a notation for Schoenberg's rational implications

I am here referring specifically to Schoenberg's 1911 theory
as introduced in his _Harmonielehre_, and not to his later
1927/34 theory articulated in the paper "Problems of Harmony"
(the latter was the basis for Partch's criticism).

I apologize for the long quotes, but want to be complete for
anyone who's interested in following this thread.

> Message 2819
> From: monz <joemonz@y...>
> Date: Sun Jan 20, 2002 4:09pm
> Subject: Re: Re: lattices of Schoenberg's rational implications
/tuning-math/messages/2819?expand=1
>
>
> Help!
>
> I set up an Excel spreadsheet to calculate the notes of
> a periodicity-block according to Gene's formula as expressed here:
>
>
> > Message 2185
> > From: "genewardsmith" <genewardsmith@j...>
> > Date: Wed Dec 26, 2001 6:25 pm
> > Subject: Re: Gene's notation & Schoenberg lattices
> > </tuning-math/message/2185>
> >
> > ...
> >
> > For any non-zero I can define a scale by calculating for 0<=n<d
> >
> > step[n] = (56/55)^round(7n/d) (33/32)^round(12n/d)
> > (64/63)^round(7n/d) (81/80)^round(-2n/d) (45/44)^round(5n/d)
>
>
>
> It worked just fine for both of these examples <snipped>,
> the 7-tone and 12-tone versions.
>
>
>
> But for the kernel I recently posted for Schoenberg ...
>
> > kernel
> >
> > 2 3 5 7 11 unison vectors ~cents
> >
> > [ 1 0 0 0 0 ] = 2:1 0
> > [-5 2 2 -1 0 ] = 225:224 7.711522991
> > [-4 4 -1 0 0 ] = 81:80 21.5062896
> > [ 6 -2 0 -1 0 ] = 64:63 27.2640918
> > [-5 1 0 0 1 ] = 33:32 53.27294323
> >
> > adjoint
> >
> > [ 12 0 0 0 0 ]
> > [ 19 1 2 -1 0 ]
> > [ 28 4 -4 -4 0 ]
> > [ 34 -2 -4 -10 0 ]
> > [ 41 -1 -2 1 12 ]
> >
> > determinant = | 12 |
>
>
> ... it doesn't work. All I get are powers of 2.
>
> Why? How can it be fixed? Do I need yet another
> independent unison-vector instead of 2:1?
>
> ********
>
> Message 2822
> From: genewardsmith <genewardsmith@j...>
> Date: Sun Jan 20, 2002 5:08pm
> Subject: Re: lattices of Schoenberg's rational implications
/tuning-math/messages/2822?expand=1
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > > determinant = | 12 |
>
> > ... it doesn't work.
>
> This determinant is why. In my example, the determinant
> had an absolute value of 1, and so we get what I call a
> "notation", meaning every 11-limit interval can be expressed
> in terms of integral powers of the basis elements. You
> have a determinant of 12, and therefore torsion. In fact,
> you map to the cyclic group C12 of order 12, and the twelveth
> power (or additively, twelve times) anything is the identity.
>
> > Why? How can it be fixed? Do I need yet another
> > independent unison-vector instead of 2:1?
>
> If you want a notation, yes. One which makes the matrix
> unimodular, ie with determinant +-1.
>
>
> ******
>
>
> Message 2900
> From: monz <joemonz@y...>
> Date: Tue Jan 22, 2002 3:34pm
> Subject: Re: Re: Minkowski reduction
> (was: ...Schoenberg's rational implications)
/tuning-math/messages/2900?expand=1
>
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@yahoogroups.com>
> > Sent: Tuesday, January 22, 2002 4:32 AM
> > Subject: [tuning-math] Re: Minkowski reduction
> > (was: ...Schoenberg's rational implications)
> >
>
> > > [monz]
> > > With variant alternate pitches written on the same line
> > > -- and thus with invariant ones on a line by themselves --
> > > these scales are combined into:
> > >
> > > 1/1
> > > 21/20 15/14
> > > 35/32 9/8
> > > 7/6 25/21 6/5
> > > 5/4
> > > 21/16
> > > 7/5 10/7
> > > 3/2
> > > 49/32 25/16 63/40
> > > 5/3 12/7
> > > 7/4
> > > 147/80 15/8
> > >
> > > ...
> > >
> > > One thing I did notice in connection with this, is that
> > > 147/80 is only a little less than 4 cents wider than 11/6,
> > > which is one of the pitches implied in Schoenberg's overtone
> > > diagram (p 23 of _Harmonielehre_) :
> > >
> > > vector ratio ~cents
> > >
> > > [ -4 1 -1 2 0 ] = 147/80 1053.2931
> > > - [ -1 -1 0 0 1 ] = 11/6 1049.362941
> > > --------------------
> > > [ -3 2 -1 2 -1 ] = 441/440 3.930158439
> > >
> > >
> > > So I know that 441/440 is tempered out.
> >
> > NO IT ISN'T! I believe it maps to 1 semitone given the set of unison
> > vectors you've put forward.
> >
> > > But I don't see
> > > how to get this as a combination of two of the other
> > > unison-vectors.
> >
> > YOU CAN'T!
>
>
> Oops... my bad. Thanks, Paul. I see it now. If "C" is Schoenberg's
> 1/1, the 147/80 is mapped to "B" but 11/6 is mapped to "Bb".
> This is precisely the note which was misprinted in the diagram in
> the English edition ... guess I accepted it for so long that I
> got confused.

