blackjack question

At Johnny Reinhard's concert (which went excellently, by the way) I
was trying to explain the origins of the Blackjack scale...

Paul or Dave, are you there...?? :)

Basically, I was trying to remember the *sequence* of its creation:

When Blackjack and the Miracle series of scales was created using the
secor generator, was the idea of having it coincide with 72-tET a
*PRECONDITION* or did it just turn out that way?? (I would presume
the former??)

Also, does the Miracle series correspond *exactly* with 72-tET or was
it *adjusted* slightly, and how much is it off? (I guess according
to the Monzo webpage it's stated that there are more than *one*
miracle generators as much as .15 cents on either side of 72-tET.
Correct?? So it never has been *specifically* defined????)

Thanks, Paul, Dave for the help. I should probably know this
already... :)

Joseph

--- In tuning@yahoogroups.com, "Joseph Pehrson
<jpehrson@r...>" <jpehrson@r...> wrote:
> At Johnny Reinhard's concert (which went excellently, by the
way) I
> was trying to explain the origins of the Blackjack scale...
>
> Paul or Dave, are you there...?? :)
>
yup -- "which piece of yours was performed, by the way? as i
recall, "blacklight" was my favorite so far . . .

> Basically, I was trying to remember the *sequence* of its
creation:
>
> When Blackjack and the Miracle series of scales was created
using the
> secor generator, was the idea of having it coincide with 72-tET
a
> *PRECONDITION*

yes -- you had decided you wanted to use 72-equal (due in no
small part to my advocacy thereof . . .)

> Also, does the Miracle series correspond *exactly* with 72-tET
or was
> it *adjusted* slightly, and how much is it off? (I guess
according
> to the Monzo webpage it's stated that there are more than
*one*
> miracle generators as much as .15 cents on either side of
72-tET.
> Correct?? So it never has been *specifically* defined????)

there are several different definitions (look up "secor", for
example: http://www.sonic-arts.org/dict/secor.htm says that
george secor's preferred size for the generator is (18/5)^(1/19) =
~116.7155940982074 cents), just as there are several types of
meantone temperament. the miracle series is much more
constrained in tuning, though, so none of them are too different
from 72-equal. the "extremes" on either side of 72-equal might
be said to correspond to 31- and 41-equal. academic music
theorists might prefer 41-equal since 21-out-of-41 is a "deep
scale" (interval vector has unique entries, with every 41-equal
interval represented at least once) . . . on the other end of the
spectrum is 31-equal, which however conflates several types of
intervals occuring in blackjack . . .

--- In tuning@yahoogroups.com, "wallyesterpaulrus

/tuning/topicId_42252.html#42253

> yup -- "which piece of yours was performed, by the way? as i
> recall, "blacklight" was my favorite so far . . .
>

***Thanks, Paul! No, I don't "pressure" Johnny to do a work of mine
every year... (just every 2 years... just joking, Johnny...)

_Blacklight_ will be performed by Dan Barrett, cellist on a Composers
Concordance concert this May 29 at NYU Loewe Theatre, just in case
you happen to be in town...

I think it's true, not surprisingly, that my Blackjack pieces are
getting "better and better..." The newest one, 3/4 completed for
soprano Meredith Borden *maybe* even transcends the others...

Anyway... Thanks for getting back so quickly on the questions!

Now, *another* one! :)

I can see from the Monzo site what you are saying about 31-tET and 41-
tET being on both "sides" of the 72-tET secor generator and, hence,
the Blackjack subset.

I find it a little peculiar that the *secor* can vary definitionally
so much that it can create 72-tET with its subsets 41-miracle, 31-
miracle and Blackjack *AS WELL AS* 41-tET and 31-tET.

Why doesn't the secor just "stay put" as a Miracle generator and then
be called something else when it generates 41-equal and 31-equal??

And how, *specifically* (I guess I had *two* questions! :) do 41-
equal and 31-equal differ from 41-tET and 31-tET. I understand that
the Miracle scales are *not* ETs..., and I can also understand what
you say about traditional scale theorists possibly perfering ETs...

Thanks, Paul!!!!

Joseph

--- In tuning@yahoogroups.com, "Joseph Pehrson
<jpehrson@r...>" <jpehrson@r...> wrote:

> Anyway... Thanks for getting back so quickly on the questions!
>
> Now, *another* one! :)
>
> I can see from the Monzo site what you are saying about 31-tET
and 41-
> tET being on both "sides" of the 72-tET secor generator and,
hence,
> the Blackjack subset.
>
> I find it a little peculiar that the *secor* can vary definitionally
> so much that it can create 72-tET with its subsets 41-miracle,
31-
> miracle and Blackjack *AS WELL AS* 41-tET and 31-tET.

well, the generator of meantone can vary much more, so that for
example it can get you 12-equal on one end and 19-equal on the
other.

> Why doesn't the secor just "stay put" as a Miracle generator
and then
> be called something else when it generates 41-equal and
31-equal??

it's still the miracle generator in all these cases because the
mapping of generators to primes remains the same. but i
suppose "secor" itself could refer to just george's favorite value.

>
> And how, *specifically* (I guess I had *two* questions! :) do 41-
> equal and 31-equal differ from 41-tET and 31-tET.

N-tET = N-tone equal temperament = N-equal.

> I understand that
> the Miracle scales are *not* ETs...

oh, you mean blackjack, canasta, and studloco? no, these
usually aren't ETs, because the 31-equal and 41-equal values of
the miracle generator are quite *extreme*, and normally it's
closer to 72-equal . . .so none of these subset scales will
themselves be an ET . . .

--- In tuning@yahoogroups.com, "wallyesterpaulrus

/tuning/topicId_42252.html#42255

>
> well, the generator of meantone can vary much more, so that for
> example it can get you 12-equal on one end and 19-equal on the
> other.
>
***Oh... I guess that makes sense, with the fifth going from 1/3
comma to 1/11th comma adjustments...etc..

