List of root-3rd-P5 triads in JI
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Original Wikitext content:
The basic structure of major and minor triads -- two stacked thirds which total to a perfect fifth -- can be generalized to produce an infinity of chords with their own distinct qualities. What follows is a list of all such chords that are possible in 47-odd-limit [[Just Intonation]], assuming a [[3_2|3/2]] perfect fifth. Wiki authors can feel free to extend this list beyond the 47-limit or leave it at that, but of course, it should be noted that a complete list would be infinite. The narrowest "third" is [[9_8|9/8]], which is decidedly not a third; and the widest "third" is [[4_3|4/3]], which ditto. Thus, the entire conceptual category of a third and then some is covered, and composers can decide for themselves what counts as a "third" and what doesn't. ||~ chord ||||~ first interval ||||~ second interval ||~ prime limit || ||~ ||~ ratio ||~ cents ||~ ratio ||~ cents ||~ || || 8:9:12 || 9/8 || 204 || 4/3 || 498 || 3 || || 30:34:45 || [[17_15|17/15]] || 217 || 45/34 || 485 || 17 || || 22:25:33 || [[25_22|25/22]] || 221 || [[33_25|33/25]] || 481 || 11 || || 36:41:54 || 41/36 || 225 || 54/41 || 477 || 41 || || 14:16:21 || [[8_7|8/7]] || 231 || [[21_16|21/16]] || 471 || 7 || || 20:23:30 || 23/20 || 242 || 30/23 || 460 || 23 || || 26:30:39 || [[15_13|15/13]] || 248 || [[13_10|13/10]] || 454 || 13 || || 32:37:48 || [[37_32|37/32]] || 251 || 48/37 || 451 || 37 || || 6:7:9 || [[7_6|7/6]] || 267 || [[9_7|9/7]] || 435 || 7 || || 40:47:60 || 47/40 || 279 || 60/47 || 423 || 47 || || 28:33:42 || 33/28 || 284 || [[14_11|14/11]] || 418 || 11 || || 22:26:33 || [[13_11|13/11]] || 289 || 33/26 || 413 || 13 || || 16:19:24 || [[19_16|19/16]] || 298 || 24/19 || 404 || 19 || || 26:31:39 || 31/26 || 305 || 39/31 || 397 || 31 || || 36:43:54 || 43/36 || 308 || 54/43 || 394 || 43 || || 10:12:15 || [[6_5|6/5]] || 316 || [[5_4|5/4]] || 386 || 5 || || 24:29:36 || 29/24 || 328 || 36/29 || 374 || 29 || || 14:17:21 || [[17_14|17/14]] || 336 || 21/17 || 366 || 17 || || 32:39:48 || [[39_32|39/32]] || 342 || [[16_13|16/13]] || 359 || 13 || || 18:22:27 || [[11_9|11/9]] || 347 || 27/22 || 355 || 11 || || 22:27:33 || 27/22 || 255 || 11/9 || 347 || 11 || || 26:32:39 || 16/13 || 359 || 39/32 || 342 || 13 || || 30:37:45 || 37/30 || 363 || 45/37 || 339 || 37 || || 4:5:6 || 5/4 || 386 || 6/5 || 316 || 5 || || 30:38:45 || 19/15 || 409 || 45/38 || 293 || 19 || || 26:33:39 || 33/26 || 413 || 13/11 || 289 || 13 || || 22:28:33 || 14/11 || 418 || 33/28 || 284 || 11 || || 18:23:27 || 23/18 || 424 || 27/23 || 278 || 23 || || 32:41:48 || [[32_41|32/41]] || 429 || 48/41 || 273 || 41 || || 14:18:21 || 9/7 || 435 || 7/6 || 267 || 7 || || 24:31:36 || 31/24 || 443 || 36/31 || 259 || 31 || || 10:13:15 || 13/10 || 454 || 15/13 || 248 || 13 || || 36:47:54 || 47/36 || 462 || 54/47 || 240 || 47 || || 26:34:39 || [[17_13|17/13]] || 464 || 39/34 || 238 || 17 || || 16:21:24 || 21/16 || 471 || 8/7 || 231 || 7 || || 22:29:33 || 29/22 || 478 || 33/29 || 224 || 29 || || 28:37:42 || 37/28 || 483 || 42/37 || 219 || 37 || || 6:8:9 || 4/3 || 498 || 9/8 || 204 || 3 ||
Original HTML content:
<html><head><title>List of root-3rd-P5 triads in JI</title></head><body>The basic structure of major and minor triads -- two stacked thirds which total to a perfect fifth -- can be generalized to produce an infinity of chords with their own distinct qualities. What follows is a list of all such chords that are possible in 47-odd-limit <a class="wiki_link" href="/Just%20Intonation">Just Intonation</a>, assuming a <a class="wiki_link" href="/3_2">3/2</a> perfect fifth. Wiki authors can feel free to extend this list beyond the 47-limit or leave it at that, but of course, it should be noted that a complete list would be infinite. The narrowest "third" is <a class="wiki_link" href="/9_8">9/8</a>, which is decidedly not a third; and the widest "third" is <a class="wiki_link" href="/4_3">4/3</a>, which ditto. Thus, the entire conceptual category of a third and then some is covered, and composers can decide for themselves what counts as a "third" and what doesn't.<br />
<br />
<table class="wiki_table">
<tr>
<th>chord<br />
</th>
<th colspan="2">first interval<br />
</th>
<th colspan="2">second interval<br />
</th>
<th>prime limit<br />
</th>
</tr>
<tr>
<th><br />
</th>
<th>ratio<br />
</th>
<th>cents<br />
</th>
<th>ratio<br />
</th>
<th>cents<br />
</th>
<th><br />
</th>
</tr>
<tr>
<td>8:9:12<br />
</td>
<td>9/8<br />
</td>
<td>204<br />
</td>
<td>4/3<br />
</td>
<td>498<br />
</td>
<td>3<br />
</td>
</tr>
<tr>
<td>30:34:45<br />
</td>
<td><a class="wiki_link" href="/17_15">17/15</a><br />
</td>
<td>217<br />
</td>
<td>45/34<br />
</td>
<td>485<br />
</td>
<td>17<br />
</td>
</tr>
<tr>
<td>22:25:33<br />
</td>
<td><a class="wiki_link" href="/25_22">25/22</a><br />
</td>
<td>221<br />
</td>
<td><a class="wiki_link" href="/33_25">33/25</a><br />
</td>
<td>481<br />
</td>
<td>11<br />
</td>
</tr>
<tr>
<td>36:41:54<br />
</td>
<td>41/36<br />
</td>
<td>225<br />
</td>
<td>54/41<br />
</td>
<td>477<br />
</td>
<td>41<br />
</td>
</tr>
<tr>
<td>14:16:21<br />
</td>
<td><a class="wiki_link" href="/8_7">8/7</a><br />
</td>
<td>231<br />
</td>
<td><a class="wiki_link" href="/21_16">21/16</a><br />
</td>
<td>471<br />
</td>
<td>7<br />
</td>
</tr>
<tr>
<td>20:23:30<br />
</td>
<td>23/20<br />
</td>
<td>242<br />
</td>
<td>30/23<br />
</td>
<td>460<br />
</td>
<td>23<br />
</td>
</tr>
<tr>
<td>26:30:39<br />
</td>
<td><a class="wiki_link" href="/15_13">15/13</a><br />
</td>
<td>248<br />
</td>
<td><a class="wiki_link" href="/13_10">13/10</a><br />
</td>
<td>454<br />
</td>
<td>13<br />
</td>
</tr>
<tr>
<td>32:37:48<br />
</td>
<td><a class="wiki_link" href="/37_32">37/32</a><br />
</td>
<td>251<br />
</td>
<td>48/37<br />
</td>
<td>451<br />
</td>
<td>37<br />
</td>
</tr>
<tr>
<td>6:7:9<br />
</td>
<td><a class="wiki_link" href="/7_6">7/6</a><br />
</td>
<td>267<br />
</td>
<td><a class="wiki_link" href="/9_7">9/7</a><br />
</td>
<td>435<br />
</td>
<td>7<br />
</td>
</tr>
<tr>
<td>40:47:60<br />
</td>
<td>47/40<br />
</td>
<td>279<br />
</td>
<td>60/47<br />
</td>
<td>423<br />
</td>
<td>47<br />
</td>
</tr>
<tr>
<td>28:33:42<br />
</td>
<td>33/28<br />
</td>
<td>284<br />
</td>
<td><a class="wiki_link" href="/14_11">14/11</a><br />
</td>
<td>418<br />
</td>
<td>11<br />
</td>
</tr>
<tr>
<td>22:26:33<br />
</td>
<td><a class="wiki_link" href="/13_11">13/11</a><br />
</td>
<td>289<br />
</td>
<td>33/26<br />
</td>
<td>413<br />
</td>
<td>13<br />
</td>
</tr>
<tr>
<td>16:19:24<br />
</td>
<td><a class="wiki_link" href="/19_16">19/16</a><br />
</td>
<td>298<br />
</td>
<td>24/19<br />
</td>
<td>404<br />
</td>
<td>19<br />
</td>
</tr>
<tr>
<td>26:31:39<br />
</td>
<td>31/26<br />
</td>
<td>305<br />
</td>
<td>39/31<br />
</td>
<td>397<br />
</td>
<td>31<br />
</td>
</tr>
<tr>
<td>36:43:54<br />
</td>
<td>43/36<br />
</td>
<td>308<br />
</td>
<td>54/43<br />
</td>
<td>394<br />
</td>
