Yahoo Tuning Groups Ultimate Backup tuning Blackjack scale as constant structure

Paul!
I always have to map out a scale on the Generalized Lattice. Here is the scale you sent which
looked like a guy hitchhiking in a very abstract way.
http://www.anaphoria.com/blackjack1.GIF

I know most put the 7 and 11 on opposite sides but i like this way better.

For my own bias it seem the 3-11 limit intervals are out of place. 11's are tricky in a 21 tone
system lying 9.65 units (log 2 of 11 times 21) so you are pushing it the opposite way down to the
9 which you can do but not ones first choice.

Anyway here is one possibility that might be of use. ha
http://www.anaphoria.com/blackjack2.GIF

Which uses the fifths where one would find it. in 21 though this pythagorean chain forms a set
taking every third member of 21.
It seems the 11th ratios just cover up the apparent inconsistencies of the chain of fifths in this
tuning where a tritone also has 12 units.

The idea that the 9/8 has 3 units and the 10/9 4 tipped me off that something might be awry. Still
it is not so big a difference.

21 is not a easy scale for Constant structures with simple ratios for the fact that 21 ET
approximates most poorly. The tolerances build up as to start to end up in places that don't make
sense. Hence the pythagorean chain. You classic low ratio intervals work better in 22 hence also
better in 22 constant Structures.

Paul Erlich wrote:

> This is intended as a gesture of respect for your musical
> philosophies, Kraig. Please give it an honest and impersonal
> evaluation.
>
> 21-tone Constant Structure
>
> Pitches Steps
> sm. lg.
> ------- ---- ----
>
> 1/1 21:20
> 21/20 50:49
> 15/14 21:20
> 9/8 64:63
> 8/7 21:20
> 6/5 55:54
> 11/9 45:44
> 5/4 21:20
> 21/16 64:63
> 4/3 21:20
> 7/5 50:49
> 10/7 21:20
> 3/2 64:63
> 32/21 21:20
> 8/5 45:44
> 18/11 22:21
> 12/7 49:48
> 7/4 22:21
> 11/6 45:44
> 15/8 21:29
> 63/32 64:63
> 2/1
>
> Note that if 22/27 is taken as an auxillary to 11/9, the scale is
> inversionally symmetrical about this pair of auxillaries.

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

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

🔗Paul Erlich <paul@stretch-music.com>

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

> For my own bias it seem the 3-11 limit intervals are out of
place. 11's are tricky in a 21 tone
> system lying 9.65 units (log 2 of 11 times 21) so you are pushing
it the opposite way down to the
> 9 which you can do but not ones first choice.

Kraig, why look at the units in 21-tone equal temperament? If I gave
you a Pythaogean diatonic scale, would you look at 7-tone equal
temperament? If I gave you the "BP diatonic" scale, would you look at
the 9th root of 3?

> Anyway here is one possibility that might be of use. ha
> http://www.anaphoria.com/blackjack2.GIF

Here the steps are no longer epimoric. I thought that mattered to
you, but if it doesn't, it sure doesn't matter to me!

> Which uses the fifths where one would find it.

Not sure what you mean by that.

In reality we'd be ignoring 225/224, 243/242, 385/384, 441/440,
540/539, and 2401/2400 and finding more fifths that the JI ratios
immediately show. But there would not be more than four notes in a
chain of fifths -- 64/63 is _not_ ignored.

> The idea that the 9/8 has 3 units and the 10/9 4

Yes, blackjack is an improper CS. Each generic interval occurs in
essentially two different sizes. Here are the mappings:

Steps Ratios
----- ------------------------------
1 {50/49, 49/48}, {22/21, 21/20}
2 {28/27, 25/24}, {16/15, 15/14}
3 12/11, 9/8
4 10/9, 8/7
5 7/6, 6/5
6 {32/27, 25/21}, 11/9
7 5/4, 9/7
8 14/11, {21/16, 55/42}
9 4/3, 11/8
10 {49/36, 15/11}, 7/5

> 21 is not a easy scale for Constant structures with simple ratios
for the fact that 21 ET
> approximates most poorly.

My point of view is that, unless you make a big deal about Rothenberg
propriety, the ET approximation is not too relevant -- you can still
get lots of consonant intervals, as you can see above.

>The tolerances build up as to start to end up in places that don't
>make sense.

Can you explain this sentence? I'd like to understand what you're
saying here.

> You classic low ratio intervals work better in 22 hence also
> better in 22 constant Structures.

