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'''311edo''' is a remarkable very high limit equal temperament, [[EDO|dividing the octave equally]] into 311 parts of 3.859 [[cent]]s each.  
{{Infobox ET}}
{{ED intro}}
 
311edo is notable for its extremely high [[consistency limit]], which provides efficient and well-tempered [[just interval]] representation relative to its size.


== Theory ==
== Theory ==
311edo is [[consistent]] through the 41-prime-limit add-73 77-odd-limit and uniquely consistent through the [[23-odd-limit]] and is a [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta gap edo]] and a [[zeta peak integer edo]]. It achieves this because except for the prime harmonics greater than 41 (but not including the prime 73 which ''is'' tuned accurately, in fact more accurately than all prior primes), all harmonics up to and including the 80th are more in-tune than out-of-tune with 311edo and 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 nearly distinctly consistent through the [[27-odd-limit]] except for [[25/24]][[~]][[26/25]], [[tempering out]] [[625/624]] ({{S|25}}), and is a [[zeta gap edo]] and a [[zeta peak integer edo]]. This is because all [[harmonic]]s up to the 42nd, and all composite harmonics up to the 80th, have no more than ±25% error. Prime 73 is also unusually accurate, more so than all smaller primes. As a result, all ratios among those harmonics are mapped consistently, with errors lower than 1.929{{c}}. This means 311edo is a ''serendipitously'' efficient temperament for approximating the [[harmonic series]] and the [[41-limit]] in general, consistently and ''simply'', given how much harmonic content it approximates/represents for its size. The next edo with a higher [[consistency limit]] is [[17461edo|17461]] ([[45-odd-limit]]), though one may prefer [[20567edo|20567]] ([[57-odd-limit]]).  


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.
311edo is also the smallest edo that is [[purely consistent]] on all the first 32 harmonics (in this case, up to the 42nd). The next edo with less maximum relative error is [[16808edo|16808]]. The smallest edo purely consistent on the first 64 harmonics is [[3159811edo|3159811]].


311edo is the 64th [[prime EDO]].
Although 311edo does not do as well as [[270edo]] in the 13-limit, it is still very accurate in the lower limits. It tempers out the [[amity comma]], 1600000/1594323, the [[lafa comma]], {{monzo| 77 -31 -12 }}, the [[vavoom comma]], {{monzo| -68 18 17 }} in the [[5-limit]]; 2401/2400 ([[breedsma]]), 65625/65536 ([[horwell comma]]), and 33554432/33480783 ([[garischisma]]) in the 7-limit; [[3025/3024]], [[4000/3993]], [[6250/6237]], [[12005/11979]], and [[19712/19683]] in the 11-limit; and 625/624, [[1575/1573]], [[2080/2079]], [[2200/2197]], [[4096/4095]], and [[4225/4224]] in the 13-limit. It allows [[petrmic chords|petrmic]] and [[nicolic chords]] in the 15-odd-limit.
 
Beyond the 13-limit, primes [[17/1|17]] and [[23/1|23]] are 311edo's first notable improvements over 270edo's approximation. It tempers out [[595/594]], [[833/832]], [[1156/1155]], [[1225/1224]], [[1275/1274]], [[2058/2057]], [[2431/2430]] in the [[17-limit]]; [[969/968]], [[1216/1215]], [[1445/1444]], [[1540/1539]], [[1729/1728]] in the [[19-limit]]; and [[760/759]], [[875/874]], [[1105/1104]], [[1197/1196]], [[1288/1287]], [[1496/1495]] in the [[23-limit]]. Their edo sum, [[581edo]], is also a very strong 23-limit temperament.
 
311edo is valuable from a psychoacoustic perspective as its step is also coincidentally above the melodic [[just-noticeable difference]], which only affirms its efficiency of interval representation.  


=== Prime harmonics ===
=== Prime harmonics ===
{{Primes in edo|311|columns=14|prec=2}}
{{Harmonics in equal|311|prec=3|columns=13}}
{{Harmonics in equal|311|prec=3|columns=13|start=14|collapsed=true|title=Approximation of prime harmonics in 311edo (continued)}}
 
=== Subsets and supersets ===
311edo is the 64th [[prime edo]], so it does not contain any nontrivial subset edos.
 
As an interval size measure, one step of 311edo is called ''gene'', named by [[Joseph Monzo]] in 2007 after [[Gene Ward Smith]]<ref>[http://tonalsoft.com/enc/g/gene.aspx Tonalsoft Encyclopedia | ''gene, 311-edo'']</ref>.


== Intervals ==
== Intervals ==
{{main| Table of 311edo intervals }}
See the collapsed table in [[#JI approximation]], or alternatively, see the draft table at [[User:Overthink/Table of 311edo intervals]].
 
== Notation ==
=== Sagittal notation ===
The [[Sagittal notation]] for 311edo uses alterations of the Promethian set. Since the apotome can be split in two, a half-sharp and a half-flat may be used.
 
<div style="text-align: center;">
{| class="wikitable"
|-
! colspan="2" | '''+ edosteps'''
! 1
! 2
! 3
! 4
! 5
! 6
! 7
! 8
! 9
! 10
! 11
! 12
! 13
! 14
! 15
! 16
! 17
! 18
! 19
! 20
! 21
! 22
! 23
! 24
! 25
! 26
! 27
! 28
! 29
! 30
|-
| rowspan="3" | Symbol
| SZ
| rowspan="3" | <big>{{sagittal||(}}</big>
| rowspan="3" | <big>{{Sagittal|)|(}}</big>
| rowspan="3" | <big>{{Sagittal|)~|}}</big>
| rowspan="3" | <big>{{Sagittal|~|(}}</big>
| rowspan="3" | <big>{{Sagittal|~~|}}</big>
| rowspan="3" | <big>{{Sagittal|/|}}</big>
| rowspan="3" | <big>{{Sagittal||)}}</big>
| rowspan="3" | <big>{{Sagittal||\}}</big>
| rowspan="3" | <big>{{Sagittal|(|}}</big>
| rowspan="3" | <big>{{Sagittal|(|(}}</big>
| rowspan="3" | <big>{{Sagittal|~|\}}</big>
| rowspan="3" | <big>{{Sagittal|//|}}</big>
| rowspan="3" | <big>{{Sagittal|/|)}}</big>
| rowspan="3" | <big>{{Sagittal|/|\}}</big>
| <big>{{Sagittal|t}}</big>
| <small>{{Sagittal||(}}{{sagittal|t}}</small>
| <small>{{Sagittal|)|(}}{{sagittal|t}}</small>
| <small>{{Sagittal|)~|}}{{sagittal|t}}</small>
| <small>{{Sagittal|~|(}}{{sagittal|t}}</small>
| <small>{{Sagittal|~~|}}{{sagittal|t}}</small>
| <small>{{Sagittal|/|}}{{sagittal|t}}</small>
| <small>{{Sagittal||)}}{{sagittal|t}}</small>
| <small>{{Sagittal||\}}{{sagittal|t}}</small>
| <small>{{Sagittal|(|}}{{sagittal|t}}</small>
| <small>{{Sagittal|(|(}}{{sagittal|t}}</small>
| <small>{{Sagittal|~|\}}{{sagittal|t}}</small>
| <small>{{Sagittal|//|}}{{sagittal|t}}</small>
| <small>{{Sagittal|/|)}}{{sagittal|t}}</small>
| <small>{{Sagittal|/|\}}{{sagittal|t}}</small>
| <small>{{Sagittal|#}}</small>
|-
| Evo
| rowspan="2" | <big>{{Sagittal|)/|\}}</big>
| <small>{{sagittal|\!/}}{{sagittal|#}}</small>
| <small>{{sagittal|\!)}}{{sagittal|#}}</small>
| <small>{{sagittal|\\!}}{{sagittal|#}}</small>
| <small>{{sagittal|~!/}}{{sagittal|#}}</small>
| <small>{{sagittal|(!(}}{{sagittal|#}}</small>
| <small>{{sagittal|(!}}{{sagittal|#}}</small>
| <small>{{sagittal|!/}}{{sagittal|#}}</small>
| <small>{{sagittal|!)}}{{sagittal|#}}</small>
| <small>{{sagittal|\!}}{{sagittal|#}}</small>
| <small>{{sagittal|~~!}}{{sagittal|#}}</small>
| <small>{{sagittal|~!(}}{{sagittal|#}}</small>
| <small>{{sagittal|)~!}}{{sagittal|#}}</small>
| <small>{{sagittal|)!(}}{{sagittal|#}}</small>
| <small>{{sagittal|!(}}{{sagittal|#}}</small>
| <small>{{sagittal|#}}</small>
|-
| Revo
| <big>{{sagittal|(|)}}</big>
| <big>{{sagittal|(|\}}</big>
| <big>{{sagittal|)||(}}</big>
| <big>{{sagittal|)~||}}</big>
| <big>{{sagittal|~||(}}</big>
| <big>{{sagittal|)||~}}</big>
| <big>{{sagittal|/||}}</big>
| <big>{{sagittal|||)}}</big>
| <big>{{sagittal|||\}}</big>
| <big>{{sagittal|~||)}}</big>
| <big>{{sagittal|(||(}}</big>
| <big>{{sagittal|~||\}}</big>
| <big>{{sagittal|//||}}</big>
| <big>{{sagittal|/||)}}</big>
| <big>{{sagittal|/||\}}</big>
|}
</div>
 
