If it's on topic, I'm also interested to know how they compare against modern steel.
Toledo steel was a very good steel, comparable to mainstream contemporary ones. It is based mostly on the content of the material and way of hardening. Now the best European steel for blades is not Spanish, but Swedish V10.
With Damascus there is a wide-spread fallacy. What is now called "true damascus" - blades based on the way of smithing of two or more steels - folding, beating, folding again,.. 32 or 256 or even thousands of layers... It has very nice looking, and can be very good (depending on the source steels and regime of hardening), but it is NOT the true damascus.
As for the last, now it is called wootz. The inner structure of wootz was created by not mechanical process, but by special regimes of hardening, when crystals of different steels are growing inside the ingot and filling it with the inner structure. Of course, composition of the steel is extremely important, too. The last can be repeated, but the secret of hardening is lost. Now you can buy only OLD blades or ingots of wootz made of OLD blades. What is the sense of the last, I don't know. But they are not expensive, in contrary to the old blades.
There existed close variants - reinventions in other countries, Russian bulat was one of them, according to legends. The Damascus steel could be extremely sharp and simultaneously, very strong - you could cut a silk scarf by letting it to fall on the blade and you could use the same blade as a belt and you could simply cut a usual sword in two.
As for samurai weapon, it is a widespread mistake, that it had good steel. On the contrary, the steel was brittle, in Japanese fencing you should never block the blow, but let it slip along the blade or to block it by back/thick side . Though this steel could be sharpen also to the extreme, and often was, and the trick with a scarf could be done with it, but it was weak and easily broken or at least dented by any direct blow. The Japanese did the most possible out of what they had, but the resources on the Japan islands are poor.
On the other side, now, while Japan masters can reach the all Mendeleev table from all the World, the best steel is made by them - blue and white steel (non-stainless ones)and the star of all - ZDP-189, with hardness 63 (other blades reach 60 at best) and stainless, too, with unknown receipt.
Japanese now make saws with hardness of 73!, which can be sharpened by diamond only, but this steel is not plastic enough to be a cold weapon steel.
An interesting professional article on the theme, much more thorough than my answer: http://www.tf.uni-kiel.de/matwis/amat/def_en/articles/vikingsword/blade_patterns_intrinsic.html
The problem is that now all these names are mixed - you can buy a "japanese damascene" for 100 dollars. Not real, of course. http://www.swordsoftheeast.com/damascus_3.aspx.
The secret of toledo swords is revealed now: http://aceros-de-hispania.com/toledo-swords.htm
An interesting personal research with links: http://tf.uni-kiel.de/matwis/amat/def_en/index.html
As it is, all three are interesting for being completely different methods of achieving a high quality of steel. Equally interesting is that they are each of high quality in different ways.
As for Tamahagane, the iron that was available in Japan was actually very poor compared to that found in Europe. It had a characteristically low carbon content, and the only way they had to increase the carbon content was to fold it into the steel during forging with a process called pattern welding. This folding did make the steel harder, and thus better able to hold an edge and less likely to bend, but it also meant that it was more brittle and more likely to chip and crack as well.
Damascus steel is something of an oddity when discussing forging technology for two reasons. Firstly, historians don't know if the descriptor refers to the smiths working the steel, where it was sold, or simply a visual similarity to textiles from the Byzantine empire. Secondly, the exact methods originally used to forge the steel in question have been lost to time. Researchers have found the various trace elements within the metal, but they don't know precisely how they were introduced. At its core, however, Damascus steel is basically another type of pattern weld. The problems this introduces are the same that are introduced with modern welding techniques, in that each weld represents a point where two pieces of metal are joined and are thus weaker than a solid piece of the same material would be.
Toledo steel, however, is separated from the other two in that it is a true alloy. Toledo steel is created by permanently mixing the iron and the other chemical additives through smelting. The difference is visibly evident by the observing the consistent appearance of Toledo steel versus the layered patterns of both Tamahagane and Damascus steel. Of course, like the other types of steel, Toledo had its drawbacks. Because of the extremely tight tolerances of the heat used during the smelting, forging, and tempering processes, the steel took an incredibly long time to create. This lead to the widespread use of daggers and shortswords, as they required less material and thus less time to create than a full-sized sword.
As for determining which of them was the best, that is completely subjective, depending on the desired characteristics, much like deciding which is better between a wrench and a hammer. A wrench is good for turning bolts, but a hammer is good for driving nails. Similarly, it can't really be said which method of producing steel is better without defining the criteria. Tamahagane could be honed to an incredibly sharp edge, but was too brittle to be used for blocking. This was fine, as most combat with katanas and similar weapons was against combatants in leather or lacquered bamboo armor. I'm not sure of the hardness characteristics of Damascus steel or the type of armor that it would have been used against, but due to the similarities it has with tamahagane, I imagine it faced similar challenges as well. Toledo steel, being produced in western Europe, would have often been used against more heavily armored opponents or against shields. For this reason, it would have been less important that the steel be able to hold a perfect edge than the tendency not to break when impacting something it couldn't cut straight through.
