Well, this isn't very advanced... but, if not math, at least it's

arithmetic...

I still don't understand how 7 of the 72-tET scale is a generator of

blackjack. It's a great concept (spooky!) since we have been

finding 7's to be very peculiar in some other instances...

Would someone please go over that again, gently??

Thanks!

Joseph

> ----- Original Message -----

> From: <jpehrson@rcn.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Tuesday, June 19, 2001 8:10 PM

>Subject: [tuning-math] 7/72 generator in blackjack

>

>

> Well, this isn't very advanced... but, if not math, at least it's

> arithmetic...

That's OK, Joe... this list is for math dummies like me, too,

as well as guys like Paul, Dave, and Graham who understand the

more esoteric stuff.

>

> I still don't understand how 7 of the 72-tET scale is a generator of

> blackjack. It's a great concept (spooky!) since we have been

> finding 7's to be very peculiar in some other instances...

>

> Would someone please go over that again, gently??

Take another look at the explanation and especially the diagram

below the graph at

http://www.ixpres.com/interval/dict/miracle.htm

Dave Keenan found the MIRACLE generator (~116.7 cents) by use

of the "brute force" approach: he had his computer perform

thousands (millions?.. billions?) of calculations and analyze

the resulting scales.

The ~116.7-cent generator came out on top as implying the

largest number of 11-limit consonances. 2^(7/72) happens

to be extremely close to the calculated MIRACLE generator

(which, I should emphasize, is only *one* possible MIRACLE

generator... there can be many, depending on the error method

selected).

The diagram on my Dictionary page shows how you cycle thru

intervals of 2^(7/72) on either side of 1/1, which in this

case really should be called 2^(0/72). Upon reaching the

10th note on either side, you've got Blackjack. Extending

to the 15th note on either side gives you Canasta.

This process is exactly analagous to constructing a meantone

cycle, except that instead of a "cycle of 5ths", you're

cycling thru the generator interval, whatever it may be.

An interesting digression: some scales can be thought of as

being constructed by more than one generator simultaneously.

Naturally, our familiar old 12-EDO is one such scale. It

can be thought of as a "cycle of 5ths" where each "5th" is

7 Semitones; here's an example centered on "D" (flats and

sharps are of course equivalent to their enharmonic cousins):

Ab - Eb - Bb - F - C - G - D - A - E - B - F# - C# - G#

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

Or it can be thought of as a cycle of Semitones:

Ab - A - Bb - B - C - C# - D - Eb - E - F - F# - G - G#

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

In either case, the generator creates a scale of 12 distinct

pitches before producing a pitch which is an exact replica

of one already existing.

-monz

http://www.monz.org

"All roads lead to n^0"

_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at http://mail.yahoo.com

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

Joseph,

As a complement to Monz's explanation, see what I just added to

http://dkeenan.com/Music/MiraclePitchChart.gif

It shows the octave as a circle divided into 72 parts. Start at D> and

follow the (new) straight line segments clockwise. You'll see each one

jumps 7/72 of an octave. When you've done that 20 times and wound up

at D<, you've generated Blackjack.

> Dave Keenan found the MIRACLE generator (~116.7 cents) by use

> of the "brute force" approach: he had his computer perform

> thousands (millions?.. billions?) of calculations and analyze

> the resulting scales.

>

> The ~116.7-cent generator came out on top as implying the

> largest number of 11-limit consonances. ...

Er, no. That all came _after_ the discovery, and merely confirmed its

"miraculous" nature (as a 7-limit or 11-limit generator, but not necc.

9-limit). I was afraid there might have been some holes in my search

strategy, but since then Graham Breed has performed a search using a

completely different method to mine, and (I think?) further confirmed

it.

Strictly speaking, the MIRACLE generator was discovered by Paul

Erlich, who extracted it from a scale that I posted, the 31-noter that

we now call Canasta. Paul then recognised that there was a more

manageable (although improper) MOS with 21 notes using the same

generator. So historically it went: Canasta - MIRACLE generator -

Blackjack - Decimal scale (although, apart from "Blackjack" we didn't

call them that immediately). But logically it goes: MIRACLE generator

- Decimal scale - Blackjack - Canasta.

So Graham,

By what figure-of-demerit and at what odd-limits can we claim that the

MIRACLE generator is the best?

Does cardinality_of_smallest_MOS_containing_a_complete_otonality

divided by exp(-(minimax_error/17c)^2) do it at 7 and 11 limits?

What's the best 9-limit generator by this FoD?

I'm sure some folks would be interested in the 13-limit result too.

