DAVE KEENAN'S MIRACLE SCALE

I've started examining Dave Keenan's 31-out-of-72 ME scale which
seems to be the "source set" for most of our proposals for Joseph
Pehrson.

It has some truly mind-blowing properties.

First of all, it has thirty-six consonant (7-limit) tetrads: eighteen
major and eighteen minor. I've seen a lot of M-out-of-N ME scales
where the number of tetrads is almost N -- but here it's _greater_
than N!

There are thirty-six consonant (5-limit) triads which are simply the
36 tetrads with their 7 identities removed.

There are twenty-five consonant (3-limit) dyads.

There are twenty-four consonant (9-limit) pentads: twelve major and
twelve minor.

There are eighteen consonant (11-limit) hexads: nine major and nine
minor.

How does the scale achieve this magic?

Answer: the scale is an MOS built from a generator which is 7/72
octave.

Stacking six of these upward gives you the 3/2. So you need a chain
of 6 to yield a 3-limit dyad. 31 - 6 = 25 -- that's why there are 25
dyads.

Stacking seven of these _downward_ gives you the 4/5. So you need a
chain of 7+6=13 to yield a 5-limit triad. 31-13=18 -- that's why
there are 18 major triads and 18 minor triads.

Stacking only two of these downward gives you the 7/8. That's why all
the triads can be completed into 7-limit tetrads.

Stacking twelve of these upward gives you the 9/4. 12+7=19, and 31-
19=12 -- that's why there are 12 major pentads and 12 minor pentads.

Stacking fifteen of these upward gives you the 11/4. 15+7=22, and 31-
22=9 -- that why there are 9 major hexads and 9 minor hexads.

It looks like the generator itself would make for a really
interesting non-octave ET, with nearly pure 5:7:8:12:18:22 hexads.
And Dave, didn't you once propose an MOS scale using a generator of
around this size, was it 115 cents, with around 10 notes?

This scale also has five 7-limit Tonality Diamonds. There are no 9-
limit Tonality Diamonds since the comma between 10/9 and 9/8 would
require a 1-step interval in 72-tET, of which this 31-tone subset has
none.

Also, as I just mentioned to Joseph in another thread, the scale has
sixteen 1-3-5-7 hexanies.

Paul,

--- In tuning@y..., paul@s... wrote:
> I've started examining Dave Keenan's 31-out-of-72 ME scale which
> seems to be the "source set" for most of our proposals for Joseph
> Pehrson.
>
> It has some truly mind-blowing properties.

This is great news, and I have a proposal: drop everything else you
are doing, no matter what it is, and devote the next forseeable
period of time composing or performing music the the wealth of
material you now have in front of you.

There is no reason *not* to take a break from all the clean-room
development and get your hands dirty making music.

To keep balance in the universe, I will commence, this evening, to
dig into your paper, which I just took out of my mailperson's hands
not 5 minutes ago, for which I thank you.

Cheers,
Jon

Here's a strange but wonderful beast to play with.

Joseph, since the generator of Dave Keenan's MIRACLE scale is 7/72,
or less than 1/11 of an octave, it struck me that a distributionally
even (not maximally even, but still "transposable") scale of 21 notes
formed from this generator would still have quite a few of the
MIRACULOUS properties!

It's not proper, but it is CS!

The scale would be:
0,2,7,9,14,16,21,23,28,30,35,37,42,44,49,51,56,58,63,65,70(,72)

Steps:
2,5,2,5,2,5,2,5,2,5,2,5,2,5,2,5,2,5,2,5,2

This scale would have 10 fewer of everything than the 31-tone MIRACLE
scale -- so there are only eight major tetrads and eight minor
tetrads and six hexanies . . . that's still quite a few!

Here's a small portion of the lattice diagram:
.
.
.
30---------0
,'/|\`. ,'/
2--------44--/-|-\-14 /
| 23--------65 \/| /
| ,'/|\`. |/,'/| `.\|/
37--/-|-\--7--/-|---49
16--------58--------28/,'
,'/|\`. |/,'/ `.\|/,'63
30--/-|-\--0--/-----42
9 |\/.51/,\/| /.\ /,'
,'/|\`. |/,'14-`.\|/--56
23--/-|-\-65--------35
/|\/ 44 \/| `.\|/,' |
/ |/,' `.\| 49 |
/ 58--------28--------70
/,' `.\ /,'
0--------42
.
.

I think a lot of the uneven 19- and 22-tone scales that Dave and I
were finding were actually "trying" to be this scale, or caught
partially between this scale and the ME 19-of-72 . . .

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

/tuning/topicId_21894.html#21894

> I've started examining Dave Keenan's 31-out-of-72 ME scale which
> seems to be the "source set" for most of our proposals for Joseph
> Pehrson.
>
> It has some truly mind-blowing properties.
>

Well, this is really important, so I'd better be sure I have the
right scale, ahem (!!)

It's the one that's outlined in:

/tuning/topicId_21636.html#21871

Correct??

> How does the scale achieve this magic?
>
> Answer: the scale is an MOS built from a generator which is 7/72
> octave.

What is it with the number 7?! This is really getting "out there...!"

We had all the just scales deviating from 19-tET by 7's and now this!

I'm hiding under the bed for the time being...

_________ _____ ______ ___
Joseph Pehrson

--- In tuning@y..., JSZANTO@A... wrote:
> Paul,
>
> --- In tuning@y..., paul@s... wrote:
> > I've started examining Dave Keenan's 31-out-of-72 ME scale which
> > seems to be the "source set" for most of our proposals for Joseph
> > Pehrson.
> >
> > It has some truly mind-blowing properties.
>
> This is great news, and I have a proposal: drop everything else you
> are doing, no matter what it is, and devote the next forseeable
> period of time composing or performing music the the wealth of
> material you now have in front of you.
>
> There is no reason *not* to take a break from all the clean-room
> development and get your hands dirty making music.
>
Yes, Jon, I remember you reacting the same way about two years ago,
when I was posting a slew of periodicity blocks.

Jon, this is a great suggestion, and I'm sure Joseph now has enough
material for a lifetime of great microtonal composition.

Hence you may be expressing frustration at seeing all this scale-
construction going on and no music.

But there is another side to what we're doing that you may not be
aware of -- the search for deeper understanding of general principles.

You see, there are some very deep multidimensional patterns
underlying all of this, but no one is going to grasp these all at
once.

By going through many particular cases, one begins to develop a feel
for the patterns and then . . . pop! . . . you'll be walking down the
street and you'll "get it" . . . something will click and you'll
suddenly understand a new overarching principle.

This is how mathematicians work.

Although Dave Keenan and I and others could stop now and make music
for the rest of our lives using a few of the scales we've created,
our particular talents and interests lead us to keep plugging away at
these creatures until something goes "click" -- or doesn't (there are
always blind alleys).

But the hope is to discover principles so that, in the future, when a
composer is looking for a scale with some particular features that
they may specify, we will be able to more quickly and surely provide
an answer. And other musical applications will suggest themselves as
the ideas crystallize and are understood in greater and greater
generality.

If you read Erv Wilson's work, particularly "D'Allessandro -- like a
hurricane", you'll begin to appreciate how profound this "popping"
process can be, and how it can provide inspiration for a lifetime of
scale-creation. One instance of this is captured in the Wilson quote
in Joe Monzo's .sig -- "I had broken through the lattice barrier".
This is a perfect example of such a conceptual "pop".

Kraig Grady has let us in on only the tip of the iceberg when it
comes to Erv Wilson's work, and it should provide materials for
generations of composers to come.

You may be aware that Harry Partch wrote the theoretical work in
_Genesis of a Music_ early in his life, and then went on to drop
theorizing for composing.

So you may wonder why someone would seemingly spend their life
developing a body of theoretical work 30 times as large as the Partch
tome. In this case, that "somebody" would be Dave Keenan and me.

