Tuning-Math Digests messages 3850 - 3874

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Message: 3850

Date: Fri, 08 Feb 2002 09:21:04

Subject: A 58 tone epimorphic scale containing Genesis

From: genewardsmith

Genesis is not epimorphic, so we should find non-CS, inconsistent
properties. We would also not be able to produce it as any kind of
block. I took the steps of Genesis which were mapped to 2 by h58,
which are 45/44, 49/48, 50/49 and 55/54, and split them in two as
follows:

45/44 = 81/80 100/99
49/48 = 245/242 121/120
50/49 = 99/98 100/99
55/54 = 100/99 121/120

I then picked the one of the two possibilites presented in each case
for filling the "gap" by choosing the option of least Tenney height,
and obtained the following:

1, 81/80, 45/44, 33/32, 21/20, 16/15, 27/25, 12/11, 11/10, 10/9, 
9/8, 8/7, 121/105, 7/6, 32/27, 6/5, 40/33, 11/9, 99/80, 5/4, 14/11,
9/7, 315/242, 21/16, 4/3, 27/20, 15/11, 11/8, 7/5, 99/70, 10/7, 16/11,
22/15, 40/27, 3/2, 32/21, 484/315, 14/9, 11/7, 8/5, 81/50, 18/11,
33/20, 5/3, 27/16, 12/7, 121/70, 7/4, 16/9, 9/5, 20/11, 11/6, 50/27,
15/8, 40/21, 64/33, 88/45, 160/81

It is readily verified that this *is* epimorphic, with map h58.

The set of steps for this scale is 121/120, 100/99, 99/98, 245/242,
81/80, 64/63, and 56/55. Geometrically it is more complex than
Genesis, with 30 verticies and 65 facets. However, a 4D parallepiped
would have 16 verticies and 8 facets so there is a limit to how simple
this can get, and I suppose Genesis is not so bad.


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Message: 3851

Date: Fri, 08 Feb 2002 18:48:53

Subject: Re: xenharmonic bridges in the 12edo comma pump (was: exactly what...)

From: genewardsmith

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

>  so our matrix for these three bridges is:
> 
>  [2 3 5] [-4     4 -1] = 81:80 syntonic comma
>          [-19/6  2  0] = 12edo==9/8 bridge  =  ~3.910001731 cents
>          [ -5/6  2 -1] = 12edo==10/9 bridge = ~17.59628787 cents
>     
> 
>  and note that these three bridges are linearly dependent.

The second bridge is just P^(1/6), where P is the Pythagorean comma.
The Pythagorean and syntonic commas together define the 12-et in the 
5-limit; one way to express that is 81/80 ^ P = 81/80 X P (where the wedge product in this case becomes the vector cross-product) =
h12, the [12, 19, 28] val. Your other bridge is W^(1/6), where
W = (81/80)^6 P^(-1). Is there anything gained by taking roots of commas? I don't see it.

>  so here's the entire list of bridges which i would say
>  are in effect for the comma pump in 12edo:
>  
>       2      3     5                               ~cents
>   
>   [ -4      4    -1   ] = 81:80 syntonic comma  = 21.5062896
>   [ -5/6    2    -1   ] = 12edo==10/9 bridge    = 17.59628787
>   [-36/11  36/11 -9/11] = 1/11cmt==10/9 bridge  = 17.59605512
>   [ -8/11   8/11 -2/11] = 1/11cmt==9/8 bridge   =  3.910234472
>   [-19/6    2     0   ] = 12edo==9/8 bridge     =  3.910001731
>   [161/66 -14/11 -2/11] = 12edo==1/11cmt bridge =  0.000232741

It seems to me the only one on this list which is important for the comma pump is 81/80, and that defines it. The others define 12-et
and 1/11 comma meantone, and have nothing to do with the pump so far as I can see.


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Message: 3852

Date: Fri, 8 Feb 2002 11:29:46

Subject: Re: xenharmonic bridges in the 12edo comma pump (was: exactly what...)

