311edo: Difference between revisions

Line 2,046:

== Detemperaments ==

~~=== Ringer scales ===~~

If for some reason you want to detemper 311edo, the most otonally simple way of doing so is via making a [[Ringer scale]]. See [[311edo/Ringer 311]] for details.

~~There are two known [[Ringer scale]]s based on~~ 311edo~~. Both include~~ the ~~complete mode 234 of the harmonic series using non-[[patent val]]s~~ of ~~311edo, which~~ is ~~believed to be the highest possible complete harmonic series mode mapped by~~ a ~~311-form. However, if one is looking for a Ringer scale,~~ [[~~217edo#Ringer_217|217edo's corresponding Ringer 217]] may be preferred due to mapping all of mode 167, which is exceptional for its size (as a~~ Ringer scale~~), just as [[311edo~~]] ~~is exceptional for its size (as an~~ [[~~edo]] interpreted as a temperament; it's unknown whether Ringer 311 possesses any special properties that are not a direct result of~~ 311edo~~'s special properties as a temperament).~~

~~====~~ Ringer 311~~[+61~~] ~~====~~

~~{{col-begin}}~~

~~{{col-break}}~~

~~Scale as chord:~~

~~936:940:941:943:944:948:950:952:954:956:958:960:962: ~~

~~964:966:968:970:972:974:976:980:982:984:986:988:990: ~~

~~992:994:996:1000:1002:1004:1006:1008:1010:1012:1016:1018:1020: ~~

~~1022:1024:1026:1028:1030:1032:1036:1038:1040:1042:1044:1048:1050: ~~

~~1052:1054:1056:1060:1063:1064:1066:1068:1070:1072:1076:1078:1080: ~~

~~1082:1084:1088:1090:1092:1096:1097:1100:1102:1104:1108:1110:1112: ~~

~~1114:1116:1120:1122:1124:1128:1130:1132:1134:1136:1140:1142:1144: ~~

~~1148:1150:1152:1156:1158:1160:1162:1164:1168:1170:1172:1176:1178: ~~

~~1180:1184:1186:1188:1192:1194:1196:1200:1202:1204:1208:1210:1212: ~~

~~1216:1218:1220:1224:1226:1228:1232:1234:1236:1240:1244:1246:1248: ~~

~~1252:1254:1256:1260:1264:1266:1268:1272:1274:1276:1280:1282:1284: ~~

~~1288:1292:1294:1296:1300:1304:1306:1308:1312:1316:1318:1320:1324: ~~

~~1326:1328:1332:1336:1338:1340:1344:1348:1350:1352:1356:1360:1362: ~~

~~1364:1368:1372:1374:1376:1380:1384:1388:1390:1392:1396:1400:1402: ~~

~~1404:1408:1412:1414:1416:1420:1424:1428:1432:1434:1436:1440:1444: ~~

~~1448:1450:1452:1456:1460:1462:1464:1468:1472:1476:1480:1484:1486: ~~

~~1488:1492:1496:1500:1504:1506:1508:1512:1516:1520:1524:1526:1528: ~~

~~1532:1536:1540:1544:1546:1548:1552:1556:1560:1564:1568:1572:1576: ~~

~~1580:1582:1584:1588:1592:1596:1600:1604:1606:1608:1612:1616:1620: ~~

~~1624:1628:1632:1636:1640:1644:1646:1648:1652:1656:1660:1664:1668: ~~

~~1672:1676:1680:1684:1688:1692:1696:1700:1702:1704:1708:1712:1716: ~~

~~1720:1724:1728:1732:1736:1740:1744:1748:1752:1756:1760:1764:1768: ~~

~~1772:1776:1780:1784:1788:1792:1796:1800:1804:1808:1812:1816:1820: ~~

~~1824:1828:1832:1836:1840:1844:1848:1852:1856:1860:1864:1868:1872~~

~~{{col-break|gap=5px}}~~

~~Reduced to mode 234:~~

