For all strict-JI fans . . .

Hello to all and this post is worth a month of not posting, so I hope
you will look at it:
...........................................................

I've been working hard on a new paper, and I've come up with some JI
intepretations of my two decatonic scales. For those unfamiliar with
them, they are meant to be the best 7-limit analogues of what the
diatonic scale is in the 5-limit. I'm only going to post 1 mode each
of three versions of each of the two scales -- you can work out the
other 9 rotations of each on your own, if you wish.

The canonical JI interpretation of the pentachordal decatonic scale
is:

Pitches Steps
1/1 21:20
21/20 160:147
8/7 21:20
6/5 10:9
4/3 21:20
7/5 15:14
3/2 16:15
8/5 15:14
12/7 21:20
9/5 10:9
2/1

Lattice:
4/3-------1/1-------3/2
\`. ,'/|\`. ,'/|\
\ 8/7-/-|-\12/7 / | \
\ | / 7/5------21/20\
\|/,' `.\|/,' `.\
8/5-------6/5-------9/5

Making use of the 50:49 comma, we can get these superparticular-step
versions . . .

Pitches Steps
1/1 15:14
15/14 16:15
8/7 21:20
6/5 10:9
4/3 21:20
7/5 15:14
3/2 16:15
8/5 15:14
12/7 21:20
9/5 10:9
2/1

Pitches Steps
1/1 21:20
21/20 16:15
28/25 15:14
6/5 10:9
4/3 21:20
7/5 15:14
3/2 16:15
8/5 15:14
12/7 21:20
9/5 10:9
2/1

The canonical JI interpretation of the symmetrical decatonic scale is:

Pitches Steps
1/1 15:14
15/14 49:45
7/6 15:14
5/4 16:15
4/3 15:14
10/7 21:20
3/2 10:9
5/3 21:20
7/4 15:14
15/8 16:15
2/1

Lattice:
5/3-------5/4------15/8
/|\`. ,'/|\`. ,'/
/ | \10/7-/-|-\15/14/
/ 7/6-------7/4 \ | /
/,' `.\|/,' `.\|/
4/3-------1/1-------3/2

Making use of the 50:49 comma, we can get these superparticular-step
versions, for those who care about such things (I'm thinking Justin
White, if you're still around):

Pitches Steps
1/1 21:20
21/20 10:9
7/6 15:14
5/4 16:15
4/3 15:14
10/7 21:20
3/2 10:9
5/3 21:20
7/4 15:14
15/8 16:15
2/1

and

Pitches Steps

1/1 15:14
15/14 10:9
25/21 21:20
5/4 16:15
4/3 15:14
10/7 21:20
3/2 10:9
5/3 21:20
7/4 15:14
15/8 16:15
2/1

There it is . . . these are periodicity blocks . . . Kraig, they're
CS, of course . . . Dan, are the superparticular-step ones
quadrivalent? Pierre, you must have special insight into this . . .
comments? Any other JI people out there . . . I want your points of
view . . . free to post your analyses. And I hope you strict-JI
people will try these . . . above all, I want to hear them used for
music . . .

One more thing . . . As most of you know, in the diatonic scale, you
need to use the 81:80 unison vector (aka syntonic comma) to avail
yourself of the scale's full triadic potential (six consonant 5-limit
triads). Well, for the pentachordal decatonic scale the 50:49 unison
vector is what is needed to avail yourself of the full tetradic
potential (six 'consonant' 7-limit tetrads) . . You need to use both
the 50:49 and 64:63 unison vectors (or one of these and 225:224) to
avail yourself of the full tetradic potential of the symmetrical
decatonic scale (eight 'consonant' 7-limit tetrads). . . Here are
fully "supplemented" versions (with enough comma variants to get all
the 'consonant' tetrads):

Pentachordal Decatonic:

1/1
21/20
8/7
6/5
4/3
7/5, 10/7
3/2
8/5
12/7, 42/25
9/5
2/1

Lattice:

7/5
/|\
/ | \
4/3-----/-1/1-\-----3/2
\`. /,'/|\`.\ ,'/|\
\ 8/7-/-|-\12/7 / | \
\ | / 7/5------21/20\
\|/,' \`.\|/,'/ `.\
8/5-----\-6/5-/-----9/5
\ | /
\|/
42/25

Symmetrical Decatonic:

1/1
21/20, 15/14
7/6, 25/21
5/4
21/16, 4/3
7/5, 10/7
3/2
5/3
7/4, 25/14
15/8, 40/21
2/1

25/21-----25/14
/|\ /|\
/ | \ / | \
/ 5/3-\---/-5/4-\----15/8
/,'/|\`.\ /,'/|\`.\ ,'/|
40/21/-|-\10/7-/-|-\15/14/ |
| / 7/6-------7/4------21/16
|/,' \`.\|/,'/ \`.\|/,'/
4/3-----\-1/1-/---\-3/2 /
\ | / \ | /
\|/ \|/
7/5------21/20

Music, music, music!!!

Welcome back, Paul E, for however long.

Regarding your decatonic scales: I am curious to know how you see them
being used, compared to more adaptive methods:

. Do you favor fixed scales because they're more applicable to (many)
acoustic instruments?

. Do you see them being used with slight adaptive techniques overtop?

. Do you favor fixed scales, with no added adaptive tuning, for
strictly musical reasons in some works?

Obviously, these questions reflect my own perspective/bias, from which
lack of adaptive tuning seems like a terrible handicap.

JdL

Hi John . . . personally, I see _major_ potential, if not outright
necessity, for adaptive tuning (in fact I'm citing you in this paper that
I'm hoping to complete today) with decatonic music . . . the post was mainly
aimed at those who will not consider anything but strict JI . . . for a
fixed-pitch instrument, the analogue to optimal meantone in the diatonic
case is remarkably close to 22-tET in the decatonic case . .

