Tuning-Math Digests messages 3750 - 3774

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Message: 3750

Date: Sat, 02 Feb 2002 01:42:26

Subject: Re: interval of equivalence, unison-vector, period

From: genewardsmith

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

> Well I confused the two things, which is completely my fault, but was 
> not helped by Graham's opinion that the thing you declined to call a 
> temperament was in fact pajara.

If you ever get around to trying 222223 with a period of 3/2 in the 22-et, tell us about it.


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Message: 3751

Date: Sat, 02 Feb 2002 10:11:29

Subject: Re: 43-edo (was: 171-EDO, Vogel (was: 7-limit MT reduced bases forets))

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> ------------
> 
> > From: monz <joemonz@y...>
> > To: <tuning-math@y...>
> > Sent: Saturday, February 02, 2002 1:50 AM
> > Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT 
reduced
> bases for ets)
> >
> > from manuel's page:
> > Stichting Huygens-Fokker: Logarithmic Interval Measures *
> >
> > >> ... Sauveur ... found 43 to be optimal
> > >> because 4 steps is almost exactly a 16/15 minor second
> > >> and 7 steps almost exactly the geometric mean of
> > >> three 9/8 and two 10/9 whole tones. The chromatic scale
> > >> contained in 43-tET is virtually identical to 1/5-comma
> > >> meantone tuning.
> 
> 
> 
> 
> 
> >
> >
> >
> > [-9  6  0]  =  3 * [-3  2  0]  (= 9:8 whole tone)
> > +
> > [ 2 -4  2]  =  2 * [ 1 -2  1]  (= 10:9 whole tone)
> > ----------
> > [-7  2  2]  (= 225:128 "augmented 6th")
> >
> >
> > [7  2  2]^(1/2)  =  [-7/2  1  1] = ~488.2687147 cents
> >
> >
> > but what significance does that have?  i don't get it
> >
> > manuel?
> 
> 
> 
> the only thing that i think i can see is some kind of
> tritone-equivalence in action, because if you ignore
> prime-factor 2 you get a mean for the 225:128 of 15:8,
> which is 2^(1/2) higher than the above interval, and
> which is the interval that is given exactly by 5 generators
> of 1/5-comma meantone
> 
> but i really don't understand what's going on

you have to take the weighted mean of three 9/8 and two 10/9 whole 
tones. that means three 9/8s "plus" two 10/9s "divided" by five, that 
is, ( (9/8)^3 * (10/9)^2 )^(1/5).


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Message: 3752

Date: Sat, 2 Feb 2002 02:52:10

Subject: Re: 43-edo (was: 171-EDO...)

From: monz

----- Original Message -----
From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Saturday, February 02, 2002 2:11 AM
Subject: [tuning-math] Re: 43-edo (was: 171-EDO, Vogel (was: 7-limit MT
reduced bases for ets))


> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > ------------
> >
> > > From: monz <joemonz@y...>
> > > To: <tuning-math@y...>
> > > Sent: Saturday, February 02, 2002 1:50 AM
> > > Subject: Re: [tuning-math] Re: 171-EDO, Vogel (was: 7-limit MT
> reduced
> > bases for ets)
> > >
> > > from manuel's page:
> > > Stichting Huygens-Fokker: Logarithmic Interval Measures *
> > >
> > > >> ... Sauveur ... found 43 to be optimal
> > > >> because 4 steps is almost exactly a 16/15 minor second
> > > >> and 7 steps almost exactly the geometric mean of
> > > >> three 9/8 and two 10/9 whole tones. The chromatic scale
> > > >> contained in 43-tET is virtually identical to 1/5-comma
> > > >> meantone tuning.
> >
> >
> > >
> > > [-9  6  0]  =  3 * [-3  2  0]  (= 9:8 whole tone)
> > > +
> > > [ 2 -4  2]  =  2 * [ 1 -2  1]  (= 10:9 whole tone)
> > > ----------
> > > [-7  2  2]  (= 225:128 "augmented 6th")
> > >
> > >
> > > [7  2  2]^(1/2)  =  [-7/2  1  1] = ~488.2687147 cents
> > >
> > >
> > > but what significance does that have?  i don't get it
> > >
> > > manuel?
> >
> >
> >
> > the only thing that i think i can see is some kind of
> > tritone-equivalence in action, because if you ignore
> > prime-factor 2 you get a mean for the 225:128 of 15:8,
> > which is 2^(1/2) higher than the above interval, and
> > which is the interval that is given exactly by 5 generators
> > of 1/5-comma meantone
> >
> > but i really don't understand what's going on
>
> you have to take the weighted mean of three 9/8 and two
> 10/9 whole tones. that means three 9/8s "plus" two 10/9s
> "divided" by five, that is, ( (9/8)^3 * (10/9)^2 )^(1/5).