234:235:941/4:943/4:236:237:475/2:238:477/2:239:479/2:240:481/2: 241:483/2:242:485/2:243:487/2:244:245:491/2:246:493/2:247:495/2: 248:497/2:249:250:501/2:251:503/2:252:505/2:253:254:509/2:255: 511/2:256:513/2:257:515/2:258:259:519/2:260:521/2:261:262:525/2: 263:527/2:264:265:1063/4:266:533/2:267:535/2:268:269:539/2:270: 541/2:271:272:545/2:273:274:1097/4:275:551/2:276:277:555/2:278: 557/2:279:280:561/2:281:282:565/2:283:567/2:284:285:571/2:286: 287:575/2:288:289:579/2:290:581/2:291:292:585/2:293:294:589/2: 295:296:593/2:297:298:597/2:299:300:601/2:301:302:605/2:303: 304:609/2:305:306:613/2:307:308:617/2:309:310:311:623/2:312: 313:627/2:314:315:316:633/2:317:318:637/2:319:320:641/2:321: 322:323:647/2:324:325:326:653/2:327:328:329:659/2:330:331: 663/2:332:333:334:669/2:335:336:337:675/2:338:339:340:681/2: 341:342:343:687/2:344:345:346:347:695/2:348:349:350:701/2: 351:352:353:707/2:354:355:356:357:358:717/2:359:360:361: 362:725/2:363:364:365:731/2:366:367:368:369:370:371:743/2: 372:373:374:375:376:753/2:377:378:379:380:381:763/2:382: 383:384:385:386:773/2:387:388:389:390:391:392:393:394: 395:791/2:396:397:398:399:400:401:803/2:402:403:404:405: 406:407:408:409:410:411:823/2:412:413:414:415:416:417: 418:419:420:421:422:423:424:425:851/2:426:427:428:429: 430:431:432:433:434:435:436:437:438:439:440:441:442: 443:444:445:446:447:448:449:450:451:452:453:454:455: 456:457:458:459:460:461:462:463:464:465:466:467:468

~~{{col-end}}~~

~~==== Ringer 311[+61, −67~~] ~~====~~

~~{{col-begin}}~~

~~{{col-break}}~~

~~Scale as chord:~~

936:940:941:943:944:948:950:952:954:956:958:960:962: 964:966:968:970:972:974:976:980:982:984:986:988:990: 992:994:996:1000:1002:1004:1006:1008:1010:1012:1016:1018:1020: 1022:1024:1026:1028:1030:1032:1036:1038:1040:1042:1044:1048:1050: 1052:1054:1056:1060:1061:1064:1066:1068:1072:1074:1076:1078:1080: 1082:1084:1088:1090:1092:1096:1097:1100:1102:1104:1108:1110:1112: 1114:1116:1120:1122:1124:1128:1130:1132:1134:1136:1140:1142:1144: 1148:1150:1152:1156:1158:1160:1162:1164:1168:1170:1172:1176:1178: 1180:1184:1186:1188:1192:1194:1196:1200:1202:1204:1208:1210:1212: 1216:1218:1220:1224:1226:1228:1232:1234:1236:1240:1244:1246:1248: 1252:1254:1256:1260:1264:1266:1268:1272:1274:1276:1280:1282:1284: 1288:1292:1294:1296:1300:1304:1306:1308:1312:1316:1318:1320:1324: 1326:1328:1332:1336:1340:1341:1344:1348:1350:1352:1356:1360:1362: 1364:1368:1372:1374:1376:1380:1384:1388:1390:1392:1396:1400:1402: 1404:1408:1412:1414:1416:1420:1424:1428:1432:1434:1436:1440:1444: 1448:1450:1452:1456:1460:1462:1464:1468:1472:1476:1480:1484:1486: 1488:1492:1496:1500:1504:1506:1508:1512:1516:1520:1524:1526:1528: 1532:1536:1540:1544:1546:1548:1552:1556:1560:1564:1568:1572:1576: 1580:1582:1584:1588:1592:1596:1600:1604:1608:1610:1612:1616:1620: 1624:1628:1632:1636:1640:1644:1646:1648:1652:1656:1660:1664:1668: 1672:1676:1680:1684:1688:1692:1696:1700:1702:1704:1708:1712:1716: 1720:1724:1728:1732:1736:1740:1744:1748:1752:1756:1760:1764:1768: 1772:1776:1780:1784:1788:1792:1796:1800:1804:1808:1812:1816:1820: 1824:1828:1832:1836:1840:1844:1848:1852:1856:1860:1864:1868:1872