The Schoenberg PBs i've been posting have been defined
entirely by commatic unison-vectors.

Paul also posted something about how i would need to include
a *chromatic* unison-vector in order to arrive at a Smithian
"notation" (... i've searched for that post but can't find it).

Well, i was thinking about this and realized that here the
441/440 is a perfect candidate for a chromatic unison-vector!

So i plugged it into my spreadsheet matrix in place of 2/1,
using the unison-vectors i derived directly from _Harmonielehre_
(rather than Gene's Minkowski-reduced ones):

kernel:

2 3 5 7 11 ratio ~cents

[-3 2 -1 2 -1] = 441:440 3.93016
[-5 2 2 -1 0] = 225:224 7.71152
[-4 4 -1 0 0] = 81:80 21.50629
[ 6 -2 0 -1 0] = 64:63 27.26409
[-5 1 0 0 1] = 33:32 53.27294

and got a unimodular adjoint (or is that unimodular inverse?):

adjoint:

[12 5 -2 19 12]
[19 8 -3 30 19]
[28 12 -5 44 28]
[34 14 -6 53 34]
[41 17 -7 65 42]

Here i see two alternative mappings to 12, in which the
only difference is h12(11)=41 or 42.

The pentatonic mapping is in there, and now there's also
one that goes to 19.

But what to make of that third column? the -h2(2)=-2
means that some form of tritone is the period, correct?
But how do i find the generator? Until i know what that is,
the other numbers don't make any sense ... do they?

And as Paul predicted, this time Gene's formula worked
like a charm, and i got the following JI PB scale:

degree ratio vector
2 3 5 7 11

( 12 2/1 [ 1 0 0 0 0] )
11 15/8 [-3 1 1 0 0]
10 16/9 [ 4 -2 0 0 0]
9 5/3 [ 0 -1 1 0 0]
8 8/5 [ 3 0 -1 0 0]
7 3/2 [-1 1 0 0 0]
6 10/7 [ 1 0 1 -1 0]
5 4/3 [ 2 -1 0 0 0]
4 5/4 [-2 0 1 0 0]
3 32/27 [ 5 -3 0 0 0]
2 9/8 [-3 2 0 0 0]
1 16/15 [ 4 -1 -1 0 0]
0 1/1 [ 0 0 0 0 0]

triangular lattice:
A E B
5:3.------.5:4-----15:8
/ \ ` F# ' / \ / \
/ \ 10:7 / \ / \
/ \ | / \ / \
Eb Bb F C G D
32:27----16:9-----4:3-------1:1------3:2-----9:8
\ / \ /
\ / \ /
\ / \ /
Db Ab
16:15----8:5

In my quest to find this notation, Paul has already
suggested that i "forget it", since Schoenberg clearly
meant for all of these unison-vectors to be tempered out
of his system.