Well, me, I'm 28. So hopefully, I will have plenty of time to
compose. Even though I have a very good ear, I'm only slowly
learning, little by little, to capture the music from my moments of
greatest inspiration, usually occuring between sleeping and waking
states.

Dave Keenan, on the other hand, is more of a mathematician than a
musician. Much more. By choosing to put his formidable mind to the
subject of tuning and microtonality, he is benefitting all composers
who choose to leave the 12-tET cage. He's probably not going to
produce any musical masterpieces anytime soon.

Similarly, a very fine composer somewhere, say in New York City,
might be fascinated with the sounds of various microtonal structures
but may be a struggling with ways to utilize them that allow his
familiar process of writing at the keyboard to be maintained, and his
goal of getting just-sounding performances from instrumentalists to
be realized. So he asks the mathematician for help.

What could be more natural than for people with different strengths
and talents to get together and collaborate on solving various
problems?

This is how society was built!

What would happen if all of a sudden all the mathematicians had to
become composers and all the composers had to become mathematicians.
I don't think that would lead to a happy time for the music listener!

Jon, I'm glad you're going to take a look at my paper, and perhaps
you'll have your own "breakthrough" and come back with a genuine
interest in the theoretical matters we're talking about.

But if not, that delete button will still be there for you. You'll
never have to look at another lattice diagram again, if you don't
want to. No harm done, no hard feelings.

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., paul@s... wrote:
>
> /tuning/topicId_21894.html#21894
>
>
> > I've started examining Dave Keenan's 31-out-of-72 ME scale which
> > seems to be the "source set" for most of our proposals for Joseph
> > Pehrson.
> >
> > It has some truly mind-blowing properties.
> >
>
> Well, this is really important, so I'd better be sure I have the
> right scale, ahem (!!)
>
> It's the one that's outlined in:
>
> /tuning/topicId_21636.html#21871
>
> Correct??

Nope -- there's no 31-tone scale mentioned in that message.

The scale in question is the following degrees of 72-tET:

0
2
5
7
9
12
14
16
19
21
23
26
28
30
33
35
37
39
42
44
46
49
51
53
56
58
60
63
65
67
70
(72)

I was about to mention that. It's the one in this message:

/tuning/topicId_21636.html#21805

I couldn't help fooling around with the unison vectors. This is a more
'friendly' set:

4 -1 0 0 80: 81
0 -2 -1 1 175:176
-2 -2 1 0 224:225
1 -1 -1 -1 384:385

Kees

----- Original Message -----
From: <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Monday, April 30, 2001 3:03 PM
Subject: [tuning] Re: DAVE KEENAN'S MIRACLE SCALE

> Nope -- there's no 31-tone scale mentioned in that message.
>
> The scale in question is the following degrees of 72-tET:
>
> 0
> 2
> 5
> 7
> 9
> 12
> 14
> 16
> 19
> 21
> 23
> 26
> 28
> 30
> 33
> 35
> 37
> 39
> 42
> 44
> 46
> 49
> 51
> 53
> 56
> 58
> 60
> 63
> 65
> 67
> 70
> (72)
>
>
> You do not need web access to participate. You may subscribe through
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>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
>

paul@stretch-music.com wrote:

> But there is another side to what we're doing that you may not be
> aware of -- the search for deeper understanding of general principles.

Don't stop what you do. It really is inspiring for composers like me (and Joseph if I might
speculate) who humbly admit to being novices in the field of tuning theory. Everyone has different
strengths. I don't think I'm any better than a mathematician just because I compose, though I
often curse mathematicians for your ability to generate such lucid creations and explanations.
That's just envy. The present 72 tet discussion however makes me more determined to understand
more deeply the general principles. You can't sing in the church choir if you don't have a church.

Best wishes.

--- In tuning@y..., "Kees van Prooijen" <kees@d...> wrote:
> I was about to mention that. It's the one in this message:
>
> /tuning/topicId_21636.html#21805
>
> I couldn't help fooling around with the unison vectors. This is a
more
> 'friendly' set:
>
> 4 -1 0 0 80: 81
> 0 -2 -1 1 175:176
> -2 -2 1 0 224:225
> 1 -1 -1 -1 384:385
>
> Kees

That almost works, Kees, but the parallelopiped produced by these
unison vectors, when mapped to 72-tET, has four changes relative to
Dave's scale. Of course, you can then transpose each of those four
notes by an 81:80 to get Dave's scale exactly.

I haven't verified if Dave's unison vectors immediately give the
correct notes in the parallelopiped, but I think that's what he was
aiming for . . .

Paul,

I guarantee that if you looked around for more 'more consonant
structures than M' type maximally even scales you'll find them. Though
nice enough, I don't think the 31-out-of-72 is really any miraculous
exception in this sense.

Anyway, nice to see you giving maximal evenness something like a good
word <G>...

For what it's worth, here's my two-term (Yasser-Kornerup) type
indexing expansion:

2/21, 1/10, 3/31, 4/41, 7/72, 11/113, 18/185, ...

And here's the Golden generator that this index is expanding towards:
1200/((21+Phi*10))*(2+Phi*1), or ~116.7725�.

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul,
>
> I guarantee that if you looked around for more 'more consonant
> structures than M' type maximally even scales you'll find them.
Though
> nice enough, I don't think the 31-out-of-72 is really any miraculous
> exception in this sense.

Well, I've looked around quite a bit . . . do you have any
suggestions?
>

>
> For what it's worth, here's my two-term (Yasser-Kornerup) type
> indexing expansion:
>
> 2/21, 1/10, 3/31, 4/41, 7/72, 11/113, 18/185, ...
>
> And here's the Golden generator that this index is expanding
towards:
> 1200/((21+Phi*10))*(2+Phi*1), or ~116.7725¢.
>
This refers, I presume, to the 7/72 oct. generator?

Yeah, well that's basically 'punishment' for using 'non-dissapearing'
unisons in a certain ET environment.
And I never associate a periodicity block with the parallelopiped produced
by it. It just signifies an equivalence relation over the infinite lattice.
(That's of course what you do yourself in getting alternative formations) It
helps me to find unisons as 'small' as possible to choose the particular
representatives of those classes, by moving along multiples of the unisons.
And as such you can always replace a unison by a suitable linear combination
of them.

Kees

----- Original Message -----
From: <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Monday, April 30, 2001 3:44 PM
Subject: [tuning] Re: DAVE KEENAN'S MIRACLE SCALE

> --- In tuning@y..., "Kees van Prooijen" <kees@d...> wrote:
> > I was about to mention that. It's the one in this message:
> >
> > /tuning/topicId_21636.html#21805
> >
> > I couldn't help fooling around with the unison vectors. This is a
> more
> > 'friendly' set:
> >
> > 4 -1 0 0 80: 81
> > 0 -2 -1 1 175:176
> > -2 -2 1 0 224:225
> > 1 -1 -1 -1 384:385
> >
> > Kees
>
> That almost works, Kees, but the parallelopiped produced by these
> unison vectors, when mapped to 72-tET, has four changes relative to
> Dave's scale. Of course, you can then transpose each of those four
> notes by an 81:80 to get Dave's scale exactly.
>
> I haven't verified if Dave's unison vectors immediately give the
> correct notes in the parallelopiped, but I think that's what he was
> aiming for . . .
>
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
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for the tuning group.
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mode.
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emails.
> tuning-help@yahoogroups.com - receive general help information.
>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
>

--- In tuning@y..., "Kees van Prooijen" <kees@d...> wrote:
> Yeah, well that's basically 'punishment' for using 'non-
dissapearing'
> unisons in a certain ET environment.

Well, Kees, I'm going to flaggelate myself now, because the
parallelopiped formed by Dave's unison vectors doesn't exactly give
his scale either.

So you were perfectly justified in giving the "simplified" unison
vectors you gave.

Now, how on earth would we derive that amazing 21-note scale from
periodicity blocks? I think once we're in a 72-tET universe, the
rules of the game change quite a bit (even though the consonant
sonorities are so close to JI).