I would have thought so before, but it seems to me I've found a
counterexample to that assertion. From my point of view, the best 22-
tone CS for the 11-limit would probably be the Orwell one, and it's
not as good as the 21-tone Blackjack.

I hope this friendly discussion will continue so that we may learn
more about one another's philosophies . . . later . . .

🔗Kraig Grady <kraiggrady@anaphoria.com>

Paul!
the reason i look at the ET ? Think of it this way, an ET is the average MOS and or CS of all
the possible ones having the same number, in this case 21. it serves as a basis to see what ratios
are available to me within a Constant structure of this many notes. In the case of Blackjack it
showed me that 12 units can be used for 3/2s instead of the 11/8 which i didn't like.

If i want a 7 tone Constant structure scale and I want to know where the 3/2 might lie i
take the the log 2 of 3 and times it 7 times. which is 4.0947.... which lets me know the obvious.
But let say I am looking for 1,557 tone CS this same method gives me 910.7866..... which will in
most cases place the 3/2 at 911 but could be at 910 in less cases.

Paul Erlich wrote:

> Kraig, why look at the units in 21-tone equal temperament? If I gave
> you a Pythaogean diatonic scale, would you look at 7-tone equal
> temperament? If I gave you the "BP diatonic" scale, would you look at
> the 9th root of 3?

>
>
> > Anyway here is one possibility that might be of use. ha
> > http://www.anaphoria.com/blackjack2.GIF
>
> Here the steps are no longer epimoric. I thought that mattered to
> you, but if it doesn't, it sure doesn't matter to me!
>
> > Which uses the fifths where one would find it.
>
> Not sure what you mean by that.
>
> In reality we'd be ignoring 225/224, 243/242, 385/384, 441/440,
> 540/539, and 2401/2400 and finding more fifths that the JI ratios
> immediately show. But there would not be more than four notes in a
> chain of fifths -- 64/63 is _not_ ignored.
>
> > The idea that the 9/8 has 3 units and the 10/9 4
>
> Yes, blackjack is an improper CS. Each generic interval occurs in
> essentially two different sizes. Here are the mappings:
>
> Steps Ratios
> ----- ------------------------------
> 1 {50/49, 49/48}, {22/21, 21/20}
> 2 {28/27, 25/24}, {16/15, 15/14}
> 3 12/11, 9/8
> 4 10/9, 8/7
> 5 7/6, 6/5
> 6 {32/27, 25/21}, 11/9
> 7 5/4, 9/7
> 8 14/11, {21/16, 55/42}
> 9 4/3, 11/8
> 10 {49/36, 15/11}, 7/5
>
> > 21 is not a easy scale for Constant structures with simple ratios
> for the fact that 21 ET
> > approximates most poorly.
>
> My point of view is that, unless you make a big deal about Rothenberg
> propriety, the ET approximation is not too relevant -- you can still
> get lots of consonant intervals, as you can see above.
>
> >The tolerances build up as to start to end up in places that don't
> >make sense.
>
> Can you explain this sentence? I'd like to understand what you're
> saying here.
>
> > You classic low ratio intervals work better in 22 hence also
> > better in 22 constant Structures.
>
> I would have thought so before, but it seems to me I've found a
> counterexample to that assertion. From my point of view, the best 22-
> tone CS for the 11-limit would probably be the Orwell one, and it's
> not as good as the 21-tone Blackjack.
>
> I hope this friendly discussion will continue so that we may learn
> more about one another's philosophies . . . later . . .
>
>

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

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

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Paul!
> the reason i look at the ET ? Think of it this way, an ET is the
average MOS and or CS of all
> the possible ones having the same number, in this case 21.

True . . .

> it serves as a basis to see what ratios
> are available to me within a Constant structure of this many notes.
In the case of Blackjack it
> showed me that 12 units can be used for 3/2s instead of the 11/8
which i didn't like.

What's wrong with having 9 units represent 11/8 in some places, and
4/3 in other places? In the diatonic scale, 2 units represents 5/4 in
some places and 6/5 in other places.

> If i want a 7 tone Constant structure scale and I want to know
where the 3/2 might lie i
> take the the log 2 of 3 and times it 7 times. which is 4.0947....
which lets me know the obvious.
> But let say I am looking for 1,557 tone CS this same method gives
me 910.7866..... which will in
> most cases place the 3/2 at 911 but could be at 910 in less cases.

Right . . . but I'm still looking for you to defend this statement in
a concrete manner:

"You classic low ratio intervals work better in 22 hence also better
in 22 constant Structures."

Yes, _on average_, 22 would be better than 21 -- but in this
_particular case_, we seem to have a 21-tone MOS/CS which, in many
respects, outdoes any conceivable 22-tone MOS/CS.

