311edo

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311edo

312edo →

Prime factorization	311 (prime)
Step size	3.85852 ¢
Fifth	182\311 (702.251 ¢)
Semitones (A1:m2)	30:23 (115.8 ¢ : 88.75 ¢)
Consistency limit	41
Distinct consistency limit	23

311edo is highly acclaimed for its large consistency limit and efficient and well-tempered just interval representation relative to its size.

Theory

311edo is consistent through the 41-odd-limit and distinctly consistent through the 23-odd-limit, and is a zeta gap edo and a zeta peak integer edo. It achieves this since all harmonics up to and including the 42nd, and all composite harmonics up to and including the 80th, are more in-tune than out-of-tune (but note prime 73 is tuned accurately, in fact more accurately than all prior primes). Thus all the ratios between those harmonics are mapped consistently – and thus with a maximum error of ~1.929¢. This means 311edo is an extremely efficient temperament for approximating the harmonic series consistently and simply, given how much harmonic content it approximates/represents for its size.

311edo is valuable from a psychoacoustic perspective as its step is also conincidentally close enough to the just-noticeable difference, which only affirms its efficiency of interval representation.

Some 41-limit commas it tempers out are 595/594, 625/624, 697/696, 703/702, 714/713, 760/759, 784/783, 820/819, 833/832, 875/874, 900/899, 925/924, 931/930, 962/961, 969/968, 1000/999, 1015/1014, 1024/1023, 1025/1024, 1036/1035, 1045/1044, 1054/1053, 1105/1104, 1148/1147, 1156/1155, 1184/1183, 1189/1188, 1190/1189, 1197/1196, 1210/1209, 1216/1215, 1225/1224, 1275/1274, 1288/1287, 1312/1311, 1332/1331, 1353/1352, 1365/1364, 1369/1368, 1444/1443, 1445/1444, 1450/1449, 1480/1479, 1496/1495, 1519/1518, 1520/1519, 1540/1539, 1596/1595, 1600/1599, 1625/1624, 1665/1664, 1666/1665, 1681/1680, 1683/1682, 1702/1701, 1729/1728, 1768/1767, 1805/1804, 1860/1859, 1886/1885, 1887/1886, 1925/1924, 2002/2001, 2016/2015, 2025/2024, 2058/2057, 2080/2079, 2091/2090, 2109/2108, 2146/2145, 2176/2175, 2185/2184, 2205/2204, 2233/2232, 2255/2254, 2295/2294, 2296/2295, 2300/2299, 2401/2400, 2431/2430, 2432/2431, 2465/2464, 2500/2499, 2542/2541, 2553/2552, 2584/2583, 2601/2600, 2625/2624, 2640/2639, 2646/2645, 2665/2664, 2737/2736, 2738/2737, 2755/2754, 2784/2783, 2850/2849, 2926/2925, and 2945/2944.

Prime harmonics

Approximation of prime harmonics in 311edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31	37	41	43
Error	Absolute (¢)	+0.000	+0.296	-0.462	-0.337	+0.451	+0.630	-0.775	-0.407	+0.665	+0.648	+0.945	-0.540	-0.767	+1.666
	Relative (%)	+0.0	+7.7	-12.0	-8.7	+11.7	+16.3	-20.1	-10.5	+17.2	+16.8	+24.5	-14.0	-19.9	+43.2
Steps (reduced)		311 (0)	493 (182)	722 (100)	873 (251)	1076 (143)	1151 (218)	1271 (27)	1321 (77)	1407 (163)	1511 (267)	1541 (297)	1620 (65)	1666 (111)	1688 (133)

Subsets and supersets

311edo is the 64th prime edo.

As an interval size measure, one step of 311edo is called gene, named after Gene Ward Smith.

Intervals

Main article: Table of 311edo intervals

Notation

Sagittal

Sagittal in textual form.

Steps	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Symbol	|(	)|(	)~|	~|(	~~|	/|	|)	|\	(|	(|(	~|\	//|	/|)	/|\	)/|\
Steps	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Symbol	(|)	(|\	)||(	)~||	~||(	)||~	/||	||)	||\	~||)	(||(	~||\	//||	/||)	/||\

Syntonic-rastmic subchroma notation

Syntonic-rastmic subchroma notation in textual form.

