Pentachords of 31edo: Difference between revisions

Line 1:

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:

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:

T111 9121 8131 7141 6151 5161 4171 3181 2191 11T1

<pre<includeonly />>

9211 8221 7231 6241 5251 4261 3271 2281 1291

T111 9121 8131 7141 6151 5161 4171 3181 2191 11T1

8311 7321 6331 5341 4351 3361 2371 1381

9211 8221 7231 6241 5251 4261 3271 2281 1291

7411 6421 5431 4441 3451 2461 1471

8311 7321 6331 5341 4351 3361 2371 1381

6511 5521 4531 3541 2551 1561

7411 6421 5431 4441 3451 2461 1471

5611 4621 3631 2641 1651

6511 5521 4531 3541 2551 1561

4711 3721 2731 1741

5611 4621 3631 2641 1651

3811 2821 1831

4711 3721 2731 1741

2911 1921

3811 2821 1831

1T11

2911 1921

9112 8122 7132 6142 5152 4162 3172 2182 1192

1T11

8212 ~~'''~~7222 6232 5242 4252 3262 2272~~'''~~ 1282

9112 8122 7132 6142 5152 4162 3172 2182 1192

7312 ~~'''~~6322 5332 4342 3352 2362~~'''~~ 1372

8212 7222 6232 5242 4252 3262 2272 1282

6412 ~~'''~~5422 4432 3442 2452~~'''~~ 1462

7312 6322 5332 4342 3352 2362 1372

5512 ~~'''~~4522 3532 2542~~'''~~ 1552

6412 5422 4432 3442 2452 1462

4612 ~~'''~~3622 2632~~'''~~ 1642

5512 4522 3532 2542 1552

3712 ~~'''~~2722~~'''~~ 1732

4612 3622 2632 1642

2812 1822

3712 2722 1732

1912

2812 1822

8113 7123 6133 5143 4153 3163 2173 1183

1912

7213 ~~'''~~6223 5233 4243 3253 2263~~'''~~ 1273

8113 7123 6133 5143 4153 3163 2173 1183

6313 ~~'''~~5323 4333 3343 2353~~'''~~ 1363

7213 6223 5233 4243 3253 2263 1273

5413 ~~'''~~4423 3433 2443~~'''~~ 1453

6313 5323 4333 3343 2353 1363

4513 ~~'''~~3523 2533~~'''~~ 1543

5413 4423 3433 2443 1453

3613 ~~'''~~2623~~'''~~ 1633

4513 3523 2533 1543

2713 1723

3613 2623 1633

1813

2713 1723

7114 6124 5134 4144 3154 2164 1174

1813

6214 ~~'''~~5224 4234 3244 2254~~'''~~ 1264

7114 6124 5134 4144 3154 2164 1174

5314 ~~'''~~4324 3334 2344~~'''~~ 1354

6214 5224 4234 3244 2254 1264

4414 ~~'''~~3424 2434~~'''~~ 1444

5314 4324 3334 2344 1354

3514 ~~'''~~2524~~'''~~ 1534

4414 3424 2434 1444

2614 1624

3514 2524 1534

1714

2614 1624

6115 5125 4135 3145 2155 1165

1714

5215 ~~'''~~4225 3235 2245~~'''~~ 1255

6115 5125 4135 3145 2155 1165

4315 ~~'''~~3325 2335~~'''~~ 1345

5215 4225 3235 2245 1255

3415 ~~'''~~2425~~'''~~ 1435

4315 3325 2335 1345

2515 1525

3415 2425 1435

1615

2515 1525

5116 4126 3136 2146 1156

1615

4216 ~~'''~~3226 2236~~'''~~ 1246

5116 4126 3136 2146 1156

3316 ~~'''~~2326~~'''~~ 1336

4216 3226 2236 1246

2416 1426

3316 2326 1336

1516

2416 1426

4117 3127 2137 1147

1516

3217 ~~'''~~2227~~'''~~ 1237

4117 3127 2137 1147

2317 1327

3217 2227 1237

1417

2317 1327

3118 2128 1138

1417

2218 1228

3118 2128 1138

1318

2218 1228

2119 1129

1318

1219

2119 1129

111T

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).

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].)

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:

111T 11T1 1T11 T111

<pre<includeonly />>

2227 2272 2722 7222

111T 11T1 1T11 T111

3334 3343 3433 4333

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.