so that's what manuel means by "geometric mean"?  i would
have never understood it that way

ok, i see it now ... but it took a little bit of work to
comprehend what's going on there ... perhaps you'd like to
reword that a bit, manuel?

isn't there a better way to say that, with a quantifiable
numeric? ... perhaps "almost exactly the geometric mean
1/5 part of the 5-tone interval composed of three 9/8 and
two 10/9 whole tones"?


anyway,



1/5-comma meantone whole-tone

= ( (9/8)^3 * (10/9)^2 )^(1/5)

= [2 3 5]^[-7/5 2/5 2/5]      (did i write that correctly?)

= ~195.3074859 cents



7 steps of 47-edo

= 2^(7/43)

= ~195.3488372 cents



The difference between them is


[2 3 5]^[336/215 -2/5 -2/5]

~0.041351317 cent  =  ~1/24 cent  =  ~1 jot



and paul, there might be a hint of an answer (from
the "new cylindrical meantones lattice" thread) as to
how to draw spirals which compare a meantone-like edo
with a fraction-of-a-comma meantone

we need to include an axis for 2 ... but hmmm ...
this pair of points would be rather far apart, since
the exponents of 3 and 5 have their signs reversed

and of course, what would including 2 mean when
the lattice is wrapped into a cylinder?

my spatial imagination can't handle this ...



-monz






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Message: 3753

Date: Sat, 02 Feb 2002 03:31:17

Subject: 7-limit MT reduced bases for ets

From: genewardsmith

9: [21/20, 27/25, 128/125]
10: [25/24, 28/27, 49/48]
12: [36/35, 50/49, 64/63]
15: [28/27, 49/48, 126/125]
19: [49/48, 81/80, 126/125]
22: [50/49, 64/63, 245/243]
27: [64/63, 126/125, 245/243]
31: [81/80, 126/125, 1029/1024]
41: [225/224, 245/243, 1029/1024]
68: [245/243, 2048/2025, 2401/2400]
72: [225/224, 1029/1024, 4375/4374]
99: [2401/2400, 3136/3125, 4375/4374]
130: [2401/2400, 3136/3125, 19683/19600]
140: [2401/2400, 5120/5103, 15625/15552]

For any prime limit, we could consider the most characteristic linear
temperament of a particular et to be the one leaving off the last
member of the MT reduced basis. It is interesting to note that the
characteristic linear temperament of 99 and 130 is the same. Of course
we can do the same for planar temperaments, etc.


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Message: 3754

Date: Sun, 03 Feb 2002 04:25:11

Subject: For Carl--5, 7, and 11-limit reduced bases for the 37-et

From: genewardsmith

5: [250/243, 262144/253125]
7: [64/63, 250/243, 686/675]
11: [55/54, 64/63, 100/99, 686/675]


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Message: 3756

Date: Sun, 03 Feb 2002 04:38:36

Subject: Re: any ideas?

From: paulerlich

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Here's something along the lines of what I'm looking for:
> 
> All maximally even subsets favor palindromic symmetry. In fact, 
these
> subsets are always the most palindromic or least skewed 
rotations or
> modes--and if they're actual palindromes, then they're also 
unique, as
> they have zero skew and therefore no inversion amongst their
> rotations.
> 
> This is the sort of a special condition that I had in mind... 
though
> I'd want it to work with any given scale, not just scales with
> Myhill's property. Any ideas?

how about the rotation where the scale's center of gravity is 
closest to 600 cents?


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Message: 3757

Date: Sun, 3 Feb 2002 20:03:02

Subject: Re: Gene's notation formula: alternate duodene?

