Tirumalai Krishnamacharya (1888-1989) is frequently referred to as the Grandfather of modern Hatha yoga.

It would appear that Krishnamacharyaya's "style" was "no style" in the sense that it predated the many different styles of Hatha yoga that would evolve later.

Four students of Krishanmarcharya would later become well known for how they later went on to teach asana:
- Krishnamacharya's son, T.V.K. Desikachar, whose style now usually referred to as Viniyoga;
- B.K.S. Iyengar, simple known as Iyengar style;
- K. Pattabhi Jois, Astanga Vinyasa;
- Indra Devi, Indra Devi Foundation.

In addition to above four, as am sure you are aware, there are now perhaps two dozen or more distinguishable Hatha yoga styles.
I cannot pretend to know which is most basic or authentic.

I have read some statements that Jois's Astanga Vinyasa style is the same method that was taught to him by Krishnamacharya.
Then again, perhaps Devi's style is authentic, since it seems that what she taught never developed a distinguishing "style name".

Following link has brief descriptions of many contemporary styles.
http://www.traditionalyogastudies.com/articles_styles.html

The biography of Krishnamacharya at wikipedia ends with this statement :

"Whether you practice the dynamic series of Pattabhi Jois, the refined alignments of B.K.S. Iyengar, the classical postures of Indra Devi, or the customized vinyasa of Viniyoga, your practice stems from one source: a five-foot, two-inch Brahmin born more than one hundred years ago in a small South Indian village." http://en.wikipedia.org/wiki/Krishnamacharya

Namaste',
dwb