-----Original Message-----
From: John A. deLaubenfels [mailto:jdl@adaptune.com]
Sent: Monday, April 02, 2001 1:18 PM
To: tuning@yahoogroups.com
Subject: [tuning] Re: For all strict-JI fans . . .

Welcome back, Paul E, for however long.

Regarding your decatonic scales: I am curious to know how you see them
being used, compared to more adaptive methods:

. Do you favor fixed scales because they're more applicable to (many)
acoustic instruments?

. Do you see them being used with slight adaptive techniques overtop?

. Do you favor fixed scales, with no added adaptive tuning, for
strictly musical reasons in some works?

Obviously, these questions reflect my own perspective/bias, from which
lack of adaptive tuning seems like a terrible handicap.

JdL

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Paul Erlich wrote,

<<There it is . . . these are periodicity blocks . . . Kraig, they're
CS, of course . . . Dan, are the superparticular-step ones
quadrivalent?>>

Hi Paul, nice to see you posting again.

None of these are quadrivalent in the sense that the syntonic diatonic
is trivalent or the Pythagorean diatonic is Myhill. Though if you were
to ignore the 225/224 you'd also dissolve the stray 2401/2400 and
would end up with some three-stepsize, trivalent possibilities.

Question...

Are there strict JI decatonics that are Myhill, trivalent, and
quadrivalent?

--Dan Stearns

Hello Paul...

--- In tuning@y..., PERLICH@A... wrote:
> Hello to all and this post is worth a month of not posting, so I
> hope you will look at it:

What, time makes the brain grow fonder? :)

And, after all that time, the first post you make is a refutation of
Kraig Grady? (I know, he must have needed it, right?) <grin>

You'll never change, Paul.

> I've been working hard on a new paper, and I've come up with some
> JI intepretations of my two decatonic scales. For those unfamiliar
> with them

[snip]

> ... I hope you strict-JI people will try these . . . above all,
> I want to hear them used for music . . .

Paul, have *you* used them for music?

Cheers,
Jon

> Music, music, music!!!

My god, he's singing Leslie Gore!....

Dan wrote,

>None of these are quadrivalent in the sense that the syntonic diatonic
>is trivalent or the Pythagorean diatonic is Myhill. Though if you were
>to ignore the 225/224 you'd also dissolve the stray 2401/2400 and
>would end up with some three-stepsize, trivalent possibilities.

Thanks, Dan. So if you quantized them to, say, 72-tET, they'd be trivalent??

I wrote,

>> Hello to all and this post is worth a month of not posting, so I
>> hope you will look at it:

Jon Szanto wrote,

> What, time makes the brain grow fonder? :)

> And, after all that time, the first post you make is a refutation of
> Kraig Grady? (I know, he must have needed it, right?) <grin>

Excuse me???

Jon, you are hallucinating again. This is not the first time this has
happened, and with the same characters. A refutation of Kraig Grady, huh?
Who dropped the acid into your coffee this morning?

>> ... I hope you strict-JI people will try these . . . above all,
>> I want to hear them used for music . . .

>Paul, have *you* used them for music?

I've used the 22-tET versions for music, such as the my last piece at the
first Microthon and my first piece at the last Microthon. I thought, as a
way of creating a bridge to some of the strict-JI folks, I'd throw this out
there. Sorry if it bothers you that I'd do so.

PERLICH@ACADIAN-ASSET.COM wrote:

>
> Lattice:
> 4/3-------1/1-------3/2
> \`. ,'/|\`. ,'/|\
> \ 8/7-/-|-\12/7 / | \
> \ | / 7/5------21/20\
> \|/,' `.\|/,' `.\
> 8/5-------6/5-------9/5
>
> Making use of the 50:49 comma, we can get these superparticular-step
> versions . . .
>
> Pitches Steps
> 1/1 15:14
> 15/14 16:15
> 8/7 21:20
> 6/5 10:9
> 4/3 21:20
> 7/5 15:14
> 3/2 16:15
> 8/5 15:14
> 12/7 21:20
> 9/5 10:9
> 2/1

Can anyone please explain how the comma is being used and what kind of comma it is? Thanks.

Hi Alison.

In this context, what I meant specifically that I was shifting one note from
the previous scale (21:20) by the 50:49 comma (hence moving it to 15:14).

However, the 50:49 comma plays the same role in this scale as the 81:80
comma plays in the diatonic scale.

To see this, go to
/tuning/files/perlich/scales/
</tuning/files/perlich/scales/> and download
</tuning/files/perlich/scales/scalattices.ZIP>
scalattices.ZIP. You'll see the familiar diatonic scale, with the 81:80
making possible six consonant triads. You'll also see the pentachordal
decatonic scale, with the 50:49 comma making possible six consonant tetrads.
I use a slightly different lattice orientation than below, but hopefully
it'll be clear. Let me know . . .

-Paul

-----Original Message-----
From: Alison Monteith [mailto:alison.monteith3@which.net]
Sent: Tuesday, April 03, 2001 4:19 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] For all strict-JI fans . . .

PERLICH@ACADIAN-ASSET.COM wrote:

>
> Lattice:
> 4/3-------1/1-------3/2
> \`. ,'/|\`. ,'/|\
> \ 8/7-/-|-\12/7 / | \
> \ | / 7/5------21/20\
> \|/,' `.\|/,' `.\
> 8/5-------6/5-------9/5
>
> Making use of the 50:49 comma, we can get these superparticular-step
> versions . . .
>
> Pitches Steps
> 1/1 15:14
> 15/14 16:15
> 8/7 21:20
> 6/5 10:9
> 4/3 21:20
> 7/5 15:14
> 3/2 16:15
> 8/5 15:14
> 12/7 21:20
> 9/5 10:9
> 2/1

Can anyone please explain how the comma is being used and what kind of comma
it is? Thanks.

