deeper analysis of Schoenberg unison-vectors

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, January 19, 2002 8:01 AM
> Subject: Re: [tuning-math] ERROR IN CARTER'S SCHOENBERG (Re: badly tuned
remote overtones)
>
>
> ... but if one is trying to ascertain
> the potential rational basis behind Schoenberg's work, how does
> one decide which unison-vectors are valid and which are not?
>
> Schoenberg was very clear about what he felt were the "overtone"
> implications of the diatonic scale (and later, the chromatic
> as well), but as I showed in my posts, the only "obvious"
> 5-limit unison-vector is the syntonic comma, and it seemed
> to me that there always needed to be *two* 5-limit unison-vectors
> in order to have a matrix of the proper size (so that it's square).
>
> (I realize that by transposition it need not be a 5-limit UV,
> but I'm not real clear on what else *could* be used, except for
> the 56:55 example Gene used.)

To make clear what I'm trying to say:

Let's begin with the unison-vectors clearly implied by
Schoenberg's 1911 diagram.

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, January 16, 2002 3:43 AM
> Subject: [tuning-math] ERROR IN CARTER'S SCHOENBERG
>
> ... the total list of unison-vectors implied by Schoenberg's
> 1911 diagram [on p 23 of the original 1911 edition of
> _Harmonielehre_, p 24 in the Carter 1978 English edition] is:
>
> Bb 11*4=44 : Bb 7*6=42 = 22:21
> F 16*4=64 : F 7*9=63 = 64:63
> F 11*6=66 : F 16*4=64 = 33:32
> F 11*6=66 : F 7*9=63 = 22:21
> A 9*9=81 :(A 20*4=80) = 81:80
> C 11*9=99 :(C 24*4=96) = 33:32
>
> But because 22:21, 33:32, and 64:63 form a dependent triplet
> (any one of them can be found by multiplying the other two),
> this does not suffice to create a periodicity-block, which
> needs another independent unison-vector.

But now let's try to find the other unison-vector we need
from Schoenberg's musical examples.

If our 1/1 is called "C", in his overtone diagram, Schoenberg calls
"Eb" the "6th overtone [= 7th harmonic] of F", so that its
ratio is 7/6.

But then Schoenberg leaves the discussion of implied 7- and
higher-limit harmonies to the later chapters, and devotes
several chapters to explaining the diatonic major scale and
its harmonies, using C as a reference pitch and C-majoras the
reference scale and key. The diagram immediately before the
one referred to above is one in which he derives the diatonic
major scale from the first 6 harmonics of F, C, and G :

5:3-----5:4----15:8
A E B
/ \ / \ / \
/ \ / \ / \
4:3-----1:1-----3:2-----9:8
F C G D

This is standard stuff, going back to Zarlino (1558).

And as everyone here knows, a description of standard
diatonic chord progressions is going to bump into the
syntonic comma wherever a II-V progression is found,
which would imply a new D on our lattice at 10/9.

According to the expanded diagram of Schoenberg's explanation
of the overtone theory on p 23 [p 24 in Carter] (going up to
the 12th harmonics), the one I refer to at the beginning, the
81:80 syntonic comma is already a part of the system anyway.
He shows A as the 5th and 10th harmonics of F and as the
9th harmonic of G, which are the ratios 81:20 and 81:40,
which in turn are the syntonic comma plus 2 "8ves" and
1 "8ve" respectively. So we already have the unison-vector
of 81:80 = [-4 4 1] included in our kernel.

Schoenberg first introduces chromatic pitches in the chapter
"Die Molltonart" [p 110-128 in the original edition,
p 95-111 in the Carter edition]: F# and G# in the context
of A-minor.

To me, his explanation clearly implies a basis somewhere
between meantone and 5-limit JI: A-minor is seen as the
relative of C-major, so the note A is ~5/3. G# is always
regarded as a "leading-tone" and is assumed to be a consonant
~5/4 above the "dominant" E, which is assumed to be ~3/2
above the tonic A.

F# is always ~5/4 above D, the "subdominant", which is
assumed to be ~4/3 above the tonic A; thus, the 10/9
version of D is the one in effect here.

So our diatonic minor-scale paradigm lattice is:

25:18----( )---25:16
F# G#
/ \ / \ / \
/ \ / \ / \
10:9----5:3-----5:4----15:8
D A E B
\ / \ / \ /
\ / \ / \ /
4:3-----1:1-----3:2
F C G

and again the syntonic comma is in effect because, according
to Schoenberg's list of available minor-key chords on p 115
[p 99 in Carter], B can also be 50/27, F# can also be 45/32,
and D can still also be 9/8.

In a tiny handful of examples Schoenberg also introduces C#
as a sharpened "3rd" (= ~5/4) in the II chord in the key of
G-major.

