# 217edo

 ← 216edo 217edo 218edo →
Prime factorization 7 × 31
Step size 5.52995¢
Fifth 127\217 (702.304¢)
Semitones (A1:m2) 21:16 (116.1¢ : 88.48¢)
Consistency limit 21
Distinct consistency limit 19
Special properties

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 21/217, 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 of prime harmonics in 217edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +0.35 +0.78 -1.08 +1.68 +0.03 +0.11 +1.10 +2.14 -1.01 -0.34
Relative (%) +0.0 +6.3 +14.2 -19.6 +30.3 +0.5 +2.1 +20.0 +38.7 -18.2 -6.1
Steps
(reduced)
217
(0)
344
(127)
504
(70)
609
(175)
751
(100)
803
(152)
887
(19)
922
(54)
982
(114)
1054
(186)
1075
(207)

## 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.

23-odd-limit intervals in 217edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/8, 16/13 0.025 0.5
19/15, 30/19 0.028 0.5
9/5, 10/9 0.085 1.5
17/13, 26/17 0.088 1.6
17/16, 32/17 0.114 2.1
17/12, 24/17 0.235 4.3
19/10, 20/19 0.321 5.8
13/12, 24/13 0.324 5.9
3/2, 4/3 0.349 6.3
19/18, 36/19 0.406 7.3
5/3, 6/5 0.434 7.8
23/22, 44/23 0.463 8.4
15/11, 22/15 0.545 9.9
19/11, 22/19 0.573 10.4
17/9, 18/17 0.585 10.6
17/10, 20/17 0.669 12.1
13/9, 18/13 0.673 12.2
9/8, 16/9 0.698 12.6
21/16, 32/21 0.735 13.3
19/12, 24/19 0.755 13.7
13/10, 20/13 0.758 13.7
21/13, 26/21 0.760 13.7
5/4, 8/5 0.783 14.2
21/17, 34/21 0.849 15.3
11/10, 20/11 0.894 16.2
11/9, 18/11 0.979 17.7
19/17, 34/19 0.991 17.9
23/15, 30/23 1.008 18.2
17/15, 30/17 1.018 18.4
23/19, 38/23 1.036 18.7
19/13, 26/19 1.079 19.5
7/4, 8/7 1.084 19.6
19/16, 32/19 1.104 20.0
15/13, 26/15 1.107 20.0
13/7, 14/13 1.109 20.1
15/8, 16/15 1.132 20.5
17/14, 28/17 1.198 21.7
11/6, 12/11 1.328 24.0
23/20, 40/23 1.357 24.5
7/6, 12/7 1.433 25.9
23/18, 36/23 1.442 26.1
21/20, 40/21 1.518 27.4
17/11, 22/17 1.564 28.3
13/11, 22/13 1.652 29.9
11/8, 16/11 1.677 30.3
9/7, 14/9 1.782 32.2
23/12, 24/23 1.791 32.4
21/19, 38/21 1.839 33.3
7/5, 10/7 1.867 33.8
23/17, 34/23 2.027 36.6
23/13, 26/23 2.115 38.2
23/16, 32/23 2.140 38.7
19/14, 28/19 2.188 39.6
15/14, 28/15 2.216 40.1
23/14, 28/23 2.306 41.7
21/11, 22/21 2.412 43.6
23/21, 42/23 2.655 48.0
11/7, 14/11 2.761 49.9
23-odd-limit intervals in 217edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
13/8, 16/13 0.025 0.5
19/15, 30/19 0.028 0.5
9/5, 10/9 0.085 1.5
17/13, 26/17 0.088 1.6
17/16, 32/17 0.114 2.1
17/12, 24/17 0.235 4.3
19/10, 20/19 0.321 5.8
13/12, 24/13 0.324 5.9
3/2, 4/3 0.349 6.3
19/18, 36/19 0.406 7.3
5/3, 6/5 0.434 7.8
23/22, 44/23 0.463 8.4
15/11, 22/15 0.545 9.9
19/11, 22/19 0.573 10.4
17/9, 18/17 0.585 10.6
17/10, 20/17 0.669 12.1
13/9, 18/13 0.673 12.2
9/8, 16/9 0.698 12.6
21/16, 32/21 0.735 13.3
19/12, 24/19 0.755 13.7
13/10, 20/13 0.758 13.7
21/13, 26/21 0.760 13.7
5/4, 8/5 0.783 14.2
21/17, 34/21 0.849 15.3
11/10, 20/11 0.894 16.2
11/9, 18/11 0.979 17.7
19/17, 34/19 0.991 17.9
23/15, 30/23 1.008 18.2
17/15, 30/17 1.018 18.4
23/19, 38/23 1.036 18.7
19/13, 26/19 1.079 19.5
7/4, 8/7 1.084 19.6
19/16, 32/19 1.104 20.0
15/13, 26/15 1.107 20.0
13/7, 14/13 1.109 20.1
15/8, 16/15 1.132 20.5
17/14, 28/17 1.198 21.7
11/6, 12/11 1.328 24.0
23/20, 40/23 1.357 24.5
7/6, 12/7 1.433 25.9
23/18, 36/23 1.442 26.1
21/20, 40/21 1.518 27.4
17/11, 22/17 1.564 28.3
13/11, 22/13 1.652 29.9
11/8, 16/11 1.677 30.3
9/7, 14/9 1.782 32.2
23/12, 24/23 1.791 32.4
21/19, 38/21 1.839 33.3
7/5, 10/7 1.867 33.8
23/17, 34/23 2.027 36.6
23/13, 26/23 2.115 38.2
23/16, 32/23 2.140 38.7
19/14, 28/19 2.188 39.6
15/14, 28/15 2.216 40.1
21/11, 22/21 2.412 43.6
11/7, 14/11 2.761 49.9
23/21, 42/23 2.875 52.0
23/14, 28/23 3.224 58.3

