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vitruvian 10-23-2007 01:42 PM

Secrets of the Dungeon Masters
 
I recently ran across some notes made a few years ago about justifying a Dungeons & Dragons-like setting with lots of subterranean catacombs and dungeons with ecologically unsound populations of monsters either trapped in rooms or wandering about, and vast quantities of treasure and magic items for the taking. Basically, it's a post-apocalyptic setting, where a highly advanced magical culture collapsed and left ruins behind. 'Adventuring' is frequently merely a polite word for 'scavenging', and a major portion of the world's economy may be based on such scavenging operations and their proceeds, especially in terms of magic items. The political and economic structure tends to be relatively feudal, since there are intermittent disruptions to the economy as more treasure troves are uncovered and massive inflation results from the influx of gold, silver, and magic goodies.

The World of the Dungeon Masters:

Once upon a time, the world was High Mana throughout, and magic was as common as dirt. In fact, raw magical energy was so freely available that even quite large enchantments were readily available (more on mechanics later).

Magic was fully industrialized; paired with actual material technology at no more than TL2 (yes, smelting iron is a post-apocalyptic invention), this resulted in a highly advanced urban civilization.

The ancients clearly made extensive use of Earth spells to build downwards, creating massive underground complexes or dungeons using Earth to Air, Shape Earth, and Earth to Stone. Essential Earth was used to make highly fertile fields for cultivation (although many of these areas are now heavily forested, trees finding the essential earth quite conducive to growth as well), as well as to make many, many artifacts of adamant and orichalchum. There seems to have been a preference in most areas to build down or into existing mountains, rather than raise edifices to the sky, although there are exceptions - many of the smaller mountains and hills in the XXXX region are now believed to actually be the remains of massive ziggurats raised as houses of worship to the sky gods.

Many of these 'dungeons' have metal torches placed at regular intervals along the smooth stone walls, enchanted with Continual Light in some instances and permanent Fire spells in others. Some merely have Glow, and there are even some examples filled with magically enchanted Sunlight. There are also many examples of 'cornucopias' or 'grails' enchanted with spells such as Purify Air, Create Air, Create Water, or even Create Food. Many of these are fully self-powered. Doors are frequently composed of dense Essential Wood banded with orichalchum; most depend on locking spells, since their physical locks of orichalchum are primitive in a mechanical sense and frequently easily picked, and some lack locks entirely. Shape Air 'fan' devices provide air circulation even dozens of levels down from the surface.

The frequent discovery of living monsters from eons past within locked rooms demonstrates that the ancients thought little of enchanting whole rooms with Suspend Time or a variation on Suspended Animation. It is speculated that many monstrous species were actually created by the ancients using the Hybridization elixir or an equivalent spell. Adventurers have also discovered hundreds if not thousands of golems of all substances and sizes, although man-sized clay golems or ushabti are the most common type. The majority of these, however, now appear to be useless since the owners capable of commanding them to act are now long dead.

Some 'dungeons' were clearly domiciles (underground castles or apartment complexes) or religious complexes; others appear to have been some kind of warehouses for the vast array of physical and magical goods possessed by the ancients. Earth magic made it quite easy for the ancients to prospect for and refine gold, silver, and jewels, and the quantities of each sometimes found are staggering. Magic items of many types are an even greater treasure - the ancient enchanters clearly were able to cast larger spells more easily than their present-day counterparts.

Secret: Many of the 'warehouse' dungeons were actually the ancient equivalent of our world's shipping warehouses, personal storage units, and animal kennels (except the ancients were able to use suspended animation in the last - many 'monsters' are actually people's abandoned pets from ancient days). Others were actually trading bazaars similar to shopping malls - again, including pet shops.

The existence of permanent Gates across vast distances and even to other planes in some instances further demonstrates the prowess of the ancients with enchantment. It is possible that the entire ancient world was connected via a gate network, making physical travel or other means of magical transportation largely unnecessary. This may explain the frequency with which smaller dungeon ruins are found in remote areas; there was no need to concentrate everybody in particular urban areas, when travel to the hinterlands could be easily accomplished and food and other goods could be gated in or magically produced on site.

