Tuning-Math Digests messages 3350 - 3374

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 4

Previous Next

3000 3050 3100 3150 3200 3250 3300 3350 3400 3450 3500 3550 3600 3650 3700 3750 3800 3850 3900 3950

3350 - 3375 -



top of page bottom of page down


Message: 3350

Date: Fri, 18 Jan 2002 22:37:35

Subject: Re: Carl's Mathworld-Complete Correspondence Theorem

From: genewardsmith

--- In tuning-math@y..., "clumma" <carl@l...> wrote:

> MCCT- For all pairs of terms [t_i, t_(i+1)], from finite
> alphabet [gamma], it is possible to select a pair of
> terms [m_i, m_(i+1)] from the Mathworld website which
> are identical.

My brother Robin was complaining to me about this, and brought up the word "pencil". It turned out he was *not* referring to

http://mathworld.wolfram.com/pencil.html *

but had picked a word at random. He was pretty triumphant when he
found out it had also been made off with by mathematicians, though I
think the mathematical use is older than those yellow things.


top of page bottom of page up down


Message: 3351

Date: Fri, 18 Jan 2002 22:40:59

Subject: Re: Carl's Mathworld-Complete Correspondence Theorem

From: clumma

>My brother Robin was complaining to me about this, and brought
>up the word "pencil". It turned out he was *not* referring to
> 
> http://mathworld.wolfram.com/pencil.html *
> 
>but had picked a word at random. He was pretty triumphant when he
>found out it had also been made off with by mathematicians, though
>I think the mathematical use is older than those yellow things.

It says 1960...?

-Carl


top of page bottom of page up down


Message: 3352

Date: Fri, 18 Jan 2002 22:46:57

Subject: Re: Carl's Mathworld-Complete Correspondence Theorem

From: paulerlich

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> >My brother Robin was complaining to me about this, and brought
> >up the word "pencil". It turned out he was *not* referring to
> > 
> > http://mathworld.wolfram.com/pencil.html *
> > 
> >but had picked a word at random. He was pretty triumphant when he
> >found out it had also been made off with by mathematicians, though
> >I think the mathematical use is older than those yellow things.
> 
> It says 1960...?
> 
> -Carl

Cremona was 1960. Desargues lived 1591-1661.


top of page bottom of page up down


Message: 3353

Date: Sat, 19 Jan 2002 09:37:24

Subject: The consonant heptads of Tweedledee and Tweedeldum

From: genewardsmith

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

> 16. Tweedledee
> 
> [3, 5, 9, 4, 1, 6, -4, 7, -8, -20]
> 
> ets 15
> 
> [55/54, 56/55, 100/99, 121/120, 126/125, 3025/3024]
> 
> [[0, -3, -5, -9, -4], [1, 2, 3, 4, 4]]
> 
> [.1329702752, 1]
> 
> a = 1.9946/15 = 159.5643303 cents
> 
> badness   255.7850727
> rms   22.12985764
> g   4.342481185

The heptad in 15-et is 2222223 in its various flavors, and this is pretty much a 15-et system, though [7,7,7,7,7,7,11] in the 53-et and
[9,9,9,9,9,9,14] in the 68-et are interesting alternatives.

> 17. Tweedledum
> 
> [3, 5, -6, 4, 1, -18, -4, -28, -8, 32]
> 
> [55/54, 64/63, 100/99, 121/120, 176/175, 385/384]
> 
> ets 7, 15, 22
> 
> [[0, -3, -5, 6, -4], [1, 2, 3, 2, 4]]
> 
> [.1357721305, 1]
> 
> a = 2.9870/22 = 162.9265567 cents
> 
> badness   262.2914819
> rms   11.79393546
> g
  6.430951940

The 22-et version of this is 3333334, but 8/59 is closer to the rms
optimal generator, and has the same 7s and 11s as the 118-et; this
version of it goes [8,8,8,8,8,8,11].


top of page bottom of page up down


Message: 3354

Date: Sat, 19 Jan 2002 09:54:15

Subject: Tweedle

From: genewardsmith

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

This one is related to Dum and Dee, so I guess it's just Tweedle.

> 11. Tweedle
> 
> [6, 10, 10, 8, 2, -1, -8, -5, -16, -12]
> 
> [50/49, 55/54, 99/98, 100/99, 121/120, 540/539, 9801/9800]
> 
> ets 22
> 
> [[0, -3, -5, -5, -4], [2, 4, 6, 7, 8]]
> 
> [.1375489239, 1/2]
> 
> a = 3.0261/22 = 165.0587086 cents
> 
> badness   238.7261371
> rms   11.89273384
> g   6.047431569


top of page bottom of page up down


Message: 3355

Date: Sat, 19 Jan 2002 07:52:11

Subject: Re: A top 20 11-limit superparticularly generated linear temperament list

From: monz

Hi Gene,

> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, January 18, 2002 8:22 PM
> Subject: [tuning-math] A top 20 11-limit superparticularly generated
linear temperament list
>
>
> I look all the 11-limit superparticulars >= 49/48 and
> found all the 11-limit linear temperaments they generated;
> there turned out to be 319 of them. The following is
> the top 20 in terms of low (logarithmically flat) badness,
> plus a special guest star "Monzo" which is what 45/44,
> 64/63 and 81/80 will give you. If Joe objects I will
> quit calling it that.