Paul Erlich wrote,

<<Well, I've looked around quite a bit . . . do you have any
suggestions?>>

It's not really an interest of mine, so no, no suggestions. I'd have
to get in there and look around first. (The 17-out-of-58 that I've
used before comes to mind, but that's a pure guess... I'd have to take
some time and look around.)

<<This refers, I presume, to the 7/72 oct. generator?>>

Well yeah, in the same sense that the Golden meantone generator refers
to 7/12.

--Dan Stearns

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

/tuning/topicId_21894.html#21903

> Here's a strange but wonderful beast to play with.
>
> Joseph, since the generator of Dave Keenan's MIRACLE scale is 7/72,
> or less than 1/11 of an octave, it struck me that a
distributionally even (not maximally even, but still "transposable")
scale of 21 notes formed from this generator would still have quite a
few of the MIRACULOUS properties!
>

Well, this is a curiosity... It looks like an hexany "mobius strip..."

________ ______ _______ ____
Joseph Pehrson

Oh, Paul! You really made my day. Never in my wildest dreams could I have
imagined to accomplish that :-)

Kees

----- Original Message -----
From: <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Monday, April 30, 2001 4:21 PM
Subject: [tuning] Re: DAVE KEENAN'S MIRACLE SCALE

> Well, Kees, I'm going to flaggelate myself now

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

/tuning/topicId_21894.html#21909

>
> Nope -- there's no 31-tone scale mentioned in that message.
>

Whoopsie! Where did Keenan hide that one??

THANKS, PAUL!!!!

> The scale in question is the following degrees of 72-tET:
>
> 0
> 2
> 5
> 7
> 9
> 12
> 14
> 16
> 19
> 21
> 23
> 26
> 28
> 30
> 33
> 35
> 37
> 39
> 42
> 44
> 46
> 49
> 51
> 53
> 56
> 58
> 60
> 63
> 65
> 67
> 70
> (72)

_________ _____ _ _____
Joseph Pehrson

Paul,

Oh heck, I tried writing a couple of replies, but didn't like any of
them! You guys are fine in my book, my message was pretty much made
up of 96.4% fun, and if what I've said in the past wasn't clear, I'll
put it in writing:

"It is perfectly fine to engage in purely theoretical research."

Your 'explanation' was such that I've saved it, and since it *is* of
interest to me, I'll write back on the subject at a later point.

> What would happen if all of a sudden all the mathematicians had to
> become composers and all the composers had to become
> mathematicians. I don't think that would lead to a happy time for
> the music listener!

Yes, but the musicians would quit bouncing checks!

> Jon, I'm glad you're going to take a look at my paper, and perhaps
> you'll have your own "breakthrough" and come back with a genuine
> interest in the theoretical matters we're talking about.

Me too. And I hope you both can spend some down time listening to
great music and have a "breakthrough" and come back with a genuine
interest in writing/playing some music. If I can't be allowed to put
off studying math, you can't put off making more music until you're
older!

> But if not, that delete button will still be there for you. You'll
> never have to look at another lattice diagram again, if you don't
> want to. No harm done, no hard feelings.

Hey, my epiphany (or "breakthrough") came with the dulcet tones of
Lou Harrison, who in a very human way extolled the virtues of scale
creation. I do not -- at all -- look askance at what you are toiling
at...

Cheers,
Jon

--- In tuning@y..., "Kees van Prooijen" <kees@d...> wrote:

/tuning/topicId_21894.html#21911

> I was about to mention that. It's the one in this message:
>
> /tuning/topicId_21636.html#21805
>

Thanks Kees!

THAT'S WHY I missed it! That post was BASICALLY about a 19-tone
scale that Dave made for me... and the 31 was at the VERY BOTTOM.

Who would think that I would read down that far!!! :)

_________ _______ _______ _____
Joseph Pehrson

--- In tuning@y..., Alison Monteith <alison.monteith3@w...> wrote:

/tuning/topicId_21894.html#21913
>
>
> paul@s... wrote:
>
> > But there is another side to what we're doing that you may not be
> > aware of -- the search for deeper understanding of general
principles.
>
> Don't stop what you do. It really is inspiring for composers like
me (and Joseph if I might speculate) who humbly admit to being
novices in the field of tuning theory. Everyone has different
> strengths. I don't think I'm any better than a mathematician just
because I compose, though I often curse mathematicians for your
ability to generate such lucid creations and explanations.
> That's just envy. The present 72 tet discussion however makes me
more determined to understand more deeply the general principles.

Thank you, Allison! That's exactly how I feel. It is INSPIRING to
watch the exchange between Keenan and Erlich... I feel I'm in the
middle of a story and the plot is always thickening!

Well, at least *I* find it entertaining; I don't know about others.

Additionally, at some point these ideas will be very USEFUL, but not
until I have a firm enough "grounding..."

Thanks for your comments, and you guessed my sentiments correctly...

________ _____ _____ ____
Joseph Pehrson

I wrote,

<<It's not really an interest of mine, so no, no suggestions. I'd have
to get in there and look around first. (The 17-out-of-58 that I've
used before comes to mind, but that's a pure guess... I'd have to take
some time and look around.)>>

Scratch that last bit, the parenthetical M-out-of-N example, as I was
thinking of something quite different there.

--Dan Stearns

--- In tuning@y..., jpehrson@r... wrote:
>
> Well, this is a curiosity... It looks like an hexany "mobius strip..."

This seems to have much more consonances, at least in the 7-limit, than any of the 19-
tone or 22-tone subsets of 72-tET that we've come up with. Dave, correct me if I'm wrong.

We can call this the "blackjack" scale due to the "surprise" of it having 21 notes.

I'm going to unsubscribe for a while. You can e-mail me (if you're reading this on the
website) by clicking on my truncated e-mail address above this message.

Oh, Neil H., if you're reading this, thanks for mailing out the 31-tone guitar (yippee!!!) --
the check is in the mail.

Cheers!

Paul,

I'm starting to try and take a look at this a bit but I had a few
questions to try and make sure that I'm clearly understanding
everything...

You wrote,

<<How does the scale achieve this magic? Answer: the scale is an MOS
built from a generator which is 7/72 octave.>>

It seems to me that this same thing can be accomplished in a simpler
ET like 41-tET or other appropriately consistent ETs like 113, right?

Even ETs that aren't all that consistent, 51-tET being a good example,
still have a generator (5/51) which when linked in the fashion you go
on to outline in the post I just quoted can accomplish the same thing,
no?

If not, please explain.

thanks,

--Dan Stearns

72er!
For the record Hanson Keyboard can be considered as having a generator of 19. Also his
keyboard can have 15 ranks as opposed to 19. see http://www.anaphoria.com/images/hebdo-key2.gif
there are more solutions that i will not be able to point out at this time. That 72 can contain
the Hebdomekontany
gives one 20 Eikosanies. Also all the hexanies are found in a crosset with the reciprocal
hexanies. for example the 1-3-5-7 hexany is found on a cross-set of the 9-11-13-15 hexany. This
scratches the surface-back to puppet land!

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

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

I wrote,

<<I'm starting to try and take a look at this a bit but I had a few
questions to try and make sure that I'm clearly understanding
everything>>

Okay, I think I pretty much answered these questions for myself. So I
just went ahead and started poking around a bit for M-out-of-Ns where
consonant identities are greater than M.

What about 19-out-of-22?

It would seem that this [16,3] ME set takes advantage of interval
class (proper) ambiguity to maximize consonances.

Anyway, this is the sort of thing I had in mind if it's correct: If
it's not, let me know.

--Dan Stearns

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

/tuning/topicId_21894.html#21908

> ...
>
> But there is another side to what we're doing that you may
> not be aware of -- the search for deeper understanding of
> general principles.

Paul, thanks *so much* for this post! It reflects my feelings
about our theoretical research *exactly*.

