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 Rithmomachia: The Game for Medieval Geeks By the in CultureWed Sep 18, 2002 at 07:20:22 AM EST Tags: Culture (all tags) In 965 Wibold, bishop of Cambrai, suggested to the local monks that they give up playing dice games and instead play "the battle between virtues and vices". In all likelihood this was the game rithmomachia, otherwise known as the ludus philosophorum. Today we are familiar with a number of ancient board games but unfortunately rithmomachia, as well as some related games Ouranomachia and Metromachia have all but disappeared.

Rithomachia was played by the intellectuals of Europe, often Church employees and most likely male. Although it is obscure today, rithomachia played an important part in the lives of this elite group. The rules were complex and convoluted and required considerable computational skill. Its players also claimed that its benefits extended well beyond numerical training - it supposedly improved the moral character of players and even gave deep insights into religious truth.

There are many different versions of the rules of play but there are many common features. There are two players who take turns moving pieces on a board, much like chess or checkers. The board is something like a chess board, though rectangular and not square, with 48 pieces, each inscribed with a number. The size of board varies with rule set but eventually it was standardized on 8 by 16.

But now the complication starts. Rithomachia was inspired by a type of mathematics made popular by Boethius. This work was dominated by various types of numerical progression: in particular arithmetic progressions (where the nth number is a+b*n for constant a and b), geometric progressions (a*b^n) and harmonic progressions (1/(a+b/n)). There were also other more complex types of progression such as multiplex progression or the progression of superparticulars. At the start of the game the initial layout of the numbered pieces, in three ranks for each side, was such that the numbers formed various types of progression. One side had odd numbers in the first rank, the other side even. So even before the game had started quite a bit of mathematical knowledge was required. And don't forget that in earlier times arabic numerals were unknown so any calculations were carried out using Roman numerals. (By the way, these proportions played an important part in medieval music theory. Consider the harmonic progression in particular.)

Pieces from the first rank could move one square, those from the second rank two and those from the third three. As a mnemonic the pieces in these ranks were sometimes circles, triangles and squares respectively (a slightly illogical mnemonic!). Like in chess, pieces could be captured, however there were many ways to capture. One way was to move a piece with a number onto another with the same number. Two pieces could take another if it was the case that if they were simultaneously to make legal moves that would land them both on the captured piece and the sum of the numbers on the two pieces sums to that of the captured one. Another way to capture was to occupy all the spaces that another piece could move to making it unable to move - this was called besieging it. Here is another example of a capture rule taken from Lever and Fulke's "The Most Noble and Auncient, and Learned Playe" published in 1563:

Of taking by cossical signs
By cossical signs: any man that hath these signs, 3, &, 33, 3&, meeting with his root in his ordinary draught that hath this sign z taketh him up, or else is taken of his, without removing into his place; except he may not take him before he remove.
Obviously there has been a little semantic drift over the centuries. There were also some much more complex capture rules requiring the pieces to be arranged in a progression.

Now we've warmed up we can get onto the conditions to achieve victory in the game. Typically a player had to line up a series of pieces in certain arrangements, often as a Boethian progression of a certain length. Whoever did this would win. There were many different choices of victory conditions based on such progressions and players would negotiate before to game to decide which were in play for a particular game. An example simple victory condition follows:

Victory of goods
Victory of goods is to take a certain number without respect of the men. As if it be covenanted that he which first taketh men amounting to the number of 100 or 200 shall have the victory.
There was even a victory condition based on simulating the armies of the Christians and Turks at war.

As you can see, this was no game for weenies and yet it was played all over Europe from Medieval times, through the Renaissance into Elizabethan times. Many rithomachia manuals still exist today and doubtless many more were originally published. It seems to me there is only one explanation: Rithomachia is a geek game and the players were predecessors of today's game playing geeks! They spent their time shut up in dark rooms hunched over books and games believing themselves to be superior to the masses because they were experts in difficult and arcane, but largely useless knowledge.

Today Boethian mathematics is almost unheard of. This is the key to the decline of rithmomachia. Boethian mathematics was highly technical but today it's abstruseness seems completely arbitrary and useless. As it was replaced by more modern approaches to mathematics the rules to rithmomachia came to seem more and more arbitrary until interest in the game completely waned. However it is worth noting that this game did have a lifetime of 500 years and was played by such illustrious luminaries as John Dee and praised by Roger Bacon.