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Yes, _on average_, 22 would be better than 21 -- but in this
> > _particular case_, we seem to have a 21-tone MOS/CS which, in
many
> > respects, outdoes any conceivable 22-tone MOS/CS.
>
> The difference in this case shows up in the more irregular step
sizes
> of the 21-tone scale--you must travel farther from 21-et to get to
> Blackjack than you need to from 22-et to get to 22-Orwell.

Exactly -- which I why I brought up Rothenberg propriety. I didn't
believe this was important to Kraig, and I thought epimoric steps
were important to him -- so he may have surprised me on two counts.

🔗Kraig Grady <kraiggrady@anaphoria.com>

Paul!
To make some things clear. I am not concerned with rules but i refer to certain properties
that give me results i like.
When working with limit JI my first concern is in compactness, in other words, using ratios that
can be used in more ways with more different intervals. That is my harmonic concern. next are
Constant structures- since then we have a melodic integrity that enables me to transpose melodic
units any where in my scale regardless of the ratios will produce a recognizable yet varied.
my formula i presented is quite useful and is not meant as a mathematical theorem. Still it gets
results and have not noticed off hand any Constant structures where this formula does not predict
the option one has.
Possibly you might be surprised to learn that i have not been using Low integer JI over the
last 6 years except in recording so old pieces. My interest has moved on to examination of Sum and
difference tone phenomenon and even Brian McLaren said he realized by the effects produced that
the accuracy of tuning i demand was necessary,

Paul Erlich wrote:

> --- In tuning@y..., genewardsmith@j... wrote:
> > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> >
> > > Yes, _on average_, 22 would be better than 21 -- but in this
> > > _particular case_, we seem to have a 21-tone MOS/CS which, in
> many
> > > respects, outdoes any conceivable 22-tone MOS/CS.
> >
> > The difference in this case shows up in the more irregular step
> sizes
> > of the 21-tone scale--you must travel farther from 21-et to get to
> > Blackjack than you need to from 22-et to get to 22-Orwell.
>
> Exactly -- which I why I brought up Rothenberg propriety. I didn't
> believe this was important to Kraig, and I thought epimoric steps
> were important to him -- so he may have surprised me on two counts.

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

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

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Paul!
> To make some things clear. I am not concerned with rules but i
refer to certain properties
> that give me results i like.
> When working with limit JI my first concern is in compactness, in
other words, using ratios that
> can be used in more ways with more different intervals. That is my
harmonic concern.

Blackjack looks quite good in this respect, rich in tetrads and
hexanies, and with a good bit of 11-limit optionally available. By
ignoring 225:224, 243:242, 385:384, 441:440, 540:539, 2401:2400,
etc., the number of ways that notes can combine to produce
consonances is staggering. If one doesn't ignore these intervals, and
considers that they change consonances to dissonances, there are
different ways of mapping Blackjack to JI (I only gave one example)
such as to preserve most of the tetrads, or most of the hexanies, or
what have you.

> next are
> Constant structures- since then we have a melodic integrity that
enables me to transpose melodic
> units any where in my scale regardless of the ratios will produce a
recognizable yet varied.

Does this entail Rothenberg propriety for you? Or is Blackjack OK in
this respect?

> Possibly you might be surprised to learn that i have not been
using Low integer JI over the
> last 6 years except in recording so old pieces.

Interesting!

> My interest has moved on to examination of Sum and
> difference tone phenomenon and even Brian McLaren said he realized
by the effects produced that
> the accuracy of tuning i demand was necessary,

Yes, combinational tones are the most sensitive to tuning of the
three psychoacoustical phenomena that favor ratios.

🔗Kraig Grady <kraiggrady@anaphoria.com>

Paul!
I am not sure of Rothenberg propriety. as it appears to have drifted from my mind as feathers
that are blown away when you shake them. is it a constant structure? the outer limit of such a
thing, there is something about that bothers me on a level i am sorry i am not able to
communicate at the moment.

Paul Erlich wrote:

> Does this entail Rothenberg propriety for you? Or is Blackjack OK in
> this respect?

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

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

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com

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

/tuning/topicId_30825.html#30825

> For my own bias it seem the 3-11 limit intervals are out of
place. 11's are tricky in a 21 tonesystem lying 9.65 units (log 2 of
11 times 21) so you are pushing it the opposite way down to the
> 9 which you can do but not ones first choice.
>
> Anyway here is one possibility that might be of use. ha
> http://www.anaphoria.com/blackjack2.GIF
>

Just out of curiousity... wasn't the fact that 11-limit intervals are
well represented by Blackjack a crucial part of the original
definition and derivation of the scale, or am I just getting confused
again??

