Adventuring is dangerous and the dangers come in many forms. If that danger is manifest as an angry ogre with bloody battle ax then the three part system of Attack Bonus, Armor Class, and Hit Points covers it. Yet, angry ogres only cover one of the endless opportunities for a Player Character to get harmed.(How exciting!)
The fire of an enraged dragon, the fire of an irritated wizard, the poison of a serpent's bite, the floor (when met from the sudden fall of a hidden pit trap.... and covered in spikes) are all other opportunities for a hero to part with life and limb. Most of these will do amazing Hit Point damage, and really there is nothing your armor can do to help. In cases of the serpent's bite the wound maybe immediate and worse: Paralysis if the DM is friendly, but possibly immediate DEATH!
When D&D was created the writers realized that another system was needful to handle sudden dangers of magic, environment, and monster. Their solution was saving throws.
In the classic B/X game saving throws were broken down into 6 categories.
Poison and Death Ray: This is a roll used when sudden death comes into play. Two examples are given by the name of the save, but any other circumstance in which a character would go from alive to dead in one instant (such as the sniping attack of an assassin, or the shock of electrocution). This a last stop save and is usually not used when there is still any other means of resistance (like getting out of the way).
Magic Wands: Most wands in the original version of D&D were essentially a type of magic gun. So the Wands saving throw is about dodging out of the way of a sudden onrushing force liken unto an arrow or a bullet. Unlike an arrow (and more like a bullet) these are forces that will pass through armor like tissue paper. This saving throw can also be applied to other similar effects such as the eye beams of a certain floating many eyed orb or a the heat ray from the headpiece of a pharaonic golem. It could also be used to resist another "canned" magic effect where the expertise of the caster is less of an issue. (This is a common use in the BECMI line of games).
Dragon Breath: This save has a name that evokes the greatest danger of D&D, but there are other less iconic, if no less deadly times it can be used. When a Dragon expels its breath of fire it encompasses the intire area. It isn't really a matter of whether or not a character will take damage, but instead a matter of how much. There are other circumstances that could demand such rolls. A tidal wave slamming a characters boat; an avalanche overwhelming a region; the fall of many yards into a pit of spikes; being in the range when burning pitch is rained down from the sky. All of the former examples and many more (magic users beware of falling farm houses) will cause massive damage. The question posed at this point is how much. The Dragon Breath save answers that.
Paralysis and Turn to Stone: This is all about a character losing control of their body. The eyes of the medusa or the touch of a ghoul can strip control of the body and render it inert. Other effects like being turned into another object (polymorphing) or being shunted to another plane could also fall into this saves bailiwick.
Spells: This is almost self explanatory. It certainly does cover the effects of Magic Users and Clerics casting spells. It also covers odd effects that have an occult origin if they aren't covered by other saves. In some cases a spell will cause effects that would match other saves (how different is a fire ball from Dragon Breath, or an Acid Arrow from a Magic wand), but the skill of the Magic User wins out and in most cases it would define the save.
The mechanics of a savings throw are simple. Each character class has a set number based on their current level. A d20 is rolled and if the number is the same or higher than the target set by the particular savings throw it is successful. In most cases this negates the effect unless the effect causes Hit Point damage, in which case the damage is halved.
If you look back ast Musings 2 you will see that HP is a measure of combat worthiness. So a character who takes damage during a save would explain why they are not a sooty stain in the same manner that a character would explain why the blow of the angry ogre didn't leave them crippled or dead. These narratives, be they ogre club or dragon breath, make up a large degree of the dynamic narration of Dungeons and Dragons (or related OSR games).
Examples:
Let's take five characters, one of each archetypal class, and a hireling in a tight spot, and have each save against the first five saves.
Bruno the Truculent, famed and indomitable Fighter, has been bitten by a poisonous Naga. The fanged human headed snake injects Bruno with its toxins of legendary potency. Bruno reals back and feels his heart stop beating, but rather than die the hero gathers his pluck and lets out a horrid battle cry. The shock of the poison meets the will of a man who has survived a thousand deaths and his heart resumes. Probably not a good day to be a naga.
Piccolo Filtchburry, who always wanted to be a bard, is in fact a Thief. On the lamb for a misunderstanding involving a counting house he has signed on with a band of adventurers to pilfer the crypt of an old man (who happened to be a wizard of dubious renown so it is totally cool). At the end of a long slog through a horrid dungeon he finds himself in front of a trapped door. After examining a trap for 10 minutes he sets to disarm it. It is an arcane affair but he is sure he has the right of it. Wrong. The yellow gem in the door he sought to disarm shoots a blinding ray, like that of a wand, at the Thief. Piccolo was expecting something of the sort and his highly trained reflexes, and healthy paranoia, have him diving to the side before it even glimmers. He hopes that the nosy magic user, who was peeking over his shoulder, was just as paranoid.
Mishan the Pious decided that ridding the world of an evil dragon and filling the coffers of his church would both be acts his Deity and his High Priest would both appreciate. His party tracked the scaled wyrm to its lair, for everyone know fighting a dragon with access to the open sky is pure foolishness. So, when sneaking down the steep incline towards it's pit he felt some confidence, until a yellow orange blast of fire came up the incline to meet his party. Before the fire engulfed him he lets out a cry to his god, who was apparently listening. for the bulky fighter in the lead and the thief before him took the brunt of dragons burning ire. Though singed Mishan alone lives of the trio. Sadly for Mishan it is only Round 1.
Razzleman the Medium is a lowly magic user, not long from his days as apprentice, and he is sure he is over his head. In a crypt crawl, the sort that tavern stories make out as an adventurers bread and butter, the zombie hordes guarding the final chamber turned out to be ghoul hordes. It seems that the local informant didn't know the difference in one undead horror and another. The young adventures are all beating a retreat when a stumble puts Razzle behind. The lead ghoul grips the young mage and his muscles begin to freeze with the chill of the grave. But Razzleman, lowly as he is, still has the makings of a wizard, and before he is paralyzed he speaks a word of unbinding. His joints loosen and he uses that moment to hurl the holy water he clutched into the wreaking face of the ghoul, which is now more inclined to clutch its face and howl. Razzle takes to his heels again.
Robert the Torchbearer hates to be called Bob, but this is the exact treatment he has received since starting out with these armored jerks. He didn't even want to do this job, but he lost a bet and took a dare. He dislikes adventurers, but a bet is sacred and if you abandon Lady Luck then she will abandon you. Lost in these thoughts, Robert "Bob" the Torchbearer didn't even see the goatee sporting man in the blood red robes. When the party of jerks that had treated him like a mule with thumbs, began dropping one after another to the ground in stupefied dazzlement, Bob just heard an almost maniacal laughter on the inside of his head. Bob knows his goddess has spared him, and maybe that scares him more than the cultist in front of him. Bob gives the thief at his feet a solid kick in the fork and makes a runner with a torch in hand and the parties ill gotten loot still in his backpack. The cultist laughs with a maniacal and almost high pitched womanly voice.
Saving Throws, like hit descriptions, aren't just "one thing". They are part luck, part reaction, part divine intervention, and partly the unknown. While they are a mechanical abstraction, after the dice are rolled they become another narrative tool. Some GMs may want to declare how the PC is saved, others may leave that description up to the Player, but either way it is another tool to make a simple die roll into a dynamic encounter.
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