Regards,

-- Dave Keenan

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

>

> Take another look at the explanation and especially the diagram

> below the graph at

> http://www.ixpres.com/interval/dict/miracle.htm

>

>

> Dave Keenan found the MIRACLE generator (~116.7 cents) by use

> of the "brute force" approach: he had his computer perform

> thousands (millions?.. billions?) of calculations and analyze

> the resulting scales.

>

> The ~116.7-cent generator came out on top as implying the

> largest number of 11-limit consonances. 2^(7/72) happens

> to be extremely close to the calculated MIRACLE generator

> (which, I should emphasize, is only *one* possible MIRACLE

> generator... there can be many, depending on the error method

> selected).

>

Hello Monz!

Thank you so much for this interesting response. Well, I read ahead,

and I guess Dave Keenan credits Paul Erlich with the discovery of the

MIRACLE generator. I guess that's how I remember it, too, looking

back at the chain (literally!) of events...

Well, from what you are saying, then, 31-EDO at with a 116.1

generator is also a MIRACLE scale, yes... of a sort?? Of course, I

had always heard of the "special" properties of 31-EDO...

Then, it is not too coincidental that a 31-tone NON-EDO using 116.7

would be "miraculous" as well... I mean it's not so miraculous that

both scales would have 31 notes, correct??

I found your "MIRACLE GENERATOR" page fascinating. I guess I really

hadn't read that one as carefully as the page that pertained

PARTICULARLY to Blackjack...

I remember when Paul Erlich did the calculus to figure out the "RMS"

or root mean square method to find the errors.

Although I can't specifically follow this in the entire, it seems

rather related to a discussion that I had with Graham Breed and John

deLaubenfels about finding "errors" by squaring things and then

taking the square root...

Is that correct? It looks as though Paul, in his calculations, is

trying to find the least errors for all the various intervals he is

considering, in squaring them and so forth, and then puts it all

nicely back in a pie (not a pie chart!) to determine the MIRACLE

generator at 116.7 cents.

Am I getting that at all...??

ANYWAY, I also very much appreciated the 72-EDO chart at the bottom

of the MIRACLE generator page... That REALLY did a good job of

breaking down how the various scales are related to the generator...

Well, the chart with the colors for the various scales did too...

It seems I'm gradually coming to a greater appreciation of this

process, finally... although I have to admit I was rather "left in

the dust" when it happened.

(In fact, I became so confused, that Paul thought I had forgotten the

entire discussion!)

Well, anyway, thanks for the help!

_________ _______ _________

Joseph Pehrson

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

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

> Joseph,

>

> As a complement to Monz's explanation, see what I just added to

> http://dkeenan.com/Music/MiraclePitchChart.gif

>

> It shows the octave as a circle divided into 72 parts. Start at D>

and follow the (new) straight line segments clockwise. You'll see

each one jumps 7/72 of an octave. When you've done that 20 times and

wound up at D<, you've generated Blackjack.

>

Hello Dave!

Thanks for this "enhancement" of your "Miracle Wheel." I understand

it better than ever, now. I did have one question, though...

It doesn't really look as though one goes around 20 times with those

lines to get blackjack...

I see a going around 10 times and then I get to D< but then there is

not a connecting line going from that to the D> which begins a

second "chain" of 10. And besides, between those two notes there are

only 4 degrees of 72, not 7.

What am I doing wrong??

Thanks!

________ _______ _______

Joseph Pehrson

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

>

> Is that correct? It looks as though Paul, in his calculations, is

> trying to find the least errors for all the various intervals he is

> considering, in squaring them and so forth, and then puts it all

> nicely back in a pie (not a pie chart!) to determine the MIRACLE

> generator at 116.7 cents.

>

> Am I getting that at all...??

Yup . . . it's identical in form to the derivation Woolhouse did (in

1835?) of the optimal meantone temperament, which turned out to have a

fifth (the generator) tempered by 7/26 of a syntonic comma. Monz has a

webpage on that . . .

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

> It doesn't really look as though one goes around 20 times with those

> lines to get blackjack...

>

> I see a going around 10 times and then I get to D< but then there is

> not a connecting line going from that to the D> which begins a

> second "chain" of 10. And besides, between those two notes there are

> only 4 degrees of 72, not 7.

>

> What am I doing wrong??

>

There is only one chain. It _starts_ at D>. It goes around the circle

once, producing 9 more notes. Then D pops up. Then 9 more notes. Then

D< is last.

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

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

>

> > It doesn't really look as though one goes around 20 times with

those

> > lines to get blackjack...

> >

> > I see a going around 10 times and then I get to D< but then there

is

> > not a connecting line going from that to the D> which begins a

> > second "chain" of 10. And besides, between those two notes there

are

> > only 4 degrees of 72, not 7.