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Hi Jon -- hope you saw my apology. I meant it.

>Hey, didn't bother me a bit! I was just curious if this was a new
>construct, different from what you've used to do/make/improvise music
>in the past -- that's all!

Yes, it's different, 'cause it's strict JI.

>Are you working too hard or stressed out?

Both. Big gig Sunday night, then stayed up all night trying to finish a
paper to send in to the MicroFest; accidentally deleted a lot of work . . .
(still finishing . . .)

Paul,

Hate this synchronous dialogue! Here's why...

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
> Hi Jon -- hope you saw my apology. I meant it.

Hah! I did after I posted the msg which you reply to below which I
deleted from the list after I read your apology, so now the thread
has a vaporized middle!

We're OK, we're OK...

> Yes, it's different, 'cause it's strict JI.

Ahhh...

> >Are you working too hard or stressed out?
>
> Both.

Well, the upside is that it will all be over soon...

Stay as well as possible,
Jon

Paul H. Erlich wrote,

<<So if you quantized them to, say, 72-tET, they'd be trivalent??>>

No, I don't think so. Just tempering out the 225/224 isn't enough. In
fact I think you'd have to tweak this scale to make it trivalent.

3224232422 in 26-tET would do the job; I mean it's trivalent, but is
it still a decatonic as you see it?

Here's that scale where the product of the small step and the
tribonacci constant equals the size of the large step.

6,2,2,10,14,26,...

0 145 239 333 506 600 745 839 1012 1106 1200
0 94 188 361 455 600 694 867 961 1055 1200
0 94 267 361 506 600 773 867 961 1106 1200
0 173 267 412 506 679 773 867 1012 1106 1200
0 94 239 333 506 600 694 839 933 1027 1200
0 145 239 412 506 600 745 839 933 1106 1200
0 94 267 361 455 600 694 788 961 1055 1200
0 173 267 361 506 600 694 867 961 1106 1200
0 94 188 333 427 521 694 788 933 1027 1200
0 94 239 333 427 600 694 839 933 1106 1200

Here's the same in 26-tET.

0 138 231 323 508 600 738 831 1015 1108 1200
0 92 185 369 462 600 692 877 969 1062 1200
0 92 277 369 508 600 785 877 969 1108 1200
0 185 277 415 508 692 785 877 1015 1108 1200
0 92 231 323 508 600 692 831 923 1015 1200
0 138 231 415 508 600 738 831 923 1108 1200
0 92 277 369 462 600 692 785 969 1062 1200
0 185 277 369 508 600 692 877 969 1108 1200
0 92 185 323 415 508 692 785 923 1015 1200
0 92 231 323 415 600 692 831 923 1108 1200

--Dan Stearns

I wrote,

<<So if you quantized them to, say, 72-tET, they'd be trivalent??>>

Dan wrote,

> No, I don't think so. Just tempering out the 225/224 isn't enough. In
> fact I think you'd have to tweak this scale to make it trivalent.

>3224232422 in 26-tET would do the job; I mean it's trivalent,

Interesting!

>but is
>it still a decatonic as you see it?

Well, if you got that from the strict-JI scales I posted, then I guess it
kinda is, but only in the same (limited) sense that diatonic scales are
expressible in 15-, 22-, 27-, and 34-tET, for example.

>Here's that scale where the product of the small step and the
>tribonacci constant equals the size of the large step.

Interesting . . . have you figured out the "tribonacci" analogue to Wilson's
Golden Horagrams, then? (Sorry, I haven't followed all your work.)

Paul H. Erlich wrote,

<<Well, if you got that from the strict-JI scales I posted,>>

No, I just tried some random three-term indexes that would still agree
with the basic 8s 2L cardinality of the decatonic scale and could
possibly be trivalent as well. [3,5,2], which is the step structure of
most of the JI scales that you gave if you were to ditch the 225/224,
wouldn't work (where "work" = trivalent) because it gives an odd
number temperament for an even numbered scale. (A while back I said
that I thought you an even numbered scale would have to have the
1:2^(1/2) in at least one of its rotations to be trivalent.)

<<have you figured out the "tribonacci" analogue to Wilson's Golden
Horagrams, then?>>

Not exactly, no.

If you take an M-out-of-N set it's always Myhill, correct? Well what
I've been doing is just carrying out this idea to three terms, or an
L-out-of-M-out-of-N set.

The problem (if you want to call it that) is that the two term
M-out-of-N sets are always Myhill, or better yet "bivalent". But this
isn't the case with the three term L-out-of-M-out-of-N sets. Here
trivalence is a special condition and not just a happy byproduct of
Myhill's property.

There's still a few kinks in what I've done here for sure. But it
works pretty much as I'd expect it to and it's inclusive
(generalized). This simply hasn't been the case with any of the other
things that I've looked at like multiple generators or three fraction
adjacent fractions, etc.

I've been hoping that someone others would jump in on this thread and
help push it out of the mud here and there, because it's apparently a
pretty hard and slippery nut to crack... I'm also hoping that some
others will jump in on Pierre's related posts as well, as I think this
would help me better bridge the gap between what we're doing.

--Dan Stearns

Dan Stearns wrote,

> If you take an M-out-of-N set it's always Myhill, correct?

Nope. Even my 10-out-of-22 sets are not Myhill.

>I'm also hoping that some
>others will jump in on Pierre's related posts as well, as I think this
>would help me better bridge the gap between what we're doing.

Yes -- it seemed Dave Keenan grasped some of what Pierre was talking about??

Paul H. Erlich wrote,

<<Nope. Even my 10-out-of-22 sets are not Myhill.>>

Well if you remember I *do* consider those scales to be Myhill and
amenable to a single generator interpretation. Now it's true that
that's my take on Myhill, so I probably shouldn't call it that, but
"Stearns' property" or something like that sounds ridiculous (well, to
me anyway) so I use Myhill, or relaxed Myhill.