Cool!  I never object to having my name on a tuning thing
(something like the old show-biz dictum "even bad publicity
is good publicity").

But ... it would be really nice if you could explain, as
only tw examples, exactly what all this means.  Since I've
already played around with these particular unison-vectors,
explaining what you did here would help me a lot to
understand the rest of your work.


>
> 19. Monzoid
>
> [1, 4, -2, -1, 4, -6, -5, -16, -16, 4]
>
> [55/54, 64/63, 81/80, 385/384]
>
> ets 5, 7
>
> [[0, -1, -4, 2, 1], [1, 2, 4, 2, 3]]
>
> [.4181947520, 1]
>
> a = 5.0183/12 = 501.8337024 cents
>
> badness   269.9708171
> rms   39.86372247
> g   3.150963571
>
> ...
>
> Number 46 Monzo
>
>
> [64/63, 81/80, 100/99, 176/175]
>
> ets 7, 12
>
> [[0, -1, -4, 2, -6], [1, 2, 4, 2, 6]]
>
> [.4190088422, 1]
>
> a = 5.0281/12 = 502.8106107 cents
>
> badness   312.5112733
> rms   28.87226550
> g   4.174754057




-monz










_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 3356

Date: Sat, 19 Jan 2002 08:01:26

Subject: Re: ERROR IN CARTER'S SCHOENBERG (Re: badly tuned remote overtones)

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, January 18, 2002 1:04 PM
> Subject: [tuning-math] ERROR IN CARTER'S SCHOENBERG (Re: badly tuned
remote overtones)
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > I think you misunderstand me, Paul.  I just mean that there's
> > probably a good chance that at least some of the time, Schoenberg
> > thought of the "Circle of 5ths" in a meantone rather than a
> > Pythagorean sense.
>
> I doubt it. For Schoenberg, the circle of 5ths closes after 12
> fifths -- which is closer to being true in Pythagorean than in most
> meantones.


I understand that, Paul ... 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.)



> > This reference to you is only meant to credit you for opening
> > my eyes to the strong meantone basis behind a good portion of
> > the "common-practice" European musical tradition.
>
> OK -- but you're confusing two completely unrelated facts -- that
> 128:125 is just in 1/4-comma meantone, and that 128:125 is one of the
> simplest unison vectors for defining a 12-tone periodicity block.


OK, I'm willing to take note of your point, but ... *why* are
these two facts "completely unrelated"?  Isn't it possible that
there *is* some relation between them that no-one has noticed
before?



-monz








_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 3357

Date: Sat, 19 Jan 2002 08:20:00

Subject: Re: A top 20 11-limit superparticularly generated linear temperament list

From: monz

> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Saturday, January 19, 2002 7:52 AM
> Subject: Re: [tuning-math] A top 20 11-limit superparticularly generated
linear temperament list
>

>
> Hi Gene,
>
> > From: genewardsmith <genewardsmith@xxxx.xxx>
> > To: <tuning-math@xxxxxxxxxxx.xxx>
> > Sent: Friday, January 18, 2002 8:22 PM
> > Subject: [tuning-math] A top 20 11-limit superparticularly generated
> linear temperament list
> >
> >
> But ... it would be really nice if you could explain, as
> only tw examples, exactly what all this means.  Since I've
> already played around with these particular unison-vectors,
> explaining what you did here would help me a lot to
> understand the rest of your work.
> >
> > Number 46 Monzo
> >
> >
> > [64/63, 81/80, 100/99, 176/175]
> >
> > ets 7, 12
> >
> > [[0, -1, -4, 2, -6], [1, 2, 4, 2, 6]]
> >
> > [.4190088422, 1]
> >
> > a = 5.0281/12 = 502.8106107 cents
> >
> > badness   312.5112733
> > rms   28.87226550
> > g   4.174754057
>
>
>
>
> -monz