>
> You see, there are some very deep multidimensional patterns
> underlying all of this, but no one is going to grasp these
> all at once.
>
> By going through many particular cases, one begins to develop
> a feel for the patterns and then . . . pop! . . . you'll be
> walking own the street and you'll "get it" . . . something
> will click and you'll suddenly understand a new overarching
> principle.
>
> ...
>
> If you read Erv Wilson's work, particularly "D'Allessandro --
> like a hurricane", you'll begin to appreciate how profound
> this "popping" process can be, and how it can provide
> inspiration for a lifetime of scale-creation. One instance
> of this is captured in the Wilson quote in Joe Monzo's .sig
> -- "I had broken through the lattice barrier".
> This is a perfect example of such a conceptual "pop".

Well, I was planning to reply to this post anyway, but now
since you mentioned me, I feel compelled.

I believe (but I'm not entirely sure) that Wilson and I are
both referring to precisely the same "lattice barrier": how
does one represent structures with more than 3 dimensions
on a 2-dimensional piece of paper or computer screen?
Anyway, that was my "barrier".

I struggled for at least a couple of years with this problem.

Those of you who have a copy of my book can see the earlier
"matrix diagrams" I created (the forerunners of my "lattice
diagrams" - eventually they will be purged from the book, so
keep those old copies as collector's items!), where prime-factors
3 and 5 could be conveniently arranged in a 2-dimensional space,
as here:
http://www.ixpres.com/interval/monzo/article/article.htm

But to include prime-factors above 5, I had to stack the
3x5 planes like steps on a ladder. I was never entirely
happy about this design.

Then suddenly one day I realized that by using a variety of
different angles, each prime-factor could have its own
unique dimension.

John Chalmers and Erv Wilson had already done this, but
I wasn't aware of their work at the time. The first two
diagrams here give an illustration:
http://www.ixpres.com/interval/chalmers/diagrams.htm

Chalmers and Wilson used more-or-less arbitrary angles
- altho Wilson's often are determined by his desire to
harness the structural possibilities of polyhedra; see:
http://www.georgehart.com/pavilion.html
for a bunch of links dealing with polyhedra.

After playing around with this for a few days, I realized
that I could impose a formula where each prime had a logical
angle for its axis, thus revealing the sort of "general
principles" mentioned here by Paul. See:
http://www.ixpres.com/interval/monzo/lattices/lattices.htm

The concepual "pop" Paul refers to was indeed a flash of
insight that struck me with a force I've felt only a few
times in my life. Chasing after this insight is the primary
reason why I myself am so interested in this theoretical
stuff.

Yes, it means that I "waste" a lot of my time on theory,
instead of making use of the talents and abilities as a
composer that other people have told me I have. But one
does with his life what one wants to, if he has that
opportunity.

Yes, I'd like to compose more music. But I suppose, to
paraphrase Partch, I'm "a composer seduced into theory".
And at this point, I don't think I can turn back.

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul,
>
> I'm starting to try and take a look at this a bit but I had a few
> questions to try and make sure that I'm clearly understanding
> everything...
>
> You wrote,
>
> <<How does the scale achieve this magic? Answer: the scale is an MOS
> built from a generator which is 7/72 octave.>>
>
> It seems to me that this same thing can be accomplished in a simpler
> ET like 41-tET or other appropriately consistent ETs like 113,
right?

41-tET wouldn't make much sense. In 41-tET, the maximum 7-limit error
is 6 cents and the maximum 11-limit error is 11 cents. But in 31-tET,
the maximum 7-limit error is again 6 cents and the maximum 11-limit
error is again 11 cents. The scale in 31-tET (i.e., all of 31-tET)
has, of course, 31 major hexads and 31 minor hexads. So it seems that
all of 31-tET is far preferable to using 31-out-of-41-tET.

However, 72-tET has a maximum 7-limit error of 3 cents and a maximum
11-limit error of 4 cents. At this level of approximation to JI, the
MIRACLE scale really stands out.

113-tET has a maximum 7-limit error of 4 cents and a maximum 11-limit
error of 5 cents. So 72-tET looks better.
>
> Even ETs that aren't all that consistent, 51-tET being a good
example,
> still have a generator (5/51) which when linked in the fashion you
go
> on to outline in the post I just quoted can accomplish the same
thing,
> no?

No. The error in the 11:4 would be over 13 cents, for example.

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> I wrote,
>
> <<I'm starting to try and take a look at this a bit but I had a few
> questions to try and make sure that I'm clearly understanding
> everything>>
>
> Okay, I think I pretty much answered these questions for myself. So
I
> just went ahead and started poking around a bit for M-out-of-Ns
where
> consonant identities are greater than M.
>
> What about 19-out-of-22?
>
> It would seem that this [16,3] ME set takes advantage of interval
> class (proper) ambiguity to maximize consonances.
>
> Anyway, this is the sort of thing I had in mind if it's correct: If
> it's not, let me know.

Here you can add three notes and greatly increase the number of
consonant structures. So it's not very "efficient", if you catch my
drift.

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:

> What about 19-out-of-22?
>
> It would seem that this [16,3] ME set takes advantage of interval
> class (proper) ambiguity to maximize consonances.
>
Actually, 19-out-of-22 is not a CS scale -- big disadvantage.

I wrote,

<<I went ahead and started poking around a bit for M-out-of-Ns where
consonant identities are greater than M. [SNIP] Anyway, this is the
sort of thing I had in mind if it's correct>>

No word from pope Paul yet... but assuming that I'm correct here, as I
expected I was able to find a few of these very easily right off the
bat and am sure that there are many more.

So I think if these types of sets are to be considered truly
"miraculous" or at least somewhat rare, then they would probably need
a stipulation something like they must be uniquely articulated thru
some limit (tonality diamond)... ?

--Dan Stearns

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
>
> > What about 19-out-of-22?
> >
> > It would seem that this [16,3] ME set takes advantage of interval
> > class (proper) ambiguity to maximize consonances.
> >
> Actually, 19-out-of-22 is not a CS scale -- big disadvantage.

Paul,

I don't think Dan was claiming that this was a very useful scale in
general, but merely that it has the property that you described as
miraculous in the post that started this thread.

Paul Erlich wrote,

<<41-tET wouldn't make much sense. In 41-tET, the maximum 7-limit
error is 6 cents and the maximum 11-limit error is 11 cents. But in
31-tET, the maximum 7-limit error is again 6 cents and the maximum
11-limit error is again 11 cents. The scale in 31-tET (i.e., all of
31-tET) has, of course, 31 major hexads and 31 minor hexads. So it
seems that
all of 31-tET is far preferable to using 31-out-of-41-tET.>>

I could argue this differently or easily give a counter example that
is line with your modified definitions here. But my point was simply
that the condition of an M-out-of-N that where consonant identities
are greater than M is not all that rare (as you seemed to be saying
before).

<<However, 72-tET has a maximum 7-limit error of 3 cents and a maximum
11-limit error of 4 cents. At this level of approximation to JI, the
MIRACLE scale really stands out. 113-tET has a maximum 7-limit error
of 4 cents and a maximum 11-limit error of 5 cents. So 72-tET looks
better.>>

Well I'm all for the most manageable (smallest) ET that will
accomplish a given goal, so even if 113 were slightly closer error
wise I'd like 72 anyway.

<<Here you can add three notes and greatly increase the number of
consonant structures. So it's not very "efficient", if you catch my
drift.>>

Again, a weak amendment I'd say! How about 19-out-of-26?

I think these 72 sets are neat. And refining the conditions that make
them unique are bound to be helpful to all who are interested.

Keep up the good work, and to all a good night!

--Dan Stearns

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

/tuning/topicId_21894.html#21984

> Those of you who have a copy of my book can see the earlier
> "matrix diagrams" I created (the forerunners of my "lattice
> diagrams" - eventually they will be purged from the book, so
> keep those old copies as collector's items!), where prime-factors
> 3 and 5 could be conveniently arranged in a 2-dimensional space,
> as here:
> http://www.ixpres.com/interval/monzo/article/article.htm
>

Hi Monz!