Secret: The ancients did indeed have greater enchanting prowess than moderns, but not because of higher levels of Magery or skill. Rather, in addition to operating in a High Mana environment where anybody could learn standard magic, the ancients made extensive use of a variation on Meditative or Devotional Magic, expending character points from study (or from bonus points, but less frequent) to buy energy. Because the mana flowed so strongly, and even common elements such as water and clay were already infused with magic, this process worked at the more favorable rate of 1 CP (usually 200 hours of 'study') per 1000 energy, as opposed to 1 CP per 25 energy as would be the case in the era of the campaign. They eventually fed so much energy into the cheap enchanted objects on which their society depended that the mana level in most areas has now been depleted to Normal or even Low.

The current society is a patchwork of feudal and oligarchic states spread across the continent, ca 1000 years after the fall of the ancients. Different peoples have different legends about what actually caused this fall, ranging from wars against dragons or giants, to a necromantic plague, to an epidemic of spell backlashes involving the summoning of demons or worse, to a simple waning of magic. Most major cities or towns have at least one large 'dungeon' in the vicinity, scavenging which formed part of the reason for the resettlement. Although many of these are mostly 'mined out', there are frequently more hidden levels to be explored even after all these centuries. In very few cases have the peoples of the modern day actually settled in the dwelling places of the ancients, owing in part to superstitious dread of their powers (this partly explains why only a few partake in 'adventuring'), but also to a pragmatic assessment of the dangers of magical boobytraps, the periodic release of monsters to wander the corridors, and the occasional infestation of undead. Not to mention that the lower levels of dungeons not infrequently end up connecting with the even *more* extensive cavern complexes of the underground races such as dwarves, gnomes, kobolds, and troglodytes, who invariably take exception to humans settling in what they consider to be their realms. More ruins are frequently discovered in the hinterlands as those areas get resettled, as well.

Secret: The underground peoples were actually 'genetically engineered' from human stock long ago to be servants in the underground complexes. After the downfall of human civilization, the slaves became the masters of their own domain, much like the Morlocks in H.G. Wells' Time Machine (well, except without the cannibalism part, except maybe for the troglodytes and some dark elves). They found that the mana level continued to be high if you dug sufficiently deep, and used Earth magic to eventually create an entire underworld in the roots of the world.

Basically, this gives the background for an archetypal D&D setting with subterranean dungeons, both in proximity to settlements and out in the hinterlands, where vast quantities of magic and treasure are 'guarded' by a variety of monsters (competitive scavenging bands from enemy races such as orcs can fill in the monster roles not satisfied by feral Suspended pets and the undead). Of course, a logical consequence of this setup would be that many communities will require 'adventurers' to purchase a license or even work for the local government directly in order to be allowed into the 'dungeon' in the first place, and will equally plan to soak the adventurers for most of their gains when (if) they re-emerge, but I don't really see that as a major problem - have fun gaming the PCs gaining admittance to the Adventurers or Treasure Hunters Guild, or negotiating with the local lord directly for a dungeon delving license, or dealing with excise and tax collector. You can also vary the extent to which magic is commonly used in everyday life on the surface, depending on just how many folks are born with Magery, whether training is widely available, and just how many non-mage magic items you actually decide to salt your dungeons with. Also note that a sort of archaeologist/scholar becomes a very viable character type - every band could use somebody to translate the writing on the wall, look out for traps, identify magic items. Mages will frequently double in this sort of role, given the utility of looking to the ancients for hints of how to advance the study of magic, but you could also have more physical Indiana Jones types.

DoctorRomulus 10-23-2007 01:54 PM

Re: Secrets of the Dungeon Masters
 
This is ...... E X C E L L E N T!!!!



Quote:

Originally Posted by vitruvian
I recently ran across some notes made a few years ago about justifying a Dungeons & Dragons-like setting with lots of subterranean catacombs and dungeons with ecologically unsound populations of monsters either trapped in rooms or wandering about, and vast quantities of treasure and magic items for the taking. Basically, it's a post-apocalyptic setting, where a highly advanced magical culture collapsed and left ruins behind. 'Adventuring' is frequently merely a polite word for 'scavenging', and a major portion of the world's economy may be based on such scavenging operations and their proceeds, especially in terms of magic items. The political and economic structure tends to be relatively feudal, since there are intermittent disruptions to the economy as more treasure troves are uncovered and massive inflation results from the influx of gold, silver, and magic goodies.

The World of the Dungeon Masters:

Once upon a time, the world was High Mana throughout, and magic was as common as dirt. In fact, raw magical energy was so freely available that even quite large enchantments were readily available (more on mechanics later).

