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311edo: Difference between revisions

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**Imported revision 239198765 - Original comment: **
 
 
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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}}
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-28 16:18:15 UTC</tt>.<br>
 
: The original revision id was <tt>239198765</tt>.<br>
311edo is notable for its extremely high [[consistency limit]], which provides efficient and well-tempered [[just interval]] representation relative to its size.
: The revision comment was: <tt></tt><br>
 
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
== Theory ==
<h4>Original Wikitext content:</h4>
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]]).  
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The //311 equal division// is a remarkable very high limit equal temperament, dividing the octave into 311 equal parts of 3.859 cents each. It is [[consistent]] through the 41 limit and uniquely consistent through the 23 limit, and is a [[The Riemann Zeta Function#Zeta EDO lists|zeta gap edo]].</pre></div>
 
<h4>Original HTML content:</h4>
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]].
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;311edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The &lt;em&gt;311 equal division&lt;/em&gt; is a remarkable very high limit equal temperament, dividing the octave into 311 equal parts of 3.859 cents each. It is &lt;a class="wiki_link" href="/consistent"&gt;consistent&lt;/a&gt; through the 41 limit and uniquely consistent through the 23 limit, and is a &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function#Zeta EDO lists"&gt;zeta gap edo&lt;/a&gt;.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
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 ===
{{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 ==
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）''] (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:Listen]]
Retrieved from "https://en.xen.wiki/w/311edo"