Any other examples of M-out-of-N sets that you'd say aren't Myhill
(and that don't fall into the "relaxed Myhill" category that I'm
talking about above)?

<<Yes -- it seemed Dave Keenan grasped some of what Pierre was talking
about??>>

Yes, and I'm hoping that he'll post something more along those lines
sometime soon... Dave?

--Dan Stearns

> Well if you remember I *do* consider those scales to be Myhill and
> amenable to a single generator interpretation.

Not the L s s s L s s s s s scales!

>Any other examples of M-out-of-N sets that you'd say aren't Myhill

The harmonic minor scale in 12-tET?

Paul H. Erlich wrote,

<<Not the L s s s L s s s s s scales!>>

Maybe I wasn't clear enough. By an M-out-of-N set I specifically mean
the M-equal in N-equal. So by a 10-out-of-22 I mean an ssLss ssLss
scale.

The beauty of this is, and this is obviously analogous to the single
generator MOS scales, that you have a built-in ordering rule for the
step structure. (See how the Harmonic minor wouldn't apply to what I'm
trying to get at here?)

--Dan Stearns

> Maybe I wasn't clear enough. By an M-out-of-N set I specifically mean
> the M-equal in N-equal.

You mean the best approximation to M-equal in N-equal?

Well, the best approximation to 7-equal in 31-equal isn't Myhill.

>The beauty of this is, and this is obviously analogous to the single
>generator MOS scales

Myhill = MOS.

Paul H. Erlich wrote,

<<You mean the best approximation to M-equal in N-equal?>>

Yes, rounded to the nearest integer.

<<Well, the best approximation to 7-equal in 31-equal isn't Myhill.>>

Hmm, I'm not sure what you mean.

0 4 9 13 18 22 27 31
0 5 9 14 18 23 27 31
0 4 9 13 18 22 26 31
0 5 9 14 18 22 27 31
0 4 9 13 17 22 26 31
0 5 9 13 18 22 27 31
0 4 8 13 17 22 26 31
0 4 9 13 18 22 27 31

Looks Myhill enough to me... ?

<<Myhill = MOS.>>

Yes, I realize this. But you see what I'm interested in here is the
expansion of this to more than two terms. Because this is a way to
make an end run around having to conquer multidimensional algebraic
labyrinths!

--Dan Stearns

Whoops, I meant the best approximation of 7-equal in 22-equal isn't Myhill.

Of course that's wrong too . . . OK, I guess you were right . . . by the
way, the term for what you're talking about is "maximally even".

-----Original Message-----
From: Paul H. Erlich
Sent: Tuesday, April 03, 2001 9:27 PM
To: 'tuning@yahoogroups.com'
Subject: RE: [tuning] For all strict-JI fans . . .

Whoops, I meant the best approximation of 7-equal in 22-equal isn't Myhill.

Paul H. Erlich wrote,

<<Whoops, I meant the best approximation of 7-equal in 22-equal isn't
Myhill.>>

Again, I'm not quite sure what it is that your thinking of here, but
here's what I get...

0 3 6 9 13 16 19 22
0 3 6 10 13 16 19 22
0 3 7 10 13 16 19 22
0 4 7 10 13 16 19 22
0 3 6 9 12 15 18 22
0 3 6 9 12 15 19 22
0 3 6 9 12 16 19 22
0 3 6 9 13 16 19 22

--Dan Stearns

Paul H. Erlich wrote,

<<by the way, the term for what you're talking about is "maximally
even".>>

Right, I brought this up with Jacky Ligon a while back in the context
of inversional symmetry.

Jacky had asked, "why do my ears tell me the inversional symmetry is
melodically useful?" And I wrote that I had noticed that maximally
even M-out-of-N sets tend towards inversional symmetry and repeating
tetrachord like structures. And perhaps these types of repeating
tetrachord like structures had had something to do with the reaction
that he was having.

I seem to remember you writing somewhat disparagingly about maximal
evenness in your 22-tET paper, is that right? It's been a while since
I've read it though and I might have that wrong.

Anyway, I think maximal evenness is a very useful concept. Especially
when it comes to organizing step structure and linking step structure
to series (as in the Fibonacci and the tribonacci etc.).

--Dan Stearns

Dan wrote,

>I wrote that I had noticed that maximally
>even M-out-of-N sets tend towards inversional symmetry and repeating
>tetrachord like structures. And perhaps these types of repeating
>tetrachord like structures had had something to do with the reaction
>that he was having.

> I seem to remember you writing somewhat disparagingly about maximal
> evenness in your 22-tET paper, is that right? It's been a while since
> I've read it though and I might have that wrong.

Right . . . though maximal evenness has been used to "derive" the diatonic
scale, clearly you don't get the diatonic scale from a maximally even
7-out-of-31 -- and yet for many purposes, the diatonic scale in 31 is close
to optimal.

And as for "tetrachord-like structures", I like them when they actually span
a 4:3. Thus, melodically, I find the modes of

L s s s L s s s s s

superior to

L s s s s L s s s s,

even though the latter can be maximally even while the former cannot.

Hi Dan,

> Are there strict JI decatonics that are Myhill, trivalent, and
> quadrivalent?

May be worth recalling that we think that there are no trivalent ones,
at least probably not, by your conjecture!

Since you made the conjecture first, before i collected the experimental evidence,
I find it pretty convincing, though it isn't conclusive of course.

This file is the one I made that lists the SCALA et modes:
/tuning/files/Robert%20Walker/n-valent_et_modes.txt

Lines beginning v___3 are trivalent, and all the ones that begin
v___3m include the midpoint 2^(1/2).

The n gives the number of notes, so for v___3m_n__6...
the number of notes in the scale is 6.

A quick inspection shows that all the trivalent ones with an even number of notes (6, or 8)
include their midpoints.

