While translating this chart, I stumbled upon this overview of the Kushikanzashi museum dedicated to hair accessories. It reminds that before Edo era, women used to wear their hair down/loosely tied (垂髪 subeshigami styles = hair tied behind and hanging down/long flowing hair).
When nihongami appeared (=the trademark Japanese bun we think of when thinking about “ancient Japan”), it was first considered casual style, so a woman would in fact nor wear an updo for formal events.
When nighongami spread, demand for combs increased, leading to a flourishing craft.
Combs also varied widely during the whole period, following the fashion and styles, from big+wide during peak Edo (which loved big hairstyles) to dainty+narrow in Meiji era (where hairstyles were much more muted).
As for the combs types depicted by Rin, you can see from left to right, top to bottom:
First column:
月形 Tsuki gata (crescent shaped) - small comb like this one became popular around 1825 after a sumptuary law banned more extravagant combs. This one was made famous by actor Iwai Hanshiro for the character Mikazuki Osen.
半月形 Hantsuki gata (half crescent shaped) - a small comb like tsuki gata, with a somehow thin and even spine and long teeth, popular during Tenpô era.
閑清形 Kansei gata (kansei shaped) - a small comb with a soft rectangular shape, popular in Tenpô era.
新牡丹形 Shinbatan gata (”young peony” shaped) - small comb with a small spine and rounded corners, fashionable after Tenpô era.
柳川櫛 Yanagawa kushi (Yanagawa comb) - a small comb made from ivory dyed red, then carved to revealed white patterns. Such combs were popular during Kôka era.
Second column:
銀覆輪櫛 Gin fukurin gushi (silver ornemental border comb) - comb with a silver framed ridge, popular during MeiwaandAn’ei eras. It is said to have been invented by famous jack of all trades Hiraga Gennai (and hence also named Gennai comb 源内櫛)
光輪形 Kourin gata (Kôrin shaped) - an ornated comb with openwork designs popular during MeiwaandAn’ei eras. The patterns were inspired by works of famous painter Ogata Kôrin hence the name
御巻用の櫛 Kushimakiyou no kushi (hair roll comb) - a long comb, big enough to wrap the entire hair around. In Hôreki era, A kyûji onna (teahouse waitress) named Minatoya Oroku who waited a teahouse in the precincts of Sensoji Temple is said to have invented this style who soon spread among the population.
硝子製の櫛 Garasusei no kushi (glass comb) - appeared in first half of 18th century. Glass was quite rare and precious then as it was a foreign good (as shown by the names used: ビードロ/bidoro from Portugese vidro or ギヤマン/giyaman from Dutch diamant). Such combs were very fragile.
角形 Kaku gata (angle shaped) - a comb with a straight rectangular shape, popular in Tenpô era.
Third column:
天丸形 Tenmaru gata (“all round” shaped) - a straight forward name for this comb which has a very large and round “yama“ (spine part) which leave plenty of room for decorations (painted, inlaid etc)
糸鬢丸形 Itobin marugata (itobin round shaped) - “Itobin” (lit. thread like sidelocks) was a men hairstyle from early Edo era (head mostly shaved except for thread like locks on temples and nape). OP wonders if the name of this comb originated from this peculiar look? It’s true the shaft kinda looks like a balding head with the tiniest bit of hair left ;)
山高形 Yamataka gata (tall mountain shaped) - a style fashionable during Kyôhô-Genbuneras. The name is pretty straight forward as it has a pretty high (=taka) spine (=yama”), which made a nice canvas for decorations.
利休形 Rikyuugata (Rikyuu shaped) - popular during Kan’en and Hôreki eras, this comb looks like a longer yamataka comb. OP doesn’t know why it was named after the Rikyû school of tea ceremony.
鎌倉形 Kamakura gata / 政子形 Masako gata (Kamakura shaped / Masako shaped) - a comb popular during MeiwaandAn’ei eras, named after the “nun shogun” Hojo Masako (wife of Minamoto no Yoritomo, the first shogun of Kamakura shogunate). Such combs were not only ornmental but also used to untangled and straighten the hair - it the go to shape when one’s think about Japanese combs.