From: monz

my generalization of Gene's periodicity-block finding formula:


> for a set of i rational unison-vectors {u1/v1, ... ui/vi},
> for any non-zero I can define a scale by calculating for 0<=n<d
> 
> step[n] = (u1/v1)^round(7n/d) (u2/v2)^round(12n/d) 
> (u3/v3)^round(7n/d) (u4/v4)^round(-2n/d) (u5/v5)^round(5n/d)



> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, February 03, 2002 5:23 PM
> Subject: [tuning-math] Re: Gene's notation formula: alternate duodene?
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > Gene, can you explain why your formula gave this
> > result instead of what i expected?  Is there a
> > "correction factor" involved?
> 
> My formula said to round to the nearest integer, but it
> didn't say what to do when two integers are equally near,
> which can happen when you have an even number of tones
> in an octave. The different rounding choices then lead
> to different blocks, which I think are equally correct.


so then i guess i'm just getting the particular results i
am because of the way Excel handles the rounding, yes?


 
> The proposal for defining blocks a while back involved
> defining a distance function designed to work with a 
> particular block problem in mind. In this case, it would give
> 
> ||q|| = max(|h12(q)|, |12 h7(q) - 7 h12(q)|, |12 h3(q) - 3 h12(q)|
> 
> If you take everything at a distance of less than six from
> the unison using this measure, and transpose to the standard
> octave (instead of the octave from 2^(-1/2) to 2^(1/2)) you
> obtain the nine note scale
> 
> 1--16/15--6/5--5/4--4/3--3/2--8/5--5/3--15/8
> 
> This is the core of the block, in every version of it.
> If you now take everything at a distance of exactly six
> from one, you get
> {10/9, 9/8, 25/18, 45/32, 64/45, 36/25, 16/9, 9/5}.
> To get a block, you add three of these to the core of the
> block, in such a way that the diameter of the resulting
> block is less than twelve: the diameter being the maximum
> of all the distances between members of the block.
>
> You can therefore add 9/5, 9/8 and 45/32, getting what you
> expected, or 64/45, 16/9, 10/9, which amounts to the same
> thing. However, you could also add 36/25, 9/5 and 9/8,
> which you didn't expect, or 16/9, 10/9, and 25/18.


in fact, i just corrected an error in my spreadsheet and
the last one is exactly what i got:


i had my spreadsheet doing an incorrect calculation because
i didn't divide the unimodular adjoint by the determinant first,
as i should have.

so, according to the way my Excel spreadsheet is handling the
rounding in Gene's formula, here's one version of the 12-tone
JI PB scale for Ellis's Duodene:


kernel:
  2  3  5       ratio    ~cents    unison-vector

[-3 -1  2]  =   25:24   70.6724269   chromatic
[ 7  0 -3]  =  128:125  41.0588584   commatic
[-4  4 -1]  =   81:80   21.5062896   commatic



adjoint:

[12  7 3]
[19 11 5]
[28 16 7]

determinant  =  | 1 |


JI periodicity-block:

   2 / 1   1200
  15 / 8   1088.268715
  16 / 9    996.0899983
   5 / 3    884.358713
   8 / 5    813.6862861
   3 / 2    701.9550009
  25 / 18   568.717426
   4 / 3    498.0449991
   5 / 4    386.3137139
   6 / 5    315.641287
  10 / 9    182.4037121
  16 / 15   111.7312853
   1 / 1      0

 
 triangular lattice:
 
           F#
          25:18
           / \
          /   \
         /     \
         D      A       E       B
       10:9----5:3-----5:4-----15:8
       / \     / \     / \     /
      /   \   /   \   /   \   /
     /     \ /     \ /     \ /
    Bb      F       C       G
   16:9----4:3-----1:1-----3:2
           / \     / \     /
          /   \   /   \   /
         /     \ /     \ /
        Db      Ab      Eb
      16:15----8:5-----6:5




-monz




 



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Message: 3758

Date: Sun, 03 Feb 2002 04:39:58

Subject: Re: For Carl--5, 7, and 11-limit reduced bases for the 37-et

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> 
wrote:
> 5: [250/243, 262144/253125]
> 7: [64/63, 250/243, 686/675]
> 11: [55/54, 64/63, 100/99, 686/675]

given the notation you've been using for these, i thought you were 
talking about 5-equal, 7-equal, and 11-equal.


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Message: 3759

Date: Sun, 3 Feb 2002 20:17:58

Subject: Re: Gene's notation formula: alternate duodene?

