Pentachords of 31edo
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The term "pentachord" may be used to refer to a scale segment in which a [[Perfect fourth|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 this page's name, the organizing figure for the 31edo tetrachords is a triangle. The 220 pentachords are analogously arranged here on a tetrahedron, as follows: <span style="font-family: Courier New,monospace;">T111 9121 8131 7141 6151 5161 4171 3181 2191 11T1</span> <span style="font-family: Courier New,monospace;">9211 8221 7231 6241 5251 4261 3271 2281 1291</span> <span style="font-family: Courier New,monospace;">8311 7321 6331 5341 4351 3361 2371 1381</span> <span style="font-family: Courier New,monospace;">7411 6421 5431 4441 3451 2461 1471</span> <span style="font-family: Courier New,monospace;">6511 5521 4531 3541 2551 1561</span> <span style="font-family: Courier New,monospace;">5611 4621 3631 2641 1651</span> <span style="font-family: Courier New,monospace;">4711 3721 2731 1741</span> <span style="font-family: Courier New,monospace;">3811 2821 1831</span> <span style="font-family: Courier New,monospace;">2911 1921</span> <span style="font-family: Courier New,monospace;">1T11</span> <span style="font-family: Courier New,monospace;">9112 8122 7132 6142 5152 4162 3172 2182 1192</span> <span style="font-family: Courier New,monospace;">8212 **7222 6232 5242 4252 3262 2272** 1282</span> <span style="font-family: Courier New,monospace;">7312 **6322 5332 4342 3352 2362** 1372</span> <span style="font-family: Courier New,monospace;">6412 **5422 4432 3442 2452** 1462</span> <span style="font-family: Courier New,monospace;">5512 **4522 3532 2542** 1552</span> <span style="font-family: Courier New,monospace;">4612 **3622 2632** 1642</span> <span style="font-family: Courier New,monospace;">3712 **2722** 1732</span> <span style="font-family: Courier New,monospace;">2812 1822</span> <span style="font-family: Courier New,monospace;">1912</span> <span style="font-family: Courier New,monospace;">8113 7123 6133 5143 4153 3163 2173 1183</span> <span style="font-family: Courier New,monospace;">7213 **6223 5233 4243 3253 2263** 1273</span> <span style="font-family: Courier New,monospace;">6313 **5323 4333 3343 2353** 1363</span> <span style="font-family: Courier New,monospace;">5413 **4423 3433 2443** 1453</span> <span style="font-family: Courier New,monospace;">4513 **3523 2533** 1543</span> <span style="font-family: Courier New,monospace;">3613 **2623** 1633</span> <span style="font-family: Courier New,monospace;">2713 1723</span> <span style="font-family: Courier New,monospace;">1813</span> <span style="font-family: Courier New,monospace;">7114 6124 5134 4144 3154 2164 1174</span> <span style="font-family: Courier New,monospace;">6214 **5224 4234 3244 2254** 1264</span> <span style="font-family: Courier New,monospace;">5314 **4324 3334 2344** 1354</span> <span style="font-family: Courier New,monospace;">4414 **3424 2434** 1444</span> <span style="font-family: Courier New,monospace;">3514 **2524** 1534</span> <span style="font-family: Courier New,monospace;">2614 1624</span> <span style="font-family: Courier New,monospace;">1714</span> <span style="font-family: Courier New,monospace;">6115 5125 4135 3145 2155 1165</span> <span style="font-family: Courier New,monospace;">5215 **4225 3235 2245** 1255</span> <span style="font-family: Courier New,monospace;">4315 **3325 2335** 1345</span> <span style="font-family: Courier New,monospace;">3415 **2425** 1435</span> <span style="font-family: Courier New,monospace;">2515 1525</span> <span style="font-family: Courier New,monospace;">1615</span> <span style="font-family: Courier New,monospace;">5116 4126 3136 2146 1156</span> <span style="font-family: Courier New,monospace;">4216 **3226 2236** 1246</span> <span style="font-family: Courier New,monospace;">3316 **2326** 1336</span> <span style="font-family: Courier New,monospace;">2416 1426</span> <span style="font-family: Courier New,monospace;">1516</span> <span style="font-family: Courier New,monospace;">4117 3127 2137 1147</span> <span style="font-family: Courier New,monospace;">3217 **2227** 1237</span> <span style="font-family: Courier New,monospace;">2317 1327</span> <span style="font-family: Courier New,monospace;">1417</span> <span style="font-family: Courier New,monospace;">3118 2128 1138</span> <span style="font-family: Courier New,monospace;">2218 1228</span> <span style="font-family: Courier New,monospace;">1318</span> <span style="font-family: Courier New,monospace;">2119 1129</span> <span style="font-family: Courier New,monospace;">1219</span> <span style="font-family: Courier New,monospace;">111T</span> 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 [[31edo MOS scales|MOS scales of 31edo]] may recognize this pentachord as belonging to Miracle[10]. 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.
