What is the name of this scale? As far as a root note goes, I think these three arrangements of it sound like they have a root.
- E, F, G, G#, B, C#, D...
- D, E, F, G, G#, B, C#...
- B, C#, D, E, F, G, G#...
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According to Ianring.com, and depending on the root:
This scale is named after Dutch composer Willem Jeths [1959–; composer's website]. It is essentially a Diminished Scale omitting the augmented 5th. Strangely, there is no documentation of why this scale was named for Willem, but at some point it wandered onto a list of scales and has been canonized by repetition. (ianring.com)
All of these scales are related to what ianring.com calls Mela Ramapriya (ianring.com, Wikipedia), but which is also known as the Petrushka (Stravinsky) scale (for reasons unclear, but presumably related to the Petrushka chord [Wikipedia], which is a subset), and the Hungarian Major scale (Wikipedia).
You have here 7 out of 8 notes of the octatonic scale. You can create this scale by alternating whole- and half-steps. Related to @PiedPiper's answer, the octatonic scale is also the combination of all the notes in any two diminsihed-seventh chords.
For the present question, take the Edim7 and Fdim7 chords...
E G Bb Db
F Ab B D
E - F - G - Ab - Bb - B - Db - D
You are missing only the Bb.
Also, because there are only seven letter names for the notes you will have to repeat one letter.