From: monz

> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, February 03, 2002 5:23 PM
> Subject: [tuning-math] Re: Gene's notation formula: alternate duodene?
>
> 
>  
> > The proposal for defining blocks a while back involved
> > defining a distance function designed to work with a 
> > particular block problem in mind. In this case, it would give
> > 
> > ||q|| = max(|h12(q)|, |12 h7(q) - 7 h12(q)|, |12 h3(q) - 3 h12(q)|
> > 
> > If you take everything at a distance of less than six from
> > the unison using this measure, and transpose to the standard
> > octave (instead of the octave from 2^(-1/2) to 2^(1/2)) you
> > obtain the nine note scale
> > 
> > 1--16/15--6/5--5/4--4/3--3/2--8/5--5/3--15/8
> > 
> > This is the core of the block, in every version of it.
> > If you now take everything at a distance of exactly six
> > from one, you get
> > {10/9, 9/8, 25/18, 45/32, 64/45, 36/25, 16/9, 9/5}.
> > To get a block, you add three of these to the core of the
> > block, in such a way that the diameter of the resulting
> > block is less than twelve: the diameter being the maximum
> > of all the distances between members of the block.
> >
> > You can therefore add 9/5, 9/8 and 45/32, getting what you
> > expected, or 64/45, 16/9, 10/9, which amounts to the same
> > thing. However, you could also add 36/25, 9/5 and 9/8,
> > which you didn't expect, or 16/9, 10/9, and 25/18.



i just realized that i mention exactly the same thing on
my webpage "Ellis's Duodene and a "best-fit" meantone"
Internet Express - Quality, Affordable Dial Up, DSL, T-1, Domain Hosting, Dedicated Servers and Colocation *


>> Note that this periodicity-block has three pitch-classes
>> which fall right on the eastern boundary: (2,-1) = 9/5,
>> (2,0) = 9/8 and (2,1) = 45/32. All three of these thus
>> have alternates a comma lower -- in other words, by the
>>  -[4,-1] = 80:81 unison-vector--, and the alternate
>> pitch-classes fall on the western boundary: (-2,0) = 16/9,
>> (-2,1) = 10/9, and (-2,2) = 25/18, respectively.
>> 
>> Also, since (-2,2) = 25/18 and (2,1) = 45/32 happen to
>> fall right on the northwest and northeast *corners* of the
>> boundary (respectively), they also have lower alternates at
>> the distance of the *other* unison-vector -[0 3] = 64:125,
>> which would place the alternates at (-2,-1) = 64/45 and
>> (2,-2) = 36/25, respectively.



you and i are talking about exactly the same structures here,
even down to exactly the same pitches.



-monz



 



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Message: 3760

Date: Sun, 03 Feb 2002 06:18:00

Subject: 686/675

From: genewardsmith

Has anyone noticed this one? I saw it turning up in the reduced bases
of 37 and 55; it is a comma of 9,10,19,27,29,36,37,46,55,56, and is
productive of a number of good temperaments.


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Message: 3761

Date: Sun, 03 Feb 2002 06:24:19

Subject: Re: 686/675

From: genewardsmith

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

> Has
anyone noticed this one? I saw it turning up in the reduced bases of
37 and 55; it is a comma of 9,10,19,27,29,36,37,46,55,56, and is
productive of a number of good temperaments.

I should add it also turned up in the survey of 7-limit temperaments.
It is (7/6)^3 / (5/4)^2.


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Message: 3762

Date: Sun, 3 Feb 2002 00:18:48

Subject: Uhrin's paper

From: monz

i would really appreciate a detailed explanation of this abstract:

"Self-affine tiles and digit sets via the geometry of numbers"
B. Uhrin 
Self-affine tiles and digit sets via the geometry of numbers *


in english

thanks



-monz



 



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Message: 3763

Date: Sun, 3 Feb 2002 23:32:28

Subject: Re: Gene's notation formula: alternate duodene?

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, February 03, 2002 10:12 PM
> Subject: [tuning-math] Re: Gene's notation formula: alternate duodene?
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > but 25:24 only occurs between notes with the same letter-name
> > and a change of accidental!  in the JI duodene, the sharps
> > and flats have distinct spellings, and 25:24 is indeed
> > functioning as a chromatic change on the same letter-name,
> > which is not what i would call a step.
> 
> conventional diatonic notation is based on a 7-tone periodicity 
> block. the duodene mixes this notation with a 12-tone periodicity 
> block.


hmmm ... that's interesting.