Magic was fully industrialized; paired with actual material technology at no more than TL2 (yes, smelting iron is a post-apocalyptic invention), this resulted in a highly advanced urban civilization.

The ancients clearly made extensive use of Earth spells to build downwards, creating massive underground complexes or dungeons using Earth to Air, Shape Earth, and Earth to Stone. Essential Earth was used to make highly fertile fields for cultivation (although many of these areas are now heavily forested, trees finding the essential earth quite conducive to growth as well), as well as to make many, many artifacts of adamant and orichalchum. There seems to have been a preference in most areas to build down or into existing mountains, rather than raise edifices to the sky, although there are exceptions - many of the smaller mountains and hills in the XXXX region are now believed to actually be the remains of massive ziggurats raised as houses of worship to the sky gods.

Many of these 'dungeons' have metal torches placed at regular intervals along the smooth stone walls, enchanted with Continual Light in some instances and permanent Fire spells in others. Some merely have Glow, and there are even some examples filled with magically enchanted Sunlight. There are also many examples of 'cornucopias' or 'grails' enchanted with spells such as Purify Air, Create Air, Create Water, or even Create Food. Many of these are fully self-powered. Doors are frequently composed of dense Essential Wood banded with orichalchum; most depend on locking spells, since their physical locks of orichalchum are primitive in a mechanical sense and frequently easily picked, and some lack locks entirely. Shape Air 'fan' devices provide air circulation even dozens of levels down from the surface.

The frequent discovery of living monsters from eons past within locked rooms demonstrates that the ancients thought little of enchanting whole rooms with Suspend Time or a variation on Suspended Animation. It is speculated that many monstrous species were actually created by the ancients using the Hybridization elixir or an equivalent spell. Adventurers have also discovered hundreds if not thousands of golems of all substances and sizes, although man-sized clay golems or ushabti are the most common type. The majority of these, however, now appear to be useless since the owners capable of commanding them to act are now long dead.

Some 'dungeons' were clearly domiciles (underground castles or apartment complexes) or religious complexes; others appear to have been some kind of warehouses for the vast array of physical and magical goods possessed by the ancients. Earth magic made it quite easy for the ancients to prospect for and refine gold, silver, and jewels, and the quantities of each sometimes found are staggering. Magic items of many types are an even greater treasure - the ancient enchanters clearly were able to cast larger spells more easily than their present-day counterparts.

Secret: Many of the 'warehouse' dungeons were actually the ancient equivalent of our world's shipping warehouses, personal storage units, and animal kennels (except the ancients were able to use suspended animation in the last - many 'monsters' are actually people's abandoned pets from ancient days). Others were actually trading bazaars similar to shopping malls - again, including pet shops.

The existence of permanent Gates across vast distances and even to other planes in some instances further demonstrates the prowess of the ancients with enchantment. It is possible that the entire ancient world was connected via a gate network, making physical travel or other means of magical transportation largely unnecessary. This may explain the frequency with which smaller dungeon ruins are found in remote areas; there was no need to concentrate everybody in particular urban areas, when travel to the hinterlands could be easily accomplished and food and other goods could be gated in or magically produced on site.

Secret: The ancients did indeed have greater enchanting prowess than moderns, but not because of higher levels of Magery or skill. Rather, in addition to operating in a High Mana environment where anybody could learn standard magic, the ancients made extensive use of a variation on Meditative or Devotional Magic, expending character points from study (or from bonus points, but less frequent) to buy energy. Because the mana flowed so strongly, and even common elements such as water and clay were already infused with magic, this process worked at the more favorable rate of 1 CP (usually 200 hours of 'study') per 1000 energy, as opposed to 1 CP per 25 energy as would be the case in the era of the campaign. They eventually fed so much energy into the cheap enchanted objects on which their society depended that the mana level in most areas has now been depleted to Normal or even Low.

