A 7-limit 12-note scale

There are four tetrads within the 12 notes of Meantone[12], and the
four detempered representatives which have the lowest maximum Hahn
distance from the unison are [0,0,-1], [0,0,0], [0,1,0] and [0,1,1],
or in terms of actual fractions, {1,7/6,7/5,7/4}, {1,5/4,3/2,7/4},
{21/20,21/16,3/2,7/4} and {9/8,21/16,3/2,15/8}. If we put these
together we get the following 10-note scale: 1, 21/20, 9/8, 7/6, 5/4,
21/16, 7/5, 3/2, 7/4, 15/8.

This has scale steps of size 15/14, 16/15, 21/20 and 28/27, with an
anomalous step of size 7/6. To convert it to an epimorphic scale, we
can fill in the gap with 15/14 * 21/20 * 28/27 = 7/6, thereby using
the same four steps, or alternatively 16/15 * 21/20 * 25/24 = 7/6.

There is only one way to fill in the gap with the first choice of
scale steps which keeps all the scale intervals within a Hahn radius
of two of the unison, 3/2--14/9--5/3--7/4. I expected this scale to be
in the Scala archives, but in fact it isn't. Alternatively we can do
it as 3/2--8/5--5/3--7/4 or even 3/2--25/16--5/3--7/4.

The scale using 28/27 has one complete 9-limit otonality, two complete
9-limit utonalities, two major tetrads, two minor tetrads, two
supermajor tetrads, two subminor tetrads, three major triads, four
minor triads, two 1-5/4-14/9 augmented triads, one 1-9/7-8/5 augmented
triad, etc etc. The inverted form, with more otonal harmony, might be
preferred: 1, 15/14, 8/7, 6/5, 9/7, 4/3, 10/7, 3/2, 8/5, 12/7, 9/5, 28/15.

>The inverted form, with more otonal harmony, might be
>preferred: 1, 15/14, 8/7, 6/5, 9/7, 4/3, 10/7, 3/2, 8/5, 12/7, 9/5, 28/15.

Care to name it?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >The inverted form, with more otonal harmony, might be
> >preferred: 1, 15/14, 8/7, 6/5, 9/7, 4/3, 10/7, 3/2, 8/5, 12/7, 9/5,
28/15.
>
> Care to name it?

For my own use, I named it meande and its inverse meandin, but I'm
open to suggestions. I take it you have not seen it before? Strange!

! meande12.scl
chord-based detempering of 7-limit meantone
12
!
21/20
9/8
7/6
5/4
21/16
7/5
3/2
14/9
5/3
7/4
15/8
2

! meandin.scl
inverted detempered 7-limit meantone
12
!
15/14
8/7
6/5
9/7
4/3
10/7
3/2
8/5
12/7
9/5
27/14
2

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> ! meande12.scl
> chord-based detempering of 7-limit meantone
> 12
> !
> 21/20
> 9/8
> 7/6
> 5/4
> 21/16
> 7/5
> 3/2
> 14/9
> 5/3
> 7/4
> 15/8
> 2

here's the triangular lattice of that one:
(hit "Reply" on Yahoo interface to view correctly)

meande12.scl
chord-based detempering of 7-limit meantone

5/3 ------ 5/4 ----- 15/8
|\ /|\ /|\
| \ / | \ / | \
14/9 ----- 7/6-\----/-7/4-\----/21/16\
'. \ /,'/ \'.\ /,'/ '.\
1/1 ------ 3/2 ------ 9/8
| / \ | /
| / \| /
7/5 ----- 21/20

-monz

>> ! meande12.scl
>> chord-based detempering of 7-limit meantone
>> 12
>> !
>> 21/20
>> 9/8
>> 7/6
>> 5/4
>> 21/16
>> 7/5
>> 3/2
>> 14/9
>> 5/3
>> 7/4
>> 15/8
>> 2
>
>here's the triangular lattice of that one:
>(hit "Reply" on Yahoo interface to view correctly)
>
>meande12.scl
>chord-based detempering of 7-limit meantone
>
>
> 5/3 ------ 5/4 ----- 15/8
> |\ /|\ /|\
> | \ / | \ / | \
>14/9 ----- 7/6-\----/-7/4-\----/21/16\
> '. \ /,'/ \'.\ /,'/ '.\
> 1/1 ------ 3/2 ------ 9/8
> | / \ | /
> | / \| /
> 7/5 ----- 21/20

Thanks monz. This is centaur with 21/16 instead of 4/3.
While 21/16 gives an extra utonal tetrad, I think in
practice having the extra fifth wins out.

