217edo
| ← 216edo | 217edo | 218edo → |
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
| 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.
| 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 |
| 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
| Periods per 8ve |
Generator* | Cents* | Associated ratio* |
Temperament |
|---|---|---|---|---|
| 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 distinct
Notation
Sagittal
217edo can be written in Sagittal using almost the entire Athenian extension (except for since it tempers out 1240029/1239040), by virtue of its apotome being equal to 21 edosteps, which is the maximum equal division of the apotome (eda) supported by Athenian. It is identical to 224edo's Sagittal notation, but it uses the 11/7C for the +6/-6 alteration instead of 55C.[1]
It shares the same exact symbol system as the Athenian notation for just intonation or Medium-precision JI notation.[1]
| Steps | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Symbol | Evo | | | | | | | | | | | | | | | | | | | | | |
| Revo | | | | | | | | | | | | |||||||||||
Ups-and-downs notation
The 5-up (quup) alteration neatly maps to the pythagorean-septimal comma.
| Steps | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
|---|---|---|---|---|---|---|---|---|---|---|---|
| Symbol | ^ | ^^ | ^^^ | v> | > | ^> | ^^> | ^^^> | v>> | >> | |
| <<<<# | ^<<<<# | vvv<<<# | vv<<<# | v<<<# | <<<# | ^<<<# | vvv<<# | vv<<# | v<<# | ||
| Steps | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |
| Symbol | ^>> | ^^>> | ^^^>> | v>>> | >>> | ^>>> | ^^>>> | ^^^>>> | v>>>> | >>>> | # |
| <<# | ^<<# | vvv<# | vv<# | v<# | <# | ^<# | vvv# | vv# | v# |
31edo-based notation
Since 217 = 31 × 7, one could base the notation on its inherited meantone fifth 126\217 (18\31) instead of its best fifth.
This could be useful when 31edo is used as a base tuning, where the whole palette of 217edo is only used to provide subtle inflections of the 31edo pitches, similar to how one might use 159edo to provide subtle corrections of 53edo pitches.
| Steps | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Symbol | ^ | ^^ | ^^^ | vvvt | vvt | vt | t | ^t | ^^t | ^^^t | v# | vv# | vvv# | # |
| v> | > | ^> | ^^> | ^^^> | v>> | >> | ^>> | ^^>> | ^^^>> | v>>> |
Scales
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)
References
[2] Ragismic microtemperaments#Brahmagupta
- ↑ 1.0 1.1 George D. Secor and David C. Keenan, Sagittal – A Microtonal Notation System, p. 11.
- ↑ Cite error: Invalid
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