=== Syntonic–rastmic subchroma notation ===
[[Syntonic–rastmic subchroma notation]] in textual form.
<div style="overflow-x: auto;">
{| class="wikitable center-all"
|-
! Steps
| 1
| 2
| 3
| 4
| 5
| 6
| 7
| 8
| 9
| 10
| 11
| 12
| 13
| 14
| 15
| 16
| 17
| 18
| 19
| 20
| 21
| 22
| 23
| 24
| 25
| 26
| 27
| 28
| 29
| 30
|-
! Symbol
| >
| /
| />
| ↑\
| ↑<
| ↑
| ↑>
| ↑/
| ↑/>
| ↑↑\
| ↑↑<
| ↑↑
| ↑↑>
| t<
| t
| t>
| #↓↓<
| #↓↓
| #↓↓>
| #↓↓/
| #↓\<
| #↓\
| #↓<
| #↓
| #↓>
| #↓/
| #\<
| #\
| #<
| #
|}
</div>
 
=== Ups and downs notation ===
[[Ups and downs notation]] uses ^ and v (up and down) to stand for 1 edostep and > and < (quip and quid) to stand for 5 edosteps. The spoken names run up, dup, trup, quup/downquip, quip, upquip, etc. >> is quipquip and >>> is tripquip. Quarter-tone accidentals can also be used for 311edo.
 
{{Ups and downs sharpness|311|true}}
 
== JI approximation ==
=== 41-odd-limit interval mappings ===
{{Q-odd-limit intervals|311|limit=41}}
 
=== Higher-limit JI ===
311edo does not maintain [[monotonicity]] in the 43-odd-limit using either mapping for 43. Therefore it may be best to consider 311edo a temperament of the 41-limit, with sporadic additional primes.
 
The 41-limit add-73 add-89 add-101 add-109 add-113 123-odd-limit is represented very close to completely [[consistent]]ly, and as aforementioned, the 77-odd-limit subset of that odd-limit is purely consistent, to which a variety of odds can be added that keep pure consistency, but for comprehensiveness and practical use as a temperament approximating the low-to-mid end of the harmonic series, we consider a larger odd-limit than that which seeks to be more complete.
 
There are 884 interval pairs in that odd limit (the 41-limit add-73 add-89 add-101 add-109 add-113 123-odd-limit), where ''pairs'' refers to that each interval has an [[octave complement]] with equal and opposite error. That odd limit can be described explicitly as the [[tonality diamond]] of {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 45, 49, 51, 55, 57, 63, 65, 69, 73, 75, 77, 81, 85, 87, 89, 91, 93, 95, 99, 101, 105, 109, 111, 113, 115, 117, 119, 121, 123}. We can also express that odd-limit as the 123-odd-limit minus only the following twelve prime odds: {43, 47, 53, 59, 61, 67, 71, 79, 83, 97, 103, 107}.
 
Of those 884 interval pairs, only 42 interval pairs (< 4.8%) are inconsistent, not mapped to the nearest interval of 311edo but to the second-nearest interval. Reduced to the lower half of the octave, these intervals, from smallest to largest, are: 101/100, 100/99, 82/81, 121/119, 119/117, 95/93, 87/85, 124/119, 85/81, 101/95, 100/93, 85/78, 93/85, 119/108, 93/82, 81/70, 138/119, 136/117, 99/85, 117/100, 95/81, 119/101, 101/85, 81/68, 140/117, 119/99, 117/95, 85/69, 100/81, 108/85, 119/93, 85/66, 156/119, 93/70, 162/119, 93/68, 119/87, 85/62, 117/85, 140/101, 164/117, 170/121.
 
Of them, only 6 interval pairs (119/117, 85/81, 93/85, 101/85, 119/93, 117/85) are more than 10% inconsistent, which is to say, all 36 of the other inconsistent intervals have less than 60% of a step of 311edo of error relative to where they are mapped in 311edo by the patent val, which is to say less than 60% [[relative interval error|relative error]], which is equal to 2.3{{cent}}. The 6 highest-error intervals mentioned instead have less than 2/3 (~66.7&) relative error.
 
The below table was generated by a simple Python 3 script to print it in plaintext using [[User: Godtone #My Python 3 code|Godtone's code]] to simplify certain steps. It should be noted that while almost all intervals shown in the table are intervals of the 123-odd-limit restricted to the aforementioned prime subgroup, the [[square-particular]]s up to [[1681/1680]] ({{S|41}}, (41/40)/(42/41)) were added manually for completeness and reference in understanding the mapping of the [[41-odd-limit]] by 311edo for the first three edosteps and the unison. The rest of the table is algorithmically generated.
 