In the Scala scales list:
/tuning/files/Robert%20Walker/n-valent_scales.txt
there's only one scale that is trivalent and has an even number of notes:
v___3md_n_12_class_5_aaaaaaaaaba_class_7_aaaaaaaaaba_ arabic1.scl | From Fortuna. Try C or G major

It also includes its midpoint (and in fact is another et mode).

One can make a trivalent scale with an even number of notes which is rational
for all its notes except for one, and I did that in one of the posts, but so far,
we don't know of any with all the notes rational.

Robert

I wrote,

<<Question... Are there strict JI decatonics that are Myhill,
trivalent, and quadrivalent?>>

Robert Walker wrote,

<<May be worth recalling that we think that there are no trivalent
ones, at least probably not, by your conjecture!Since you made the
conjecture first, before i collected the experimental evidence, I find
it pretty convincing, though it isn't conclusive of course.>>

Hi Robert,

I still think this is the correct. However, if you allow for an
alternate non-octave periodicity whose midpoint is a rational then the
problem is easily solved.

Here's a strict JI interpretation of the [6,2,2] trivalent decatonic
example I gave yesterday where P = 4:9

1/1 16/15 15/13 16/13 45/32 3/2 8/5 45/26 24/13 128/65 9/4

Now if you wanted to pull this back so that the periodicity was still
for all intents and purposes an octave (albeit a stretched one) and
the trivalent scale a JI scale, you could let P = 144:289 and have
something on the order of

1/1 304/289 19/17 19/16 4/3 17/12 76/51 19/12 16/9 17/9 289/144

--Dan Stearns

I wrote,

<<Now if you wanted to pull this back so that the periodicity was
still for all intents and purposes an octave (albeit a stretched one)
and the trivalent scale a JI scale, you could let P = 144:289 and have
something on the order of>>

Here's another, and more decatonic like, [6,2,2] trivalent JI example

1/1 128/119 8/7 17/14 4/3 17/12 32/21 34/21 16/9 17/9 289/144

--Dan Stearns

"Paul H. Erlich" wrote:

> Hi Alison.
>
> In this context, what I meant specifically that I was shifting one note from
> the previous scale (21:20) by the 50:49 comma (hence moving it to 15:14).
>
> However, the 50:49 comma plays the same role in this scale as the 81:80
> comma plays in the diatonic scale.
>
> To see this, go to
> /tuning/files/perlich/scales/
> </tuning/files/perlich/scales/> and download
> </tuning/files/perlich/scales/scalattices.ZIP>
> scalattices.ZIP. You'll see the familiar diatonic scale, with the 81:80
> making possible six consonant triads. You'll also see the pentachordal
> decatonic scale, with the 50:49 comma making possible six consonant tetrads.
> I use a slightly different lattice orientation than below, but hopefully
> it'll be clear. Let me know . . .
>
> -Paul
>
> Thanks for the reply. I'll get on to the tuning files straight away.

Best Wishes.

Hi Paul and JI fans!

Being very busy I will comment at moment only on the two following Paul's
decatonic scales, reserving to complete later :

<<

Pitches Steps
1/1 15:14
15/14 16:15
8/7 21:20
6/5 10:9
4/3 21:20
7/5 15:14
3/2 16:15
8/5 15:14
12/7 21:20
9/5 10:9
2/1

Pitches Steps
1/1 21:20
21/20 10:9
7/6 15:14
5/4 16:15
4/3 15:14
10/7 21:20
3/2 10:9
5/3 21:20
7/4 15:14
15/8 16:15
2/1

>>

Why these two?

In the Paul's examples, almost all tones are within the odd 21-limit with
very few exceptions containing odd 25, while almost all steps, with only
two exceptions, are in the ordered set

<21/20 16/15 15/14 10/9>

Since I want to contribute with an algebraic approach I would like emphasis
generic aspects and then avoid at begining the few exceptions that would
imply a higher complexity. So what is both common and simple will not be
masked. Afterwards, exceptions analysis could be made showing clearly the
complexity rising implied.

It is why I retained these two scales having not the noted exceptions.

----------

I can confirm these two Paul's scales are modes in sense of the gammier
theory, meaning (approximately) there exist a simple math structure in
which these modes are strictly appropriate. Since the term "proper mode"
has already another microtonal sense, I could perhaps use the expression
"sui generis mode".

Besides, rather than to explain abstractly what are gammiers I will use in
large sense the term "system" implying only the necessity to strictly
integrate in a generic way all these well-defined objects : primes, steps,
unison vectors, srutis, basis, coordinates, odd generators, diamonds,
modes, degree functions, sruti functions, lattices, modal representations,
etc.

----------

Q. How much systems are thus possible within these following limits?

a) prime 7-limit -- using primal basis <2 3 5 7>
b) non-degenerate -- discarding <2 3 5>, <2 3 7>, <2 3>, etc.
c) odd 21-limit -- eliminating odd > 21

A. There exist exactly 16 such systems.

That may seem arbitrary so if someone wanted to verify, I list here the
minimal generator of these systems with the amount of tones generated and
the number of degrees in octave. I recall that the tones are obtained
filing matrices similar to Partch's diamonds but using the following
generators rather than complete N-limit standard sequences.

Generator Tones Degrees

1 3 5 7 15 17 6
1 3 7 9 15 19 7
1 3 5 7 9 15 23 7
1 3 5 9 21 19 7
1 5 7 9 21 21 7
3 5 7 9 21 17 6
1 3 5 7 9 21 23 7
1 3 7 15 21 17 6
1 5 7 15 21 17 6
3 5 7 15 21 17 6
3 5 9 15 21 17 6
1 3 5 9 15 21 23 7
1 7 9 15 21 21 7
1 3 7 9 15 21 23 7
5 7 9 15 21 17 6
1 3 5 7 9 15 21 27 10

As we can see, there exist (within the given limits) 7 hexatonic systems, 8
heptatonic systems and only one decatonic system, the last one. Besides, if
we extend to odd 25-limit, we can find many other decatonic systems but
none having the basis <2 3 5 7>.