But, more than once in _Harmonielehre_, Schoenberg did
indeed allude to a rational basis which might underlie the
compositions from his "free atonality" period, so i'm
very interested in examining that rational basis.

So, guys, am i on the right track with this one?

Paul, how does this scale compare with the PB you would
find by your method using these criteria?

-monz

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Take 2 ...

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 11:43 AM
> Subject: [tuning-math] a notation for Schoenberg's rational implications
>
>
> I am here referring specifically to Schoenberg's 1911 theory
> as introduced in his _Harmonielehre_,
> <... etc>
>
>
> Well, i was thinking about this and realized that here the
> 441/440 is a perfect candidate for a chromatic unison-vector!
>
> So i plugged it into my spreadsheet matrix in place of 2/1,
> using the unison-vectors i derived directly from _Harmonielehre_
> (rather than Gene's Minkowski-reduced ones):
>
> <snipped matrix details>
>
>
> triangular lattice:
> A E B
> 5:3.------.5:4-----15:8
> / \ ` F# ' / \ / \
> / \ 10:7 / \ / \
> / \ | / \ / \
> Eb Bb F C G D
> 32:27----16:9-----4:3-------1:1------3:2-----9:8
> \ / \ /
> \ / \ /
> \ / \ /
> Db Ab
> 16:15----8:5

Here's another JI PB for comparison, derived from 45:44,
which i had originally interpreted as a commatic
unison-vector -- thanks to the misprint i discovered in
the English translation (B instead of Bb for 11th/F),
and which led me down the wrong path for years -- but
which can now be used as a chromatic unison-vector.

kernel
2 3 5 7 11 ratio ~cents

[-2 2 1 0 -1] = 45:44 38.9057732
[-5 2 2 -1 0] = 225:224 7.7115230
[-4 4 -1 0 0] = 81:80 21.5062896
[ 6 -2 0 -1 0] = 64:63 27.2640918
[-5 1 0 0 1] = 33:32 53.2729432

adjoint

[12 -7 -2 7 12]
[19 -11 -3 11 19]
[28 -16 -5 16 28]
[34 -20 -6 19 34]
[41 -24 -7 24 42]

determinant = | 1 |

We still have the same two mappings to 12, as well as
the one to -2 (which i don't understand).

But this time, the mappings to 5 and 19 disappear and
are replaced by two "diatonic" mappings to 7, which
differ only in their mapping of prime-factor 7: the
first one sends it to the 20th degree, which is a "7th",
and the second sends it to the 19th degree, which is
a "6th".

Am i right about this?

And the JI periodicity-block scale derived from this
has only one note different from the one in my last post,
and that is the "tritone", which is 64:45 here instead
of 10:7 :

triangular lattice:
A E B
5:3------5:4-----15:8
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
Eb Bb F C G D
32:27----16:9-----4:3-----1:1------3:2-----9:8
/ \ / \ /
/ \ / \ /
/ \ / \ /
Gb Db Ab
64:45----16:15----8:5

-monz

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

> The Schoenberg PBs i've been posting have been defined
> entirely by commatic unison-vectors.
>
> Paul also posted something about how i would need to include
> a *chromatic* unison-vector in order to arrive at a Smithian
> "notation" (... i've searched for that post but can't find it).
> Well, i was thinking about this and realized that here the
> 441/440 is a perfect candidate for a chromatic unison-vector!
> So i plugged it into my spreadsheet matrix in place of 2/1,
> using the unison-vectors i derived directly from _Harmonielehre_
> (rather than Gene's Minkowski-reduced ones):
>
>
> kernel:
>
> 2 3 5 7 11 ratio ~cents
>
> [-3 2 -1 2 -1] = 441:440 3.93016
> [-5 2 2 -1 0] = 225:224 7.71152
> [-4 4 -1 0 0] = 81:80 21.50629
> [ 6 -2 0 -1 0] = 64:63 27.26409
> [-5 1 0 0 1] = 33:32 53.27294
>
>
> and got a unimodular adjoint (or is that unimodular inverse?):
>
> adjoint:
>
> [12 5 -2 19 12]
> [19 8 -3 30 19]
> [28 12 -5 44 28]
> [34 14 -6 53 34]
> [41 17 -7 65 42]
>
>
> Here i see two alternative mappings to 12, in which the
> only difference is h12(11)=41 or 42.
>
> The pentatonic mapping is in there, and now there's also
> one that goes to 19.
>
> But what to make of that third column? the -h2(2)=-2
> means that some form of tritone is the period, correct?

that's generator, not period, i believe.