Original HTML content:
<html><head><title>Pentachords of 31edo</title></head><body>The term "pentachord" may be used to refer to a scale segment in which a <a class="wiki_link" href="/Perfect%20fourth">perfect fourth</a> is divided into four steps. This follows the usage of <a class="wiki_link" href="/Paul%20Erlich">Paul Erlich</a> (see <a class="wiki_link" href="/omnitetrachordality">omnitetrachordality</a>), and is a generalization of the classical "<a class="wiki_link" href="/tetrachord">tetrachord</a>," a division of the perfect fourth into three steps. <a class="wiki_link" href="/31edo">31edo</a>'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 <a class="wiki_link" href="/Tricesimoprimal%20Tetrachordal%20Tesseract">Tricesimoprimal Tetrachordal Tesseract</a>. In spite of this page's name, the organizing figure for the 31edo tetrachords is a triangle. The 220 pentachords are analogously arranged here on a tetrahedron, as follows:<br /> <br /> <span style="font-family: Courier New,monospace;">T111 9121 8131 7141 6151 5161 4171 3181 2191 11T1</span><br /> <span style="font-family: Courier New,monospace;">9211 8221 7231 6241 5251 4261 3271 2281 1291</span><br /> <span style="font-family: Courier New,monospace;">8311 7321 6331 5341 4351 3361 2371 1381</span><br /> <span style="font-family: Courier New,monospace;">7411 6421 5431 4441 3451 2461 1471</span><br /> <span style="font-family: Courier New,monospace;">6511 5521 4531 3541 2551 1561</span><br /> <span style="font-family: Courier New,monospace;">5611 4621 3631 2641 1651</span><br /> <span style="font-family: Courier New,monospace;">4711 3721 2731 1741</span><br /> <span style="font-family: Courier New,monospace;">3811 2821 1831</span><br /> <span style="font-family: Courier New,monospace;">2911 1921</span><br /> <span style="font-family: Courier New,monospace;">1T11</span><br /> <br /> <span style="font-family: Courier New,monospace;">9112 8122 7132 6142 5152 4162 3172 2182 1192</span><br /> <span style="font-family: Courier New,monospace;">8212 <strong>7222 6232 5242 4252 3262 2272</strong> 1282</span><br /> <span style="font-family: Courier New,monospace;">7312 <strong>6322 5332 4342 3352 2362</strong> 1372</span><br /> <span style="font-family: Courier New,monospace;">6412 <strong>5422 4432 3442 2452</strong> 1462</span><br /> <span style="font-family: Courier New,monospace;">5512 <strong>4522 3532 2542</strong> 1552</span><br /> <span style="font-family: Courier New,monospace;">4612 <strong>3622 2632</strong> 1642</span><br /> <span style="font-family: Courier New,monospace;">3712 <strong>2722</strong> 1732</span><br /> <span style="font-family: Courier New,monospace;">2812 1822</span><br /> <span style="font-family: Courier New,monospace;">1912</span><br /> <br /> <span style="font-family: Courier New,monospace;">8113 7123 6133 5143 4153 3163 2173 1183</span><br /> <span style="font-family: Courier New,monospace;">7213 <strong>6223 5233 4243 3253 2263</strong> 1273</span><br /> <span style="font-family: Courier New,monospace;">6313 <strong>5323 4333 3343 2353</strong> 1363</span><br /> <span style="font-family: Courier New,monospace;">5413 <strong>4423 3433 2443</strong> 1453</span><br /> <span style="font-family: Courier New,monospace;">4513 <strong>3523 2533</strong> 