> > Why doesn't the secor just "stay put" as a Miracle generator
> and then be called something else when it generates 41-equal and
> 31-equal??
>
> it's still the miracle generator in all these cases because the
> mapping of generators to primes remains the same. but i
> suppose "secor" itself could refer to just george's favorite value.
>

***"Mapping of generators to primes??" I'm not following... :(

> >
> > And how, *specifically* (I guess I had *two* questions! :) do 41-
> > equal and 31-equal differ from 41-tET and 31-tET.
>
> N-tET = N-tone equal temperament = N-equal.
>

***He, he... that's funny, Paul. No, I knew *that* much! :)

What I meant is how the Miracle blackjack, canasta and studloco
differ from 21-tET, 31-tET and 41-tET. I guess that, certainly in
Blackjack you have *two main* sizes of tones, alternating.

Do canasta and studloco exhibit similar irregular patternings??

I find the Monzo chart hard to see, since he just keeps *adding* the
various Miracle pitches to, say, Blackjack, Canasta and Studloco, so
the particular patternings of *each* given scale are hard to see...
follow??

Thanks, Paul!!!!

JP

--- In tuning@yahoogroups.com, "Joseph Pehrson
<jpehrson@r...>" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus
>
> /tuning/topicId_42252.html#42255
>
> >
> > well, the generator of meantone can vary much more, so that
for
> > example it can get you 12-equal on one end and 19-equal on
the
> > other.
> >
> ***Oh... I guess that makes sense, with the fifth going from 1/3
> comma to 1/11th comma adjustments...etc..
>
right . . . in all cases you're approximating the basic
consonances the same way, it's just that you're trading off their
errors of approximation differently . . favoring the minor third in
the former case and favoring the fifth in the latter . . .

> > > Why doesn't the secor just "stay put" as a Miracle generator
> > and then be called something else when it generates
41-equal and
> > 31-equal??
> >
> > it's still the miracle generator in all these cases because the
> > mapping of generators to primes remains the same. but i
> > suppose "secor" itself could refer to just george's favorite
value.
> >
>
> ***"Mapping of generators to primes??" I'm not following... :(

well, let's start with the meantone case, which is simpler. the two
primes you care about are 3 and 5. you approximate the 3 with
*one* generator, and you approximate 5 with *four* generators.
so the mapping from generators to primes in meantone is [1,4] .
from this mapping you can also read off the fact that the major
sixth, or 5/3, is mapped by 4-1=3 generators . . . now by taking
inversions you have all the 5-limit consonances. as long as the
generator was roughly in the meantone range, you're
approximating all the consonances pretty well (though there will
always be some errors in at least some of the consonances).

now in the miracle case, the primes you care about are 3, 5, 7,
and 11 . . . and the mapping is [6, -7, 2, 15] i think . . . taking
these numbers and their differences, you can see how all the
11-limit consonances are mapped . . . again, there will be certain
unavoidable errors, but as long as the generator is close to the
"secor", all these consonances will be approximated pretty well .
. .

if you varied the generator so much that some consonant interval
was approximated better by a *different* (and not much larger)
number of generators than implied by the miracle mapping, then
you probably wouldn't consider it a miracle generator anymore . .
. definitely no longer a "secor" . . .

>
> What I meant is how the Miracle blackjack, canasta and
studloco
> differ from 21-tET, 31-tET and 41-tET. I guess that, certainly in
> Blackjack you have *two main* sizes of tones, alternating.
>
> Do canasta and studloco exhibit similar irregular
>patternings??

yes, just like the diatonic scale, they are "MOS" scales . . . if you
take the blackjack scale and divide all the 5/72 oct. steps into
either 2+3 or 3+2 (all of them the same way), you'll send up with
a mode of canasta, in which all the steps are 2/72 oct. or 3/72
oct. . . . then if you divide all the 3s into 1+2 or 2+1, you get
studloco, where all the steps are 2 or 1 . . .

--- In tuning@yahoogroups.com, "wallyesterpaulrus

/tuning/topicId_42252.html#42258

> >
> > ***"Mapping of generators to primes??" I'm not following... :(
>

***Thanks, Paul. I don't know why that terminology "buzzed my brain"
the first time around... it makes sense...

> well, let's start with the meantone case, which is simpler. the two
> primes you care about are 3 and 5. you approximate the 3 with
> *one* generator, and you approximate 5 with *four* generators.
> so the mapping from generators to primes in meantone is [1,4] .
> from this mapping you can also read off the fact that the major
> sixth, or 5/3, is mapped by 4-1=3 generators

***That's kinda fascinating how that arithmetic works out..., and I
can see right here on the keyboard how that would be the case, and
quite simple...

. . . again, there will be certain
> unavoidable errors, but as long as the generator is close to the
> "secor", all these consonances will be approximated pretty well .

***Yes, I see what you mean about similar mappings of the primes.
>
> >
> > Do canasta and studloco exhibit similar irregular
> >patternings??
>
> yes, just like the diatonic scale, they are "MOS" scales . . . if
you take the blackjack scale and divide all the 5/72 oct. steps into
> either 2+3 or 3+2 (all of them the same way), you'll send up with
> a mode of canasta, in which all the steps are 2/72 oct. or 3/72
> oct. . . . then if you divide all the 3s into 1+2 or 2+1, you get
> studloco, where all the steps are 2 or 1 . . .

***So Blackjack is alternating 2 and 5 (/72) and canasta alternates 2
and 3 (/72) and studloco alternates 2 and 1 (/72)...

I suppose there are points in canasta and studloco where
the "overlap" occurs and two *consecutive* "small intervals" fall
together, just like in Blackjack??

Tx!