> >

> > What am I doing wrong??

> >

> There is only one chain. It _starts_ at D>. It goes around the

circle

> once, producing 9 more notes. Then D pops up. Then 9 more notes.

Then

> D< is last.

Thanks, Paul... I just started on the wrong note... :)

________ _______ ______

Joseph Pehrson

> ----- Original Message -----

> From: <jpehrson@rcn.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Wednesday, June 20, 2001 7:04 PM

> Subject: [tuning-math] Re: 7/72 generator in blackjack

>

>

> Well, from what you are saying, then, 31-EDO at with a 116.1

> generator is also a MIRACLE scale, yes... of a sort?? Of course, I

> had always heard of the "special" properties of 31-EDO...

>

> Then, it is not too coincidental that a 31-tone NON-EDO using 116.7

> would be "miraculous" as well... I mean it's not so miraculous that

> both scales would have 31 notes, correct??

Yes, Joe, you're on the right track. If I understand him correctly,

Graham considers 31-EDO to be a MIRACLE temperament "of a sort".

But that ~0.7-cent error accumulates further down the chain to

produce intervals that are rather farther away from JI than the

72-EDO-based MIRACLE scales.

The MIRACLE *generator* is the magical element in these tunings.

You can generate a number of different-sized scales from it

and they will all have essentially the same harmonic properties,

the differences being simply a matter of the gaps in the smaller

scales.

So the 10-tone version *is* a very useable scale for a composer

who wants many JI implications with a tiny pitch-set. You've

become convinced that Blackjack (21-tone MIRACLE) will do the

trick for you, and I've decided to focus on Canasta (31-tone

MIRACLE). The 41-tone MIRACLE scale is yet another scale

offering even *more* JI implications, and Graham has recommended

one a bit larger than that (was it 46?). And of course, the

full 72-EDO set closes the possibilities of the 2^(7/72) generator.

> ANYWAY, I also very much appreciated the 72-EDO chart at the bottom

> of the MIRACLE generator page... That REALLY did a good job of

> breaking down how the various scales are related to the generator...

Yes, quite a bit of the MIRACLE discussion was elaborated via

Graham's decimal notation, which is (I think) a lot harder to

grasp for a "regular" performer/composer microtonalist who's

used to using deviations from 12-EDO, altho from a theoretical

perspective decimal *is* more elegant to describe these tunings.

So even for me, 72-EDO notation makes it easier to understand.

(I'm just sorry that I'm so devoted to my own 72-EDO notation that

now it goes up against the version everyone else has decided to use.)

> It seems I'm gradually coming to a greater appreciation of this

> process, finally... although I have to admit I was rather "left in

> the dust" when it happened.

>

> (In fact, I became so confused, that Paul thought I had forgotten the

> entire discussion!)

You're not alone, Joe. I too was quite mystified during the first

week or so of the MIRACLE discussion, and it was only when I realized

the theoretical *AND* practical importance of it (i.e, how easily it

can be mapped to a Ztar) that I devoted some serious study to it and

began to understand.

The MIRACLE tunings, especially Canasta for me, solve a lot of the

riddles and problems I have been facing in dealing with large

extended JI systems - specifically, how to map so many damn pitches

to a playable instrument. MIRACLE solves the problem by distributing

the small errors so well that a very small pitch-set can represent

a huge number of JI structures.

I've begun making a lattice of the implications of the Canasta

scale (to go on my Canasta page), but it's got so many ratios

on it that I fear it will be another "spaghetti lattice"...

and I'm only up to 7-limit, haven't even plotted 11 yet!

If you think of using the 2^(7/72) generator to create extended

(i.e., >12) scales in the same way that you can use a tempered

(i.e., narrowed) meantone "5th" or the wider-than-12-EDO Pythagorean

"5th" to create >12 "extended" cycles, I think that will help

make the whole process clearer.

The meantone and Pythagorean "5ths" don't close the cycle at the

13th note, because the 1st and 13th are separated by a small

interval which is exactly or approximately one of the commas.

Similarly, if you call your origin 0 and create an 11-tone

MIRACLE scale, the notes at either end (-5 and +5 generators

away from the origin) will be separated by 2^(2/72), or

33&1/3 cents.

This is a bit bigger than a "comma", but the process is the same.

By continuing to extend the cycle beyond these 11 notes, you

get pairs of pitches all separated by 2^(2/72) - this is exactly

why the Blackjack scale has L=5 s=2 (in terms of 72-EDO degrees).