Steps	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Symbol	>	/	/>	↑\	↑<	↑	↑>	↑/	↑/>	↑↑\	↑↑<	↑↑	↑↑>	t<	t
Steps	16	17	18	19	20	21	22	23	24	25	26	27	28	29	30
Symbol	t>	#↓↓<	#↓↓	#↓↓>	#↓↓/	#↓\<	#↓\	#↓<	#↓	#↓>	#↓/	#\<	#\	#<	#

Ups and downs notation

One possible notation uses / and \ (lifts and drops) to stand for 5 edosteps. Double is abbreviated as "dub-":

0\311 = P1 = perfect unison

1\311 = ^1 = up unison

2\311 = ^^1 = dup unison

3\311 = vv/1 = dudlift unison

4\311 = v/1 = downlift unison

5\311 = /1 = lift unison

6\311 = ^/1 = uplift unison

7\311 = ^^/1 = duplift unison

8\311 = vv//1 = dud-dublift unison

9\311 = v//1 = down-dublift unison

10\311 = //1 = dublift unison

11\311 = ^//1 = up-dublift unison = vv\\m2 = dud-dubdropminor second

12\311 = ^^//1 = dup-dublift unison = v\\m2 = down-dubdropminor second

13\311 = \\m2 =  dubdropminor second

14\311 = ^\\m2 =  up-dubdropminor second

15\311 = ^^\\m2 =  dup-dubdropminor second

16\311 = vv\m2 =  duddropminor second

17\311 = v\m2 =  downdropminor second

18\311 = \m2 =  dropminor second

19\311 = ^\m2 =  updropminor second

20\311 = ^^\m2 =  dupdropminor second

21\311 = vvm2 =  dudminor second

22\311 = vm2 =  downminor second

23\311 = m2 =  minor second

24\311 = ^m2 = upminor second

25\311 = ^^m2 =  dupminor second

26\311 = vv/m2 =  dudliftminor second

27\311 = v/m2 =  downliftminor second

28\311 = /m2 =  liftminor second

29\311 = ^/m2 =  upliftminor second

30\311 = ^^/m2 =  dupliftminor second

31\311 = vv\~2 =  duddropmid second

32\311 = v\~2 =  downdropmid second

33\311 = \~2 =  dropmid second

34\311 = ^\~2 = updropmid second

35\311 = ^^\~2 = dupdropmid second

36\311 = vv~2 =  dudmid second

37\311 = v~2 =  downmid second

38\311 = ~2 =  mid second

etc.

Regular temperament properties

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
				Absolute (¢)	Relative (%)
2.3	[493 -311⟩	[⟨311 493]]	-0.0933	0.0933	2.42
2.3.5	1600000/1594323, [-59 5 22⟩	[⟨311 493 722]]	+0.0040	0.1573	4.08
2.3.5.7	2401/2400, 65625/65536, 1600000/1594323	[⟨311 493 722 873]]	+0.0331	0.1453	3.76
2.3.5.7.11	2401/2400, 3025/3024, 4000/3993, 19712/19683	[⟨311 493 722 873 1076]]	+0.0004	0.1454	3.77
2.3.5.7.11.13	625/624, 1575/1573, 2080/2079, 2200/2197, 2401/2400	[⟨311 493 722 873 1076 1151]]	-0.0280	0.1472	3.81
2.3.5.7.11.13.17	595/594, 625/624, 833/832, 1156/1155, 1575/1573, 2200/2197	[⟨311 493 722 873 1076 1151 1271]]	+0.0031	0.1561	4.05
2.3.5.7.11.13.17.19	595/594, 625/624, 833/832, 969/968, 1156/1155, 1216/1215, 1575/1573	[⟨311 493 722 873 1076 1151 1271 1321]]	+0.0146	0.1492	3.87
2.3.5.7.11.13.17.19.23	595/594, 625/624, 760/759, 833/832, 875/874, 969/968, 1105/1104, 1156/1155	[⟨311 493 722 873 1076 1151 1271 1321 1407]]	-0.0033	0.1496	3.88

Rank-2 temperaments

Table of rank-2 temperaments by generator

Periods per 8ve	Generator*	Cents*	Associated Ratio*	Temperaments
1	10\311	38.59	45/44	Hemitert
1	11\311	42.44	40/39	Humorous
1	17\311	65.59	27/26	Luminal
1	20\311	77.17	256/245, 23/22	Tertiaseptal / tertiaseptia
1	22\311	84.89	21/20	Amicable / amical / amorous
1	29\311	111.90	16/15	Vavoom
1	35\311	135.05	27/25	Superlimmal
1	43\311	165.92	11/10	Satin
1	67\311	258.52	[-32 13 5⟩	Lafa
1	88\311	339.55	243/200	Paramity
1	91\311	351.13	49/40	Newt
1	108\311	416.72	14/11	Unthirds
1	129\311	497.75	4/3	Gary
1	133\311	513.18	35/26	Trinity
1	142\311	547.92	48/35	Calamity
1	143\311	551.77	11/8	Emkay
1	155\311	598.08	572/405	Vydubychi

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

Detemperaments

Ringer scales

There are two known Ringer scales based on 311edo. Both consistently map the complete mode 234 of the harmonic series using non-patent vals of 311edo, which is believed to be the highest possible complete harmonic series mode mapped by a 311-form.

Ringer 311[+61]

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

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

Ringer 311[+61, −67]

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

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:

Music

Eliora

• Etude in C, Op. 1, No. 1 (2022)

Tee Teck Wei

• Baoyu（𨰻𨰻） (2023) – for electric organs tuned in 311edo