From: monz

> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Thursday, February 07, 2002 6:00 AM
> Subject: [tuning-math] xenharmonic bridges in the 12edo comma pump (was:
exactly what...)
>
> ...
>
>  so here's the entire list of bridges which i would say
>  are in effect for the comma pump in 12edo:
>
>       2      3     5                               ~cents
>
>   [ -4      4    -1   ] = 81:80 syntonic comma  = 21.5062896
>   [ -5/6    2    -1   ] = 12edo==10/9 bridge    = 17.59628787
>   [-36/11  36/11 -9/11] = 1/11cmt==10/9 bridge  = 17.59605512
>   [ -8/11   8/11 -2/11] = 1/11cmt==9/8 bridge   =  3.910234472
>   [-19/6    2     0   ] = 12edo==9/8 bridge     =  3.910001731
>   [161/66 -14/11 -2/11] = 12edo==1/11cmt bridge =  0.000232741


oops ... that really should say "are in effect for the note D
for the comma pump in 12edo".  i didn't examine the bridges
which are in effect for any of the other notes.


-monz






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Message: 3853

Date: Fri, 8 Feb 2002 11:36:17

Subject: Re: xenharmonic bridges in the 12edo comma pump

From: monz

> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, February 08, 2002 10:48 AM
> Subject: [tuning-math] Re: xenharmonic bridges in the 12edo comma pump
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> >  so our matrix for these three bridges is:
> > 
> >  [2 3 5] [-4     4 -1] = 81:80 syntonic comma
> >          [-19/6  2  0] = 12edo==9/8 bridge  =  ~3.910001731 cents
> >          [ -5/6  2 -1] = 12edo==10/9 bridge = ~17.59628787 cents
> >     
> > 
> >  and note that these three bridges are linearly dependent.
> 
> The second bridge is just P^(1/6), where P is the Pythagorean comma.
> The Pythagorean and syntonic commas together define the 12-et in
> the 5-limit; one way to express that is 81/80 ^ P = 81/80 X P
> (where the wedge product in this case becomes the vector cross-product)
> = h12, the [12, 19, 28] val. Your other bridge is W^(1/6), where
> W = (81/80)^6 P^(-1). Is there anything gained by taking roots
> of commas? I don't see it.
> 
> >  so here's the entire list of bridges which i would say
> >  are in effect for the comma pump in 12edo:
> >  
> >       2      3     5                               ~cents
> >   
> >   [ -4      4    -1   ] = 81:80 syntonic comma  = 21.5062896
> >   [ -5/6    2    -1   ] = 12edo==10/9 bridge    = 17.59628787
> >   [-36/11  36/11 -9/11] = 1/11cmt==10/9 bridge  = 17.59605512
> >   [ -8/11   8/11 -2/11] = 1/11cmt==9/8 bridge   =  3.910234472
> >   [-19/6    2     0   ] = 12edo==9/8 bridge     =  3.910001731
> >   [161/66 -14/11 -2/11] = 12edo==1/11cmt bridge =  0.000232741
> 
> It seems to me the only one on this list which is important for
> the comma pump is 81/80, and that defines it. The others define
> 12-et and 1/11 comma meantone, and have nothing to do with the
> pump so far as I can see.


thanks for all of that, Gene.  i'm not sure what any of this means,
but my feeling is that all these small intervals (and many others
which i did not list) have something to do with what we're hearing
when composers intentionally mix the properties of different
tuning systems like this, so they're worth looking at.



-monz


 




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Message: 3854

Date: Fri, 08 Feb 2002 22:31:27

Subject: Re: A 58 tone epimorphic scale containing Genesis

From: paulerlich

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

>Genesis is not epimorphic, so we should find non-CS, inconsistent 
>properties. We would also not be able to produce it as any kind of 
>block.

i thought my original question made it clear that we were to take 
11/10 and 20/11 as auxillaries, not as part of the block.


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Message: 3855

Date: Fri, 08 Feb 2002 22:44:53

Subject: Re: A 58 tone epimorphic scale containing Genesis

From: genewardsmith

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

> i thought my original question made it clear that we were to take 
> 11/10 and 20/11 as auxillaries, not as part of the block.

I don't know how to take something as an auxillary, but we can certainly look at the scale we get by leaving 11/10 and 20/11 out of
Genesis. Is this what you mean? Perhaps you could tell us again what you are actually looking for.


