[For the uninitiated, the octatonic scale refers, in fact, to three symmetrical scales, or rather three transpositions of a single symmetrical scale, consisting of eight notes rather than the "normal" seven of the heptatonic scalar universe (major and minor scales, plus the church modes and other scales too numerous to mention). Two versions (if you will) of the octatonic scale consist of an alternating half-step whole-step (that is, semitone-tone or minor second-major second) pattern, the remaining consists of an alternating whole-step half-step (tone-semitone or major second-minor second). Naturally, it cannot operate under the same rules regarding the hierarchy of pitches and the preparation and resolution of dissonant harmonies as one might expect in more "ordinary" scales.]
Each of the three transpositions of the octatonic (or diminished) scale contain four triads that in an ordinary heptatonic (or tonal) context otherwise would function as tonic (I) chords from whence systemic tonal gravity would emerge. Within this octatonic modal universe, all four of these triads exist in both their major (Ionian, Lydian, Mixolydian) and minor (Aeolian, Dorian, Phrygian) manifestations. Hence, when writing of the presence of these triads in the octatonic system, I will refer to each pair of the four as a triad class. For instance, in the alternating whole-step half-step octatonic scale starting and ending on C, four triad classes arise -- those of F major (and minor), A♭ major (and minor), B major (and minor), and D major (and minor). (Regarding the B major triad in this scale, the enharmonic spelling B-E♭-G♭, though an unorthodox spelling of the chord, arises naturally, as does the atypical enharmonic spelling of D major as D-G♭-A.)
Observe that the roots of each triad class, isolated and stacked consecutively, coalesce into a diminished seventh chord. The above example, the alternating whole-step half-step octatonic scale starting and ending on C, provides us with an F diminished seventh -- its constituent pitches being the roots of each of the triad classes emerging from the scale (F, A♭, B, D). This comes as no surprise. It is widely understood that the symmetrically alternating character of the octatonic scale, in all three possible transpositions, yields two interlocking diminished seventh chords. From this phenomenon, the octatonic scale derives its alternative name, the diminished scale. The diminished seventh chord, consisting of three stacked minor thirds (F - A♭, A♭- B, B - D) and, from another vantage point, two overlapping tritones (diminished fifths or augmented fourths; F - B, A♭ - D), is a vagrant chord. Thus, throughout the common practice period, coming to a close roughly around the First World War, consensus had it that composers handle the diminished seventh chord with the utmost discretion.
By its very nature, the diminished seventh chord trespasses the logically home-bound and internally-coherent system of functional tonal harmony underlying all of Western art music from the late Renaissance madrigal to the final twilight of pre-world-war Romanticism. In describing the diminished seventh chord as a vagrant chord, the theorist acknowledges the simple reality that it can resolve onto four different triad classes. The F diminished seventh chord [and its "inverted" diminished sevenths starting on G♯ / A♭, B / C♭, and D] counts among its possible "proper" (that is, common-practice tonal) resolutions the triad classes with roots F♯, A, C, and D♯ / E♭. With its four possible cadential turns, the diminished seventh chord does not carry strong tonal gravity, it does not suggest any single tonic as ordinarily does the dominant seventh chord. Using the diminished seventh chord and employing proper voice-leading principles, one may link together keys most distant from one another on the circle of fifths -- for instance, the F diminished seventh chord might permit "rapid transport" between a C triad and an F♯ triad, the two triads' roots divided by a tritone (augmented fourth or diminished fifth), the most distant intervallic relationship in the tonal hierarchy arising from the overtone series. Hence, the diminished seventh suspends the sense of rootedness in a compositional matrix, conveying a sense of wandering and homelessness. It is popularly a chord frequently associated with stress and anxiety, and surely its "atonal" character contributes to this conception.
Curiously, the diminished seventh outwardly resembles the dominant seventh chord (V7), that basic dissonant harmony most magnetically attracted to the tonic triad (I) and so of the utmost consequence in the tonal system. Where the diminished seventh differs from the dominant seventh chord is in its sharpening of one of the degrees of the latter. Hence, the F diminished seventh chord (F - A♭ - B - D) is but a semitone away from the E dominant seventh (E - G♯ - B - D), G dominant seventh (G - B - D - F), B♭ dominant seventh ( B♭ - D - F - A♭, or the A♯ enharmonic equivalent: A♯ - C𝄪 - E♯ - G♯), and C♯ dominant seventh (C♯ - E♯ - G♯ - B, or the D♭ enharmonic equivalent: D♭ - F - A♭ - C♭). These semitonal alterations that produce four possible dominant seventh triads themselves spell out a diminished seventh chord when, again, one stacks the roots of each. The example above yields an E diminished seventh chord (E - G - A♯ - C♯, or E - G - B♭ - D♭).
Now, let us "derive" the F diminished seventh chord from the C alternating whole-step half-step octatonic scale. The resolutions of each of the possible dominant seventh chords from which the F diminished seventh chord is but a semitonal alteration away yields a series of triads not present in this octatonic transposition. The E dominant seventh resolves to A major or minor, the G dominant seventh resolves to C major or minor, the B♭ / A♯ dominant seventh resolves to E♭ major or minor, the C♯ / D♭ dominant seventh resolves to F♯ major or minor. Consider that the following pitches exist in the C alternating whole-step half-step octatonic scale -- C, D, D♯ / E♭, E♯ / F, F♯ / G♭, G♯ / A♭, A, B / C♭. Although the triadic root pitches A, C, D♯ / E♭, and F♯ / G♭ exist here, the triadic classes A, C, D♯ / E♭, and F♯ / G♭ are absent, supplanted in the scale by major and minor G♯ / A♭, B / C♭, D, and E♯ / F. To "discover" the triadic resolutions of the F diminished seventh's "dominant seventh semitonal alternations" within an octatonic scale, we must discuss another transposition, the C alternating half-step whole-step octatonic scale, consisting of the following pitch class -- C, C♯ / D♭, D♯ / E♭, E, F♯ / G♭, G, A, A♯ / B ♭. From this scale derivation of two interlocking diminished seventh triads -- C diminished seventh (C, D♯ / E♭, F♯ / G♭, A) and C♯ / D♭ diminished seventh (C♯ / D♭, E, G, A♯ / B ♭) -- occurs. And the triads that exist in this octatonic transposition? A, C, D♯ / E♭, and F♯ / G♭.
Remember that there is no F diminished seventh chord in the C alternating half-step whole-step octatonic scale as there is in the C alternating whole-step half-step octatonic scale. The former transposition boasts two interlocking diminished seventh chords as the basis for its pitch class (F diminished seventh chord and F♯ diminished seventh chord), the latter boasts two (C diminished seventh chord and C♯ diminished seventh chord). Shared between these two transpositions is the C or F♯ diminished seventh chord (regardless of whether said chord begins on C or F♯, it constitutes an identical pitch class). Yet it is arguably the "other" diminished seventh chord, the one in which they differ, that determines their inherent modal quality. There is much to be explored here in the future, insofar as it may guide compositional procedures in the octatonic tonal universe towards a greater sense of structural unity and coherence and, therefore, of more lasting expressive power.