> And as Paul predicted, this time Gene's formula worked
> like a charm,

can you remind me what you're referring to?

> and i got the following JI PB scale:
>
> degree ratio vector
> 2 3 5 7 11
>
> ( 12 2/1 [ 1 0 0 0 0] )
> 11 15/8 [-3 1 1 0 0]
> 10 16/9 [ 4 -2 0 0 0]
> 9 5/3 [ 0 -1 1 0 0]
> 8 8/5 [ 3 0 -1 0 0]
> 7 3/2 [-1 1 0 0 0]
> 6 10/7 [ 1 0 1 -1 0]
> 5 4/3 [ 2 -1 0 0 0]
> 4 5/4 [-2 0 1 0 0]
> 3 32/27 [ 5 -3 0 0 0]
> 2 9/8 [-3 2 0 0 0]
> 1 16/15 [ 4 -1 -1 0 0]
> 0 1/1 [ 0 0 0 0 0]
>
>
>
> triangular lattice:
> A E B
> 5:3.------.5:4-----15:8
> / \ ` F# ' / \ / \
> / \ 10:7 / \ / \
> / \ | / \ / \
> Eb Bb F C G D
> 32:27----16:9-----4:3-------1:1------3:2-----9:8
> \ / \ /
> \ / \ /
> \ / \ /
> Db Ab
> 16:15----8:5
>
>
>
> In my quest to find this notation, Paul has already
> suggested that i "forget it", since Schoenberg clearly
> meant for all of these unison-vectors to be tempered out
> of his system.
>
> But, more than once in _Harmonielehre_, Schoenberg did
> indeed allude to a rational basis which might underlie the
> compositions from his "free atonality" period, so i'm
> very interested in examining that rational basis.
>
> So, guys, am i on the right track with this one?
>
> Paul, how does this scale compare with the PB you would
> find by your method using these criteria?

what criteria? first of all, i have no idea what you did above, as
you have too many unison vectors to define a PB.

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> Take 2 ...
>
>
> > From: monz <joemonz@y...>
> > To: <tuning-math@y...>
> > Sent: Sunday, February 03, 2002 11:43 AM
> > Subject: [tuning-math] a notation for Schoenberg's rational
implications
> >
> >
> > I am here referring specifically to Schoenberg's 1911 theory
> > as introduced in his _Harmonielehre_,
> > <... etc>
> >
> >
> > Well, i was thinking about this and realized that here the
> > 441/440 is a perfect candidate for a chromatic unison-vector!
> >
> > So i plugged it into my spreadsheet matrix in place of 2/1,
> > using the unison-vectors i derived directly from _Harmonielehre_
> > (rather than Gene's Minkowski-reduced ones):
> >
> > <snipped matrix details>
> >
> >
> > triangular lattice:
> > A E B
> > 5:3.------.5:4-----15:8
> > / \ ` F# ' / \ / \
> > / \ 10:7 / \ / \
> > / \ | / \ / \
> > Eb Bb F C G D
> > 32:27----16:9-----4:3-------1:1------3:2-----9:8
> > \ / \ /
> > \ / \ /
> > \ / \ /
> > Db Ab
> > 16:15----8:5
>
>
>
> Here's another JI PB for comparison, derived from 45:44,
> which i had originally interpreted as a commatic
> unison-vector -- thanks to the misprint i discovered in
> the English translation (B instead of Bb for 11th/F),
> and which led me down the wrong path for years -- but
> which can now be used as a chromatic unison-vector.
>
>
> kernel
> 2 3 5 7 11 ratio ~cents
>
> [-2 2 1 0 -1] = 45:44 38.9057732
> [-5 2 2 -1 0] = 225:224 7.7115230
> [-4 4 -1 0 0] = 81:80 21.5062896
> [ 6 -2 0 -1 0] = 64:63 27.2640918
> [-5 1 0 0 1] = 33:32 53.2729432
>
>
> adjoint
>
> [12 -7 -2 7 12]
> [19 -11 -3 11 19]
> [28 -16 -5 16 28]
> [34 -20 -6 19 34]
> [41 -24 -7 24 42]
>
>
> determinant = | 1 |
>
>
> We still have the same two mappings to 12, as well as
> the one to -2 (which i don't understand).
>
> But this time, the mappings to 5 and 19 disappear and
> are replaced by two "diatonic" mappings to 7, which
> differ only in their mapping of prime-factor 7: the
> first one sends it to the 20th degree, which is a "7th",
> and the second sends it to the 19th degree, which is
> a "6th".
>
>
> Am i right about this?