 
> if an interval takes you from one pitch to another _within
> the block_, that interval is *not* a unison vector of that
> block. if it takes you between two adjacent, in pitch, notes
> in the block, it is a step vector of the block.


a h  ! ! !    thanks, Paul, now it's clear as a mountain stream.

the "within the block" bit is the key that unlocked that
puzzle for me.

but now i'm really curious -- why is it necessary to put a
step-vector into the kernel to derive a notation.  intuitively,
it makes sense to me, even as i ask it, but i'd still like
a good explanation of how and why it works.



-monz


 



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Message: 3764

Date: Sun, 03 Feb 2002 09:15:08

Subject: Re: Uhrin's paper

From: genewardsmith

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> i would really appreciate a detailed explanation of this abstract:
> 
> "Self-affine tiles and digit sets via the geometry of numbers"
> B. Uhrin 
> Self-affine tiles and digit sets via the geometry of numbers *

Are you interested in this abstract in particular, or self-affine tilings, or what, exactly?


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Message: 3765

Date: Sun, 3 Feb 2002 20:41:12

Subject: Re: a notation for Schoenberg's rational implications

From: monz

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


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


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

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



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


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



-monz






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Message: 3766

Date: Sun, 03 Feb 2002 09:36:34

Subject: Re: Uhrin's paper

From: genewardsmith

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> i would really appreciate a detailed explanation of this abstract:
> 
> "Self-affine tiles and digit sets via the geometry of numbers"
> B. Uhrin 
> Self-affine tiles and digit sets via the geometry of numbers *

R^n is real n-dimensional space, a compact set in such a space is a
closed and bounded set, and an expanding matrix is one with all of its
eigenvalues greater than one in absolute value. This talk comes from a
curious generalization that Jeff Lagarias came up with, which
generalizes the idea of a base b expansion to where the base is a
matrix. What's the connection to music? I could probably get Jeff to
send me a reprint of what he has done if we can't get it off the web.


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Message: 3767

Date: Sun, 3 Feb 2002 20:49:32

Subject: Re: Gene's notation formula: alternate duodene?

From: paulerlich

To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Sunday, February 03, 2002 8:37 PM
Subject: [tuning-math] Re: Gene's notation formula: alternate duodene?


> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > 
> > 
> > for a set of rational unison-vectors {u1/v1, ... un/vn},
> > for any non-zero I can define a scale by calculating for 0<=n<d
> > 
> > step[n] = (u1/v1)^round(7n/d) (u2/v2)^round(12n/d) 
> > (u3/v3)^round(7n/d) (u4/v4)^round(-2n/d) (u5/v5)^round(5n/d)
> 
> you really think this is a correct generalization?


no, i was responding to your previous post and realized
that it's not.  it still has numbers in it which come from
a particular homomorphism.

i'm working on a real generalization, and it will be in
my other post.


-monz


 



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Message: 3768

Date: Sun, 3 Feb 2002 23:45:15

Subject: Re: Gene's PB formula, generalized

From: monz

> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, February 03, 2002 10:18 PM
> Subject: [tuning-math] Re: Gene's PB formula, generalized
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > for a set of i rational unison-vectors {u1/v1, u2/v2,... ui/vi},
> > where {hx, hy, ...hq} is the top row of the unimodular adjoint
> > of the kernel matrix of the unison-vectors,
> 
> This should be h1, h2, ... hi to correspond to your unison vectors.

absolutely.


> Also, u1/v1 is a step vector, and the matrix therefore is not just
> the kernel, but a set of generators for the kernel plus a step vector.


right, got it.


> "Unimodular adjoint" should just be "matrix inverse", and you should
> note that since u1/v1 is a step vector, the matrix is unimodular,
> and hence is invertible to an integral matrix.


ok, thanks.  how's this? :


where M is the matrix composed of a set of i rational
vectors {u1/v1, u2/v2,... ui/vi} in which u1/v1 is a
step-vector and {u2/v2 ... ui/vi} are commatic unison-vector
generators of the kernel, and where {h1, h2, ...hi} is the
top row of M^-1,

for any non-zero a scale can be defined by calculating
for 0 <= n < d   :

  step[n] = (u1/v1)^round(h1(2)*n/d) * (u2/v2)^round(h2(2)*n/d)
  * ... (ui/vi)^round(hi(2)*n/d) .


(i think we should describe what n and d are.)