So, to answer the original question of a comparison between the three kinds of steel, I doubt that you would be able to find anything that addressed it without doing so in the context of making blades, as steel that was used for other purposes was typically of a different formulation. And, to answer the second question, contemporary formulas have been invented which surpass all three in just about any category you could think of. After all, there have been thousands of years of technological advances since the invention of these methods, depending on which you are referring to.
A portion of one of the ancient Damascus steel blades was removed and examined, the molecular structure was made up of a series of carbon nanotubes around iron nanowires. It's unknown as to how this was achieved, and hasn't been replicated by modern humans. It's sometimes referred to as Wootz steel. Ref
Modern attempts at replicating Damascus steel involves stacking alternating sheets of high carbon and medium carbon into a bar, heating the bar and twisting the bar over, before folding it back on itself and hammering flat again. This could be repeated several times, less than 10 I would presume. Too many folds will begin to undo the work. like a croissant
As far as I know, the Japanese folded steel is exaggerated. the "pig iron" that was available would need to be strengthened by hammering to remove slag impurities, and thus would become folded over itself several times, but it was still only "wrought iron" at that point, and was still inferior to "crucible steel" which was only discovered in the 17 century by the Europeans, but had been used in Asia much earlier. From the 8th century, the vikings may have learned the crucible steel techniques from the middle east, but 'forgot' them by some time around the 11th century. Look up the "Ulfberht" swords.
Something I learned recently was that the Katana's strength came from it's use of wrought iron as a body for the sword, with an edge made of high carbon steel to allow it to have strength while cutting, but maintain flexibility and avoid snapping.
So long story short, ancient Damascus steel exceeds modern steel hypothetically.
Japanese steel was not particularly advanced, so unfortunately it would be very much outperformed by modern steel. The superior engineering of the sword structure was the Katana's great strength.
And I do not know about Toledo steel, so I'll have to do some research
A couple of the claims written in the answers above need to be discussed with counter points;
The ancient Japanese samurai sword that we are talking about was specifically developed to address the brittle nature of steel that time, no it was not brittle by yesterday's standard or today's.
Precisely because of the brittle issue in steel, the katana was developed using a technique consisting of three types of metal, the center was soft, that was sandwiched by harder folded steel
The folding does more then simply homogenize the carbon content or knock out the carbon, in fact carbon was added between layers as the smith thought needed, they had an experienced feel for too much or too little carbon during their forging process
Folding does more then remove impurities, the folds seem to add a level of strength and pliability, or "toughness", a grain is produced that could also be manipulated to add differentiating strengths to different areas in the blade, like alternating the grain of hardwood. I am not sure when differentiating the grain is added to the process, or how many and which smiths actually did it.
After that folding and sandwiching process, the folding then is not the same as done today, there were layers of other products added between folds with indigenous ingredians, this "recipe" is probably lost, though some smiths in japan claim to still have it carried down.
The edge was case hardened further in a process known as "clay tempering", different types of clay insulating the blade to different degrees to produce different levels of hardness in the final forging process.
The edge of the blade had to be hard to maintain that sharpness, combining the different types of metal in the body and edge is now called "lamination", there were different methods for combining and welding the three types of metal, I some far more elaborate then others, an authentic katana is NOT simply folded steel, it's folded AND laminated, AND clay tempered.
That painstaking procedure was specifically designed so the entire sword would not be brittle as claimed in above answers, the lore reports, in the first battle using this "new method" of lamination, not one sword returned broken, it was not only the folded steel that produced the superior toughness, the "recipe, the nano tubes, the lamination. Most reproductions today do not use a multi steel lamination, though "some" of the more expensive reproductions might.
Additionally, metallurgists have performed forensics and discovered the original katana did indeed enjoy "nanotubes, nanowires" and "impurities" like tungsten and vanadium, "impurities" that made the steel both hard and flexible, stronger then one might get using unfolded unlamenated modern carbon steel. (I believe swedish powdered steel has tungsten and vanadium added in their forging process.)