i think so -- gene should check.

> And the JI periodicity-block scale derived from this

again, i'm baffled. how do you get a PB when you have one too many
UVs?

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 8:34 PM
> Subject: [tuning-math] Re: a notation for Schoenberg's rational
implications
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > The Schoenberg PBs i've been posting have been defined
> > entirely by commatic unison-vectors.
> >
> > Paul also posted something about how i would need to include
> > a *chromatic* unison-vector in order to arrive at a Smithian
> > "notation" (... i've searched for that post but can't find it).
> > Well, i was thinking about this and realized that here the
> > 441/440 is a perfect candidate for a chromatic unison-vector!
> > So i plugged it into my spreadsheet matrix in place of 2/1,
> > using the unison-vectors i derived directly from _Harmonielehre_
> > (rather than Gene's Minkowski-reduced ones):
> >
> >
> > kernel:
> >
> > 2 3 5 7 11 ratio ~cents
> >
> > [-3 2 -1 2 -1] = 441:440 3.93016
> > [-5 2 2 -1 0] = 225:224 7.71152
> > [-4 4 -1 0 0] = 81:80 21.50629
> > [ 6 -2 0 -1 0] = 64:63 27.26409
> > [-5 1 0 0 1] = 33:32 53.27294

> > And as Paul predicted, this time Gene's formula worked
> > like a charm,
>
> can you remind me what you're referring to?

i wish i could find it -- i've been searching like mad.
i had posted a question about gene's formula, and he responded
that i had to have a determinant of +/-1 in order to obtain
a "notation".

i asked about how to do this, and you posted something about
needing to include a *chromatic* unison-vector in order to
get the "notation". that's the post i can't find now.
but as you can see, it does work.

> > Paul, how does this scale compare with the PB you would
> > find by your method using these criteria?
>
> what criteria? first of all, i have no idea what you did above,
> as you have too many unison vectors to define a PB.

i think if you replace the 441:440 with 2:1, you'll be
able to derive the periodicity-block using your method.
that was the one that i added in this time to get the
"notation".

-monz

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

> i think if you replace the 441:440 with 2:1, you'll be
> able to derive the periodicity-block using your method.

well . . . i don't use 2:1 explicitly . . .

> that was the one that i added in this time to get the
> "notation".

so the PB comes _without_ using 441:440?

anyway, how strange to call that a chromatic unison vector. are you
sure you got that from me?

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 8:57 PM
> Subject: [tuning-math] Re: a notation for Schoenberg's rational
implications
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > i think if you replace the 441:440 with 2:1, you'll be
> > able to derive the periodicity-block using your method.
>
> well . . . i don't use 2:1 explicitly . . .
>
> > that was the one that i added in this time to get the
> > "notation".
>
> so the PB comes _without_ using 441:440?

well, no ... when i use 2:1 instead of 441:440, i can
recover the set of homorphisms, but i only get "octaves"
when i try to find the PB pitches.

as i said, i have to use a chromatic unison-vector in
addition to all the commatic ones, in order to get the
full PB.