JP

P.S. I still wish Joe Monzo would have had a separate graph for each
of the Miracle scales, rather than combining the colors, so we could
see the patternings of the scales. (I know Joe is generally
extremely busy... or busy biking... :)

--- In tuning@yahoogroups.com, "Joseph Pehrson
<jpehrson@r...>" <jpehrson@r...> wrote:

>
> ***So Blackjack is alternating 2 and 5 (/72) and canasta
alternates 2
> and 3 (/72) and studloco alternates 2 and 1 (/72)...

well, it's not a strict alternation, if you work it out . . . for example
you'll see that canasta has about twice as many "2s" as "3s" --
since it has all the "2s" from the blackjack scale plus additional
"2s" deriving from splitting the "5s" into "2+3" . . .

> I suppose there are points in canasta and studloco where
> the "overlap" occurs and two *consecutive* "small intervals" fall
> together, just like in Blackjack??

not really . . . you don't find this in the diatonic scale or most other
MOSs either . . . one MOS scale that does exhibit this sort of
pattern is the bohlen/pierce scale of 9 tones out of the 13 equal
divisions of the 3:1, iirc . . .

--- In tuning@yahoogroups.com, "wallyesterpaulrus

/tuning/topicId_42252.html#42263

> >
> > ***So Blackjack is alternating 2 and 5 (/72) and canasta
> alternates 2 and 3 (/72) and studloco alternates 2 and 1 (/72)...
>
> well, it's not a strict alternation, if you work it out . . . for
example you'll see that canasta has about twice as many "2s" as "3s" -
-
> since it has all the "2s" from the blackjack scale plus additional
> "2s" deriving from splitting the "5s" into "2+3" . . .
>

***Hmmm, well that makes sense, Paul. It would seem then, at least
from *my* perspective, that canasta might be harder for me to work
with than Blackjack, even though Joe Monzo has been encouraging it,
since I've already had some trouble with the 33 cent pitches in
Blackjack being rather similar...

JP

hi paul and Joe,

> From: <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, February 09, 2003 5:43 PM
> Subject: [tuning] Re: blackjack question
>

> --- In tuning@yahoogroups.com, "Joseph Pehrson
> <jpehrson@r...>" <jpehrson@r...> wrote:
>
> > I understand that the Miracle scales are *not* ETs...
>
> oh, you mean blackjack, canasta, and studloco? no, these
> usually aren't ETs, because the 31-equal and 41-equal values of
> the miracle generator are quite *extreme*, and normally it's
> closer to 72-equal . . .so none of these subset scales will
> themselves be an ET . . .

i was going to respond to Joe that he seems to be
restricting the name "Miracle" to 72edo and its subsets.

but now i'm a bit confused by paul's remark. i know
that 21edo does not work as blackjack, but it's my
understanding that 31edo *can* function as canasta,
and 41edo *can* function as studloco ... even admitting
that the "preferred" versions of canasta and studloco are
more along the lines of the 72edo subsets. paul?

> From: <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, February 09, 2003 6:00 PM
> Subject: [tuning] Re: blackjack question
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus
>
> /tuning/topicId_42252.html#42255
>
> >
> > well, the generator of meantone can vary much more, so that for
> > example it can get you 12-equal on one end and 19-equal on the
> > other.
> >
> ***Oh... I guess that makes sense, with the fifth going from 1/3
> comma to 1/11th comma adjustments...etc..

the largest meantone generator is actually the 12edo "5th"
of 700 cents, which is just a hair larger than the true
1/11-comma meantone "5th" of ~699.9998836 cents.

the term "meantone" properly refers to the "whole-step"
of 1/4-comma meantone, which is exactly midway in
size between the two JI "whole-steps" of 10/9 and 9/8.

the smallest possible "meantone" generator is thus that
of 1/2-comma meantone, because that tuning's "whole-step"
or "tone" is exactly the ratio 10/9, which is the
small-end boundary of any possible "mean"-tone.

the generator of 1/2-comma meantone is ~691.2018561
cents. so the meantone generators fall between that
and 12edo's 700 cent generator, so the range of variation
in meantone generator size is ~8.798143933 cents.

here's what paul is referring to:

meantones with generators smaller than 1/3-comma
have rarely found much favor, so if we consider that
to be our lower boundary, its close relative 19edo
has a generator just slightly smaller at ~694.7368421
cents. this would make our range of variation
~5.263157895 cents.

> What I meant is how the Miracle blackjack, canasta and
> studloco differ from 21-tET, 31-tET and 41-tET.

Joe, again, here you're using "Miracle" to refer strictly
to 72edo, which is not the standard usage. "MIRACLE"
(i use all-capitals because it's an acronym) refers to
the whole family of tunings with a generator size
anywhere from ~117.0731707 cents (in 41edo) to
~116.1290323 cents (in 31edo).

this is a range of variation in generator size of
~0.944138474 cents. note how much smaller it is
than the range of variation in meantone generator size.

what you *really* mean is: "how the *72-tET* blackjack,
canasta, and studloco differ from 21-tET, 31-tET, and 41-tET".

technically, blackjack and canasta exist in 72, 41, and 31edo
varieties, and studloco exists in both 72 and 41edo.
i'll write more about these variations below.

> I guess that, certainly in
> Blackjack you have *two main* sizes of tones, alternating.
>
> Do canasta and studloco exhibit similar irregular patternings??
>
> I find the Monzo chart hard to see, since he just keeps *adding* the
> various Miracle pitches to, say, Blackjack, Canasta and Studloco, so
> the particular patternings of *each* given scale are hard to see...
> follow??

i can discuss each of these by referring you to the
respective entries in my Tuning Dictionary.

blackjack
---------

http://sonic-arts.org/monzo/blackjack/blackjack.htm

in the 72edo version of blackjack (the one Joe is
using), the large step (L) is 5 degrees of 72edo
and the small step (s) is 2 degrees, so L = 5/2s
or L = 2.5s .

in the 31edo version of blackjack, L is 2 degrees
of 31edo and s is 1 degree, so L = 2s.

in the 41edo version of blackjack, L is 3 degrees
of 41edo and s is 1 degree, so L = 3s.