"One is the true Damascus steel, or Wootz steel, which is a high carbon alloy with tremendous edge retention possibly due to its composition of carbon nanotubes and carbide nanowires, with a wavy surface texture originating from the crystalline structure of alloy metals such as tungsten and vanadium - elements that occur naturally in iron ore from southern India - to the surface during the manufacturing process"
The curve in the katana isn't cut or bent by the swordsmith, it was a happy consequence of the "new" forging process, the bend that occurs when the different steels are cooled in the clay tempering process, though the experienced smith could and would influence the nature of the curve.
I have no idea how an authentic ancient katana would compare to a sword made from Swedish powdered steel, I don't think anyone who possesses an ancient Katana would be willing to donate it for destruction comparison.
However tests have been conducted comparing the long sword of the era vs the katana, in the test I am posting, the katana is a "modern" made katana that does NOT use lamination.
Nor does the metal in this youtube have the nano tubes and nano wires that added more toughness and durability to the steel, in other words, it's not likely the samurai sword in the demo is as good a weapon as the ancient katana, but the same could be true about the broadsword
The katana wins every test in this modern comparison, it wins as a slashing weapon, as a penetrating weapon, as a weapon against slicing leather armor, as a weapon slicing metal armor, as a weapon piercing straight through armor.
the "myth" of the katana is surely romanticized and exaggerated, yet those playing down the quality of the sword have under estimated the superior metalurgic qualities the katana enjoyed.
In any event, though current technology with swedish powdered steel might produce a blade with over all better qualities then produced for the samurai, the skill making the weapon and tool was and is wonderful
This question is, I think, impossible to answer definitively because steel coming from a particular area will vary greatly in quality and be largely dependent on the care taken in making it, not the place or culture of its origin. In fact, the same steelmaker can turn out relatively low-quality or high-quality steel depending on how much effort he puts into it.
Quality of Steel
Defining the quality of steel is difficult to do because depending on the use of the steel and the way it is forged, very different characteristics are desirable. There are a few common denominators, though. Having any phosphorus, sulfur or mercury is always very bad. Another big factor is the consistency of the steel. Consistency can always be improved by working the bloom longer, so it will vary greatly depending on how much work the steelmaker does. For one customer they may toil long to make the steel good, but another just gets a fast billet.
One example of "high-quality" steel is that coming from certain Swedish ores. The advantage of these ores is that they are low-phosphorus and have vanadium in them. Vanadium tends to bind to any free nitrogen or oxygen present and scour it. Presence of either of these gases weakens the steel.
Toledo and Damascus
The first thing to note is that Toledo and Damascus are mainly noted for their methods of making blades, not steel, and both were similar to each other. The quality of a sword is determined by complex aspects of forging and tempering, not so much as the quality of the steel. Toledo followed the traditions of Damascus and used the same basic methods of using welded fillets to compose the blade. Both places are know for using complex methods of tempering which were secret.
The steel used in both places could have come from a wide range of sources, but would of been of variable quality.
The quality of old Japanese weapons is a perfect example of how even very primitive methods can yield super high quality steel as long as you put the work into it. A well-made historical Japanese sword can easily cut any comparable European or Arab sword in half, but this is because of the temper and forging, not because of the quality of the steel.
Comparison to Modern Steel
Overall, modern steel is much higher quality than ancient steel because it is more consistent and uses alloying elements like chromium, vanadium and manganese scientifically, whereas in olden days the presence of those metals was more accidental. Note, though, that common modern steel is not necessarily better. Often ordinary steel is made from scrap containing all kinds of contaminants and the mill just does the minimum to meet spec. This results in a sort of mediocre average quality. Ancient steel, pure and carefully made, can exceed such steel in quality. However, when a modern steelmaker takes care and only uses pure materials and careful processes the result will be far better than ancient steel.
Some informative answers particularly the respondent who wrote about the making of historical katanas which accurately described the process as lamination not pattern welding. Whoever wrote that doesn't have a clue about Japanese swordsmithing. The remarkable thing about those ancient swords was that despite having poor quality ore to work with they developed a technique that more than made up for that. Can any other sword in the world cut through five bodies in one stroke? There are swords in Japanese museum have been tested as being able to do that (as gruesome as that is to imagine that is a lot of meat and bone to cut through) if the sword was not both extremely sharp and resistant to breaking it would not be able to do that. I have no doubt the same sword could cut a Toledo steel sword but I doubt a Toledo steel sword could cut a katana. The laminated design of the katana made it resistant to breakage and very sharp. The worst that would happen in blocking to a well made katana is a chipped edge.
I don't know if it will help clarify, but using some of my own-read knowledge, it would be the safest to say that the Japanese sword (to be exact, during the mainstream Sengoku period) were made with processes that DID involve pattern welding or lamination (there may be cases of both for all we know). Do note that these mostly only held true for actual Samurai who could afford them and not the average footsoldier or Ashigaru who mostly used swords made from cheaper steel/construction methologies, using materials like homogenous steel.