Incidentally there was a recent resurgence in the play of rithmomachia when it was released as a shareware computer game called Ambush.

Unlike chess it was probably not played in Iraq and the pieces were typically not made of sapphire.

References

• The Philosopher's Game, Ann E Moyer, University of Michigan Press (also contains the entire text of the rithmomachia Lever and Fulke's 1563 manual)
• Das mittelalterliche Zahlenkampfspiel, Arno Borst, Carl Winter Universitätsverlag

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 Rithmomachia: The Game for Medieval Geeks | 67 comments (50 topical, 17 editorial, 0 hidden)
 Wow (4.60 / 20) (#14) by XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX on Wed Sep 18, 2002 at 01:04:40 AM EST

 It puts one in touch with one's common humanity to know that even centuries a ago, in a society totally different from our own, there were groups of intelligent young men sitting around not getting laid.
 Celibacy (4.20 / 5) (#35) by ucblockhead on Wed Sep 18, 2002 at 12:39:12 PM EST

 Well, these were monks...they weren't supposed to get laid. ----------------------- This is k5. We're all tools - duxup[ Parent ]
 O Fortuna (none / 0) (#43) by the on Wed Sep 18, 2002 at 02:33:33 PM EST

 But you do know Carmina Burana don't you? I guess the Goliards didn't have much time for Rithmomachia. -- The Definite Article [ Parent ]
 Well... (5.00 / 1) (#45) by ucblockhead on Wed Sep 18, 2002 at 02:52:38 PM EST

 The key word is "supposed to"...'tis true, they probably got laid more than the average geek. ----------------------- This is k5. We're all tools - duxup[ Parent ]
 I voted for this post because.... (3.25 / 4) (#18) by vile on Wed Sep 18, 2002 at 02:26:01 AM EST

 I think the nick 'the' rules. ~ The money is in the treatment, not the cure.
 +1 FP (2.00 / 3) (#19) by thatwhichfalls on Wed Sep 18, 2002 at 02:41:58 AM EST

 Excellent - never heard of it before today and now I can't wait to play it.
 The game (4.33 / 3) (#20) by BloodmoonACK on Wed Sep 18, 2002 at 02:46:45 AM EST

 It would have been nice had you mentioned in the article that the game is flawed. According to the first link in the article, one side has an advantage. Obviously, this would not let the same kind of competition exist that is around in Chess...once you learned the rules you can essentially always win. Perhaps this had something to do with the games decline. "It's like declaring a 'war on crime' and then claiming every (accused) thief is an 'enemy combatant'." - Hizonner
 All non-random turn-based games are "flawed&q (4.50 / 4) (#22) by Amorsen on Wed Sep 18, 2002 at 05:05:31 AM EST

 One side can always force a draw, or if a draw is not possible in that game, a win. Chess is "flawed" in the same way. Currently it is not known whether it is black or white in chess that has the advantage. [ Parent ]
 White has (2.75 / 4) (#24) by GaussZ88 on Wed Sep 18, 2002 at 08:24:00 AM EST

 because if black had a winningstrategy you could make the first move with white and then just play like having never made the move thus respectivly acting like black (and having the winningstrategy). This argument is applieable to most deterministic games I think. Though the good thing about chess is that the possibilitys are so vast that no one can calculate the winningstrategy. It would be interesting to know if there is an easy to calculate winningstrategy in Rithmomachia. [ Parent ]
 Not so simple (5.00 / 3) (#30) by p3d0 on Wed Sep 18, 2002 at 10:51:27 AM EST

 Your argument is flawed. It assumes that (1) the first step of black's winning strategy is known to white before white makes its first move, (2) the first step of black's strategy is a legal move for white, (3) white's making this move is not harmful to white's position, and (4) that white's making this move prevents black from using the winning strategy. Consider this imaginary game: each player, X and O, starts with one piece on a board like the following (and I hope the ASCII art works...):    -   |X|    -   / \  -   - | | | |  -   -   \ /    -   |O|    - Each player can move his piece to any adjacent square. If the other player's piece is already on that square, it is captured and the capturing player wins. X goes first. Clearly, whichever of the two available squares X chooses, O will choose the same square and X will lose. Assumptions 1, 3, and 4 above are invalid in this case. Obviously this is a trivial game, but it demonstrates that your argument is flawed, so how can you be sure it applies to chess? -- Patrick Doyle My comments do not reflect the opinions of my employer.[ Parent ]
 Right (1.00 / 1) (#68) by GaussZ88 on Sat Nov 30, 2002 at 09:29:54 AM EST