(Joe, thanks to your questions, i've updated my blackjack
page to include new graphics of 31 and 41edo versions
of blackjack.)

canasta
-------

http://sonic-arts.org/dict/canasta.htm

the graphics on that page show that the 41 and
72edo versions of canasta also have irregular
interval patterns.

as the generator size gets closer and closer to
that of 31edo, the patterning becomes less and
less irregular.

the "most irregular" patterning of canasta is that
of the 41edo version, where L is 2 degrees of 41edo
and s is 1 degree, or L = 2s .

in the 72edo version of canasta, L is 3 degrees
of 72edo and s is 2 degrees, so L is 3/2 the size
of s, or L = 1.5s .

in the 31edo version, of course, all 31 degrees
of canasta are exactly the same distance apart,
so L = s .

studloco
--------

http://sonic-arts.org/dict/studloco.htm

the 41edo version of studloco has steps which are
all equally-spaced, so L = s .

in the 72edo version of studloco, L is 2 degrees of
72edo and s is 1 degree, so L = 2s .

-monz

wallyesterpaulrus wrote:
> now in the miracle case, the primes you care about are 3, 5, 7, > and 11 . . . and the mapping is [6, -7, 2, 15] i think . . . taking > these numbers and their differences, you can see how all the > 11-limit consonances are mapped . . . again, there will be certain > unavoidable errors, but as long as the generator is close to the > "secor", all these consonances will be approximated pretty well . > . .

[6, -7, -2, 15]

As you wrote it, two secors approximate 7:4.

There are other equal temperaments in the Miracle family. The tuning I prefer, with 11:8 just, is roughly 185-equal. Near that is 113-equal (which is consistent!) where the balance is towards 9:7. There's also 103-equal where the tuning pushes towards 31-equal, so blackjack is closer to being proper. It works fairly well up to the 17-limit, but you probably wouldn't be using miracle that far.

Graham

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

/tuning/topicId_42252.html#42271

>
> i was going to respond to Joe that he seems to be
> restricting the name "Miracle" to 72edo and its subsets.

***Hi Monz,

Well, yes, I was under the impression that that was the definition!
If it *isn't* than the MIRACLE set is quite large, obviously both
including our 72-tET derivatives and "standard" 31-equal and such
like.

Do Paul and Dave agree to this definition? (Paul will read this, but
I will send Dave an e-mail so he looks this over...)

> here's what paul is referring to:
>
> meantones with generators smaller than 1/3-comma
> have rarely found much favor, so if we consider that
> to be our lower boundary, its close relative 19edo
> has a generator just slightly smaller at ~694.7368421
> cents. this would make our range of variation
> ~5.263157895 cents.

***Thanks so much, Monz, for your comments on meantone. However, I'm
a little confused with this. Doesn't 1/4-comma meantone have a
generator *smaller* than 1/3-comma meantone?? And *that* found great
favor, yes?? Or am I missing something here...
>

> Joe, again, here you're using "Miracle" to refer strictly
> to 72edo, which is not the standard usage. "MIRACLE"
> (i use all-capitals because it's an acronym) refers to
> the whole family of tunings with a generator size
> anywhere from ~117.0731707 cents (in 41edo) to
> ~116.1290323 cents (in 31edo).
>

***I would love to have this definition confirmed by *all parties*
involved...

You know, Monz, you may want to rewrite the MIRACLE pages a bit so
it's very clear that the "standard" 31-equal is part of this, etc.
It's kind of hidden in the subtext right now... so it seems to me...

>>
>
> canasta
> -------
>
> http://sonic-arts.org/dict/canasta.htm
>
> the graphics on that page show that the 41 and
> 72edo versions of canasta also have irregular
> interval patterns.
>

***I would say it looks "mighty" irregular. In fact, on first
appraisal, I would think it difficult to use, since the large steps
are in slightly "weird" places... Hmmm.

Thanks, Monz!!!

Joe

--- In tuning@yahoogroups.com, "Joseph Pehrson
<jpehrson@r...>" <jpehrson@r...> wrote:

> ***Hmmm, well that makes sense, Paul. It would seem then,
at least
> from *my* perspective, that canasta might be harder for me to
work
> with than Blackjack, even though Joe Monzo has been
encouraging it,
> since I've already had some trouble with the 33 cent pitches in
> Blackjack being rather similar...
>
> JP

understood . . . and we *did* originally try to keep the smallest
steps of all the just scales above 35 cents, so this is a bit of a
"defect" (though it makes your music microtonal by any definition
of the word) . . .

joseph, if you're itching to try a different scale within 72-equal, i
highly recommend 9-equal. tune it up and try a few different
inharmonic timbres until you find one that "fits". . . as you can see
on the first graph on monz's equal temperament page, 9 is
almost off the chart (it's on the lower left), which is why you need
inharmonic timbres for acceptable harmony, but it is a member
of a large number of temperament family -- in fact every single
interval in 9-equal can be used as a generator for a family of
coherent scales. best of all, 9-equal is even simpler than
12-equal, yet sounds strikingly different . . .