This, however, is dependent on each specific sword (saying all swords made and used by soldiers within the time period of one year were all made with one specific process would be extreme). Swordsmiths or at least the most popular ones had their own "School/Style" which uses each their own unique recipes or methods in forging the metal into a blade and turning it into a weapon.
Now, the TS did ask about steel quality and not sword quality. I cannot vouch for any of the three, but it would be better to just compare the basic toughness of the three steels' hardest raw products. To clarify, for example, tamahagane is one of a few products of a Japanese method of smelting with the tatara.
The tamagahane itself isn't the only steel composing the hardest parts of a katana's blade thus resulting to its famed cutting edge. But it certainly is a component, combined with the nabe-gane (higher carbon steel) which at this point DOES involve the infamous folding process to make the kawa-gane which will be the hard cutting edge. As far as I know there are sayings that 16-18 times of folding made the best quality of hard steel for the cutting edge (16 is stated in the wikipedia page). So we can basically say that tamahagane itself isn't the final steel to compare but would be the kawa-gane as it is the main edge of the katana.
As for the katana's curve and its composition, the curve is attributed to it (the entire blade/product) being made from multiple steel/metals that have different hardness, ductility, and malleability, plus the clay tempering or other tempering methods used. There are traditionally-forged Japanese blades which have warped at a bad angle because of mistakes in the cooling process or in the final metals used to form the finished blade.
Now some people might be under the illusion that katana or the tamahagane is really great (or popularized in both fantasy and reality) because of its mere hard properties. The truth is actually not in that one steel made from tamahagane, but in the construction of the blade itself.
I will put links on some youtube videos so people can see the actual tests already made before (not mine ofc). What actually makes a katana made from tamahagane endure breakage as a cutting weapon is not merely the cutting edge's hardness, but the ability of the hard and soft metal parts to absorb/abate/adapt to impact and "vibration" when cutting stuff.
You can see in this video how cutting through meat and bone with a katana is smooth because the metals in the blade don't really have to absorb much impact/vibration: time warp katana: www.youtube.com/watch?v=iH_oLEllyvg
Here on these two videos, a single (the same iirc) katana made by the same traditional swordswith (the white-haired old Japanese guy on the video) demonstrates the quality of the steels I'm talking about. On the one with the .45 magnum pistol, the katana visibly "vibrates" upon absorbing the bullet's impact while letting it cut the projectile in two. This is thanks to the softer metal normally on the core and back of the blade.
Meanwhile, when the katana was put under the continuous stress of a stream of .50BMG machinegun bullets, where the hard metals supported by the more ductile soft metals couldn't absorb/vibrate enough to cope under impact (you can see on the slow-motion review how the blade remained still and instead chipped/broke), that there is a limit to the kawa-gane's hardness despite its extreme sharpness and high quality toughness. katana vs .45magnum: youtube /watch?v=OBFlYwluqMk the same katana vs .50BMG machinegun rapidfire: youtube /watch?v=YPQ1W3qobys
Do note though that the kawa-gane withstood at least four to six shots before the core/center of the blade with the softer steel was hit by the seventh shot which caused the sword to instantly snap while vibrating side-to-side because of the impact.
To end this post of mine, here's a video of a modern forged Japanese katana (the second in line from the top of all 3 swords), compared to a real sword of the samurai period (the really old-looking one at the bottom). Note the modern one forged for Iaito practice not showing the signs of multiple kinds of steel laminated/pattern welded together, but does have a real tempered hamon.
The real sword at the bottom has the distinct "line" in the middle of the blade which gives the edge made of kawa-gane distinction (sadly the edge was worn-out enough that the hamon is barely visible). That line is probably where the kawa-gane made of tamahagane and nabe-gane join the rest of the other soft-steels.
Japanese sword comparison: youtube /watch?v=I5zecWQQ2zU
For more information regarding the forging of tamahagane or the processes involved in such methods, you can mainly read these wiki pages which have pretty much the same content as with other sources that only vary in the smith's/writer's own methodologies/preferences.
In conclusion, blades made from damascus or toledo steel would probably yield the same results if the basic idea of a tough and sharp steel cutting edge supported by a softer steel which absorbs impact/vibration. The only difference would be in the hardness of the raw material when put under the same strain (as with the machine gun test). Obviously, hardness on the scale will be the determining factor of whether or not the edge will chip in that test. Pardon my "partial" links as this is my 1st post.
en.wikipedia.org/wiki/Tamahagane /wiki/Katana /wiki/Japanese_swordsmithing /wiki/Laminated_steel_blade /wiki/Pattern_welding