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Osaka Maiko Yachiyo II, 1915
Myōseki (inherited names) are often used when naming a new Maiko (Apprentice Geisha) as a mark of respect for a former Geiko (Geisha) and as a means of carrying the cachet of celebrity down through generations.

The geimei (professional name) of a meigi (famous geisha) who has retired from the profession is given to a promising young maiko, but with the suffix ‘the second’ added to it. This can carry on ad infinitum i.e. ‘the third, the fourth, etc.’ with successive generations. This hereditary naming system applies to almost all artisan professions in Japan, including Tayuu (Courtesans) and Kabuki Artists. source: Blue Ruin