🔗jpehrson@rcn.com

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

/tuning/topicId_30825.html#30825

> Anyway here is one possibility that might be of use. ha
> http://www.anaphoria.com/blackjack2.GIF
>
> Which uses the fifths where one would find it. in 21 though this
pythagorean chain forms a set
> taking every third member of 21.
> It seems the 11th ratios just cover up the apparent inconsistencies
of the chain of fifths in this
> tuning where a tritone also has 12 units.
>
> The idea that the 9/8 has 3 units and the 10/9 4 tipped me off that
something might be awry. Still
> it is not so big a difference.
>

I'm not quite so certain I'm understanding this. Is Kraig basically
taking the 11-limit intervals out of Blackjack and substituting
instead further members on a chain of fifths that are close to the 11-
limit intervals??

And, of course, that would change the *big tempered* Blackjack
lattice entirely, no? So that everything wouldn't work right in a
big "mesh..."

Or am *I* "meshed" on that...

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_30825.html#30836

>
> Steps Ratios
> ----- ------------------------------
> 1 {50/49, 49/48}, {22/21, 21/20}
> 2 {28/27, 25/24}, {16/15, 15/14}
> 3 12/11, 9/8
> 4 10/9, 8/7
> 5 7/6, 6/5
> 6 {32/27, 25/21}, 11/9
> 7 5/4, 9/7
> 8 14/11, {21/16, 55/42}
> 9 4/3, 11/8
> 10 {49/36, 15/11}, 7/5
>

So, these "multiple intervals" are, essentially the "inflections"
made by the small steps in Blackjack... yes?

🔗jpehrson@rcn.com

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

/tuning/topicId_30825.html#30844

> Paul!
> the reason i look at the ET ? Think of it this way, an ET is the
average MOS and or CS of all the possible ones having the same
number, in this case 21. it serves as a basis to see what ratios
> are available to me within a Constant structure of this many notes.
In the case of Blackjack it showed me that 12 units can be used for
3/2s instead of the 11/8 which i didn't like.
>

But Blackjack is, essentially, a *lot* different from 21-tET, isn't
it?... with intervals of 5 secors and intervals of only 2 secors...

Can a comparison with 21-tET really describe anything of value?

If it can, I need further explanation about it...

Thanks!

🔗graham@microtonal.co.uk

Kraig:
> > For my own bias it seem the 3-11 limit intervals are out of
> place. 11's are tricky in a 21 tonesystem lying 9.65 units (log 2 of
> 11 times 21) so you are pushing it the opposite way down to the
> > 9 which you can do but not ones first choice.
> >
> > Anyway here is one possibility that might be of use. ha
> > http://www.anaphoria.com/blackjack2.GIF

Joseph P:
> Just out of curiousity... wasn't the fact that 11-limit intervals are
> well represented by Blackjack a crucial part of the original
> definition and derivation of the scale, or am I just getting confused
> again??

The thing about Blackjack is that it works quite well even if you ignore
the 11s. I think Paul's rationalisation might be better without them, but
not the way Kraig did it because that stops being a Blackjack
approximation (and probably stops it being CS with it). The septimal
neutral third is 60:49, and it's actually closer to half a 3:2 than 11:9.
That follows from 2401:2400 being smaller than 243:242.

I do have a Blackjack rationalisation with more 11-ness, and I'll upload
it to tuning-math whenever I can be bothered. Blackjack's really more
interesting if you leave the approximations as approximations.

Oh, I have decided on my ideal secor. It's 80*log2(2.75) or 116.755
cents. If you don't want the intervals of 11, make that 116.792 cents.

Graham

🔗jpehrson@rcn.com

🔗Kraig Grady <kraiggrady@anaphoria.com>

Joseph!
I haven't been following the blackjack in entirely. But my problem with the 11s is where they
occur (in this particular JI version)
on the extremes. the thing is if one maps the 11/8 to the same interval as the 4/3 you can replace
any 4/3 with an I1/8. I have problems with this in that the 4/3 differs from the 11/8 by almost
1/21th of an octave. Hence why i said it was pushing it!

jpehrson@rcn.com wrote:

> > Anyway here is one possibility that might be of use. ha
> > http://www.anaphoria.com/blackjack2.GIF
> >
>
> Just out of curiousity... wasn't the fact that 11-limit intervals are
> well represented by Blackjack a crucial part of the original
> definition and derivation of the scale, or am I just getting confused
> again??

