> </tuning-math/message/44>

> From: "monz" <joemonz@y...>

> Date: Sun May 27, 2001 5:55 pm

> Subject: Re: Fwd: optimizing octaves in MIRACLE scale.

>

> (My quotes of Schoenberg are from the English translation

> of _Harmonielehre_ by Roy Carter, and the page numbers

> refer to that edition.)

>

>

> Schoenberg [p 23] posits the existences of two "forces", one

> pulling downward and one pulling upward around the tonic,

> which he illustrates as: F <- C -> G and likens to resistance

> against gravity. In mathematical terms, he is referring to

> the harmonic relationships of 3^-1 and 3^1, respectively.

>

>> [Schoenberg, p 24:]

>>

>> ...thus it is explained how the scale that finally emerged

>> is put together from the most important components of a

>> fundamental tone and its nearest relatives. These nearest

>> relatives are just what gives the fundamental tone stability;

>> for it represents the point of balance between their opposing

>> tendencies. This scale appears as the residue of the properties

>> of the three factors, as a vertical projection, as addition:

>

>

> Schoenberg then presents a diagram of the overtones and the

> resulting scale, which I have adaptated, adding the partial-numbers

> which relate all the overtones together as a single set:

>

> b-45

> g-36

> e-30

> d-27

> c-24

> a-20

> g-18 g-18

> f-16

> c-12 c-12

> f-8

>

>

> f c g a d e b

> 8 12 18 20 27 30 45

>

I will now lattice these pitches, using as nomenclature

for the notes my ASCII 72-EDO notation; legend:

- + ~cents alteration from 12-EDO

b # 100 [i.e., 12-EDO]

v ^ 50

< > 33&1/3

- + 16&2/3

no accidental 0 [12-EDO]

Here is a "standard triangular" 5-limit lattice

of this diatonic scale:

20--- 30--- 45

A- E- B-

/ \ / \ / \

/ \ / \ / \

8 ---12--- 18--- 27

F C G D

Look familiar? It should.

An ASCII representation of a Monzo lattice

of this scale looks like this:

D

/27

B- /

/45'-._ /

/ G

/ /18

E- /

30'-._ /

/ ' C

/ /12

A- /

20'-._ /

' F

8

>

>> [Schoenberg:]

>>

>> Adding up the overtones (omitting repetitions) we get the seven

>> tones of our scale. Here they are not yet arranged consecutively.

>> But even the scalar order can be obtained if we assume that the

>> further overtones are also in effect. And that assumption is

>> in fact not optional; we must assume the presence of the other

>> overtones. The ear could also have defined the relative pitch

>> of the tones discovered by comparing them with taut strings,

>> which of course become longer or shorter as the tone is lowered

>> or raised. But the more distant overtones were also a

>> dependable guide. Adding these we get the following:

>

>

>

> Schoenberg then extends the diagram to include the

> following overtones:

>

> fundamental partials

>

> F 2...12, 16

> C 2...11

> G 2...12

>

> (Note, therefore, that he is not systematic in his employment

> of the various partials.)

>

>

> Again, I adapt the diagram by adding partial-numbers:

>

> d-108

> c-99

> b-90

> a-81

> g-72

> f-66

> f-64

> (f-63)

> e-60

> d-54 d-54

> c-48 c-48

> b-45

> b-44

> (bb-42)

> a-40

> g-36 g-36 g-36

> f-32

> e-30

> (eb-28)

> d-27

> c-24 c-24

> a-20

> g-18 g-18

> f-16

> c-12 c-12

> f-8

>

>

> (eb) (bb)

> c d e f g a b c d e f g a b c d

> [44] [64]

> (28) (42) [66]

> 24 27 30 32 36 40 45 48 54 60 63 72 81 90 99 108

>

>

> (Note also that Schoenberg was unsystematic in his naming

> of the nearly-1/4-tone 11th partials, calling 11th/F by the

> higher of its nearest 12-EDO relatives, "b", while calling

> 11th/C and 11th/G by the lower, "f" and "c" respectively.

> This, ironically, is the reverse of the actual proximity

> of these overtones to 12-EDO: ~10.49362941, ~5.513179424,

> and ~0.532729432 Semitones, respectively).

>

>

> The partial-numbers are also given for the resulting scale

> at the bottom of the diagram, showing that 7th/F (= eb-28)

> is weaker than 5th/C (= e-30), and 7th/C (= bb-42) is weaker

> than 5th/G (= b-45).

>

> Also note that 11th/F (= b-44), 16th/F (= f-64) and 11th/C

> (= f-66) are all weaker still, thus I have included them in

> square brackets. These overtones are not even mentioned by

> Schoenberg.