Say, can Musica generate ASCII lattices? That would be
a great feature. You rotate your scale in the 3-D viewer,
and then print to ASCII for posting to a list.

-Carl

>>> The inverted form, with more otonal harmony, might be
>>> preferred: 1, 15/14, 8/7, 6/5, 9/7, 4/3, 10/7, 3/2, 8/5,
>>> 12/7, 9/5, 28/15.
>>
>> Care to name it?
>
>For my own use, I named it meande and its inverse meandin, but
>I'm open to suggestions. I take it you have not seen it before?
>Strange!
>
>! meandin.scl
>inverted detempered 7-limit meantone
>12
>!
>15/14
>8/7
>6/5
>9/7
>4/3
>10/7
>3/2
>8/5
>12/7
>9/5
>27/14
>2

So the 28/15 was a typo? Would that have influenced your
scale archive search?

The 27/14 is not within Hahn distance 2 of the unison...

! 10/7----------15/14
! /:\ /:\
! / : \ / : \
! / : \ / : \
! 4/3-----------1/1-----------3/2 \
! \`. / ,'/ \`. \ / ,'/ \`. \
! \ '. /.' / \ '.\ /.' / \ '.\
! \ 8/7--/-----\-12/7--/-----\--9/7----------27/14
! \ : / \ : / \ :
! \ : / \ : / \ :
! \:/ \:/ \:
! 8/5-----------6/5-----------9/5

But if we move the unison...

! 5/3-----------5/4
! /:\ /:\
! / : \ / : \
! / : \ / : \
! 14/9-----------7/6-----------7/4 \
! \`. / ,'/ \`. \ / ,'/ \`. \
! \ '. /.' / \ '.\ /.' / \ '.\
! \ 4/3--/-----\--1/1--/-----\--3/2-----------9/8
! \ : / \ : / \ :
! \ : / \ : / \ :
! \:/ \:/ \:
! 28/15----------7/5----------21/20

Now we're back to centaur with a 28/15 instead of a 15/8, which
is of course the 225:224 again.

-Carl

hi Carl,

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> >> ! meande12.scl
> >> chord-based detempering of 7-limit meantone
> >> 12
> >> !
> >> 21/20
> >> 9/8
> >> 7/6
> >> 5/4
> >> 21/16
> >> 7/5
> >> 3/2
> >> 14/9
> >> 5/3
> >> 7/4
> >> 15/8
> >> 2
> >
> >here's the triangular lattice of that one:
> >(hit "Reply" on Yahoo interface to view correctly)
> >
> >meande12.scl
> >chord-based detempering of 7-limit meantone
> >
> >
> > 5/3 ------ 5/4 ----- 15/8
> > |\ /|\ /|\
> > | \ / | \ / | \
> >14/9 ----- 7/6-\----/-7/4-\----/21/16\
> > '. \ /,'/ \'.\ /,'/ '.\
> > 1/1 ------ 3/2 ------ 9/8
> > | / \ | /
> > | / \| /
> > 7/5 ----- 21/20
>
> Thanks monz. This is centaur with 21/16 instead of 4/3.
> While 21/16 gives an extra utonal tetrad, I think in
> practice having the extra fifth wins out.

guess what? ... the whole reason i drew that lattice
was because i had a feeling that this resembled
Centaur. glad it helped you to confirm the connection.

i remember the discussion we had back at the end of
1998 on 12-tone 7-limit tunings which gave maximum
number of consonant chords.