By the time you reach a cycle bounded by -15 and +15 generators,

you've filled out the "octave" pitch-space pretty evenly, hence

the Canasta scale.

Extending it to one more note on either side would give a pitch

separated from the one on the other end by only 2^(1/72), or

16&2/3 cents. So now all MIRACLE scales above cardinality 31

will have pairs of pitches separated by *that* interval.

Then finally the 72nd generator closes the cycle... in other

words, at that point the separation of pitches on either end

finally reduces to 2^(0/72), which is a unison.

-monz

http://www.monz.org

"All roads lead to n^0"

_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at http://mail.yahoo.com

In-Reply-To: <9gpf0f+i66g@eGroups.com>

> By what figure-of-demerit and at what odd-limits can we claim that the

> MIRACLE generator is the best?

11-limit by most figures of demerit, although that is ignoring some

obviously too complex scales.

7-limit it's second by the default FOD, whatever that was. 9-limit it

doesn't score so well.

> Does cardinality_of_smallest_MOS_containing_a_complete_otonality

> divided by exp(-(minimax_error/17c)^2) do it at 7 and 11 limits?

> What's the best 9-limit generator by this FoD?

13/41, consistent with 19 and 22 note scales. Schismic is second.

Miracle does really badly in the 7-limit with this measure. See

etc.

> I'm sure some folks would be interested in the 13-limit result too.

Right. I've added, but not checked, that.

<http://x31eq.com/temper.html>

It should be easily hackable now. Although there's a lot of code in

writetemper.py, you should be able to work out how to supply different

limits, or figures of demerit.

Obviously, you don't need the bit that uploads to my website.

Graham

31-EDO is "MIRACLE" in exactly the same way that 12-tET is meantone.

i.e. Not really. At best borderline, with large deviations from JI.

We have the following approximate analogy

Meantone MIRACLE

----------------

12-EDO 31-EDO

31-EDO 72-EDO

19-EDO 41-EDO

> <http://x31eq.com/temper.html>

Your page says "11 limit" where it should be "13 limit".

On Wed, 20 Jun 2001 23:20:51 -0700, "monz" <joemonz@yahoo.com> wrote:

>Yes, quite a bit of the MIRACLE discussion was elaborated via

>Graham's decimal notation, which is (I think) a lot harder to

>grasp for a "regular" performer/composer microtonalist who's

>used to using deviations from 12-EDO, altho from a theoretical

>perspective decimal *is* more elegant to describe these tunings.

I've also found it useful as a keyboard notation for Blackjack on a

traditional keyboard -- it's easy to set up a 2-octave keyboard mapping

with the "naturals" on the white keys and the sharpened notes on the black

keys, a lot easier than remembering which key is mapped to which

combination of 72-TET accidentals.

--

see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--

hmiller (Herman Miller) "If all Printers were determin'd not to print any

@io.com email password: thing till they were sure it would offend no body,

\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

--- In tuning-math@y..., Herman Miller <hmiller@I...> wrote:

> On Wed, 20 Jun 2001 23:20:51 -0700, "monz" <joemonz@y...> wrote:

>

> >Yes, quite a bit of the MIRACLE discussion was elaborated via

> >Graham's decimal notation, which is (I think) a lot harder to

> >grasp for a "regular" performer/composer microtonalist who's

> >used to using deviations from 12-EDO, altho from a theoretical

> >perspective decimal *is* more elegant to describe these tunings.

>

> I've also found it useful as a keyboard notation for Blackjack on a

> traditional keyboard -- it's easy to set up a 2-octave keyboard

mapping

> with the "naturals" on the white keys and the sharpened notes on the

black

> keys, a lot easier than remembering which key is mapped to which

> combination of 72-TET accidentals.

Herman,

Could you describe this keyboard mapping in more detail? The main

problem I'm having here is that there are 11 "unnaturals" in Blackjack

but only 10 black keys in 2 octaves (although Manuel argued for 11

naturals and 10 unnaturals in Blackjack).

-- Dave Keenan

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

> Similarly, if you call your origin 0 and create an 11-tone

> MIRACLE scale, the notes at either end (-5 and +5 generators

> away from the origin) will be separated by 2^(2/72), or

> 33&1/3 cents.

>

> This is a bit bigger than a "comma", but the process is the same.

> By continuing to extend the cycle beyond these 11 notes, you

> get pairs of pitches all separated by 2^(2/72) - this is exactly

> why the Blackjack scale has L=5 s=2 (in terms of 72-EDO degrees).

>

Thank you Monz, for this. This is the first time I've started to

understand it. After the first "decimal" cycle in blackjack, the

_second_ cycle around has pitches all 1/6 tone from the original

cycle...