Quite frankly, I believe this article could use a little "gentle
introduction." The basic ideas of the vectoral expressions, I
believe, are not really all that difficult in their fundamental
presentation... and I believe such a demonstration, as we have done
the last couple of weeks on this list, would GREATLY benefit people
trying to understand your fascinating theories!

best,

_________ _______ _______ _____
Joseph Pehrson

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

> Chalmers and Wilson used more-or-less arbitrary angles
> - altho Wilson's often are determined by his desire to
> harness the structural possibilities of polyhedra; see:
> http://www.georgehart.com/pavilion.html
> for a bunch of links dealing with polyhedra.

Rather than having an initial desire to harness the structural
possibilities of polyhedra, I feel these properties have relevance to
Wilson's lattices as a natural result of using a lattice diagram that
shows each consonant interval as a direct connection.

Paul!
Well said. One of the important features of microtonality are new structural possibilities!

paul@stretch-music.com wrote:

>
> Rather than having an initial desire to harness the structural
> possibilities of polyhedra, I feel these properties have relevance to
> Wilson's lattices as a natural result of using a lattice diagram that
> shows each consonant interval as a direct connection.
> s/

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

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

Well, Dan,when I looked at these ME M-out-of-N scales a while back, I was looking for CS
scales that both improved the harmonies quite a bit over M-tET, and had about as many
consonant tetrads as notes. Although I found quite a few of these, such as 19-out-of-31, I didn't
see any that had considerably more tetrads than notes. 31-out-of-72 was the first I've seen with
this property (its complement, 41-out-of-72, should be another one, and should even have
almost as many _hexads_ as notes . . .).

I guess the matter could be quantified in terms of maximizing some function related to the
number of consonant tetrads (or N-ads) and inversely related to the number of notes and the
size of the errors.

The obvious paradigm for this seach is the diatonic scale, which has almost as many consonant
triads (6) as notes (7).

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

/tuning/topicId_21894.html#22014

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> /tuning/topicId_21894.html#21984
>
>
> > Those of you who have a copy of my book can see the earlier
> > "matrix diagrams" I created (the forerunners of my "lattice
> > diagrams" - eventually they will be purged from the book, so
> > keep those old copies as collector's items!), where prime-factors
> > 3 and 5 could be conveniently arranged in a 2-dimensional space,
> > as here:
> > http://www.ixpres.com/interval/monzo/article/article.htm
> >
>
> Hi Monz!
>
> Quite frankly, I believe this article could use a little "gentle
> introduction." The basic ideas of the vectoral expressions, I
> believe, are not really all that difficult in their fundamental
> presentation... and I believe such a demonstration, as we have done
> the last couple of weeks on this list, would GREATLY benefit people
> trying to understand your fascinating theories!

Oh my goodness...

That paper *was* supposed to be the "gentle introduction"
to my theories! It was what I had originally wanted to publish
back in 1995, but it kept growing and became my book, which
is still growing...

I don't know if I'd know where to begin. Anyone else care
to give it a try? Paul?...

-monz
http://www.monz.org
"All roads lead to n^0"

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

/tuning/topicId_21894.html#22017

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > Chalmers and Wilson used more-or-less arbitrary angles
> > - altho Wilson's often are determined by his desire to
> > harness the structural possibilities of polyhedra; see:
> > http://www.georgehart.com/pavilion.html
> > for a bunch of links dealing with polyhedra.
>
> Rather than having an initial desire to harness the
> structural possibilities of polyhedra, I feel these properties
> have relevance to Wilson's lattices as a natural result of
> using a lattice diagram that shows each consonant interval
> as a direct connection.

OK, I can buy that... a result rather than an intention.

As you're aware, Paul, I sometimes use the "triangular"
style of lattice (which "shows each consonant interval
as a direct connection") myself.

So while I'm here, maybe I should add that the reason I
stick so adamantly to my usual lattice formula - while you
and others argue that the "each consonant interval" method
is better - is that it allows me to graph *any* (and I do
mean *any*) historical tuning as a subset of a huge imaginary
lattice that contains them all.

This is important for my purposes, and my lattice formula
is the only one I've ever come across that will do it.

-monz
http://www.monz.org
"All roads lead to n^0"

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

> So while I'm here, maybe I should add that the reason I
> stick so adamantly to my usual lattice formula - while you
> and others argue that the "each consonant interval" method
> is better - is that it allows me to graph *any* (and I do
> mean *any*) historical tuning as a subset of a huge imaginary
> lattice that contains them all.
>
> This is important for my purposes, and my lattice formula
> is the only one I've ever come across that will do it.

You've got to be kidding.

You guys have been moving really fast but I'll still try to
wedge a word in wedge-wise...

I've been experimenting with a 'tool' at home, if I can call
it that. I wrote a program which inputs "diatonic strings"
(BBaBBBa as an example of a familiar one) and the program
walks all the possible values of B and a in cents against
a 'psuedo-harmonic-entropy' graph, looking for local minima,
where a scale will have 'maximum concordance' (with all the
caveats, simpler intervals with less accuracy may
sometimes be preferred to complex ones with higher accuracy
and visca-versa).

There will be more on this later. So I came home and plugged
in the string

BaaBaaBaaBaaBaaBaaBaaBaaBaaBaaa

took the results to excel and the best result according to
my current criteria are B=49c a=33.8c. The 'complexity'
minimum is at 49 and the error minimum is at 50 (the '3'
in 72tet). My overall weighting lands on 49.

For what its worth, the RI buckets my program puts
the scale (as I wrote it above) into is

'1/1 '33/32 '21/20 '15/14 '11/10 '9/8
'8/7 '20/17 '6/5 '11/9 '29/23 '9/7
'17/13 '31/23 '11/8 '7/5 '23/16 '22/15 '3/2
'20/13 '11/7 '8/5 '28/17 '27/16 '12/7
'30/17 '9/5 '11/6 '17/9 '25/13 '47/24

I hope to write more about this program to this
forum after my weekend, but thought I'd at least chime
in with this 'me too!'

Bob Valentine

--- In tuning@y..., BVAL@I... wrote:

> I hope to write more about this program to this
> forum after my weekend, but thought I'd at least chime
> in with this 'me too!'

Thanks. As you know, I feel this approach is only valid if you're
going to use the scale against a drone, fixed at the pitch of the
tonic of the mode you've chosen. What interests me more are the
intervals between pairs of notes in the scale, the tonic not
necessarily being a member of each pair. That approach would treat
all modes of the scale as identical (which yours doesn't).

What do you get if you use the mirror-symmetrical mode of the Miracle
scale instead?

On Wed, 02 May 2001 19:20:34 -0000 "monz" <joemonz@yahoo.com> writes:

> Then suddenly one day I realized that by using a variety of
> different angles, each prime-factor could have its own
> unique dimension.
>
> John Chalmers and Erv Wilson had already done this, but
> I wasn't aware of their work at the time. The first two
> diagrams here give an illustration:
> http://www.ixpres.com/interval/chalmers/diagrams.htm
>
> Chalmers and Wilson used more-or-less arbitrary angles
> - altho Wilson's often are determined by his desire to
> harness the structural possibilities of polyhedra; see:
> http://www.georgehart.com/pavilion.html
> for a bunch of links dealing with polyhedra.
>
> After playing around with this for a few days, I realized
> that I could impose a formula where each prime had a logical
> angle for its axis, thus revealing the sort of "general
> principles" mentioned here by Paul. See:
> http://www.ixpres.com/interval/monzo/lattices/lattices.htm

I guess this is very similar to what I discussed in my brief thread with
Paul two weeks ago about multidimensional scaling solutions. I hope to
post more about the use and theory of multidimensional scaling soon for
those not familiar with it. Since my search through my local public
library and community college library catalogs revealed no hits on
various subject keywords related to it, I imagine not very many people
here are familiar with the subject (although most of the subjects on this
list also come up with very few hits in library databases or on the web).