{| class="mw-collapsible mw-collapsed wikitable center-1 center-2 center-3"
|+ style="font-size: 105%; white-space: nowrap;" | Table of 311edo intervals
|-
! #
! Cents
! Marks
! Approximate Intervals<ref group="note">Odd harmonics and subharmonics are in '''bold''', inconsistent intervals in ''italics''</ref>
|-
| 0
| 0.0
| P1
| '''1/1''', [[1681/1680|S41]], [[1600/1599|S40]], [[1444/1443|S38]], [[1369/1368|S37]], [[1225/1224|S35 = S49*S50]], [[1156/1155|S34]], [[1024/1023|S32]], [[900/899|S30]], ''[[784/783|S28]]'', ''[[625/624|S25]]''
|-
| 1
| 3.85
|
| ''[[1521/1520|S39]]'', ''[[1296/1295|S36]]'', ''[[1089/1088|S33]]'', ''[[961/960|S31]]'', [[841/840|S29]], [[729/728|S27]], [[676/675|S26 = S13/S15]], [[576/575|S24]], [[529/528|S23]], [[484/483|S22]], [[441/440|S21 = 441/440]], [[400/399|S20 = 400/399]], [[361/360|S19 = 361/360]], ''[[289/288|S17 = 289/288]]''
|-
| 2
| 7.71
|
| ''[[324/323|S18 = 324/323]]'', [[256/255|S16 = 256/255]], [[243/242|S9/S11 = 243/242]], [[225/224|S15 = 225/224]], [[196/195|S14 = 196/195]], ''[[170/169]]''
|-
| 3
| 11.57
|
| [[169/168|S13 = 169/168]], [[144/143|S12 = 144/143]], [[171/170]]
|-
| 4
| 15.43
|
| [[124/123]], [[121/120]], [[120/119]], [[117/116]], [[116/115]], [[115/114]], [[114/113]], [[113/112]], [[112/111]], [[111/110]], [[110/109]], [[109/108]], [[105/104]], [[102/101]], ''[[100/99]]''
|-
| 5
| 19.29
|
| ''[[101/100]]'', [[99/98]], [[96/95]], [[93/92]], [[92/91]], [[91/90]], [[90/89]], [[89/88]], [[88/87]], [[85/84]], ''[[82/81]]''
|-
| 6
| 23.15
|
| [[81/80]], [[78/77]], [[77/76]], [[76/75]], [[75/74]], [[74/73]], [[73/72]], [[70/69]]
|-
| 7
| 27.0
|
| [[69/68]], [[66/65]], '''[[65/64]]''', '''[[64/63]]''', [[63/62]], [[123/121]], ''[[119/117]]''
|-
| 8
| 30.86
| sd2
| ''[[121/119]]'', [[117/115]], [[58/57]], [[115/113]], [[57/56]], [[113/111]], [[56/55]], [[111/109]], [[55/54]]
|-
| 9
| 34.72
|
| [[52/51]], [[51/50]], [[101/99]], [[50/49]], [[49/48]], ''[[95/93]]''
|-
| 10
| 38.58
|
| [[93/91]], [[46/45]], [[91/89]], [[45/44]], [[89/87]]
|-
| 11
| 42.44
|
| ''[[87/85]]'', [[42/41]], [[124/121]], [[41/40]], [[40/39]], [[119/116]]
|-
| 12
| 46.3
|
| [[39/38]], [[116/113]], [[77/75]], [[115/112]], [[38/37]], [[113/110]], [[75/73]], [[112/109]], [[37/36]]
|-
| 13
| 50.16
|
| [[36/35]], [[35/34]], [[104/101]], [[34/33]]
|-
| 14
| 54.01
|
| [[101/98]], '''[[33/32]]''', [[98/95]], [[65/63]], '''[[32/31]]''', [[95/92]]
|-
| 15
| 57.87
| sA1
| [[31/30]], [[123/119]], [[92/89]], [[91/88]], [[121/117]], [[30/29]], [[119/115]]
|-
| 16
| 61.73
|
| [[88/85]], [[117/113]], [[29/28]], [[115/111]], [[57/55]], [[85/82]], [[113/109]], [[28/27]]
|-
| 17
| 65.59
|
| [[109/105]], [[27/26]], [[80/77]], [[105/101]]
|-
| 18
| 69.45
|
| [[26/25]], [[77/74]], '''[[128/123]]''', [[51/49]], [[76/73]], [[126/121]], [[25/24]]
|-
| 19
| 73.31
|
| ''[[124/119]]'', [[99/95]], [[73/70]], [[121/116]], [[24/23]], [[119/114]], [[95/91]]
|-
| 20
| 77.17
|
| [[117/112]], [[93/89]], [[116/111]], [[23/22]], [[114/109]], [[91/87]], [[68/65]], [[113/108]]
|-
| 21
| 81.02
|
| [[89/85]], [[22/21]], [[109/104]], [[65/62]], ''[[85/81]]''
|-
| 22
| 84.88
|
| [[21/20]], [[104/99]], [[41/39]]
|-
| 23
| 88.74
| m2
| [[81/77]], [[101/96]], [[121/115]], [[20/19]], [[119/113]], [[98/93]]
|-
| 24
| 92.6
|
| [[39/37]], [[58/55]], [[77/73]], [[96/91]], [[115/109]], [[19/18]]
|-
| 25
| 96.46
|
| [[93/88]], [[130/123]], [[37/35]], [[92/87]], [[55/52]], '''[[128/121]]''', [[73/69]]
|-
| 26
| 100.32
|
| [[18/17]], [[89/84]], [[124/117]], [[123/116]], [[35/33]]
|-
| 27
| 104.18
|
| [[87/82]], [[52/49]], [[121/114]], [[69/65]], [[120/113]], '''[[17/16]]'''
|-
| 28
| 108.03
|
| ''[[101/95]]'', [[117/110]], [[116/109]], [[33/31]], [[115/108]], [[82/77]], [[49/46]]
|-
| 29
| 111.89
|
| [[81/76]], '''[[16/15]]''', [[111/104]], [[95/89]]
|-
| 30
| 115.75
| A1
| [[78/73]], [[109/102]], [[31/29]], [[108/101]], [[77/72]], [[123/115]]
|-
| 31
| 119.61
|
| [[91/85]], [[121/113]], [[15/14]], [[119/111]], [[74/69]]
|-
| 32
| 123.47
|
| [[44/41]], [[117/109]], [[73/68]], [[102/95]], [[29/27]], [[130/121]], ''[[100/93]]''
|-
| 33
| 127.33
|
| '''[[128/119]]''', [[99/92]], [[113/105]], [[14/13]]
|-
| 34
| 131.18
|
| '''[[69/64]]''', [[124/115]], [[55/51]], [[96/89]], [[41/38]], [[109/101]], [[68/63]], [[95/88]]
|-
| 35
| 135.04
|
| [[27/25]], [[121/112]], [[40/37]], [[119/110]]
|-
| 36
| 138.9
|
| [[92/85]], [[13/12]]
|-
| 37
| 142.76
|
| [[89/82]], [[38/35]], [[101/93]], [[63/58]], [[88/81]], [[113/104]], [[25/23]]
|-
| 38
| 146.62
| N2
| [[87/80]], [[62/57]], [[99/91]], [[37/34]], [[123/113]], [[49/45]], [[110/101]], ''[[85/78]]''
|-
| 39
| 150.48
|
| [[109/100]], [[121/111]], [[12/11]], [[119/109]], [[95/87]]
|-
| 40
| 154.34
|
| [[130/119]], [[82/75]], '''[[35/32]]''', '''[[128/117]]'''
|-
| 41
| 158.19
|
| ''[[93/85]]'', [[81/74]], [[104/95]], [[23/21]], [[126/115]], [[80/73]], [[57/52]], [[34/31]]
|-
| 42
| 162.05
|
| [[124/113]], [[45/41]], [[101/92]], [[56/51]], [[123/112]], [[89/81]], [[100/91]], [[111/101]]
|-
| 43
| 165.91
|
| [[11/10]], [[120/109]], [[109/99]], [[98/89]], [[76/69]], ''[[119/108]]''
|-
| 44
| 169.77
|
| [[54/49]], [[75/68]], '''[[32/29]]''', [[85/77]]
|-
| 45
| 173.63
|
| [[116/105]], [[21/19]], [[136/123]], [[115/104]], [[73/66]]
|-
| 46
| 177.49
| d3
| [[31/28]], [[72/65]], [[113/102]], [[41/37]], [[51/46]], [[112/101]]
|-
| 47
| 181.35
|
| [[132/119]], [[81/73]], [[91/82]], [[101/91]], [[111/100]], [[121/109]], [[10/9]]
|-
| 48
| 185.2
|
| [[109/98]], [[99/89]], [[89/80]], [[69/62]], '''[[128/115]]''', [[49/44]]
|-
| 49
| 189.06
|
| [[39/35]], [[126/113]], [[29/26]], [[77/69]]
|-
| 50
| 192.92
|
| [[124/111]], [[19/17]], [[123/110]], [[104/93]], [[85/76]], [[113/101]]
|-
| 51
| 196.78
|
| [[28/25]], [[121/108]], [[65/58]], [[102/91]], [[37/33]]
|-
| 52
| 200.64
|
| [[46/41]], [[101/90]], [[55/49]], '''[[64/57]]''', [[73/65]], [[82/73]], [[91/81]], [[100/89]], [[136/121]]
|-
| 53
| 204.5
| M2
| '''[[9/8]]''', [[98/87]]
|-
| 54
| 208.36
|
| [[62/55]], [[115/102]], [[44/39]], [[123/109]], [[114/101]], [[35/31]]
|-
| 55
| 212.21
|
| [[96/85]], [[87/77]], [[113/100]], [[26/23]], [[95/84]], [[112/99]]
|-
| 56
| 216.07
|
| [[77/68]], [[111/98]], '''[[128/113]]''', [[17/15]]
|-
| 57
| 219.93
|
| ''[[93/82]]'', [[101/89]], [[42/37]], [[109/96]], [[92/81]], [[25/22]]
|-
| 58
| 223.79
|
| [[108/95]], [[58/51]], [[91/80]], [[124/109]], [[33/29]], [[140/123]], [[74/65]], [[115/101]], [[41/36]]
|-
| 59
| 227.65
|
| [[57/50]], [[65/57]], [[138/121]], '''[[73/64]]''', [[89/78]], [[105/92]], [[113/99]]
|-
| 60
| 231.51
|
| '''[[8/7]]''', [[119/104]]
|-
| 61
| 235.36
| sd3
| [[87/76]], [[63/55]], [[55/48]], [[102/89]]
|-
| 62
| 239.22
|
| [[39/34]], [[109/95]], [[101/88]], [[132/115]], [[31/27]], [[116/101]], [[85/74]], [[100/87]]
|-
| 63
| 243.08
|
| [[23/20]], [[130/113]], [[84/73]], [[38/33]]
|-
| 64
| 246.94
|
| [[121/105]], [[98/85]], [[113/98]], '''[[128/111]]''', [[15/13]]
|-
| 65
| 250.8
|
| [[52/45]], [[89/77]], [[126/109]], '''[[37/32]]''', [[140/121]]
|-
| 66
| 254.66
|
| ''[[81/70]]'', [[22/19]], [[117/101]], [[95/82]], [[73/63]], [[51/44]], [[80/69]]
|-
| 67
| 258.52
|
| ''[[138/119]]'', [[29/25]], [[65/56]], [[101/87]], [[36/31]], [[115/99]], ''[[136/117]]''
|-
| 68
| 262.37
| sA2
| [[93/80]], [[57/49]], [[121/104]], '''[[64/55]]''', [[85/73]]
|-
| 69
| 266.23
|
| ''[[99/85]]'', [[7/6]]
|-
| 70
| 270.09
|
| [[132/113]], [[111/95]], [[104/89]], [[90/77]], [[76/65]]
|-
| 71
| 273.95
|
| ''[[117/100]]'', [[48/41]], [[89/76]], [[130/111]], [[41/35]], [[116/99]], '''[[75/64]]''', [[109/93]], [[34/29]], ''[[95/81]]''