Here is a triangular 5-limit lattice of this expanded

scale, showing 7- and 11-limit ratios in () and [] which

are near in pitch to the 5-limit ones as physically close

to them on the diagram. * indicates notes which are not

exact 8ve-equivalents even tho Schoenberg implied that

they are.

*81*----- ------90

40------60--(42)[44]45

20--(28)30------

A- E- B-

/ \ / \ / \

/ \ / \ / \

/ \ / \ / \

63[64][66]-*99*------72-----108

32------48-------36------54

16------24-------18------27

8------12------- ------

F C G D

And here is an ASCII-fied 11-limit Monzo lattice

showing where all of these pitches actually fall

according to my lattice formula, when prime-factored:

C^

/ \99

/ \

/ \ A

F^ \ /81

/ \[66] \ /

/ \ \ /

/ \ \ D

Bb^ \ \ /27 54 108

\[44] \ B- \ /

\ \/90-._ \ /

\ /\45 ' G ---------------F<

\ / \ /18 36 72 /63

\ E- \ / /

\/60-._ \ / /

/\30 ' C ---------------Bb<

/ \ /12 24 48 /(42)

A- \ / /

40'-._ \ / /

20 ' F ---------------Eb<

8 16 32 [64] (28)

Of course, this description of the scale is only valid

where "C" is the "tonic". Given Schoenberg's ideas about

"pantonality", it will be difficult if not impossible

to determine what the "tonic" is. Perhaps an interval

analysis of the composition can reveal something. Of

course, the tonic will be dynamically shifting all the

time.

Also, this explanation was intended to describe the

rational implications of the *diatonic* scale, allowing

only a few of the chromatic pitches. For the full

chromatic scale, Schoenberg's later 13-limit system

presented in "Problems of Harmony" must be consulted.

Could anyone out there do some periodicity-block

calculations on this theory and say something about that?

Altho I analyze this tuning exactly according to Schoenberg's

analysis (i.e., 8ve-specific), he himself considered it

to represent 8ve-invariant pitches. His unison-vectors

are 33:32, 45:44 64:63, and 81:80. His unusual explanation

of Eb< (28) and Bb< (42) also makes 16:15 some type of

special interval, if not a unison-vector.

-monz

http://www.monz.org

"All roads lead to n^0"

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

>

> Could anyone out there do some periodicity-block

> calculations on this theory and say something about that?

It's pretty clear that Schoenberg's theory implies a 12-tone

periodicity block.

I was looking again at the post I sent here regarding the

math of Schoenberg's tuning ideas:

/tuning-math/messages/516?expand=1

I constructed matrices of the unison-vectors mentioned

by Schoenberg. The matrix of the later 13-limit system

(explained by Schoenberg in 1934) expectedly has a determinant

of 12. But interestingly, the 11-limit matrix (as described

in 1911 in _Harmonielehre_) has a determinant of 7.

I found this interesting first of all because Schoenberg,

in _Harmonielehre_, defines the 7-tone diatonic scale in

familiar 5-limit terms, then introduces the 11-limit ratios

in an attempt to explain the origin of the chromatic notes.

But his explanation is somewhat vague and incomplete, and

introduces a notational inconsistency about which I say

more below, so it's not really a surprise that the determinant

of this periodicity-block is not 12.

For the unison-vectors, I specified the relationships between

pairs of notes which Schoenberg described as equivalent.

Unison-vector matrices:

1911 _Harmonielehre_ 11-limit system

( 1 0 0 1 ) = 33:32

(-2 0 -1 0 ) = 64:63

( 4 -1 0 0 ) = 81:80

( 2 1 0 -1 ) = 45:44

Determinant = 7

1934 _Problems of Harmony_ 13-limit system

(-2 0 0 -1 1 ) = 104:99

( 2 0 -1 0 1 ) = 117:112

(-2 0 -1 0 0 ) = 64:63

( 4 -1 0 0 0 ) = 81:80

( 2 1 0 -1 0 ) = 45:44

Determinant = 12

The 13-limit system gives me no surprises. But the 11-limit

system is intriguing. I have noted many times (as in my book

and in that post) that Schoenberg was inconsistent in his

naming of the pitches of this system.

From my tuning-math post:

>> (Note also that Schoenberg was unsystematic in his naming

>> of the nearly-1/4-tone 11th partials, calling 11th/F by the

>> higher of its nearest 12-EDO relatives, "b", while calling

>> 11th/C and 11th/G by the lower, "f" and "c" respectively.

>> This, ironically, is the reverse of the actual proximity

>> of these overtones to 12-EDO: ~10.49362941, ~5.513179424,

>> and ~0.532729432 Semitones, respectively).