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> hi paul and Joe,
>
>
> > From: <wallyesterpaulrus@y...>
> > To: <tuning@yahoogroups.com>
> > Sent: Sunday, February 09, 2003 5:43 PM
> > Subject: [tuning] Re: blackjack question
> >
>
> > --- In tuning@yahoogroups.com, "Joseph Pehrson
> > <jpehrson@r...>" <jpehrson@r...> wrote:
> >
> > > I understand that the Miracle scales are *not* ETs...
> >
> > oh, you mean blackjack, canasta, and studloco? no, these
> > usually aren't ETs, because the 31-equal and 41-equal
values of
> > the miracle generator are quite *extreme*, and normally it's
> > closer to 72-equal . . .so none of these subset scales will
> > themselves be an ET . . .
>
>
> i was going to respond to Joe that he seems to be
> restricting the name "Miracle" to 72edo and its subsets.
>
> but now i'm a bit confused by paul's remark. i know
> that 21edo does not work as blackjack, but it's my
> understanding that 31edo *can* function as canasta,
> and 41edo *can* function as studloco ... even admitting
> that the "preferred" versions of canasta and studloco are
> more along the lines of the 72edo subsets. paul?

well that's why i said "usually" above. is the diatonic scale
7-equal? "usually" not.

p.s. it is possible to use fifths wider than 700 cents, or narrower
than 691 cents, for a meantone mapping -- the consonances
just won't be as accurate, but i don't see a clear boundary there
beyond which the tuning immediately fails to be meantone . . .
the meantone line is, after all, infinitely long, in the first graph on
your equal temperament page . . .

--- In tuning@yahoogroups.com, "Joseph Pehrson
<jpehrson@r...>" <jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> /tuning/topicId_42252.html#42271
>
> >
> > i was going to respond to Joe that he seems to be
> > restricting the name "Miracle" to 72edo and its subsets.
>
> ***Hi Monz,
>
> Well, yes, I was under the impression that that was the
definition!
> If it *isn't* than the MIRACLE set is quite large, obviously both
> including our 72-tET derivatives and "standard" 31-equal and
such
> like.

31-equal is *really* extreme. i might say that the analogues of
21-equal, 31-equal and 41-equal for miracle would be 5-equal,
7-equal and 12-equal for meantone. so consider 31-equal to be
a miracle tuning if you consider 7-equal to be a meantone tuning
(i don't care if you say yes or no, just be consistent :) )

>
> Do Paul and Dave agree to this definition? (Paul will read this,
but
> I will send Dave an e-mail so he looks this over...)
>
> > here's what paul is referring to:
> >
> > meantones with generators smaller than 1/3-comma
> > have rarely found much favor, so if we consider that
> > to be our lower boundary, its close relative 19edo
> > has a generator just slightly smaller at ~694.7368421
> > cents. this would make our range of variation
> > ~5.263157895 cents.
>
> ***Thanks so much, Monz, for your comments on meantone.
However, I'm
> a little confused with this. Doesn't 1/4-comma meantone have
a
> generator *smaller* than 1/3-comma meantone??

well, the fifth is bigger, but the fourth is smaller, than those of
1/3-comma meantone.

> And *that* found great
> favor, yes??

?? 1/4-comma found great favor because it has less total error,
or lower maximum error, than 1/3-comma, considering all the
5-limit consonances . . .

> > Joe, again, here you're using "Miracle" to refer strictly
> > to 72edo, which is not the standard usage. "MIRACLE"
> > (i use all-capitals because it's an acronym) refers to
> > the whole family of tunings with a generator size
> > anywhere from ~117.0731707 cents (in 41edo) to
> > ~116.1290323 cents (in 31edo).
> >
>
> ***I would love to have this definition confirmed by *all parties*
> involved...

i am loathe to draw a strict boundary in this manner . . . but see
above . . .

>
> You know, Monz, you may want to rewrite the MIRACLE pages a
bit so
> it's very clear that the "standard" 31-equal is part of this, etc.
> It's kind of hidden in the subtext right now... so it seems to me...

would you mention 7-equal in a page on meantone? i don't think
so . . .

> >
> > canasta
> > -------
> >
> > http://sonic-arts.org/dict/canasta.htm
> >
> > the graphics on that page show that the 41 and
> > 72edo versions of canasta also have irregular
> > interval patterns.
> >
>
> ***I would say it looks "mighty" irregular. In fact, on first
> appraisal, I would think it difficult to use, since the large steps
> are in slightly "weird" places... Hmmm.

the 10 large steps are as evenly spaced around the octave as
possible, corresponding to the "decimal scale" on which graham
breed's miracle notation is based . . . so what's "weird" about
them?

--- In tuning@yahoogroups.com, "wallyesterpaulrus

/tuning/topicId_42252.html#42277

<>
> understood . . . and we *did* originally try to keep the smallest
> steps of all the just scales above 35 cents, so this is a bit of a
> "defect" (though it makes your music microtonal by any definition
> of the word) . . .

***Well, the way *I'm* looking at it, it's simply a "feature," not
necessarily a "bug!" :) Something to be careful about, since if I'm
trying for a just sonority and use the *wrong* pitch... and there, of
course, would be such in this type of scale, I should *know* about it!

But, as I've mentioned previously on one of these threads, the
exciting aspect of Blackjack for *me* is the fact that it *can* be so
consonant on lower-limit quasi-JI and still have "teeny-tiny" "added"
pitches...

>
> joseph, if you're itching to try a different scale within 72-equal,
i highly recommend 9-equal.

***Thanks for the tip. I'll try it out.

Joseph

--- In tuning@yahoogroups.com, "wallyesterpaulrus

/tuning/topicId_42252.html#42281
>
> the 10 large steps are as evenly spaced around the octave as
> possible, corresponding to the "decimal scale" on which graham
> breed's miracle notation is based . . . so what's "weird" about
> them?

***I guess, Paul, what I would like to see on the Monzo webpages are
*separate* charts of canasta and studloco so I could see the
patterning...

JP

--- In tuning@yahoogroups.com, "wallyesterpaulrus

/tuning/topicId_42252.html#42277
>
> joseph, if you're itching to try a different scale within 72-equal,
i
> highly recommend 9-equal. tune it up and try a few different
> inharmonic timbres until you find one that "fits". . . as you can
see
> on the first graph on monz's equal temperament page, 9 is
> almost off the chart (it's on the lower left), which is why you
need
> inharmonic timbres for acceptable harmony, but it is a member
> of a large number of temperament family -- in fact every single
> interval in 9-equal can be used as a generator for a family of
> coherent scales. best of all, 9-equal is even simpler than
> 12-equal, yet sounds strikingly different . . .