 You are right :) It can not. [ Parent ]
 Actually... (4.00 / 1) (#25) by Yosho on Wed Sep 18, 2002 at 09:45:28 AM EST

 In Go, there's a rule called komi that states that in exchange for black going first, white get an extra 5.5 points. Supposedly that's enough to make it even. ;-) [ Parent ]
 Not In Go (4.66 / 3) (#27) by DarkZero on Wed Sep 18, 2002 at 09:58:45 AM EST

 Some time in the last fifty years or so, a rule called "komi" was introduced to the three to four thousand year old game of Go. Since black moves first, the komi rule gives white five and a half moku (points) to start with, whereas black starts with zero moku. While it may strike people as blasphemy to change the rules of one of the oldest games in recorded history four thousand years after its birth, it fixed the entire game of Go. Black used to win far more than fifty percent of the time because it went first, but after the komi rule, the game seems to have finally become even, with neither side having a real advantage. It also removed the possibility of a drawn game, because someone always wins by at least a half moku. Obviously this couldn't be implemented in Chess, because Chess is based on capturing a single piece instead of acquiring the most points through capturing the most enemy pieces or the most territory, but it looks like it could possibly be implemented in rithmomachia, because rithmomachia works on a point based system. Then again, as I look further into the rules of rithmomachia, it appears that the two sides, rather than being equal sides, of which one always takes the first turn, are much more like the different races in StarCraft or WarCraft. Interesting... [ Parent ]
 Still "flawed" (none / 0) (#32) by zakalwe on Wed Sep 18, 2002 at 11:09:43 AM EST

 That doesn't change whether or not there is still some "perfect game" that allows the person who has the right starting colour to always win.  If you could map out all the possible decisions that could be made (a huge task even in chess, and orders of magnitude greater for Go), either a specific player must be able to win always, or either player can force a draw (eg Tic-tac-toe). In practice however, mapping out all the correct decisions is far beyond our current capacity even in chess, so the advantage is purely a theoretical one. [ Parent ]
 "Flawed" (none / 0) (#33) by DarkZero on Wed Sep 18, 2002 at 11:46:34 AM EST

 In practice however, mapping out all the correct decisions is far beyond our current capacity even in chess, so the advantage is purely a theoretical one. Is a minute, imperceivabe, untestable flaw really a flaw? Since so far no known human being has been able to map out every possible move in Go in their mind (the amount of possibilities can range well into thousands just thinking two or three steps ahead in the game) and no programmer or group of programmers has been able to create a Go program that can regularly best even a mid-level player, no one has ever actually seen this flaw in action. And if no one has seen it in action or is even able to think through all of the possible moves in Go to predict the occurrence of such a flaw, it might as well not exist, if it even truly does to begin with. ...or either player can force a draw (eg Tic-tac-toe). The komi rule actually negates the possibility of a draw. Since one of white's starting moku is a HALF moku, which is absolutely impossible to attain otherwise in the game, someone must always win by at least one half moku. I'm guessing that this was included in the role because of an abuse of the containment tactics available in Go, which could just keep anyone from scoring for the entire game and force a draw. Now, with the komi rule in place, black cannot focus on containment for the entire game because they would lose by five and a half moku, and white cannot pull a containment tactic off either because they cannot make the first move and thus orient the entire game around boxing their opponent in. [ Parent ]
 A flaw by any other name.. (none / 0) (#40) by Kwil on Wed Sep 18, 2002 at 01:31:39 PM EST

 Is a minute, imperceivabe, untestable flaw really a flaw? You answer your own question: Since so far no known human Just because we haven't found it, doesn't mean it's not there. You might as well say that the original release of Windows had no flaws.. not until some bugger started finding them. That Jesus Christ guy is getting some terrible lag... it took him 3 days to respawn! -NJ CoolBreeze[ Parent ]
 No perfect game of GO. (none / 0) (#57) by Kintanon on Thu Sep 19, 2002 at 09:49:10 AM EST

