311edo: Difference between revisions

Line 1,841:

=== Interval mappings ===

=== Zeta peak index ===

{| class="wikitable center-all"

|-

! colspan="3" | Tuning

! colspan="3" | Strength

! colspan="2" | Closest edo

! colspan="2" | Integer limit

|-

! ZPI

! Steps per octave

! Step size (cents)

! Height

! Integral

! Gap

! Edo

! Octave (cents)

! Consistent

! Distinct

|-

| [[2293zpi]]

| 311.004029926555

| 3.85847090239759

| 13.066467

| 1.895394

| 22.370880

| 311edo

| 1199.98445064565

| 42

| 25

|}

== Regular temperament properties ==

Line 1,848:

Line 1,879:

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal 8ve stretch (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning error

|-

Line 1,857:

Line 1,888:

| {{monzo| 493 -311 }}

| {{mapping| 311 493 }}

| ~~−0~~.0933

| −0.0933

| 0.0933

| 2.42

Line 1,885:

Line 1,916:

| 625/624, 1575/1573, 2080/2079, 2200/2197, 2401/2400

| {{mapping| 311 493 722 873 1076 1151 }}

| ~~−0~~.0280

| −0.0280

| 0.1472

| 3.81

Line 1,906:

Line 1,937:

| 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

| −0.0033

| 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 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.

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 ===

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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

|-

! Periods per 8ve

! Periods per 8ve

! Generator*

! Cents*

! Associated ratio*

! Associated ratio*

! Temperaments

|-

Line 2,027:

Line 2,057:

| [[Vydubychi]]

|}

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

<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|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.

~~== Zeta properties ==~~

~~=== Zeta peak index ===~~

~~{| class="wikitable"~~

|-

~~! colspan="3" | Tuning~~

~~! colspan="3" | Strength~~

~~! colspan="2" | Closest EDO~~

~~! colspan="2" | Integer limit~~

|-

~~! ZPI~~

~~! Steps per octave~~

~~! Step size (cents)~~

~~! Height~~

~~! Integral~~

~~! Gap~~

~~! EDO~~

~~! Octave (cents)~~

~~! Consistent~~

~~! Distinct~~

|-

~~| [[2293zpi]]~~

~~| 311.004029926555~~

~~| 3.85847090239759~~

~~| 13.066467~~

~~| 1.895394~~

~~| 22.370880~~

~~| 311edo~~

~~| 1199.98445064565~~

~~| 42~~

~~| 25~~

|}

== Detemperaments ==

@@ Line 1,841: / Line 1,841: @@
 === Interval mappings ===
 {{Q-odd-limit intervals|311|limit=41}}
+=== Zeta peak index ===
+{| class="wikitable center-all"
+|-
+! colspan="3" | Tuning
+! colspan="3" | Strength
+! colspan="2" | Closest edo
+! colspan="2" | Integer limit
+|-
+! ZPI
+! Steps per octave
+! Step size (cents)
+! Height
+! Integral
+! Gap
+! Edo
+! Octave (cents)
+! Consistent
+! Distinct
+|-
+| [[2293zpi]]
+| 311.004029926555
+| 3.85847090239759
+| 13.066467
+| 1.895394
+| 22.370880
+| 311edo
+| 1199.98445064565
+| 42
+| 25
+|}
 == Regular temperament properties ==
@@ Line 1,848: / Line 1,879: @@
 ! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br />8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve stretch (¢)
 ! colspan="2" | Tuning error
 |-
@@ Line 1,857: / Line 1,888: @@
 | {{monzo| 493 -311 }}
 | {{mapping| 311 493 }}
-| &minus;0.0933
+| −0.0933
 | 0.0933
 | 2.42
@@ Line 1,885: / Line 1,916: @@
 | 625/624, 1575/1573, 2080/2079, 2200/2197, 2401/2400
 | {{mapping| 311 493 722 873 1076 1151 }}
-| &minus;0.0280
+| −0.0280
 | 0.1472
 | 3.81
@@ Line 1,906: / Line 1,937: @@
 | 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 }}
-| &minus;0.0033
+| −0.0033
 | 0.1496
 | 3.88
 |}
-et 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 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.
-et 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 ===
@@ Line 1,919: / Line 1,949: @@
 |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
 |-
-! Periods<br />per 8ve
+! Periods<br>per 8ve
 ! Generator*
 ! Cents*
-! Associated<br />ratio*
+! Associated<br>ratio*
 ! Temperaments
 |-
@@ Line 2,027: / Line 2,057: @@
 | [[Vydubychi]]
 |}
-<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|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.
-== Zeta properties ==
-=== Zeta peak index ===
-{| class="wikitable"
-|-
-! colspan="3" | Tuning
-! colspan="3" | Strength
-! colspan="2" | Closest EDO
-! colspan="2" | Integer limit
-|-
-! ZPI
-! Steps per octave
-! Step size (cents)
-! Height
-! Integral
-! Gap
-! EDO
-! Octave (cents)
-! Consistent
-! Distinct
-|-
-| [[2293zpi]]
-| 311.004029926555
-| 3.85847090239759
-| 13.066467
-| 1.895394
-| 22.370880
-| 311edo
-| 1199.98445064565
-| 42
-| 25
-|}
 == Detemperaments ==