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Message: 3856

Date: Fri, 08 Feb 2002 23:31:33

Subject: Re: A 58 tone epimorphic scale containing Genesis

From: paulerlich

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> 
wrote:
> 
> > i thought my original question made it clear that we were to take 
> > 11/10 and 20/11 as auxillaries, not as part of the block.
> 
> I don't know how to take something as an auxillary, but we can 
certainly look at the scale we get by leaving 11/10 and 20/11 out of
> Genesis. Is this what you mean?

yes, or whatever choice is best. there should be only 41 notes inside 
the parallelepiped and whatever other shape you may happen to 
investigate (what's the 4-dimensional analogue of a 3-d rhombic 
dodecahedron or a 2-d hexagon? i feel that should be the most general 
case, and certain vertices may coincide in specific instances, 
causing some edges/faces/cells to disappear).


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Message: 3857

Date: Sat, 09 Feb 2002 09:33:47

Subject: Genesis Minus as a block

From: genewardsmith

I undertook a proceedure to determine if "Genesis Minus", by which I
mean Genesis less 11/10 and 20/11, is a block by my understanding of
what a block is. That understanding is that a block is epimorphic and
convex, where by "convex" I mean it is convexly closed: every lattice
point contained in the convex hull of the octave equivalence classes
of the scale are already in the scale. Equivalently, a block is
epimorphic and has the property that there exists a norm such that no
point can be added to the block without increasing its diameter. It
seems Genesis Minus is a block by this definition (I hope Joe notices
I said the word "definition".) 

I downloaded and ran the "qhull" program, which gave me a set of
inequalities defining the convex hull, which I converted into a vector
space norm. By this norm, Genesis Minus is within a radius of one of
the unison, whereas the nearest comma seems to be 385/384 at a
distance of three. The Voroni cells of the lattice of 41-et commas
using this distance measure give the appropriate tiling of the 4D
space.

Here are the distances from unity of the tones of Genesis Minus
according to this distance measure:

1   0
81/80   1.
33/32   1.
21/20   1.
16/15   1.
12/11   1.
10/9   1.
9/8   .5196155385
8/7   1.
7/6   1.
32/27   1.
6/5   1.
11/9   1.
5/4   1.
14/11   1.
9/7   1.
21/16   1.
4/3   .0392310770
27/20   1.
11/8   1.
7/5   1.
10/7   1.
16/11   1.
40/27   1.
3/2   .0392310770
32/21   1.
14/9   1.
11/7   1.
8/5   1.
18/11   1.
5/3   1.
27/16   1.
12/7   1.
7/4   1.
16/9   .5196155385
9/5   1.
11/6   1.
15/8   1.
40/21   1.
64/33   1.
160/81   1.


I'll run some more computations; computing this distance measure is a
little slow.


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Message: 3859

Date: Mon, 11 Feb 2002 00:35:55

Subject: h72 = h31 + h41

From: monz

hi Gene,

in an old post, you wrote:


Yahoo groups: /tuning-math/message/1168 *

> tuning-math Message 1168
> From:  genewardsmith@j... 
> Date:  Fri Oct 5, 2001  6:18 am
> Subject:  Re: 3rd-best 11-limit temperament
>
>
> . . .  I've been meaning to suggest that Manuel 
> consider putting into Scala a routine to calculate
> Gen(m, n, p) and Mos(n,m,p) for two ets m and n and
> a prime limit p; in case m and n are not relatively
> prime this needs to be adjusted by working inside of
> the interval of repetition. Of course one can also
> think of this in terms of the ets generated by linear
> combinations of hm and hn, as for instance
> h53 = h22 + h31 and h72 = h31 + h41.



by "h72 = h31 + h41", do you mean the following?