|-
| 72
| 277.81
|
| [[88/75]], [[115/98]], [[27/23]], '''[[128/109]]''', [[74/63]]
|-
| 73
| 281.67
|
| [[87/74]], [[20/17]], [[113/96]], [[73/62]], ''[[119/101]]''
|-
| 74
| 285.53
|
| [[33/28]], [[112/95]], [[46/39]], [[105/89]], [[85/72]]
|-
| 75
| 289.38
|
| [[124/105]], [[13/11]], [[136/115]], [[123/104]], [[110/93]]
|-
| 76
| 293.24
| m3
| [[58/49]], [[45/38]], [[77/65]], [[109/92]], '''[[32/27]]'''
|-
| 77
| 297.1
|
| [[121/102]], [[89/75]], [[108/91]], [[146/123]], '''[[19/16]]''', [[120/101]], [[82/69]]
|-
| 78
| 300.96
|
| ''[[101/85]]'', [[44/37]], [[113/95]], [[69/58]], [[119/100]], [[144/121]], [[25/21]]
|-
| 79
| 304.82
|
| ''[[81/68]]'', [[87/73]], [[31/26]], [[130/109]], [[68/57]], [[105/88]], [[37/31]]
|-
| 80
| 308.68
|
| [[117/98]], [[92/77]], [[49/41]], [[104/87]], [[55/46]], ''[[140/117]]''
|-
| 81
| 312.54
|
| [[91/76]], [[109/91]], [[115/96]], [[121/101]]
|-
| 82
| 316.39
|
| [[6/5]], ''[[119/99]]''
|-
| 83
| 320.25
| A2
| [[101/84]], [[89/74]], '''[[77/64]]''', [[148/123]], [[136/113]], [[65/54]], [[112/93]]
|-
| 84
| 324.11
|
| [[88/73]], [[41/34]], [[76/63]], [[111/92]], [[146/121]], [[35/29]]
|-
| 85
| 327.97
|
| [[99/82]], [[93/77]], [[29/24]], [[110/91]], [[75/62]], [[98/81]]
|-
| 86
| 331.83
|
| [[121/100]], [[144/119]], [[23/19]], [[132/109]], [[109/90]], [[63/52]], [[40/33]]
|-
| 87
| 335.69
|
| [[91/75]], [[108/89]], [[17/14]], [[113/93]]
|-
| 88
| 339.54
|
| [[62/51]], [[45/37]], [[73/60]], [[28/23]], [[123/101]], [[95/78]]
|-
| 89
| 343.4
|
| '''[[39/32]]''', '''[[128/105]]''', [[89/73]], [[50/41]], [[111/91]]
|-
| 90
| 347.26
|
| [[116/95]], [[138/113]], [[11/9]], [[148/121]]
|-
| 91
| 351.12
| N3
| [[104/85]], [[93/76]], [[60/49]], [[109/89]], [[49/40]], [[136/111]], [[38/31]]
|-
| 92
| 354.98
|
| [[92/75]], [[146/119]], [[27/22]], [[124/101]], [[70/57]], [[113/92]]
|-
| 93
| 358.84
|
| [[91/74]], [[123/100]], '''[[16/13]]''', ''[[85/69]]''
|-
| 94
| 362.7
|
| ''[[117/95]]'', [[101/82]], [[69/56]], [[90/73]], [[37/30]], [[95/77]], ''[[100/81]]''
|-
| 95
| 366.55
|
| [[121/98]], [[21/17]], [[152/123]], [[110/89]], [[89/72]], [[68/55]], [[115/93]]
|-
| 96
| 370.41
|
| [[99/80]], [[26/21]], [[109/88]], [[140/113]], [[57/46]], [[119/96]], [[150/121]]
|-
| 97
| 374.27
|
| [[31/25]], [[36/29]], [[113/91]], [[77/62]], [[41/33]]
|-
| 98
| 378.13
|
| [[87/70]], [[46/37]], [[148/119]], [[51/41]], [[56/45]]
|-
| 99
| 381.99
| d4
| [[81/65]], [[91/73]], [[96/77]], [[101/81]], [[111/89]], [[116/93]], [[126/101]], [[136/109]], [[146/117]]
|-
| 100
| 385.85
|
| '''[[5/4]]'''
|-
| 101
| 389.71
|
| [[154/123]], [[144/115]], [[124/99]], [[119/95]], [[114/91]], [[109/87]]
|-
| 102
| 393.56
|
| [[69/55]], '''[[64/51]]''', [[123/98]], [[113/90]], [[152/121]], [[49/39]]
|-
| 103
| 397.42
|
| [[93/74]], [[44/35]], [[39/31]], [[112/89]], [[73/58]], [[34/27]]
|-
| 104
| 401.28
|
| [[63/50]], [[92/73]], [[121/96]], [[150/119]], [[29/23]], [[140/111]], [[111/88]], [[82/65]]
|-
| 105
| 405.14
|
| [[101/80]], [[24/19]], [[115/91]], [[91/72]], [[110/87]], [[148/117]]
|-
| 106
| 409.0
| M3
| [[62/49]], '''[[81/64]]''', [[138/109]], [[19/15]], '''[[128/101]]'''
|-
| 107
| 412.86
|
| [[52/41]], [[33/26]], [[146/115]], [[113/89]], [[80/63]]
|-
| 108
| 416.72
|
| ''[[108/85]]'', [[89/70]], [[117/92]], [[14/11]]
|-
| 109
| 420.57
|
| [[121/95]], [[93/73]], [[144/113]], [[65/51]], [[116/91]], [[51/40]], [[88/69]], [[37/29]]
|-
| 110
| 424.43
|
| [[152/119]], [[23/18]], ''[[119/93]]''
|-
| 111
| 428.29
|
| [[87/68]], '''[[32/25]]''', [[105/82]], [[73/57]], [[114/89]], '''[[41/32]]''', [[50/39]]
|-
| 112
| 432.15
|
| [[109/85]], [[77/60]], [[95/74]], [[104/81]], [[113/88]], [[140/109]]
|-
| 113
| 436.01
|
| [[9/7]], [[148/115]], [[130/101]], [[112/87]], ''[[85/66]]''
|-
| 114
| 439.87
| sd4
| [[58/45]], [[156/121]], [[49/38]], [[89/69]], [[40/31]]
|-
| 115
| 443.72
|
| [[31/24]], [[146/113]], [[115/89]], [[84/65]], '''[[128/99]]''', [[75/58]], [[119/92]]
|-
| 116
| 447.58
|
| [[22/17]], [[123/95]], [[101/78]], [[136/105]], [[57/44]], [[35/27]]
|-
| 117
| 451.44
|
| [[48/37]], [[109/84]], [[74/57]], [[100/77]], [[113/87]], [[152/117]]
|-
| 118
| 455.3
|
| [[13/10]], [[160/123]], [[121/93]], [[95/73]], [[82/63]]
|-
| 119
| 459.16
|
| [[99/76]], [[116/89]], [[73/56]], [[30/23]]
|-
| 120
| 463.02
|
| [[124/95]], [[111/85]], '''[[64/49]]''', [[81/62]], [[98/75]], [[115/88]], [[132/101]], [[17/13]]
|-
| 121
| 466.88
| sA3
| [[89/68]], [[72/55]], [[55/42]], [[148/113]], [[38/29]]
|-
| 122
| 470.73
|
| ''[[156/119]]'', [[101/77]], '''[[21/16]]''', [[130/99]]
|-
| 123
| 474.59
|
| [[46/35]], [[117/89]], [[96/73]], [[121/92]], [[146/111]], [[25/19]], [[154/117]]
|-
| 124
| 478.45
|
| [[54/41]], [[112/85]], [[29/22]], [[120/91]], [[91/69]], [[95/72]]
|-
| 125
| 482.31
|
| [[33/25]], [[144/109]], [[37/28]], [[152/115]], [[115/87]], [[119/90]], [[160/121]], [[41/31]]
|-
| 126
| 486.17
|
| [[45/34]], [[49/37]], [[102/77]]
|-
| 127
| 490.03
|
| [[126/95]], [[65/49]], [[69/52]], [[73/55]], [[150/113]], [[77/58]], '''[[85/64]]'''
|-
| 128
| 493.89
|
| ''[[93/70]]'', [[101/76]], [[109/82]], [[113/85]], [[117/88]], [[121/91]]
|-
| 129
| 497.74
| P4
| '''[[4/3]]'''
|-
| 130
| 501.6
|
| [[123/92]], [[119/89]]
|-
| 131
| 505.46
|
| [[99/74]], [[91/68]], [[87/65]], [[162/121]], [[154/115]], [[75/56]], [[146/109]]
|-
| 132
| 509.32
|
| [[114/85]], [[55/41]], [[51/38]], [[98/73]]
|-
| 133
| 513.18
|
| [[121/90]], [[160/119]], [[39/29]], [[152/113]], [[113/84]], [[74/55]], [[109/81]], [[35/26]], [[136/101]]
|-
| 134
| 517.04
|
| [[101/75]], [[66/49]], '''[[128/95]]''', [[31/23]], [[120/89]], [[89/66]], [[85/63]]
|-
| 135
| 520.9
|
| [[27/20]], [[104/77]], [[77/57]], [[50/37]], [[123/91]], [[73/54]], [[119/88]]
|-
| 136
| 524.75
| A3
| [[23/17]], [[111/82]], [[88/65]], [[65/48]], [[42/31]], [[164/121]]
|-
| 137
| 528.61
|
| [[99/73]], [[156/115]], [[19/14]], [[148/109]], [[110/81]]
|-
| 138
| 532.47
|
| '''[[87/64]]''', [[121/89]], [[34/25]], [[49/36]]
|-
| 139
| 536.33
|
| ''[[162/119]]'', [[109/80]], [[124/91]], [[154/113]], [[15/11]]
|-
| 140
| 540.19
|
| [[116/85]], [[101/74]], [[56/41]], [[138/101]], [[41/30]], [[160/117]], ''[[119/87]]''
|-
| 141
| 544.05
|
| ''[[93/68]]'', [[26/19]], [[115/84]], [[89/65]], [[152/111]], [[63/46]], [[100/73]], [[37/27]], ''[[85/62]]''
|-
| 142
| 547.9
|
| [[48/35]], [[70/51]], [[136/99]]
|-
| 143
| 551.76
|
| '''[[11/8]]''', [[150/109]], '''[[128/93]]''', [[95/69]]
|-
| 144
| 555.62
| sA4
| ''[[117/85]]'', [[62/45]], [[113/82]], [[164/119]], [[51/37]], [[91/66]], [[40/29]]
|-
| 145
| 559.48
|
| [[69/50]], [[156/113]], [[29/21]], [[105/76]], [[76/55]], [[123/89]], [[170/123]], [[112/81]]
|-
| 146
| 563.34
|
| [[101/73]], [[18/13]], ''[[140/101]]''
|-
| 147
| 567.2
|
| [[104/75]], [[154/111]], [[111/80]], [[68/49]], [[168/121]], [[25/18]]
|-
| 148
| 571.06
|
| [[132/95]], [[57/41]], [[146/105]], '''[[89/64]]''', [[121/87]], '''[[32/23]]'''
|-
| 149
| 574.91
|
| [[39/28]], [[124/89]], [[46/33]], [[152/109]], [[113/81]]
|-
| 150
| 578.77
|
| [[81/58]], [[88/63]], [[95/68]], [[102/73]], [[109/78]], [[123/88]], [[130/93]]
|-
| 151
| 582.63
|
| [[7/5]], ''[[164/117]]''
|-
| 152
| 586.49
| d5
| [[115/82]], [[108/77]], [[101/72]], [[87/62]], [[80/57]], [[73/52]], ''[[170/121]]''
|-
| 153
| 590.35
|
| [[52/37]], '''[[45/32]]''', '''[[128/91]]''', [[38/27]]
|-
| 154
| 594.21
|
| [[69/49]], [[162/115]], [[31/22]], [[148/105]], [[55/39]]
|-
| 155
| 598.07
|
| [[24/17]], [[113/80]], [[89/63]], [[154/109]], [[65/46]], [[41/29]], [[140/99]]
