311edo: Difference between revisions

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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 [[~~The Riemann Zeta Function and Tuning #Zeta EDO lists|~~zeta gap edo]] and a [[~~The Riemann Zeta Function and Tuning #Zeta EDO lists|~~zeta peak integer edo]]. It achieves this since all [[harmonic]]s 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 [[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 [[harmonic]]s 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.

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 [[comma]]s it [[tempering out|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.

Some 41-limit [[comma]]s it [[tempering out|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 ===

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| [[Emkay]]

|-

|1

| 1

| 155\311

| 598.08

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| [[Vydubychi]]

|}

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

== Detemperaments ==

=== Ringer scales ===

There are two known [[Ringer scale]]s based on 311edo. Both consistently map 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.

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

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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] ====

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[[~~Category~~:~~311edo|~~ ]]

== Music ==

; [[Eliora]]

* [https://www.youtube.com/watch?v=GYzCOpwfTrg ''Etude in C'', Op. 1, No. 1] (2022)

; [[Tee Teck Wei]]

* [https://www.youtube.com/watch?v=HqShkc6Fl30 ''Baoyu（𨰻𨰻）''] (2023) – for electric organs tuned in 311edo

~~== Music ==~~

~~:'''[[Eliora]]'''~~

* ''[https://www.youtube.com/watch?v=GYzCOpwfTrg Etude in C, Op. 1, No. 1]''

~~:'''Tee Teck Wei'''~~

* ''[https://www.youtube.com/watch?v=HqShkc6Fl30 Baoyu（𨰻𨰻）for electric organs tuned in 311edo]''

~~~~

~~[[Category:Equal divisions of the octave|###]] ~~

~~[[Category:Prime EDO]]~~

[[Category:Listen]]

@@ Line 3: / Line 3: @@
 edo is highly acclaimed for its large consistency limit and efficient and well-tempered just interval representation relative to its size.
 == Theory ==
-edo is [[consistent]] through the 41-odd-limit and distinctly consistent through the [[23-odd-limit]], and is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta gap edo]] and a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta peak integer edo]]. It achieves this since all [[harmonic]]s 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.
+edo 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 [[harmonic]]s 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.
-edo 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.
+edo 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 [[comma]]s it [[tempering out|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.
+Some 41-limit [[comma]]s it [[tempering out|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 ===
@@ Line 323: / Line 324: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 424: / Line 425: @@
 | [[Emkay]]
 |-
-|1
+| 1
 | 155\311
 | 598.08
@@ Line 430: / Line 431: @@
 | [[Vydubychi]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 == Detemperaments ==
 === Ringer scales ===
 There are two known [[Ringer scale]]s based on 311edo. Both consistently map 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.
 ==== Ringer 311[+61] ====
 {{col-begin}}
@@ Line 468: / Line 471: @@
 :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}}
@@ Line 480: / Line 484: @@
 {{col-end}}
-[[Category:311edo| ]]
+== Music ==
+; [[Eliora]]
+* [https://www.youtube.com/watch?v=GYzCOpwfTrg ''Etude in C'', Op. 1, No. 1] (2022)
+; [[Tee Teck Wei]]
+* [https://www.youtube.com/watch?v=HqShkc6Fl30 ''Baoyu（𨰻𨰻）''] (2023) – for electric organs tuned in 311edo
-== Music ==
-:'''[[Eliora]]'''
-* ''[https://www.youtube.com/watch?v=GYzCOpwfTrg Etude in C, Op. 1, No. 1]''
-:'''Tee Teck Wei'''
-* ''[https://www.youtube.com/watch?v=HqShkc6Fl30 Baoyu（𨰻𨰻）for electric organs tuned in 311edo]''
-<!-- main article -->
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
-[[Category:Prime EDO]]
 [[Category:Listen]]