Quinticular comma

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 height (log2 nd)	40.8719
Weil height (log2 max(n, d))	40.8746
Wilson height (sopfr(nd))	124
Harmonic entropy (Shannon, [math]\displaystyle{ \sqrt{nd} }[/math])	~1.29923 bits
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

• Glacier comma – another example of quintic-particular comma