|-
| 156
| 601.92
|
| [[99/70]], [[58/41]], [[92/65]], [[109/77]], [[126/89]], [[160/113]], [[17/12]]
|-
| 157
| 605.78
|
| [[78/55]], [[105/74]], [[44/31]], [[115/81]], [[98/69]]
|-
| 158
| 609.64
|
| [[27/19]], '''[[91/64]]''', '''[[64/45]]''', [[37/26]]
|-
| 159
| 613.5
| A4
| ''[[121/85]]'', [[104/73]], [[57/40]], [[124/87]], [[144/101]], [[77/54]], [[164/115]]
|-
| 160
| 617.36
|
| ''[[117/82]]'', [[10/7]]
|-
| 161
| 621.22
|
| [[93/65]], [[176/123]], [[156/109]], [[73/51]], [[136/95]], [[63/44]], [[116/81]]
|-
| 162
| 625.08
|
| [[162/113]], [[109/76]], [[33/23]], [[89/62]], [[56/39]]
|-
| 163
| 628.93
|
| '''[[23/16]]''', [[174/121]], '''[[128/89]]''', [[105/73]], [[82/57]], [[95/66]]
|-
| 164
| 632.79
|
| [[36/25]], [[121/84]], [[49/34]], [[160/111]], [[111/77]], [[75/52]]
|-
| 165
| 636.65
|
| ''[[101/70]]'', [[13/9]], [[146/101]]
|-
| 166
| 640.51
|
| [[81/56]], [[123/85]], [[178/123]], [[55/38]], [[152/105]], [[42/29]], [[113/78]], [[100/69]]
|-
| 167
| 644.37
| sd5
| [[29/20]], [[132/91]], [[74/51]], [[119/82]], [[164/113]], [[45/31]], ''[[170/117]]''
|-
| 168
| 648.23
|
| [[138/95]], '''[[93/64]]''', [[109/75]], '''[[16/11]]'''
|-
| 169
| 652.09
|
| [[99/68]], [[51/35]], [[35/24]]
|-
| 170
| 655.94
|
| ''[[124/85]]'', [[54/37]], [[73/50]], [[92/63]], [[111/76]], [[130/89]], [[168/115]], [[19/13]], ''[[136/93]]''
|-
| 171
| 659.8
|
| ''[[174/119]]'', [[117/80]], [[60/41]], [[101/69]], [[41/28]], [[148/101]], [[85/58]]
|-
| 172
| 663.66
|
| [[22/15]], [[113/77]], [[91/62]], [[160/109]], ''[[119/81]]''
|-
| 173
| 667.52
|
| [[72/49]], [[25/17]], [[178/121]], '''[[128/87]]'''
|-
| 174
| 671.38
|
| [[81/55]], [[109/74]], [[28/19]], [[115/78]], [[146/99]]
|-
| 175
| 675.24
| d6
| [[121/82]], [[31/21]], [[96/65]], [[65/44]], [[164/111]], [[34/23]]
|-
| 176
| 679.09
|
| [[176/119]], [[108/73]], [[182/123]], [[37/25]], [[114/77]], [[77/52]], [[40/27]]
|-
| 177
| 682.95
|
| [[126/85]], [[132/89]], [[89/60]], [[46/31]], '''[[95/64]]''', [[49/33]], [[150/101]]
|-
| 178
| 686.81
|
| [[101/68]], [[52/35]], [[162/109]], [[55/37]], [[168/113]], [[113/76]], [[58/39]], [[119/80]], [[180/121]]
|-
| 179
| 690.67
|
| [[73/49]], [[76/51]], [[82/55]], [[85/57]]
|-
| 180
| 694.53
|
| [[109/73]], [[112/75]], [[115/77]], [[121/81]], [[130/87]], [[136/91]], [[148/99]]
|-
| 181
| 698.39
|
| [[178/119]], [[184/123]]
|-
| 182
| 702.25
| P5
| '''[[3/2]]'''
|-
| 183
| 706.1
|
| [[182/121]], [[176/117]], [[170/113]], [[164/109]], [[152/101]], ''[[140/93]]''
|-
| 184
| 709.96
|
| '''[[128/85]]''', [[116/77]], [[113/75]], [[110/73]], [[104/69]], [[98/65]], [[95/63]]
|-
| 185
| 713.82
|
| [[77/51]], [[74/49]], [[68/45]]
|-
| 186
| 717.68
|
| [[62/41]], [[121/80]], [[180/119]], [[174/115]], [[115/76]], [[56/37]], [[109/72]], [[50/33]]
|-
| 187
| 721.54
|
| [[144/95]], [[138/91]], [[91/60]], [[44/29]], [[85/56]], [[41/27]]
|-
| 188
| 725.4
|
| [[117/77]], [[38/25]], [[111/73]], [[184/121]], [[73/48]], [[178/117]], [[35/23]]
|-
| 189
| 729.26
|
| [[99/65]], '''[[32/21]]''', [[154/101]], ''[[119/78]]''
|-
| 190
| 733.11
| sd6
| [[29/19]], [[113/74]], [[84/55]], [[55/36]], [[136/89]]
|-
| 191
| 736.97
|
| [[26/17]], [[101/66]], [[176/115]], [[75/49]], [[124/81]], '''[[49/32]]''', [[170/111]], [[95/62]]
|-
| 192
| 740.83
|
| [[23/15]], [[112/73]], [[89/58]], [[152/99]]
|-
| 193
| 744.69
|
| [[63/41]], [[146/95]], [[186/121]], [[123/80]], [[20/13]]
|-
| 194
| 748.55
|
| [[117/76]], [[174/113]], [[77/50]], [[57/37]], [[168/109]], [[37/24]]
|-
| 195
| 752.41
|
| [[54/35]], [[88/57]], [[105/68]], [[156/101]], [[190/123]], [[17/11]]
|-
| 196
| 756.27
|
| [[184/119]], [[116/75]], '''[[99/64]]''', [[65/42]], [[178/115]], [[113/73]], [[48/31]]
|-
| 197
| 760.12
| sA5
| [[31/20]], [[138/89]], [[76/49]], [[121/78]], [[45/29]]
|-
| 198
| 763.98
|
| ''[[132/85]]'', [[87/56]], [[101/65]], [[115/74]], [[14/9]]
|-
| 199
| 767.84
|
| [[109/70]], [[176/113]], [[81/52]], [[148/95]], [[120/77]], [[170/109]]
|-
| 200
| 771.7
|
| [[39/25]], '''[[64/41]]''', [[89/57]], [[114/73]], [[164/105]], '''[[25/16]]''', [[136/87]]
|-
| 201
| 775.56
|
| ''[[186/119]]'', [[36/23]], [[119/76]]
|-
| 202
| 779.42
|
| [[58/37]], [[69/44]], [[80/51]], [[91/58]], [[102/65]], [[113/72]], [[146/93]], [[190/121]]
|-
| 203
| 783.27
|
| [[11/7]], [[184/117]], [[140/89]], ''[[85/54]]''
|-
| 204
| 787.13
|
| [[63/40]], [[178/113]], [[115/73]], [[52/33]], [[41/26]]
|-
| 205
| 790.99
| m6
| '''[[101/64]]''', [[30/19]], [[109/69]], '''[[128/81]]''', [[49/31]]
|-
| 206
| 794.85
|
| [[117/74]], [[87/55]], [[144/91]], [[182/115]], [[19/12]], [[160/101]]
|-
| 207
| 798.71
|
| [[65/41]], [[176/111]], [[111/70]], [[46/29]], [[119/75]], [[192/121]], [[73/46]], [[100/63]]
|-
| 208
| 802.57
|
| [[27/17]], [[116/73]], [[89/56]], [[62/39]], [[35/22]], [[148/93]]
|-
| 209
| 806.43
|
| [[78/49]], [[121/76]], [[180/113]], [[196/123]], '''[[51/32]]''', [[110/69]]
|-
| 210
| 810.28
|
| [[174/109]], [[91/57]], [[190/119]], [[99/62]], [[115/72]], [[123/77]]
|-
| 211
| 814.14
|
| '''[[8/5]]'''
|-
| 212
| 818.0
| A5
| [[117/73]], [[109/68]], [[101/63]], [[93/58]], [[178/111]], [[162/101]], [[77/48]], [[146/91]], [[130/81]]
|-
| 213
| 821.86
|
| [[45/28]], [[82/51]], [[119/74]], [[37/23]], [[140/87]]
|-
| 214
| 825.72
|
| [[66/41]], [[124/77]], [[182/113]], [[29/18]], [[50/31]]
|-
| 215
| 829.58
|
| [[121/75]], [[192/119]], [[92/57]], [[113/70]], [[176/109]], [[21/13]], [[160/99]]
|-
| 216
| 833.44
|
| [[186/115]], [[55/34]], [[144/89]], [[89/55]], [[123/76]], [[34/21]], [[196/121]]
|-
| 217
| 837.29
|
| ''[[81/50]]'', [[154/95]], [[60/37]], [[73/45]], [[112/69]], [[164/101]], ''[[190/117]]''
|-
| 218
| 841.15
|
| ''[[138/85]]'', '''[[13/8]]''', [[200/123]], [[148/91]]
|-
| 219
| 845.01
|
| [[184/113]], [[57/35]], [[101/62]], [[44/27]], [[119/73]], [[75/46]]
|-
| 220
| 848.87
| N6
| [[31/19]], [[111/68]], [[80/49]], [[178/109]], [[49/30]], [[152/93]], [[85/52]]
|-
| 221
| 852.73
|
| [[121/74]], [[18/11]], [[113/69]], [[95/58]]
|-
| 222
| 856.59
|
| [[182/111]], [[41/25]], [[146/89]], '''[[105/64]]''', '''[[64/39]]'''
|-
| 223
| 860.45
|
| [[156/95]], [[202/123]], [[23/14]], [[120/73]], [[74/45]], [[51/31]]
|-
| 224
| 864.3
|
| [[186/113]], [[28/17]], [[89/54]], [[150/91]]
|-
| 225
| 868.16
|
| [[33/20]], [[104/63]], [[180/109]], [[109/66]], [[38/23]], [[119/72]], [[200/121]]
|-
| 226
| 872.02
|
| [[81/49]], [[124/75]], [[91/55]], [[48/29]], [[154/93]], [[164/99]]
|-
| 227
| 875.88
|
| [[58/35]], [[121/73]], [[184/111]], [[63/38]], [[68/41]], [[73/44]]
|-
| 228
| 879.74
| d7
| [[93/56]], [[108/65]], [[113/68]], [[123/74]], '''[[128/77]]''', [[148/89]], [[168/101]]
|-
| 229
| 883.6
|
| ''[[198/119]]'', [[5/3]]
|-
| 230
| 887.45
|
| [[202/121]], [[192/115]], [[182/109]], [[152/91]]
|-
| 231
| 891.31
|
| ''[[117/70]]'', [[92/55]], [[87/52]], [[82/49]], [[77/46]], [[196/117]]
|-
| 232
| 895.17
|
| [[62/37]], [[176/105]], [[57/34]], [[109/65]], [[52/31]], [[146/87]], ''[[136/81]]''
|-
| 233
| 899.03
|
| [[42/25]], [[121/72]], [[200/119]], [[116/69]], [[190/113]], [[37/22]], ''[[170/101]]''
|-
| 234
| 902.89
|
| [[69/41]], [[101/60]], '''[[32/19]]''', [[123/73]], [[91/54]], [[150/89]], [[204/121]]
|-
| 235
| 906.75
| M6
| '''[[27/16]]''', [[184/109]], [[130/77]], [[76/45]], [[49/29]]
|-
| 236
| 910.61
|
| [[93/55]], [[208/123]], [[115/68]], [[22/13]], [[105/62]]
|-
| 237
| 914.46
|
| [[144/85]], [[178/105]], [[39/23]], [[95/56]], [[56/33]]
|-
| 238
| 918.32
|
| ''[[202/119]]'', [[124/73]], [[192/113]], [[17/10]], [[148/87]]
|-