 You quite obviously have no understanding of the game of Go. There is not a set web of moves that can be made to always win at Go. I will garauntee this. The reason is because unless your opponent cooperates with you the board position will never ever ever ever ever stay in the configuration necessary for that assured victory. Assured victory requires that your opponent be less skilled than you are, no other option will do. Even with perfect knowledge of the game will not give you a set list of 150 stone placements, which when made in order, will win the game for you regardless of where your opponent places. Kintanon [ Parent ]
 Yes there is (none / 0) (#58) by zakalwe on Thu Sep 19, 2002 at 11:15:07 AM EST

 No matter how difficult, Go is entirely deterministic, and so there must be some method of a guaranteed win for one of the players (For some games there may be a forced draw for either player, or a perpetual stalemate if the game has no limit on this.) Even with perfect knowledge of the game will not give you a set list of 150 stone placements, which when made in order, will win the game for you regardless of where your opponent places.This may be the source of the misunderstanding.  I'm not saying there is some static series of moves that guarantee a win - only that for every move the opponent makes, there is some move that a perfect player can make to guarantee their win (assuming they are in the advantaged position)  Hence, a perfect player will always win given the right starting position. [ Parent ]
 Doesn't matter really (none / 0) (#61) by zocky on Thu Sep 19, 2002 at 01:46:42 PM EST

 Of course any deterministic game (and turn-based board games are deterministic) can be played perfectly by considering all the possible moves. It's just a matter of scale. Tic-tac-toe is small and simple - there are just 9! = 362880 ways to lay 9 piecese on 9 squares. The number of games is much smaller - first divide by 8 to take into account all symmetries = 45360. That's even before taking into account all the games that end in 5th, 6th, 7th or 8th move. People can actually compute the whole game of Tic-tac-toe. OTOH, the number of possible games in chess or go is huge. I'm not going to calculate it now, but it could easily be more than the number of atoms in the universe mulitplied by the number of seconds since the Big Bang. So while in theory it can be done, there is no way to actually compute the whole chess or go game, let alone expect a human brain to do it. So there can't be a perfect chess or go player and the game isn't flawed. I haven't studied rithmomachia, but it looks complex enough. But there is, of course, the entirely different problem of unbeatable tactics (as opposed to strategy). Rules of some games do allow players to follow a very simple tactic and win or draw everytime. This applies just as well as tic-tac-toe. Who cares about tens of thousands of combinations when there's a simple unbeatable tactic: 10 If there's a winning move, win 20 Elsif there's a winning move for your opponent, block it 30 Elsif the middle is empty, put your piece in the middle 40 Else Play anything As somebody mentioned, rules of go were changed to prevent unbeatable tactics. In chess there are no unbeatalbe tactis since the number of combinations is instantly too large to calculate - after just one move by both players there can be 400 different positions, none of them symmetrical, because chess is not symmetrical. Almost all opening moves (apart from knights) increase the number of possible further moves, so we can safely claim that there are at least 160.000 possible combinations in the first two moves, which is far more than the whole game of tic-tac-toe. So, does anybody know whether there are unbeatable tactics in rithmomachia? ---I mean, if coal can be converted to energy, then couldn't diamonds?[ Parent ]
 Deterministic Tactics, Go, Go-Moku, and Chess (none / 0) (#66) by kaibutsu on Wed Sep 25, 2002 at 07:52:23 AM EST

