Gene's 10985/10976 or (14/13)^3 vs. 5/4 as cantonisma

This is a quick executive summary on a longer post in which I
propose the name cantonisma for the comma described in 2010 by
Gene Ward Smith equal to the difference between 5/4 and (14/13)^3
at 10985/10976 (1.419 cents).

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The cantonisma arises in a minutely retuned expansion of Gene's
Cantonpenta scale, his original version being a tempering of his
just 12-note Canton tuning in 271-EDO, with 14/13 (+7 fifths)
virtually just at 128.413 cents, and a fifth at 704.059 cents.

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My minute retuning was to set the fifth at precisely (224/13)^1/7
or 704.043 cents for a just 14/13 (128.298 cents), then expanding
Gene's 12-note Cantonpenta into a 17-MOS system which, if spelled
as Gb-A#, has the symmetries of Cantonpenta if D is the 1/1 (the
point of symmetry, with 8 fifths down and 8 fifths up).

By expanding this 17-MOS to a 29-MOS, and then placing two 29-MOS
chains at 58.786 cents apart to achieve a just 12:13:14 division
(with 14/13 and 7/6 pure, and thus also 13/12), I arrived at a
58-note rank-3 system where +21 fifths produces a just (14/13)^3
or 2744/2197 at 384.895 cents, a cantonisma narrow of 5/4 -- some
half a cent more accurate than the schismatic approximation at
8192/6561 (384.360 cents).

This 2744/2197 approximation at 384.9 cents in 16 locations, plus
another 16 locations where a mapping of -13 fifths (447.446
cents) less the spacing generator of 58.786 cents produces a
major third at 388.660 cents (2.347 cents wide of 5/4), result
in 5/4 approximations within 2.35 cents of just at 32 of 58
locations.

For this 58-note system I propose the name Cantonpentamint-58,
the "-mint," as in Peppermint, implying a rank-3 system with two
MOS chains spaced for a just 7/6 from tone (+2 fifths) plus spacing.
Here this means (2, 704.043, 58.786), as compared with Peppermint
at (2, 704.096, 58.680).

Gene's Canton and Cantonpenta are of special interest because of
their 12-note structure with discontinuous chains of fifths. While Cantonpenta in 271-EDO has a fifth almost identical to
Keenan Pepper's Noble Fifth tuning (704.096 cents) and the rank-3
Peppermint, Gene's discontinuous 12-note structure makes his
concept quite unique, and very creative!

However, my special interest in this briefer post is the
cantonisma at 10985/10976 itself, a comma noted by Gene in 2010
which is beautifully exemplified by a minutely retuned and then
expanded version of his Cantonpenta.

! cantonpentamint58.scl
!
rank-3 variant on Gene Ward Smith's Cantonpenta with just 12:13:14
58
!
27.51001
48.51128
58.78570
79.78697
107.29698
128.29824
138.57267
176.80952
187.08394
208.08521
235.59522
256.59649
266.87091
287.87218
315.38219
336.38346
346.65788
384.89473
395.16916
416.17043
443.68043
464.68170
474.95613
495.95739
523.46740
544.46867
554.74309
575.74436
603.25437
624.25564
634.53006
672.76691
683.04134
704.04261
731.55261
752.55388
762.82831
783.82957
811.33958
832.34085
842.61527
880.85213
891.12655
912.12782
939.63783
960.63910
970.91352
991.91479
1019.42480
1040.42606
1050.70049
1088.93734
1099.21176
1120.21303
1147.72304
1168.72431
1178.99873
2/1

With many thanks,

Margo Schulter
mschulter@...

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> This is a quick executive summary on a longer post in which I
> propose the name cantonisma for the comma described in 2010 by
> Gene Ward Smith equal to the difference between 5/4 and (14/13)^3
> at 10985/10976 (1.419 cents).

Thanks for the name! I've added it to the Xenwiki comma list. Tempering out the cantonisma is a feature 224, 270 and 494, among others, have in common.