> Say, can Musica generate ASCII lattices? That would be
> a great feature. You rotate your scale in the 3-D viewer,
> and then print to ASCII for posting to a list.

not yet. very doubtful that that feature will make
it into version 1.0, but we'll definitely include it
soon in a future release.

right now we're beginning work on the staff-notations.
when that's finished the beta release will be out,
within the next couple of months.

however ... i *did* use Musica to help me create
this one! with a bit of rotation, the triangular
lattice made by the software looks exactly like
the ASCII one i drew!

-monz

>> >meande12.scl
>> >chord-based detempering of 7-limit meantone
>> >
>> > 5/3 ------ 5/4 ----- 15/8
>> > |\ /|\ /|\
>> > | \ / | \ / | \
>> >14/9 ----- 7/6-\----/-7/4-\----/21/16\
>> > '. \ /,'/ \'.\ /,'/ '.\
>> > 1/1 ------ 3/2 ------ 9/8
>> > | / \ | /
>> > | / \| /
>> > 7/5 ----- 21/20
>>
//
>> Say, can Musica generate ASCII lattices? That would be
>> a great feature. You rotate your scale in the 3-D viewer,
>> and then print to ASCII for posting to a list.
>
>not yet. very doubtful that that feature will make
>it into version 1.0, but we'll definitely include it
>soon in a future release.

Cool.

>however ... i *did* use Musica to help me create
>this one! with a bit of rotation, the triangular
>lattice made by the software looks exactly like
>the ASCII one i drew!

Actually, there's an error in your drawing -- the line
connecting 7/6, 7/4 and 21/16 should be in front of the
lines connecting the triads around 1/1. Also the lines
connecting 7/5 and 21/20 to 7/4 and 21/16 should be
frontmost. So you can see how nice an automated ASCII
feature would be!

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> Thanks monz. This is centaur with 21/16 instead of 4/3.
> While 21/16 gives an extra utonal tetrad, I think in
> practice having the extra fifth wins out.

Not if you are trying to maximize tetrads.

>> Thanks monz. This is centaur with 21/16 instead of 4/3.
>> While 21/16 gives an extra utonal tetrad, I think in
>> practice having the extra fifth wins out.
>
>Not if you are trying to maximize tetrads.

As I said, you loose a tetrad. But it's utonal, and of course
in Centaur the 4/3 works over the G chord just fine. In fact
it seems better as far as traditional forms go to have the
septimal 7th over the IV (blues) than the V chord (classical).

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >> >meande12.scl
> >> >chord-based detempering of 7-limit meantone
> >> >
> >> > 5/3 ------ 5/4 ----- 15/8
> >> > |\ /|\ /|\
> >> > | \ / | \ / | \
> >> >14/9 ----- 7/6-\----/-7/4-\----/21/16\
> >> > '. \ /,'/ \'.\ /,'/ '.\
> >> > 1/1 ------ 3/2 ------ 9/8
> >> > | / \ | /
> >> > | / \| /
> >> > 7/5 ----- 21/20
> >>
> //
> >> Say, can Musica generate ASCII lattices? That would be
> >> a great feature. You rotate your scale in the 3-D viewer,
> >> and then print to ASCII for posting to a list.
> >
> >not yet. very doubtful that that feature will make
> >it into version 1.0, but we'll definitely include it
> >soon in a future release.
>
> Cool.
>
> >however ... i *did* use Musica to help me create
> >this one! with a bit of rotation, the triangular
> >lattice made by the software looks exactly like
> >the ASCII one i drew!
>
> Actually, there's an error in your drawing -- the line
> connecting 7/6, 7/4 and 21/16 should be in front of the
> lines connecting the triads around 1/1. Also the lines
> connecting 7/5 and 21/20 to 7/4 and 21/16 should be
> frontmost. So you can see how nice an automated ASCII
> feature would be!

hmm ... yes, i see that. it happened because the Musica
triangular geometry is slightly different from what
we are forced to use in ASCII.

that's another thing that will be updated in future
releases ... eventually the user will be able to create
any kind of geometry desired. for release 1.0 we had
to hard-code some choices.

anyway, i was also going to say that even tho Musica
can't make ASCII lattices yet, the user *can* take a
screenshot, save it as a .gif, and upload it to
tuning-files. i plan to start doing a lot of this
myself as work on the software itself eases up a bit.