| 239
| 922.18
|
| [[63/37]], '''[[109/64]]''', [[46/27]], [[196/115]], [[75/44]]
|-
| 240
| 926.04
|
| ''[[162/95]]'', [[29/17]], [[186/109]], '''[[128/75]]''', [[99/58]], [[70/41]], [[111/65]], [[152/89]], [[41/24]], ''[[200/117]]''
|-
| 241
| 929.9
|
| [[65/38]], [[77/45]], [[89/52]], [[190/111]], [[113/66]]
|-
| 242
| 933.76
|
| [[12/7]], ''[[170/99]]''
|-
| 243
| 937.62
| sd7
| [[146/85]], '''[[55/32]]''', [[208/121]], [[98/57]], [[160/93]]
|-
| 244
| 941.47
|
| ''[[117/68]]'', [[198/115]], [[31/18]], [[174/101]], [[112/65]], [[50/29]], ''[[119/69]]''
|-
| 245
| 945.33
|
| [[69/40]], [[88/51]], [[126/73]], [[164/95]], [[202/117]], [[19/11]], ''[[140/81]]''
|-
| 246
| 949.19
|
| [[121/70]], '''[[64/37]]''', [[109/63]], [[154/89]], [[45/26]]
|-
| 247
| 953.05
|
| [[26/15]], '''[[111/64]]''', [[196/113]], [[85/49]], [[210/121]]
|-
| 248
| 956.91
|
| [[33/19]], [[73/42]], [[113/65]], [[40/23]]
|-
| 249
| 960.77
|
| [[87/50]], [[148/85]], [[101/58]], [[54/31]], [[115/66]], [[176/101]], [[190/109]], [[68/39]]
|-
| 250
| 964.63
| sA6
| [[89/51]], [[96/55]], [[110/63]], [[152/87]]
|-
| 251
| 968.48
|
| [[208/119]], '''[[7/4]]'''
|-
| 252
| 972.34
|
| [[198/113]], [[184/105]], [[156/89]], '''[[128/73]]''', [[121/69]], [[114/65]], [[100/57]]
|-
| 253
| 976.2
|
| [[72/41]], [[202/115]], [[65/37]], [[123/70]], [[58/33]], [[109/62]], [[160/91]], [[51/29]], [[95/54]]
|-
| 254
| 980.06
|
| [[44/25]], [[81/46]], [[192/109]], [[37/21]], [[178/101]], ''[[164/93]]''
|-
| 255
| 983.92
|
| [[30/17]], '''[[113/64]]''', [[196/111]], [[136/77]]
|-
| 256
| 987.78
|
| [[99/56]], [[168/95]], [[23/13]], [[200/113]], [[154/87]], [[85/48]]
|-
| 257
| 991.63
|
| [[62/35]], [[101/57]], [[218/123]], [[39/22]], [[204/115]], [[55/31]]
|-
| 258
| 995.49
| m7
| [[87/49]], '''[[16/9]]'''
|-
| 259
| 999.35
|
| [[121/68]], [[89/50]], [[162/91]], [[73/41]], [[130/73]], '''[[57/32]]''', [[98/55]], [[180/101]], [[41/23]]
|-
| 260
| 1003.21
|
| [[66/37]], [[91/51]], [[116/65]], [[216/121]], [[25/14]]
|-
| 261
| 1007.07
|
| [[202/113]], [[152/85]], [[93/52]], [[220/123]], [[34/19]], [[111/62]]
|-
| 262
| 1010.93
|
| [[138/77]], [[52/29]], [[113/63]], [[70/39]]
|-
| 263
| 1014.79
|
| [[88/49]], '''[[115/64]]''', [[124/69]], [[160/89]], [[178/99]], [[196/109]]
|-
| 264
| 1018.64
|
| [[9/5]], [[218/121]], [[200/111]], [[182/101]], [[164/91]], [[146/81]], [[119/66]]
|-
| 265
| 1022.5
| A6
| [[101/56]], [[92/51]], [[74/41]], [[204/113]], [[65/36]], [[56/31]]
|-
| 266
| 1026.36
|
| [[132/73]], [[208/115]], [[123/68]], [[38/21]], [[105/58]]
|-
| 267
| 1030.22
|
| [[154/85]], '''[[29/16]]''', [[136/75]], [[49/27]]
|-
| 268
| 1034.08
|
| ''[[216/119]]'', [[69/38]], [[89/49]], [[198/109]], [[109/60]], [[20/11]]
|-
| 269
| 1037.94
|
| [[202/111]], [[91/50]], [[162/89]], [[224/123]], [[51/28]], [[184/101]], [[82/45]], [[113/62]]
|-
| 270
| 1041.8
|
| [[31/17]], [[104/57]], [[73/40]], [[115/63]], [[42/23]], [[95/52]], [[148/81]], ''[[170/93]]''
|-
| 271
| 1045.65
|
| '''[[117/64]]''', '''[[64/35]]''', [[75/41]], [[119/65]]
|-
| 272
| 1049.51
|
| [[174/95]], [[218/119]], [[11/6]], [[222/121]], [[200/109]]
|-
| 273
| 1053.37
| N7
| ''[[156/85]]'', [[101/55]], [[90/49]], [[226/123]], [[68/37]], [[182/99]], [[57/31]], [[160/87]]
|-
| 274
| 1057.23
|
| [[46/25]], [[208/113]], [[81/44]], [[116/63]], [[186/101]], [[35/19]], [[164/89]]
|-
| 275
| 1061.09
|
| [[24/13]], [[85/46]]
|-
| 276
| 1064.95
|
| [[220/119]], [[37/20]], [[224/121]], [[50/27]]
|-
| 277
| 1068.81
|
| [[176/95]], [[63/34]], [[202/109]], [[76/41]], [[89/48]], [[102/55]], [[115/62]], '''[[128/69]]'''
|-
| 278
| 1072.66
|
| [[13/7]], [[210/113]], [[184/99]], '''[[119/64]]'''
|-
| 279
| 1076.52
|
| ''[[93/50]]'', [[121/65]], [[54/29]], [[95/51]], [[136/73]], [[218/117]], [[41/22]]
|-
| 280
| 1080.38
|
| [[69/37]], [[222/119]], [[28/15]], [[226/121]], [[170/91]]
|-
| 281
| 1084.24
| d8
| [[230/123]], [[144/77]], [[101/54]], [[58/31]], [[204/109]], [[73/39]]
|-
| 282
| 1088.1
|
| [[178/95]], [[208/111]], '''[[15/8]]''', [[152/81]]
|-
| 283
| 1091.96
|
| [[92/49]], [[77/41]], [[216/115]], [[62/33]], [[109/58]], [[220/117]], ''[[190/101]]''
|-
| 284
| 1095.81
|
| '''[[32/17]]''', [[113/60]], [[130/69]], [[228/121]], [[49/26]], [[164/87]]
|-
| 285
| 1099.67
|
| [[66/35]], [[232/123]], [[117/62]], [[168/89]], [[17/9]]
|-
| 286
| 1103.53
|
| [[138/73]], '''[[121/64]]''', [[104/55]], [[87/46]], [[70/37]], [[123/65]], [[176/93]]
|-
| 287
| 1107.39
|
| [[36/19]], [[218/115]], [[91/48]], [[146/77]], [[55/29]], [[74/39]]
|-
| 288
| 1111.25
| M7
| [[93/49]], [[226/119]], [[19/10]], [[230/121]], [[192/101]], [[154/81]]
|-
| 289
| 1115.11
|
| [[78/41]], [[99/52]], [[40/21]]
|-
| 290
| 1118.97
|
| ''[[162/85]]'', [[124/65]], [[208/109]], [[21/11]], [[170/89]]
|-
| 291
| 1122.82
|
| [[216/113]], [[65/34]], [[174/91]], [[109/57]], [[44/23]], [[111/58]], [[178/93]], [[224/117]]
|-
| 292
| 1126.68
|
| [[182/95]], [[228/119]], [[23/12]], [[232/121]], [[140/73]], [[190/99]], ''[[119/62]]''
|-
| 293
| 1130.54
|
| [[48/25]], [[121/63]], [[73/38]], [[98/51]], '''[[123/64]]''', [[148/77]], [[25/13]]
|-
| 294
| 1134.4
|
| [[202/105]], [[77/40]], [[52/27]], [[210/109]]
|-
| 295
| 1138.26
|
| [[27/14]], [[218/113]], [[164/85]], [[110/57]], [[222/115]], [[56/29]], [[226/117]], [[85/44]]
|-
| 296
| 1142.12
| sd8
| [[230/119]], [[29/15]], [[234/121]], [[176/91]], [[89/46]], [[238/123]], [[60/31]]
|-
| 297
| 1145.98
|
| [[184/95]], '''[[31/16]]''', [[126/65]], [[95/49]], '''[[64/33]]''', [[196/101]]
|-
| 298
| 1149.83
|
| [[33/17]], [[101/52]], [[68/35]], [[35/18]]
|-
| 299
| 1153.69
|
| [[72/37]], [[109/56]], [[146/75]], [[220/113]], [[37/19]], [[224/115]], [[150/77]], [[113/58]], [[76/39]]
|-
| 300
| 1157.55
|
| [[232/119]], [[39/20]], [[80/41]], [[121/62]], [[41/21]], ''[[170/87]]''
|-
| 301
| 1161.41
|
| [[174/89]], [[88/45]], [[178/91]], [[45/23]], [[182/93]]
|-
| 302
| 1165.27
|
| ''[[186/95]]'', [[96/49]], [[49/25]], [[198/101]], [[100/51]], [[51/26]]
|-
| 303
| 1169.13
| sA7
| [[108/55]], [[218/111]], [[55/28]], [[222/113]], [[112/57]], [[226/115]], [[57/29]], [[230/117]], ''[[238/121]]''
|-
| 304
| 1172.99
|
| ''[[234/119]]'', [[242/123]], [[124/63]], '''[[63/32]]''', '''[[128/65]]''', [[65/33]], [[136/69]]
|-
| 305
| 1176.84
|
| [[69/35]], [[144/73]], [[73/37]], [[148/75]], [[75/38]], [[152/77]], [[77/39]], [[160/81]]
|-
| 306
| 1180.7
|
| ''[[81/41]]'', [[168/85]], [[87/44]], [[176/89]], [[89/45]], [[180/91]], [[91/46]], [[184/93]], [[95/48]], [[196/99]], ''[[200/101]]''
|-
| 307
| 1184.56
|
| ''[[99/50]]'', [[101/51]], [[208/105]], [[216/109]], [[109/55]], [[220/111]], [[111/56]], [[224/113]], [[113/57]], [[228/115]], [[115/58]], [[232/117]], [[119/60]], [[240/121]], [[123/62]]
|-
| 308
| 1188.42
|
|
|-
| 309
| 1192.28
|
|
|-
| 310
| 1196.14
|
|
|-
| 311
| 1200.0
| P8
| '''[[2/1]]'''
|}
<references group="note" />
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3
| {{monzo| 493 -311 }}
| {{mapping| 311 493 }}
| −0.0933
| 0.0933
| 2.42
|-
| 2.3.5
| 1600000/1594323, {{monzo| -59 5 22 }}
| {{mapping| 311 493 722 }}
| +0.0040
| 0.1573
| 4.08
|-
| 2.3.5.7
| 2401/2400, 65625/65536, 1600000/1594323
| {{mapping| 311 493 722 873 }}
| +0.0331
| 0.1453
| 3.76
|-
| 2.3.5.7.11
| 2401/2400, 3025/3024, 4000/3993, 19712/19683
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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
| {{mapping| 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
| {{mapping| 311 493 722 873 1076 1151 1271 1321 1407 }}
| −0.0033
| 0.1496
| 3.88
|}
 
