217edo: Difference between revisions

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
Error	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
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	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

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)

@@ Line 15: / Line 15: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| class="wikitable center-4 center-5 center-6"
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal 8ve <br>Stretch (¢)
+! colspan="2" | Tuning Error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
 |-
 | 2.3
 | {{monzo| 344 -217 }}
 | {{mapping| 217 344 }}
-| &minus;0.110
+| -0.110
 | 0.1101
 | 1.99
@@ Line 27: / Line 35: @@
 | {{monzo| 8 14 -13 }}, {{monzo| 32 -7 -9 }}
 | {{mapping| 217 344 504 }}
-| &minus;0.186
+| -0.186
 | 0.1398
 | 2.53
@@ Line 34: / Line 42: @@
 | 3136/3125, 4375/4374, 823543/819200
 | {{mapping| 217 344 504 609 }}
-| &minus;0.043
+| -0.043
 | 0.2757
 | 4.99
@@ Line 41: / Line 49: @@
 | 441/440, 3136/3125, 4000/3993, 4375/4374
 | {{mapping| 217 344 504 609 751 }}
-| &minus;0.131
+| -0.131
 | 0.3034
 | 5.49
@@ Line 48: / Line 56: @@
 | 364/363, 441/440, 676/675, 3136/3125, 4375/4374
 | {{mapping| 217 344 504 609 751 803 }}
-| &minus;0.111
+| -0.111
 | 0.2808
 | 5.08
@@ Line 55: / Line 63: @@
 | 364/363, 441/440, 595/594, 676/675, 1156/1155, 3136/3125
 | {{mapping| 217 344 504 609 751 803 887 }}
-| &minus;0.099
+| -0.099
 | 0.2616
 | 4.73
@@ Line 62: / Line 70: @@
 | 343/342, 364/363, 441/440, 476/475, 595/594, 676/675, 1216/1215
 | {{mapping| 217 344 504 609 751 803 887 922 }}
-| &minus;0.119
+| -0.119
 | 0.2504
 | 4.53
@@ Line 69: / Line 77: @@
 | 343/342, 364/363, 392/391, 441/440, 476/475, 507/506, 595/594, 676/675
 | {{mapping| 217 344 504 609 751 803 887 922 982 }}
-| &minus;0.158
+| -0.158
 | 0.2610
 | 4.72
-{{comma basis end}}
+|}
 * 217et has lower relative errors than any previous equal temperaments in the 19- and 23-limit. It is the first to beat [[72edo|72]] in the 19-limit and [[193edo|193]] in the 23-limit. The next equal temperament that does better in either subgroup is [[243edo|243e]] for absolute error and [[270edo|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.
@@ Line 78: / Line 86: @@
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| class="wikitable center-all left-5"
+|+Table of rank-2 temperaments by generator
+! Periods<br>per 8ve
+! Generator*
+! Cents*
+! Associated<br>Ratio*
+! Temperaments
 |-
 | 1
@@ Line 147: / Line 161: @@
 |-
 | 7
-| 94\217<br />(1\217)
+| 94\217<br>(1\217)
-| 519.82<br />(5.53)
+| 519.82<br>(5.53)
-| 27/20<br />(325/324)
+| 27/20<br>(325/324)
 | [[Brahmagupta]]
 |-
 | 31
-| 90\217<br />(1\217)
+| 90\217<br>(1\217)
-| 497.70<br />(5.53)
+| 497.70<br>(5.53)
-| 4/3<br />(243/242)
+| 4/3<br>(243/242)
 | [[Birds]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 == Scales ==
@@ Line 179: / Line 193: @@
 >>> 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)]
+>>> 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]
+>>> 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]]))
+   print(str(i)+'th: ',':'.join([ str(h) for h in r31s2[i] ]))
 th:  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
 th:  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
@@ Line 192: / Line 206: @@
 >>> # 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")
+   print(str(i)+'th:\n')
-   worstneji(r31s2[i], 9)
+   worstneji(r31s2[i],9)
-   print("\n" * 2)
+   print('\n'*2)
 </syntaxhighlight>
 [[Category:Arch]]
 [[Category:Birds]]
 [[Category:Cotoneum]]

217edo: Difference between revisions

Revision as of 11:22, 16 November 2024

Contents

Theory

Prime harmonics

Approximation to JI

Selected just intervals

Regular temperament properties

Rank-2 temperaments

Scales

Detemperaments

Ringer 217

Deriving 31nejis

Navigation menu

217edo: Difference between revisions

Revision as of 11:22, 16 November 2024

Theory

Prime harmonics

Approximation to JI

Selected just intervals

Regular temperament properties

Rank-2 temperaments

Scales

Detemperaments

Ringer 217

Deriving 31nejis

Navigation menu

Search