OK, I gave this a whirl thru my spreadsheet and this is
what I got:


kernel

  2  3  5  7 11   unison vectors   ~cents

[ 1  0  0  0  0 ]  =    2:1     0
[ 4  0 -2 -1  1 ]  =  176:175   9.864608166
[ 2 -2  2  0 -1 ]  =  100:99   17.39948363
[ 6 -2  0 -1  0 ]  =   64:63   27.2640918
[-4  4 -1  0  0 ]  =   81:80   21.5062896

adjoint

[ 0  0  0 -0  0 ]
[ 0  1  1 -1  0 ]
[ 0  4  4 -4  0 ]
[ 0 -2 -2  2  0 ]
[ 0  6  6 -6  0 ]

determinant = | 0 |


mapping of Ets (top row above) to Uvs

[ 1    1/3    2/3   -2/3  0 ]
[ 4  2&2/3  2&2/3 -2&2/3  0 ]
[ 2  1&1/3  1&1/3 -1&1/3  0 ]
[ 6  4      4     -4      0 ]
[-4 -2&2/3 -2&2/3  2&2/3  0 ]


I don't really understand what this is saying either.

(Many of the "0"s were actually given by Excel as
tiny numbers such as "2.22045 * 10^-16", which is
what it actually gave as the determinant.)



-monz







_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 3358

Date: Sat, 19 Jan 2002 10:05:03

Subject: Re: Schoenberg's 1927/34 "Problems of Harmony" theory

From: monz

Graham, Gene, Paul,



Can you please verify that what I said here is correct,
or fix and explain if it's not?


> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, January 18, 2002 2:52 AM
> Subject: [tuning-math] Schoenberg's 1927/34 "Problems of Harmony" theory
>
>
> ...
>
> > [Schoenberg] 1934 _Problems of Harmony_ 13-limit system
>
> ...
>
> matrix
>
> [  1  0  0  0  0 ]  =    2:1
> [ -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
>
>
> adjoint:
>
> [ 12  0  0   0  0  0 ]
> [ 19 -1  1  -1  1  1 ]
> [ 28 -4  4  -4 -8  4 ]
> [ 34  2 -2 -10 -2 -2 ]
> [ 42 -6  6  -6 -6 -6 ]
> [ 44  4  8  -8 -4 -4 ]
>
>
> OK, I see that the first column-vector gives a typical
> 12-EDO mapping.  Interestingly, now the 11th harmonic
> is mapped to 42 degrees of 12-EDO -- if "C" is n^0, this
> is "F#", the opposite of how Schoenberg mapped it in 1911
> (as "F"), and indeed this is exactly how Schoenberg now
> notates 11 in "Problems of Harmony".  And the "new" 13th
> harmonic is mapped to the 44th degree ("Ab"), which again
> is how Schoenberg notates it.
>
>
> As for the other column-vectors:
>
> I can see that all of them map 3 = 1 generator, the "5th",
> typical of both meantone and Pythagorean.
>
> Columns 2, 3, 4, and 6 map 5 = 4 generators, also typical
> of meantone, and the 5th column maps 5 = -8 generators,
> typical of a Pythagorean-based schismic temperament.
>
> Columns 2, 3, 5, and 6 map 7 to -2 generators, the "minor 7th",
> the closest approximation in Pythagorean.  Column 4 maps
> 7 = 10 generators, the "augmented 6th", which is a typical
> meantone mapping.
>
> Columns 2, 3, and 4 map 11 = 6 generators, the "tritone" or
> "augmented 4th", a meantone-like approximation.  Columns
> 5 and 6 map 11 = -6 generators, the Pythagorean "diminished 5th",
> again only an approximation.
>
> Columns 2, 5, and 6 map 13 = -4 generators, the "minor 6th",
> a meantone-like approximation, and columns 3 and 4 map
> 13 = 8 generators, the "augmented 5th", a Pythagorean
> approximation.


----------

I appreciated Graham's explanation of this ...


> > adjoint
> >
> > [ 12  0   0   0   0 ]
> > [ 19 -1   0   0   3 ]
> > [ 28 -4   0   0   0 ]
> > [ 34  2   0 -12  -6 ]
> > [ 41  1  12   0  -3 ]
> >
> > determinant = | 12 |
>
> See the third column has all zeros except for a 12 right at the bottom.
> That means, trivially, it has a greatest common divisor of 12.  (We don't
> count zeros in the gcd.)  As a rule of thumb, the GCD of a column tells
you
> how many *equal* steps the equivalence interval is being divided into.  In
> this case, we're dividing the octave into 12 equal steps.
>
> Dividing the octave this way is the same as defining a new period to be a
> fraction of the original equivalence interval.  Divide the whole column
> through by the GCD, and you get the mapping within the period.  In this
> case, that gives [0 0 0 0 1].  The first zero tells you that the octave
> takes the same value as it does in 12-equal.  Not much of a surprise.  The
> next four zeros tell you that 3:1, 5:1 and 7:1 are also taken from
12-equal.
> And the 1 in the last column tells you that 11:1 is one generator step
away
> from it's value in 12-equal.  For 11:1 to be just, you'd have a 51 cent
> generator.
>
> This is a fairly trivial example, and not much use as a temperament.  But
> you could realise it by having two keyboards tuned a quartertone apart.
To
> play an 11-limit otonality, you'd have C-E-G-Bb-D on one keyboard, and F
on
> the other.