> anyway, how strange to call that a chromatic unison vector.
> are you sure you got that from me?

n o ! i didn't get t h a t from you. i got the
necissity of having a chromatic unison-vector from you.

i got the idea to use 441:440 as a chromatic unison-vector
when i derived it from the Minkowski-reduced version of the
Schoenberg PB which Gene and you calculated, and you
pointed out to me that it was not a commatic unison-vector
because in fact the pitches separated by it did involve a
change of accidental.

so i replaced the 2:1 i had in my matrix with that, and
_voil�_! -- out came the PB!

but as you can see from my subsequent post, i believe
there's more validity to Schoenberg's actual theory in
using 45:44 as a chromatic unison-vector instead, and
it does result in a scale which has one different note.

-monz

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

> >
> > so the PB comes _without_ using 441:440?
>
>
> well, no ... when i use 2:1 instead of 441:440,

not a good idea to use 2:1 as a unison vector, especially if you're
trying to do thing's gene's way.

> as i said, i have to use a chromatic unison-vector in
> addition to all the commatic ones, in order to get the
> full PB.
>
>
> > anyway, how strange to call that a chromatic unison vector.
> > are you sure you got that from me?
>
>
> n o ! i didn't get t h a t from you. i got the
> necissity of having a chromatic unison-vector from you.

this sure isn't an example of where there's such a necessity, or even
such a possibility.

> i got the idea to use 441:440 as a chromatic unison-vector
> when i derived it from the Minkowski-reduced version of the
> Schoenberg PB which Gene and you calculated, and you
> pointed out to me that it was not a commatic unison-vector
> because in fact the pitches separated by it did involve a
> change of accidental.

it's a chromatic unison vector with respect to a 7-tone PB in an MOS
tuning, not with respect to a 12-tone PB in an equal tuning. In the
latter it functions as a step vector -- but is a very unlikely choice
for one.

> so i replaced the 2:1 i had in my matrix with that, and
> _voilà_! -- out came the PB!

you should clarify with gene what goes into the formula you're using.
i expect he'll confirm that what you're putting in is not a unison
vector (or in the kernel).

> but as you can see from my subsequent post, i believe
> there's more validity to Schoenberg's actual theory in
> using 45:44 as a chromatic unison-vector instead, and
> it does result in a scale which has one different note.

of course all these scales are just representing 12-et, so there's no
point in 'fretting' over single notes. remember that the
parallelogram construction is arbitrary, and differences in it and
other constructions are only meaningful when you're not tempering out
all the unison vectors. in order to get the intervals to work the way
schoenberg intended, you have to temper out all the unison vectors.

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

> it's a chromatic unison vector with respect to a 7-tone PB in an MOS
> tuning, not with respect to a 12-tone PB in an equal tuning. In the
> latter it functions as a step vector -- but is a very unlikely choice
> for one.

It worked, though.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > it's a chromatic unison vector with respect to a 7-tone PB in an
MOS
> > tuning, not with respect to a 12-tone PB in an equal tuning. In
the
> > latter it functions as a step vector -- but is a very unlikely
choice
> > for one.
>
> It worked, though.

well clearly the formula wasn't depending on it to any great extent.
no ratios of 11 or multiples of 11 even made it into the block.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, February 03, 2002 10:14 PM
> Subject: [tuning-math] Re: a notation for Schoenberg's rational
implications
>
>
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> >
> > > it's a chromatic unison vector with respect to a
> > > 7-tone PB in an MOS tuning, not with respect to a
> > > 12-tone PB in an equal tuning. In the latter it
> > > functions as a step vector -- but is a very unlikely
> > > choice for one.
> >
> > It worked, though.
>
> well clearly the formula wasn't depending on it to any
> great extent. no ratios of 11 or multiples of 11 even
> made it into the block.

yes, i was curious about the 11-limit PB without 7 that Gene
derived for Schoenberg way back around Christmas ... if you
recall, i posted a question then wondering why there were no
7-limit ratios.

so i was very surprised here to see that the PB that i
finally derived, as the one which i think represents the
first 184 pages of _Harmonielehre_ the best, is only 5-limit!

-monz

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