Material

Material (被召物 （マテリアル）Himeshimono (Materiaru)) is the term used by humans to refer to the beings residing on the other side.

Principles
Materials are called into the human world by summoners by having them possess their vessel through the Blood Sign ceremony. Summoners achieve this by using their Blood-Signs to carve out their names. Materials are divided into three classes: Regulation, Divine, and Unexplored. The upper two classes cannot be summoned through normal means. A Divine-class Material can be summoned in a battle after summoning 100 Regulation-classes, and an Unexplored-class can be summoned after summoning 50 Divine classes or by perfectly hitting the specific Petals spelling their name into the Spots in order. Materials can defy the laws of physics and achieve results not even a large army could, and they cannot be harmed by conventional methods such as thermobaric warheads or laser bombardments. Normal weapons are useless once a summoning battle starts and summoners call in a Material, as they cannot harm a Material or a summoner defended by their protective circle. Thus, the only counter is for another summoner to step up and summon their own Material, as the fact that no resident of the human world can defeat a resident of the other world, even if they are the weakest Material, is an absolute rule.

The cost of a Material is determined by the number of Petals needed to summon it, and its sound range is determined by the number of small, middle, and high sounds (The lowest sounds do not influence the sound range). The sound ranges have a circular relationship much like rock-paper-scissors. Low beats high, high beats middle, middle beats low. Unless there is a large difference in the cost levels of the two Materials, that circular relationship cannot be overcome with brute force. The minimum cost gap needed to overcome a disadvantage in sound range is of 10.

However, there a few exceptions to these rules for Divine-class Materials who have special relationships in their original myths which overturn their differences in cost and/or sound range. The Materials, normally masses of pure fighting insincts, can be lead to panic or fear and try avoid the fight. For example, Fenrir and Odin or Beelzebub and Lucifer, Caution is needed when crossing mythological boundaries, such as between Norse and Greek.

Regulation-class
Regulation-class Materials are the man-made frameworks put together as stepping stones to reach the Divine-class. They are the Materials every summoner has to start a battle with. Regulation-classes were created using secret life-creating techniques such as the alchemical homunculi or Kabbalic golems.

Other unnamed Regulation-class materials have been summoned in the story: "a stuffed animal holding a bloody axe", "a giant stag beetle that's legs and pincers have been torn off by a mischievous child and who has wheels and razor blades attached in their place", "a massive gear that can roll around on its own", "a giant wolf with a metal jaw", "a giant serpent wrapped in flames", "a giant fish that tears through the sky", "a queen bee with a human face", "a giant umbrella with fangs covering the outer edge", "a goddess statue filled with murderous gimmicks", "a great serpent made of several chains", "an automaton created by sewing together human skin", "an old tome covered in countless eyeballs", "a gear with a giant smile plastered on the side", "a dinosaur not found in any encyclopedia", "a winged alligator", "a giant face made of stone", "a jellyfish giving off a putrid stench", "a giant suit of armor", "a burning metal ball" "a giant swollen eyeball", "countless wolves connected by the single shadow they share", "a giant rotting dog", "a man-eating plant with countless blades instead of petals that launches steel seeds with the force of bullets" and "a beautiful Western doll with round and fat maggots wriggling in the cracks covering its body".

Divine-class
Divine-class Materials are the gods and monsters of legend. They cannot be summoned through normal means. Unlike the Regulation-class Materials, who were created by humans, Divine-class Materials are individual beings who appeared in the world on their own.

Other Divine-classes described in passing are a sea dragon (Famous in the Pacific, with an 8-letter name that begins with T, related to the creation myth of the vast ocean, the legends claim it had created the land and the islands), and the archangels of a certain monotheistic religion.

Unexplored-class
Unexplored-class Materials are the beings that lurk beyond the gods that were discovered by mankind when the Blood Sign ceremony was developed. Like Divine-class Materials, they cannot be summoned through normal means. Though summoners can freely call even the gods of legend and use them for their own purposes with 100% certainty, some worship the Unexplored-class that lies beyond the Divine-class. The three Unexplored-class Materials at the peak of the low, middle and high sounds are known as the Three (大三角 Ōsankaku, lit. "Large Triangle"), who handle the Award system and are said to take extremely good care of humanity and freely lend them power.

The most powerful Unexplored-class is the White Queen, who doesn't belong to any of the sound ranges. She's also the most popular and desired Material, her popularity exceeds even the Three in their respective factions and reaches the level of worship and prayer, even by summoners who can freely control the gods. For example the Meinokawa shrine unofficially worships the White Queen as their second god, and the sculptures and plates of the White Queen are one of Lu Niang-Lan's most popular products.

All Unexplored-class Materials have female forms. The reason for this is unknown to humans. The Wicked Green Woman eventually reveals to Kyousuke that Unexplored-class Materials are the antropomorphic personification of the laws and elements of the world, being based on colors that represent fire, water, wind, earth, time, life and death, good and bad deeds, etc. The world takes shape because each of the Unexplored-classes supports one of these elements. Their role is to manage heaven and keep the gears of the other world running. However, the Wicked Green Woman also revealed to him that the Unexplored-class had another purpose, which also explains their female forms and assigned colors: they were initially born to oppose the White Queen, which was considered too dangerous of an existence, hoping her power would be reduced to nothing if the laws supporting her power and existence were removed, one color at a time, allowing the White Queen to be sealed. However, despite all Unexplored-classes working together against her, they all failed their mission during the Secret War and submitted to the White Queen's overwhelming power, losing all hope to oppose her.

Special Material
The Black Maw that Swallows All (nu – lp – eu – bf – zuh – ei – jkv – iu – a – xw) is the worst of all monsters. When a summoner breaks a Taboo, they will lose control of their Material, which will be forcibly transformed into the Black Maw, a vortex of pitch-black slime that opens a maw filled with countless fangs, and will devour the summoner. As pointed out by Kyousuke Shiroyama, the Black Maw has the same summoning cost as the White Queen and you can get the Queen's name by rearranging the Black Maw's name. This is because the White Queen and the Black Maw are one and the same; the White Queen is both the most desired material and the most hated.