 Well, the rules change to Go isn't to take care of unbeatable tactics, per se, but rather to take care of a common point advantage for the player who goes first. Giving a few points to the black player wouldn't fix Go if it suffered from an "unbeatable tactic." An example of a game that _does_ suffer from an unbeatable tactic is Go-Moku, which is a kind of poor-man's Go. Played on the same 19x19 board with the same pieces, the objective is to create a 5-in-a-row chain of your pieces. It can be shown (try it!) that black can always win. Real Go doesn't at all suffer from such a problem. So what's going on here? With Go (and Go-Moku) the possible board positions are massive, uncomputable even. But the case with Go-Moku is such that we can step back from the system somehow and see that there is this sequence of moves that will always win. In fact, it doesn't really matter that Go-Moku is on a 19x19 board rather than an infinite board, while board size is very much a consideration for Go. I _think_ the distinction lies purely in the victory conditions; there's not enough 'shades of grey' in Go-Moku to tell a good game from a bad one. You either win or you lose, and it's very hard to quantify how much you've won by. This problem also extends to Chess to a lesser extent, but is pretty non-existant in Go. This counting of territory means that there are huge variations in how you win as well as how you've gotten to a winning position. (This is a BIG Paraenthesis: probably not worth reading if you're not into computational Go. Anyway, sometime in the early eighties, a couple of doctoral students found a way to apply a Game Theory principle called 'chilling' to Go. The basic idea is that in certain end-game positions, the player who knows how to chill will get the last point, and therefore win a game that is, going into this particular kind of board position, tied. So Go isn't entirely safe from we conniving mathematicians after all, though the chinks we have found in her armor have been few and far between.) -kaibutsu[ Parent ]
 Uh, this is true for all games of skill (none / 0) (#63) by Kintanon on Fri Sep 20, 2002 at 10:52:06 AM EST

 Otherwise it wouldn't be a game of skill, it would be a game of luck. Wouldn't it? A game is not flawed simply because greater skill results in a win every time. That's what makes it a game of skill. Kintanon [ Parent ]
 True. (none / 0) (#64) by zakalwe on Fri Sep 20, 2002 at 11:39:53 AM EST

 Yes - "Flaw" is in quotes because I'm using it only in the sense that the original post used it.  In practical terms, its pretty irrelevant, and will be unless/until computers grow powerful enough to calculate all the possibilities. In a purely theoretical sense, you could win at Go without even knowing the rules, provided you had a big book containing the optimum move to make for every possible position (including score.)  In real terms, it isn't much of a flaw for the purely practical reason that creating such a book is beyond anyone's abilities.  If we ever do "solve" Chess or Go, there may be some diminished interest, but they'll still be enjoyable games between non-assisted humans. It might actually be possible to create games of skill where this isn't possible, where the game involves some form of information hiding.  In games like this, playing an optimum strategy may reveal more about yourself than is gained (at least if the other players could detect your strategy).  Most games like this aren't pure skill (Poker is the obvious one), but you could arguably claim RoShamBo as one.  Here, no perfect game exists because the optimum play depends on your oponents strategy, in ways you may not be able to account for. [ Parent ]
 Heh, Got me on the last couple of lines... (none / 0) (#65) by Kintanon on Fri Sep 20, 2002 at 04:27:01 PM EST

 As I as reading the comment I almost jumped the gun and posted, "Aha! Poker is the perfect game of skill!" >:) Then I finished reading and you already mentioned how it fulfills the requirements for a game of skill, and makes it perfect by being partially random. So, yeah, if you did have a list of every optimal move for every situation, you'd be an unbeatable Go player... I think that's one of those 'problems' with a game that people just make up because they don't like the game for some other reason. Especially since it isn't humanly possible.>:) Kintanon [ Parent ]
 What about (none / 0) (#41) by kallisti on Wed Sep 18, 2002 at 01:35:22 PM EST

 A game in which if the players play flawlessly, then the game continues on, but a mistake leads into an area where skillful play on the part of one player can lead to victory. Would you still consider that a flawed game? If so, I have no idea what an unflawed game could possibly look like. And, with allowance of repetition, is it possible to create such a game. Define a game state as an ID, a letter and two sets of state IDs. The left set tells what state player "left" can move to, the other are the moves for player "right". A state looks like this A: BC|D In which left can move to state B or C and right can only move to D. A set with no values, such as Z:| means the moving player loses. The game starts in state A: A: BC | BC B: A | A C: D | D D: | So, at the start either player can choose B or C, choosing B leads back to A, choosing C leads to their death. Obviously, this game is way too simple, but it is entirely possible to imagine a much larger space in which the traps are more subtle. A game contructed this way has no way to force a win, but requires perfect play in order to continue. This would be a perfectly fair game of skill. Now, if you remove cycles or claim that repeats are draws, then you would be correct. [ Parent ]
 A draw (1.00 / 1) (#50) by Peaker on Wed Sep 18, 2002 at 04:49:54 PM EST

 is noone's advantage. If one side can force a draw, the game is balanced. All deterministic games with no option of draw are flawed, those in which a win cannot be forced, like Chess probably is, are balanced. This is why this game is flawed (can force a win), while Chess probably isn't (can only force a draw). [ Parent ]
 It's tricky to give a definitive answer actually (4.00 / 2) (#26) by the on Wed Sep 18, 2002 at 09:48:55 AM EST