-monz


 



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Message: 3769

Date: Sun, 3 Feb 2002 09:45:08

Subject: Re: Uhrin's paper

From: monz

> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, February 03, 2002 1:36 AM
> Subject: [tuning-math] Re: Uhrin's paper
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > 
> > i would really appreciate a detailed explanation of this abstract:
> > 
> > "Self-affine tiles and digit sets via the geometry of numbers"
> > B. Uhrin 
> > Self-affine tiles and digit sets via the geometry of numbers *
> 
> R^n is real n-dimensional space, a compact set in such a space
> is a closed and bounded set, and an expanding matrix is one
> with all of its eigenvalues greater than one in absolute value.
> This talk comes from a curious generalization that Jeff Lagarias
> came up with, which generalizes the idea of a base b expansion
> to where the base is a matrix.


thanks, gene ... but i still don't understand

what's an eigenvalue?



> What's the connection to music?


ah ... that's for   y o u   to tell   m e   !!!


i'm just hunting down stuff that discusses lattices,
so that i can learn more about them


BTW ... i took a look at the first couple of chapters from
the Conway/Sloane book you recommended (_Sphere Packings..._),
but don't understand much of that either     :(



-monz



 



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Message: 3770

Date: Sun, 3 Feb 2002 23:46:41

Subject: Re: Gene's notation formula: alternate duodene?

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, February 03, 2002 11:36 PM
> Subject: [tuning-math] Re: Gene's notation formula: alternate duodene?
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > but now i'm really curious -- why is it necessary to put a
> > step-vector into the kernel to derive a notation.
> 
> because otherwise, you'd never get past the unison, to the second, 
> third, etc.!


right, that makes sense ... but how does only one step-vector
give you the whole scale?  all the other steps can be derived
from that and the commas, apparently.   ?


-monz


 



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Message: 3771

Date: Sun, 03 Feb 2002 18:35:41

Subject: Re: For Carl--5, 7, and 11-limit reduced bases for the 37-et

From: clumma

> 5: [250/243, 262144/253125]
> 7: [64/63, 250/243, 686/675]
> 11: [55/54, 64/63, 100/99, 686/675]

Thanks!

-Carl


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Message: 3772

Date: Sun, 3 Feb 2002 21:03:55

Subject: Gene's PB formula, generalized (was: a notation for Schoenberg's...)

From: monz

> From: paulerlich <paul@xxxxxxxxxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, February 03, 2002 8:36 PM
> Subject: [tuning-math] Re: a notation for Schoenberg's rational
implications
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> > And the JI periodicity-block scale derived from this
>
> again, i'm baffled. how do you get a PB when you have one too many
> UVs?


i'm using the formula Gene posted, which i generalized to:


> for a set of i rational unison-vectors {u1/v1, ... ui/vi},
> for any non-zero I can define a scale by calculating for 0<=n<d
>
> step[n] = (u1/v1)^round(7n/d) (u2/v2)^round(12n/d)
> (u3/v3)^round(7n/d) (u4/v4)^round(-2n/d) (u5/v5)^round(5n/d)


but oops! ... i realize now that this is still not entirely
generalized.

all those numbers (7,12,7,-2,5) are from a particular set of
homomorphisms (the first Schoenberg PB Gene calculated, back
around Christmas), and need to be replaced by variables.

i don't really know what to call them, so i'll just make this
do: {hv, hw, hx, hy, hz}.  it's the top row of numbers in the
adjoint (or is it a unimodular inverse?) of the kernel.


so the generalized formula really is:


for a set of i rational unison-vectors {u1/v1, u2/v2,... ui/vi},
where {hx, hy, ...hq} is the top row of the unimodular adjoint
of the kernel matrix of the unison-vectors, for any non-zero
I can define a scale by calculating for 0 <= n < d   :

step[n] = (u1/v1)^round(hx*n/d) * (u2/v2)^round(hy*n/d)
* ... (ui/vi)^round(hq*n/d) .



-monz









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Message: 3773

Date: Sun, 3 Feb 2002 11:43:33

Subject: a notation for Schoenberg's rational implications

From: monz

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


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



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



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

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



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

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


kernel:

  2  3  5  7 11       ratio     ~cents

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


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

adjoint:

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


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

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

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



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

 degree  ratio       vector
                  2  3  5  7 11

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



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



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

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

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

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



-monz








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Message: 3774

Date: Sun, 3 Feb 2002 21:11:16

Subject: Re: a notation for Schoenberg's rational implications

From: monz

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


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

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


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


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

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

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


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



-monz









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