217edo: Difference between revisions

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== Theory ==

217edo is a strong [[19-limit]] system, the smallest [[consistency|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 [[5/1|5th]] and [[7/1|7th]] [[harmonic]]s with [[31edo]] (217 = 7 × 31), as well as the [[11/9]] interval (supporting the [[31-comma temperaments #Birds|birds temperament]]). However, compared to 31edo, its [[patent val]] differ on the mappings for [[3/1|3]], [[11/1|11]], [[13/1|13]], [[17/1|17]] and [[19/1|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 complement]]s.

217edo is a strong [[19-limit]] system, the smallest [[consistency|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 [[5/1|5th]] and [[7/1|7th]] [[harmonic]]s with [[31edo]] ({{nowrap|217 {{=}} 7 × 31}}), as well as the [[11/9]] interval (supporting the [[31-comma temperaments #Birds|birds temperament]]). However, compared to 31edo, its [[patent val]] differ on the mappings for [[3/1|3]], [[11/1|11]], [[13/1|13]], [[17/1|17]] and [[19/1|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 complement]]s.

The equal temperament [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }}, and the [[escapade comma]], {{monzo| 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]].

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| {{monzo| 344 -217 }}

| {{mapping| 217 344 }}

| -0.110

| −0.110

| 0.1101

| 1.99

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| {{monzo| 8 14 -13 }}, {{monzo| 32 -7 -9 }}

| {{mapping| 217 344 504 }}

| -0.186

| −0.186

| 0.1398

| 2.53

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| 3136/3125, 4375/4374, 823543/819200

| {{mapping| 217 344 504 609 }}

| -0.043

| −0.043

| 0.2757

| 4.99

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| 441/440, 3136/3125, 4000/3993, 4375/4374

| {{mapping| 217 344 504 609 751 }}

| -0.131

| −0.131

| 0.3034

| 5.49

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| 364/363, 441/440, 676/675, 3136/3125, 4375/4374

| {{mapping| 217 344 504 609 751 803 }}

| -0.111

| −0.111

| 0.2808

| 5.08

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| 364/363, 441/440, 595/594, 676/675, 1156/1155, 3136/3125

| {{mapping| 217 344 504 609 751 803 887 }}

| -0.099

| −0.099

| 0.2616

| 4.73

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| 343/342, 364/363, 441/440, 476/475, 595/594, 676/675, 1216/1215

| {{mapping| 217 344 504 609 751 803 887 922 }}

| -0.119

| −0.119

| 0.2504

| 4.53

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| 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 }}

| -0.158

| −0.158

| 0.2610

| 4.72

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|-

| 7

| 94\217 (1\217)

| 94\217 (1\217)

| 519.82 (5.53)

| 519.82 (5.53)

| 27/20 (325/324)

| 27/20 (325/324)

| [[Brahmagupta]]

|-

| 31

| 90\217 (1\217)

| 90\217 (1\217)

| 497.70 (5.53)

| 497.70 (5.53)

| 4/3 (243/242)

| 4/3 (243/242)

| [[Birds]]

|}

<nowiki />* [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if ~~it is~~ distinct

<nowiki />* [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==

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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

</pre>

==== Deriving 31nejis ====

This section shows how one can programmatically derive the 7 possible 31nejis aforementioned through use of [[User:Godtone]]'s [[User:Godtone#My_Python_3_code|copyleft Python 3 code]]:

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print('\n'*2)

</syntaxhighlight>

[[Category:Arch]]

[[Category:Birds]]

[[Category:Cotoneum]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is a strong [[19-limit]] system, the smallest [[consistency|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 [[5/1|5th]] and [[7/1|7th]] [[harmonic]]s with [[31edo]] (217 = 7 × 31), as well as the [[11/9]] interval (supporting the [[31-comma temperaments #Birds|birds temperament]]). However, compared to 31edo, its [[patent val]] differ on the mappings for [[3/1|3]], [[11/1|11]], [[13/1|13]], [[17/1|17]] and [[19/1|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 complement]]s.
+edo is a strong [[19-limit]] system, the smallest [[consistency|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 [[5/1|5th]] and [[7/1|7th]] [[harmonic]]s with [[31edo]] ({{nowrap|217 {{=}} 7 × 31}}), as well as the [[11/9]] interval (supporting the [[31-comma temperaments #Birds|birds temperament]]). However, compared to 31edo, its [[patent val]] differ on the mappings for [[3/1|3]], [[11/1|11]], [[13/1|13]], [[17/1|17]] and [[19/1|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 complement]]s.
 The equal temperament [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }}, and the [[escapade comma]], {{monzo| 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]].
@@ Line 29: / Line 29: @@
 | {{monzo| 344 -217 }}
 | {{mapping| 217 344 }}
-| -0.110
+| −0.110
 | 0.1101
 | 1.99
@@ Line 36: / Line 36: @@
 | {{monzo| 8 14 -13 }}, {{monzo| 32 -7 -9 }}
 | {{mapping| 217 344 504 }}
-| -0.186
+| −0.186
 | 0.1398
 | 2.53
@@ Line 43: / Line 43: @@
 | 3136/3125, 4375/4374, 823543/819200
 | {{mapping| 217 344 504 609 }}
-| -0.043
+| −0.043
 | 0.2757
 | 4.99
@@ Line 50: / Line 50: @@
 | 441/440, 3136/3125, 4000/3993, 4375/4374
 | {{mapping| 217 344 504 609 751 }}
-| -0.131
+| −0.131
 | 0.3034
 | 5.49
@@ Line 57: / Line 57: @@
 | 364/363, 441/440, 676/675, 3136/3125, 4375/4374
 | {{mapping| 217 344 504 609 751 803 }}
-| -0.111
+| −0.111
 | 0.2808
 | 5.08
@@ Line 64: / Line 64: @@
 | 364/363, 441/440, 595/594, 676/675, 1156/1155, 3136/3125
 | {{mapping| 217 344 504 609 751 803 887 }}
-| -0.099
+| −0.099
 | 0.2616
 | 4.73
@@ Line 71: / Line 71: @@
 | 343/342, 364/363, 441/440, 476/475, 595/594, 676/675, 1216/1215
 | {{mapping| 217 344 504 609 751 803 887 922 }}
-| -0.119
+| −0.119
 | 0.2504
 | 4.53
@@ Line 78: / Line 78: @@
 | 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 }}
-| -0.158
+| −0.158
 | 0.2610
 | 4.72
@@ Line 163: / Line 163: @@
 |-
 | 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]]
 |}
-<nowiki />* [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Scales ==
@@ Line 190: / Line 190: @@
 :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
 </pre>
 ==== Deriving 31nejis ====
 This section shows how one can programmatically derive the 7 possible 31nejis aforementioned through use of [[User:Godtone]]'s [[User:Godtone#My_Python_3_code|copyleft Python 3 code]]:
@@ Line 212: / Line 213: @@
    print('\n'*2)
 </syntaxhighlight>
 [[Category:Arch]]
 [[Category:Birds]]
 [[Category:Cotoneum]]