27, 28, 29, and 34

27-et

The 27-et is a good choice for the 7-limit, and shares with the 12
(and 22) et the property of having the septimal comma, 64/63, "in the
kernel", as we say on the math group, so that 21/16 is identified
with 4/3. The subdominant tone as a part of the V7 is actually not
bad, since the fifths are sharp and therefore the fourths are flat.

I've mentioned the 6/27 generator before, and it's a nice one, giving
us a 12-note scale of pattern (2223)x3 and a 15-note scale of pattern
(22221)x3; since it has a 7-complexity of 9 we do have quite a few
complete tetrads. We also have a 9/27 generator, giving us a (12)x9
18-note scale. If someone wants to call the 12-note scale a
temperament and see how that works, I say go for it.

The 7/27 generator has a 7-complexity of 10, and we might like the 11-
note scale with pattern 11511511515 or the 15-note scale with pattern
111411141114114 with that. It should be noted that the lack of
Myhill's property won't actually give you musical cooties, and you
can use 1114115115114, 1114114115114 or 1114111411141113, etc if you
like!

We can go for the circle of fifths/fourths if we want also, and in
this case that means a 7-complexity of 11 and some scales that ought
to make even Margo sit up in her chair—-a Goth diatonic with pattern
5515551, and a sore-temperament with pattern 114141141414.

28-et

The 28-et uses, old, recycled fifths from the 7-et which at over 16
cents are really quite flat, but it has excellent major thirds and
decent 7s and 11s, though since the 11 is sharp it doesn't work as
well as it might. What interests me about it the most is the
8/28 = 2/7 generator, which gives us scales with the remarkably low
7-complexity of 4, for instance (2221)x4 at 16 notes and (223)x4 at
12 notes, which once again is worth looking at as a sort of exotic
12-tone temperament as well as a scale.

29-et

This division was first considered by someone who was aiming at the
31-et, and missed; here I am talking about it deliberately! It's
about the same for the 5-limit and decidedly in better tune for the
7-limit than the 12-et, and if people can stand the 12-et they can
stand this one too, though flatness rather than sharpness may be a
problem for ears attuned to the 12-et.

With a complexity of 7, the 3/29 generator is worth consideration; it
gives us a 9-note scale with pattern 333333335 and one of 10 steps
that tries hard to be the 10-et and doesn't quite make it:
3333333332.

34-et

The 34-et 7, 15.88 cents flat, is so much worse that its triads that
there is a tendency to view it as a triad-machine, without much value
in the 7-limit or beyond. However, the low 7-complexity of 6 for the
9/34 generator belies that notion. And we might note the lousy 7s
aren't nearly as lousy as those of the 12-et! It has a 7-note scale
with pattern 2727277, a nice 11-note scale for fans of 11-note
scales, with pattern 22522522525, and a 15-note scale with pattern
222322232223223.

--- In tuning@y..., genewardsmith@j... wrote:

> 28-et
>
> The 28-et uses, old, recycled fifths from the 7-et which at over 16
> cents are really quite flat, but it has excellent major thirds and
> decent 7s and 11s, though since the 11 is sharp it doesn't work as
> well as it might. What interests me about it the most is the
> 8/28 = 2/7 generator, which gives us scales with the remarkably low
> 7-complexity of 4, for instance (2221)x4 at 16 notes and (223)x4 at
> 12 notes, which once again is worth looking at as a sort of exotic
> 12-tone temperament as well as a scale.

What fascinates me about 28-tET is (a) the ability to modulate between four different 7-tET
systems, and (b) the ability to modulate between seven different octatonic systems. The
octatonic scale, sLsLsLsL, is a very important scale in jazz, Bloch, Stravinsky, Badings, and film
music. It's rich in triads and tetrads.
>
> 29-et
>
> This division was first considered by someone who was aiming at the
> 31-et, and missed; here I am talking about it deliberately! It's
> about the same for the 5-limit and decidedly in better tune for the
> 7-limit than the 12-et, and if people can stand the 12-et they can
> stand this one too,

Well, if you're talking about diatonic music, you're assuming that inflating the comma, which is
"supposed" to vanish, to over 40 cents, isn't a problem!
>
though flatness rather than sharpness may be a
> problem for ears attuned to the 12-et.
>
> With a complexity of 7,

At what odd limit?

