For Kraig and Joseph -- two 7-limit BlackJust scales

Please take a look at

/tuning/files/perlich/scales/blackjust1.gif
(Keenan's block, 1/1 = G in new standard key)

and

/tuning/files/perlich/scales/blackjust2.gif
(my block, 1/1 = D in new standard key)

Kraig, just focus in on a single block (they're all the same) in each of these and give me your evaluation. Many other versions are possible, for example ones with more of the just hexanies, but let me know what you think of these two for now.

Joseph, both of these are identical to your tempered blackjack lattice, except that certain intervals are shown in gray because they are out-of-tune by

225:224, 1029:1024, or 33075:32768 (first lattice)
225:224, 1029:1024, or 2401:2400 (second lattice)

In other words, the gray intervals are the "wolves" of the JI tunings.

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_30942.html#30942

> Please take a look at
>
>
/tuning/files/perlich/scales/blackjust1.g
if
> (Keenan's block, 1/1 = G in new standard key)
>
> and
>
>
/tuning/files/perlich/scales/blackjust2.g
if
> (my block, 1/1 = D in new standard key)
>
>
> Kraig, just focus in on a single block (they're all the same) in
each of these and give me your evaluation. Many other versions are
possible, for example ones with more of the just hexanies, but let me
know what you think of these two for now.
>
>
> Joseph, both of these are identical to your tempered blackjack
lattice, except that certain intervals are shown in gray because they
are out-of-tune by
>
> 225:224, 1029:1024, or 33075:32768 (first lattice)
> 225:224, 1029:1024, or 2401:2400 (second lattice)
>
> In other words, the gray intervals are the "wolves" of the JI
tunings.

Got it! Well, these are fascinating.

Why did you decide to do one centering on "D" also?? That's still in
the new "standard key" of C-G-D-A, yes?

Thanks!

JP

Paul!
For now, I can say I like them both from a harmonic viewpoint! the closer and more compact the better as it allows each note to occur in more than one way/meaning. I have to look at the scale flatten out also for repeating tetrachords or
what have you what have you, sometimes i will forgo a strict harmonic symmetry to have such things, but then you then have two scales if you just consider the inversion. This just what i like musically, and to each his own and most of all, what
one is planning or is doing.

If you have more hexanies up your sleeve let's have them!

I like this lattice and how the uncolored arms are the closest equivalents as such things can be useful compositionally. Where did the template come from?

paul@stretch-music.com wrote:

> Please take a look at
>
> /tuning/files/perlich/scales/blackjust1.gif
> (Keenan's block, 1/1 = G in new standard key)
>
> and
>
> /tuning/files/perlich/scales/blackjust2.gif
> (my block, 1/1 = D in new standard key)
>
> Kraig, just focus in on a single block (they're all the same) in each of these and give me your evaluation. Many other versions are possible, for example ones with more of the just hexanies, but let me know what you think of these two for now.

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., jpehrson@r... wrote:

> Why did you decide to do one centering on "D" also??

1/1 ends up closer to the center of the block that way.

> That's still in
> the new "standard key" of C-G-D-A, yes?

Yes.

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:

> I have to look at the scale flatten out also for repeating
tetrachords or
> what have you what have you,

I think Dave Keenan found some repeating tetrachords . . . Dave?

>sometimes i will forgo a strict harmonic symmetry to have such
>things,

Me too -- witness my pentachordal vs. symmetrical decatonic scales.
We think very much alike!

> If you have more hexanies up your sleeve let's have them!

groups.yahoo.com/group/tuning/files/perlich/scales/blackjust3.gif
1/1 is mapped to F in the _old_ standard "key"

The errors in the gray intervals here are 225:224, 2401:2400, and
16875:16807 -- 7.7 cents, 0.7 cents, and 7.0 cents. These could
hardly be called wolves! So in addition to the four just hexanies,
you get two pretty decent ones (not to mention 16 just to pretty
decent tetrads).

> I like this lattice and how the uncolored arms are the closest
>equivalents as such things can be useful compositionally. Where did
>the template come from?