-monz

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> So the 28/15 was a typo? Would that have influenced your
> scale archive search?

No and no. I ran the program through something which does a minimax
Tenney height; the mode you probably will want is one with a major or
minor tetrad on the tonic; your second version below moved to 4/3
would be a good choice. It doesn't make any difference to Scala, which
checks all the modes and inverted modes.

> The 27/14 is not within Hahn distance 2 of the unison...
>
> ! 10/7----------15/14
> ! /:\ /:\
> ! / : \ / : \
> ! / : \ / : \
> ! 4/3-----------1/1-----------3/2 \
> ! \`. / ,'/ \`. \ / ,'/ \`. \
> ! \ '. /.' / \ '.\ /.' / \ '.\
> ! \ 8/7--/-----\-12/7--/-----\--9/7----------27/14
> ! \ : / \ : / \ :
> ! \ : / \ : / \ :
> ! \:/ \:/ \:
> ! 8/5-----------6/5-----------9/5
>
> But if we move the unison...
>
> ! 5/3-----------5/4
> ! /:\ /:\
> ! / : \ / : \
> ! / : \ / : \
> ! 14/9-----------7/6-----------7/4 \
> ! \`. / ,'/ \`. \ / ,'/ \`. \
> ! \ '. /.' / \ '.\ /.' / \ '.\
> ! \ 4/3--/-----\--1/1--/-----\--3/2-----------9/8
> ! \ : / \ : / \ :
> ! \ : / \ : / \ :
> ! \:/ \:/ \:
> ! 28/15----------7/5----------21/20
>
> Now we're back to centaur with a 28/15 instead of a 15/8, which
> is of course the 225:224 again.

I didn't remark on the fact that tempering it in marvel would give us
some additional chords, since this seems always to be true. A marvel
version of centaur/meandin I've put in my own archives; once I did
that, I found that marvel tempering the Malcolm monochord gives us
something which is inverse equal, which means there are a lot more
marvel equivalences out there.

A nifty addition to Scala would be something which could check for
equivalent JI scales up to equivalence by a temperament. One way to do
that would be to input a list of vals, apply it to all of the
transpostions and inversions of the scale, and check if the result is
the same as the same list of vals applied to any other scales of the
same size and of prime limit no greater than the temperament in question.

Anyway, here is centaur in marvel if anyone wants it.

! centmarv.scl
1/4-kleismic marvel tempered centaur/meandin
12
!
84.467193
200.054240
268.798786
384.385833
499.972880
584.440073
700.027120
768.771666
884.358713
968.825906
1084.412953
1200.000000

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> >Not if you are trying to maximize tetrads.
>
> As I said, you loose a tetrad. But it's utonal, and of course
> in Centaur the 4/3 works over the G chord just fine. In fact
> it seems better as far as traditional forms go to have the
> septimal 7th over the IV (blues) than the V chord (classical).

If you are going to get practical on me, and if you care about triads
as well as tetrads, the obvious thing to do is to temper via marvel.
If you *don't* care about triads you may as well use meandin.

The marvel tempering has six major triads and five minor triads, all
less than a schisma off true north, but it makes no improvements in
the tetrad situation.

>> So the 28/15 was a typo? Would that have influenced your
>> scale archive search?
>
>No and no. I ran the program through something which does a minimax
>Tenney height; the mode you probably will want is one with a major or
>minor tetrad on the tonic; your second version below moved to 4/3
>would be a good choice. It doesn't make any difference to Scala,
which
>checks all the modes and inverted modes.

