All the Pentatonics in 10EDO

In 12EDO, the most interesting scales are (I think) the ones that use about half the notes: 5-, 6- and 7-note scales. In 10EDO, then, it makes sense to look at 5-note scales at least, and perhaps those with 4 and 6 notes. Today we'll look at all the available pentatonics in 10EDO and some relationships between them.


10EDO is the tuning you get by dividing an octave into 10 equal parts instead of the usual 12. Its notes will be described by the digits from 0 to 9 inclusive, with 0 being whatever you take to be your reference note (in 10EDO, as in normal 12EDO, notes separated by an octave are considered the same pitch class).

The intervals will also be described by numbers. So the interval 3 is the interval from note 0 to note 3. A scale will be described as a list of pitches in square brackets, e.g. [0, 1, 4, 7, 8], or as a sequence of intervals, e.g. 13312. In general we will use the latter notation. Notice that in this notation, the sum of all the intervals must always be 10.

As usual on this blog, we're most interested in "scale groups" -- collections of scales that are all modes of each other. We will only consider two scales to be different if they belong to different scale groups. Similarly, if two scales are identical except for transposition -- e.g. [0, 1, 4, 7, 8] and [1, 2, 5, 8, 9] -- they will of course be considered "the same scale".

I'll use the following terminology for intervals in 10EDO: semitone, tone, third, fourth, tritone, fifth, sixth, flat seventh, seventh. Mostly this is to keep some sanity when transferring these ideas to the keyboard; do remember that all these intervals (except the tritone) are different from their more familiar 12EDO cousins.

The Scales

One of the nicest scales in 10EDO is 22222, the "ten-equal whole tone" scale. This is the only symmetrical pentatonic in 10EDO (which is one more than 12EDO has!).

Introducing the interval 3 (a "third") requires us to "borrow" a semitone from one of the tones in the previous example. Since there are four remaining semitones we could borrow from, we obtain four new scales: 31222, 32122, 32212, 32221.

Repeating this procedure creates six different scales with two thirds, one tone and two semitones: 33211, 33121, 33112, 31321, 31312, 32311. We can't do it again because; you can't fit a pentatonic scale with three thirds into 10EDO because 3 + 3 + 3 = 9, leaving us two notes short but with only one semitone left in the octave.

We can use a similar logic expanding one of the intervals in 22222 to a fourth by borrowing from two tones, giving us five more pentatonics: 42211, 42121, 41212, 41122, 41221. Again, we can't have a pentatonic with two fourths; 4 + 4 = 8, leaving two semitones for three notes. But we can have a 4 and a 3, which produces four more: 43111, 41311, 41131, 41113.

Beyond this we reach the scales containing a leap of a fifth, which are 52111, 51211, 51121, 51112, and after that only one scale is possible: 61111, the scale that's just five consecutive notes separated by semitones. Unless I've miscounted, that's all the pentatonics in 10EDO.


So that's 24 pentatonics; not that many, really, but still a lot to digest all at once. So we'd like to find some relationships between them. Since 10 - 5 = 5, the complement of a pentatonic in 10EDO is another pentatonic, so that's one interesting relationship: which pentatonics are complements of each other, and which are their own complements?

Another natural one is inverses. Which pentatonics are their own inverses, and which invert to their complements? Are there any that invert to something completely different? The answer is yes.

Here's the complete picture. For each pentatonic I give its complement and its inverse. If either matches the original it's marked (O). If the inverse matches the complement, the inverse is marked (C).

Original     Complement    Inverse

22222        22222 (O)     22222 (O)

31222        32221         32221 (C)
32122        32212         32212 (C)
32212        32122         32122 (C)
32221        31222         31222 (C)

33211        41212         33112    
33121        33121 (O)     33121 (O)
33112        42121         33211    
31321        31312         31312 (C)
31312        31321         31321 (C)
32311        41221         32311 (O)

42211        41122         41122 (C)
42121        33112         41212    
41212        33211         42121    
41122        42211         42211 (C)
41221        32311         41221 (O)

43111        51121         41113    
41311        41131         41131 (C)
41131        41311         41311 (C)
41113        51211         43111    

52111        51112         51112 (C)
51211        41113         51121    
51121        43111         51211    
51112        52111         52111 (C)

61111        61111 (O)     61111 (O)

Scales that are inverses or complements of each other have an important structural relationship, at least in the music I play where these are significant features. So as a first try at bringing some order to them I've arranged them into groups along those lines:

Singletons (each is its own complement and its own inverse):

Pairs (each is the complement or inverse of the other, and the inverse or complement of itself):

        31222        32221         
        32122        32212         
        31321        31312
        32311        41221
        42211        41122
        41311        41131
        52111        51112

Quartets (complements and inverses are all different):
        33211        41212         33112    42121
        43111        51121         41113    51211 

Now it's time to sit with these for a while and see which ones sound promising to explore further.