I did it in Paint. It's like the ones in _The Forms Of Tonality_
(there you see diatonic and decatonic lattices with a similar display
of periodicity) . . . I've been using it for various blackjack
lattices posted here for months now . . .

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., jpehrson@r... wrote:
>
> > Why did you decide to do one centering on "D" also??
>
> 1/1 ends up closer to the center of the block that way.

I note that that block is highly asymetrical. Why didn't you use the
mirror image of it, which would have 1/1 = G? I count 5 otonal and 5
utonal tetrads, so that wouldn't change.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> > I have to look at the scale flatten out also for repeating
> tetrachords or
> > what have you what have you,
>
> I think Dave Keenan found some repeating tetrachords . . . Dave?

For Blackjack in its entirety, there is only one note where identical
tetrachords meet, and that's the point of symmetry of the scale B[. Of
course various subsets may be tetrachordal with other points of
conjunction.

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> > > I have to look at the scale flatten out also for repeating
> > tetrachords or
> > > what have you what have you,
> >
> > I think Dave Keenan found some repeating tetrachords . . . Dave?

Some 10 and 7 note tetrachordal subsets are shown in
/tuning/topicId_27221.html#27221
notated in the key centered on C.

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> Why didn't you use the
> mirror image of it, which would have 1/1 = G? I count 5 otonal and
5
> utonal tetrads, so that wouldn't change.

If you'd like to make the change, go for it.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> > Why didn't you use the
> > mirror image of it, which would have 1/1 = G? I count 5 otonal and
> 5
> > utonal tetrads, so that wouldn't change.
>
> If you'd like to make the change, go for it.

Not really. I don't really see the point of these rational-isations of
Blackjack, which ignore most of its useful properties and limit
themselves to considering 7-limit consonances. So many such
rational-isations are possible, each with its own raison d'etre. But
if Kraig finds even one of them interesting, then I guess it is a
worthwhile excercise.

The mirror-image block I referred to is
1/1 = G
21/20
16/15
28/25
8/7
6/5
49/40
5/4
21/16
4/3
7/5
10/7
3/2
32/21
8/5
49/30
12/7
7/4
147/80
28/15
49/25

I find it very interesting that Kraig finds knowledge of the near-just
intervals compositionally useful. Even folks that insist on
rational-number scales, like to have a scale that makes maximum use of
commas like the 224:225. That's kind of where this all started for me,
with Carl Lumma's search for such a 12-note scale.

In answer to Joseph's question "Aren't 11-limit intervals part of the
deal": One of the reasons Miracle temperament is such a miracle is
because if you temper to distribute the 224:225 (7-limit), you
essentially get the 384:385 (11-limit) for free, and vice versa.

Regards,
-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> Not really. I don't really see the point of these rational-isations
of
> Blackjack, which ignore most of its useful properties and limit
> themselves to considering 7-limit consonances. So many such
> rational-isations are possible, each with its own raison d'etre.
But
> if Kraig finds even one of them interesting, then I guess it is a
> worthwhile excercise.
>
[snip]
>
> I find it very interesting that Kraig finds knowledge of the near-
just
> intervals compositionally useful.

Well that kind of contradicts your statement that the rational-
izations "ignore most of the useful properties", doesn't it? At
least, you must admit, when I lattice the just scales as I did,
showing the repeating blocks. It's actually very easy to visually
show which "commas" create wolves where in those lattices, though I
didn't want to reveal _too_ much :)

> Even folks that insist on
> rational-number scales, like to have a scale that makes maximum use
of
> commas like the 224:225.

Yup. And both Carl and Kraig seem to feel it's valuable to have just
intervals in some places and larger errors in others, rather than
spreading out the errors over all the intervals.

> In answer to Joseph's question "Aren't 11-limit intervals part of
the
> deal": One of the reasons Miracle temperament is such a miracle is
> because if you temper to distribute the 224:225 (7-limit), you
> essentially get the 384:385 (11-limit) for free, and vice versa.

