Symmetrical scales of 88cET
Within the frame of 88cET, taking a pattern such as 3+1 produces a scale which is a subset of the chromatic 88cET. This particular example could be considered a moment of symmetry scale with a period of 4 and a generator of 3 or 1 (which are inversions in modulo 4). This page attempts to list many of the simplest scales that can be derived in this manner.
1
Period: 1
Degrees (also Ubiquitous tones): 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
Impossible tones: none
Note: If 1 degree is taken as the period, then the full chromatic 88cET gamut is arrived at. Having only one step size, this is not an MOS. It is somewhat arbitrary to stop counting at 41, but that is the convention here.
2
Period: 2
Degrees (also Ubiquitous tones): 0 2 4 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40
Impossible tones: 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Note: If 2 degrees is taken as the period, we arrive at 176cET (nearly the 4th root of 3/2). Having only one step size, this is not an MOS. Half the intervals of 88cET are available all the time (we use the phrase "Ubiquitous tones" for this); the other half is available never ("Impossible tones"). Andrew Heathwaite has nicknamed this scale "Stride."
3
Period: 3
Degrees (also Ubiquitous tones): 0 3 6 9 12 15 18 21 24 27 30 33 36 39
Impossible tones: 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41
Note: One third of the intervals of 88cET are available all the time; the remaining two thirds are available never.
2+1 / 1+2
Period: 3
Generator: 1 or 2
Degrees Mode 1 (2+1): 0 2 3 5 6 8 9 11 12 14 15 17 18 20 21 23 24 26 27 29 30 32 33 35 36 38 39 41
Degrees Mode 2 (1+2): 0 1 3 4 6 7 9 10 12 13 15 16 18 19 21 22 24 25 27 28 30 31 33 34 36 37 39 40
Ubiquitous tones: 0 3 6 12 15 18 21 24 27 30 33 36 39
Common tones: 1 2 4 5 7 8 10 11 13 14 16 17 19 20 22 23 25 26 28 29 31 32 34 35 37 38 40 41
Impossible tones: none
Note: Here, the set of "Ubiquitous tones" refers to the intervals available in all modes. The set of "Common tones" refers to intervals that are available in this scale, but not in every mode. (For instance, Mode 1 contains degree 1 and no degree 2, and Mode 2 contains degree 2 and no degree 1 -- so degrees 1 and 2 are Common tones but not Ubiquitous tones.) There are no "Impossible tones" in this scale.
4
Period: 4
Degrees (also Ubiquitous tones): 0 4 8 12 16 20 24 28 32 36 40
Impossible tones: 1 2 3 5 6 7 9 10 11 13 14 15 17 18 19 21 22 23 25 26 27 29 30 31 33 34 35 37 38 39 41
3+1 / 1+3
Period: 4
Generator: 1 or 3
Degrees Mode 1 (3+1): 0 3 4 7 8 11 12 15 16 19 20 23 24 27 28 31 32 35 36 39 40
Degrees Mode 2 (1+3): 0 1 4 5 8 9 12 13 16 17 20 21 24 25 28 29 32 33 36 37 40 41
Ubiquitous tones: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
Common tones: 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39
Impossible tones: 2 6 10 14 18 22 26 30 34 38
Note: Andrew Heathwaite has nicknamed this scale "Snake".
5
Period: 5
Degrees (also Ubiquitous tones): 0 5 10 15 20 25 30 35 40
Impossible tones: 1 2 3 4 6 7 8 9 11 12 13 14 16 17 18 19 21 22 23 24 26 27 28 29 31 32 33 34 36 37 38 39 41
4+1 / 1+4
Period: 5
Generator: 1 or 4
Degrees Mode 1 (4+1): 0 4 5 9 10 14 15 19 20 24 25 29 30 34 35 39 40
Degrees Mode 2 (1+4): 0 1 5 6 10 11 15 16 20 21 25 26 30 31 35 36 40 41
Ubiquitous tones: 0 5 10 15 20 25 30 35 40
Common tones: 1 4 6 9 11 14 16 19 21 24 26 29 31 34 36 39 41
Impossible tones: 2 3 7 8 12 13 17 18 22 23 27 28 32 33 37 38
3+2 / 2+3
Period: 5
Generator: 2 or 3
Degrees Mode 1 (3+2): 0 3 5 8 10 13 15 18 20 23 25 28 30 33 35 38 40
Degrees Mode 2 (2+3): 0 2 5 7 10 12 15 17 20 22 25 27 30 32 35 37 40
Ubiquitous tones: 0 5 10 15 20 25 30 35 40
Common tones: 2 3 7 8 12 13 17 18 22 23 27 28 32 22 37 38
Impossible tones: 1 4 6 9 11 14 16 19 21 24 26 29 31 34 36 39 41
Note: The Impossible tones and Common tones for the past two scale families are switched, while the Ubiquitous tones stay the same.
2+2+1 / 2+1+2 / 1+2+2
Period: 5
Generator: 2 or 3
Degrees Mode 1 (2+2+1): 0 2 4 5 7 9 10 12 14 15 17 19 20 22 24 25 27 29 30 32 34 35 37 39 40
Degrees Mode 2 (2+1+2): 0 2 3 5 7 8 10 12 13 15 17 18 20 22 23 25 27 28 30 32 33 35 37 38 40
Degrees Mode 3 (1+2+2): 0 1 3 5 6 8 10 11 13 15 16 18 20 21 23 25 26 28 30 31 33 35 36 38 40 41
Ubiquitous tones: 0 5 10 15 20 25 30 35 40
Common tones: 1 2 3 4 6 7 8 9 11 12 13 14 16 17 18 19 21 22 23 24 26 27 28 29 31 32 33 34 36 37 38 39 4
Impossible tones: none