I'm reading about some jazz stuff, and get told there are ony two whole tone scales, and three diminished scales. The whole tones are pretty obvious - once you get past the second semitone, you're into a mode of the original one. But - I thought there were the whole/half and the half/whole diminished scales, so what's the third one? Unless the writer conflated scales and arpeggios, whereby the notes of say, Co, C♯o and Do are all different, but when we arrive at E♭o, that's a mode of the first, Co?
There are three unique whole-half (or half-whole) diminished scales starting a half-step apart from each other. Each one has an equivalent half-whole (or whole-half) diminished scale starting a whole step (or half step) above it.
Equivalences between whole-half and half-whole diminished scales Whole-Half scale | Half-Whole scale starting pitch | starting pitch ------------------------------------- C, Eb, Gb, or A | D, F, Ab, or B C#, E, G, or Bb | C, Eb, Gb, or A D, F, Ab, or B | C#, E, G, or Bb
only two whole tone scales, and three diminished scales.
They just mean that when you account from transposing the scales there are only 2 unique sets of tones for the whole tone scale, and 3 for a diminished scale.
But they are overlooking there are two diminished (or octatonic) scales.
In terms of interval structure there is one whole tones scale - all whole steps, and two octatonic scales - one alternating whole and half steps, the other alternating half and whole steps.
When you transpose a whole tone scale by a whole tone, you get the same set of tones.
When you transpose a diminished/octatonic scale by a minor third, you get the same set of tones.
If you take the rotations, or modes, of the scales instead of transpose them, something funny happens. All rotations of a whole tone scale produce the same set of tones just like transposing by a whole step. But, with the diminished scale, every other rotation produced the same set of tones or the "opposite" diminished scale, for example take second rotation of a W/H diminished scale and you get a H/W diminished scale.
If, for some reason, you treat the first and second rotation of a diminished scale to be the same thing, then there are only 3 diminished scales.
If you treat octatonic scales as just embellishments of a diminished seventh chord, then I guess it sort of makes sense to treat W/H and H/W as the same.
If you realize that W/H octatonic has a perfect fourth above the tonic, and H/W has diminished sixth - enharmonically equivalent to a perfect fifth - that's a significant structural difference, and W/H and H/W are different scales. With H/W you can construct major, minor, and diminished triads above the tonic.
Coincidentally, I recently was looking at my scale syllabus from a certain Jamey, for a dominant seventh flat nine chord. He gives the diminished scale, but only the H/W version. That makes sense, because it provides the flat nine. But, that also underscores the two diminished scales are not necessary to be treated as the same.
As far as I’m concerned there are 12 whole tone scales, 12 W/H diminished scales and 12 H/W diminished scales. The two diminished scales I mentioned are separate entities. I’ve never agreed with the “there are only 2 whole tone scales” etc. explanation. Of course several of the scales share notes but I approach them as individual entities. Just as I don’t think of E Phrygian as a C scale from E to E I don’t think of a E whole tone scale as a C whole tone scale from E to E. The tonal center of a scale should have some importance, even if it shares notes with 4 or 6 other scales.
Based on his logic of “only 2 whole tone scales”, the writer is definitely thinking of a single diminished scale in 3 chromatic positions, C, C#, D and saying everything else just comes from there.