Phaotic comma: Difference between revisions

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The '''phaotic comma''', or '''phaotisma''' (from Ancient Greek ''pháos'' 'light'), is an [[unnoticeable comma|unnoticeable]] [[13-limit]] (also 2.3.5.13 subgroup) comma with a ratio of '''256000/255879''', and a size of approximately 0.82 [[cent]]s. It is significant in that it is the interval separating a stack of two [[81/80|meantone commas]] from [[40/39]], as well as the interval separating an octave-reduced stack of three instances of [[27/20]] from [[16/13]].
The '''phaotic comma''', or '''phaotisma''' (from Ancient Greek ''pháos'' 'light'), is an [[unnoticeable comma|unnoticeable]] [[13-limit]] ([[2.3.5.13 subgroup]]) comma with the ratio '''256000/255879''', and a size of approximately 0.82{{cent}}. It separates a stack of two [[81/80|syntonic commas]] from [[40/39]], or three [[27/20]] acute fourths from [[16/13]] (up an octave). It factors into {{nowrap|([[4096/4095]])([[4375/4374]])}} or {{nowrap|([[6656/6561]])([[507/500]])<sup>&minus;1</sup>}}.


== Temperaments ==
== Temperaments ==

Revision as of 22:11, 29 November 2024

Interval information
Ratio 256000/255879
Factorization 211 × 3-9 × 53 × 13-1
Monzo [11 -9 3 0 0 -1
Size in cents 0.818472¢
Names phaotic comma,
phaotisma
Color name s3uy31, Sathutriyo comma
FJS name [math]\displaystyle{ \text{A1}^{5,5,5}_{13} }[/math]
Special properties reduced
Tenney norm (log2 nd) 35.9309
Weil norm (log2 max(n, d)) 35.9316
Wilson norm (sopfr(nd)) 77
Comma size unnoticeable
S-expression S79⋅S802
Open this interval in xen-calc

The phaotic comma, or phaotisma (from Ancient Greek pháos 'light'), is an unnoticeable 13-limit (2.3.5.13 subgroup) comma with the ratio 256000/255879, and a size of approximately 0.82 ¢. It separates a stack of two syntonic commas from 40/39, or three 27/20 acute fourths from 16/13 (up an octave). It factors into (4096/4095)(4375/4374) or (6656/6561)(507/500)−1.

Temperaments

Tempering it out splits 40/39 into two equal halves, each representing 81/80. It leads to a form of phaotismic temperament in the full 13-limit and phaotic temperament in the 2.3.5.13 subgroup.

See also