Quinticular comma: Difference between revisions

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~~'''1419857/1417176''', the~~ '''quinticular comma''' is a 2.3.17 [[subgroup]] comma ~~that is~~ the difference between five [[18/17]]s and ~~one~~ [[4/3]]. It is tempered out in following EDOs:

The '''quinticular comma''' is a 2.3.17 [[subgroup]] comma of measuring 3.27 [[cent]]s, with frequency ratio of 1419857/1417176. It forms the difference between five semitones of [[18/17]] and a perfect fourth, [[4/3]]. It is a quintic-particular{{idiosyncratic}} interval, that is, of the form ((''n''+5)/''n'') / ((''n''+3)/(''n''+2))5 = (S(''n''+1)×S(''n''+2)2) / (S(''n''+3)2×S(''n''+4)).

* 12''n'' ~~family (up to~~ 120)

* 12''n''+1 ~~family (between~~ 109 ~~and~~ 337)

== Temperaments ==

Tempering out this comma in the 2.3.17 subgroup leads to the '''quinticular temperament''', which is related to a number of [[Cluster MOS|cluster temperaments]], such as [[quintaleap]], [[quindromeda]], and [[quintaschis]].

This comma is tempered out in following EDOs:

{| class="wikitable center-1"

|-

! EDO group

! Quinticular EDOs

|-

| 12''n'' ||{{Optimal ET sequence|12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 2772, 2784}}

|-

| 12''n''+1 ||{{Optimal ET sequence|109, 121, 133, 145, 157, 169, 181, 193, 205, 217, 229, 241, 253, 265, 277, 289, 301, 313, 325, 337}}

|-

| 12''n''+2 ||{{Optimal ET sequence|326, 338, 350, 362, 374, 386, 398, 410, 422, 434, 446, 458, 470, 482, 494, 506, 518, 530, 542, 554}}

|-

| 12''n''+3 ||{{Optimal ET sequence|555, 567, 579, 591, 603, 615, 627, 639, 651, 663, 675, 687, 699, 711, 723, 735, 747, 759, 771, 783}}

|-

| 12''n''+4 ||{{Optimal ET sequence|772, 784, 796, 808, 820, 832, 844, 856, 868, 880, 892, 904, 916, 928, 940, 952, 964, 976, 988, 1000}}

|-

| 12''n''+5 ||{{Optimal ET sequence|1001, 1013, 1025, 1037, 1049, 1061, 1073, 1085, 1097, 1109, 1121, 1133, 1145, 1157, 1169, 1181, 1193, 1205, 1217, 1229}}

|-

| 12''n''+6 ||{{Optimal ET sequence|1230, 1242, 1254, 1266, 1278, 1290, 1302, 1314, 1326, 1338, 1350, 1362, 1374, 1386, 1398, 1410, 1422, 1434, 1446}}

|-

| 12''n''+7 ||{{Optimal ET sequence|1495, 1507, 1519, 1531, 1543, 1555, 1567, 1579, 1591, 1603, 1615, 1627, 1639, 1651, 1663}}

|-

| 12''n''+8 ||{{Optimal ET sequence|1748, 1760, 1772, 1784, 1796, 1808, 1820, 1832, 1844, 1856, 1868, 1880, 1892}}

|-

| 12''n''+9 ||{{Optimal ET sequence|2001, 2013, 2025, 2037, 2049, 2061, 2073, 2085, 2097, 2109}}

|-

| 12''n''+10 ||{{Optimal ET sequence|2254, 2266, 2278, 2290, 2302, 2314, 2326, 2338}}

|-

| 12''n''+11 ||{{Optimal ET sequence|2507, 2519, 2531, 2543, 2555}}

|}

=== Quinticular ===

[[Subgroup]]: 2.3.17

[[Mapping]]: {{mapping|1 2 5|0 -5 -11}}

: Mapping generators: ~2, ~18/17

[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~18/17 = 99.5839

{{Optimal ET sequence|legend=1|12, 133, 145, 157, 169, 181, 193, 205, 217, 229, 241, 735, 976, 1217, 1458g}}

[[Badness]]: 0.002442

== Etymology ==

The word ''quinticular'' was introduced by [[User:Xenllium|Xenllium]] in 2025.

The word ''quinticular'' was introduced by [[User:Xenllium|Xenllium]] in 2025. It is a contraction of ''quintans'' (Latin for "one fifth", splitting a fourth in five) and ''particular'' into a single word.