One of the key uses of multidimensional scaling is to reveal subtle
relationships of data points in large sets. The previously mentioned
studies by Krumhansel and Kessler, 1982 (really, I will give more info on
this later, I promise), and by Paul Erlich conclude that relationships
amongst key signatures (interval classes in Paul's analysis) in 12tET can
be summarized in four dimensions. Krumhansel's interpretation was that
one two-dimensional projection of the four-dimensional solution reveals
the circle of fifths relationships amongst key signatures, and another
two-dimensional projection reveal parallel major-key/minor-key
relationships. Looking at the second graph in her book "Cognitive
Foundations of Musical Pitch," pg 43, I see that major keys are grouped
closely by major thirds (Db F A, D F# Bb, etc.), with adjacent trios of
minor keys (db f a, d f# bb, etc.) in four sets progressing by semitones,
in a circle, for a total of 24 key signatures. It seems to me that the
first graph presents 3-limit relationships, and the second group presents
5-limit relationships.

A key feature of multidimensional scaling is the "stress value." The
computer program takes as input a matrix containing similarity values for
all pairs of data points under consideration (in Krumhansel's study, the
correlation coefficients of pairs of key signatures, a 24 x 24 matrix).
The program calculates distances amongst the points using the Pythagorean
Theorem, and determines a number of dimensions that does the least
violence to the data. The greater the number of dimensions, the lower
the stress value, and vice versa. It is my hypothesis that the best
representation of sets of intervals will have a dimensionality
corresponding to the limit value of the system. In other words, an
11-limit system would need to be represented in 11 dimensions for a low
stress value. However, I am new at this line of inquiry, so please don't
consider me an authority on the subject.

The "popping" sensation you are describing seems to relate to the problem
of visualizing data patterns in more than three dimensions. Fortunately,
this is a problem encountered by vast numbers of scientists,
philosophers, and mystics; consequently, there seems to be quite a few
more books on this subject. I just checked one out from the library
entitled "Beyond the Third Dimension: Geometry, Computer Graphics, and
Higher Dimensions" by Thomas F Banchoff, copyright 1990 by Scientific
American Library. It is helping me quite a bit.

By the way, Monz, I have started to dig into your on-line dictionary, and
I'm finding it very helpful. I know that more than one of us is
benefiting by your "[seduction] with theory." I also, however, wish I
had more time for music.

Sincerely,
Kurt

--- In tuning@y..., Kurt S Nelson <kurtnelson2@j...> wrote:

/tuning/topicId_21894.html#22050

> I guess this is very similar to what I discussed in my brief
> thread with Paul two weeks ago about multidimensional scaling
> solutions. <...etc.>

Thanks for this informative post, Kurt.

> By the way, Monz, I have started to dig into your on-line
> dictionary, and I'm finding it very helpful. I know that
> more than one of us is benefiting by your "[seduction] with
> theory."

Glad the Dictionary is useful to you. One reason I sacrifice
so much of my time for this stuff is because of my addiction
to it, but another reason is because it really used to be
hard to find info about tuning theory, and contributing all
of this makes it much easier for others to get involved.

> I also, however, wish I had more time for music.

As I wrote recently, "So many tunings... so little time..."

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning@y..., Kurt S Nelson <kurtnelson2@j...> wrote:
>
> A key feature of multidimensional scaling is the "stress value."
The
> computer program takes as input a matrix containing similarity
values for
> all pairs of data points under consideration (in Krumhansel's
study, the
> correlation coefficients of pairs of key signatures, a 24 x 24
matrix).
> The program calculates distances amongst the points using the
Pythagorean
> Theorem, and determines a number of dimensions that does the least
> violence to the data.

More precisely, you specify a number of dimensions, and the program
calculates a set of positions for the points such that the inverse
distances (or some other function of the distances you may specify)
mirror the similarity values (or some function thereof) with minimum
sum-of-squared error.

Some programs may also include a heuristic for determining the "best"
number of dimensions. More dimensions can never hurt your fit, but
fewer dimensions simplify the picture.

> The greater the number of dimensions, the lower
> the stress value, and vice versa. It is my hypothesis that the best
> representation of sets of intervals will have a dimensionality
> corresponding to the limit value of the system. In other words, an
> 11-limit system would need to be represented in 11 dimensions for a
low
> stress value. However, I am new at this line of inquiry, so please
don't
> consider me an authority on the subject.

I think we pretty much know how this will fall out. For a JI scale,
you need one dimension for a 3-limit system, two dimensions for a 5-
limit system, three for 7-limit, four for 11-limit, etc. For a
tempered scale, you need as many additional dimensions as there are
unison vectors. For example, Krumhansl's 4-dimensional torus comes
from the two-dimensional 5-limit JI lattice being curved around to
meet itself at both the syntonic comma and the diesis. A 5-limit
meantone system would be only 3-dimensional because only the syntonic
comma is used to "wrap" the 2-d 5-limit plane into a cylinder.

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

/tuning/topicId_21894.html#22049

> --- In tuning@y..., BVAL@I... wrote:
>
> > I hope to write more about this program to this
> > forum after my weekend, but thought I'd at least chime
> > in with this 'me too!'
>
> Thanks. As you know, I feel this approach is only valid if you're
> going to use the scale against a drone, fixed at the pitch of the
> tonic of the mode you've chosen. What interests me more are the
> intervals between pairs of notes in the scale, the tonic not
> necessarily being a member of each pair. That approach would treat
> all modes of the scale as identical (which yours doesn't).
>

I've been thinking about this kind of thing too, in light of the
Erlich just 19 scale #3 in Just Intonation as contrasted with, still,
19-tET. It is true, a drone would really greatly increase the
potential of a just scale, but the question is whether it would fit
into one's compositional style. For me, particularly if it went on
for some time, it most probably would not, which is why I was
interested in the diadic comparison study...

___________ ________ _______ ______
Joseph Pherons

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

> I've been thinking about this kind of thing too, in light of the
> Erlich just 19 scale #3 in Just Intonation as contrasted with,
still,
> 19-tET. It is true, a drone would really greatly increase the
> potential of a just scale,

In some cases, yes, but not in this particular case. The 19-tone just
scale #3 has a lot of consonances of consonances, and consonances of
consonances of consonances, which would be quite dissonant if you
held the 1/1 as a drone against them.

Note: this reply was meant to answer Joseph Pehrson as best I know
his thinking and should not be taken out of context.

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

/tuning/topicId_21894.html#22056

> --- In tuning@y..., jpehrson@r... wrote:
>
> > I've been thinking about this kind of thing too, in light of the
> > Erlich just 19 scale #3 in Just Intonation as contrasted with,
> still,
> > 19-tET. It is true, a drone would really greatly increase the
> > potential of a just scale,
>
> In some cases, yes, but not in this particular case. The 19-tone
just scale #3 has a lot of consonances of consonances, and
consonances of consonances of consonances, which would be quite
dissonant if you held the 1/1 as a drone against them.
>
> Note: this reply was meant to answer Joseph Pehrson as best I know
> his thinking and should not be taken out of context.

Oh! Of course... got it. Thanks Paul...
_______ ______ __ ______
Joseph Pehrson

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

/tuning/topicId_21894.html#22043

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > So while I'm here, maybe I should add that the reason I
> > stick so adamantly to my usual lattice formula - while you
> > and others argue that the "each consonant interval" method
> > is better - is that it allows me to graph *any* (and I do
> > mean *any*) historical tuning as a subset of a huge imaginary
> > lattice that contains them all.
> >
> > This is important for my purposes, and my lattice formula
> > is the only one I've ever come across that will do it.
>
> You've got to be kidding.

C'mon, Paul, *you* know I'm not kidding. In my book, I discuss
tuning systems with various prime factors all the way up to 499
(and possibly far beyond that).