Have you noticed that when you first presented this scale it
contained a 28/15, while when you later gave the scl
representation this had been switched to 27/14? Of course I
know that Scala checks all modes, but it dosen't check for
typos last i checked.

>> But if we move the unison...
>>
>> ! 5/3-----------5/4
>> ! /:\ /:\
>> ! / : \ / : \
>> ! / : \ / : \
>> ! 14/9-----------7/6-----------7/4 \
>> ! \`. / ,'/ \`. \ / ,'/ \`. \
>> ! \ '. /.' / \ '.\ /.' / \ '.\
>> ! \ 4/3--/-----\--1/1--/-----\--3/2-----------9/8
>> ! \ : / \ : / \ :
>> ! \ : / \ : / \ :
>> ! \:/ \:/ \:
>> ! 28/15----------7/5----------21/20
>>
>> Now we're back to centaur with a 28/15 instead of a 15/8, which
>> is of course the 225:224 again.
>
>I didn't remark on the fact that tempering it in marvel would
>give us some additional chords, since this seems always to be
>true.

Is Marvel the 7-limit version of Miracle? I assume it eats
the 225:224 which is at issue here.

>Anyway, here is centaur in marvel if anyone wants it.
>
>! centmarv.scl
>1/4-kleismic marvel tempered centaur/meandin
>12
>!
>84.467193
>200.054240
>268.798786
>384.385833
>499.972880
>584.440073
>700.027120
>768.771666
>884.358713
>968.825906
>1084.412953
>1200.000000

This is very closely related to Dave's wafso-just scale.

-Carl

> >> So the 28/15 was a typo? Would that have influenced your
> >> scale archive search?
> >
> >No and no. I ran the program through something which does a
> >minimax Tenney height;

Also, you must not have disclosed this step -- you said
you were trying to minimize Hahn distance from the unison.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> Is Marvel the 7-limit version of Miracle? I assume it eats
> the 225:224 which is at issue here.

Marvel is 225/224-planar, or in the 11-limit, 225/224 and 385/384. It
is of great practical significance when looking at scales since you
can pretty well take it as given that you will get marvel
equivalencies out the wazoo. Hence, your 5-limit scale has 7-limit
implications, and your 7-limit scale has more chords in it than you
thought it did. If you distribute the 225/224 you get within a
schisma, which is pretty good going.

> This is very closely related to Dave's wafso-just scale.

Let me guess--it's tempered in 225/224. I think Dave may have invented
the idea.

>> This is very closely related to Dave's wafso-just scale.
>
>Let me guess--it's tempered in 225/224. I think Dave may
>have invented the idea.

Actually it was my idea, at least this time around, which is
why some authors call these scales Fokker-Lumma or Lumma-Fokker
scales. But Dave developed the method (very similar to TOP) of
tempering out a single comma in a scale.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >> This is very closely related to Dave's wafso-just scale.
> >
> >Let me guess--it's tempered in 225/224. I think Dave may
> >have invented the idea.
>
> Actually it was my idea, at least this time around, which is
> why some authors call these scales Fokker-Lumma or Lumma-Fokker
> scales. But Dave developed the method (very similar to TOP) of
> tempering out a single comma in a scale.

A Lumma-Fokker is a marvel tempered Fokker block, or a marvel-tempered
anything?

>> >> This is very closely related to Dave's wafso-just scale.
>> >
>> >Let me guess--it's tempered in 225/224. I think Dave may
>> >have invented the idea.
>>
>> Actually it was my idea, at least this time around, which is
>> why some authors call these scales Fokker-Lumma or Lumma-Fokker
>> scales. But Dave developed the method (very similar to TOP) of
>> tempering out a single comma in a scale.
>
>A Lumma-Fokker is a marvel tempered Fokker block, or a marvel-tempered
>anything?

Any 12-tone epimorphic scale with 225:224 tempered out.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> Any 12-tone epimorphic scale with 225:224 tempered out.

My classification of 12-note 5-limit Fokker blocks was done with
several ideas in mind, one being that these give Lumma type scales
when tempered. They have a unique 5-limit representative, which can be
thought of as the primary form of the scale, and used to describe a
class of JI scales which are equivalent by marvel.

