Decatonic scale

A decatonic scale is a ten note musical scale. If the notes are ordered, a decatonic set has 3,628,800 permutations,[3] however, in twelve tone equal temperament only six unordered ten note sets exist, 10-1—10-6:

  1. Play 10-1  (all pitch classes but t & e)
  2. Play 10-2  (all but 9 & e)
  3. Play 10-3  (all but 8 & e)
  4. Play 10-4  (all but 7 & e)
  5. Play 10-5  (all but 6 & e)
  6. Play 10-6  (all but 5 & e)
Four-semitone tritone scale[1] Play 
Symmetrical decatonic scale: "symmetric pattern [between steps] ½-½-1-½-½|½-½-1-½-½"[2] Play 

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.

Four-semitone tritone scale/Messiaen's 7th mode Play 
Dominant seventh raised ninth vs. dominant seventh split third chord. Play 

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♭.[1] 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.[1] However, pitch sets containing more than seven notes become increasingly similar to each other.[4]

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.[5] Play 

References
  1. Dziuba, Mark (2014). The Ultimate Guitar Scale Bible: 130 Useful Scales for Improvisation, p.48. Alfred Music. ISBN 9781470625757.
  2. Rechberger, Herman (2018). Scales and Modes Around the World: The complete guide to the scales and modes of the world, p.43. Fennica Gehrman Ltd. ISBN 9789525489286.
  3. Wyble, Jimmy (2011). Concepts for the Classical and Jazz Guitar, p.2. Mel Bay. ISBN 9781610656207.
  4. 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 new intervals can be added." "On the other hand, as sonorities are projected beyond the six-tone series they tend to lose their individuality. All seven-tone series, for example, contain all 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."
  5. Just Intonation Network (2002). 1/1: The Quarterly Journal of the Just Intonation Network, Volume 11, p.17. Just Intonation Network: San Francisco, CA.
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