 Because there are so many rule variants and players choose which rules they are going to use before play. I'm sure it could swing either way depending on the rule set. -- The Definite Article [ Parent ]
 More resources (5.00 / 5) (#23) by Davidicus on Wed Sep 18, 2002 at 08:01:45 AM EST

 If you are interested in period/midevil games, a friend of mine has a site up about it, as he researches them.: rules for many games at: http://www.waks.org/game-hist/game-rules.html links to more resources at: http://www.waks.org/game-hist/
 geekus arcanus (5.00 / 2) (#28) by eudas on Wed Sep 18, 2002 at 10:09:13 AM EST

 "hey spent their time shut up in dark rooms hunched over books and games believing themselves to be superior to the masses because they were experts in difficult and arcane, but largely useless knowledge. " then there is the other perspective, that which states that the masses over time have somehow come to regard themselves equal to those who master difficult and arcane knowledge. eudas "We're placing this wood in your ass for the good of the world" -- mrgoat
 Hush, brother (5.00 / 2) (#29) by the on Wed Sep 18, 2002 at 10:18:53 AM EST

 then there is the other perspective, that which states that the masses over time have somehow come to regard themselves equal to those who master difficult and arcane knowledge. Shhh...>looks over shoulder<...these are dangerous times and such opinions are not for speaking aloud. -- The Definite Article [ Parent ]
 missing something (none / 0) (#39) by NFW on Wed Sep 18, 2002 at 01:31:14 PM EST

 You left out a largely useless. the masses over time have somehow come to regard themselves equal to those who master difficult and arcane, but largely useless knowledge. The masses ain't wrong, at least not in this case. Everybody has useless hobbies, and they are no cause for delusions of grandeur. -- Got birds? [ Parent ]
 no kidding (4.00 / 2) (#42) by adequate nathan on Wed Sep 18, 2002 at 02:33:01 PM EST

 Look at those Esperanto guys.Nathan "For me -- ugghhh, arrgghh."-Canadian Prime Minister Jean Chrétien, in Frank magazine, Jan. 20th 2003Join the petition: Rusty! Make dumped stories & discussion public![ Parent ]
 largely useless (5.00 / 1) (#51) by eudas on Wed Sep 18, 2002 at 06:39:15 PM EST

 actually i left that out intentionally because it is part of the masses' attempt to raise themselves by describing arcane knowledge as 'useless'. eudas "We're placing this wood in your ass for the good of the world" -- mrgoat[ Parent ]
 you with words play (5.00 / 2) (#55) by adequate nathan on Thu Sep 19, 2002 at 01:12:45 AM EST

 Knowledge arcane both useless and useful may be. Some knowledge arcane useful is; knowledge the tax-codes of in April much money might you save. Some knowledge arcane useful is in potential; abstract algebra to understand useless is if in a vacuum your knowledge is, but to other things a gateway it might be (too esoteric it is for applications most areas of life within, but useless it is not because some people a use for it do find.)Some knowledge arcane in principle useless is. Fall into this category the majority of g**k knowledge does. Fools they are who the Borg-cube-vs-super-stardestroyer debate. Perfidious they are who this with real knowledge conflate.Nathan "For me -- ugghhh, arrgghh."-Canadian Prime Minister Jean Chrétien, in Frank magazine, Jan. 20th 2003Join the petition: Rusty! Make dumped stories & discussion public![ Parent ]
 Cube vs Star Destroyer (none / 0) (#59) by eudas on Thu Sep 19, 2002 at 11:36:51 AM EST

 I will agree with you, debating Star Trek vs Star Wars is pretty silly and useless. eudas "We're placing this wood in your ass for the good of the world" -- mrgoat[ Parent ]
 Who cares, Go rules anyway (3.50 / 2) (#31) by Fen on Wed Sep 18, 2002 at 10:57:24 AM EST

