Definition of Iron, Steel, and Cast Iron as a basic introduction
Iron. Chemical symbol "Fe".
Iron is a general word used to describe metals that have pure iron as their main constituent. Sometimes the word iron is used as simile of something hard like an "iron lady".
Most iron wares around us are not made of chemically pure iron but are alloys, the most important of which is Carbon. Carbon is a big factor in understanding the difference between Iron, Steel and Cast iron. Adding some carbon to chemically pure iron makes steel. Add even more and you'll make cast iron

Technically, iron means just that, chemically pure iron. Without carbon, iron is very soft and ductile.
Iron become softer upon heating. So a smith can change its shape by hammering but it can never be hardened by heat treatment.
Steel is an alloy of iron and carbon. The amount of carbon dictates whether a steel is hard or it is tough.
Adding Carbon makes the iron harder. The more carbon the harder the steel. Carbon content in steel  usually falls a range between 0.3 ~ 1.5 %  by volume.
Iron /carbon alloys within this range are called steels.
Steels can be forged and hardened by heat treatment. The high carbon steel is harder than the low carbon steel. Because carbon content is critical to hardening, the effect of heat treatment is big on high carbon steel, and small on low carbon steels.
In other words, high carbon steels are more sensitive to tempering work. High carbon steels undergo  structural changes when heated and cooled rapidly making them useful for items that require degrees of hardness.
High carbon steels appear some different phases that come from the difference of the situation in tempering work. The result of relation between sensitive steel and good tempering work makes subtle appearances on blade surface.

Cast Iron
Carbon content over 1.5 % make iron alloys brittle, non-ductile and unable be worked by hammering. Also these alloys can't be hardened by heat treatment.
Such metals are easy to melt, and easy to break by hammering. Therefore they are used only for casting work.
Therefore we can see that iron, steel, and cast iron form a family based on the absence or degree of carbon content.
Although iron is not used for blades it can be used for tsuba and other fittings like fuchi/kashira, kuirikata, and kojiri etc.
Low carbon steel is used for various kinds of tools and arms.
Most good tsubas are made of  low carbon steel. Some "dubious" tsuba are made of cast iron. (=> Tsuba of cast iron)
High carbon steel is used for cutting edges of blades and tools.
Usually a blade is made of combination with some kinds of steel. (=> construction)
Sword smiths can use either iron or cast iron as ingredients to produce their own steels. They control the carbon content of the steel using their forge. We call this home-made steel "Oroshi-gane".

Purity of materials for making blades and tsuba
In the modern age there are numerous kinds of alloy steels. Other metals are added to the iron to produce special properties for example stainless steel. We call these steels "alloy steels" to differentiate them from "pure carbon steels". (Although this term is strange in exact meaning of the word, I will use it. Please see note at bottom.)
Even commercial Carbon steels have trace amounts of other metals, that is why we don't call them "pure" carbon steels. Japanese swordsmiths never use these modern steels for blades or tsuba and a Japanese blade can not be made with these metals.
TAMA-HAGANE is the main material used in making a Japanese blade. It is a primitive and pure steel made using a traditional Japanese furnace, the TATARA.
The materials for OROSHI-GANE must also be pure of course.
The next step, fold welding, can remove slags and bubbles, but can't remove metal contamination.
Therefore the materials must be as pure a steel as possible.
A Japanese blade must be made from this "pure" steel. Pure steel, simply iron and carbon is a reflection of the beauty of nature.
The value of the Japanese blade lies in its purity. (=> Material of Japanese blades)
Some other terms about iron/steel
pure iron =  the most pure iron we can get by the top technology today. It can be one of the materials for oroshigane.
sponge iron = one form of pure iron. It looks like a sponge because it has many bubbles inside of it.
electrolytic iron = another kind of pure iron. It is made by electrolysis from iron salts.
TETSU = iron
HAGANE = carbon steel
ZUKU = cast iron
IMONO = cast iron ware
TAMA-HAGANE = a pure steel made by TATARA method.
OROSHI-GANE = a pure steel that is made by sword smith themselves.

I have received some advise from a Western smith who is knowledgeable in this field.
It is a general word used to describe metals that have pure iron as their main constituent. In metallurgy we call these metals ferrous alloys.
Cast iron
(Carbon content over 1.5 % make iron alloys brittle.)
Not strictly true. 1.5 -2.5 % C is white cast and is extremely hard being FeC.
You need to get out to 3.2%C + to get into grey casts iron which aren't hard
Purity of Tamahagane
(In the modern age there are numerous kinds of alloy steels.)
These are steels in the exact meaning of the word. Carbon steels are just the most basic type.
(Pure carbon steel)
I've tried to reflect your ideas on the 'purity' of tamahagane although technically I have problems with this word. All ferrous alloys are 'Steel' in the western sense and are therefore alloys of iron and carbon. The normal usage here would be 'simple' carbon steels but I don't think the connotation is what you want.

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