... and would appreciate some insight into how it would apply
to my description of the 1927/34 13-limit system shown at the top
of this post.

Also, Graham's explanation here is of the matrix which I believe
is a good candidate for modeling Schoenberg's 1911 theory.  Can
someone please expand on what Graham says and show what it might
have to do with Schoenberg's adoption of 12-EDO as his preferred tuning?

I've shown that Schoenberg rejected microtonality on practical grounds
(lack of instruments, impact on his own financial situation, etc.),
but his acceptance of 12-EDO eventually became so strong that,
in addition to the known numerological motives behind many of his
choices, my hunch is that there's probably a strong aesthetic motive
as well, and an explanation like this would help to formulate that.


Just trying to make sure that I understand this stuff.
Thanks.



-monz







_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 3359

Date: Sat, 19 Jan 2002 11:40:56

Subject: deeper analysis of Schoenberg unison-vectors

From: monz

> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> 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@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> 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







_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 3360

Date: Sat, 19 Jan 2002 20:06:58

Subject: Re: deeper analysis of Schoenberg unison-vectors

From: joemonz

--- 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


top of page bottom of page up down


Message: 3361

Date: Sat, 19 Jan 2002 20:56:56

Subject: Re: deeper analysis of Schoenberg unison-vectors

From: joemonz

--- 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


top of page bottom of page up down


Message: 3362

Date: Sat, 19 Jan 2002 20:56:22

Subject: Re: A top 20 11-limit superparticularly generated linear temperament list

From: genewardsmith

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

> But ... it would be really nice if you could explain, as
> only tw examples, exactly what all this means.  Since I've
> already played around with these particular unison-vectors,
> explaining what you did here would help me a lot to
> understand the rest of your work.
> 
> 
> >
> > 19. Monzoid

This was number 19 on the list, in terms of a badness measure.

> > [1, 4, -2, -1, 4, -6, -5, -16, -16, 4]

This is the "wedgie".
When standardized, there is a unique wedgie corresponding to each
(non-torsion, non-equal) temperment. This is a linear temperament
wedgie; in the 11-limit the planar temperament wedgies also have ten
dimensions; however this wedgie is computed from three unisons or two
ets, whereas a planar would be computed from two unisons or three ets.

> > [55/54, 64/63, 81/80, 385/384]

These are all the 11-limit superparticulars equal to or less than
49/48 which are commas of the temperament--meaning they are tempered
out. Since there are four of them, there is a linear dependency, but
we can generate Monzoid from three independent ones.


> > ets 5, 7

These are "standard" ets, which round off to the nearest integer when
mapping primes; 12 is not on the list, but h5+h7 would be if I listed
anything "nonstandard".

> > [[0, -1, -4, 2, 1], [1, 2, 4, 2, 3]]

This is the period matrix, in a way easier to print than as a
4x2 matrix. The first list is the first column, giving maps to primes
of the generator, the second column is the octaves.

> > [.4181947520, 1]


> > a = 5.0183/12 = 501.8337024 cents

These are the two generators, the second being merely an octave, and
the first being a slightly sharp fourth.

> > badness   269.9708171
> > rms   39.86372247
> > g   3.150963571

"Badness" is the flat badness measure, "rms" is an average value for
how much, in cents, the 11-limit consonances are off (40 cents!), and
g is the average number of generator steps to get to a consonance (a
mere 3.)


> > Number 46 Monzo
> >
> >
> > [64/63, 81/80, 100/99, 176/175]

45/44 does not appear only because 45/44 > 49/48, which I used as a
cut-off.

> > ets 7, 12

This time, the "standard" h12 12-et map makes its appearance.

> > [[0, -1, -4, 2, -6], [1, 2, 4, 2, 6]]
> >
> > [.4190088422, 1]
> >
> > a = 5.0281/12 = 502.8106107 cents

A small difference in the size of the optimal generator, because Monzo
maps 11 differently than Monzoid.

> > badness   312.5112733
> > rms   28.87226550
> > g   4.174754057

The different 11-map makes the 11-limit more accurate, but it takes
more steps on average because 11 maps to -6 and not 1.


top of page bottom of page up down


Message: 3363

Date: Sat, 19 Jan 2002 00:09:37

Subject: Re: Hi Dave K.

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> Thanks, but I don't think RMS will work. That implies a Euclidean 
> metric, but a "taxicab" metric seems to be what we want here.