***Well, using the relatively new "MIDI relay" function of SCALA this
exercise was a snap. I did some improvisation in this scale and I
must admit it is quite remarkable! Very xenharmonic, that's for
sure...

I see on your "accuracy" chart, Paul, that it emulates 3-limit and 5-
limit, although not to a terrific accuracy. I can certainly *hear*
these sonorities, though, when I'm "messing around" in the improv...

Joseph

hi paul,

> From: <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, February 10, 2003 9:59 AM
> Subject: [tuning] Re: blackjack question
>
>
> >
> > [me, monz]
> > i was going to respond to Joe that he seems to be
> > restricting the name "Miracle" to 72edo and its subsets.
> >
> > but now i'm a bit confused by paul's remark. i know
> > that 21edo does not work as blackjack, but it's my
> > understanding that 31edo *can* function as canasta,
> > and 41edo *can* function as studloco ... even admitting
> > that the "preferred" versions of canasta and studloco are
> > more along the lines of the 72edo subsets. paul?
>
> well that's why i said "usually" above. is the diatonic scale
> 7-equal? "usually" not.
>
> p.s. it is possible to use fifths wider than 700 cents,
> or narrower than 691 cents, for a meantone mapping --
> the consonances just won't be as accurate, but i don't
> see a clear boundary there beyond which the tuning
> immediately fails to be meantone . . . the meantone
> line is, after all, infinitely long, in the first graph
> on your equal temperament page . . .

well, OK ... i was assuming that 10/9 and 9/8
represented the boundaries of what would be called
a "tone", but i guess that's not *necessarily* the
case.

i haven't played much with the tunings, but by
the graphs i have on my canasta and blackjack
pages it seems to me that the 31edo version of
blackjack and the 41edo version of canasta are
viable.

it seems to me that 31edo-blackjack and 41edo-canasta
would have enough in common structurally with the
"standard" 72edo versions that they'd be *recognizable*
as blackjack and canasta, which i see as an important
point in this discussion.

Joe, i'd be interested in your tuning up and trying out
31edo-blackjack and posting some comments. you've
played around with 72edo-blackjack enough now that
i think you have a better grasp of its *practical*
peculiarities than anyone else around here.

-monz

hi Joe,

> From: <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, February 10, 2003 11:53 AM
> Subject: [tuning] Re: blackjack question
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus
>
> /tuning/topicId_42252.html#42281
> >
> > the 10 large steps are as evenly spaced around
> > the octave as possible, corresponding to the
> > "decimal scale" on which graham breed's miracle
> > notation is based . . . so what's "weird" about
> > them?
>
> ***I guess, Paul, what I would like to see on the
> Monzo webpages are *separate* charts of canasta and
> studloco so I could see the patterning...

i guess you still haven't looked here:

http://sonic-arts.org/dict/canasta.htm

http://sonic-arts.org/dict/studloco.htm

-monz

--- In tuning@yahoogroups.com, "Joseph Pehrson <jpehrson@r...>"
<jpehrson@r...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus
>
> /tuning/topicId_42252.html#42277
> >
> > joseph, if you're itching to try a different scale within 72-
equal,
> i
> > highly recommend 9-equal. tune it up and try a few different
> > inharmonic timbres until you find one that "fits". . . as you can
> see
> > on the first graph on monz's equal temperament page, 9 is
> > almost off the chart (it's on the lower left), which is why you
> need
> > inharmonic timbres for acceptable harmony, but it is a member
> > of a large number of temperament family -- in fact every single
> > interval in 9-equal can be used as a generator for a family of
> > coherent scales. best of all, 9-equal is even simpler than
> > 12-equal, yet sounds strikingly different . . .
>
>
> ***Well, using the relatively new "MIDI relay" function of SCALA
this
> exercise was a snap. I did some improvisation in this scale and I
> must admit it is quite remarkable! Very xenharmonic, that's for
> sure...
>
> I see on your "accuracy" chart, Paul, that it emulates 3-limit and
5-
> limit, although not to a terrific accuracy. I can certainly *hear*
> these sonorities, though, when I'm "messing around" in the improv...
>
> Joseph

now we're talkin'! and, i'm sure that you'll find that these
sonorities are far more acceptable with certain timbres than with
others . . .

two of my favorite scales in 9-equal are pelog (try the 1 1 3 1 3
mode and the 3 1 1 3 1 mode, each of which contains one "major triad"
and one "minor triad") and augmented (2 1 2 1 2 1, complete with
three "major triads" and three "minor triads"). mixing these two very
different effects in one piece of music is sure to spin people's
heads -- and it can all be notated a la sims!

--- In tuning@yahoogroups.com, "Joseph Pehrson <jpehrson@r...>"
<jpehrson@r...> wrote:

> I see on your "accuracy" chart, Paul, that it emulates 3-limit and
5-
> limit, although not to a terrific accuracy.

joseph -- perhaps you didn't see the triangle hidden under the two
dashes -- it also emulates 7-limit at about that same level of
accuracy!

in fact, gene ward smith has proposed using 9-equal as a basis for
the notation of 7-limit music.

why? partly because a chain of nine 7:6s comes awfully close to
closing on itself (two octaves higher, if you don't octave-reduce).

the "comma" at the end of this chain is 40353607:40310784, or 1.8
cents.

thus the 9-equal 7:6 deviates from just intonation by only one-ninth
this amount, or 0.2 cents!!

so you could write a piece in 9-equal, and if you notated it in
cents, you could claim it was in just intonation, since it would look
identical to a chain of 7:6s . . .

Hi Joseph,

I'm afraid I agree with almost everything Paul, Monz and Graham have
written in this thread. Where they disagree on minor points I usually
agree with the last to post.