~~{{col-break|gap=5px}}~~

~~Reduced to mode 234:~~

234:235:941/4:943/4:236:237:475/2:238:477/2:239:479/2:240:481/2: 241:483/2:242:485/2:243:487/2:244:245:491/2:246:493/2:247:495/2: 248:497/2:249:250:501/2:251:503/2:252:505/2:253:254:509/2:255: 511/2:256:513/2:257:515/2:258:259:519/2:260:521/2:261:262:525/2: 263:527/2:264:265:1061/4:266:533/2:267:268:537/2:269:539/2:270: 541/2:271:272:545/2:273:274:1097/4:275:551/2:276:277:555/2:278: 557/2:279:280:561/2:281:282:565/2:283:567/2:284:285:571/2:286: 287:575/2:288:289:579/2:290:581/2:291:292:585/2:293:294:589/2: 295:296:593/2:297:298:597/2:299:300:601/2:301:302:605/2:303: 304:609/2:305:306:613/2:307:308:617/2:309:310:311:623/2:312: 313:627/2:314:315:316:633/2:317:318:637/2:319:320:641/2:321: 322:323:647/2:324:325:326:653/2:327:328:329:659/2:330:331: 663/2:332:333:334:335:1341/4:336:337:675/2:338:339:340:681/2: 341:342:343:687/2:344:345:346:347:695/2:348:349:350:701/2: 351:352:353:707/2:354:355:356:357:358:717/2:359:360:361: 362:725/2:363:364:365:731/2:366:367:368:369:370:371:743/2: 372:373:374:375:376:753/2:377:378:379:380:381:763/2:382: 383:384:385:386:773/2:387:388:389:390:391:392:393:394: 395:791/2:396:397:398:399:400:401:402:805/2:403:404:405: 406:407:408:409:410:411:823/2:412:413:414:415:416:417: 418:419:420:421:422:423:424:425:851/2:426:427:428:429: 430:431:432:433:434:435:436:437:438:439:440:441:442: 443:444:445:446:447:448:449:450:451:452:453:454:455: 456:457:458:459:460:461:462:463:464:465:466:467:468:

~~{{col-end}}~~

== Music ==

@@ Line 2,046: / Line 2,046: @@
 == Detemperaments ==
-=== Ringer scales ===
+If for some reason you want to detemper 311edo, the most otonally simple way of doing so is via making a [[Ringer scale]]. See [[311edo/Ringer 311]] for details.
-There are two known [[Ringer scale]]s based on 311edo. Both include the complete mode 234 of the harmonic series using non-[[patent val]]s of 311edo, which is believed to be the highest possible complete harmonic series mode mapped by a 311-form. However, if one is looking for a Ringer scale, [[217edo#Ringer_217|217edo's corresponding Ringer 217]] may be preferred due to mapping all of mode 167, which is exceptional for its size (as a Ringer scale), just as [[311edo]] is exceptional for its size (as an [[edo]] interpreted as a temperament; it's unknown whether Ringer 311 possesses any special properties that are not a direct result of 311edo's special properties as a temperament).
-==== Ringer 311[+61] ====
-{{col-begin}}
-{{col-break}}
-Scale as chord:
-:940:941:943:944:948:950:952:954:956:958:960:962:<br />
-:966:968:970:972:974:976:980:982:984:986:988:990:<br />
-:994:996:1000:1002:1004:1006:1008:1010:1012:1016:1018:1020:<br />
-:1024:1026:1028:1030:1032:1036:1038:1040:1042:1044:1048:1050:<br />
-:1054:1056:1060:1063:1064:1066:1068:1070:1072:1076:1078:1080:<br />
-:1084:1088:1090:1092:1096:1097:1100:1102:1104:1108:1110:1112:<br />
-:1116:1120:1122:1124:1128:1130:1132:1134:1136:1140:1142:1144:<br />
-:1150:1152:1156:1158:1160:1162:1164:1168:1170:1172:1176:1178:<br />
-:1184:1186:1188:1192:1194:1196:1200:1202:1204:1208:1210:1212:<br />
-:1218:1220:1224:1226:1228:1232:1234:1236:1240:1244:1246:1248:<br />
-:1254:1256:1260:1264:1266:1268:1272:1274:1276:1280:1282:1284:<br />
-:1292:1294:1296:1300:1304:1306:1308:1312:1316:1318:1320:1324:<br />
-:1328:1332:1336:1338:1340:1344:1348:1350:1352:1356:1360:1362:<br />
-:1368:1372:1374:1376:1380:1384:1388:1390:1392:1396:1400:1402:<br />
-:1408:1412:1414:1416:1420:1424:1428:1432:1434:1436:1440:1444:<br />
-:1450:1452:1456:1460:1462:1464:1468:1472:1476:1480:1484:1486:<br />
-:1492:1496:1500:1504:1506:1508:1512:1516:1520:1524:1526:1528:<br />
-:1536:1540:1544:1546:1548:1552:1556:1560:1564:1568:1572:1576:<br />
-:1582:1584:1588:1592:1596:1600:1604:1606:1608:1612:1616:1620:<br />
-:1628:1632:1636:1640:1644:1646:1648:1652:1656:1660:1664:1668:<br />
-:1676:1680:1684:1688:1692:1696:1700:1702:1704:1708:1712:1716:<br />
-:1724:1728:1732:1736:1740:1744:1748:1752:1756:1760:1764:1768:<br />
-:1776:1780:1784:1788:1792:1796:1800:1804:1808:1812:1816:1820:<br />
-:1828:1832:1836:1840:1844:1848:1852:1856:1860:1864:1868:1872
-{{col-break|gap=5px}}
-Reduced to mode 234:
-:235:<b>941/4</b>:<b>943/4</b>:236:237:<b>475/2</b>:238:<b>477/2</b>:239:<b>479/2</b>:240:<b>481/2</b>:<br />241:<b>483/2</b>:242:<b>485/2</b>:243:<b>487/2</b>:244:245:<b>491/2</b>:246:<b>493/2</b>:247:<b>495/2</b>:<br />248:<b>497/2</b>:249:250:<b>501/2</b>:251:<b>503/2</b>:252:<b>505/2</b>:253:254:<b>509/2</b>:255:<br /><b>511/2</b>:256:<b>513/2</b>:257:<b>515/2</b>:258:259:<b>519/2</b>:260:<b>521/2</b>:261:262:<b>525/2</b>:<br />263:<b>527/2</b>:264:265:<b>1063/4</b>:266:<b>533/2</b>:267:<b>535/2</b>:268:269:<b>539/2</b>:270:<br /><b>541/2</b>:271:272:<b>545/2</b>:273:274:<b>1097/4</b>:275:<b>551/2</b>:276:277:<b>555/2</b>:278:<br /><b>557/2</b>:279:280:<b>561/2</b>:281:282:<b>565/2</b>:283:<b>567/2</b>:284:285:<b>571/2</b>:286:<br />287:<b>575/2</b>:288:289:<b>579/2</b>:290:<b>581/2</b>:291:292:<b>585/2</b>:293:294:<b>589/2</b>:<br />295:296:<b>593/2</b>:297:298:<b>597/2</b>:299:300:<b>601/2</b>:301:302:<b>605/2</b>:303:<br />304:<b>609/2</b>:305:306:<b>613/2</b>:307:308:<b>617/2</b>:309:310:311:<b>623/2</b>:312:<br />313:<b>627/2</b>:314:315:316:<b>633/2</b>:317:318:<b>637/2</b>:319:320:<b>641/2</b>:321:<br />322:323:<b>647/2</b>:324:325:326:<b>653/2</b>:327:328:329:<b>659/2</b>:330:331:<br /><b>663/2</b>:332:333:334:<b>669/2</b>:335:336:337:<b>675/2</b>:338:339:340:<b>681/2</b>:<br />341:342:343:<b>687/2</b>:344:345:346:347:<b>695/2</b>:348:349:350:<b>701/2</b>:<br />351:352:353:<b>707/2</b>:354:355:356:357:358:<b>717/2</b>:359:360:361:<br />362:<b>725/2</b>:363:364:365:<b>731/2</b>:366:367:368:369:370:371:<b>743/2</b>:<br />372:373:374:375:376:<b>753/2</b>:377:378:379:380:381:<b>763/2</b>:382:<br />383:384:385:386:<b>773/2</b>:387:388:389:390:391:392:393:394:<br />395:<b>791/2</b>:396:397:398:399:400:401:<b>803/2</b>:402:403:404:405:<br />406:407:408:409:410:411:<b>823/2</b>:412:413:414:415:416:417:<br />418:419:420:421:422:423:424:425:<b>851/2</b>:426:427:428:429:<br />430:431:432:433:434:435:436:437:438:439:440:441:442:<br />443:444:445:446:447:448:449:450:451:452:453:454:455:<br />456:457:458:459:460:461:462:463:464:465:466:467:468
-{{col-end}}
-==== Ringer 311[+61, &minus;67] ====
-{{col-begin}}
-{{col-break}}
-Scale as chord:
-:940:941:943:944:948:950:952:954:956:958:960:962:<br />964:966:968:970:972:974:976:980:982:984:986:988:990:<br />992:994:996:1000:1002:1004:1006:1008:1010:1012:1016:1018:1020:<br />1022:1024:1026:1028:1030:1032:1036:1038:1040:1042:1044:1048:1050:<br />1052:1054:1056:1060:1061:1064:1066:1068:1072:1074:1076:1078:1080:<br />1082:1084:1088:1090:1092:1096:1097:1100:1102:1104:1108:1110:1112:<br />1114:1116:1120:1122:1124:1128:1130:1132:1134:1136:1140:1142:1144:<br />1148:1150:1152:1156:1158:1160:1162:1164:1168:1170:1172:1176:1178:<br />1180:1184:1186:1188:1192:1194:1196:1200:1202:1204:1208:1210:1212:<br />1216:1218:1220:1224:1226:1228:1232:1234:1236:1240:1244:1246:1248:<br />1252:1254:1256:1260:1264:1266:1268:1272:1274:1276:1280:1282:1284:<br />1288:1292:1294:1296:1300:1304:1306:1308:1312:1316:1318:1320:1324:<br />1326:1328:1332:1336:1340:1341:1344:1348:1350:1352:1356:1360:1362:<br />1364:1368:1372:1374:1376:1380:1384:1388:1390:1392:1396:1400:1402:<br />1404:1408:1412:1414:1416:1420:1424:1428:1432:1434:1436:1440:1444:<br />1448:1450:1452:1456:1460:1462:1464:1468:1472:1476:1480:1484:1486:<br />1488:1492:1496:1500:1504:1506:1508:1512:1516:1520:1524:1526:1528:<br />1532:1536:1540:1544:1546:1548:1552:1556:1560:1564:1568:1572:1576:<br />1580:1582:1584:1588:1592:1596:1600:1604:1608:1610:1612:1616:1620:<br />1624:1628:1632:1636:1640:1644:1646:1648:1652:1656:1660:1664:1668:<br />1672:1676:1680:1684:1688:1692:1696:1700:1702:1704:1708:1712:1716:<br />1720:1724:1728:1732:1736:1740:1744:1748:1752:1756:1760:1764:1768:<br />1772:1776:1780:1784:1788:1792:1796:1800:1804:1808:1812:1816:1820:<br />1824:1828:1832:1836:1840:1844:1848:1852:1856:1860:1864:1868:1872
-{{col-break|gap=5px}}
-Reduced to mode 234:
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-{{col-end}}
 == Music ==