The current society is a patchwork of feudal and oligarchic states spread across the continent, ca 1000 years after the fall of the ancients. Different peoples have different legends about what actually caused this fall, ranging from wars against dragons or giants, to a necromantic plague, to an epidemic of spell backlashes involving the summoning of demons or worse, to a simple waning of magic. Most major cities or towns have at least one large 'dungeon' in the vicinity, scavenging which formed part of the reason for the resettlement. Although many of these are mostly 'mined out', there are frequently more hidden levels to be explored even after all these centuries. In very few cases have the peoples of the modern day actually settled in the dwelling places of the ancients, owing in part to superstitious dread of their powers (this partly explains why only a few partake in 'adventuring'), but also to a pragmatic assessment of the dangers of magical boobytraps, the periodic release of monsters to wander the corridors, and the occasional infestation of undead. Not to mention that the lower levels of dungeons not infrequently end up connecting with the even *more* extensive cavern complexes of the underground races such as dwarves, gnomes, kobolds, and troglodytes, who invariably take exception to humans settling in what they consider to be their realms. More ruins are frequently discovered in the hinterlands as those areas get resettled, as well.

Secret: The underground peoples were actually 'genetically engineered' from human stock long ago to be servants in the underground complexes. After the downfall of human civilization, the slaves became the masters of their own domain, much like the Morlocks in H.G. Wells' Time Machine (well, except without the cannibalism part, except maybe for the troglodytes and some dark elves). They found that the mana level continued to be high if you dug sufficiently deep, and used Earth magic to eventually create an entire underworld in the roots of the world.

Basically, this gives the background for an archetypal D&D setting with subterranean dungeons, both in proximity to settlements and out in the hinterlands, where vast quantities of magic and treasure are 'guarded' by a variety of monsters (competitive scavenging bands from enemy races such as orcs can fill in the monster roles not satisfied by feral Suspended pets and the undead). Of course, a logical consequence of this setup would be that many communities will require 'adventurers' to purchase a license or even work for the local government directly in order to be allowed into the 'dungeon' in the first place, and will equally plan to soak the adventurers for most of their gains when (if) they re-emerge, but I don't really see that as a major problem - have fun gaming the PCs gaining admittance to the Adventurers or Treasure Hunters Guild, or negotiating with the local lord directly for a dungeon delving license, or dealing with excise and tax collector. You can also vary the extent to which magic is commonly used in everyday life on the surface, depending on just how many folks are born with Magery, whether training is widely available, and just how many non-mage magic items you actually decide to salt your dungeons with. Also note that a sort of archaeologist/scholar becomes a very viable character type - every band could use somebody to translate the writing on the wall, look out for traps, identify magic items. Mages will frequently double in this sort of role, given the utility of looking to the ancients for hints of how to advance the study of magic, but you could also have more physical Indiana Jones types.


Dalillama 10-23-2007 03:13 PM

Re: Secrets of the Dungeon Masters
 
Quote:

Originally Posted by vitruvian

They eventually fed so much energy into the cheap enchanted objects on which their society depended that the mana level in most areas has now been depleted to Normal or even Low.

This is probably the actual cause of their downfall right here.
Very good writeup.

vitruvian 10-23-2007 03:32 PM

Re: Secrets of the Dungeon Masters
 
Further thoughts:

Add necropoli to the list of dungeon types. Unfortunately, one of the nails in the coffin of Ancient society (or should that be crowbar to the nails in the coffin) was a Final War in which necromantic spells of mass destruction were used and the undead were weaponized, resulting in whole cities of the dead full of wandering zombies, ghouls, and vampires. Add in that members of the oligarchy on either side opted for lichdom, soul jarring, or investment in soul golems over a natural lifespan or other means of life extension, and there's a lot of stuff from the past that can come back to (literally) bite the current society.

The Ancients actually had very few mages per se - Magery was not a highly selected trait while the mana level was High. Instead, almost all magic was done using the rules for Meditative Magic, requiring only a modicum of skill in the Ancient version of Ritual Magic and the specific College path under which a specific working fell. At 1000 energy per CP, and the usual rate of 200 hours for CP during self-'study', this works out to 5 energy per man-hour put into an enchantment by somebody with the correct training, or 40 energy per day per participant with a short 8-hour work day.