72-udo (unequal division of the octave),
the combination of 31edo and 41edo 

~cents

(1200)
1170.731707
1161.290323
1141.463415
1122.580645
1112.195122
1083.870968
1082.926829
1053.658537
1045.16129
1024.390244
1006.451613
 995.1219512
 967.7419355
 965.8536585
  936.5853659
  929.0322581
  907.3170732
  890.3225806
  878.0487805
  851.6129032
  848.7804878
  819.5121951
  812.9032258
  790.2439024
  774.1935484
  760.9756098
  735.483871
  731.7073171
  702.4390244
  696.7741935
  673.1707317
  658.0645161
  643.902439
  619.3548387
  614.6341463
  585.3658537
  580.6451613
  556.097561
  541.9354839
  526.8292683
  503.2258065
  497.5609756
  468.2926829
  464.516129
  439.0243902
  425.8064516
  409.7560976
  387.0967742
  380.4878049
  351.2195122
  348.3870968
  321.9512195
  309.6774194
  292.6829268
  270.9677419
  263.4146341
  234.1463415
  232.2580645
  204.8780488
  193.5483871
  175.6097561
  154.8387097
  146.3414634
  117.0731707
  116.1290323
   87.80487805
   77.41935484
   58.53658537
   38.70967742
   29.26829268
    0


and can you explain a little more fully what
you wrote there?


-monz



 



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Message: 3860

Date: Mon, 11 Feb 2002 01:22:06

Subject: secondary generator definition

From: monz

there was a need to supplement the definition of "generator",
which my Dictionary had as a generator of a scale, with a
secondary definition describing the generator of a kernel.

Gene, if you already posted your definition of "generator",
i'm sorry i can't recall it now.

this is what i added:

Internet Express - Quality, Affordable Dial Up, DSL, T-1, Domain Hosting, Dedicated Servers and Colocation *

>> In periodicity-block theory, there are small intervals
> called unison-vectors, a select few of which are able to
> generate a kernel, which in JI is the periodicity-block
> enclosing a finite set of ratios on the lattice.


corrections, additions, etc.



-monz


 



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Message: 3861

Date: Mon, 11 Feb 2002 09:44 +0

Subject: Re: h72 = h31 + h41

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <004101c1b2d7$1f67e5a0$af48620c@xxx.xxx.xxx>
monz wrote:

> by "h72 = h31 + h41", do you mean the following?
> 
> 
> 72-udo (unequal division of the octave),
> the combination of 31edo and 41edo 

No, that's "the nearest-prime mapping of 72edo which is consistent with 
the nearest-prime mappings of 31edo and 41edo."  If you know how many 
steps an interval approximates to in 31edo and 41edo, you can add them 
together to get the approximation to 72edo where this equation holds.


                         Graham


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Message: 3862

Date: Mon, 11 Feb 2002 10:06:40

Subject: Re: h72 = h31 + h41

From: genewardsmith

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> by "h72 = h31 + h41", do you mean the following?
> 
> 
> 72-udo (unequal division of the octave),
> the
combination of 31edo and 41edo 

Not at all. I start by assuming the temperaments I am looking at are
both regular and consistent, and regard them as defined by mappings
from JI to abstract "notes" consisting of generator steps, and the
tuning of the temperament by a mapping in turn of the "notes" to "tones" which are real numbers. In the case of the 11-limit,
"h31" is the mapping defined b sending 2 to 31, 3 to 49, 5 to 72,
7 to 87 and 11 to 107. BY definition h31(a*b) = h31(a) + h31(b), so
this defines a mapping from any 11-limit interval to an integer. This
one-dimensional mapping I call a "val"; it is in a sense the dual
concept to interval. There is likewise a [41,65,95,115,142] mapping I
call h41, and a [72,114,167,202,249] mapping I call h72. Denoting by
"g+h" the mapping which sends a to g(a)+h(a), we have h72=h31+h41; in
terms of the mappings above regarded as column vectors, this is vector
addition.


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Message: 3863

Date: Mon, 11 Feb 2002 11:32:37

Subject: Re: h72 = h31 + h41

From: monz

hi Graham and Gene,

> From: <graham@xxxxxxxxxx.xx.xx> 
> To: <tuning-math@xxxxxxxxxxx.xxx> 
> Sent: Monday, February 11, 2002 1:44 AM 
> Subject: [tuning-math] Re: h72 = h31 + h41 
>
>
> In-Reply-To: <004101c1b2d7$1f67e5a0$af48620c@xxx.xxx.xxx>
> monz wrote:
> 
> > by "h72 = h31 + h41", do you mean the following?
> > 
> > 
> > 72-udo (unequal division of the octave),
> > the combination of 31edo and 41edo 
>
> No, that's "the nearest-prime mapping of 72edo which is consistent with 
> the nearest-prime mappings of 31edo and 41edo." If you know how many 
> steps an interval approximates to in 31edo and 41edo, you can add them 
> together to get the approximation to 72edo where this equation holds.