== See also ==

* [[Glacier comma]] – another example of quintic-particular comma

[[Category:Commas named for their regular temperament properties]]

@@ Line 6: / Line 6: @@
 }}
-'''1419857/1417176''', the '''quinticular comma''' is a 2.3.17 [[subgroup]] comma that is the difference between five [[18/17]]s and one [[4/3]]. It is tempered out in following EDOs:
+The '''quinticular comma''' is a 2.3.17 [[subgroup]] comma of measuring 3.27 [[cent]]s, with frequency ratio of 1419857/1417176. It forms the difference between five semitones of [[18/17]] and a perfect fourth, [[4/3]]. It is a quintic-particular{{idiosyncratic}} interval, that is, of the form ((''n''+5)/''n'') / ((''n''+3)/(''n''+2))<sup>5</sup> = (S(''n''+1)×S(''n''+2)<sup>2</sup>) / (S(''n''+3)<sup>2</sup>×S(''n''+4)).
-* 12''n'' family (up to 120)
-* 12''n''+1 family (between 109 and 337)
+== Temperaments ==
+Tempering out this comma in the 2.3.17 subgroup leads to the '''quinticular temperament''', which is related to a number of [[Cluster MOS|cluster temperaments]], such as [[quintaleap]], [[quindromeda]], and [[quintaschis]].
+This comma is tempered out in following EDOs:
+{| class="wikitable center-1"
+|-
+! EDO <br>group
+! Quinticular EDOs
+|-
+| 12''n'' ||{{Optimal ET sequence|12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 2772, 2784}}
+|-
+| 12''n''+1 ||{{Optimal ET sequence|109, 121, 133, 145, 157, 169, 181, 193, 205, 217, 229, 241, 253, 265, 277, 289, 301, 313, 325, 337}}
+|-
+| 12''n''+2 ||{{Optimal ET sequence|326, 338, 350, 362, 374, 386, 398, 410, 422, 434, 446, 458, 470, 482, 494, 506, 518, 530, 542, 554}}
+|-
+| 12''n''+3 ||{{Optimal ET sequence|555, 567, 579, 591, 603, 615, 627, 639, 651, 663, 675, 687, 699, 711, 723, 735, 747, 759, 771, 783}}
+|-
+| 12''n''+4 ||{{Optimal ET sequence|772, 784, 796, 808, 820, 832, 844, 856, 868, 880, 892, 904, 916, 928, 940, 952, 964, 976, 988, 1000}}
+|-
+| 12''n''+5 ||{{Optimal ET sequence|1001, 1013, 1025, 1037, 1049, 1061, 1073, 1085, 1097, 1109, 1121, 1133, 1145, 1157, 1169, 1181, 1193, 1205, 1217, 1229}}
+|-
+| 12''n''+6 ||{{Optimal ET sequence|1230, 1242, 1254, 1266, 1278, 1290, 1302, 1314, 1326, 1338, 1350, 1362, 1374, 1386, 1398, 1410, 1422, 1434, 1446}}
+|-
+| 12''n''+7 ||{{Optimal ET sequence|1495, 1507, 1519, 1531, 1543, 1555, 1567, 1579, 1591, 1603, 1615, 1627, 1639, 1651, 1663}}
+|-
+| 12''n''+8 ||{{Optimal ET sequence|1748, 1760, 1772, 1784, 1796, 1808, 1820, 1832, 1844, 1856, 1868, 1880, 1892}}
+|-
+| 12''n''+9 ||{{Optimal ET sequence|2001, 2013, 2025, 2037, 2049, 2061, 2073, 2085, 2097, 2109}}
+|-
+| 12''n''+10 ||{{Optimal ET sequence|2254, 2266, 2278, 2290, 2302, 2314, 2326, 2338}}
+|-
+| 12''n''+11 ||{{Optimal ET sequence|2507, 2519, 2531, 2543, 2555}}
+|}
+=== Quinticular ===
+[[Subgroup]]: 2.3.17
+[[Mapping]]: {{mapping|1 2 5|0 -5 -11}}
+: Mapping generators: ~2, ~18/17
+[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000, ~18/17 = 99.5839
+{{Optimal ET sequence|legend=1|12, 133, 145, 157, 169, 181, 193, 205, 217, 229, 241, 735, 976, 1217, 1458g}}
+[[Badness]]: 0.002442
 == Etymology ==
-The word ''quinticular'' was introduced by [[User:Xenllium|Xenllium]] in 2025.
+The word ''quinticular'' was introduced by [[User:Xenllium|Xenllium]] in 2025. It is a contraction of ''quintans'' (Latin for "one fifth", splitting a fourth in five) and ''particular'' into a single word.
+== See also ==
+* [[Glacier comma]] – another example of quintic-particular comma
 [[Category:Commas named for their regular temperament properties]]