So altogether up to this point we have this lattice:

50:27---25:18---25:24---25:16
B F# C# G#
\ / \ / \ / \
\ / \ / \ / \
10:9----5:3-----5:4----15:8---45:32
D A E B F#
\ / \ / \ / \ /
\ / \ / \ / \ /
4:3-----1:1-----3:2-----9:8
F C G D

These are the only pitches implied in any of Schoenberg's
explanations until the chapter "Modulation" [p 169-198 in
the original edition, p 150-174 in the Carter edition]. Thus,
excluding the prefatory chapters on aesthetics, about 1/3
of _Harmonielehre_ devoted to discussion of this simple
harmonic paradigm.

On p 184 [p 161 in Carter], music example number 110,
we see a D# in a musical example for the first time in
_Harmonielehre_. The first chord is a C-major triad, or
I in the key of C-major, which Schoenberg also designates
simultaneously as VI in E-minor. The second chord is a V
in E-minor, which is a B-major triad, and so its ~5/4 is
D# 75/64 :

50:27---25:18---25:24---25:16---75:64
B F# C# G# D#
\ / \ / \ / \ / \
\ / \ / \ / \ / \
10:9----5:3-----5:4----15:8---45:32
D A E B F#
\ / \ / \ / \ /
\ / \ / \ / \ /
4:3-----1:1-----3:2-----9:8
F C G D

So comparing this D# 75/64 with our Eb 7/6, now we finally
have a canditate for another 7-limit unison-vector, namely
225:224 = [-5 2 2 -1] .

So as of p 184 in _Harmonielehre_, we can construct as system
valid for Schoenberg's theories, as follows:

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 |

mapping of ETs to UVs

[ 12 -7 12 0 12 ]
[ 0 1 0 1 -2 ]
[ 0 0 0 0 1 ]
[ 0 0 0 1 0 ]
[ 0 0 1 0 0 ]

This last matrix shows that 12-ET maps all of the
unison-vectors except 225:224 to 0 or 12 (i.e., unison),
correct?

And that the last three do not temper out the 81:80, 64:63,
and 33:32 respectively, correct?

Further illumation would be appreciated.

-monz

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

> Message 2798
> From: "monz" <joemonz@y...>
> Date: Sat Jan 19, 2002 2:40 pm
> Subject: deeper analysis of Schoenberg unison-vectors
>
> ...
>
> So as of p 184 in _Harmonielehre_, we can construct as system
> valid for Schoenberg's theories, as follows:
>
> 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 |
>
>
> mapping of ETs to UVs
>
> [ 12 -7 12 0 12 ]
> [ 0 1 0 1 -2 ]
> [ 0 0 0 0 1 ]
> [ 0 0 0 1 0 ]
> [ 0 0 1 0 0 ]
>
>
> This last matrix shows that 12-ET maps all of the
> unison-vectors except 225:224 to 0 or 12 (i.e., unison),
> correct?
>
> And that the last three do not temper out the 81:80, 64:63,
> and 33:32 respectively, correct?
>
>
> Further illumation would be appreciated.

Specificially: what is that second line saying? It looks
like 225:224 and 64:63 are 1 step, and 33:32 is -2 steps.
What tuning is that? Would these be an example of a mapping
to two keyboards where 1 is 12-tET, and the second is
mistuned by some amount that renders ~1/6-tone (i.e., an
amount which makes 7-limit harmonies accurate) as 1 step,
and ~1/4-tone (i.e., to make 11-limit harmonies accurate)
as -2 steps ?