What I found is that eliminating this inconsistency, i.e.,

calling 11th/F a "Bb" instead of "B", also destroys the

periodicity-block aspect of this system. The 45:44 unison-vector

which results from calling that note "B" is *necessary*

in order to define the 7-tone periodicity-block. Calling it

a "Bb" removes the 45:44 UV and replaces it with the 22:21 UV

already found as a result of combining the existing 33:32

and 64:63 UVs for the various "F"s. Thus, the matrix has

only 3 UVs and lacks the remaining one which is necessary

to define a PB.

But why do I get a determinant of 7 for the 11-limit system?

Schoenberg includes Bb and Eb as 7th harmonics in his description,

which gives a set of 9 distinct pitches. But even when

I include the 15:14 unison-vector, I still get a determinant

of -7. And if I use 16:15 instead, then the determinant

is only 5.

Can someone explain what's going on here, and what candidates

may be found for unison-vectors by extending the 11-limit system,

in order to define a 12-tone periodicity-block? Thanks.

(and Merry Christmas to all)

love / peace / harmony ...

-monz

http://www.monz.org

"All roads lead to n^0"

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

> Can someone explain what's going on here, and what candidates

> may be found for unison-vectors by extending the 11-limit system,

> in order to define a 12-tone periodicity-block? Thanks.

See if this helps;

We can extend the set {33/32,64/63,81/80,45/44} to an 11-limit notation in various ways, for instance

<56/55,33/32,65/63,81/80,45/44>^(-1) = [h7,h12,g7,-h2,h5]

where g7 differs from h7 by g7(7)=19. Using this, we find the corresponding block is

(56/55)^n (33/32)^round(12n/7) (64/63)^n (81/80)^round(-2n/12)

(45/44)^round(5n/7), or 1-9/8-32/27-4/3-3/2-27/16-16/9; the Pythagorean scale. We don't need anything new to find a 12-note scale; we get

1--16/15--9/8--32/27--5/4--4/3--16/11--3/2--8/5--5/3--19/9--15/8

or variants, the variants coming from the fact that 12 is even, by using 12 rather than 7 in the denominator.

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

> 1--16/15--9/8--32/27--5/4--4/3--16/11--3/2--8/5--5/3--19/9--15/8

This should be 16/9 rather than 19/9, of course.

Help!

I set up an Excel spreadsheet to calculate the notes of

a periodicity-block according to Gene's formula as expressed here:

> Message 2164

> From: "genewardsmith" <genewardsmith@j...>

> Date: Wed Dec 26, 2001 3:27 am

> Subject: Re: lattices of Schoenberg's rational implications

/tuning-math/message/2164

>

>

> We can extend the set {33/32,64/63,81/80,45/44} to an

> 11-limit notation in various ways, for instance

>

> <56/55,33/32,65/63,81/80,45/44>^(-1) = [h7,h12,g7,-h2,h5]

>

> where g7 differs from h7 by g7(7)=19. Using this,

> we find the corresponding block is

>

> (56/55)^n (33/32)^round(12n/7) (64/63)^n (81/80)^round(-2n/12)

> (45/44)^round(5n/7), or 1-9/8-32/27-4/3-3/2-27/16-16/9;

> the Pythagorean scale. We don't need anything new to find

> a 12-note scale; we get

>

> 1--16/15--9/8--32/27--5/4--4/3--16/11--3/2--8/5--5/3--16/9--15/8

>

> or variants, the variants coming from the fact that 12 is even,

> by using 12 rather than 7 in the denominator.

> 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,

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?

-monz

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--- 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.

----- Original Message -----

From: genewardsmith <genewardsmith@juno.com>

To: <tuning-math@yahoogroups.com>

Sent: Sunday, January 20, 2002 2:08 PM

Subject: [tuning-math] Re: lattices of Schoenberg's rational implications

> --- 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.

OK ... I still have lots to learn about torsion.

> In fact, you map to the cyclic group C12 of order 12,

Huh?

> and the twelveth [_sic_: twelfth] power (or additively,

> twelve times) anything is the identity.

OK, I can follow that.

> > 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.

So what's the secret to finding that?

And so then is there or is there not any value in calculating

the kernel which has 2:1 in it? If yes, then what? If no,

then why not?

Don't we need determinants <-1 and >1 in order to have a

denominator with which to find the JI scale?

-monz

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

> > If you want a notation, yes. One which makes the matrix

> > unimodular, ie with determinant +-1.

>

>

> So what's the secret to finding that?

Forget it. I don't know why you want to bother with Gene's "notation"

here. The "notation" would allow you to specify just ratios

unambiguously. However, I think the only reality for Schoenberg's

system is a tuning where there is ambiguity, as defined by the kernel

<33/32, 64/63, 81/80, 225/224>. BTW, is this Minkowski-reduced?