They also had a common necromantic ritual for extracting energy from the sacrifice of a nonsentient animal, and even more from human sacrifice in those cultures that allowed it (mainly of prisoners from hostile nations). Consider the Ancient culture as being TL 2(+6), with TL 8 level wealth. That makes each CP in Signature Gear worth half of starting wealth, or $10k. Since each CP is also 'worth' 1000 energy, that makes each $10 worth of animal (or other material sacrifice, if allowed by the nation's gods) equal to an energy point; clearly, oxen and fatted calves will be worth more than chickens or mice. Figure a sacrificed prisoner of war's 'value' based either on dollar value as a slave, or if an important personage, based on CPs devoted to status, appearance, Magery, or similar advantages. The shaman-priests of a given nation's gods could also garner bonus CP that were immediately plowed back into magical energy by performing other acts that were pleasing to those gods, from planting a grove of trees to raising a particularly tall building to creating a special statue in the god's image (and possibly investing the earned energy in making the statue animate, in which case the animating spirit would be an aspect of the god).

Adventurers could find a huge cache of adamant golems, standing motionless in formation. Unfortunately, their owner *is* still around somewhere, whether as a lich, by possessing one of the PCs or an Ally, or just being in a suspension room that the PCs foolishly open. Or, the Ancients might have used a command device for their golems with a specific sigil on it, which becomes the MacGuffin that many rival parties are after, since he who commands the Adamant Army rules the world.

Giants, like dwarves and the other underground peoples, were magically created from human stock. They are manavores to feed the permanent Enlarge spells that forms a part of their bloodline, which partially explains why they tend to live in remote areas where the mana level never declined as much. The Gigantomachy was but one of the events that spelled the end for the Ancients.

Dragons were originally bred as war-mounts, but gained too much intelligence and also rebelled. They too are manavores more comfortable in remote areas. The Dracomachy was probably more damaging than the Gigantomachy to Ancient culture, as a number of large underground cities were taken over by the most powerful wyrms, who may still to this day be nesting and feeding off the accumulated magical energies of their treasuries. Yes, lots of dungeons with dragons on the bottom level, resting on a pile of shining gold (each and every coin enchanted with Glow) and other magical knick-knacks.

Magical books of the Ancients are not terribly useful to modern-day mages, both because they are very few in number (most common spell rituals were actually transmitted orally), and because the form of magic has changed a great deal, from everybody doing Meditative Magic to only those with Magery spending their own FP on individual spell skills and regular Quick & Dirty or Slow & Sure enchantment. Even if modern-day mages can learn the trick of Ancient enchantment, energy is only available in most places at the 1 CP per 25 energy rate, which works out to 1 energy per 8-hour day, about the same as Slow & Sure enchantment anyway. Ancient mages who somehow survive to the present day will face similar restrictions, and either need to learn the new ways of magic or depend on their troves of enchanted items. Mages are still terribly interested in the past, however.

The closest thing the current society has to the Ancient way of doing things is the structure of certain temples, or the agricultural association called the Grange, where ceremonies involving multiple wizard-priests and scores of lay participants are used to cast large spells such as Resurrection or Bless Crops.

younglorax 10-23-2007 03:38 PM

Re: Secrets of the Dungeon Masters
 
Wow. That is all around fantastic.

Also, I really like Mana as a natural resource.

vitruvian 10-23-2007 03:57 PM

Re: Secrets of the Dungeon Masters
 
Quote:

At 1000 energy per CP, and the usual rate of 200 hours for CP during self-'study', this works out to 5 energy per man-hour put into an enchantment by somebody with the correct training, or 40 energy per day per participant with a short 8-hour work day. Consider the Ancient culture as being TL 2(+6), with TL 8 level wealth. That makes each CP in Signature Gear worth half of starting wealth, or $10k. Since each CP is also 'worth' 1000 energy, that makes each $10 worth of animal (or other material sacrifice, if allowed by the nation's gods) equal to an energy point; clearly, oxen and fatted calves will be worth more than chickens or mice.
That also means that each energy point worth $10, which gives a nice cost mechanism for enchanted items, and a nice way of determining the average citizen's income, if they were to keep the full proceeds of their labor. Figure each person would be earning 40 energy, $400 for each day's labor, if they kept it all; this equates to earning Average starting wealth of $20,000 every ten weeks. Assume about a 50% 'tax' rate in terms of how wealth gets passed up the food chain and/or your employer fails to share all the profits, and that becomes $200 per day, 20 weeks to recoup starting wealth, and the income for a standard 'day enchanter' of Average Wealth is ($20k x 52/20) = ~$52k. This implies a slightly higher effective income for TL 2(+6) than for TL8 proper, but they have lots of stuff to spend it on, and this does ignore vacations and other inefficiencies. By comparison, a standard worker golem of clay costs 800 energy, or $8k, before markup, costing about 5 weeks income from someone of average wealth, about like a cheap car or similar. Markup for different types of magic items would depend on the scarcity of people who knew the College (and optionally technique) for the specific spell.

