7&14 temperament - 14 out of 35

Blackwood[10] has become my favorite scale. The cool thing about it is
that no matter where you go, you can modulate by a 3/2, and the 3/2's
are unbelievably bright and colorful. Blackwood[10] is the "diatonic"
version, and blackwood[15] is the "chromatic" version, where you can
take the "sound" of something like a melodic or harmonic minor scale,
and symmetricize it Blackwood style. Furthermore, if you start
stacking 5-limit major triads (generating maj 9 chords, and major #11
chords, etc), you end up hitting the root after 5 fifths by
definition, and the whole thing sounds perfectly consonant. It's
really a remarkable scale.

So I thought to continue this concept by coming up with the next best
closed cycle of fifths, which is 7-tet*, and then throwing in a second
contorted 7-tet chain such that you get decent 5-limit harmony. So, in
the parlance of regular mapping, it's the 5-limit temperament that
eliminates 2187/2048.

Punching this number into Graham's uv finder for some reason causes it
to crash, but due to hax0ring I got the page here:
http://x31eq.com/cgi-bin/rt.cgi?ets=7_21&error=5.0&limit=5

A good tuning for this is 14 out of 35, where the mode is 1 4 1 4 1 4
1 4 1 4 1 4 1 4. The period is 1/7 oct or 171 cents, and the generator
is 34 cents. 3/2 is 686 cents, 5/4 is 377 cents, and 3/2 is 309 cents.

This also makes for a remarkably simple notation - just use the
numbers 1-7 (or A-G if you want) and the accidentals + and -, which
denote a comma as per HEWM notation, and represent movement by a 34
cent generator. So a major triad would be 1 3+ 5, and a minor triad
would be 1 3- 5, etc. So the 14-note "major" MOS here would be 1 1+ 2
2+ 3 3+ 4 4+ 5 5+ 6 6+ 7 7+, and its "minor" version 1 2- 2 3- 3 4- 4
5- 5 6- 6 7- 7 1-'.

This has some interesting properties even outside of the ones it
shares with blackwood. One is that there are higher-limit implications
as well, and decent 8:9:10:11:12 pentads over any root where a major
triad exists, but that since this temperament is inconsistent in the
9-limit, the "just" 8:9 approximation to use here is going to differ
from the one you'll use if dealing with linked 5-limit triads (like a
major 9 chord). Another is that the interval classes aren't set up the
same way that blackwood's are - the "just" 9/8 shares a class with
6/5, and the "just" 5/4 shares a class with a 480 cent 5-tet fourth
(!). Having two fourths angers my OCD, but makes for interesting
"minor add 4" chords with the flat fourth. Whether these are
approximating 16:19:21:24 or something else is beyond me, but they
sound nice anyway.

The 21-note MOS, which is 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1,
has so many higher-limit harmonies (and always transposable by 3/2!)
that it made my brain explode. It's 4:30 AM and I have to go to bed,
but I think it's incredible. It's the "chromatic" version, but instead
of sounding like something analogous to the meantone "chromatic"
scale, it sounds like something analogous to the half-whole octatonic
scale (the kind you use over dominant 7 chords). There are
Stravinskian-sounding higher-limit sonorities everywhere you look. If
we had all started with this scale, the word "chromatic" would mean
something completely different to us. Algorithm: come up with some
really cool higher-limit chord, and then transpose it up the scale,
and listen to how the interval classes change.

I couldn't find much information on this temperament - has it been
fully explored? Are there any other cool features I'm missing? Does it
have a name other than 7&14?

-Mike

*The next number after this, as we all know, is a closed cycle of 12
fifths, which is compton temperament. Perhaps all of the hip modal
stuff that modern jazz guys are doing stems from their intuitive use
of 12-equal as compton temperament, whereas the blues treat it as a
dominant temperament, and classical music treats it as meantone.

Mike Battaglia <battaglia01@...> wrote:

> So I thought to continue this concept by coming up with
> the next best closed cycle of fifths, which is 7-tet*,
> and then throwing in a second contorted 7-tet chain such
> that you get decent 5-limit harmony. So, in the parlance
> of regular mapping, it's the 5-limit temperament that
> eliminates 2187/2048.
>
> Punching this number into Graham's uv finder for some
> reason causes it to crash, but due to hax0ring I got the
> page here:
> http://x31eq.com/cgi-bin/rt.cgi?ets=7_21&error=5.0&limit=5

Oh, right. I've fixed it now. Fixed in the sense that it
gives you a 3-limit 7-note equal temperament, anyway. And
in the 5-limit, a contorted 7-equal.

> I couldn't find much information on this temperament -
> has it been fully explored? Are there any other cool
> features I'm missing? Does it have a name other than 7&14?

I see Jamesbond does a similar job in the 7-limit.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> > I couldn't find much information on this temperament -
> > has it been fully explored? Are there any other cool
> > features I'm missing? Does it have a name other than 7&14?
>
> I see Jamesbond does a similar job in the 7-limit.

The dicot family page is positively infested with 7s and 14s, but this doesn't belong. I could put an apotome family page up and call this battaglia.

On Wed, Dec 29, 2010 at 11:25 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> > > I couldn't find much information on this temperament -
> > > has it been fully explored? Are there any other cool
> > > features I'm missing? Does it have a name other than 7&14?
> >
> > I see Jamesbond does a similar job in the 7-limit.
>
> The dicot family page is positively infested with 7s and 14s, but this doesn't belong. I could put an apotome family page up and call this battaglia.

I was actually thinking "whitewood temperament," haha. It's a cool
scale, but something of a novelty. If there's a "battaglia"
temperament I'd rather save it for something with less error, maybe if
I discover something cool while weeding through all of these subgroup
temperaments.

