Tunnel commas

A xenharmonic bridge is a p-limit comma where the exponent of p
is +-1. I'll call a o-odd-limit comma where the product of the prime
factors greater than three is a o-limit-consonance a "tunnel comma";
this somewhat goofy name serves to suggest they share an affinty with
bridge commas.

Tunnel commas are few in number, but they divide the range of tunings
for the fifth up into regions in a significant and highly irregular
way.

Below is a list of the significant 15-limit or less tunnel commas.
Listed first is the comma, and then the no-threes consonance reduced
to the range 1 < z < sqrt(2). Then I give the tuning for the fifth
for which the interval is pure (eigentuning), and then the least
squares tuning. Hence, for 81/80 I give the 1/4-comma tuning, where
the interval 5/4 is pure, and then the Woolhouse tuning. The commas
are listed in order of size of the least squares fifth, from sharp to
flat.

Comma: interval, eigentuning, least squares tuning

64/63: 8/7, 715.587, 711.692

416/405: 13/10, 713.553, 711.368

40/39: 13/10, 745.786, 711.252

896/891: 14/11, 704.377, 703.925

352/351: 13/11, 703.597, 703.117

5120/5103: 7/5, 702.915, 702.889

32805/32768: 5/4, 701.711, 701.728

81/80: 5/4, 696.578, 696.165

1701/1664: 14/13, 694.340, 695.325

1053/1024: 16/13, 689.868, 693.157

45/44: 11/10, 682.502, 691.806

We have an isolated range around 81/80, the meantone region (31, 50,
etc.) We have a less isolated region around 32805/32768, by itself it
is schismatic (53, 65, 118, 171) but put it together with the
slightly sharper 5120/5103 region and you have the garibaldi region
(41, 53, 94.) 896/891 bears the same relationship to 14/11 as 81/80
does to 5/4, and 46, which has nearly pure 14/11s, is analogous in
the same way to 31. Instead of four fifths leading to a 14/11,
352/351 is about three fourths leading to a 13/11. The two are quite
close in tuning, but even closer to 352/351 is the 5120/5103 fifth,
which is about six fifths defining a 10/7. The three commas define a
planar temperment, which can be extended to various linear
temperaments, but which devolve to leapday or 13-limit garibaldi if
we want a fifth as the generator and an octave as the period. This
very interesting range is associated with 46 and the multiples of 29:
58 and 87.

There is now a large gap up to the sharp systems associated with 64/63
(dominant seventh comma) and the curious twin commas 416/405 and
40/39, which act almost in the same way, being mirroed in the comma
(416/405)/(40/39) = 676/675 = (26/15)^2/3.

On the flat side are commas we can associate with 19 and 26, if that
turns you on. But the most striking thing to me in the above is the
41-46 range of tunings, and especially 46 and leapday. The 17/34/68
complex, despite having its horn tooted a lot recently, doesn't seem
as interesting as the 29/58/87 complex from this point of view.