Note that Ancient society didn't have a big break in the costs of items at the limit for Quick & Dirty casting, since they didn't use it - it's a flat $10 per energy point, modified by any factors for markup. Therefore, the distribution of cheap, weak items vs. more powerful items doesn't have a big breakpoint either, or at least not any more than the distribution of cheap vs. expensive items in your home. Assuming that there were individuals that were Filthy Rich or Multimillionaires and could pay whole cohorts of enchanters to make their toys, there should be some pretty wild enchantments lying around - which is borne out by the existence of Suspend Time with Link enchantments on 10 foot by 10 foot rooms (probably amortized by the renting entity over time), or gates starting at tens of thousands of points of energy.

The Ancients probably needed some kind of way to make magical energy and money more easily fungible than by barter of specific enchanted items. One way might have been to invent an exchange medium that actually could contain magical energy, similar to the stuff described in 3e GURPS Egypt, paut. What would be the most logical medium, do you think? Why, gold, silver, and jewels. There didn't even need to be a means of extracting the magic from these items, apart from destroying them or converting them to dross (lead for gold, tin for silver, quartz or rock for jewels). Figure each small gold coin or sun is 10 energy, $100, each silver penny or moon is 1 energy, $10, and the worth of jewels is determined by their carats. The modern Powerstone spell is an extension of this principle where the jewels reabsorb mana over time, while the old coinage is the equivalent of manastones.

Now, what are the impacts if all those treasuries of gold retained all that magical energy? Aside from dragons liking their treasure hoards even more, that is.

Omniplex 10-23-2007 04:01 PM

Re: Secrets of the Dungeon Masters
 
This is probably the one of the most excellent setting Idea's I've ever seen. It provides a perfect explanation to all of the strange mainstays of popular fantasy roleplaying, that is, crawling through dungeons to fight monsters and loot the magic items that they inexplicably guard. wow You wouldn't mind if I use this for a game sometime would you?

Kirby 10-23-2007 04:09 PM

Re: Secrets of the Dungeon Masters
 
No question, this is a heaping helping of excellence. I'm not so fond of the orthodox GURPS enchanting rules, so I suspect I'd change the economic calculations somewhat, but it gives me all kinds of glee to imagine a bizarre manatech civilization degenerating into an apparently irrational collection of heterogeneous elements like the world of the old D&D cartoon.

The world would absolutely have to be culturally nonsensical, with pockets of whatever cultures one needed imported from other universes or created by hypermages out of whole cloth, ignoring sociological and historical norms just as much as the rest of the world ignores ecological and architectural norms.

D&D as magical postapocalypse. Eminently stealable; I salute you, Vitruvian.

--K

vitruvian 10-23-2007 04:16 PM

Re: Secrets of the Dungeon Masters
 
Quote:

You wouldn't mind if I use this for a game sometime would you?
Please do - it doesn't seem like I'm getting a campaign together anytime soon, and even if I do this probably comes after Discworld Supers. Just be sure to credit me if you write it up on a website or anything like that.

Elements of this came together when I went to my personal storage unit and read _Glasshouse_ by Charles Stross on the same weekend.

Hmmm... just realized that this gives a great mechanism for transforming gold pieces back into CP. Should it use the old or the new conversion rate? In the present day, it would be 1 CP per 25 energy, not per 1000. Maybe if the gold pieces each contained 25 energy instead of 10 - yep, 1 CP per gold piece. Even *more* like old-style D&D!

Also, keep the same CP to $ ratio, and this makes even more clear that the post-apocalyptic feudal society has only a fraction of the wealth of the Ancients, and why dungeon-delving is the preferred profession of those seeking to make their fortunes in an otherwise class-stratified society.

Kirby 10-23-2007 04:30 PM

Re: Secrets of the Dungeon Masters
 
Quote:

Originally Posted by vitruvian
yep, 1 CP per gold piece. Even *more* like old-style D&D!

Okay, I just dissed on your economics a few minutes ago, but I'm prepared to eat crow: The above correspondence is just lucid, crystalline beauty. One couldn't refuse that equation.

--K


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