Pentachords of 31edo

The term "pentachord" may be used to refer to a scale segment in which a perfect fourth is divided into four steps. This follows the usage of Paul Erlich (see omnitetrachordality), and is a generalization of the classical "tetrachord," a division of the perfect fourth into three steps. 31edo's perfect fourth can be found at 13 degrees, and measures approximately 503.226¢. A complete list of the 66 tetrachords of 31edo can be found at Tricesimoprimal Tetrachordal Tesseract. In spite of that page's name, the organizing figure for the 31edo tetrachords is a triangle, not a tesseract. The 220 pentachords are analogously arranged here on a tetrahedron:

```T111 9121 8131 7141 6151 5161 4171 3181 2191 11T1
9211 8221 7231 6241 5251 4261 3271 2281 1291
8311 7321 6331 5341 4351 3361 2371 1381
7411 6421 5431 4441 3451 2461 1471
6511 5521 4531 3541 2551 1561
5611 4621 3631 2641 1651
4711 3721 2731 1741
3811 2821 1831
2911 1921
1T11
9112 8122 7132 6142 5152 4162 3172 2182 1192
8212 7222 6232 5242 4252 3262 2272 1282
7312 6322 5332 4342 3352 2362 1372
6412 5422 4432 3442 2452 1462
5512 4522 3532 2542 1552
4612 3622 2632 1642
3712 2722 1732
2812 1822
1912
8113 7123 6133 5143 4153 3163 2173 1183
7213 6223 5233 4243 3253 2263 1273
6313 5323 4333 3343 2353 1363
5413 4423 3433 2443 1453
4513 3523 2533 1543
3613 2623 1633
2713 1723
1813
7114 6124 5134 4144 3154 2164 1174
6214 5224 4234 3244 2254 1264
5314 4324 3334 2344 1354
4414 3424 2434 1444
3514 2524 1534
2614 1624
1714
6115 5125 4135 3145 2155 1165
5215 4225 3235 2245 1255
4315 3325 2335 1345
3415 2425 1435
2515 1525
1615
5116 4126 3136 2146 1156
4216 3226 2236 1246
3316 2326 1336
2416 1426
1516
4117 3127 2137 1147
3217 2227 1237
2317 1327
1417
3118 2128 1138
2218 1228
1318
2119 1129
1219
111T
```

Note that the "T" in some of the pentachords above is short for "ten" and represents an interval of 10 degrees of 31edo (10\31). The pentachords in boldface are the ones which exclude the 1-degree interval (diesis); this may be a desirable constraint for melodic considerations. There are 56 pentachords of this type. Further limiting the set to those pentachords that exclude both the diesis and the 2-degree interval gives us a list of four quasi-equal pentachords that form the core of this figure: 4333, 3433, 3343 and 3334. (Those familiar with the MOS scales of 31edo may recognize this pentachord as belonging to Miracle[10].)

A further subset of interest might include those pentachords with only two step sizes, as these can be found in moment of symmetry scales. (Of course, other pentachords may be found as subsets of MOS scales.) This relatively small list with 12 members includes:

```111T 11T1 1T11 T111
2227 2272 2722 7222
3334 3343 3433 4333
```

It should be mentioned that these divisions can be applied in any instance where a span of 13 units is divided into four groups. For instance, these may be interpreted as every possible tetrad within one octave of 13edo, or one tritave of BP, etc.