Re: [tuning] Re 3,9,17,19 based scales

Hi John,

Thanks for the citations. The way you describe this--by building a
16:17:18:19 series segment up from 1/1 and 4/3 and a corresponding
subharmonic series down from 3/2 and 2/1--is exactly the way I went
about it:

19/12-----19/16
3/2-`.----9/8 `.
17/12---`.17/16 `.
4/3-------1/1-------3/2
`. 32/17`.---24/17
`.16/9----`.-4/3
32/19-----24/19

Note that you could also get an interesting 11-limit version of the
chromatic gamut by doing the same with an 8:9:10:11 series segment:

5/3---------5/4
/ \' . . '/ \
/ \ . '' ./ \
3/2--11/6---9/8--11/8
/' .. ' \ /' .. ' \
/. ' ' . \ /. ' ' . \
4/3---------1/1---------3/2
\' . . '/ \' . . '/
\ . '' ./ \ . '' ./
16/11-16/9--12/11--4/3
\ /' .. ' \ /
\ /. ' ' . \ /
8/5---------6/5

take care,

--Dan Stearns

----- Original Message -----
From: "John Chalmers" <JHCHALMERS@UCSD.EDU>
To: "tuning" <tuning@yahoogroups.com>
Sent: Saturday, May 25, 2002 1:47 PM
Subject: [tuning] Re 3,9,17,19 based scales

> Xmaxmixmax and Dan Stearns: I located my old notes on 19-limit
> approximations to the chromatic gamut discussed in Tuning Digests
2055
> and 2056. Unfortunately, I had misremembered the scale I mentioned
in an
> earlier post (2057?) unless there is yet another one lurking in the
back
> issues of the Journal of the Acoustic Society of America in the late
> 40's and 50's. In any case, the reference to the scale I had in mind
is
> T. E. Simonton. 1953. Integral Ratio Chromatic Temperament. JASA
25(6):
> 1167. I'm not quite sure of the date as my old index card is very
faded
> -- it could be 1951 or even 1957. The scale is below and includes
> ratios of 5, being based on the sequence (15:)16:17:18:19:20 as
1/1
> 17/16 9/8 19/16 5/4 4/3 17/12 3/2 19/12 5/3 16/9 17/9 2/1.
>
> Replacing the 5/4 with 24/19 and the 5/3 with 32/19 produces Dan
> Stearn's scale in TD 2056. One can construct this scale by building
the
> harmonic series 16:17:18:19 up from 1/1 and 4/3 and the
corresponding
> subharmonc series down from 3/2 and 2/1. This procedure generates
both
> of the tritone-flanking intervals 17/12 and 24/17, which differ by 6
> cents. Probably either could be used, or one might just as well use
600
> cents (square root of 2) in place of either for an error of only 3
> cents.
>
> A somewhat similar scale is Wronski's, described on page 105 of
Jocelyn
> Godwin's book "Music and the Occult." Its ratios are 1/ 1 17/16 9/8
> 85/72 5/4 4/3 17/12 51/32 27/16 85/48 17/9 2/1.
>
> I discoverd that both of the above scales and 4 variants were in one
of
> my JiCalc stacks (JCC) for the Macs. Two 7-tone scales based on
similar
> premises have also been constructed by Dan and me and are in the
stack
> as well.
>
> --John
>
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