But isn't it a bit strange that a generator that is 7 units of 72-

tET should create FIVES... and that 5+2, the small interval = 7.

I'm not entirely understanding why that is...

??

Thanks!

________ _____ ______

Joseph Pehrson

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

> 31-EDO is "MIRACLE" in exactly the same way that 12-tET is

meantone.

> i.e. Not really. At best borderline, with large deviations from JI.

>

> We have the following approximate analogy

>

> Meantone MIRACLE

> ----------------

> 12-EDO 31-EDO

> 31-EDO 72-EDO

> 19-EDO 41-EDO

Thanks, Dave! This is very interesting...

______ _____ ______

Joseph Pehrson

In-Reply-To: <9gu46b+b1l3@eGroups.com>

Dave Keenan wrote:

> 31-EDO is "MIRACLE" in exactly the same way that 12-tET is meantone.

> i.e. Not really. At best borderline, with large deviations from JI.

>

> We have the following approximate analogy

>

> Meantone MIRACLE

> ----------------

> 12-EDO 31-EDO

> 31-EDO 72-EDO

> 19-EDO 41-EDO

That could be continued

Meantone MIRACLE

----------------

26-EDO 21-EDO

7-EDO 10-EDO

5-EDO 11-EDO

although 7 and 5 may be the wrong way round, depending on which way you

look at it from. They show these sections of the scale tree:

7 5

12

19

26 31

11 10

21

31

41

72

Graham

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

> But isn't it a bit strange that a generator that is 7 units of 72-

> tET should create FIVES... and that 5+2, the small interval = 7.

>

> I'm not entirely understanding why that is...

>

> ??

It's very simple.

72 divided by 7 is 10, with a remainder of 2.

So after 10 generators in the cycle, you're 2 units short of where

you started.

Another generator later, you're 7-2 = 5 units beyond where you

started, and 2 units short of the second note in the chain.

Similarly, you'll divide each of the 7-unit steps of the first cycle

into a large (5-unit) step and a small (2-unit) step as you go around

the second cycle.

When you've completed the second cycle, you've completed the

blackjack scale.

When you've completed the third cycle, you've completed the canasta

scale.

When you've completed the fourth cycle, you've completed the MIRACLE-

41 scale.

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

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

>

> > But isn't it a bit strange that a generator that is 7 units of

72-

> > tET should create FIVES... and that 5+2, the small interval = 7.

> >

> > I'm not entirely understanding why that is...

> >

> > ??

>

> It's very simple.

>

> 72 divided by 7 is 10, with a remainder of 2.

>

> So after 10 generators in the cycle, you're 2 units short of where

> you started.

>

> Another generator later, you're 7-2 = 5 units beyond where you

> started, and 2 units short of the second note in the chain.

>

> Similarly, you'll divide each of the 7-unit steps of the first

cycle into a large (5-unit) step and a small (2-unit) step as you go

around the second cycle.

>

> When you've completed the second cycle, you've completed the

> blackjack scale.

Of course... thanks Paul... that is an easy concept. I wonder if

I'd been able to figure it out if I'd continued to puzzle over it...

Well, anyway, you saved me some time... thanks!

_________ ______ ______

Joseph Pehrson

On Fri, 22 Jun 2001 03:04:34 -0000, "Dave Keenan" <D.KEENAN@UQ.NET.AU>

wrote:

>Could you describe this keyboard mapping in more detail? The main

>problem I'm having here is that there are 11 "unnaturals" in Blackjack

>but only 10 black keys in 2 octaves (although Manuel argued for 11

>naturals and 10 unnaturals in Blackjack).

0^ 1^ 2^ 3^ 4^ 5^ 6^ 7^ 8^ 9^

0 1 2 2 3 4 5 5 6 7 7 8 9 0

0v

C# Eb F# G# Bb C# Eb F# G# Bb

C D E F G A B C D E F G A B C

I also have a 24-note mapping with the duplicate notes mapped to more

useful ones:

0^ 1^ 2^ 3^ 4^ 5^ 6^ 7^ 8^ 9^

0 1 2 3 4 5 6 7 8 9 0

2v 5v 7v 0v

It's not a contiguous MIRACLE scale, but it just happens to work out

nicely, since 2v and 5v complete the 4:5:6:7:9:11 chord over 0.

--

see my music page ---> ---<http://www.io.com/~hmiller/music/index.html>--

hmiller (Herman Miller) "If all Printers were determin'd not to print any

@io.com email password: thing till they were sure it would offend no body,

\ "Subject: teamouse" / there would be very little printed." -Ben Franklin