Yes of course. Sorry. Just replace every ocurrence of "rms" with 
"taxicab" in what I wrote.


top of page bottom of page up down


Message: 3364

Date: Sat, 19 Jan 2002 21:34:47

Subject: Re: deeper analysis of Schoenberg unison-vectors

From: genewardsmith

--- 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?


top of page bottom of page up down


Message: 3365

Date: Sat, 19 Jan 2002 04:22:05

Subject: A top 20 11-limit superparticularly generated linear temperament list

From: genewardsmith

I look all the 11-limit superparticulars >= 49/48 and found all the
11-limit linear temperaments they generated; there turned out to be
319 of them. The following is the top 20 in terms of low
(logarithmically flat) badness, plus a special guest star "Monzo"
which is what 45/44, 64/63 and 81/80 will give you. If Joe objects I
will quit calling it that. 

"Arabic", by the way, came in #32 but clearly would be much higher if
we forgot about 7. Things to note are temperaments which don't seem to
have much to do with "good" ets and temperaments which are close
relatives of other temperaments. The first on the list,
Hemiennealimmal, could certainly claim to be able to produce authentic
Partch tunings of the 11-limit, and it would be interesting to check
in how many keys 72 notes tempered in this way could play the Partch
43-tone scale to extreme accuracy. It is also interesting to note how
many of the top temperaments are compatible with 72.

1. Hemiennealimmal

[36, 54, 36, 18, 2, -44, -96, -68, -145, -74]

[2401/2400, 3025/3024, 4375/4374, 9801/9800]

ets 72, 198, 270, 342, 612

[[0, 2, 3, 2, 1], [18, 12, 17, 34, 54]]

[.4591217954, 1/18]

a = 33.0568/72 = 280.9825/612 = 550.9491544 cents

badness   78.02778100
rms   .1987978829
g   36.


2. Miracle

[6, -7, -2, 15, -25, -20, 3, 15, 59, 49]

[225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400, 3025/3024]

ets 10, 31, 41, 72

[[0, 6, -7, -2, 15], [1, 1, 3, 3, 2]]

[.9722688696e-1, 1]

a = 7.0003/72 = 116.6722643 cents

badness   125.5016755
rms   1.901465778
g   12.35198075


3. Octoid

[24, 32, 40, 24, -5, -4, -45, 3, -55, -71]

[540/539, 3025/3024, 4375/4374, 9801/9800]

ets 72, 80, 152, 224, 296

[[0, -3, -4, -5, -3], [8, 16, 23, 28, 31]]

[.1383934690, 1/8]

a = 9.9643/72 = 31.0001/224 = 166.0721626

badness   147.3854996
rms   .7687062948
g   23.42160176


4. Undecimal augmented fifth

[12, 22, -4, -6, 7, -40, -51, -71, -90, -3]

[385/384, 441/440, 3025/3024, 4375/4374, 9801/9800]

ets 26, 46, 72, 118, 190

[[0, 6, 11, -2, -3], [2, -1, -3, 7, 9]]

a = 25.0090/72 = 416.8172169 cents

[.3473476807, 1/2]

badness   169.9769111
rms   1.249416902
g   19.06380265


5.

[12, 34, 20, 30, 26, -2, 6, -49, -48, 15]

[243/242, 441/440, 540/539, 2401/2400, 9801/9800]

ets 58, 72, 130

[[0, -6, -17, -10, -15], [2, 4, 7, 7, 9]]

[.6933142420e-1, 1/2]

a = 4.9919/72 = 83.1977090 cents

badness   179.9856041
rms   1.462301383
g   17.95231779


6.

[12, -2, 20, -6, -31, -2, -51, 52, -7, -86]

[225/224, 385/384, 540/539, 9801/9800]

ets 22, 50, 72, 94

[[0, 6, -1, 10, -3], [2, 1, 5, 2, 8]]

[.1806533524, 1/2]

a = 13.0070/72 = 216.7840228 cents

badness   195.0280356
rms   1.584514315
g   17.95231779


7. Orwell

[7, -3, 8, 2, -21, -7, -21, 27, 15, -22]

[99/98, 121/120, 176/175, 225/224, 385/384, 540/539]

ets 9, 22, 31, 53

[[0, 7, -3, 8, 2], [1, 0, 3, 1, 3]]

[.2262038561, 1]

a = 11.9888/53 = 271.4446272 cents

badness   210.4954018
rms   5.548614670
g   8.860022575


8. Semihemimeantone

[4, 16, 9, 10, 16, 3, 2, -24, -32, -3]

[81/80, 99/98, 121/120, 243/242, 441/440, 540/539, 2401/2400]

ets 31

[[0, -4, -16, -9, -10], [1, 3, 8, 6, 7]]

[.3548751316, 1]

a = 11.0011/31 = 425.8501579 cents

badness   218.9540099
rms   6.965622568
g   7.914724072


9.