[[Category:31edo]]

[[Category:~~pentachords~~]]

[[Category:Pentachords]]

@@ Line 1: / Line 1: @@
-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:
+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:
- T111 9121 8131 7141 6151 5161 4171 3181 2191 11T1
+<pre<includeonly />>
-8221 7231 6241 5251 4261 3271 2281 1291
+T111 9121 8131 7141 6151 5161 4171 3181 2191 11T1
-7321 6331 5341 4351 3361 2371 1381
+8221 7231 6241 5251 4261 3271 2281 1291
-6421 5431 4441 3451 2461 1471
+7321 6331 5341 4351 3361 2371 1381
-5521 4531 3541 2551 1561
+6421 5431 4441 3451 2461 1471
-4621 3631 2641 1651
+5521 4531 3541 2551 1561
-3721 2731 1741
+4621 3631 2641 1651
-2821 1831
+3721 2731 1741
-1921
+2821 1831
-T11
+1921
-8122 7132 6142 5152 4162 3172 2182 1192
+T11
-'''7222 6232 5242 4252 3262 2272''' 1282
+8122 7132 6142 5152 4162 3172 2182 1192
-'''6322 5332 4342 3352 2362''' 1372
+<span style="font-weight: bolder;">7222 6232 5242 4252 3262 2272</span> 1282
-'''5422 4432 3442 2452''' 1462
+<span style="font-weight: bolder;">6322 5332 4342 3352 2362</span> 1372
-'''4522 3532 2542''' 1552
+<span style="font-weight: bolder;">5422 4432 3442 2452</span> 1462
-'''3622 2632''' 1642
+<span style="font-weight: bolder;">4522 3532 2542</span> 1552
-'''2722''' 1732
+<span style="font-weight: bolder;">3622 2632</span> 1642
-1822
+<span style="font-weight: bolder;">2722</span> 1732
+1822
-7123 6133 5143 4153 3163 2173 1183
-'''6223 5233 4243 3253 2263''' 1273
+7123 6133 5143 4153 3163 2173 1183
-'''5323 4333 3343 2353''' 1363
+<span style="font-weight: bolder;">6223 5233 4243 3253 2263</span> 1273
-'''4423 3433 2443''' 1453
+<span style="font-weight: bolder;">5323 4333 3343 2353</span> 1363
-'''3523 2533''' 1543
+<span style="font-weight: bolder;">4423 3433 2443</span> 1453
-'''2623''' 1633
+<span style="font-weight: bolder;">3523 2533</span> 1543
-1723
+<span style="font-weight: bolder;">2623</span> 1633
+1723
-6124 5134 4144 3154 2164 1174
-'''5224 4234 3244 2254''' 1264
+6124 5134 4144 3154 2164 1174
-'''4324 3334 2344''' 1354
+<span style="font-weight: bolder;">5224 4234 3244 2254</span> 1264
-'''3424 2434''' 1444
+<span style="font-weight: bolder;">4324 3334 2344</span> 1354
-'''2524''' 1534
+<span style="font-weight: bolder;">3424 2434</span> 1444
-1624
+<span style="font-weight: bolder;">2524</span> 1534
+1624
-5125 4135 3145 2155 1165
-'''4225 3235 2245''' 1255
+5125 4135 3145 2155 1165
-'''3325 2335''' 1345
+<span style="font-weight: bolder;">4225 3235 2245</span> 1255
-'''2425''' 1435
+<span style="font-weight: bolder;">3325 2335</span> 1345
-1525
+<span style="font-weight: bolder;">2425</span> 1435
+1525
-4126 3136 2146 1156
-'''3226 2236''' 1246
+4126 3136 2146 1156
-'''2326''' 1336
+<span style="font-weight: bolder;">3226 2236</span> 1246
-1426
+<span style="font-weight: bolder;">2326</span> 1336
+1426
-3127 2137 1147
-'''2227''' 1237
+3127 2137 1147
-1327
+<span style="font-weight: bolder;">2227</span> 1237
+1327
-2128 1138
-1228
+2128 1138
+1228
-1129
+1129
-T
+T
+</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).
-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].)
+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:
-T 11T1 1T11 T111
+<pre<includeonly />>
-2272 2722 7222
+T 11T1 1T11 T111
-3343 3433 4333
+2272 2722 7222
+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.
 [[Category:31edo]]
-[[Category:pentachords]]
+[[Category:Pentachords]]