There are of course an infinite number of possible miracle tunings
since the generator can take on any of a continuous range of values.

In one sense there is no sharp cutoff of narrowest or widest miracle
generator. It's a matter of taste. However for convenience we usually
take the cutoff points to occur where, if we go beyond it we find that
some length of generator chain other than the ones based on the [6,
-7, -2, 15] prime mapping, manages to produce a better approximation
of some consonance.

But we'll get different answers for this depending whether we want 7,
9 or 11 limit consonances, and depending whether we are using
Blackjack, Canasta or Studloco (limiting the length of the rogue chain).

I calculated one of these once, I don't remember which one (prob.
7-limit blackjack), but I remember 3/31 oct was right on one boundary
and the other boundary actually went a little beyond 4/41 oct.

I have to disagree with Paul's analogy. I don't think 31-ET is as bad
a miracle tuning as 7-ET is a meantone tuning (unless maybe if you're
talking 11-limit). At least for 7-limit miracle I'd go more for this
analogy.

Meantone Miracle
5 10
7 21
12 31
19 41
31 72

--- In tuning@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:

> I have to disagree with Paul's analogy. I don't think 31-ET is as
bad
> a miracle tuning as 7-ET is a meantone tuning (unless maybe if
you're
> talking 11-limit).

it's not nearly as bad in terms of error size, of course. but it's
actually worse in at least the following senses:

the pentatonic scale in meantone is like the canasta scale in miracle
because each is the first MOS with any complete chords.

> At least for 7-limit miracle I'd go more for this
> analogy.
>
> Meantone Miracle
> 5 10
> 7 21
> 12 31
> 19 41
> 31 72

fair enough. joseph can print this out if he wishes. however, i would
make the point that while 12-equal conflates only one pair of
intervals in the 7-tone meantone scale -- the diminished fifth and
the augmented fourth -- 31-equal conflates about a third of all
blackjack intervals with one another . . . including notably the 300
vs. 317-cent intervals. not that this conflation is a bad thing . . .

i would also point out that once you've reached 31-equal, there are
*better* (simpler) generator-mappings to the 11 (9?)-limit complete
chord than miracle -- namely meantone itself -- while meantone is
undeniably the simplest generator-mapping for the complete 5-limit
chord in 12-equal.

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

> blackjack intervals with one another . . . including notably the
300
> vs. 317-cent intervals. not that this conflation is a bad
thing . . .

. . . though it might be rather disturbing for someone who's worked
out a system or composition where these pairs of intervals must be
distinguished (say one member of a pair is used as a consonance and
another as a dissonance).

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

/tuning/topicId_42252.html#42291

>
> Joe, i'd be interested in your tuning up and trying out
> 31edo-blackjack and posting some comments. you've
> played around with 72edo-blackjack enough now that
> i think you have a better grasp of its *practical*
> peculiarities than anyone else around here.
>

***Hi Monz,

If you could please give me the "pitch numbers" or "scale degrees" of
the 21 note 31edo Blackjack, maybe I can do this... (Maybe I could
figure it out myself, but it would take some time...)

Thanks!

Joe

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

/tuning/topicId_42252.html#42298

>
> i guess you still haven't looked here:
>
> http://sonic-arts.org/dict/canasta.htm
>
> http://sonic-arts.org/dict/studloco.htm
>

***Sorry, Monz... I see you "done did it..."

Joe

--- In tuning@yahoogroups.com, "wallyesterpaulrus

/tuning/topicId_42252.html#42299
>
> now we're talkin'! and, i'm sure that you'll find that these
> sonorities are far more acceptable with certain timbres than with
> others . . .
>
> two of my favorite scales in 9-equal are pelog (try the 1 1 3 1 3
> mode and the 3 1 1 3 1 mode, each of which contains one "major
triad"
> and one "minor triad") and augmented (2 1 2 1 2 1, complete with
> three "major triads" and three "minor triads"). mixing these two
very
> different effects in one piece of music is sure to spin people's
> heads -- and it can all be notated a la sims!

***I agree this is all pretty amazing. After more Blackjack work, I
think I'll head in this direction for a "change..."

Thanks, Paul!!!

JP

--- In tuning@yahoogroups.com, "wallyesterpaulrus

/tuning/topicId_42252.html#42324

<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Joseph Pehrson <jpehrson@r...>"
> <jpehrson@r...> wrote:
>
> > I see on your "accuracy" chart, Paul, that it emulates 3-limit
and
> 5-
> > limit, although not to a terrific accuracy.
>
> joseph -- perhaps you didn't see the triangle hidden under the two
> dashes -- it also emulates 7-limit at about that same level of
> accuracy!
>

***Yes, this is true. It was difficult to see on the chart... That's
great news!

JP

--- In tuning@yahoogroups.com, "Dave Keenan <d.keenan@u...>"

/tuning/topicId_42252.html#42326

<d.keenan@u...> wrote:
> Hi Joseph,
>
> I'm afraid I agree with almost everything Paul, Monz and Graham have
> written in this thread. Where they disagree on minor points I
usually
> agree with the last to post.
>

***Well, surely it's the "freshest opinion" in any case... :) Thanks
for the response, and I'm actually happy to learn that MIRACLE has
such a broad definition. I guess that makes sense, since all these
scales seem to have "special properties" and the difference in size
of the generators seems particularly small...

Thanks again!

Joseph

My position on this has always been that 31 to 41 is the typical miracle range, and 19 to 12 is the typical meantone range. That's what I say on my website -- and I also mention 7-equal on the meantone page.

wallyesterpaulrus wrote:
> the pentatonic scale in meantone is like the canasta scale in miracle > because each is the first MOS with any complete chords.

If the complete chords are 5-limit for meantone and 11-limit for miracle. But comparing 5-limit with 11-limit temperaments is going to give strange results.