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

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

🔗Kraig Grady <kraiggrady@anaphoria.com>

JP!
remember i wasn't trying to keep blackjack , I was looking at the JI 21 tone Constant
structure and making what would be a conservative version of the scale. After all why would we
want 11s in the farthest tones from our center? not the way we hear.

jpehrson@rcn.com wrote:

> I'm not quite so certain I'm understanding this. Is Kraig basically
> taking the 11-limit intervals out of Blackjack and substituting
> instead further members on a chain of fifths that are close to the 11-
> limit intervals??
>
> And, of course, that would change the *big tempered* Blackjack
> lattice entirely, no? So that everything wouldn't work right in a
> big "mesh..."
>
> Or am *I* "meshed" on that...
>
> JP

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

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

🔗Paul Erlich <paul@stretch-music.com>

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

Yes . . .

> And, of course, that would change the *big tempered* Blackjack
> lattice entirely, no? So that everything wouldn't work right in a
> big "mesh..."
>
> Or am *I* "meshed" on that...

Well, we started from a _just_ version of Blackjack, in which many
of the lattice connections are broken already -- but typically by
intervals less than 10 cents. Kraig moved some pitches by even more
than that, so now the connection with the original lattice is
tenuous at best . . .

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_30825.html#30836
>
> >
> > Steps Ratios
> > ----- ------------------------------
> > 1 {50/49, 49/48}, {22/21, 21/20}
> > 2 {28/27, 25/24}, {16/15, 15/14}
> > 3 12/11, 9/8
> > 4 10/9, 8/7
> > 5 7/6, 6/5
> > 6 {32/27, 25/21}, 11/9
> > 7 5/4, 9/7
> > 8 14/11, {21/16, 55/42}
> > 9 4/3, 11/8
> > 10 {49/36, 15/11}, 7/5
> >
>
> So, these "multiple intervals" are, essentially the "inflections"
> made by the small steps in Blackjack... yes?

No -- the members of each pair are about 50 cents apart. 50 cents is
the "chromatic unison" in Blackjack, while 33 cents is the small
step.

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Joseph!
> I haven't been following the blackjack in entirely. But my problem
with the 11s is where they
> occur (in this particular JI version)
> on the extremes. the thing is if one maps the 11/8 to the same
interval as the 4/3 you can replace
> any 4/3 with an I1/8. I have problems with this in that the 4/3
differs from the 11/8 by almost
> 1/21th of an octave. Hence why i said it was pushing it!

I could easily change those three 11-limit ratios to 7-limit ones
that would be appropriate. Then Graham would be happy, Kraig might
be happy, and we could show it as it occurs in the lattice Joseph is
used to . . . none of this will change the fact that the chromatic
unison vector in Blackjack is 50 cents (36:35 for example). But
honestly, I don't see why this bothers Kraig so much -- in the
diatonic scale, 5/4 and 6/5 are mapped to the same generic interval
(the third) and they are a chromatic semitone (25:24, usually 70-100
cents) apart . . .

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> > Joseph!
> > I haven't been following the blackjack in entirely. But my
problem
> with the 11s is where they
> > occur (in this particular JI version)
> > on the extremes. the thing is if one maps the 11/8 to the same
> interval as the 4/3 you can replace
> > any 4/3 with an I1/8. I have problems with this in that the 4/3
> differs from the 11/8 by almost
> > 1/21th of an octave. Hence why i said it was pushing it!
>
> I could easily change those three 11-limit ratios to 7-limit ones
> that would be appropriate. Then Graham would be happy, Kraig might
> be happy, and we could show it as it occurs in the lattice Joseph
is
> used to . . . none of this will change the fact that the chromatic
> unison vector in Blackjack is 50 cents (36:35 for example). But
> honestly, I don't see why this bothers Kraig so much -- in the
> diatonic scale, 5/4 and 6/5 are mapped to the same generic
interval
> (the third) and they are a chromatic semitone (25:24, usually
70-100
> cents) apart . . .

I anticipate Kraig's reply -- 1/7 of an octave, rather than 1/21st
of an octave, would be an appropriate standard to compare the
chromatic semitone to.

So basically, Kraig doesn't seem to like improper scales . . . but
I'll try a 7-limit version and see if he likes it any better . . .

Joseph, if you have mathematical questions, no matter how basic, you
might want to consider posting them to

tuning-math@yahoogroups.com . . .

