly, but also expands on the fact that Bungie uses different myths, religions, and archetypes to define the world of Destiny. I believe this to be true, but I think that a lot of the Vex background is influenced by Greek mythology, the Cabal influenced by ancient Rome, and the Hive influenced by ancient Egypt. While I would love to provide finer details to this belief, that's not what this post is about. But the religious diversity can be contributed to the same moral guidelines the Freemasons follow.
TL;DR
Pythagoras and Alpha Lupi
As seen in the symbolism behind the 47th problem of Euclid, Pythagoras had a modest influence on the Freemasons, as both were "dedicated to the study of mathematics/geometry and logical reasoning as opposed to religious dogma" (previous link, Influence on Other Groups). It's even said that Freemasonry emerged from the Pythagorean Brotherhood, those that followed the teachings and lifestyle of Pythagoras. A fascinating discovery was made when Pythagoras began applying mathematics to music and music theory. The note that a given string made when plucked was directly proportionate to the length of the string. Pythagoras also had another theory, (Musica Universalis)[https://en.wikipedia.org/wiki/Musica_universalis], or Harmony of the Spheres, or also, Music of the Spheres. You may recognize that last name as the title of a song co-written for Destiny by Paul McCartney. If you click the link and watch the video, he confirms that Destiny's soundtrack is indeed lydian with a flat b7, pointed out by u/wordofcrota. If I'm not mistaken, you can also hear the Oracle song within Music of the Spheres. In relevance to the Alpha Lupi mosaic, as u/A_Cryptarch detailed in this post and this post, the Oracle spawn points in the Vault of Glass very closely resemble an Alpha Lupi mosaic flipped on its axis. To this, I have three theories:
The Oracle song in Music of the Spheres is the correct version, while an incorrect version plays in the Vault. We would have to transpose the Oracle Song in Music of the Spheres and kill the appropriate Oracles in the correct order corresponding to the "right" song.
Using the theory of Musica Universalis, the idea that "mathematical relationships express qualities or "tones" of energy which manifest in numbers, visual angles, shapes and sounds – all connected within a pattern of proportion," (link) to determine the correct kill order of the Oracles, possibly in tandem with the Alpha Lupi mosaic.
Seven raid weapon produces a different pitch when shot, and each Oracle must be killed with the appropriate weapon at the appropriate time, i.e. the original spawn order of the Oracles. (This has almost nothing to do with the information I have brought forth, but I have been theorizing about it for a while) This could be easily debunked if we analyzed the sounds each weapon makes when shot, much like what u/realcoolioman did with the Oracles.
I appreciate your time and I'll see ya in the future