<td>43<br />
</td>
</tr>
<tr>
<td>10:12:15<br />
</td>
<td><a class="wiki_link" href="/6_5">6/5</a><br />
</td>
<td>316<br />
</td>
<td><a class="wiki_link" href="/5_4">5/4</a><br />
</td>
<td>386<br />
</td>
<td>5<br />
</td>
</tr>
<tr>
<td>24:29:36<br />
</td>
<td>29/24<br />
</td>
<td>328<br />
</td>
<td>36/29<br />
</td>
<td>374<br />
</td>
<td>29<br />
</td>
</tr>
<tr>
<td>14:17:21<br />
</td>
<td><a class="wiki_link" href="/17_14">17/14</a><br />
</td>
<td>336<br />
</td>
<td>21/17<br />
</td>
<td>366<br />
</td>
<td>17<br />
</td>
</tr>
<tr>
<td>32:39:48<br />
</td>
<td><a class="wiki_link" href="/39_32">39/32</a><br />
</td>
<td>342<br />
</td>
<td><a class="wiki_link" href="/16_13">16/13</a><br />
</td>
<td>359<br />
</td>
<td>13<br />
</td>
</tr>
<tr>
<td>18:22:27<br />
</td>
<td><a class="wiki_link" href="/11_9">11/9</a><br />
</td>
<td>347<br />
</td>
<td>27/22<br />
</td>
<td>355<br />
</td>
<td>11<br />
</td>
</tr>
<tr>
<td>22:27:33<br />
</td>
<td>27/22<br />
</td>
<td>255<br />
</td>
<td>11/9<br />
</td>
<td>347<br />
</td>
<td>11<br />
</td>
</tr>
<tr>
<td>26:32:39<br />
</td>
<td>16/13<br />
</td>
<td>359<br />
</td>
<td>39/32<br />
</td>
<td>342<br />
</td>
<td>13<br />
</td>
</tr>
<tr>
<td>30:37:45<br />
</td>
<td>37/30<br />
</td>
<td>363<br />
</td>
<td>45/37<br />
</td>
<td>339<br />
</td>
<td>37<br />
</td>
</tr>
<tr>
<td>4:5:6<br />
</td>
<td>5/4<br />
</td>
<td>386<br />
</td>
<td>6/5<br />
</td>
<td>316<br />
</td>
<td>5<br />
</td>
</tr>
<tr>
<td>30:38:45<br />
</td>
<td>19/15<br />
</td>
<td>409<br />
</td>
<td>45/38<br />
</td>
<td>293<br />
</td>
<td>19<br />
</td>
</tr>
<tr>
<td>26:33:39<br />
</td>
<td>33/26<br />
</td>
<td>413<br />
</td>
<td>13/11<br />
</td>
<td>289<br />
</td>
<td>13<br />
</td>
</tr>
<tr>
<td>22:28:33<br />
</td>
<td>14/11<br />
</td>
<td>418<br />
</td>
<td>33/28<br />
</td>
<td>284<br />
</td>
<td>11<br />
</td>
</tr>
<tr>
<td>18:23:27<br />
</td>
<td>23/18<br />
</td>
<td>424<br />
</td>
<td>27/23<br />
</td>
<td>278<br />
</td>
<td>23<br />
</td>
</tr>
<tr>
<td>32:41:48<br />
</td>
<td><a class="wiki_link" href="/32_41">32/41</a><br />
</td>
<td>429<br />
</td>
<td>48/41<br />
</td>
<td>273<br />
</td>
<td>41<br />
</td>
</tr>
<tr>
<td>14:18:21<br />
</td>
<td>9/7<br />
</td>
<td>435<br />
</td>
<td>7/6<br />
</td>
<td>267<br />
</td>
<td>7<br />
</td>
</tr>
<tr>
<td>24:31:36<br />
</td>
<td>31/24<br />
</td>
<td>443<br />
</td>
<td>36/31<br />
</td>
<td>259<br />
</td>
<td>31<br />
</td>
</tr>
<tr>
<td>10:13:15<br />
</td>
<td>13/10<br />
</td>
<td>454<br />
</td>
<td>15/13<br />
</td>
<td>248<br />
</td>
<td>13<br />
</td>
</tr>
<tr>
<td>36:47:54<br />
</td>
<td>47/36<br />
</td>
<td>462<br />
</td>
<td>54/47<br />
</td>
<td>240<br />
</td>
<td>47<br />
</td>
</tr>
<tr>
<td>26:34:39<br />
</td>
<td><a class="wiki_link" href="/17_13">17/13</a><br />
</td>
<td>464<br />
</td>
<td>39/34<br />
</td>
<td>238<br />
</td>
<td>17<br />
</td>
</tr>
<tr>
<td>16:21:24<br />
</td>
<td>21/16<br />
</td>
<td>471<br />
</td>
<td>8/7<br />
</td>
<td>231<br />
</td>
<td>7<br />
</td>
</tr>
<tr>
<td>22:29:33<br />
</td>
<td>29/22<br />
</td>
<td>478<br />
</td>
<td>33/29<br />
</td>
<td>224<br />
</td>
<td>29<br />
</td>
</tr>
<tr>
<td>28:37:42<br />
</td>
<td>37/28<br />
</td>
<td>483<br />
</td>
<td>42/37<br />
</td>
<td>219<br />
</td>
<td>37<br />
</td>
</tr>
<tr>
<td>6:8:9<br />
</td>
<td>4/3<br />
</td>
<td>498<br />
</td>
<td>9/8<br />
</td>
<td>204<br />
</td>
<td>3<br />
</td>
</tr>
</table>
</body></html>