deeply curious,

-monz

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

> Message 2798
> From: "monz" <joemonz@y...>
> Date: Sat Jan 19, 2002 2:40 pm
> Subject: deeper analysis of Schoenberg unison-vectors
>
>
>
> Let's begin with the unison-vectors clearly implied by
> Schoenberg's 1911 diagram.
>
>
> > From: monz <joemonz@y...>
> > To: <tuning-math@y...>
> > Sent: Wednesday, January 16, 2002 3:43 AM
> > Subject: [tuning-math] ERROR IN CARTER'S SCHOENBERG
> >
> > ... the total list of unison-vectors implied by Schoenberg's
> > 1911 diagram [on p 23 of the original 1911 edition of
> > _Harmonielehre_, p 24 in the Carter 1978 English edition] is:
> >
> > Bb 11*4=44 : Bb 7*6=42 = 22:21
> > F 16*4=64 : F 7*9=63 = 64:63
> > F 11*6=66 : F 16*4=64 = 33:32
> > F 11*6=66 : F 7*9=63 = 22:21
> > A 9*9=81 :(A 20*4=80) = 81:80
> > C 11*9=99 :(C 24*4=96) = 33:32
> >
> > But because 22:21, 33:32, and 64:63 form a dependent triplet
> > (any one of them can be found by multiplying the other two),
> > this does not suffice to create a periodicity-block, which
> > needs another independent unison-vector.
>
>
> But now let's try to find the other unison-vector we need
> from Schoenberg's musical examples.
>
> If our 1/1 is called "C", in his overtone diagram, Schoenberg calls
> "Eb" the "6th overtone [= 7th harmonic] of F", so that its
> ratio is 7/6.
>
> But then Schoenberg leaves the discussion of implied 7- and
> higher-limit harmonies to the later chapters, and devotes
> several chapters to explaining the diatonic major scale and
> its harmonies, using C as a reference pitch and C-majoras the
> reference scale and key. The diagram immediately before the
> one referred to above is one in which he derives the diatonic
> major scale from the first 6 harmonics of F, C, and G :
>
>
> 5:3-----5:4----15:8
> A E B
> / \ / \ / \
> / \ / \ / \
> 4:3-----1:1-----3:2-----9:8
> F C G D
>
>
> This is standard stuff, going back to Zarlino (1558).
>
> And as everyone here knows, a description of standard
> diatonic chord progressions is going to bump into the
> syntonic comma wherever a II-V progression is found,
> which would imply a new D on our lattice at 10/9.

I thought it worth pointing out that from the very beginning
of his descriptions of the diatonic scale, the 81:80 must
be tempered out, so that the proper lattice for at least the
first 184 pages of _Harmonielehre_ would be a cylindrical
meantone-based one.

> So comparing this D# 75/64 with our Eb 7/6, now we finally
> have a canditate for another 7-limit unison-vector, namely
> 225:224 = [-5 2 2 -1] .

Also worth pointing out: 224/224 is neither a divisor
nor product of any of the other potential unison-vectors
<22:21, 33:32, 63:64, 81:80>, thus it satisfies the condition
we need for the unison-vector we're seeking, namely, that
it be independent of all the others.

> mapping of ETs to UVs
>
> [ 12 -7 12 0 12 ]
> [ 0 1 0 1 -2 ]
> [ 0 0 0 0 1 ]
> [ 0 0 0 1 0 ]
> [ 0 0 1 0 0 ]

What does the -7 mean in the first row? It's telling us
something significant about how 12-tET handles 225:224 in
this kernel, but what?

-monz

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

> > Bb 11*4=44 : Bb 7*6=42 = 22:21
> > F 16*4=64 : F 7*9=63 = 64:63
> > F 11*6=66 : F 16*4=64 = 33:32
> > F 11*6=66 : F 7*9=63 = 22:21
> > A 9*9=81 :(A 20*4=80) = 81:80
> > C 11*9=99 :(C 24*4=96) = 33:32
> >
> > But because 22:21, 33:32, and 64:63 form a dependent triplet
> > (any one of them can be found by multiplying the other two),
> > this does not suffice to create a periodicity-block, which
> > needs another independent unison-vector.

What it creates, in fact, is Monzoid, as shown by the fact that the above list is compatible with both h5 and h7. Taking triples, we find that the Monzoid wedgie results from (81/80,63/63,33/32),
(81/80,64/63,22/21), and (64/63,33/32,22/21).

> mapping of ETs to UVs
>
> [ 12 -7 12 0 12 ]
> [ 0 1 0 1 -2 ]
> [ 0 0 0 0 1 ]
> [ 0 0 0 1 0 ]
> [ 0 0 1 0 0 ]

What is this?

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

> Also worth pointing out: 224/224 is neither a divisor
> nor product of any of the other potential unison-vectors
> <22:21, 33:32, 63:64, 81:80>, thus it satisfies the condition
> we need for the unison-vector we're seeking, namely, that
> it be independent of all the others.

For independence, you need that any product to rational powers won't give you 225/224, or equivalently, that any product to integral powers will not give you any power of 225/224. You can check this with your linear algebra package, by taking the rank (or in this case, also the determinant) of the matrix of row vectors.

> Message 2798
> From: "monz" <joemonz@y...>
> Date: Sat Jan 19, 2002 2:40 pm
> Subject: deeper analysis of Schoenberg unison-vectors
/tuning-math/message/2798
>
> ...
>
> So as of p 184 in _Harmonielehre_, we can construct as system
> valid for Schoenberg's theories, as follows:
>
> 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 |

Then I had something after this, about which Gene asked
(and rightly so, as will be seen):

> Message 2802
> From: "genewardsmith" <genewardsmith@j...>
> Date: Sat Jan 19, 2002 4:34 pm
> Subject: Re: deeper analysis of Schoenberg unison-vectors
/tuning-math/message/2802
>
>
> > mapping of ETs to UVs
> >
> > [ 12 -7 12 0 12 ]
> > [ 0 1 0 1 -2 ]
> > [ 0 0 0 0 1 ]
> > [ 0 0 0 1 0 ]
> > [ 0 0 1 0 0 ]
>
> What is this?