In the case of centaur/meandin, the 5-limit scale is New Albion aka
Indian12, so these would be in the New Albion familty.

The mapping giving the projection of marvel to the 5-limit is
[m2, m3, m5] where m2 = <1 0 0 5 -12|, m3 = <0 1 0 2 -1|,
m5 = <0 0 1 2 -3|. If q is any 11-limit interval, then the mapping is
given by marvproj(q) = 2^m2(q) * 3^m3(q) * 5^m5(q). The mapping is
simply the Hermite reduction of any three vals which are a basis for
the vals of marvel, such as 19, 22, and 31; and in general Hermite
reduction is very useful for finding these, since it will always give
it if it exists, and tell you if it does not. Even when it does not we
need not dispair; 2401/2400, with 7^4 in the numerator, will not
project to the 5-limit but there is a single 3 dividing the
denominator and so it projects to no-threes scales.

>> Any 12-tone epimorphic scale with 225:224 tempered out.
>
>My classification of 12-note 5-limit Fokker blocks was done with
>several ideas in mind, one being that these give Lumma type scales
>when tempered.

Yes, I recall.

>They have a unique 5-limit representative, which can be thought
>of as the primary form of the scale, and used to describe a
>class of JI scales which are equivalent by marvel.

I'd rather think of the primary forms in the 7-limit. Hahn
distance and/or Tenney height might be used to canonize one
7-limit version in each class. It would be nice to account
for the following scales in this manner:

! 12_max7.scl
!
32 7-limit dyads in 12 notes, Paul Hahn.
12
!
21/20
7/6
6/5
5/4
4/3
7/5
3/2
8/5
5/3
7/4
28/15
2/1
!

! 12_prism.scl
!
225:224 scale by Carl Lumma.
12
!
16/15
28/25
7/6
5/4
4/3
7/5
112/75
8/5
5/3
7/4
28/15
2/1
!

! 12_class.scl
!
31 dyads covered by 4 tetrads (7-limit).
12
!
21/20
35/32
6/5
5/4
21/16
7/5
3/2
25/16
42/25
7/4
15/8
2/1
!
! Erv Wilson, 1969.

! 12_centaur.scl
!
Excellent 7-limit scale, Kraig Grady 1982.
12
!
21/20
9/8
7/6
5/4
4/3
7/5
3/2
14/9
5/3
7/4
15/8
2/1
!

-Carl

> In the case of centaur/meandin,

By the way, you still haven't answered my question as
to which of the two versions of meandin that you've
given is the real one.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> >They have a unique 5-limit representative, which can be thought
> >of as the primary form of the scale, and used to describe a
> >class of JI scales which are equivalent by marvel.
>
> I'd rather think of the primary forms in the 7-limit. Hahn
> distance and/or Tenney height might be used to canonize one
> 7-limit version in each class.

Why? It seems my method is simpler. The Hahn reduced version could of
course be found, but it's more work, and if you reduce to the 5-limit
you can compare to the 5-limit Fokker blocks.

It would be nice to account
> for the following scales in this manner:

By "account for" do you mean give the Hahn reduction, give the 5-limit
reduction, both, or neither?

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> > In the case of centaur/meandin,
>
> By the way, you still haven't answered my question as
> to which of the two versions of meandin that you've
> given is the real one.

Meandin is the inverse of meande, in any of its modes, and the Scala
scl file I gave should be correct. Is the other version actually
different, and what is it?

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> I'd rather think of the primary forms in the 7-limit. Hahn
> distance and/or Tenney height might be used to canonize one
> 7-limit version in each class. It would be nice to account
> for the following scales in this manner:

Prism is diadie1, aka lumm5r, in the 5-limit projection, but you
already knew that.