I could show now that the two Paul's decatonic scales are well sui generis
mode in the last decatonic 27-tone system. (It would be better to do that
later when I will find time to produce graphical elements). The goal here
is only to show the inverse algebraic way : how to describe the algebraic
system corresponding to these Paul's scales? That will exhibit thus their
deep consistency.

----------

Let us forget for the moment we knowed a decatonic system and let us derive
it from the decatonic modes listed. Using one or another mode will result
to the same system so we will use arbitrary the first :

<1 15/14 8/7 6/5 4/3 7/5 3/2 8/5 12/7 9/5 2>

Let m be a valid mode. How, operatively, the steps can be obtained? First
we have to order the tones and then differentiate the successive tones. Let
us write that

tG(m) = d'(m)

It means the tonal generator tG of the mode m (meaning the steps of m) is
obtained applying successively

the ordering operator ' and then
the differentiate operator d (e.g. d <7 9 15> = <9/7 5/3>)

[ I want to post eventually precise definitions
and examples about the 4 operators ' d e p
hoping that context here and intuition are
sufficient for understanding. ]

Since our decatonic mode m is already ordered we may apply d directly to m.

tG(m) = d'(m)
= d <1 15/14 8/7 6/5 4/3 7/5 3/2 8/5 12/7 9/5 2>

= <15/14 16/15 21/20 15/14 16/15 15/14 21/20 10/9>

Now, applying yet ' which order and keep only one copy of each interval we
obtain the ordered tonal generator 'tG

'tG(m) = 'd'(m) = <21/20 16/15 15/14 10/9>

It is so easy that most do that without thinking : order, calculate the
intervals, reorder (keeping one copy of each interval). Since I will use
such operators, we may thus advance in the algebra with the same easyness.

How can we obtained the unison vectors U of any system where m is a sui
generis mode? The unison vectors are simply increments of steps. So we may
write

U(m) = d'(tG) = d'd'(m)

Here we have

U(m) = d <21/20 16/15 15/14 10/9>

= <64/63 225/224 28/27>

Two distinct systems may have the same unison vectors. To distinguish
between systems, we have to add at unison vectors (or "modal srutis") the
minimal step (or "tonal sruti") obtaining so the srutal basis sB which
characterize specifically the system used.

(e.g. With the same unison vectors 25/24 and 81/80 we have
the Zarlino system having the minimal step 16/15 and
the Hindu system having the minimal step 256/243.)

Rather than using d'd' to obtain U we use e'd' to obtain the srutal basis
sB. The differentiate operator e is a variant of d keeping the first
element. So

sB(m) = e <21/20 16/15 15/14 10/9>

= <21/20 64/63 225/224 28/27>

We hold there the essential system tool.

----------

Any system has two types of interval classes, the degrees and the srutis
which determine two types of periodicity associated with octave circularity.

(e.g. the Hindu system has a 7 periodicity for steps amount
SA RI GA ... and a 22 periodicity for srutis amount.)

So we can associate to any system two functions, the D(X) and S(X)
functions (including variants modulo 2), which permit to calculate for any
interval X its value in degree (or total amount of steps) and its value in
total amount of srutis (or sruti degree).

These functions are easily derived from the srutal basis sB. Indeed, we can
read these functions directly in a matrix linking the basis sB and the
basis <2 3 5 7>.

Before to calculate these functions it would be probably useful to say few
words about representing coordinates and matrices in basis.

----------

I give first an idea (for those who are not familiar with matrices and
vectors) of what are a basis and what are the coordinate vectors of an
interval in a basis.

The set of primes <2 3 5 7> is a basis. It means that there exist a unique
combination of each prime amount producing a given interval.

21/20 = <2 3 5 7> (-2,1,-1,1) = (3)(7) / (2)(2)(5)
16/15 = <2 3 5 7> (4,-1,-1,0) = (2)(2)(2)(2) / (3)(5)

These amount of primes (-2,1,-1,1) and (4,-1,-1,0) are the coordinate
vectors of 21/20 and 16/15 in the basis <2 3 5 7>. In a lattice
representation, we use such vectors.

----------

Let us call primal basis pB the ordered basis <2 3 5 7> and let X be any
interval in the group G where the generic element is

2^x 3^y 5^z 7^t

and where the elements (x,y,z,t) are relative integers. G may be noted

pB ZxZxZxZ or pB Z^4

for any interval may be expressed as

<2 3 5 7> (x,y,z,t)

-- These equivalent notations may also be used

(x)
<2 3 5 7> (y) == <2 3 5 7> *(x y z t)
(z)
(t)
e.g.

2/1 = <2 3 5 7> (1,0,0,0)

(0)
5/1 = <2 3 5 7> (0)
(1)
(0)

28/15 = <2 3 5 7> *(2 -1 -1 1)

So any interval of G corresponds to a unique coordinate vector Xp =
(x,y,z,t) and we say that the vector Xp represents the interval X in the
basis pB.

Since we will use here three distinct basis we have to distinguish between
their distinct coordinate vectors (for a same given interval). We will use

Xp = (xp1, xp2, xp3, xp4)

rather than

Xp = (x,y,z,t)

when there is ambiguity with other basis.

The tonal generator tG is a basis in the Paul's scale underlying system. It
is why we will note it now tB. The tonal generator is not always a basis :
its dimension has to be equal to the primal basis dimension to be a basis.

