Quinticular comma: Difference between revisions

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Created page with "{{Infobox interval |Ratio = 1419857/1417176 |Name = quinticular comma |Color name = s17o<sup>5</sup>4, saquinso 4th |Comma = yes }} '''1419857/1417176''', the '''quinticular comma''' is a 2.3.17 subgroup comma that is the difference between five 18/17s and one 4/3. It is tempered out in following EDOs: * 12''n'' family (up to 120) * 12''n''+1 family (between 109 and 337) == Etymology == The word ''quinticular'' was introduced by Xenllium..."
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Quintic-particular is an idiosyncratic term
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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))<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 ==
== 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]]
[[Category:Commas named for their regular temperament properties]]

Latest revision as of 12:43, 23 May 2025

Interval information
Ratio 1419857/1417176
Subgroup monzo 2.3.17 [-3 -11 5⟩
Size in cents 3.272038¢
Name quinticular comma
Color name s17o54, saquinso 4th
FJS name [math]\displaystyle{ \text{5d4}^{17,17,17,17,17} }[/math]
Special properties reduced
Tenney norm (log2 nd) 40.8719
Weil norm (log2 max(n, d)) 40.8746
Wilson norm (sopfr(nd)) 124
Comma size unnoticeable
Open this interval in xen-calc

The quinticular comma is a 2.3.17 subgroup comma of measuring 3.27 cents, 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 term] 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)).

Temperaments

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

This comma is tempered out in following EDOs:

EDO
group 		Quinticular EDOs
12n 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 2772, 2784
12n+1 109, 121, 133, 145, 157, 169, 181, 193, 205, 217, 229, 241, 253, 265, 277, 289, 301, 313, 325, 337
12n+2 326, 338, 350, 362, 374, 386, 398, 410, 422, 434, 446, 458, 470, 482, 494, 506, 518, 530, 542, 554
12n+3 555, 567, 579, 591, 603, 615, 627, 639, 651, 663, 675, 687, 699, 711, 723, 735, 747, 759, 771, 783
12n+4 772, 784, 796, 808, 820, 832, 844, 856, 868, 880, 892, 904, 916, 928, 940, 952, 964, 976, 988, 1000
12n+5 1001, 1013, 1025, 1037, 1049, 1061, 1073, 1085, 1097, 1109, 1121, 1133, 1145, 1157, 1169, 1181, 1193, 1205, 1217, 1229
12n+6 1230, 1242, 1254, 1266, 1278, 1290, 1302, 1314, 1326, 1338, 1350, 1362, 1374, 1386, 1398, 1410, 1422, 1434, 1446
12n+7 1495, 1507, 1519, 1531, 1543, 1555, 1567, 1579, 1591, 1603, 1615, 1627, 1639, 1651, 1663
12n+8 1748, 1760, 1772, 1784, 1796, 1808, 1820, 1832, 1844, 1856, 1868, 1880, 1892
12n+9 2001, 2013, 2025, 2037, 2049, 2061, 2073, 2085, 2097, 2109
12n+10 2254, 2266, 2278, 2290, 2302, 2314, 2326, 2338
12n+11 2507, 2519, 2531, 2543, 2555

Quinticular

Subgroup: 2.3.17

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: 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 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

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