65/19 atom: Difference between revisions

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'''272629233/272629760''', the '''65/19&nbsp;atom''', is an unnoticeable comma in the 2.3.5.13.19 subgroup. It is the amount by which the 13-comma [[6656/6561]] ("tetris comma") exceeds the 19/5-comma [[41553/40960]] (an [[apotome]] above [[19/16]] lowered by [[5/4]]). Their difference is small enough that [[sagittal]] accidentals equate the 13-comma with the 19/5 comma: the most accurate possible representation of the 13th harmonic in sagittal accidentals (as three octaves above a <code>)/||| P5</code>) accrues this 0.0033{{cent}} error.
'''272629760/272629233''', the '''65/19&nbsp;atom''', is a [[19-limit]] (2.3.5.13.19 subgroup) [[unnoticeable comma]]. It represents the tiny interval between the pythagorean comma [[pythagorean comma|3<sup>12</sup>/2<sup>19</sup>]] and the wilsorma [[65/64]] lowered by the 19-schisma [[513/512]].


Alternatively, the 65/19&nbsp;atom is the difference between the 1/19&nbsp;comma [[19683/19456]] ([[pythagorean comma|3<sup>12</sup>/2<sup>19</sup>]] lowered by [[513/512]]) and the wilsorma [[65/64]].
Notably, the 65/19&nbsp;atom is also the amount by which the 13-comma [[6656/6561]] ("tetris comma") exceeds the 19/5-comma [[41553/40960]] (an [[apotome]] above [[19/16]] lowered by [[5/4]]). Their difference is small enough that the untempered Promethian (high precision) and Herculean (ultra precision) [[sagittal]] accidental systems equate the 13-comma with the 19/5 comma: the most accurate possible representation of the [[13/1|13th harmonic]] in JI sagittal accidentals at these precision levels (as three octaves above a {{nowrap|{{sagittal| /||| }}P5}}) accrues this 0.0033{{cent}} error.
 
[[Category:Commas named systematically]]

Latest revision as of 16:02, 3 March 2025

Interval information
Ratio 272629760/272629233
Factorization 222 × 3-15 × 5 × 13 × 19-1
Monzo [22 -15 1 0 0 1 0 -1⟩
Size in cents 0.003346521¢
Name 65/19 atom
Color name ss19u3oy2, sasa-nuthoyo 2nd
FJS name [math]\displaystyle{ \text{d2}^{5,13}_{19} }[/math]
Special properties reduced
Tenney height (log2 nd) 56.0447
Weil height (log2 max(n, d)) 56.0447
Wilson height (sopfr(nd)) 126
Comma size unnoticeable
Open this interval in xen-calc

272629760/272629233, the 65/19 atom, is a 19-limit (2.3.5.13.19 subgroup) unnoticeable comma. It represents the tiny interval between the pythagorean comma 312/219 and the wilsorma 65/64 lowered by the 19-schisma 513/512.

Notably, the 65/19 atom is also the amount by which the 13-comma 6656/6561 ("tetris comma") exceeds the 19/5-comma 41553/40960 (an apotome above 19/16 lowered by 5/4). Their difference is small enough that the untempered Promethian (high precision) and Herculean (ultra precision) sagittal accidental systems equate the 13-comma with the 19/5 comma: the most accurate possible representation of the 13th harmonic in JI sagittal accidentals at these precision levels (as three octaves above a ⁠ ⁠P5) accrues this 0.0033 ¢ error.

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