! centr.scl
Marvel projection to the 5-limit of centaur
12
!
135/128
9/8
75/64
5/4
4/3
45/32
3/2
25/16
5/3
225/128
15/8
2

! hahnmaxr.scl
Paul Hahn's 12_hahn7 marvel projected to the 5-limit
12
!
135/128
75/64
6/5
5/4
4/3
45/32
3/2
8/5
5/3
225/128
15/8
2

! classr.scl
Marvel projection to the 5-limit of class
12
!
135/128
1125/1024
6/5
5/4
675/512
45/32
3/2
25/16
27/16
225/128
15/8
2

! prismr.scl
Marvel projection to the 5-limit of prism
12
!
16/15
9/8
75/64
5/4
4/3
45/32
3/2
8/5
5/3
225/128
15/8
2

>> >They have a unique 5-limit representative, which can be thought
>> >of as the primary form of the scale, and used to describe a
>> >class of JI scales which are equivalent by marvel.
>>
>> I'd rather think of the primary forms in the 7-limit. Hahn
>> distance and/or Tenney height might be used to canonize one
>> 7-limit version in each class.
>
>Why? It seems my method is simpler. The Hahn reduced version could
>of course be found, but it's more work, and if you reduce to the
>5-limit you can compare to the 5-limit Fokker blocks.

Because then people who want 7-limit untempered JI can have it.

>> It would be nice to account
>> for the following scales in this manner:
>
>By "account for" do you mean give the Hahn reduction, give the
>5-limit reduction, both, or neither?

It would be nice to know which are equivalent under marvel,
then for each of the remaining unique scales find the
hahn-reduced forms. Marvelized version of your 5-limit blocks

majraj2
majsyn3
ragisyn1
syndia4
syndie
tertiadie3
thirds

might be considered as well.

-Carl

>> By the way, you still haven't answered my question as
>> to which of the two versions of meandin that you've
>> given is the real one.
>
>Meandin is the inverse of meande, in any of its modes, and the Scala
>scl file I gave should be correct. Is the other version actually
>different, and what is it?

>! meandin.scl
>inverted detempered 7-limit meantone
>12
>!
>15/14
>8/7
>6/5
>9/7
>4/3
>10/7
>3/2
>8/5
>12/7
>9/5
>27/14
>2

When you first gave this you had 28/15 in place of 27/14.
Judging by the appearance on the lattice, the above version
is the inverse of meande and the version with 28/15 in
place of 27/14 is not a mode of either. Can you confirm
or deny?

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> It would be nice to know which are equivalent under marvel,
> then for each of the remaining unique scales find the
> hahn-reduced forms. Marvelized version of your 5-limit blocks

The Hahn-reduction will be with respect to some choice of 1/1, so
dealing with that is an added complication. The 5-limit version is
unique and easy to compute. However, the mere fact that the 7-limit is
harder makes it more of a challege and hence perhaps more interesting.
We could step through the scale and Hahn-reduce at each degree, for
instance, but now I wonder why stop there? Another approach would be
to start from the chords of the marvel-tempered version, and then try
to maximize the ones we keep. Then we have to decide what "maximize"
means--would we detemper New Albion to Centaur and call it maximal, or
is Meandin maximal?

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:

> When you first gave this you had 28/15 in place of 27/14.
> Judging by the appearance on the lattice, the above version
> is the inverse of meande and the version with 28/15 in
> place of 27/14 is not a mode of either. Can you confirm
> or deny?

Yes, but it seems like a viable scale also. It's 5-limit reduction is
not like anything I have listed. The 5-limit Fokker blocks are turing
out less useful in classifing these than I thought they would be. It
would be interesting to see the lattice diagrams for the 5-limit
reductions of some of these scales. Maybe some dedicated soul will
write a utility to draw these some day, or we can wait for Monz to
finally arrive with a program.

>> When you first gave this you had 28/15 in place of 27/14.
>> Judging by the appearance on the lattice, the above version
>> is the inverse of meande and the version with 28/15 in
>> place of 27/14 is not a mode of either. Can you confirm
>> or deny?
>
>Yes,

Thank you! Try to understand my frustration in this thread!

-Carl