* 311et has lower relative errors than any previous equal temperaments in the 23-limit and beyond. In the 23-limit it beats [[282edo|282]] and is bettered by [[373edo|373g]] in terms of absolute error, and by [[581edo|581]] in terms of relative error.
* 311et is also notable in the 17- and 19-limit, with lower absolute errors than any previous equal temperaments, beating [[270edo|270]] in both subgroups and is bettered by [[354edo|354]] in the 17-limit, and by [[400edo|400]] in the 19-limit.
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>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
| 23/22
| [[Tertiaseptal]] / tertiaseptia
|-
| 1
| 22\311
| 84.89
| 21/20
| [[Amicable]] / amical / amorous
|-
| 1
| 26\311
| 100.32
| 675/637
| [[Heptacot]]
|-
| 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
| {{Monzo| -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]]
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 
=== Commas ===
Some 41-limit [[comma]]s 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.
 
== Scales ==
=== MOS scales ===
''See: [[User:BudjarnLambeth/311edo MOS scales]].''
 
=== Mode 16 of the harmonic series ===
311edo accurately approximates the mode 16 of [[harmonic series]].
 
{| class="wikitable center-all"
|-
! Overtones
! 16
! 17
! 18
! 19
! 20
! 21
! 22
! 23
! 24
|-
! JI ratios
| 1/1
| 17/16
| 9/8
| 19/16
| 5/4
| 21/16
| 11/8
| 23/16
| 3/2
|-
! …in cents
| 0
| 104.955
| 203.910
| 297.513
| 386.314
| 470.781
| 551.318
| 628.274
| 701.955
|-
! Degrees in 311edo
| 0
| 27
| 53
| 77
| 100
| 122
| 143
| 163
| 182
|-
! …in cents
| 0
| 104.180
| 204.502
| 297.106
| 385.852
| 470.740
| 551.768
| 628.939
| 702.251
|}
 