Something I got from Graham. (I've been searching for an
hour in both the tuning-math and tuning archives, and in
my private emails, looking for it, and unfortunately can't
find it!) He explained how the adjoint shows the mapping,
and included this after it.

Here's how it works:

... Oh no! My bad! That last one, the "mapping"
matrix, was supposed to look like this:

[ 1 0 0 0 0 ]
[ 0 1 0 0 0 ]
[ 0 0 1 0 0 ]
[ 0 0 0 1 0 ]
[ 0 0 0 0 1 ]

I don't know what happened to give me that wrong matrix,
and I'm glad Gene asked about it, because otherwise I wouldn't
have realized that I made an error. Anyway, this is how it works:

Look again at the first two matrices (the kernel and its
adjoint). I divide the number from each successive row of
the left column of the adjoint by the determinant, to get
the proper numbers of the inverse, then multiply each of those
quotients by each respective number in the top row of the
kernel, add all of those products, and put the sum in the
top row of the left column of the new "mapping" matrix.

Then I go thru the left column of the inverse again,
this time multiplying each row of that column by each
number in the second row of the kernel, and put that sum
down in the second row of the left column of the "mapping matrix".
And so on for all the other rows of that column.

Then repeat the same procedure for the second column-vector
of the adjoint; the third column-vector of the adjoint; etc.

So, using this example, the top row of the kernel is
the vector for 2:1 = [ 1 0 0 0 0 ], so the left column
of the inverse multiplied by this row gives (12/12)*1
and everything else times zero, so the sum is 1, which
is set down as the first number of the left column.

The next operation multiplies the left column of the
inverse with the second row of the kernel:

[ 12/12 ] * [-5 2 2 -1 0 ]
[ 19/12 ]
[ 28/12 ]
[ 34/12 ]
[ 41/12 ]

= -5 + 19/6 + 28/6 - 17/6 + 0 = 0

So a zero is set down in the second row of the left column.

And so on.

According to what I remember Graham saying, correlating
each row of the "mapping" matrix with the corresponding
row of the kernel, each column of this "mapping" matrix
shows which unison-vector is not tempered out by the
temperament shown in the corresponding column of the adjoint.

Thus, the "1" in the top row of the left column shows that
the 12-tET does not temper out the 2:1 (?), the "1" in the
second row of the second column shows that the

[ 0 ]
[ 1 ]
[ 4 ]
[ -2 ]
[ -1 ]

temperament does not temper out the 225:224, etc.

Gene or Graham, can you explain what's going on here?

And Paul: based on my summaries of _Harmonielehre_, do you
agree with me that this PB accurately describes Schoenberg's
theory up to at least p 184 of that book?

-monz

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Hi Gene,

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, January 19, 2002 1:40 PM
> Subject: [tuning-math] Re: deeper analysis of Schoenberg unison-vectors
>
>
> --- In tuning-math@y..., "joemonz" <joemonz@y...> wrote:
>
> > Also worth pointing out: 224/224 is neither a divisor
> > nor product of any of the other potential unison-vectors
> > <22:21, 33:32, 63:64, 81:80>, thus it satisfies the condition
> > we need for the unison-vector we're seeking, namely, that
> > it be independent of all the others.
>
> For independence, you need that any product to rational
> powers won't give you 225/224, or equivalently, that any
> product to integral powers will not give you any power of
> 225/224.

Can you explain this in a little more detail, by using
examples relevant to the Schoenberg PB I presented?

> You can check this with your linear algebra package,

Don't have one ... I do all this on an Excel spreadsheet.

> by taking the rank (or in this case, also the determinant)
> of the matrix of row vectors.

By "matrix of row vectors" you mean the kernel, right?
And what's the "rank"?

-monz

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>
> > Message 2798
> > From: "monz" <joemonz@y...>
> > Date: Sat Jan 19, 2002 2:40 pm
> > Subject: deeper analysis of Schoenberg unison-vectors
> /tuning-math/message/2798
> >
> > ...
> >
> > So as of p 184 in _Harmonielehre_, we can construct as system
> > valid for Schoenberg's theories, as follows:
> >
> > 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 |

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, January 20, 2002 12:08 AM
> Subject: Re: [tuning-math] deeper analysis of Schoenberg unison-vectors

UV map

> [ 1 0 0 0 0 ]
> [ 0 1 0 0 0 ]
> [ 0 0 1 0 0 ]
> [ 0 0 0 1 0 ]
> [ 0 0 0 0 1 ]

