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

1 scales

Thread (2 messages)

From: Margo Schulter (2013-08-19)
Subject: 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).

<http://tech.groups.yahoo.com/group/tuning/message/89597>

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.

<http://tech.groups.yahoo.com/group/tuning/message/96595>

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@...

From: genewardsmith (2013-09-05)
Subject: Re: Gene's 10985/10976 or (14/13)^3 vs. 5/4 as cantonisma

--- 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.