chords rising out of the fifth degree, the dominant ninth chord is not a real chord formation. Jean-Philippe Rameau explained the diminished seventh chord as a dominant seventh chord whose supposed fundamental bass is borrowed from the sixth degree in minor, raised a semitone producing a stack of minor thirds. Thus, in C, the dominant seventh is G7 (G–B–D–F) and the sixth degree borrowed from the minor scale produces A–B–D–F. In his Treatise on Harmony, he observed that three minor thirds and an augmented second make up a chord where the augmented second is such that "the ear is not offended" by it. He may have been talking of the augmented second in quarter-comma meantone, a tuning he favored, which is close to the just septimal minor third of 7:6. Function Most common functions The most common form of the diminished seventh chord is that rooted on the leading tone – for example, in the key of C, the chord (B–D–F–A) – so its other constituents are the , , and (flat submediant) scale degrees. These notes occur naturally in the harmonic minor scale. But this chord also appears in major keys, especially after the time of J.S. Bach, where it is borrowed from the parallel minor. The chord possesses a dominant function when rooted on the leading tone (otherwise it doesn't, but can serve other functions - see below), and this is most straightforwardly shown when the root of a dominant seventh chord is omitted. The remaining third, fifth and seventh of that chord form a diminished triad (whose new root is the third of the former chord), to which a diminished seventh can be added. Thus, in C (major or minor), a dominant seventh chord consisting of G–B–D–F can be replaced by a diminished seventh chord B–D–F–A. In jazz harmony, a combination of the original chord with its substitute (with G in the bass and A simultaneously in an upper voice) yields the seven flat-9 chord, which intensifies the dominant function of either a diminished seventh or dominant seventh chord. Other transformations of this kind facilitate a variety of substitutions and modulations: any of the four notes in a diminished seventh chord are raised by a semitone, that raised note is then the flat-seventh of a half-diminished seventh chord. Similarly, if any of the four notes in the diminished seventh chord are lowered by a semitone, that lowered