So in other words, the way Gene would write it:

h12(225/224) = h12(81/80) = h12(64/63) = h12(33/32) = 0
h12(2/1) = 1

But how do you label those other four columns? Well, for
the time being, I'll call them h0, g0, f0, and e0, respectively
from left to right, so that:

h0(2/1) = h0(81/80) = h0(63/64) = h0(33/32) = 0 , h0(225/224) = 1

g0(2/1) = g0(225/224) = g0(63/64) = g0(33/32) = 0 , g0(81/80) = 1

f0(2/1) = f0(225/224) = f0(81/80) = f0(33/32) = 0 , f0(64/63) = 1

e0(2/1) = e0(225/224) = e0(81/80) = e0(64/63) = 0 , e0(33/32) = 1

So, the 2nd and 4th column-vectors in the adjoint (h0 and f0,
respectively) define two versions of meantone:

- one (h0) in which 7 maps to the "minor 7th" = -2 generators,
and which tempers out all the UVs except 225/224;

- one (f0) in which 7 maps to the "augmented 6th" = +10 generators,
and which tempers out all the UVs except 64/63;

and both of which map 11 to the "perfect 4th" = -1 generator.

But what about the 3rd and 5th column-vectors in the adjoint
(g0 and e0, respectively)? What tunings are they? I don't get it.

And what relevance to these other mappings have to Schoenberg's
theory?

-monz

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Hi Graham and Gene,

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, January 20, 2002 2:39 AM
> Subject: Re: [tuning-math] deeper analysis of Schoenberg unison-vectors
>
>
> >
> > > Message 2798
> > > From: "monz" <joemonz@y...>
> > > Date: Sat Jan 19, 2002 2:40 pm
> > > Subject: deeper analysis of Schoenberg unison-vectors
> > /tuning-math/message/2798
> > >
> > > ...
> > >
> > > So as of p 184 in _Harmonielehre_, we can construct as system
> > > valid for Schoenberg's theories, as follows:
> > >
> > > 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 |
>
>
> > From: monz <joemonz@yahoo.com>
> > To: <tuning-math@yahoogroups.com>
> > Sent: Sunday, January 20, 2002 12:08 AM
> > Subject: Re: [tuning-math] deeper analysis of Schoenberg unison-vectors
>
> UV map
>
> > [ 1 0 0 0 0 ]
> > [ 0 1 0 0 0 ]
> > [ 0 0 1 0 0 ]
> > [ 0 0 0 1 0 ]
> > [ 0 0 0 0 1 ]
>
>
> So in other words, the way Gene would write it:
>
> h12(225/224) = h12(81/80) = h12(64/63) = h12(33/32) = 0
> h12(2/1) = 1
>
>
> But how do you label those other four columns? Well, for
> the time being, I'll call them h0, g0, f0, and e0, respectively
> from left to right, so that:
>
> h0(2/1) = h0(81/80) = h0(63/64) = h0(33/32) = 0 , h0(225/224) = 1
>
> g0(2/1) = g0(225/224) = g0(63/64) = g0(33/32) = 0 , g0(81/80) = 1
>
> f0(2/1) = f0(225/224) = f0(81/80) = f0(33/32) = 0 , f0(64/63) = 1
>
> e0(2/1) = e0(225/224) = e0(81/80) = e0(64/63) = 0 , e0(33/32) = 1
>
>
> So, the 2nd and 4th column-vectors in the adjoint (h0 and f0,
> respectively) define two versions of meantone:
>
> - one (h0) in which 7 maps to the "minor 7th" = -2 generators,
> and which tempers out all the UVs except 225/224;
>
> - one (f0) in which 7 maps to the "augmented 6th" = +10 generators,
> and which tempers out all the UVs except 64/63;
>
> and both of which map 11 to the "perfect 4th" = -1 generator.
>
>
> But what about the 3rd and 5th column-vectors in the adjoint
> (g0 and e0, respectively)? What tunings are they? I don't get it.
>
> And what relevance to these other mappings have to Schoenberg's
> theory?

OK, the 5th column is like the one you already explained to
me before, where 11 is mapped to a note 1 generator more than
the 12-tET value, like on a second keyboard tuned a quarter-tone
higher. So I understand that.

The most I can do with the 3rd column is this: the GCD is 2,
so that's equivalent to dividing the 8ve in half, right?
Which makes the tritone the interval of equivalence? So if
I divide the whole column by 2, I get [0 1 -2 -2 -1]. So
does this tell me how many generators away from 12-tET this
tuning maps 3, 5, 7, and 11? And exactly what *is* the generator?

Thanks.