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> I could easily change those three 11-limit ratios to 7-limit ones
> that would be appropriate. Then Graham would be happy, Kraig might
> be happy, and we could show it as it occurs in the lattice Joseph is
> used to . . . none of this will change the fact that the chromatic
> unison vector in Blackjack is 50 cents (36:35 for example). But
> honestly, I don't see why this bothers Kraig so much -- in the
> diatonic scale, 5/4 and 6/5 are mapped to the same generic interval
> (the third) and they are a chromatic semitone (25:24, usually 70-100
> cents) apart . . .

I have an idea that there might be more to this "Constant Structure"
idea. For instance, the reason for that particular name has yet to be
explained to my satisfaction. I'm wondering if certain groups of
scales can be said to _have_ a constant structure.

Clearly, having the same number of notes per octave is a requirement
for belonging to the same "constant structure" class, but it isn't
sufficient. I guess that members of the same class must be able to be
continuously transformed into each other without ever ceasing to be a
Constant Structure in the Scala - Constant Structure Margin - sense.

The point of all this is that one must recognise that while e.g 22-tET
and Indian JI shruti scales have the same "structure", Blackjack and
21-tET do not. They don't belong to the same constant structure class.
They do not _have_ the same constant structure.

Just a thought.

🔗genewardsmith@juno.com

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

> Clearly, having the same number of notes per octave is a
requirement
> for belonging to the same "constant structure" class, but it isn't
> sufficient. I guess that members of the same class must be able to
be
> continuously transformed into each other without ever ceasing to be
a
> Constant Structure in the Scala - Constant Structure Margin - sense.

It seems to me that if two scales both represent each interval by the
same number of steps, they do so because they both belong to the
class of a particular val (since they should be consistent with how
they represent the prime intervals.) Hence they can be transformed
continuously from one to another by the simple expedient of morphing
linearly. If the definition of constant structure allows something
weaker than belonging to a particular val, I'd recommed dumping it.

As far as I can see, Blackjack and the 21-et do have the same
constant structure.

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > I could easily change those three 11-limit ratios to 7-limit ones
> > that would be appropriate. Then Graham would be happy, Kraig
might
> > be happy, and we could show it as it occurs in the lattice Joseph
is
> > used to . . . none of this will change the fact that the
chromatic
> > unison vector in Blackjack is 50 cents (36:35 for example). But
> > honestly, I don't see why this bothers Kraig so much -- in the
> > diatonic scale, 5/4 and 6/5 are mapped to the same generic
interval
> > (the third) and they are a chromatic semitone (25:24, usually 70-
100
> > cents) apart . . .
>
> I have an idea that there might be more to this "Constant
Structure"
> idea. For instance, the reason for that particular name has yet to
be
> explained to my satisfaction. I'm wondering if certain groups of
> scales can be said to _have_ a constant structure.

Well, as long as we're talking to Kraig, we might as well stick to
his definition:

/tuning/topicId_5244.html#5244

Note that the enharmonic scale is listed as an example.

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_30825.html#30889

> --- In tuning@y..., jpehrson@r... wrote:
> > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> >
> > /tuning/topicId_30825.html#30836
> >
> > >
> > > Steps Ratios
> > > ----- ------------------------------
> > > 1 {50/49, 49/48}, {22/21, 21/20}
> > > 2 {28/27, 25/24}, {16/15, 15/14}
> > > 3 12/11, 9/8
> > > 4 10/9, 8/7
> > > 5 7/6, 6/5
> > > 6 {32/27, 25/21}, 11/9
> > > 7 5/4, 9/7
> > > 8 14/11, {21/16, 55/42}
> > > 9 4/3, 11/8
> > > 10 {49/36, 15/11}, 7/5
> > >
> >
> > So, these "multiple intervals" are, essentially the "inflections"
> > made by the small steps in Blackjack... yes?
>
> No -- the members of each pair are about 50 cents apart. 50 cents
is the "chromatic unison" in Blackjack, while 33 cents is the small
> step.

Hi Paul...

Could you please run this "chromatic unison" bit by me again... I'm
not getting it...

Is this the "reoccurrance" of the same pitch at a different part of
the lattice??

Thanks!

🔗jpehrson@rcn.com

🔗Kraig Grady <kraiggrady@anaphoria.com>

Paul!
This is Erv's definition.
This clarifies my internal confusion. When we speak of Constant structures or MOS, we are focusing
on the melodic properties of the scale. as a JI scale it was the harmonic that seemed out of
place. Hence i was looking at both as we do and found the harmonic construct a bit on the edges..