> fair enough. joseph can print this out if he wishes. however, i would > make the point that while 12-equal conflates only one pair of > intervals in the 7-tone meantone scale -- the diminished fifth and > the augmented fourth -- 31-equal conflates about a third of all > blackjack intervals with one another . . . including notably the 300 > vs. 317-cent intervals. not that this conflation is a bad thing . . .

And in 72-equal, blackjack is an improper scale. Whereas neither the diatonic nor pentatonic scales are improper in typical meantones. So blackjack obviously isn't like the diatonic scale! 7 out of 12 and 21 out of 31 are both proper scales. Most of the conflation in blackjack involves rare intervals anyway, so 1/3 is over-dramatising.

> i would also point out that once you've reached 31-equal, there are > *better* (simpler) generator-mappings to the 11 (9?)-limit complete > chord than miracle -- namely meantone itself -- while meantone is > undeniably the simplest generator-mapping for the complete 5-limit > chord in 12-equal.

I don't like this argument -- I thought it was agreed that one equal temperament can belong to more than one linear temperament family. As you're choosing a different sized generator, it's comparing apples and oranges.

Better mappings for the same generator are more important. But it's what happens the other side of the equal temperament that's important. The alternative 5-limit mapping for 12-equal is better when you perturb it in the schismic direction, so 12-equal is one extreme of a typical meantone. It also has a simpler 7-limit generator-mapping than typical meantones, but perturb it in the meantone direction and the typical mapping is still best, so 12-equal is one extreme of what counts as a meantone.

Various 11-limit intervals are conflated in 31-equal. Perturb the miracle tuning towards 72-equal, and most of them will slightly improve in tuning (I think 11:9 is an exception). So you might be able to find a more accurate mapping for the miracle generator, but only by going outside the typical miracle range. Which confirms 31-equal as being one extreme of that range -- not the extended range in which melodic patterns are roughly the same, which is from 10-equal to 21-equal (or arguably from 10 to 31).

Graham

Graham Breed <graham@microtonal.co.uk> writes:

{{Various 11-limit intervals are conflated in 31-equal. Perturb the
miracle tuning towards 72-equal, and most of them will slightly improve
in tuning (I think 11:9 is an exception). So you might be able to find
a more accurate mapping for the miracle generator, but only by going
outside the typical miracle range.}}

The 200th row of the Farey sequence between 3/31 and 4/41 is this:

[3/31, 19/196, 16/165, 13/134, 10/103, 17/175, 7/72, 18/185, 11/113,
15/154, 19/195, 4/41]

For 5 and 7 limit harmonies, we can slightly improve on 7/72 by using
17/175 instead, which in fact I've done in my Blackjack pieces. For
11-limit harmonies, 7/72 is the obvious choice--using 39/401 or 46/473
instead hardly seems worth it. The optimal range is closely focused on 72
through the 11-limit, and beyond that we need to ask which extension of
miracle we are considering.

>

Paul!
One could also keep it in JI and claim it was ET. kinda what happened with western music, or the
music of the Chopi or Thailand.
The thing is JI is more than ratios, it is a way of thinking about structure and organization. i
don't know of any JI practioners who would organize their material in such a way, but time will
tell. Possibly we get to the level of a paradox like weather something is a wave or particle, but
how we define it leads to different things. This "psychological" difference is as important as any
acoustical study on perception.

>
> From: "wallyesterpaulrus <wallyesterpaulrus@yahoo.com>" <wallyesterpaulrus@yahoo.com>
> Subject: Re: blackjack question
>
>
> the "comma" at the end of this chain is 40353607:40310784, or 1.8
> cents.
>
> thus the 9-equal 7:6 deviates from just intonation by only one-ninth
> this amount, or 0.2 cents!!
>
> so you could write a piece in 9-equal, and if you notated it in
> cents, you could claim it was in just intonation, since it would look
> identical to a chain of 7:6s . . .
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

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

> The alternative 5-limit mapping for 12-equal is better when you
perturb
> it in the schismic direction,

you'd never see this if you were just looking at a diatonic scale.

> so 12-equal is one extreme of a typical
> meantone. It also has a simpler 7-limit generator-mapping than
typical
> meantones, but perturb it in the meantone direction

how far? i'm not following.

> and the typical
> mapping is still best, so 12-equal is one extreme of what counts as
a
> meantone.

i have no problem with using a 701-cent generator for a meantone
mapping.

Hi Paul!

I was looking at blackjackintmat3.xls again, which I had totally
forgotten about (being so concerned with the next pitch-indicating
generation).

I notice that most of the ratios of 4:5 have a 3 cent error listed.

I thought you had told me before that Blackjack only had an error of
2 cents at the 5-limit.

I'm confused...

Thanks!

Joseph

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> Hi Paul!
>
> I was looking at blackjackintmat3.xls again, which I had totally
> forgotten about (being so concerned with the next pitch-indicating
> generation).
>
> I notice that most of the ratios of 4:5 have a 3 cent error listed.

all of them should.

> I thought you had told me before that Blackjack only had an error of
> 2 cents at the 5-limit.

then i would have been wrong. unless i was talking rms error or
something.

--- In tuning@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:

/tuning/topicId_42252.html#45461

> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
> wrote:
> > Hi Paul!
> >
> > I was looking at blackjackintmat3.xls again, which I had totally
> > forgotten about (being so concerned with the next pitch-
indicating
> > generation).
> >
> > I notice that most of the ratios of 4:5 have a 3 cent error
listed.
>
> all of them should.
>
> > I thought you had told me before that Blackjack only had an
error of
> > 2 cents at the 5-limit.
>
> then i would have been wrong. unless i was talking rms error or
> something.

***Well, I printed that out from one of your posts, but I'm not
going to "hunt and peck" for it at this point. I'll just use the
colored chart that I forgot about. Personally, I'm not bothered by
a 3 cent error... Seems pretty tiny to me...

Joseph