 Oldest game, best game. --Self.
 Let me go ahead and pre-empt the thread... (4.33 / 3) (#34) by graal on Wed Sep 18, 2002 at 12:06:49 PM EST

 ...that will inevitably follow this post. Chess people: CHESS! Go people: GO! Chess people: CHESS! Go people: GO! Chess people: CHESS! Go people: GO! Chess people: CHESS! Go people: GO! Chess people: CHESS! Go people: GO! and so on. -- For Thou hast commanded, and so it is, that every inordinate affection should be its own punishment. -- St. Augustine (Confessions, i)[ Parent ]
 Eccentric person: (4.00 / 1) (#36) by roystgnr on Wed Sep 18, 2002 at 12:45:44 PM EST

 GESS! [ Parent ]
 Go is awesome but that makes this game no less fun (none / 0) (#38) by Accuracy on Wed Sep 18, 2002 at 01:15:02 PM EST

 I do agree Go is the best game ever, and a better game than this... but even us Go players are always looking for other new and interesting games, at least all of the ones I know. [ Parent ]
 Unrelated nitpick (4.83 / 6) (#37) by opalhawk on Wed Sep 18, 2002 at 12:47:11 PM EST

 (By the way, these proportions played an important part in medieval music theory. Consider the harmonic progression in particular.) This phrase is a little confusing. There is currently to my knowledge no evidence that the mathematics of harmonic progression were mapped out or understood in medieval times. In 1300 ad the ideal of Pythagorean tuning was introduced, perhaps that is what you are referring to. Since this system required that there be some irregular intervals between notes, True non-key / mode harmonic progression was not available. Everything seems to have been notated completely without the benefit of our modern analysis. In about 1580 Equal temperament was introduced. This system corrected for the loss of notes that occurred in the Pythagorean tuning system, and allowed us to write music that was easily transposed to any compatible key or mode. There is still no evidence that Roman numerals were used as a mode of harmonic analysis until The beginning of the Baroque period with J.S. Bach in 1685. 1685 is when Bach was born, but it is considered the beginning of the Baroque period. Some of Bach's older contemporaries were already using Roman numeral chordal analysis at the end of the Renaissance period, but Bach is considered to be the Champion Composer of this system because of what it did to aide his work in the new field of counterpoint. Okay, I doubt anyone else even cares about what I just said, but I feel better. ~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~* One must still have chaos in oneself to be able to give birth to a dancing star. -Nietzsche ~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*
 Thanks for the extra detail (5.00 / 1) (#44) by the on Wed Sep 18, 2002 at 02:43:57 PM EST

 The details of Boethian mathematics are pretty obscure, even to a modern day mathematician. I'd probably have to do a lot more research than I have time for to actually track down how Boethius related his various progressions to music theory. I can imagine it was highly complex given the description of pythagorean tuning here. I must write some code some time to generate these tones and experiment a bit. -- The Definite Article [ Parent ]
 That would be awesome... (none / 0) (#46) by opalhawk on Wed Sep 18, 2002 at 03:22:57 PM EST

 If you do ever end up writing that code, I would be interested in looking at some of your experimentation. I have messed around with period instruments, and directed a medieval/ Renaissance choir for a short time, but I have never delved into the little things that make that tuning system work. Most scholars who try to put it into practical application don't bother to tune to a system that they can't redily replicate on modern instruments like the Piano. I speculate that there was a different quality involved in the sound under the Pythagorean paradigm. ~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~* One must still have chaos in oneself to be able to give birth to a dancing star. -Nietzsche ~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~* [ Parent ]
 I'm only thinking of doing the most trivial thing (5.00 / 1) (#47) by the on Wed Sep 18, 2002 at 03:29:22 PM EST

 Just writing some code to play a sequence of notes at given frequencies. I just figured out how to write software to play synthesized audio samples on my Powerbook and was already messing about with very simple digital filters so playing scales with notes at specific frequencies should be easy. My knowledge of music is pretty minimal. -- The Definite Article [ Parent ]
 well... (none / 0) (#48) by opalhawk on Wed Sep 18, 2002 at 04:36:25 PM EST