Let us use this coherent notation

X (any interval)

pB (primal Basis) -- the primes
tB (tonal Basis) -- the steps
sB (srutal Basis) -- the srutis

Xp (coordinates of X in primal Basis) -- amounts of each prime
Xt (coordinates of X in tonal Basis) -- amounts of each step
Xs (coordinates of X in srutal Basis) -- amounts of each sruti

Xp = (xp1, xp2, xp3, xp4)
Xt = (xt1, xt2, xt3, xt4)
Xs = (xs1, xs2, xs3, xs4)

tMp (matrix representing the tonal Basis in primal Basis)
pMt (matrix representing the primal Basis in tonal Basis)
sMp (matrix representing the srutal Basis in primal Basis)
pMs (matrix representing the primal Basis in srutal Basis)

So we may write clearly

X = pB Xp = tB Xt = sB Xs

tB = pB tMp pB = tB pMt tMp pMt = I
sB = pB sMp pB = sB pMs sMp pMs = I

X = pB sMp Xs = pB tMp Xt

It is short and probably sour but I have not time now for details.

----------

Now let us calculate the D(X) and S(X) functions starting with sB. We have

sB = pB sMp
(-2 6 -5 2)
<21/20 64/63 225/224 28/27> = <2 3 5 7> ( 1 -2 2 -3)
(-1 0 2 0)
( 1 -1 -1 1)

And since pMs is the inverse of sMp we have by inversion

(10 16 23 28)
pMs = ( 7 11 16 19)
( 5 8 12 14)
( 2 3 5 6)

Assuming that the interval X is represented by (x,y,z,t) in the primal
basis <2 3 5 7>, we can read the D function in the first line

D(X) = 10x + 16y + 23z + 28t

exhibiting the 10 periodicity of steps amount in octave. And we can obtain
almost so easily the sruti function summing each column

S(X) = 24x + 38y + 56z + 67t

exhibiting the 24 periodicity of srutis amount in octave.

We can then apply the octave circularity obtaining

D(X mod 2) = (6y + 3z + 8z) mod 10
X(X mod 2) = (14y + 8z + 19t) mod 24

Since all that seem very abstract let us see on a schema their
corresponding concrete coincidences :

(z) (y) (t) (x)

0 1 2 (3) 4 5 (6) 7 (8) 9 (10) (Degrees)

0 5 (8) 10 (14) 16 (19) (24) (Srutis)

(H5) (H3) (H7) (H2) (Harmonics)
S7 S3 S5 (Subs)

A last remark about the tonal basis tB. Its representation pMt in <2 3 5 7>
and its inverse tMp are

(-2 4 -1 1)
pMt = ( 1 -1 1 -2)
(-1 -1 1 1)
( 1 0 -1 0)

( 3 5 7 9)
pMs = ( 2 3 4 5)
( 3 5 7 8)
( 2 3 5 6)

We can obtain the basis [10 16 23 28] giving the D(X) function summing each
column of pMs but this method is not universal since the tonal generator is
not always a basis : in that case pMs do not exist.

----------

I apologize for a too quick and incomplete comment but at the moment it is
even impossible for me to predict when I will have time to complete.

For me, the musical approach and the algebraic approach are antithetic. The
algebra tend to englobe in the genericity while the musician tend to
exploit the particularities. Happily, for without creative counterpart laws
and systems are dead things. Chaos is the reciprocal dead pole. I think
that Art and all what is living use continuously what is resisting and what
is malleable to maintain a dynamic equilibrium.

Hoping generators, matrices, functions, . . . be useful to tame the
resistant part in the decatonic scales proposed by Paul.

Pierre Lamothe

Hi Pierre and thanks for your reply!

>How can we obtained the unison vectors U of any system where m is a sui
>generis mode? The unison vectors are simply increments of steps. So we >may
>write

> U(m) = d'(tG) = d'd'(m)

>Here we have

> U(m) = d <21/20 16/15 15/14 10/9>

> = <64/63 225/224 28/27>

That's a very interesting derivation. Kees van Prooijen derived the
decatonic scales from the set of unison vectors

<64/63 50/49 49/48>

But this of course amounts to the same thing since

50/49 * 63/64 = 225/224

and

49/48 * 64/63 = 28/27

In fact, in the new paper I just sent to the MicroFest, I show that the
decatonic scale can be defined by any two of the three _commatic_ unison
vectors

(50/49, 64/63, 225/224)

and any one of the three _chromatic_ unison vectors

(25/24, 28/27, 49/48).

Note that the three commatic unison vectors are all smaller than the three
chromatic unison vectors, which are all smaller than the scale's step sizes:

(10/9, 15/14, 16/15, 21/20).

The diatonic analogue to this is that the diatonic scale can be defined by
the _commatic_ unison vector

(81/80)

and either of the two _chromatic_ unison vectors

(25/24, 135/128).

Note that the commatic unison vector is smaller than the chromatic unison
vectors, which are all smaller than the scale's step sizes:

(9/8, 10/9, 16/15).

>Two distinct systems may have the same unison vectors. To distinguish
>between systems, we have to add at unison vectors (or "modal srutis") >the
>minimal step (or "tonal sruti") obtaining so the srutal basis sB which
>characterize specifically the system used.

> (e.g. With the same unison vectors 25/24 and 81/80 we have
> the Zarlino system having the minimal step 16/15 and
> the Hindu system having the minimal step 256/243.)

The unison vectors 25/24 and 81/80 will always lead to a 7-tone scale -- so
what is your "Hindu" 7-tone scale with minimal step 256/243?

>Assuming that the interval X is represented by (x,y,z,t) in the primal
>basis <2 3 5 7>, we can read the D function in the first line

> D(X) = 10x + 16y + 23z + 28t

>exhibiting the 10 periodicity of steps amount in octave. And we can >obtain
>almost so easily the sruti function summing each column

> S(X) = 24x + 38y + 56z + 67t

>exhibiting the 24 periodicity of srutis amount in octave.

