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The earliest systematic treatment of the octatonic scale was in Edmond de Polignac's unpublished treatise "Étude sur les successions alternantes de tons et demi-tons (Et sur la gamme dite majeure-mineure)" (Study of the Succession of Alternating Whole Tones and Semitones (and of the so-called Major-Minor Scale)) from c. 1879, [1] which ...
This leads to scales with translational symmetry which include the octatonic scale (also known as the symmetric diminished scale; its mirror image is known as the inverse symmetric diminished scale [citation needed]) and the two-semitone tritone scale:
List of musical scales and modes Name Image Sound Degrees Intervals Integer notation # of pitch classes Lower tetrachord Upper tetrachord Use of key signature usual or unusual ; 15 equal temperament
These scales contain all three notes of a minor triad: the root, a minor third (rather than the major third, as in a major triad or major scale), and a perfect fifth (rather than the diminished fifth, as in a diminished scale or half diminished scale).
Its tonic chord is a diminished triad (B dim = B dim 5 min 3 = B D F, in the Locrian mode using the white-key diatonic scale with starting note B, corresponding to a C major scale starting on its 7th tone). This mode's diminished fifth and the Lydian mode's augmented fourth are the only modes that contain a tritone as a note in their modal scale.
Because of the repetition of the interval pattern after only two notes, each note in the scale can be the root in another symmetric diminished scale. For example, the C diminished scale of the half-step-first type, has the same notes as the half-step-first E ♭ diminished scale as well as the whole-step-first D ♭ diminished scale.
Scales are typically listed from low to high pitch. Most scales are octave-repeating, meaning their pattern of notes is the same in every octave (the Bohlen–Pierce scale is one exception). An octave-repeating scale can be represented as a circular arrangement of pitch classes, ordered by increasing (or decreasing) pitch class.
In major scales, a diminished triad occurs only on the seventh scale degree. For instance, in the key of C, this is a B diminished triad (B, D, F). Since the triad is built on the seventh scale degree, it is also called the leading-tone triad. This chord has a dominant function.