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

However, I think the only reality for Schoenberg's

> system is a tuning where there is ambiguity, as defined by the kernel

> <33/32, 64/63, 81/80, 225/224>. BTW, is this Minkowski-reduced?

Nope. The honor belongs to <22/21, 33/32, 36/35, 50/49>.

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

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

>

> However, I think the only reality for Schoenberg's

> > system is a tuning where there is ambiguity, as defined by the

kernel

> > <33/32, 64/63, 81/80, 225/224>. BTW, is this Minkowski-reduced?

>

> Nope. The honor belongs to <22/21, 33/32, 36/35, 50/49>.

Awesome. So this suggests a more compact Fokker parallelepiped

as "Schoenberg PB" -- here are the results of placing it in different

positions in the lattice (you should treat the inversions of these as

implied):

0 1 1

84.467 21 20

203.91 9 8

315.64 6 5

386.31 5 4

470.78 21 16

617.49 10 7

701.96 3 2

786.42 63 40

933.13 12 7

968.83 7 4

1088.3 15 8

0 1 1

119.44 15 14

203.91 9 8

315.64 6 5

386.31 5 4

470.78 21 16

617.49 10 7

701.96 3 2

786.42 63 40

933.13 12 7

968.83 7 4

1088.3 15 8

0 1 1

119.44 15 14

155.14 35 32

301.85 25 21

386.31 5 4

470.78 21 16

617.49 10 7

701.96 3 2

772.63 25 16

884.36 5 3

968.83 7 4

1088.3 15 8

0 1 1

84.467 21 20

155.14 35 32

266.87 7 6

386.31 5 4

470.78 21 16

582.51 7 5

701.96 3 2

737.65 49 32

884.36 5 3

968.83 7 4

1053.3 147 80

Gene, Paul, Graham,

> Message 2850

> From: "paulerlich" <paul@s...>

> Date: Sun Jan 20, 2002 10:48 pm

> Subject: Re: lattices of Schoenberg's rational implications

/tuning-math/message/2850

>

>

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

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

> >

> > > However, I think the only reality for Schoenberg's

> > > system is a tuning where there is ambiguity, as defined by

> > > the kernel <33/32, 64/63, 81/80, 225/224>. BTW, is this

> > > Minkowski-reduced?

> >

> > Nope. The honor belongs to <22/21, 33/32, 36/35, 50/49>.

>

> Awesome. So this suggests a more compact Fokker parallelepiped

> as "Schoenberg PB" -- here are the results of placing it in different

> positions in the lattice (you should treat the inversions of these as

> implied):

>

> <tables of scales snipped>

Paul, thanks!!! This *is* awesome! Just what I've been

waiting and hoping for since that posting on Christmas day.

Actually, it's something I've been trying to achieve since

about 1988 or so. Fantastic. Thanks for your help too, Gene,

and for your explanations, Graham.

Lattices and webpages to follow soon!!

And then, some serious work on retuning Schoenberg MIDIs!

-monz

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returning to an old subject ...

during a big discussion i instigated concerning

possible periodicity-blocks which might describe

Schoenberg's 1911 12-tET theory as posited in his

_Harmonielehre_,

--- In tuning-math@yahoogroups.com, "genewardsmith"

<genewardsmith@j...> wrote:

> Message 2848

> From: "genewardsmith" <genewardsmith@j...>

> Date: Sun Jan 20, 2002 7:20 pm

> Subject: Re: lattices of Schoenberg's rational implications

>

>

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

>

> > However, I think the only reality for Schoenberg's

> > system is a tuning where there is ambiguity, as defined

> > by the kernel <33/32, 64/63, 81/80, 225/224>. BTW,

> > is this Minkowski-reduced?

>

> Nope. The honor belongs to <22/21, 33/32, 36/35, 50/49>.

a few messages after that,

Paul Erlich posted four different possible lattices

based on those unison-vectors.

i've made a rectangular-style lattice based on those

same four unison-vectors, using the Tonalsoft software.

this is what the software gave me:

~cents.. ratio

0000.000 1/1

0084.467 21/20

0231.174 8/7

0266.871 7/6

0386.314 5/4

0498.045 4/3

0582.512 7/5

0701.955 3/2

0813.686 8/5

0884.359 5/3

0968.826 7/4

1115.533 40/21

here is a screen-shot of the actual Tonalsoft lattice:

/tuning-math/files/monz/

tuning-math-2848-minkowski-reduced-schoenberg-12et.jpg

... delete the line-break in that URL, or use this one:

-monz