Paul Erlich wrote:

>
> Well, as long as we're talking to Kraig, we might as well stick to
> his definition:
>
> /tuning/topicId_5244.html#5244
>
> Note that the enharmonic scale is listed as an example.
>

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

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

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_30825.html#30889
>
>
> > --- In tuning@y..., jpehrson@r... wrote:
> > > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > >
> > > /tuning/topicId_30825.html#30836
> > >
> > > >
> > > > Steps Ratios
> > > > ----- ------------------------------
> > > > 1 {50/49, 49/48}, {22/21, 21/20}
> > > > 2 {28/27, 25/24}, {16/15, 15/14}
> > > > 3 12/11, 9/8
> > > > 4 10/9, 8/7
> > > > 5 7/6, 6/5
> > > > 6 {32/27, 25/21}, 11/9
> > > > 7 5/4, 9/7
> > > > 8 14/11, {21/16, 55/42}
> > > > 9 4/3, 11/8
> > > > 10 {49/36, 15/11}, 7/5
> > > >
> > >
> > > So, these "multiple intervals" are, essentially
the "inflections"
> > > made by the small steps in Blackjack... yes?
> >
> > No -- the members of each pair are about 50 cents apart. 50 cents
> is the "chromatic unison" in Blackjack, while 33 cents is the small
> > step.
>
> Hi Paul...
>
> Could you please run this "chromatic unison" bit by me again... I'm
> not getting it...

Let's look at the usual diatonic scale again. Each generic interval
comes in two sizes:

Steps Ratios
------------ ------------------------------
1 (second) m2: 16/15; M2: {10/9, 9/8}
2 (third) m3: 6/5; M3: 5/4
3 (fourth) p4: 4/3; A4: {25/18, 45/32}

The difference between the two sizes of each generic interval is a
chromatic unison (aka augmented first, augmented prime, chromatic
semitone) whose ratio is {25/24 or 135/128}.

> Is this the "reoccurrance" of the same pitch at a different part of
> the lattice??

No -- that's a _comma_, or commatic unison vector -- this is not
distinguished at all in the musical notation. It's the interval
between ratios within the same set of {}s above:

Difference between 10/9 and 9/8 is 81/80
Difference between 25/18 and 45/32 is 81/80
Difference between 25/24 and 135/128 is 81/80

🔗jpehrson@rcn.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_30825.html#30923

>
> Let's look at the usual diatonic scale again. Each generic interval
> comes in two sizes:
>
> Steps Ratios
> ------------ ------------------------------
> 1 (second) m2: 16/15; M2: {10/9, 9/8}
> 2 (third) m3: 6/5; M3: 5/4
> 3 (fourth) p4: 4/3; A4: {25/18, 45/32}
>
> The difference between the two sizes of each generic interval is a
> chromatic unison (aka augmented first, augmented prime, chromatic
> semitone) whose ratio is {25/24 or 135/128}.
>

Thanks Paul... I see where you're going with this and now how it was
relating to the "constant structure" exploration...

Thanks!

🔗Pierre Lamothe <plamothe@aei.ca>

Hi Dave,

You wrote:

<< ... this "Constant Structure" idea. For instance, the reason for that particular name has yet to be explained to my satisfaction. >>

Maybe what follows will not be at your satisfaction but may contribute to explain that name. I have shown in Gammes et principes Hellegouarch that Wilson CS implies Hellegouarch Quotient-group, Fokker Periodicity Block and my Congruity axiom. While Quotient-Group and Periodicity Block concern infinite space, CS and Congruity concern finite space. However the two last concepts imply their extension at an infinite space, so they can become equivalent to the first two concepts.

CS means there is no ambiguity under tonic rotation. So translating a form like the following 0 1 2 3 4 5 6

. . . V . . . . .
0 * * 5 2 6 * 0 .
. * * 3 0 4 1 * .
. 0 * * * * * * U
. . . . . 0 . . .

(such that one of them touch the 0 at unison) the no_ambiguity_condition implies an epimorphism having the property D(XY) = D(X) + D(Y), which is forcely extendable at infinite.

. . . V . . . . .
0 4 1 5 2 6 3 0 .
. 2 6 3 0 4 1 5 .
. 0 4 1 5 2 6 3 U
. . . . . 0 . . .

The term Constant seems appropriate since it holds the periodicity concept : a CS structure implies an underlying sublattice which is the kernel, here <U V>Z2, of the morphism.