There's no other way I know of to represent the harmonic
relationships of *all* of them uniquely as subsets of one huge
all-inclusive set, other than my lattice formula.

I agree with the benefits you and others have emphasized about
using other lattice formulas, but they're always for systems
of small dimensionality, usually 4-dimensional at most.

In my analyses, I have to represent the following prime-factors:

most historical JI systems: 3, 5, 7, 11, 13

Ptolemy: 3, 5, 7, 11, 23

Ben Johnston: all of the above plus 17, 19, 23, 29, 31

Ezra Sims: all of the above plus 37

Franz Richter Herf: all of the above plus 47

John Dowland: 3, 7, 11, 17, 31, 211

La Monte Young: 3, 7, 29, 31, 59, 61, 67, 71, 113, 127, 131,
137, 139, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271,
277, 281, 283

Boethius: 3, 19, 499

Fabio of Colonna: 3, 5, 7, 11, 13, 17, 23, 29, 37, 41, 43, 53,
67, 83, 101, 131, 163, 197, 1039, 1787, 3617

(My analysis of Colonna is still tentative; some of the numbers
may be wrong... more research necessary.)

That's 46 different prime-dimensions if Colonna is included, or
36 without him. And please note: this list does not include
my speculative interpretations such as that of Marchetto, etc.
All these ratios were *required* by these composers/theorists.
If I were to include analyses of McLaren's rational tunings,
I'd need even more.

Please tell me about a better way to represent a 46-dimensional
structure than the one I've found.

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> --- In tuning@y..., paul@s... wrote:
>
> /tuning/topicId_21894.html#22043
>
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > > So while I'm here, maybe I should add that the reason I
> > > stick so adamantly to my usual lattice formula - while you
> > > and others argue that the "each consonant interval" method
> > > is better - is that it allows me to graph *any* (and I do
> > > mean *any*) historical tuning as a subset of a huge imaginary
> > > lattice that contains them all.
> > >
> > > This is important for my purposes, and my lattice formula
> > > is the only one I've ever come across that will do it.
> >
> > You've got to be kidding.
>
>
> C'mon, Paul, *you* know I'm not kidding. In my book, I discuss
> tuning systems with various prime factors all the way up to 499
> (and possibly far beyond that).
>
> There's no other way I know of to represent the harmonic
> relationships of *all* of them uniquely as subsets of one huge
> all-inclusive set, other than my lattice formula.
>
First of all, that rules out all non-JI systems. Non-JI systems
(which I believe have been more important than JI systems, except for
Pythagorean) are not uniquely represented by your lattice, because
the fundamental theorem of arithmetic only applies when the exponents
of the primes are integers.

Secondly, there is at least one other way I know you know of, mapping
the primes to a semicircle rather than a circle.

But of course there are an infinite number of ways to do it, for
example one could use the golden angle times the ordinal number of
the prime. Any one-to-one mapping of primes to angles will do, and
any set of lengths will do too (as long as no overlaps result).

>
> I agree with the benefits you and others have emphasized about
> using other lattice formulas, but they're always for systems
> of small dimensionality, usually 4-dimensional at most.

I don't know what you mean.

If you mean showing all the consonances, this is a completely
separate issue. You could show all the consonances on your current
lattice diagrams if you wanted to, by using dotted lines to connect
pairs of notes separated by a 5:3, etc.

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

/tuning/topicId_21894.html#22058

> Please tell me about a better way to represent a 46-dimensional
> structure than the one I've found.

Well... there's also the variant of my formula that is based
on 180 degrees instead of 360. So I have two good formulas.

By all means, inform me of others.

-monz
http://www.monz.org
"All roads lead to n^0"

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> /tuning/topicId_21894.html#22058
>
> > Please tell me about a better way to represent a 46-dimensional
> > structure than the one I've found.
>
>
> Well... there's also the variant of my formula that is based
> on 180 degrees instead of 360. So I have two good formulas.
>
> By all means, inform me of others.

As you now know, there are an infinite number of ways to do it.
Here's what I would do.

First, the 5-limit lattice would be represented exactly as in the
second lattice at http://www.kees.cc/tuning/lat_perbl.html.

Then I'd use multidimensional scaling on the four points in a 1:3:5:7
tetrad, constraining the 1:3:5 tetrad to be represented as above, to
determine the best length and angle for the 7:4 (and concurrently,
the 7:5 and 7:6, whether you choose to represent those with lines or
not) on the 2-d plane. I posted the result of doing this a long time
ago, back when the list was on the Mills server.

Having fixed those, I'd do the same for 11.

Then 13.

Then 17.

And so on.

The semicircle method is of course much simpler in that you
immediately can see what fraction of an octave a particular prime
interval represents. But I'd still quibble with your lengths, being
more likely to use log(p) rather than p for the length of the prime p.

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

/tuning/topicId_21894.html#22059

> Secondly, there is at least one other way I know you know of,
> mapping the primes to a semicircle rather than a circle.

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

/tuning/topicId_21894.html#22060

> Well... there's also the variant of my formula that is based
> on 180 degrees instead of 360. So I have two good formulas.

OK... I see we're online at the same time, and stumbling over
each other's posts.

[Paul again:]

> First of all, that rules out all non-JI systems. Non-JI
> systems (which I believe have been more important than JI
> systems, except for Pythagorean) are not uniquely represented
> by your lattice, because the fundamental theorem of arithmetic
> only applies when the exponents of the primes are integers.

Of course, this is a defect of my formula. But it's only in
the past few years (since being on the Tuning List) that I've
become interested in non-JI tunings. _JustMusic: A New Harmony_
was, as its title indicates, intended to be an exploration of
*rational* tuning systems.

While I do mention non-JI tunings in my book, if I were to try
to include them in a substantial way it would entail a complete
rewriting, which this book certainly doesn't need (again).

My analyses of non-JI systems will simply have to go into
a subsequent book. ;-)

Also, I might note that, while it's true that, as you say,
"non-JI systems ... are not uniquely represented by [my]
lattice", they *can* still be represented on my lattice,
albeit not uniquely, and to me that still portrays useful
information on how they relate to the JI tunings they
approximate.

For example, I've plotted meantones onto my lattices, and one
can see how strings of 5:4's line up in 1/4-comma meantone,
and how strings of 6:5's line up in 1/3-comma meantone, etc.
It's true that there are other ways to plot these tunings,
but the plottings I used *do* relate them visually to their
JI cousins.

> But of course there are an infinite number of ways to do it,
> for example one could use the golden angle times the ordinal
> number of the prime. Any one-to-one mapping of primes to
> angles will do, and any set of lengths will do too (as long
> as no overlaps result).

I understand the second sentence to be true, but am pretty
foggy about what you describe in the first. Can you elaborate?

Thanks for the input.

-monz
http://www.monz.org
"All roads lead to n^0"

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

> > But of course there are an infinite number of ways to do it,
> > for example one could use the golden angle times the ordinal
> > number of the prime. Any one-to-one mapping of primes to
> > angles will do, and any set of lengths will do too (as long
> > as no overlaps result).
>
>
> I understand the second sentence to be true, but am pretty
> foggy about what you describe in the first. Can you elaborate?
>
The idea would be to mirror the way many plants and flowers grow.
First, a single "bud" pops out of the center. After the first bud has
grown a bit, the next "bud" pops out at the golden angle (222.49
degrees). Then the next bud pops out at the golden angle relative to
that. And so on. Each bud would represent a different prime. This
would ensure that, if you used a lot of different primes in your
lattice, their lines would stay as far as possible from one another
and near-overlaps would be avoided.

Yet another attempt to kill off a certain subject heading.

Bob Valentine wrote:

> I've been experimenting with a 'tool' at home, if I can call
> it that. I wrote a program which inputs "diatonic strings"
> (BBaBBBa as an example of a familiar one) and the program
> walks all the possible values of B and a in cents against
> a 'psuedo-harmonic-entropy' graph, looking for local minima,
> where a scale will have 'maximum concordance' (with all the
> caveats, simpler intervals with less accuracy may
> sometimes be preferred to complex ones with higher accuracy
> and visca-versa).

