A decatonic scale is a ten note musical scale. If the notes are ordered, a decatonic set has 3,628,800 permutations,Wyble, Jimmy (2011). Concepts for the Classical and Jazz Guitar, p. 2. Mel Bay. . however, in twelve tone equal temperament only six unordered ten note sets exist, 10-1—10-6:
(all pitch classes but t & e) (all but 9 & e) (all but 8 & e) (all but 7 & e) (all but 6 & e) (all but 5 & e)
Given that two of the notes from the chromatic scale are missing and only two whole tones are possible, all 10-note scales are cohemitonic scales.
The four-semitone tritone scale (set 10-6) is a decatonic scale consisting of four semitones, a whole tone, four semitones, and a whole tone (four semitones a tritone apart): 0,1,2,3,4,6,7,8,9,10. This may be related to the seven notes of the diatonic scale as, "1 2 2 3 3 4 5 6 6 7," and thus spelled, on C, as C, D, D, E, E, F, G, A, A, B.Dziuba, Mark (2014). The Ultimate Guitar Scale Bible: 130 Useful Scales for Improvisation, p. 48. Alfred Music. . This is a mode of Olivier Messiaen's seventh mode of limited transposition; it has six transpositions, like the tritone, and five modes (the same pitch classes with C, D, D, E, or E taken as the first scale step or tonic). This allows a dominant seventh chord to be built upon the tonic and a seventh sharp nine chord, and allows the tonic chord to have an altered ninth, eleventh, and thirteenth factors. However, pitch sets containing more than seven notes become increasingly similar to each other.Hanson, Howard. (1960) Harmonic Materials of Modern Music, p. 33. New York: Appleton-Century-Crofts. LOC 58-8138. "When the projection [of the perfect fifth] is carried beyond seven tones, no intervals can be added." "On the other hand, as sonorities are projected beyond the six-tone series they tend to lose their individuality. seven-tone series, for example, contain of the six basic intervals, and difference in their proportion decreases as additional tones are added ... Such patterns tend to lose their identity, producing a monochromatic effect with its accompanying lack of the essential element of contrast."
A decatonic scale that has been used or considered by Kyle Gann and La Monte Young in 13-limit just intonation is 1/1, 12/11, 32/27, 9/7, 4/3, 132/91, 3/2, 18/11, 16/9, 176/91, and 2/1.Just Intonation Network (2002). 1/1: The Quarterly Journal of the Just Intonation Network, Volume 11, p. 17. Just Intonation Network: San Francisco.
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