Pentachords of 31edo: Difference between revisions
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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 | 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{{cent}}. 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: | ||
<pre<includeonly />> | |||
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 <span style="font-weight: bolder;">7222 6232 5242 4252 3262 2272</span> 1282 | |||
7312 <span style="font-weight: bolder;">6322 5332 4342 3352 2362</span> 1372 | |||
6412 <span style="font-weight: bolder;">5422 4432 3442 2452</span> 1462 | |||
5512 <span style="font-weight: bolder;">4522 3532 2542</span> 1552 | |||
4612 <span style="font-weight: bolder;">3622 2632</span> 1642 | |||
3712 <span style="font-weight: bolder;">2722</span> 1732 | |||
2812 1822 | |||
1912 | |||
8113 7123 6133 5143 4153 3163 2173 1183 | |||
7213 <span style="font-weight: bolder;">6223 5233 4243 3253 2263</span> 1273 | |||
6313 <span style="font-weight: bolder;">5323 4333 3343 2353</span> 1363 | |||
5413 <span style="font-weight: bolder;">4423 3433 2443</span> 1453 | |||
4513 <span style="font-weight: bolder;">3523 2533</span> 1543 | |||
3613 <span style="font-weight: bolder;">2623</span> 1633 | |||
2713 1723 | |||
1813 | |||
7114 6124 5134 4144 3154 2164 1174 | |||
6214 <span style="font-weight: bolder;">5224 4234 3244 2254</span> 1264 | |||
5314 <span style="font-weight: bolder;">4324 3334 2344</span> 1354 | |||
4414 <span style="font-weight: bolder;">3424 2434</span> 1444 | |||
3514 <span style="font-weight: bolder;">2524</span> 1534 | |||
2614 1624 | |||
1714 | |||
6115 5125 4135 3145 2155 1165 | |||
5215 <span style="font-weight: bolder;">4225 3235 2245</span> 1255 | |||
4315 <span style="font-weight: bolder;">3325 2335</span> 1345 | |||
3415 <span style="font-weight: bolder;">2425</span> 1435 | |||
2515 1525 | |||
1615 | |||
5116 4126 3136 2146 1156 | |||
4216 <span style="font-weight: bolder;">3226 2236</span> 1246 | |||
3316 <span style="font-weight: bolder;">2326</span> 1336 | |||
2416 1426 | |||
1516 | |||
4117 3127 2137 1147 | |||
3217 <span style="font-weight: bolder;">2227</span> 1237 | |||
2317 1327 | |||
1417 | |||
3118 2128 1138 | |||
2218 1228 | |||
1318 | |||
2119 1129 | |||
1219 | |||
111T | |||
</pre> | |||
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). | 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 [[ | 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 [[List of MOS scales in 31edo|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: | 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: | ||
<pre<includeonly />> | |||
111T 11T1 1T11 T111 | |||
2227 2272 2722 7222 | |||
3334 3343 3433 4333 | |||
</pre> | |||
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. | 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. | ||
[[Category:31edo]] | [[Category:31edo]] | ||
[[Category: | [[Category:Pentachords]] | ||
Latest revision as of 18:44, 7 July 2025
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.