217edo

217 equal divisions of the octave (abbreviated 217edo or 217ed2), also called 217-tone equal temperament (217tet) or 217 equal temperament (217et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 217 equal parts of about 5.53 ¢ each. Each step represents a frequency ratio of 2^1/217, or the 217th root of 2.

Theory

217edo is a strong 19-limit system, the smallest distinctly consistent in the 19-odd-limit and consistent to the 21-odd-limit as well as the no-23 31-odd-limit. It shares the same 5th and 7th harmonics with 31edo (217 = 7 × 31), as well as the 11/9 interval (supporting the birds temperament). However, compared to 31edo, its patent val differ on the mappings for 3, 11, 13, 17 and 19 – in fact, this edo has a very accurate 13th harmonic, as well as the 19/15 interval. It can also be used in the 23-limit. The only inconsistently mapped intervals in the 23-odd-limit are 23/14, 23/21, and their octave complements.

The equal temperament tempers out the parakleisma, [8 14 -13⟩, and the escapade comma, [32 -7 -9⟩ in the 5-limit; 3136/3125, 4375/4374, 10976/10935 and 823543/819200 in the 7-limit; 441/440, 4000/3993, 5632/5625, and 16384/16335 in the 11-limit; 364/363, 676/675, 1001/1000, 1575/1573, 2080/2079 and 4096/4095 in the 13-limit; 595/594, 833/832, 936/935, 1156/1155, 1225/1224, 1701/1700 in the 17-limit; 343/342, 476/475, 969/968, 1216/1215, 1445/1444, 1521/1520 and 1540/1539 in the 19-limit. It allows minor minthmic chords, werckismic chords, and sinbadmic chords in the 13-odd-limit, in addition to island chords and nicolic chords in the 15-odd-limit. It provides the optimal patent val for the 11- and 13-limit arch and the 11- and 13-limit cotoneum.

Prime harmonics

Approximation to JI

Selected just intervals

The following tables show how 23-odd-limit intervals are represented in 217edo. Prime harmonics are in bold; inconsistent intervals are in italics.

Regular temperament properties

• 217et has lower relative errors than any previous equal temperaments in the 19- and 23-limit. It is the first to beat 72 in the 19-limit and 193 in the 23-limit. The next equal temperament that does better in either subgroup is 243e for absolute error and 270 for relative error.
• 23-limit is not the subgroup it does the best, with the no-23 29- and 31-limit approximated even better.
• It is also prominent in the 17-limit, with a lower absolute error than any previous equal temperaments, beating 183 and superseded by 224.

Rank-2 temperaments

* octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

Scales

• Cotoneum5
• Cotoneum7
• Cotoneum12
• Cotoneum17
• Cotoneum29
• Cotoneum41

Detemperaments

Ringer 217

217edo is the basis for an exceptional Ringer scale that maps an unusually very large amount of the harmonic series (without having to omit any other harmonics) compared to other edos in this size range. Specifically, it maps the complete mode 167 of the harmonic series, corresponding to the entire 333-odd-limit. As 217 = 31 * 7, this can be used to derive 7 possible 31nejis.

167:168:337/2:169:339/2:170:341/2:171:687/4:172:173:347/2:174:349/2:175:351/2:176:353/2:177:178:357/2:179:359/2:180:361/2:181:182:365/2:183:367/2:184:369/2:185:186:373/2:187:375/2:188:189:379/2:190:191:383/2:192:385/2:193:194:389/2:195:196:393/2:197:395/2:198:199:399/2:200:401/2:201:202:203:813/4:204:409/2:205:206:413/2:207:208:417/2:209:210:421/2:211:212:425/2:213:214:429/2:215:216:217:435/2:218:219:439/2:220:221:443/2:222:223:224:449/2:225:226:227:455/2:228:229:459/2:230:231:232:465/2:233:234:469/2:235:236:237:238:239:479/2:240:241:483/2:242:243:244:245:491/2:246:247:248:497/2:249:250:251:252:505/2:253:254:255:256:257:515/2:258:259:260:261:262:263:527/2:264:265:266:267:535/2:268:269:270:271:272:273:274:549/2:275:276:277:278:279:280:281:563/2:282:283:284:285:286:287:288:289:290:291:292:293:294:589/2:295:296:297:298:299:300:301:302:303:304:305:306:307:308:309:310:311:312:313:314:315:316:317:318:319:320:321:322:323:324:325:326:327:328:329:330:331:332:333:334

Deriving 31nejis

This section shows how one can programmatically derive the 7 possible 31nejis aforementioned through use of User:Godtone's copyleft Python 3 code:

>>> r217text = '[paste the above Ringer 217 data here]'
>>> r217=toneji(r217text) # r217
>>> r31s = [ [r217[7*i+j] for i in range(31)]+[r217[j]*2] for j in range(7) ]
>>> r31s2 = [ toneji(':'.join([ str(h) for h in r31 ]),True) for r31 in r31s ]
>>> for i in range(7):
  print(str(i)+'th: ',':'.join([ str(h) for h in r31s2[i] ]))
0th:  274:280:286:293:299:306:313:320:327:334:342:350:358:366:374:383:392:400:409:418:428:438:448:458:468:479:490:500:512:524:535:548
1th:  351:359:367:375:384:393:401:410:420:429:439:449:459:469:480:491:502:514:526:536:549:562:574:588:600:614:628:642:656:672:687:702
2th:  301:308:315:322:329:337:344:352:360:368:376:385:394:402:412:421:430:440:450:460:470:482:492:504:515:527:538:550:563:576:589:602
3th:  258:264:270:276:282:289:295:302:309:316:323:330:338:346:353:361:369:378:386:395:404:413:422:432:442:452:462:472:483:494:505:516
4th:  227:232:237:242:248:253:259:265:271:277:283:290:296:303:310:317:324:331:339:347:354:362:370:379:388:396:406:414:424:434:443:454
5th:  416:425:435:444:455:465:476:486:497:508:520:532:544:556:568:582:594:608:622:636:650:664:680:696:712:728:744:760:778:796:813:832
6th:  213:218:223:228:233:239:244:249:255:261:267:273:279:285:292:298:305:312:319:326:333:341:349:357:365:373:382:390:399:408:417:426
>>> # using the below code can be used to show that only the 0th and 1th 31nejis are mapped correctly by 31edo's patent val
>>> for i in range(7): # (output omitted to avoid spam)
  print(str(i)+'th:\n')
  worstneji(r31s2[i],9)
  print('\n'*2)