semineutral-17 tunings: 36-EDO MOS, MET-24

Hello, all.

Thanks, Mike, for your discussion of the sensi temperament, which
gave me an idea for semineutral, 17-note MOS mode of 36-EDO that
can also be implemented in some unequal tunings like MET-24.

While sensi is based on the idea of a generator around 17/46
octave (443.478 cents) or 31/24 (443.081 cents) equal to about
half of a 5/3 major third (884.359 cents), semineutral goes for
something a bit more in the neutral direction, as the name
suggests.

Actually, semineutral is a bit of a pun. The idea is to
approximate a JI division such as 63:81:104 (1/1-9/7-104/63 or
0-435.084-867.792 cents), where two large major thirds averaging
just a tad smaller than 9/7 form a large neutral third at
somewhere around 104/63 or the slightly smaller 33/20 (866.959
cents). This means that the third will average out somewhere
around 433 cents.

Thus it's "semineutral" both because each large major third is
equal to exactly or approximately half of the neutral sixth; and
because that sixth is itself "semineutral," with some submajor
qualities -- almost another kind of major sixth, but more active
than 5/3, as 22/13 or the like is on the other side of the
5-limit.

In sensi, the large major third is actually the generator of the
whole system. In semineutral, however, the related but less
ambitious goal is to get a fair number of 63:81:104 sonorities
and lots of those near-9/7 major thirds, along with lots of
perfect fifths and fourths, and septimal and neutral intervals
generally.

In 36-EDO, this plays out as a 17-note MOS with 15 small steps at
66.7 cents (the excellent thirdtone of this system) and 2 large
steps of 100 cents. We get 6 locations for 0-433.3-866.7 cents,
the semineutral sonority itself; 11 locations for 9/7 at 1.751
cents narrow; and 14 locations for 3/2.

<http://www.bestII.com/~mschulter/semineutral_36-ED2.scl>

! semineutral_36-ED2.scl
!
Semineutral tuning in 36-EDO, 0-433.33-866.67 cents
17
!
66.66667
133.33333
200.00000
266.66667
366.66666
433.33333
500.00000
566.66667
633.33333
700.00000
766.66667
866.66667
933.33333
1000.00000
1066.66667
1133.33333
2/1

Here's a version in MET-24 with 6 locations for our semineutral
sonority with two large major thirds forming an approximate 33/20
or 104/63 at 866.0 or 867.2 cents, and 11 for near-9/7 major
thirds in a smaller size of 427.7 cents and larger ones of 438.3
or 439.5 cents. However, we have only 13 regular fifths near 3/2,
as opposed to 14 in the 36-EDO version. The set is a Constant
Structure (CS for short), meaning that a given interval is always
formed from the same number of steps -- although not necessarily
vice versa!

<http://www.bestII.com/~mschulter/met24-semineutral17_Fsharp.scl>

! met24-semineutral17_F#.scl
!
17-CS semineutral sixth from two large major thirds (~63:81:104)
17
!
57.42187
139.45313
207.42187
264.84375
370.31250
427.73437
496.87500
577.73438
635.15625
704.29687
761.71875
867.18750
924.60937
992.57812
1074.60937
1132.03125
2/1

The 36-EDO version is an MOS tuning, and ideal for a maximum
number of supraminor and submajor intervals (e.g. thirds at 333.3
and 366.7 cents); the MET-24 version has a greater variety of
neutral intervals, but is a bit less regular.

Both versions offer, as one attraction in the neutral range,
approximations of Phi (833.090 cents), a point of maximum
complexity or metastability as Dave Keenan and I have sometimes
described it. In the 36-EDO tuning, we get 10 locations at
833.333 cents, virtually identical to Phi; in MET-24, we have
seven locations at 829.688 cents, not too far from Phi and very
close to 21/13 (830.253 cents).

With many thanks to Mike, and best New Year's wishes,

Margo

On Thu, Dec 27, 2012 at 4:23 AM, Margo Schulter <mschulter@...> wrote:
>
> Thus it's "semineutral" both because each large major third is
> equal to exactly or approximately half of the neutral sixth; and
> because that sixth is itself "semineutral," with some submajor
> qualities -- almost another kind of major sixth, but more active
> than 5/3, as 22/13 or the like is on the other side of the
> 5-limit.

Hi Margo - I've got a half-written reply to your posts that I hope to
send off tomorrow, but while this is here - I note that this seems to
be related to what we're calling "squares" temperament, where two
9/7's make up a 13/8 (or maybe an 18/11, depending on the subgroup
temperament). This might be a very nice subgroup version of that.

I also see you mentioned 33/20, which is a favorite interval of mine
as it appears on the chord 1/1-3/2-5/2-7/2-11/3-11/2-99/16-33/4
between the 33/4 and 5/2 (as 33/10). If you bring the 33/4 down to
33/8 and split the 33/20 in half as you describe in your temperament,
you get a very nice neutral sixth sound, somewhat close to 13/8 but a
bit sharper. I find this chord to be very interesting, as it mixes
together two completely different sounds in my head; the chord I wrote
above has a certain sort of very sultry sound to it, which for no
particular reason I hear as purple (assuming 1/1 is C), whereas the
neutral sixth is suddenly nice and cloudy and gray. It's like
discovering that sweet and sour go together well, or something...