Right, but to Kraig, at least in the rotation I initially gave, the
21-tone MOS didn't make "organic" enough a use of the 11-limit.
Probably the 31-tone MOS would satisfy him in that respect,
especially if I could draw an 11-limit lattice with all the repeating
blocks in the appropriate places (yikes!).

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > I find it very interesting that Kraig finds knowledge of the near-
> just
> > intervals compositionally useful.
>
> Well that kind of contradicts your statement that the rational-
> izations "ignore most of the useful properties", doesn't it?

Not really, although it goes a little way towards it.

> At
> least, you must admit, when I lattice the just scales as I did,
> showing the repeating blocks. It's actually very easy to visually
> show which "commas" create wolves where in those lattices,

Yes. Excellent.

> though I
> didn't want to reveal _too_ much :)

I'm sorry, I don't get this joke.

> > In answer to Joseph's question "Aren't 11-limit intervals part of
> the
> > deal": One of the reasons Miracle temperament is such a miracle is
> > because if you temper to distribute the 224:225 (7-limit), you
> > essentially get the 384:385 (11-limit) for free, and vice versa.
>
> Right, but to Kraig, at least in the rotation I initially gave, the
> 21-tone MOS didn't make "organic" enough a use of the 11-limit.

And I totally agree with him. I'm guessing that if you rationalise 7's
you make the 11's unusable and vice versa. Only by tempering do you
get to use them both. But I haven't checked this.

> Probably the 31-tone MOS would satisfy him in that respect,
> especially if I could draw an 11-limit lattice with all the
repeating
> blocks in the appropriate places (yikes!).

I think we can do one for Blackjack, and find a suitable 11-limit
block to rational-ise, if we just leave out ratios of 5. Margo likes
tunings without 5s. Kraig likes 7:9:11s.

Here's one that maximises 1:3:7:9:11s. It has four of them joined by
common notes.
E[ :B[ :Db^:F] :A
|
Db^:Ab^:Bv :Eb^:G<
|
F< :C< :D> :G< :Bb^
|
D> :A> :C :E> :G#v

1/1 = C<
49/48
77/72
12/11
8/7
7/6
27/22
96/77
21/16
4/3
108/77
63/44
3/2
49/32
77/48
18/11
12/7
7/4
11/6
144/77
21/11

By the way, according to RMS error minimisation, it looks like
Blackjack benefits from widening the octaves by up to 0.8 of a cent,
depending on which intervals you care about.

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> > though I
> > didn't want to reveal _too_ much :)
>
> I'm sorry, I don't get this joke.

Sorry -- I meant ;)

> > Right, but to Kraig, at least in the rotation I initially gave,
the
> > 21-tone MOS didn't make "organic" enough a use of the 11-limit.
>
> And I totally agree with him. I'm guessing that if you rationalise
7's
> you make the 11's unusable and vice versa. Only by tempering do you
> get to use them both. But I haven't checked this.

You know, the Eikosany doesn't really have much over Blackjack in the
11-limit, except for the "each-tetrad-is-unique" biftiness.

> > Probably the 31-tone MOS would satisfy him in that respect,
> > especially if I could draw an 11-limit lattice with all the
> repeating
> > blocks in the appropriate places (yikes!).
>
> I think we can do one for Blackjack, and find a suitable 11-limit
> block to rational-ise, if we just leave out ratios of 5. Margo
likes
> tunings without 5s. Kraig likes 7:9:11s.
>
> Here's one that maximises 1:3:7:9:11s. It has four of them joined
by
> common notes.
> E[ :B[ :Db^:F] :A
> |
> Db^:Ab^:Bv :Eb^:G<
> |
> F< :C< :D> :G< :Bb^
> |
> D> :A> :C :E> :G#v
>
> 1/1 = C<
> 49/48
> 77/72
> 12/11
> 8/7
> 7/6
> 27/22
> 96/77
> 21/16
> 4/3
> 108/77
> 63/44
> 3/2
> 49/32
> 77/48
> 18/11
> 12/7
> 7/4
> 11/6
> 144/77
> 21/11

So now I'll have to sit down and make a lattice of this . . .