Is there a name for the 8&16 temperament I was discussing in the other
thread either? Maybe superdiminished or semidiminished or something
like that? Or maybe this is just a 13-limit extension of diminished
temperament, which itself works beautifully as a 2.3.5.7.17.19
subgroup temperament...

-Mike

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

>I could put an apotome family page up and call this battaglia.
>

It's here:

http://xenharmonic.wikispaces.com/Apotome+family

If Mike doesn't like the name battaglia temperament, I'll change it.

Here's dicot:

http://xenharmonic.wikispaces.com/Dicot+family

On 12/29/2010 4:44 AM, Mike Battaglia wrote:

> I couldn't find much information on this temperament - has it been
> fully explored? Are there any other cool features I'm missing? Does it
> have a name other than 7&14?

I don't know a name for this -- although it has some connection with Easley Blackwood's 21-ET etude, it hasn't had much attention even in the 5 limit version. I don't recall seeing it in any higher limit lists.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>

> > The dicot family page is positively infested with 7s and 14s, but this doesn't belong. I could put an apotome family page up and call this battaglia.
>
> I was actually thinking "whitewood temperament," haha.

Done.

Mike wrote :

> A good tuning for this is 14 out of 35, where the mode is 1 4 1 4 1 4
> 1 4 1 4 1 4 1 4. The period is 1/7 oct or 171 cents, and the generator
> is 34 cents. 3/2 is 686 cents, 5/4 is 377 cents, and 3/2 is 309 cents.

6/5.

> The 21-note MOS, which is 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1 1 3 1,
> has so many higher-limit harmonies (and always transposable by 3/2!)
> that it made my brain explode. It's 4:30 AM and I have to go to bed,
> but I think it's incredible. It's the "chromatic" version, but instead
> of sounding like something analogous to the meantone "chromatic"
> scale, it sounds like something analogous to the half-whole octatonic
> scale (the kind you use over dominant 7 chords). There are
> Stravinskian-sounding higher-limit sonorities everywhere you look. If
> we had all started with this scale, the word "chromatic" would mean
> something completely different to us. Algorithm: come up with some
> really cool higher-limit chord, and then transpose it up the scale,
> and listen to how the interval classes change.
>
> I couldn't find much information on this temperament - has it been
> fully explored? Are there any other cool features I'm missing? Does it
> have a name other than 7&14?

Funnily, I arrived to a quasi-similar tuning some years ago, from a
complete other way :
I am using (sqrt(17) + 9) /8 = 1.6403882032022 or 856.84472846532
cents as generator, the solution of the differentially coherent 9x -
4x2 = 4 Dagbhocar algorithm (it is one point that can be found on my -
c diagramm in my article on Differential coherence, if you have it
printed).
It has the rare property to generate a quasi-7-edo, with a fifth of
1.4865061711801 = 686.31054306924 cents, and a Thaï-like tone of
1.1035192688772 = 170.53418539608 cents (the equivalent to your period).
In order to be effective the differential coherence needs a second
generator of a pure 9/8 to transpose the first sequence, and here
comes the equivalent of your generator, between this 9/8 and the Thaï
tone =
203.9100017 - 170.53418539608 cents = 33.37581613 cents.
But this new row of the transposed 7-edo requires eventually his own
transposition by 9/8, that sums up to 21 notes and so on to arrive to
a 35-MOS .
So you see this is a very close version of your tuning, with perhaps
more complexity as a temperament but with the -c property in addition.
The reason I was interesting in this double-7 quasi-edo tuning was
for one thing the very circular Thaï scales it proposes, but mainly
because the Dagbhocar sequence has his own fractal waveform which
provides ideal timbres for the tuning.
It suggests MANY convergent rational sequences, among which this one
I like very much :
231 95 39 1 105 689 4521 (octave-reduced)
The other row would be the same thing * by 9, and an example of the -
c is :
144 - 105 = 39
I like very much this sequence because it passes by 2^9 and the
symmetry is impressive - it figurates for example a 4096/4095 schisma
of difference between 64/39 and 105/64.
With more unequal intervals you can start with 559 (13*43), instead
of ending with 4521(3*11*137) :
559 231 95 39 1 105 689, or onn the contrary you can use the
algebraïc solution of course all the way.

Note that 1.6403882032022 is the square of (sqrt(17) + 1)/4 =
1.2807764064 which is Daghboc, having its own very strong fractal
waveform, appearing in my list as a correct solution for Sidi
temperament, of which I don't know much except that it should
normally lead to a quasi 14-edo.

- - - - - - - -
Jacques

A variant of this system here would be to use 3/2 instead of 9/8 as
2nd generator. Same properties and pure fifths in addition, but many
more notes - I am not sure if I like it.
If it were to have a 70 quasi-edo, then doing the same thing but from
the 14 quasi-edo of Dagbhoc
2x^2 - 2 = x = (sqrt(17) + 1)/4 = 1.2807764064044 or 428.42236423272
cents along with 3/2 as 2nd generator would be a better solution as
it would cumulate Dagbhoc and Dagbhocar properties.
- - - - - - -
Jacques

I wrote :

> In order to be effective the differential coherence needs a second
> generator of a pure 9/8 to transpose the first sequence, and here
> comes the equivalent of your generator, between this 9/8 and the
> Thaï tone =
> 203.9100017 - 170.53418539608 cents = 33.37581613 cents.
> But this new row of the transposed 7-edo requires eventually his
> own transposition by 9/8, that sums up to 21 notes and so on to
> arrive to a 35-MOS .