## Regular temperament properties

Subgroup Comma List Mapping Optimal 8ve
Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [344 -217 [217 344]] -0.110 0.1101 1.99
2.3.5 [8 14 -13, [32 -7 -9 [217 344 504]] -0.186 0.1398 2.53
2.3.5.7 3136/3125, 4375/4374, 823543/819200 [217 344 504 609]] -0.043 0.2757 4.99
2.3.5.7.11 441/440, 3136/3125, 4000/3993, 4375/4374 [217 344 504 609 751]] -0.131 0.3034 5.49
2.3.5.7.11.13 364/363, 441/440, 676/675, 3136/3125, 4375/4374 [217 344 504 609 751 803]] -0.111 0.2808 5.08
2.3.5.7.11.13.17 364/363, 441/440, 595/594, 676/675, 1156/1155, 3136/3125 [217 344 504 609 751 803 887]] -0.099 0.2616 4.73
2.3.5.7.11.13.17.19 343/342, 364/363, 441/440, 476/475, 595/594, 676/675, 1216/1215 [217 344 504 609 751 803 887 922]] -0.119 0.2504 4.53
2.3.5.7.11.13.17.19.23 343/342, 364/363, 392/391, 441/440, 476/475, 507/506, 595/594, 676/675 [217 344 504 609 751 803 887 922 982]] -0.158 0.2610 4.72
• 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

Table of rank-2 temperaments by generator
Periods
per 8ve
Generator* Cents* Associated
Ratio*
Temperaments
1 3\217 16.59 100/99 Quincy
1 5\217 27.65 64/63 Arch
1 9\217 49.77 36/35 Hemiquindromeda
1 10\217 55.30 16875/16384 Escapade
1 18\217 99.54 18/17 Quintagar / quintoneum / quinsandra
1 30\217 165.90 11/10 Satin
1 33\217 182.49 10/9 Mitonic / mineral
1 57\217 315.21 6/5 Parakleismic / paralytic
1 86\217 475.58 320/243 Vulture
1 90\217 497.70 4/3 Cotoneum
1 101\217 558.53 112/81 Condor
7 94\217
(1\217)
519.82
(5.53)
27/20
(325/324)
Brahmagupta
31 90\217
(1\217)
497.70
(5.53)
4/3
(243/242)
Birds

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

## 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)
```