> the 3/29 generator is worth consideration; it
> gives us a 9-note scale with pattern 333333335 and one of 10 steps
> that tries hard to be the 10-et and doesn't quite make it:
> 3333333332.
>
> 34-et
>
> The 34-et 7, 15.88 cents flat, is so much worse that its triads that
> there is a tendency to view it as a triad-machine, without much value
> in the 7-limit or beyond. However, the low 7-complexity of 6 for the
> 9/34 generator belies that notion. And we might note the lousy 7s
> aren't nearly as lousy as those of the 12-et! It has a 7-note scale
> with pattern 2727277, a nice 11-note scale for fans of 11-note
> scales, with pattern 22522522525, and a 15-note scale with pattern
> 222322232223223.

34-tET is inconsistent in the 7-limit, but consistency starts to become a red herring when you
have this many notes per octave. Can you go into more detail on the harmonic resources of the
7- and 11-note scales you mention?

I wrote,

> --- In tuning@y..., genewardsmith@j... wrote:
>
> > 28-et
> >
> > The 28-et uses, old, recycled fifths from the 7-et which at over 16
> > cents are really quite flat, but it has excellent major thirds and
> > decent 7s and 11s, though since the 11 is sharp it doesn't work as
> > well as it might. What interests me about it the most is the
> > 8/28 = 2/7 generator, which gives us scales with the remarkably low
> > 7-complexity of 4, for instance (2221)x4 at 16 notes and (223)x4 at
> > 12 notes, which once again is worth looking at as a sort of exotic
> > 12-tone temperament as well as a scale.
>
> What fascinates me about 28-tET is (a) the ability to modulate between four different 7-tET
> systems, and (b) the ability to modulate between seven different octatonic systems. The
> octatonic scale, sLsLsLsL, is a very important scale in jazz, Bloch, Stravinsky, Badings, and
film
> music. It's rich in triads and tetrads.

(b) is actually equivalent to your observation, Gene, since the octatonic scale in 28-tET is
(25)x4. I posted this observation many years ago on rec.music.compose. A couple of years
ago, I told Dan Stearns that I'd like to explore all the ETs from 26 to 31, inclusive (30-tET allows
for 10-tET and 15-tET to be explored), on guitar. Given that 26, 27, 28, and 30 have
somewhat out-of-tune fifths, and that the nylon-string guitar is both very permissive for
out-of-tune fifths and the best type of guitar for Mark Rankin's interchangeable fingerboards, I
think I'm going to implement this with six fingerboards (Mark usually provides four but I'm sure
he can be "flexible").

22-tET is still my favorite, though . . . Gene, why don't you do a "22 and 31" post?

--- In tuning@y... </group/tuning/post?
protectID=189075234009078072015097190036129>, "Paul Erlich"
<paul@s... </group/tuning/post?
protectID=197166044078146233033082190>> wrote:

> > With a complexity of 7,
>
> At what odd limit?

7-limit

> 34-tET is inconsistent in the 7-limit, but consistency starts to
become a red herring when you
> have this many notes per octave. Can you go into more detail on the
harmonic resources of the
> 7- and 11-note scales you mention?

I could, but they are circle-of-minor-thirds scales of a kind you
know about already. We have 5/19 < 9/34 < 4/15, and 9/34 is a lot
like 5/19, though on his web page Keenan seemed uncertain if it
should be included. The 9/34 generator can also be described as the
19-15 system (19 white keys and 15 black keys); all of these have
49/48 and 126/125 in the kernel, so that 8/7 and 7/6 are represented
by the same note, and 25/21, which is a 5/3 up followed by 5/7 down,
counts as a minor third. The 9/34 generator is a 1/4-kleisma
temperament, if that is marginal I don't know what to say about 4/15,
which is over 1/2 kleisma. However, the 15-et still smoothes things
out so far as scale step size goes. So far as triads go, the sharp 15
tends to cancel the flat 19, but it doesn't help with the 7: the 19
has it 408 relative cents flat, the 15 132 rc flat, and so the 34 has
it the sum of the two, or 408 rc flat. (To get absolute cents from
relative cents, just divide by the number of steps in the octave of
the scale division.)

You might find the following 7-note PB interesting; it's based on
commas of 49/48 and 126/125, with a chroma of 10/9:

1--7/6--6/5--7/5--10/7--5/3--12/7--(2)

The 15-et represents 50/49 and 36/35 by one step and 7/6 by three
steps, so it is a little more regular than 19, which does it by four
vs. one step. The 34-et of course is in between; the small steps are
2/34 and the 7/6 is 7/34.