[9, 5, -3, 7, -13, -30, -20, -21, -1, 30]

ets 15, 31, 46

[121/120, 126/125, 176/175, 385/384, 441/440, 3025/3024]

[[0, 9, 5, -3, 7], [1, 1, 2, 3, 3]]

[.6494333856e-1, 1]

a = 2.0132/31 = 77.9320062 cents

badness   223.3668950
rms   4.418576095
g   10.52547929


10.

[1, -1, 3, 4, -4, 2, 3, 10, 13, 1]

[55/54, 56/55, 99/98, 3025/3024]

ets 5

[[0, -1, 1, -3, -4], [1, 2, 2, 4, 5]]

[.3798204598, 1]

11.0148/29 = 455.7845520 cents

badness   235.8100854
rms   44.34125247
g   2.725540575


11.

[6, 10, 10, 8, 2, -1, -8, -5, -16, -12]

[50/49, 55/54, 99/98, 100/99, 121/120, 540/539, 9801/9800]

ets 22

[[0, -3, -5, -5, -4], [2, 4, 6, 7, 8]]

[.1375489239, 1/2]

a = 3.0261/22 = 165.0587086 cents

badness   238.7261371
rms   11.89273384
g   6.047431569


12. Nonkleismic

[10, 9, 7, 25, -9, -17, 5, -9, 27, 46]

[126/125, 176/175, 243/242, 441/440, 540/539, 2401/2400]

ets 31, 58, 89

[[0, 10, 9, 7, 25], [1, -1, 0, 1, -3]]

[.2584558979, 1]

a = 23.0026/89 = 310.1470775 cents

badness   240.3019988
rms   3.316530191
g   13.06303399


13. Magic

[5, 1, 12, -8, -10, 5, -30, 25, -22, -64]

[100/99, 225/224, 385/384, 540/539]

ets 19, 22, 41

[[0, 5, 1, 12, -8], [1, 0, 2, -1, 6]]

[.3172615104, 1]

13.0077/41 = 380.7138126 cents

badness   242.7224832
rms   4.730404304
g   10.62006188


14. Septimal

[0, 0, 7, 0, 0, 11, 0, 16, 0, -24]

[55/54, 81/80, 100/99, 121/120, 243/242]

ets 7

[[0, 0, 0, -1, 0], [7, 11, 16, 21, 24]]

[.2141802354, 1/7]

a = 257.0162824 cents

badness   245.8506632
rms   22.63634705
g   4.183300133


15. Meanertone

[1, 4, 3, -1, 4, 2, -5, -4, -16, -13]

[55/54, 56/55, 81/80, 3025/3024]

ets 5

[[0, -1, -4, -3, 1], [1, 2, 4, 4, 3]]

[.4194849382, 1]

a = 13.0040/31 = 503.3819256 cents

badness   252.8666930
rms   47.54854253
g   2.725540575


16. Tweedledee

[3, 5, 9, 4, 1, 6, -4, 7, -8, -20]

ets 15

[55/54, 56/55, 100/99, 121/120, 126/125, 3025/3024]

[[0, -3, -5, -9, -4], [1, 2, 3, 4, 4]]

[.1329702752, 1]

a = 1.9946/15 = 159.5643303 cents

badness   255.7850727
rms   22.12985764
g   4.342481185


17. Tweedledum

[3, 5, -6, 4, 1, -18, -4, -28, -8, 32]

[55/54, 64/63, 100/99, 121/120, 176/175, 385/384]

ets 7, 15, 22

[[0, -3, -5, 6, -4], [1, 2, 3, 2, 4]]

[.1357721305, 1]

a = 2.9870/22 = 162.9265567 cents

badness   262.2914819
rms   11.79393546
g   6.430951940

18. Pentoid

[2, 3, 1, -2, 0, -4, -10, -6, -15, -9]

[49/48, 56/55, 99/98, 385/384]

ets 4, 5, 9

[[0, -2, -3, -1, 2], [1, 2, 3, 3, 3]]

[.2183480607, 1]

a = 5.0220/23 = 262.0176727 cents

badness   267.0829245
rms   40.16092708
g   3.116774888


19. Monzoid

[1, 4, -2, -1, 4, -6, -5, -16, -16, 4]

[55/54, 64/63, 81/80, 385/384]

ets 5, 7

[[0, -1, -4, 2, 1], [1, 2, 4, 2, 3]]

[.4181947520, 1]

a = 5.0183/12 = 501.8337024 cents

badness   269.9708171
rms   39.86372247
g   3.150963571


20. Catakleismic

[6, 5, 22, -21, -6, 18, -54, 37, -66, -135]

[225/224, 385/384, 540/539, 4375/4374]

ets 19, 72

[[0, 6, 5, 22, -21], [1, 0, 1, -3, 9]]

