12-tone tunings for 7-limit harmony (for Joseph Pehrson et al)

According to Vogel's _On the Relations of Tone_ and Lindley and
Turner-Smith's _Mathematical Models of Musical Scales_, many 19th and
early 20th century composers were trying to imply 7-limit harmony
using 12-tET notation and instruments. The dominant seventh chord was
used to "imply" 4:5:6:7, and the half-diminished seventh for
1/7:1/6:1/5:1/4. Unfortunately, 12-tET makes errors of up to 33 cents
in the intervals in these chords. Fortunately, these chords are
available at all positions in the scale, making for a total of 24 (12
dominant sevenths and 12 half-dimished sevenths).

It is interesting to consider ways of modifying 12-tET (retaining a
rough scale of "semitones") to improve some of these tetrads. There
is a tradeoff, though -- the better the good tetrads, the fewer of
them we can have.

The first way is to take the subset of 22-tET I call the "symmetrical
dodecatonic scale". This is much like the file erl.kbm Joseph Pehrson
mentioned except that F is raised by 1 step of 22-tET. So
the "semitones" are all 2 steps of 22-tET, except that between F and
F#, and between B and C, they're only 1 step of 22-tET. Then we get
the following tetrads with all intervals within 17 cents of JI:

4:5:6:7:
D F# A C
A C# E G
E G# B D
G# C D# F#
D# G A# C#
A# D F G#

1/7:1/6:1/5:1/4:
A C D# G
E G A# D
B D F A
D# F# A C#
A# C# E G#
F G# B D#

So, by restricting ourselves to 12 tetrads instead of 24, we can cut
the maximum error with respect to JI in half.

With this many tetrads, I wouldn't be surprised if we could find some
significant passages of tetradic music, by say Wagner or Stravinsky,
that if suitably transposed would come out sounding very smooth in
this tuning. It doesn't hurt that all diminished seventh chords in
this tuning sound very nice as well.

In the next tuning, we will cut the number of tetrads in
half again, down to 6, but the maximum error with respect to JI will
be reduced by almost a factor of 10. This is of course the Fokker-
Lumma-Keenan scale, (lumma.scl, fokker_12t.scl, and fokker_12t2.scl
in the Scala archive are all good examples of this scale, with very
tiny differences between them).
Mapped so that 1/1 = C, we get the following tetrads within 2 cents
of JI:

4:5:6:7:
C E G Bb
Db F Ab B
F A C Eb

1/7:1/6:1/5:1/4:
C Eb Gb Bb
Db E G B
Ab B D Gb

I'm not sure if one could find a great example in the repertory of an
excerpt that features just these tetrads (or a transposition thereof)
and none others, but perhaps one could compose a new example that
featured the near-just tetrads of this scale.

Finally, if one is a stickler for exact JI, one can only have 4 7-
limit tetrads in a roughly even 12-tone scale. There are many ways to
do this (including several ways of "de-tempering" the Fokker-Lumma-
Keenan scale). Here is the lattice diagram of one from Paul Hahn,
Mills Tuning Digest 1598 (12/1/98):

42:25------21:20-------21:16
\'-. .-'/ \'-. .-'/ \'-.
\ 6:5--/---\--3:2--/---\-15:8
\ /|\ / \ /|\ / \ /|
\ | / \ | / \ |
/ \|/ \ / \|/ \ / \|
/ 7:5---------7:4--------35:32
/.-' '-.\ /.-' '-.\ /.-'
8:5---------1:1---------5:4

which you will note has two hexanies in addition to the four tetrads.
An equally good scale is obtained by substituting 25:15 for 8:5 and
105:64 for 42:25:

21:20-------21:16------105:64
.-'/ \'-. .-'/ \'-. .-'/
6:5--/---\--3:2--/---\-15:8 /
|\ / \ /|\ / \ /|\ /
| / \ | / \ | /
|/ \ / \|/ \ / \|/ \
7:5---------7:4--------35:32 \
'-.\ /.-' '-.\ /.-' '-.\
1:1---------5:4--------25:16

Thanks to the other Paul!