You can distill the above chord down to 1/1-5/4-n6-33/16, where n6 is
the neutral sixth directly between 5/4 and 33/16. This makes the 33/16
on the outside sound completely random to my ears, but I can hear it
as "connecting" to the tonic by throwing 11/8 in there as well, making
it 1/1-5/4-11/8-n6-33/16. I would expect that in live performances,
musicians would tend to pitch the n6 down towards 13/8 a bit to
increase the resonance there.

Making it a bit sharper and bisecting the 33/20 allows you to modulate
in some interesting ways, such as by turning the n6 and 33/16 into the
"7" and "9" of a 4:5:6:7:9 chord, meaning the lowest two notes (e.g.
the 5/4 dyad) drop down by 12/11. I find this to be a very pleasant
chord progression which is, in a mathematical sense, characteristic of
the temperament which you suggest here. You can also just drop the
lowest note and keep the second note as is, which temporarily leads to
the interval between the two lowest notes being 15/11. You can treat
this as a suspension which can then resolve down to a 5/4 above the
new tonic - but, as 15/11 is so close to 11/8, and since this is an
otherwise straightforwardly otonal chord, you might find that the
whole thing has a very resonant sound to it, as though it might not
need to resolve anywhere at all! Tempering out 121/120 makes the
equivalence between 15/11 and 11/8 official.

I'll finish responding to your other posts, but in the meantime, as I
noted you said were particularly fascinated more by interesting chord
progressions than a universal theory of music, perhaps you might find
the above to be a good example of the sort of "practical" uses made
possible by this temperament, among (of course) other things.

-Mike

PS to Gene: I still find this temperament easier to make sense of than
Orwell, unfortunately, but thanks for the try on XA :(

> Hi Margo - I've got a half-written reply to your posts that I hope
> to send off tomorrow, but while this is here - I note that this
> seems to be related to what we're calling "squares" temperament,
> where two 9/7's make up a 13/8 (or maybe an 18/11, depending on the
> subgroup temperament). This might be a very nice subgroup version
> of that.

Hi, Mike.

Especially given all you have to deal with for that other reply
<grin>, I thought I'd just offer a brief comment on the matter of
sensi and squares, and congratulate you for some incredibly creative
uses of 33/20 and more in extended chords of a kind quite beyond my
experience (where four voices is usually "a very rich texture").

Briefly, my guess is that semineutral-17 in its 36-EDO version with a
major third at 13\36 (433.333 cents) and a large neutral sixth at
26\36 (866.667 cents) would be in what you termed the "No man's land
between sensi and squares," somewhere between 5\14 (428.571 cents) and
4\11 (436.364 cents).

</tuning/topicId_104408.html#104408>

I say "would be," because my 17-MOS of 36-EDO does not actually use
13\36 as the generator -- but there is an abstract temperament which
does use a generator at or around this size, and does thus inhabit the
"no-man's land" you were discussing in April.

In its 36-EDO version, it has a generator of 13\36 or 433.333 cents, a
large neutral third at 26\36 or 866.667 cents, and a near-4/3 fourth
from 15 generators (195\36, 6500.000 cents) less 5 octaves. With a
generator of 13\36, we get a 31-MOS with 26 fifths at 700 cents.

This abstract temperament, whether named "semineutral" or something
else, is distinct in its mapping of 128/3 to 15 generators; compare
squares with a mapping of 8/3 to 4 generators (e.g. 6\17, 24\17), or
sensi with its mapping of 6/1 to 7 generators (e.g. 17\46, 119\46).

What the TOP or POTE is, I'm not sure, but 13\36 is one obvious
implementation.

With warmest thanks,

Margo

Alright, starting to catch up on some old posts I had to let hang...
starting here:

On Thu, Dec 27, 2012 at 10:39 PM, Margo Schulter <mschulter@...>
wrote:
>
> Hi, Mike.
>
> Especially given all you have to deal with for that other reply
> <grin>, I thought I'd just offer a brief comment on the matter of
> sensi and squares, and congratulate you for some incredibly creative
> uses of 33/20 and more in extended chords of a kind quite beyond my
> experience (where four voices is usually "a very rich texture").

Thanks; I just really like 33/16, for some reason.. :)

> This abstract temperament, whether named "semineutral" or something
> else, is distinct in its mapping of 128/3 to 15 generators; compare
> squares with a mapping of 8/3 to 4 generators (e.g. 6\17, 24\17), or
> sensi with its mapping of 6/1 to 7 generators (e.g. 17\46, 119\46).

So, we can get this 2.3.7.11/5 temperament:

http://x31eq.com/cgi-bin/rt.cgi?ets=36_277p&limit=2_3_7_11%2F5

Adding 121/120 gives us this really complex 11-limit temperament

http://x31eq.com/cgi-bin/rt.cgi?ets=36ce_61d&limit=11

But I wonder if there's a simpler way to map 5 and 11.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia wrote:

> Adding 121/120 gives us this really complex 11-limit temperament
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=36ce_61d&limit=11
>
> But I wonder if there's a simpler way to map 5 and 11.

Orwell. :)