Yahoo Tuning Groups Ultimate Backup tuning An interesting non-just non-equal temperament

I was playing around with some tunings the other day, and noticed that if I
detune the major and minor thirds slightly, I can get a close approximation
to 7/4 by going up a major third and down two minor thirds (which would be
125/72 in JI). So instead of the usual 225/224, this scale is based on
making the 126/125 vanish. Specifically, the major third is 388 cents (1.7
cents sharp) and the minor third is 312 cents (3.6 cents flat), resulting
in a perfect fifth of 700 cents (2 cents flat) and an augmented sixth of
964 cents (4.8 cents flat). Here's a good 12-note subset of this tuning,
which has a number of complete tetrads and triads. (Can anyone find a
better 12-note subset?)

A#
C / \ G
/ \
B-----F#----C#
\ Ab/ \ Eb/ \
\ / \ / \
D-----A-----E
\ / \ / \
\ / \ / A#\
F-----C-----G
\ / \ /
B \ / F#\ / C#
Ab----Eb

Another interesting thing about this scale is that the "fifths" G-D and E-B
are only about 14 cents flat, which produces noticeable beating, but not
nearly as unpleasant as the syntonic comma in traditional JI. Furthermore,
this scale has a good approximation of 11/8, from taking a major third down
and three minor thirds up (resulting in an interval of 548 cents, only 3.3
cents flat). In the 12-note subset, this 11/8 interval is found between A#
and Eb.

I've also been playing with 7-note "modes" of this scale; in fact, the mode
which led me to discover this tuning is Ab A C C# Eb E F#. Another
interesting mode, which reminds me of some middle-eastern music, is G Ab A#
C D Eb F.

--
see my music page ---> +--<http://www.io.com/~hmiller/music/music.html>--
Thryomanes /"If all Printers were determin'd not to print any
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Here's a reply from Dave Keenan to a message I sent him about the new scale
I've been playing with lately:

>>This is a description of a scale I sent to the tuning list. Paul Erlich
>>suggested that I should contact you, since you've done a lot of work with
>>similar scales. Have you heard of anything like this before?
>
>No. I haven't seen this one before. Thanks. I love these things!
>
>I can however describe it as a third (and previously unsuspected) member of
>a class of 12-tone 7-limit tunings for which I don't have a name. The other
>two members are a 12 of 31-tET (or nearby meantone) that I found (I really
>just converted an existing Just scale to 31-tET/meantone) and a tuning
>where the 225/224 disappears due to Carl Lumma (and we later learned that
>Fokker had described it earlier in Just terms, rather than tempered).
>
>So in mine we distribute the 81/80 and in Lumma's we distribute the
>225/224. Yours is right in between with its distributed 126/125. Beautiful!
>So it is also intermediate in its tradeoff between the number of available
>7-limit harmonies and their accuracy.
>
>Actually, everything I said above applies not to the 12-tone subset you
>gave, but the ones whose lattices appear below. I think you will find they
>have more 7-limit harmonies. At least they have one more tetrad.
>
> D#----A#
> F / \ C / G
> / \ /
> B-----F#
>Db \ Ab/ \
> \ / \
> D-----A-----E
> \ / \ / \
> \ / D#\ / A#\
> F-----C-----G
> / \ /
> / B \ / F#
> Db----Ab
>
>We still have E:B and G:D as mild wolves and we still have a good
>approximation to an 8:11, but now it is D#:Ab. Note that the lattice could
>also be:
>
> D#----A#
> F / \ C /
> / \ /
>E-----B-----F#
> \ Db/ \ Ab/ \
> \ / \ / \
> G-----D-----A
> \ / \
> \ / D#\ A#
> F-----C
> / \ /
> E / B \ / F#
> Db----Ab
>
>but one would normally prefer to have more otonal tetrads than utonal. C:G
>and A:E are now the mild wolves. An obvious possibility is to tune the E
>and G halfway between the two results above, so we get 6 tetrads, with two
>of them, and two triads, having much worse fifths than the others, but no
>longer being wolves. This is getting closer to the 31-tET/meantone case.
>
>What this class of tunings (mine, Lumma's and yours) have in common, apart
>from all being proper 12-tone 7-limit scales, is that if notes are mapped
>to the nearest 31-tET notes, they all come out as the same mode, which, as
>a chain of fifths, looks like:
>
>Db Ab . . F C G D A E B F# . . D# A#
>
>So they all have the 3 wolves A#:F, F#:Db and Ab:D#. Lumma's has an
>additional wolf D:A as the price of all the remaining 7-limit intervals
>getting to within 2 cents of Just (which really *is* Just in my book).
>Yours has two additional (but mild-mannered) wolves, or no wolves but 4 bad
>fifths, and can get the remaining 7-limit intervals within 4.6 cents (the
>tempering you describe has a 5:7 which is 6.5 cents narrow).
>
>I may have considered 126/125 for this purpose and rejected it. I think I
>decided that the errors of 4.6 cents were too close to what was available
>from the meantone case. I may have been too hasty.
>
>It would be a worthwhile project to make some tables comparing the
>properties (errors in intervals, numbers of tetrads, triads, consonances)
>of the three tunings (and maybe also a related 7-limit Just tuning). The
>specific tempering chosen for each should be the result of the same kind of
>optimisation e.g. min RMS-error, min max-absolute-error, min
>max-absolute-beat-rate (my current favourite). Or maybe all three optima
>should be examined in all three tunings. Unfortunately I don't have time to
>pursue this at present. Let me know if you ever do.
>
>Feel free to post the above to the tuning list.
>
>Regards,
>
>-- Dave Keenan
>http://dkeenan.com

An interesting non-just non-equal temperament

🔗Herman Miller <hmiller@xx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Herman Miller <hmiller@xx.xxxx>