[.2639230436, 1]

a = 19.0025/72 = 316.7076522 cents

badness   271.0589693
rms   1.697136764
g   20.98979344


Number 46 Monzo


[64/63, 81/80, 100/99, 176/175]

ets 7, 12

[[0, -1, -4, 2, -6], [1, 2, 4, 2, 6]]

[.4190088422, 1]

a = 5.0281/12 = 502.8106107 cents

badness   312.5112733
rms   28.87226550
g   4.174754057


top of page bottom of page up down


Message: 3366

Date: Sat, 19 Jan 2002 21:40:07

Subject: Re: deeper analysis of Schoenberg unison-vectors

From: genewardsmith

--- 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.


top of page bottom of page up down


Message: 3367

Date: Sat, 19 Jan 2002 23:35:22

Subject: Re: A top 20 11-limit superparticularly generated linear temperament list

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
Sorry about cutting your statement/question up as follows but I want 
to address the accuracy and complexity points separately.

> The first on the list, 
Hemiennealimmal, could certainly claim to be able to produce authentic 
Partch tunings of the 11-limit, ... to extreme accuracy.
>

Yes. But even Partch didn't require such accuracy. I understand that 
he couldn't tell the difference between his scale and either 41-tET or 
72-tET versions of it. So the extreme accuracy doesn't mitigate the 
badness of the extreme complexity, and Miracle leaves Hemiennealimmal 
for dead with any reasonable badness measure that relates to human 
beings.

> ... and it would be interesting to check 
in how many keys 72 notes tempered in this way could play the Partch 
43-tone scale ...
>

I think the answer is zero. But the question seems fairly irrelevant 
of any temperament, since no-one I know wants to have as many as 72 
notes per octave on a keyboard or fretboard and no composer I know 
wants to have to deal with that many notes (choosing always some more 
manageable subset). Also I don't find it likely that anyone would want 
to play Partch's scale in more than one "key" per piece. 

A more relevant question is how many notes of a given temperament does 
it take to include _one_ version of Partch's scale (without 
conflating any notes)? Since Partch's scale contains the 11-limit 
diamond, if I'm reading your cryptic lists of numbers correctly, the 
answer for hemiennealimmal cannot be less than (2*3+1)*18 = 126. The 
answer for Miracle is (2*22+1)*1 = 45.

> 1. Hemiennealimmal
> 
> [36, 54, 36, 18, 2, -44, -96, -68, -145, -74]
> 
> [2401/2400, 3025/3024, 4375/4374, 9801/9800]
> 
> ets 72, 198, 270, 342, 612
> 
> [[0, 2, 3, 2, 1], [18, 12, 17, 34, 54]]
> 
> [.4591217954, 1/18]
> 
> a = 33.0568/72 = 280.9825/612 = 550.9491544 cents
> 
> badness   78.02778100
> rms   .1987978829
> g   36.
> 
> 
> 2. Miracle
> 
> [6, -7, -2, 15, -25, -20, 3, 15, 59, 49]
> 
> [225/224, 243/242, 385/384, 441/440, 540/539, 2401/2400, 3025/3024]
> 
> ets 10, 31, 41, 72
> 
> [[0, 6, -7, -2, 15], [1, 1, 3, 3, 2]]
> 
> [.9722688696e-1, 1]
> 
> a = 7.0003/72 = 116.6722643 cents
> 
> badness   125.5016755
> rms   1.901465778
> g   12.35198075


top of page bottom of page up down


Message: 3368

Date: Sun, 20 Jan 2002 01:45:43

Subject: Re: A top 20 11-limit superparticularly generated linear temperament list

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> Yes.
But even Partch didn't require such accuracy. 

If I remember correctly, he did require such accuracy. He considered
the fifth of 53-et to be good enough, and the third to be close, but
no cigar. The fifth is flat by .068 cents, and the major third is flat
1.408 cents. The rms for Hemiennealimmal is 1/5 cent, so it is in this
range and well towards the small side. Hence it seems reasonable to
conclude that using this tuning would be authentically Partch. 

Of course, there's the question of how accurately his own instruments
were tuned, which would give another handle on what authenticity would
mean in this context.

I understand that 
> he couldn't tell the difference between his scale and either 41-tET
or 
> 72-tET versions of it. 

The question is one of his requirements, not his hearing.

> I think the answer is zero. 