-monz

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monz wrote:

> The most I can do with the 3rd column is this: the GCD is 2,
> so that's equivalent to dividing the 8ve in half, right?
> Which makes the tritone the interval of equivalence? So if
> I divide the whole column by 2, I get [0 1 -2 -2 -1]. So
> does this tell me how many generators away from 12-tET this
> tuning maps 3, 5, 7, and 11? And exactly what *is* the generator?

The house terminology is that you have a period of tritone, but the
interval of equivalence is still an octave. As for the generator, well,
either entry of 1 will map to it.

Graham

Hi Graham,

> From: <graham@microtonal.co.uk>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, January 20, 2002 12:27 PM
> Subject: [tuning-math] Re: deeper analysis of Schoenberg unison-vectors
>
>
> monz wrote:
>
> > The most I can do with the 3rd column is this: the GCD is 2,
> > so that's equivalent to dividing the 8ve in half, right?
> > Which makes the tritone the interval of equivalence? So if
> > I divide the whole column by 2, I get [0 1 -2 -2 -1]. So
> > does this tell me how many generators away from 12-tET this
> > tuning maps 3, 5, 7, and 11? And exactly what *is* the generator?
>
> The house terminology is that you have a period of tritone, but the
> interval of equivalence is still an octave.

OK, sorry ... I realize that I should have made that distinction
myself. But ... what *is* that distinction? Does "period of tritone"
mean that some form of tritone is the generator?

> As for the generator, well, either entry of 1 will map to it.

Hmmm ... but the signs are opposite, which I think is why I'm confused.
If the mapping of both 3 and 11 showed "1", then it would be more
understandable: 3 and 11 both map to the generator, which is somewhere
in the vicinity of a tritone ... that makes sense to me. But here
we have 1 and -1, respectively. I called this column g0, so from
the adjoint, we have g0(3)=2 and g0(11)=-4. Does the sign not
matter because the tritone splits the interval of equivalence
exactly in half?

And can you also explain how 5 and 7 both map to the same
number of generators in this case?

And yet again I ask:

> And what relevance to these other mappings have to Schoenberg's
> theory?

Still confused,

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> So in other words, the way Gene would write it:
>
> h12(225/224) = h12(81/80) = h12(64/63) = h12(33/32) = 0
> h12(2/1) = 1

Actually, I'd write it h12(2) = 12; by definition, in fact,
hn(2) = n.

> But what about the 3rd and 5th column-vectors in the adjoint
> (g0 and e0, respectively)? What tunings are they? I don't get it.

g0 is Twintone, aka Paultone. e0 sends everything to 0 mod 12, and is not a temperament.

> And what relevance to these other mappings have to Schoenberg's
> theory?

(1) It is consistent with twintone as well as meantone, and so is a 12-et theory

(2) It sends all your commas to 0 mod 12, so again it is a 12-et theory.

Hi Gene,

Regarding:

> So as of p 184 in _Harmonielehre_, we can construct as system
> valid for Schoenberg's theories, as follows:
>
> 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 |
>
>
> UV map
>
> [ 1 0 0 0 0 ]
> [ 0 1 0 0 0 ]
> [ 0 0 1 0 0 ]
> [ 0 0 0 1 0 ]
> [ 0 0 0 0 1 ]
>

Gene replied to my questions:

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, January 20, 2002 1:49 PM
> Subject: [tuning-math] Re: deeper analysis of Schoenberg unison-vectors
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > So in other words, the way Gene would write it:
> >
> > h12(225/224) = h12(81/80) = h12(64/63) = h12(33/32) = 0
> > h12(2/1) = 1
>
> Actually, I'd write it h12(2) = 12; by definition, in fact,
> hn(2) = n.

Right, I understand that, but it's for the values in the
*adjoint*. This is for the values in the "UV map" matrix,
which simply shows which UVs are tempered out and which
are not.

> > But what about the 3rd and 5th column-vectors in the
> > adjoint (g0 and e0, respectively)? What tunings are they?
> > I don't get it.
>
> g0 is Twintone, aka Paultone.

I still don't know what that is, and need to do some studying.

> e0 sends everything to 0 mod 12, and is not a temperament.

So then exactly what *is* that column-vector telling us?
Simply that this is 12-tET?

> > And what relevance to these other mappings have to
> > Schoenberg's theory?
>
> (1) It is consistent with twintone as well as meantone,
> and so is a 12-et theory
>
> (2) It sends all your commas to 0 mod 12, so again it
> is a 12-et theory.

Ah ... a few flickers illuminate the dark cave!

Two new questions, then:

1) Where is it written that 12-tET is consistent with both
meantone and twintone? This kind of stuff needs to be
in my Dictionary.

2) How can we see that "it sends all your commas to 0 mod 12"?
Is that what the adjoint's 5th column-vector ("e0") is saying?

Thanks!

