Decatonic Scales in Music Theory
A decatonic scale is a musical scale consisting of ten distinct notes. While the mathematical possibilities for ordering these notes are vast—with 3,628,800 possible permutations for a decatonic set—the actual number of unique sets is much smaller when applied to standard tuning systems.
In twelve-tone equal temperament (the standard system of tuning where an octave is divided into twelve equal semitones), only six unordered ten-note sets exist, categorized as 10-1 through 10-6. Because these scales omit only two notes from the full chromatic scale, they are all classified as cohemitonic scales, meaning they contain chromatic semitones in all positions of the scale.
Key Facts
- A decatonic scale contains ten notes.
- In twelve-tone equal temperament, there are only six unique unordered ten-note sets (10-1 to 10-6).
- All ten-note scales in this system are cohemitonic.
- The four-semitone tritone scale is a specific decatonic scale (set 10-6) related to Olivier Messiaen's work.
- Decatonic scales can also be constructed using 13-limit just intonation, a system based on whole-number ratios.
The Four-Semitone Tritone Scale
One of the most notable decatonic scales is the four-semitone tritone scale (set 10-6). This scale is structured with four semitones, followed by a whole tone, another four semitones, and a final whole tone. This creates a symmetrical pattern where the four semitones are separated by a tritone (an interval of six semitones).
In terms of pitch classes, this scale is represented as 0, 1, 2, 3, 4, 6, 7, 8, 9, 10. When spelled starting on C, the notes are C, D♭, D♮, E♭, E♮, F♯, G, A♭, A♮, and B♭.
![Four-semitone tritone scale[1] Playⓘ](/images/1f/3a/1f3a1b8c6f855e48b054ebee66bda96aff18f8058d7b10290a3d34e7d2353e2b.png)
Relationship to Messiaen and Modes
This scale is a mode of Olivier Messiaen's seventh mode of limited transposition. Like the tritone interval itself, it has six possible transpositions. It also features five distinct modes, depending on which note (C, D♭, D, E♭, or E) is treated as the tonic or first scale step.
From a harmonic perspective, this structure allows for the construction of a dominant seventh chord on the tonic, as well as a seventh sharp nine chord. Furthermore, it enables the tonic chord to incorporate altered ninth, eleventh, and thirteenth factors.
Symmetry and Just Intonation
Some decatonic scales are defined by their symmetrical patterns. For example, a symmetrical decatonic scale may follow a repeating step pattern of ½–½–1–½–½.
![Symmetrical decatonic scale: "symmetric pattern [between steps] ½–½–1–½–½|½–½–1–½–½"[2] Playⓘ](/images/bf/dc/bfdc832a3fb71493de9fe25c2825b43e4a9f8864c5b25919f18e30f38986b0af.png)
Beyond equal temperament, composers such as Kyle Gann and La Monte Young have utilized decatonic scales within 13-limit just intonation (a tuning system using ratios of integers up to 13). A specific example of such a scale uses the following ratios: 1/1, 12/11, 32/27, 9/7, 4/3, 132/91, 3/2, 18/11, 16/9, 176/91, and 2/1.
Summary of Decatonic Scale Characteristics
| Scale Type | Tuning System | Key Characteristic | Notable Association |
|---|---|---|---|
| Standard Decatonic | 12-Tone Equal Temperament | 6 unordered sets (10-1 to 10-6) | Cohemitonic nature |
| Four-Semitone Tritone | 12-Tone Equal Temperament | Symmetrical tritone separation | Olivier Messiaen |
| Just Intonation Decatonic | 13-Limit Just Intonation | Integer-ratio based intervals | Kyle Gann / La Monte Young |
Frequently Asked Questions
What is a decatonic scale?
A decatonic scale is any musical scale that contains ten distinct notes within its octave.
Why are all ten-note scales in equal temperament cohemitonic?
Because they only omit two notes from the twelve-note chromatic scale, they inevitably contain chromatic semitones in all positions.
How many transpositions does the four-semitone tritone scale have?
The four-semitone tritone scale has six transpositions.
What is the relationship between the four-semitone tritone scale and Olivier Messiaen?
The four-semitone tritone scale is a mode of Messiaen's seventh mode of limited transposition.
What is 13-limit just intonation in the context of decatonic scales?
It is a tuning system based on whole-number ratios (up to 13) rather than equal divisions of the octave, used by composers like Kyle Gann and La Monte Young to create specific decatonic sets.