1543</span><br /> <span style="font-family: Courier New,monospace;">3613 <strong>2623</strong> 1633</span><br /> <span style="font-family: Courier New,monospace;">2713 1723</span><br /> <span style="font-family: Courier New,monospace;">1813</span><br /> <br /> <br /> <span style="font-family: Courier New,monospace;">7114 6124 5134 4144 3154 2164 1174</span><br /> <span style="font-family: Courier New,monospace;">6214 <strong>5224 4234 3244 2254</strong> 1264</span><br /> <span style="font-family: Courier New,monospace;">5314 <strong>4324 3334 2344</strong> 1354</span><br /> <span style="font-family: Courier New,monospace;">4414 <strong>3424 2434</strong> 1444</span><br /> <span style="font-family: Courier New,monospace;">3514 <strong>2524</strong> 1534</span><br /> <span style="font-family: Courier New,monospace;">2614 1624</span><br /> <span style="font-family: Courier New,monospace;">1714</span><br /> <br /> <span style="font-family: Courier New,monospace;">6115 5125 4135 3145 2155 1165</span><br /> <span style="font-family: Courier New,monospace;">5215 <strong>4225 3235 2245</strong> 1255</span><br /> <span style="font-family: Courier New,monospace;">4315 <strong>3325 2335</strong> 1345</span><br /> <span style="font-family: Courier New,monospace;">3415 <strong>2425</strong> 1435</span><br /> <span style="font-family: Courier New,monospace;">2515 1525</span><br /> <span style="font-family: Courier New,monospace;">1615</span><br /> <br /> <span style="font-family: Courier New,monospace;">5116 4126 3136 2146 1156</span><br /> <span style="font-family: Courier New,monospace;">4216 <strong>3226 2236</strong> 1246</span><br /> <span style="font-family: Courier New,monospace;">3316 <strong>2326</strong> 1336</span><br /> <span style="font-family: Courier New,monospace;">2416 1426</span><br /> <span style="font-family: Courier New,monospace;">1516</span><br /> <br /> <span style="font-family: Courier New,monospace;">4117 3127 2137 1147</span><br /> <span style="font-family: Courier New,monospace;">3217 <strong>2227</strong> 1237</span><br /> <span style="font-family: Courier New,monospace;">2317 1327</span><br /> <span style="font-family: Courier New,monospace;">1417</span><br /> <br /> <span style="font-family: Courier New,monospace;">3118 2128 1138</span><br /> <span style="font-family: Courier New,monospace;">2218 1228</span><br /> <span style="font-family: Courier New,monospace;">1318</span><br /> <br /> <span style="font-family: Courier New,monospace;">2119 1129</span><br /> <span style="font-family: Courier New,monospace;">1219</span><br /> <br /> <span style="font-family: Courier New,monospace;">111T</span><br /> <br /> <br /> 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).<br /> <br /> The pentachords in boldface are the ones which exclude the 1-degree interval (<a class="wiki_link" href="/diesis">diesis</a>); 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 <a class="wiki_link" href="/31edo%20MOS%20scales">MOS scales of 31edo</a> may recognize this pentachord as belonging to Miracle[10].<br /> <br /> 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 <a class="wiki_link" href="/13edo">13edo</a>, or one <a class="wiki_link" href="/tritave">tritave</a> of <a class="wiki_link" href="/BP">BP</a>, etc.</body></html>