Mark Gould's 53edo analogue of blackjack (was: 31edo version of blackjack)

hi Mark,

> From: "Mark Gould" <mark.gould@argonet.co.uk>
> To: "Tuning List" <tuning@yahoogroups.com>
> Sent: Saturday, November 02, 2002 6:25 AM
> Subject: [tuning] Re: 31edo version of blackjack
>
>
> Looking at the 21 from 31 blackjack,
>
> <snip>
>
> An equaivalent in 53 I can also determine:
>
> 0,2,5,7,10,12,15,17,20,22,25,27,30,32,35,37,40,42,45,47,51,(0

that's a rough analogy to blackjack, but 53edo really
doesn't work as a blackjack (i.e., MIRACLE) tuning.

except for one pitch (the highest one, 2^(51/53)),
this tuning can be viewed as a chain with a generator
of 2^(5/53) = ~113.2075472 cents, which goes from
0 to +20 generators.

that one foreign note is ~11&5/9 Semitones, and would
appear as either -11 or +42 generators -- far out of
the range of the 0...+20 chain -- whereas the one which
does occur in this chain as +10 generators or 2^(50/53)
is ~11&1/3 Semitones. neither of these is really close
to the true blackjack value of 11&2/3 Semitones.

the other alternative is to use +21 generators,
which is 2^(52/53) = ~11&7/9 Semitones. but both
choices are equally far from the true blackjack note
of 11&2/3 (= 11&6/9) Semitones.

a key feature of MIRACLE tunings is the vanishing of
"ampersand's comma", the interval which represents
3^7 * 5^6 on the "8ve"-equivalent 5-limit lattice.

if you look at both of the lattices (72edo and 31edo)
which i've recently added to my blackjack page
<http://sonic-arts.org/monzo/blackjack/blackjack.htm>
you'll see that ampersand's comma vanishes in both of
those tunings.

if you play around with the "tiling applet" on my
bingo-card-lattice page
<http://sonic-arts.org/dict/bingo.htm#tile-applet>
you'll see that the only other tuning (of those for
which i've made tiling-lattices so far) which tempers
out ampersand's comma is 41edo. other candidates
(as you can see from the stationary lattices on that
page) are 10, 11, 20, and 21edo (along with 31 and 41).
there are others which i haven't latticed.

in 53edo, however, ampersand's comma is represented
by 1 degree. 31 and 41 are considered to give the
lower and upper limits, respectively, of the MIRACLE
generator (the secor). from my blackjack page:

>> "Any tuning with a generator in the range between
>> 116.1 cents (= 31-EDO) and 117.8 cents (= 0.7 cents
>> larger than 41-EDO) has MIRACLE properties ..."

at ~113.2 cents, your 2^(5/53) generator doesn't
qualify as a secor.

-monz
"all roads lead to n^0"

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

>
> a key feature of MIRACLE tunings is the vanishing of
> "ampersand's comma", the interval which represents
> 3^7 * 5^6 on the "8ve"-equivalent 5-limit lattice.
>
> if you look at both of the lattices (72edo and 31edo)
> which i've recently added to my blackjack page
> <http://sonic-arts.org/monzo/blackjack/blackjack.htm>
> you'll see that ampersand's comma vanishes in both of
> those tunings.
>
> if you play around with the "tiling applet" on my
> bingo-card-lattice page
> <http://sonic-arts.org/dict/bingo.htm#tile-applet>
> you'll see that the only other tuning (of those for
> which i've made tiling-lattices so far) which tempers
> out ampersand's comma is 41edo. other candidates
> (as you can see from the stationary lattices on that
> page) are 10, 11, 20, and 21edo (along with 31 and 41).
> there are others which i haven't latticed.
>
> in 53edo, however, ampersand's comma is represented
> by 1 degree.

that's it! excellent explanation, monz! hopefully, we will someday
have 7-limit bingo cards up there, and the whole thing could be
explained more easily and forcefully (that is, with much simpler --
*closer* -- commas) -- in MIRACLE tunings, 225:224, 1029:1024, and
consequently their difference 2401:2400 all vanish (giving scales
like blackjack their harmonic and melodic properties). 53-equal is
not such a tuning system.

Thanks for the comments on the 53 version.

I was taking the pattern

2 1 2 1 etc to 1 1 pattern and expanding it to 3 2. They aren't equivalent
for the rules everyone puts up for miracle. I was just looking at the
structure in raw EDO steps.

Mark

--- In tuning@y..., Mark Gould <mark.gould@a...> wrote:
> Thanks for the comments on the 53 version.
>
> I was taking the pattern
>
> 2 1 2 1 etc to 1 1 pattern and expanding it to 3 2. They aren't
equivalent
> for the rules everyone puts up for miracle. I was just looking at
the
> structure in raw EDO steps.

the end of the pattern still wasn't right, i think. look again!

hi Mark,

> From: "Mark Gould" <mark.gould@argonet.co.uk>
> To: <tuning@yahoogroups.com>
> Sent: Monday, November 04, 2002 11:33 PM
> Subject: [tuning] 53 EDO imitation of Blackjack - Not!
>
>
> Should have been a 54 EDO version:
>
> 0,2,5,7,10,12,15,17,20,22,25,27,30,32,35,37,40,42,45,47,50,52,(0
>
> Thanks for spotting that one Paul.
>
> Mark - wishing he could count sometimes...
>
>
> Message: 9
> Date: Mon, 04 Nov 2002 21:26:39 -0000
> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> Subject: Re: Mark Gould's 53edo analogue of blackjack
> (was: 31edo version of blackjack)
> --- In tuning@y..., Mark Gould <mark.gould@a...> wrote:
>
> > Thanks for the comments on the 53 version.
> >
> > I was taking the pattern
> >
> > 2 1 2 1 etc to 1 1 pattern and expanding it to 3 2.
> > They aren't equivalent for the rules everyone puts
> > up for miracle. I was just looking at the structure
> > in raw EDO steps.
>
> the end of the pattern still wasn't right, i think. look again!
>

i made a big deal about that top note in my post, too:

> From: "monz" <monz@attglobal.net>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, November 02, 2002 10:44 AM
> Subject: [tuning] Mark Gould's 53edo analogue of blackjack
> (was: 31edo version of blackjack)
>
>
> that's a rough analogy to blackjack, but 53edo really
> doesn't work as a blackjack (i.e., MIRACLE) tuning.
>
> except for one pitch (the highest one, 2^(51/53)),
> this tuning can be viewed as a chain with a generator
> of 2^(5/53) = ~113.2075472 cents, which goes from
> 0 to +20 generators.
>
> that one foreign note is ~11&5/9 Semitones, and would
> appear as either -11 or +42 generators -- far out of
> the range of the 0...+20 chain -- whereas the one which
> does occur in this chain as +10 generators or 2^(50/53)
> is ~11&1/3 Semitones. neither of these is really close
> to the true blackjack value of 11&2/3 Semitones.
>
> the other alternative is to use +21 generators,
> which is 2^(52/53) = ~11&7/9 Semitones. but both
> choices are equally far from the true blackjack note
> of 11&2/3 (= 11&6/9) Semitones.

-monz