The collection of notes in a single scale can have multiple modes with different names, where each note in the scale is the root of a different mode. (See Diatonic Modes & Chords for a detailed explanation.) For example C Ionian (Major) and C Mixolydian have the same root and share six notes but have different parent scales and differ by one note (F v. F#). C Ionian (Major) and G Mixolydian share the same seven notes (same parent scale) but a different root (C v. G).
For chromatic-cluster-free scales that span the octave and are made entirely of semitone and whole-tone (two-semitone) steps, there are only four possible scale types: the hexatonic (six-note) whole-tone scale, the heptatonic (seven-note) diatonic and acoustic scales, and the octatonic (eight-note) diminished scale. These scales, coupled with the chromatic scale, are the foundation of jazz theory and the focus of Harmonious.
Important additional scales are the harmonic major and harmonic minor scales with their seven modes apiece; the hexatonic double augmented scale (with two modes); and the heptatonic diminished scales, subsets of the octatonic diminished scale. Together with the whole-tone, diatonic, acoustic, and diminished (octatonic) these scales are the basis of every possible chromatic-cluster-free chord or scale (Tymoczko 2011 chapter 4).
Other notable scales include: the diatonic complement, the bluesy pentatonic scale or “black keys;” the acoustic scale complement, the dominant ninth chord played as a scale; modes of the melodic minor ♭2 scale and modes of the Hungarian minor scale; and blues and bebop scales, formed from other cluster-free scales and modes by adding an extra grace note or passing tone (for rhythmic purposes, or to add chromatic flair).
Outside of the western tradition are many scales that can be played on 12-TET instruments: the Hindustani thaats and Carnatic ragas, many of which are represented among the scales listed above. Also, Arabic or Middle-Eastern traditions have some scales that can be played in 12-TET, including many maqam but also the Arab tone system known as gadwal uses 24-TET, or quarter tones (half of a semitone, or fifty cents).