-monz

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

> 1) Where is it written that 12-tET is consistent with both
> meantone and twintone? This kind of stuff needs to be
> in my Dictionary.

12-et is the only thing consistent with both twintone and meantone. The twintone wedgie is tt = [-2,4,4,-2,-12,11] and the meantone wedgie is mt = [-1,-4,-10,-12,13,-4]. We can take wedge products of the wedgie with either vals (leading to intervals) or intervals (leading to vals); in particular we have

mt ^ 64/63 = h12

tt ^ 81/80 = -h12

(according to the basis I used in my program). Hence h12 is the only thing which works with both of them.

Taking wedge products of wedgies with vals gives us stuff like

mt ^ h22 = 225/224

tt ^ h15 = 63/64

and so forth.

> 2) How can we see that "it sends all your commas to 0 mod 12"?
> Is that what the adjoint's 5th column-vector ("e0") is saying?

Right--it sends everything to 0 but 33/32, and it sends that to 12.

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

> OK, the 5th column is like the one you already explained to
> me before, where 11 is mapped to a note 1 generator more than
> the 12-tET value, like on a second keyboard tuned a quarter-tone
> higher.

Hmm . . . quarter-tones should _not_ figure into an analysis of a 12-
tone periodicity block. Gene, am I missing something?

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

> Hmm . . . quarter-tones should _not_ figure into an analysis of a 12-
> tone periodicity block. Gene, am I missing something?

Quarter-tones have nothing to do with it--it's telling us that 11 is being mapped inconsistently.

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

> > > > 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

Here is the contents of the Fokker parallelepiped defined by these
UVs, at one (arbitrary) position in the lattice:

cents numerator denominator
84.467 21 20
203.91 9 8
315.64 6 5
386.31 5 4
498.04 4 3
590.22 45 32
701.96 3 2
813.69 8 5
905.87 27 16
996.09 16 9
1088.3 15 8
1200 2 1

It's pretty clear that most of the consonances will straddle across
different instances of the PB, rather than being contained mostly
within this set of JI pitches.

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> Hi Graham,
>
>
> > From: <graham@m...>
> > To: <tuning-math@y...>
> > Sent: Sunday, January 20, 2002 12:27 PM
> > Subject: [tuning-math] Re: deeper analysis of Schoenberg unison-
vectors
> >
> >
> > monz wrote:
> >
> > > The most I can do with the 3rd column is this: the GCD is 2,
> > > so that's equivalent to dividing the 8ve in half, right?
> > > Which makes the tritone the interval of equivalence? So if
> > > I divide the whole column by 2, I get [0 1 -2 -2 -1]. So
> > > does this tell me how many generators away from 12-tET this
> > > tuning maps 3, 5, 7, and 11? And exactly what *is* the
generator?
> >
> > The house terminology is that you have a period of tritone, but
the
> > interval of equivalence is still an octave.
>
>
> OK, sorry ... I realize that I should have made that distinction
> myself. But ... what *is* that distinction? Does "period of
tritone"
> mean that some form of tritone is the generator?

The 1/2-octave can be thought of as a generator in the same way that
1 octave can be thought of as a generator in the usual cases, say
meantone for example. Normally we refer to the _other_ generator as
_the_ generator, and 1/2-octave or 1 octave or 1/n octave as the
period. It's the interval of repetition.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > 1) Where is it written that 12-tET is consistent with both
> > meantone and twintone? This kind of stuff needs to be
> > in my Dictionary.
>
> 12-et is the only thing consistent with both twintone and meantone.

I noticed that quite a long time ago. Practicing my 22-tET guitar is
a great lesson in twintone that I can apply back on my 12-tET guitar.
Same for 31-tET and meantone. Both guitars make me better at 12-tET.
Of course, 12-tET always sounds out-of-tune afterwards.

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

> > > But what about the 3rd and 5th column-vectors in the
> > > adjoint (g0 and e0, respectively)? What tunings are they?
> > > I don't get it.
> >
> > g0 is Twintone, aka Paultone.
>
>
> I still don't know what that is

http://www-math.cudenver.edu/~jstarret/22ALL.pdf

Especially the section, "Tuning the Decatonic Scale", page 22.

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

> I noticed that quite a long time ago. Practicing my 22-tET guitar is
> a great lesson in twintone that I can apply back on my 12-tET guitar.
> Same for 31-tET and meantone. Both guitars make me better at 12-tET.
> Of course, 12-tET always sounds out-of-tune afterwards.

The one problem with microtonality is that it does this.

Have you ever considered the [34, 54, 79, 96] version of twintone? You mentioned playing with someone who had a 34-et guitar, I think. It came up when I was playing with the twintone wedgie, since
tt ^ 4375/4374 = [34, 54, 79, 96]. I looked for more things I might have overlooked in this way, but didn't find much:
ennealimmal ^ 2048/2025 = h126, and
[-16,-2,-5,-6,-37,34] ^ 64/63 = h37.