 That will still be a neat experiment. You should be able to find some Cantus firmi (I think that is the plural of Cantus Firmus...) on the net. That would give you a better idea of how the tuning system was meant to work, what all it was built to facilitate. And who knows, even with a minimal knowledge of music, you could still turn out to be our next neo-medieval minimalist composer ;)Sorry... bad music humor... anyway, good luck! ~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~* One must still have chaos in oneself to be able to give birth to a dancing star. -Nietzsche ~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~*~* [ Parent ]
 BLASPHEMY! (5.00 / 1) (#49) by notcarlos on Wed Sep 18, 2002 at 04:37:17 PM EST

 [...] and directed a medieval/ Renaissance choir [...] Keep the Humanists out of the choir! Boo! He will destroy you like an academic ninja.-- Rating on Rate My Professors.com [ Parent ]
 Nitpick nitpick (none / 0) (#56) by x31eq on Thu Sep 19, 2002 at 06:54:42 AM EST

 This phrase is a little confusing. There is currently to my knowledge no evidence that the mathematics of harmonic progression were mapped out or understood in medieval times... Well, it seems to have confused you ;-) The harmonic progression as it applies to Boethius is the numbers 1, 1/2, 1/3, 1/4, 1/5, ... These tell you how to measure a string to imitate "natural" (of a one-dimensional resonator) overtones. In about 1580 Equal temperament was introduced. This system corrected for the loss of notes that occurred in the Pythagorean tuning system, and allowed us to write music that was easily transposed to any compatible key or mode. 1580 may be some theoretical date. Equal temperament (or close enough given the random errors) was in practical use for fretted instruments some time before that. But not used on keyboard instruments until much later (say 1850 depending on how equal you want it). I don't see any notes lost in Pythagorean tuning. The flaw in 16th century terms is that the thirds are tuned poorly. The solution is meantone, which transposes the same as Pythagorean. Equal temperament does allow free transposition within a closed set of notes, but you don't need that for Common Practice harmonic progressions. Anything you can write in staff notation can be played in meantone (or Pythagorean for that matter, although it might not sound right). See Margo Schulter's FAQ if you want more on this. Okay, I doubt anyone else even cares about what I just said, but I feel better. Hah! Well, you're definitely wrong there! [ Parent ]
 Herman Hesse and Rithmomachia (5.00 / 1) (#52) by sapalong on Wed Sep 18, 2002 at 07:31:19 PM EST

 Did anyone ever read the wonderful book 'Magister Ludi' (also known as 'The Glass Bead Game') by Hesse? It is about intellectual monks in Europe playing a game that sounds somewhat similar to this, although encompassing more varied intellectual pursuits than just arithmetic progressions. Hesse described it such that it seemed to be about the highest expression of the intellectual mind. The monastaries where culled for the brightest youngsters to be taken off and instructed in the ways of the game.I wonder if the inspiriation for the book came from Rithmomachia? A good read regardless.
 I think there's no doubt about it (none / 0) (#53) by the on Wed Sep 18, 2002 at 07:54:05 PM EST

 Though a google on "glass bead game" and rithomachia or even glasperlenspiel and rithmomachia turns up nothing. I think a good article could be written on this subject - though not for K5. I'll ask my German friend who read the Arno Borst book and pointed out the existence of the game to me many years ago. -- The Definite Article [ Parent ]
 There was recently a decent writeup of... (none / 0) (#60) by graal on Thu Sep 19, 2002 at 01:35:14 PM EST

 ...Das Glasperlenspiel here. -- For Thou hast commanded, and so it is, that every inordinate affection should be its own punishment. -- St. Augustine (Confessions, i)[ Parent ]
 this is driving me mad (4.00 / 1) (#62) by mikpos on Fri Sep 20, 2002 at 10:23:55 AM EST

 Is it rithomachia or rithmomachia?
 Rithmomachia (none / 0) (#67) by the on Wed Sep 25, 2002 at 09:06:15 PM EST

 My fingers get like that sometimes. They decide they'll consistently misspell a word. They don't even tell my brain about it. Annoying sometimes. About the madness. I hope it gets better. Sorry. -- The Definite Article [ Parent ]
 Rithmomachia: The Game for Medieval Geeks | 67 comments (50 topical, 17 editorial, 0 hidden)
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