Neato. (I think that's the 1970's Australian version of "Way Cool")

> There will be more on this later. So I came home and plugged
> in the string
>
> BaaBaaBaaBaaBaaBaaBaaBaaBaaBaaa
>
> took the results to excel and the best result according to
> my current criteria are B=49c a=33.8c. The 'complexity'
> minimum is at 49 and the error minimum is at 50 (the '3'
> in 72tet). My overall weighting lands on 49.

So you're saying that the optimum value for the generator B+a+a is
between 116.6 and 116.7c, but favouring the lower value.

It sounds like you're only runing this to the nearest cent on B. Could
you add a decimal place?

-- Dave Keenan

Hey Dave Keenan.

Can you re-run your 11-limit and 7-limit searches for multiple chains
of fifths? Perhaps there's another "MIRACLE SCALE" out there waiting
to be exploited, and perhaps it will turn out to be expressible in 72-
tET (where so many potential generators are going to lead to a cycle
of less than 72 notes) or some other composite ET.

I wrote:

> Can you re-run your 11-limit and 7-limit searches for multiple
chains
> of fifths?

Of course I meant multiple chains of _generators_, not necessarily
fifths, separated by a 1/2 octave, 1/3 octave, etc.

Hey guys, can we please kill off this embarrassing shouting subject
heading. Most of the threads going on under it aren't even about it
anymore.

--- In tuning@y..., paul@s... wrote:
> Hey Dave Keenan.
>
> Can you re-run your 11-limit and 7-limit searches for multiple
chains
> of fifths? Perhaps there's another "MIRACLE SCALE" out there waiting
> to be exploited, and perhaps it will turn out to be expressible in
72-
> tET (where so many potential generators are going to lead to a cycle
> of less than 72 notes) or some other composite ET.

You mean where the chains are 1/n of an octave apart for small n? Why
limit it to chains of fifths. I'd look at chains of anything from 0 to
600c.

It would take me a lot of time to make up spreadsheets to do each set
of chains. I currently have sheets for 1,2 and 3. I haven't figured
out how to make it so I can just type in the number of chains. But I
think I can give a convincing argument that there aren't any.

There's no point in looking for these generators unless we know which
7-limit or 11-limit intervals are within 5.4 cents of which m/n of an
octave, for small n (using meantone error as the standard for
quasi-justness).

e.g. We know that when you're willing to allow erorrs up to 17.5c it
is worth looking at 2, 3 and 4 such chains (a 1/2, 1/3 and 1/4 octave
apart), because 5:7, 4:5, and 5:6 respectively are so approximated.

We're looking for scales with about 31 notes or less, so there's no
point in considering more than 12 chains. We're not going to get a
generator where all 7 (or 11) limit intervals are within 5.4c and
those that are approximated on a single chain don't span more than 2
generators (i.e. 3 notes) (3*12=36). Agreed?

So here are the 11-limit candidates I've found up to n=12 (and some
near misses).

1/6 oct = 8:9 -3.9c
1/7 oct = 10:11 +6.4c
2/7 oct = 9:11 -4.6c
1/8 oct = 11:12 -0.6c
2/9 oct = 6:7 -0.2c
4/11 oct = 7:9 +1.3c
5/11 oct = 8:11 -5.9c
5/12 oct = 3:4 +2.0c

Notice that you have to go to 6 chains before you get anything useful.

At the 7-limit you have to go to 9 chains to get anything useful.

With 9 chains we get 6:7 (and only 6:7) _between_ the chains. We
wouldn't want more than 4 notes (3 generators) in each chain. Can we
get 2:3,4:5,5:6,4:7,5:7 all to be approximated within 5.4c by no more
than a chain of 3 of any generator. Seems pretty unlikely, but if you
really want me to search for that with my single chain search, let me
know.

At the 11 limit, with 6 or more chains we'd need 13 of the fourteen
11-limit intervals to be approximated on a single chain by no more
than 5 generators. Again, this seems incredibly unlikely to me. What
do you think?

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> With 9 chains we get 6:7 (and only 6:7) _between_ the chains. We
> wouldn't want more than 4 notes (3 generators) in each chain. Can we
> get 2:3,4:5,5:6,4:7,5:7 all to be approximated within 5.4c by no
more
> than a chain of 3 of any generator. Seems pretty unlikely, but if
you
> really want me to search for that with my single chain search, let
me
> know.
>
> At the 11 limit, with 6 or more chains we'd need 13 of the fourteen
> 11-limit intervals to be approximated on a single chain by no more
> than 5 generators. Again, this seems incredibly unlikely to me. What
> do you think?

Oops. No. It's not that simple. But can anyone else come up with an
argument so we don't have to do the full search?

-- Dave Keenan

>
> From: paul@stretch-music.com
> Subject: Re: DAVE KEENAN'S MIRACLE SCALE
>
>
> > I hope to write more about this program to this
> > forum after my weekend, but thought I'd at least chime
> > in with this 'me too!'
>
> Thanks. As you know, I feel this approach is only valid if you're
> going to use the scale against a drone, fixed at the pitch of the
> tonic of the mode you've chosen. What interests me more are the
> intervals between pairs of notes in the scale, the tonic not
> necessarily being a member of each pair. That approach would treat
> all modes of the scale as identical (which yours doesn't).
>

Ahah, I forgot to mention, this program calculates a minimum
while considerring all rotations (modes) of the scale. I only
showed the RI interpretation for the one I typed in, but the
palindromatic version was included in the calculation.

I could look and see how it compares as a modal complexity minimum.

Bob

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:
>
> Ahah, I forgot to mention, this program calculates a minimum
> while considerring all rotations (modes) of the scale.

Oh good.

One other point -- I'd be wary of inconsistency -- in 72-tET, this may occur when using ratios of
19 or higher odd numbers. In particular, if one interval falls into the bucket for a:b, and another
interval falls into the bucket for b:c, you should make sure the bucket a:c contains no interval
different from the "sum" of the two first intervals. Otherwise, a scale that seems good for dyads
might fall apart when trying to construct chords.

--- In tuning@yahoogroups.com, paul@s... wrote:

/tuning/topicId_21894.html#21938

> --- In tuning@y..., jpehrson@r... wrote:
> >
> > Well, this is a curiosity... It looks like an hexany "mobius
strip..."
>
> This seems to have much more consonances, at least in the 7-limit,
than any of the 19-
> tone or 22-tone subsets of 72-tET that we've come up with. Dave,
correct me if I'm wrong.
>
> We can call this the "blackjack" scale due to the "surprise" of it
having 21 notes.
>
> I'm going to unsubscribe for a while. You can e-mail me (if you're
reading this on the
> website) by clicking on my truncated e-mail address above this
message.
>
> Oh, Neil H., if you're reading this, thanks for mailing out the 31-
tone guitar (yippee!!!) --
> the check is in the mail.
>
> Cheers!

***Thanks, Paul! So it was *you* after all. Actually, I had read
this post and missed that (it was late...)

Joseph

hi Joe,

> From: "Joseph Pehrson" <jpehrson@rcn.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, July 09, 2003 10:10 AM
> Subject: [tuning] Re: A new CS, DE 21-out-of-72 proposal for Joseph
Pehrson
>
>
> --- In tuning@yahoogroups.com, paul@s... wrote:
>
> /tuning/topicId_21894.html#21938
>
> > --- In tuning@y..., jpehrson@r... wrote:
> >
>
> <snip>
>
> >
> > We can call this the "blackjack" scale due to the
> > "surprise" of it having 21 notes.
>
> <snip>
>
> ***Thanks, Paul! So it was *you* after all.
> Actually, I had read this post and missed that
> (it was late...)

so, then the information in my webpages *is* correct.
(i'm usually pretty good about documenting my research...)

-monz