{| class="wikitable center-all"
|-
! Overtones
! 25
! 26
! 27
! 28
! 29
! 30
! 31
! 32
|-
! JI ratios
| 25/16
| 13/8
| 27/16
| 7/4
| 29/16
| 15/8
| 31/16
| 2/1
|-
! …in cents
| 772.627
| 840.528
| 905.865
| 968.826
| 1029.577
| 1088.269
| 1145.036
| 1200
|-
! Degrees in 311edo
| 200
| 218
| 235
| 251
| 267
| 282
| 297
| 311
|-
! …in cents
| 771.704
| 841.158
| 906.752
| 968.489
| 1030.23
| 1088.1
| 1145.98
| 1200
|}
 
The scale in adjacent steps is 27, 26, 24, 23, 22, 21, 20, 19, 18, 18, 17, 16, 16, 15, 15, 14. Three interval pairs are conflated: {{nowrap|25/24 ~ 26/25|28/27 ~ 29/28}}, and {{nowrap|30/29 ~ 31/30}}.
 
== Detemperaments ==
The most otonally simple way of detempering 311edo is a [[Ringer scale]]. See [[311edo/Ringer 311]] for details.
 
== Music ==
; [[Eliora]]
* [https://www.youtube.com/watch?v=GYzCOpwfTrg ''Etude in C'', Op. 1, No. 1] (2022)
 
; [[Francium]]
* "From the Ground" from ''Scoop'' (2024) – [https://open.spotify.com/track/1f6bIxfJ2BOdNaYomqOMYs Spotify] | [https://francium223.bandcamp.com/track/from-the-ground Bandcamp] | [https://www.youtube.com/watch?v=7Hg1A7F1-Wc YouTube]
* "Translator Server Error" from ''Naughty Girl Era'' (2024) – [https://open.spotify.com/track/7h7rrd7iQCbrzYvstSzla0 Spotify] | [https://francium223.bandcamp.com/track/translator-server-error Bandcamp] | [https://www.youtube.com/watch?v=XD3WoUVgc_M YouTube]
* "Vermin Supreme" from ''The Scallop Disco Accident'' (2025) – [https://open.spotify.com/track/7CfDZrVpfvXJqlMgdufcc7 Spotify] | [https://francium223.bandcamp.com/track/vermin-supreme Bandcamp] | [https://www.youtube.com/watch?v=62uK_ykpmh4 YouTube]
* "Love Is Just a Flying Pig Going to a Funeral." from ''Random Sentences'' (2025) – [https://open.spotify.com/track/12YjJv0nmd8URmNGQZJrqu Spotify] | [https://francium223.bandcamp.com/track/love-is-just-a-flying-pig-going-to-a-funeral Bandcamp] | [https://www.youtube.com/watch?v=ty-W_UIBE5c YouTube]
* "kumturd" from ''wiloliquy'' (2025) – [https://open.spotify.com/track/6Oh0vTTepdUOM2uJORj5dM Spotify] | [https://francium223.bandcamp.com/track/kumturd Bandcamp] | [https://www.youtube.com/watch?v=9evKRneZV0g YouTube]
* "Is That An Albino Duck?" from ''Questions, Vol. 2'' (2025) – [https://open.spotify.com/track/5PJcYEi4lCwisz9XZUtLRR Spotify] | [https://francium223.bandcamp.com/track/is-that-an-albino-duck Bandcamp] | [https://www.youtube.com/watch?v=-kH4PNezV1M YouTube]
* "Don't Worry About Me" from ''Don't'' (2025) – [https://open.spotify.com/track/50RsAwtgZczThSqV6mjmsu Spotify] | [https://francium223.bandcamp.com/track/dont-worry-about-me Bandcamp] | [https://www.youtube.com/watch?v=7euMpGAcI14 YouTube]
 
; [[Tee Teck Wei]]
* [https://www.youtube.com/watch?v=HqShkc6Fl30 ''Baoyu（𨰻𨰻）''] ([https://www.bilibili.com/video/BV1ph4y1f75h Bilibili video of the same work of him/her for China mainland users who can't access Youtube normally]) (2023) &ndash; for electric organs tuned in 311edo
 
== External links ==
* [http://tonalsoft.com/enc/g/gene.aspx gene, 311-edo] on [[Tonalsoft Encyclopedia]]
 
== References ==


[[Category:Theory]]
[[Category:Listen]]
[[Category:Equal divisions of the octave]]
[[Category:311edo| ]] <!-- main article -->
[[Category:Prime EDO]]
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