> By the way, according to RMS error minimisation, it looks like
> Blackjack benefits from widening the octaves by up to 0.8 of a cent,

I told Monz a 1-cent-per-octave stretched 72-tET would be the best
stretched 72-tET for this.

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

/tuning/topicId_30942.html#31032

> I find it very interesting that Kraig finds knowledge of the near-
just intervals compositionally useful. Even folks that insist on
> rational-number scales, like to have a scale that makes maximum use
of commas like the 224:225. That's kind of where this all started for
me, with Carl Lumma's search for such a 12-note scale.

Hmmm... *my* understanding is that Blackjack, realized in 72-tET is
actually "juster" on the overall than true "justness..." Yes??

I'm not sure I'm a fan yet of "BlackJust..." That's working a
bit "blackward" isn't it?? :)

>
> In answer to Joseph's question "Aren't 11-limit intervals part of
the deal": One of the reasons Miracle temperament is such a miracle
is because if you temper to distribute the 224:225 (7-limit), you
> essentially get the 384:385 (11-limit) for free, and vice versa.
>

That's interesting... Why is that again??

Thanks!

JP

Dave & Joseph!
I meant to comment on this before.

> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> /tuning/topicId_30942.html#31032
>
> > I find it very interesting that Kraig finds knowledge of the near-
> just intervals compositionally useful.

When ever i have a tuning I use these places as areas of tension, in fact without this contrast a
composition would be dull. It is not unlike the use of dissonance in development sections.
What i need is not consonance (except at the end) but tension. But unlike what we have in 12
ET i need a tension that is not ambiguous.

>
>
> joseph wrote
> I'm not sure I'm a fan yet of "BlackJust..." That's working a
> bit "blackward" isn't it?? :)

I like that there was a 21 tone constant structure. that was about all as i hadn't run across one
before

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> Hmmm... *my* understanding is that Blackjack, realized in 72-tET is
> actually "juster" on the overall than true "justness..." Yes??

Hoo boy. Saying things like this can get you into trouble around here.
:-) But I agree. Although I'd put it like this. Blackjack is JI and it
has way more just 7-limit intervals and chords than any strictly
rational-number based (RI) tuning with no more notes.

> I'm not sure I'm a fan yet of "BlackJust..." That's working a
> bit "blackward" isn't it?? :)

Me neither. But each to their own.

> > In answer to Joseph's question "Aren't 11-limit intervals part of
> the deal": One of the reasons Miracle temperament is such a miracle
> is because if you temper to distribute the 224:225 (7-limit), you
> > essentially get the 384:385 (11-limit) for free, and vice versa.
> >
>
> That's interesting... Why is that again??

Because god made it that way. ;-) Which is to say that this is
ultimately unanswerable. One could maybe point to various other
properties of the numbers, and that _may_ be enlightening. But why do
those other properties exist? Either you go 'round in circles or you
admit that ultimately no one knows. By the way, I'm an atheist mystic
(today), in case anyone was wondering. :-)

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> Hmmm... *my* understanding is that Blackjack, realized in 72-tET
is
> actually "juster" on the overall than true "justness..." Yes??

Well, what we've found is that (if you don't want to use 72-tET but
do want to use JI) you can tune Blackjack in JI, and it will work
OK -- all the 7-limit consonant intervals will be either just, 0.7
cents out, 7.0 cents out, or 7.7 cents out.

> I'm not sure I'm a fan yet of "BlackJust..." That's working a
> bit "blackward" isn't it?? :)

LOL!

Well, it's a matter of opinion. Do you want to have all the consonant
intervals out-of-tune by the same amount always, or do you want to
have variety in them, such that they may be just, 0.7 cents out, 7.0
cents out, or 7.7 cents out depending on where they occur in the
scale.

> > In answer to Joseph's question "Aren't 11-limit intervals part of
> the deal": One of the reasons Miracle temperament is such a miracle
> is because if you temper to distribute the 224:225 (7-limit), you
> > essentially get the 384:385 (11-limit) for free, and vice versa.
> >
>
> That's interesting... Why is that again??

Oof -- that's a harder question better asked on tuning-math. It's
nice that you're curious, though!