Not sure what the musical meaning of this is -- as you know, the decatonic
scales fall most comfortably into a 22-tone periodicity -- since in that
system, the commatic unison vectors (50/49, 64/63, 225/224) are still
unisons, while the chromatic unison vectors (25/24, 28/27, 49/48) become one
step. Similarly, the diatonic scale falls rather comfortably into a 12-tone
periodicity -- since in that system, the commatic unison vector (81/80) is
still a unison, while the chromatic unison vectors (25/24, 135/128) become
one step. So I'm not sure what the 24 would represent. Please take that as
encouragement to show me!

Anyhow, this was all very interesting, and I'm beginning to get the hang of
your language!

-Paul

Paul H. Erlich wrote,

<<maximal evenness has been used to "derive" the diatonic scale,
clearly you don't get the diatonic scale from a maximally even
7-out-of-31>>

No, of course. But you do get the diatonic from a 7-out-of-12 and I'm
sure that you'd probably get the near QCM 31 diatonic from a
7-out-of-12-out-of-19-out-of-31 as well.

But that's probably neither here nor there anyway... I mean maximal
evenness is obviously not good for everything, or even many things,
but it is good for quite a few things... and it's those types of
things that most interest me about it.

BTW, here's some interesting JI (or better yet RI) interpretations of
the [6,2,2] decatonic without the empirical tweaking to make it
trivalent across a 1:2, or near 1:2.

If you take the [6,2,2] as a straight L-out-of-M-out-of-N set, then
you get a trivalent variant of Messiaen's 10-tone scale of limited
transposability. If this in turn were taken to two-dimensions you
could use the 5-limit lattice as a template for an intersting rational
interpretation.

If P = 12:17 one could take the two-dimensions as

5/4
/
/
1/1--13/12

Or as

17/14
/
/
1/1--17/16

The first follows from a 17/12 15/12 13/12 and the second from a 17/12
17/14 17/16.

Here's the first as a "lattice" (the parenthetical ratio shows the
analogy to the 81/80 comma -- a 2197/2160).

(180/169)---15/13------5/4
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
17/13------1/1------13/12

Here's the second (the analogy to the comma is a 2048/2023 here).

(128/119)----8/7------17/14
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
4/3-------1/1------17/16

If P = 17:24 one could take the two-dimensions as

21/17
/
/
1/1--18/17

Or as

6/5
/
/
1/1--24/23

The first follows from a 24/17 21/17 18/17 and the second from a 24/17
24/20 24/23.

Here's the first as a lattice (the analogy to the comma is a
2023/1944).

(119/108)----7/6------21/17
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
4/3-------1/1------18/17

Here's the second (here the analogy to the comma is a 12167/11520).

(529/480)---23/20------6/5
\ / \ / \
\ / \ / \
\ / \ / \
\ / \ / \
23/17------1/1------24/23

I would think that some of these scales with their near just 7-limit
tetrads could also be seen as trivalent (or relaxed trivalent) RI
variations of your symmetric decatonic.

--Dan Stearns

--- In tuning@y..., "D.Stearns" <STEARNS@C...> wrote:
> Paul H. Erlich wrote,
>
> <<maximal evenness has been used to "derive" the diatonic scale,
> clearly you don't get the diatonic scale from a maximally even
> 7-out-of-31>>
>
> No, of course. But you do get the diatonic from a 7-out-of-12 and I'm
> sure that you'd probably get the near QCM 31 diatonic from a
> 7-out-of-12-out-of-19-out-of-31 as well.

Yes, and even 7-out-of-12-out-of-31 or 7-out-of-19-out-of-31. That's known as second-order
maximal evenness, and there are two distinct ways of defining it (differing as to, for example,
what the possible examples of ME 7-out-of-12-out-of-22 would be, with potential comparisons
to the Indian scales . . . this should all be in the archives).

Hi Paul!

I will explain later (when???) on specific post about sruti system but for
now perhaps this image worth tausend words :

http://www.aei.ca/~plamothe/pix/24srutis.gif

Pierre

Sorry, there was an error in the image title. It' now OK. If the word
decatonic is not there then refresh.

http://www.aei.ca/~plamothe/pix/24srutis.gif

--- In tuning@y..., Pierre Lamothe <plamothe@a...> wrote:
>
> Hi Paul!
>
> I will explain later (when???) on specific post about sruti
system but for
> now perhaps this image worth tausend words :
>
> http://www.aei.ca/~plamothe/pix/24srutis.gif
>
> Pierre

I don't understand -- for example, how can 9/8 and 10/9 both be
#4 if 81:80 is not a unison vector in this system?

Paul,

What are unison vectors?

First sense :

In the infinite group G generated by a primal base pB of rank N the unison
vectors are first an infinite subgroup U determining the infinite group G/U
of interval classes and the finite subgroup G/2U of degrees in octave
circularity.

Second sense :

The N-1 elements of a basis generating U.

-----

Here 9/8 and 10/9 are in two distinct interval classes (degrees). We have

D(10/9) = 1 and D(9/8) = 2

So there not exist any element K in U = <64/63 225/224 28/27> Z^3 such that

9/8 = K 10/9

The tone 81/80 is not a unison vector but an element of classe 1, the class
of steps while unison vectors are elements of class 0.

And we have by group isomorphism

D(9/8) = D(81/80) + D(10/9)

2 = 1 + 1

Pierre

--- In tuning@y..., Pierre Lamothe <
plamothe@a...> wrote:

> Here 9/8 and 10/9 are in two distinct interval classes (degrees). We have
>
> D(10/9) = 1 and D(9/8) = 2
>
> So there not exist any element K in U = <64/63 225/224 28/27> Z^3 such that
>
> 9/8 = K 10/9
>
> The tone 81/80 is not a unison vector but an element of classe 1, the class
> of steps while unison vectors are elements of class 0.

Exactly. Hence, it would seem that 9/8 and
10/9 shouldn't both correspond to #4 out of
24 . . . right? (Not that I know why you have
24).