Here it is:

1--81/80--33/32--21/20--16/15--12/11--11/10--10/9--9/8--8/7--7/6
32/27--6/5--11/9--5/4--14/11--9/7--21/16--4/3--27/20--11/8--7/5 
10/7--16/11--40/27--3/2--32/21--14/9--11/7--8/5--18/11--5/3--27/16
12/7--7/4--16/9--9/5--20/11--11/6--15/8--40/21--64/33--160/81--(2)

This is mapped to

[[0, 5, 3, 1, -5, 1, -2, -1, 4, -2, 0, -6, -1, -3, 3, 1, 2, 4, -2, 3,
1, -1, 1, -1, -3, 2, -4, -2, -1, -3, 3, 1, 6, 0, 2, -4, 1, 2, -1, 5, 
-1, -3, -5]
, [0, -41, -24, -7, 43, -6, 19, 11, -30, 20, 4, 54, 13, 30, -19, -2, 
-10, -26, 24, -17, 0, 17, 1, 18, 35, -6, 44, 28, 20, 37, -12, 5, -36,
14, -2, 48, 7, -1, 24, -25, 25, 42, 59]]

by Hemiennealimmal, so you are right, it won't fit--the first
generator ranging from -6 to 6. 

But the question seems fairly irrelevant 
> of any temperament, since no-one I know wants to have as many as 72 
> notes per octave on a keyboard or fretboard and no composer I know 
> wants to have to deal with that many notes (choosing always some
more 
> manageable subset).

This is assuming that you must be using a keyboard or fretboard.
Partch, after all, *did* have 43 actual tones per octave in play, so I
don't see how this theory holds up.

 Also I don't find it likely that anyone would want 
> to play Partch's scale in more than one "key" per piece. 

Even if they did not, tuning Partch's scale in this way would give you
some equivalences for free (deriving from 2401/2400, 3025/3024,
4375/4374 and 9801/9800) which would make tempering Partch's 43 tones
in this way a perfectly reasonable option.


top of page bottom of page up down


Message: 3369

Date: Sun, 20 Jan 2002 02:51:44

Subject: A comparison of Partch's scale in RI and Hemiennealimmal

From: genewardsmith

RI:

edges: 18, 32, 64, 88

connectivity: 0, 0, 0, 2


Hemiennealimmal:

edges: 64, 106, 159, 219

connectivity: 0, 0, 0, 4


The numbers are edges/connectivity in the 5, 7, 9 and 11-limits. I
conclude that a great deal is gained by tempering in this way, and
nothing significant is conceded in terms of quality of intonation. Of
course, 72-et would do much better yet, but then some concessions will
have been made.


top of page bottom of page up down


Message: 3370

Date: Sun, 20 Jan 2002 06:55:42

Subject: maple presentation?

From: clumma

Gene, have you seen this?

Article Information *

-Carl


top of page bottom of page up down


Message: 3371

Date: Sun, 20 Jan 2002 00:08:37

Subject: Re: deeper analysis of Schoenberg unison-vectors

From: monz

> Message 2798 
> From:  "monz" <joemonz@y...> 
> Date:  Sat Jan 19, 2002  2:40 pm
> Subject:  deeper analysis of Schoenberg unison-vectors
Yahoo groups: /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
Yahoo groups: /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



 



_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 3372

Date: Sun, 20 Jan 2002 00:27:34

Subject: more questions about adjoints and mappings

From: monz

Here's a simple example: Ellis's Duodene


kernel

   2  3  5   unison vectors   ~cents

[  1  0  0 ]  =    2:1     1200
[ -4  4 -1 ]  =   81:80      21.5062896
[  7  0 -3 ]  =  128:125     41.05885841


adjoint

[ -12  0  0 ]
[ -19 -3  1 ]
[ -28  0  4 ]

determinant  =  | -12 |


"mapping" of UVs

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


So here, I can see that the h12 mapping does not
temper out the 2:1 ... and I still don't understand
what that means.  Is it simply because any "8ve"-based
ET must include 2:1 by definition?

I can also see that the third column of the adjoint
specifies some kind of meantone, which tempers out
the 2:1 and the 81:80, but not the diesis 128:125.
Is there a way to tell what flavor of meantone it is?

And the middle column of the adjoint specifies some
temperament which does not temper out the 81:80.
But can someone explain what kind of tuning this is?



-monz


 



_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 3373

Date: Sun, 20 Jan 2002 00:30:30

Subject: Re: deeper analysis of Schoenberg unison-vectors

From: monz

Hi Gene,


> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> 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


 





_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 3374

Date: Sun, 20 Jan 2002 02:39:56

Subject: Re: deeper analysis of Schoenberg unison-vectors

From: monz

> 
> > Message 2798 
> > From:  "monz" <joemonz@y...> 
> > Date:  Sat Jan 19, 2002  2:40 pm
> > Subject:  deeper analysis of Schoenberg unison-vectors
> Yahoo groups: /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@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> 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


 



_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up

Previous Next

3000 3050 3100 3150 3200 3250 3300 3350 3400 3450 3500 3550 3600 3650 3700 3750 3800 3850 3900 3950

3350 - 3375 -

top of page