In-Reply-To: <005201c1a1f5$e389c180$af48620c@dsl.att.net>
monz wrote:

> Hmmm ... but the signs are opposite, which I think is why I'm confused.
> If the mapping of both 3 and 11 showed "1", then it would be more
> understandable: 3 and 11 both map to the generator, which is somewhere
> in the vicinity of a tritone ... that makes sense to me. But here
> we have 1 and -1, respectively. I called this column g0, so from
> the adjoint, we have g0(3)=2 and g0(11)=-4. Does the sign not
> matter because the tritone splits the interval of equivalence
> exactly in half?

You can either have 3:1 or 11:1, tritone reduced, as the generator. Or
you could have either (3:1 the same as 1:11) or (1:3 the same as 11:1).
The latter makes sense, because 4:3 and 11:8 are already within the
tritone.

> And can you also explain how 5 and 7 both map to the same
> number of generators in this case?

5:4 is 386 cents, and 7:4 is 969 cents. Tritone-reduced, 7:4 is 369
cents. 386 and 369 cents are close enough to be approximated equal.

Note you can also go to <http://microtonal.co.uk/temper/>, choose the
"temperaments from unison vectors" option (without the 2:1) and plug in
your unison vectors. It happens to give generators larger than an octave
currently, but that's not important.

> And yet again I ask:
>
> > And what relevance to these other mappings have to Schoenberg's
> > theory?

That's for to you to work out.

Graham

Hi Paul,

Here is the earlier PB you calculated, before Minkowski reduction:

> Message 2839
> From: paulerlich <paul@s...>
> Date: Sun Jan 20, 2002 9:05pm
> Subject: Re: deeper analysis of Schoenberg unison-vectors
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > 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
>
> Here is the contents of the Fokker parallelepiped defined by these
> UVs, at one (arbitrary) position in the lattice:
>
> cents numerator denominator
> 84.467 21 20
> 203.91 9 8
> 315.64 6 5
> 386.31 5 4
> 498.04 4 3
> 590.22 45 32
> 701.96 3 2
> 813.69 8 5
> 905.87 27 16
> 996.09 16 9
> 1088.3 15 8
> 1200 2 1

I later wrote, about the Minkowski-reduced PB:

> Message 2862
> From: monz <joemonz@y...>
> Date: Mon Jan 21, 2002 3:02pm
> Subject: Re: Minkowski reduction (was: ...Schoenberg's rational
implications)
>
> ...
>
> My first question is: this is a 7-limit periodicity-block,
> so can you explain how the two 11-limit unison-vectors disappeared?
> I've been trying to figure it out but don't see it.

And I pose the same question here. I've added and subtracted
all pairs of UVs in this kernel, and I don't see anything that
should eliminate 11 from the PB. *Please* explain!
(I understand intuitively how it works, but I don't see it
happening in these numbers.)

-monz

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--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:

> Have you ever considered the [34, 54, 79, 96] version of twintone?
>You mentioned playing with someone who had a 34-et guitar, I think.

The 34-tET version of twintone? It's better than 12, but much worse
than 22.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
>
> > Have you ever considered the [34, 54, 79, 96] version of twintone?
> >You mentioned playing with someone who had a 34-et guitar, I think.

> The 34-tET version of twintone? It's better than 12, but much worse
> than 22.

Depends on what you are using the 7-limit stuff to do, I would think--it is sweeter so far as the 5-limit goes.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...>
wrote:
> >
> > > Have you ever considered the [34, 54, 79, 96] version of
twintone?
> > >You mentioned playing with someone who had a 34-et guitar, I
think.
>
> > The 34-tET version of twintone? It's better than 12, but much
worse
> > than 22.
>
> Depends on what you are using the 7-limit stuff to do, I would
think--it is sweeter so far as the 5-limit goes.

Then we're talking diaschismic, not twintone. Or if they're the same
thing, we need another word for "paultone".

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

> > > > Have you ever considered the [34, 54, 79, 96] version of
> twintone?
> > > >You mentioned playing with someone who had a 34-et guitar, I
> think.

> > Depends on what you are using the 7-limit stuff to do, I would
> think--it is sweeter so far as the 5-limit goes.
>
> Then we're talking diaschismic, not twintone. Or if they're the same
> thing, we need another word for "paultone".

Why? I mean the linear temperament [-2,4,4,-2,-12,11]. Add 245/243 to
the mix and you have 22-et, add 4375/4374 instead and you have 34-et in the version where 7/4 is 19.4 cents sharp, rather than being flat.
But they are both twintone.