. . . . . . 0 . . . . . . 0 . . . 0 .
. . . 0 . . . . . . 0 . . . . . . . .
0 . . . . . . 0 . . . . . . 0 . . . 0
. . . . 0 . . . . . . 0 . . . . . . .
. 0 . . . . . . V . . . . . . 0 . . .
. . . . . 0 . . . * * * 0 . . . . . .
. . 0 . . . . . . 0 * * * . . . 0 . .
. . . . . . 0 . . . . . . U . . . . .
. . . 0 . . . . . . 0 . . . . . . 0 .
0 . . . . . . 0 . . . . . . 0 . . . .
. . . . 0 . . . . . . 0 . . . . . . 0
. 0 . . . . . . 0 . . . . . . 0 . . .

Regards,

Pierre

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

--- In tuning@y..., genewardsmith@j... wrote:
> As far as I can see, Blackjack and the 21-et do have the same
> constant structure.

You're right. And please forget this version of "constant structure".
It's nonsense. I'm embarrassed now, that I wrote that message.

I now think the property I was groping after was simply the one of
belonging to the same temperament. Where the temperament boundaries
are drawn at the point where, given a limited number of notes in the
chain(s)/plane(s) etc. the number of generators giving the best
approximation to some P-limit ratio, changes.

So am I right in saying that, taken as a 22-note 5-limit planar
temperament, 22-tET and 22 shrutis are the same, but taken as a 21
note 7-limit (or 11-limit) linear temperament 21-tET and blackjack are
not? Never mind, it isn't important.

-- Dave Keenan

🔗Paul Erlich <paul@stretch-music.com>

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

> but taken as a 21
> note 7-limit (or 11-limit) linear temperament 21-tET and blackjack
are
> not?

My answer is that the question is ill-posed, because 21-tET is not
consistent in the 7-limit. You would have to specify how 21-tET were
to be used in order to be able to answer the question.

BTW, I found a couple of Donald E. Hall papers and Hall mentions a
concept equivalent to consistency in one of them ('85). He uses a
probability (of approximating consonances, compared with a
probability of approximating random intervals) measure to evaluate
ETs, and if a triad is consistent, he only considers a minimum set of
two independent intervals; but if the tuning is inconsistent, the
probabilities of all three of the component consonant intervals are
used. Very strange.

🔗genewardsmith@juno.com

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

> > but taken as a 21
> > note 7-limit (or 11-limit) linear temperament 21-tET and
blackjack
> are
> > not?

> My answer is that the question is ill-posed, because 21-tET is not
> consistent in the 7-limit. You would have to specify how 21-tET
were
> to be used in order to be able to answer the question.

If Miracle is defined by <225/224,243/242,385/384>, then it should
have all of these in the kernel. This is true of h10 and h31, so the
val in question must be b21 = h31-h10, or

[21]
[33]
[49]
[59]
[72]

This differs from h21 because h21(11)=73, and b21(11)=72. If we
mean "b21" when we say "21-tET" the answer to the question is "yes";
otherwise, it is "no" in the 11-limit. However, the two are identical
in the 7-limit, so there the answer is "yes" without qualification.

This is an example of why it is useful to get away from tET thinking
and worries about consistency, and use vals instead.

🔗Paul Erlich <paul@stretch-music.com>

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., genewardsmith@j... wrote:
>
> > > However, the two are identical
> > > in the 7-limit, so there the answer is "yes" without
> qualification.
>
> > How can you say that? It all depends on which set of 7-limit
> > intervals you use the best approximations of!
>
> What does that have to do with it? Best approximations really does
> not work as a way of thinking about any temperament, equal or
> otherwise.

So then how can you say what the map is between 21-tET and 7-limit
intervals, "without qualification"?

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > So then how can you say what the map is between 21-tET and 7-
limit
> > intervals, "without qualification"?
>
> I already gave it: g21(2)=21, g21(3)=33, g21(5)=49, g21(7)=59,
> g21(11)=72. It *must* be this if you are talking about Miracle,
> though of course not in general.

Sir, step back a bit -- you've lost the thread of this discussion.
You said that not in the 11-limit but in the 7-limit, h21 has the
same map as MIRACLE-21, "without qualification"! Do you now care to
qualify this statement?

🔗genewardsmith@juno.com

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@y..., genewardsmith@j... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> > Sir, step back a bit -- you've lost the thread of this
discussion.
> > You said that not in the 11-limit but in the 7-limit, h21 has the
> > same map as MIRACLE-21, "without qualification"! Do you now care
to
> > qualify this statement?
>
> In the 7-limit, h21 = g21, so there's no qualification. The correct
> 21-et map for Miracle happens to be what you get by rounding log2
(p)
> to the nearest integer, in other words.

And you said it doesn't matter which consonant intervals you choose.
But what if someone chose to go for the best approximation of 5/3?
Didn't you say that this was the most important interval in Miracle,
or something like that?