> From: genewardsmith <genewardsmith@xxxx.xxx> 
> To: <tuning-math@xxxxxxxxxxx.xxx> 
> Sent: Monday, February 11, 2002 2:06 AM 
> Subject: [tuning-math] Re: h72 = h31 + h41 
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > by "h72 = h31 + h41", do you mean the following?
> > 
> > 
> > 72-udo (unequal division of the octave),
> > the combination of 31edo and 41edo 
>
> Not at all. I start by assuming the temperaments I am looking
> at are both regular and consistent, and regard them as defined
> by mappings from JI to abstract "notes" consisting of generator
> steps, and the tuning of the temperament by a mapping in turn
> of the "notes" to "tones" which are real numbers. In the
> case of the 11-limit, "h31" is the mapping defined b sending
> 2 to 31, 3 to 49, 5 to 72, 7 to 87 and 11 to 107. BY definition
> h31(a*b) = h31(a) + h31(b), so this defines a mapping from any
> 11-limit interval to an integer. This one-dimensional mapping
> I call a "val"; it is in a sense the dual concept to interval.
> There is likewise a [41,65,95,115,142] mapping I call h41, and
> a [72,114,167,202,249] mapping I call h72. Denoting by "g+h"
> the mapping which sends a to g(a)+h(a), we have h72=h31+h41;
> in terms of the mappings above regarded as column vectors,
> this is vector addition. 


so, then does this express what you're saying? :

[ 31   41   72] [2 3 5 7 11]
[ 49   65  114]
[ 72   95  167]
[ 87  115  202]
[107  142  249]


if it does, then i think i understand what you're doing.

and can you explain what you mean by 'a "val" ... is in
a sense the dual concept to interval' ?  i don't get it.



-monz


 



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Message: 3864

Date: Mon, 11 Feb 2002 15:09:32

Subject: Clough / Engebretsen / Kochavi -- MTS article

From: monz

hi Gene,


if you haven't yet read this, i strongly urge you
to do so.  i think you'll get a lot out of it

... and then, hopefully, you'll explain it to me!   :)



John Clough, Nora Engebretsen, and Jonathan Kochavi
"Scales, Sets, and Interval Cycles: A Taxonomy"
_Music Theory Spectrum_, vol 21 #1, Spring 1999, p 74-104



ABSTRACT

>> Recent studies in the theory of scales by Agmon, Balzano,
>> Carey and Clampitt, Clough and Douthett, Clough and Myerson,
>> and Gamer have in common the central role of the interval
>> cycle.  Based on scale features defined in these studies,
>> and an additional feature called *distributional evenness*
>> defined here, a taxonomy is proposed for pitch-class sets
>> (pcsets) the [_sic_: that] that correspond to interval cycles
>> or to certain conjunctions thereof.  Pairwise implicative
>> relationships among the features are explored.  Of 20 sets
>> of features that are consistent with these relationships,
>> 13 are found to be instantiated by actual pcsets and 7 others
>> are shown to be incapable of instantiation.  Most instantiated
>> feature-sets correspond to infinite classes of pcsets which
>> are shown to be enumerable; one such feature-set is found
>> to be uniquely realized (up to transposition) in the usual
>> diatonic pcset.


-monz




 



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Message: 3865

Date: Tue, 12 Feb 2002 12:49 +0

Subject: Re: h72 = h31 + h41

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <00cb01c1b332$dcdd1e60$af48620c@xxx.xxx.xxx>
monz wrote:

> so, then does this express what you're saying? :
> 
> [ 31   41   72] [2 3 5 7 11]
> [ 49   65  114]
> [ 72   95  167]
> [ 87  115  202]
> [107  142  249]

Not really.  The multiplication's the wrong way round.  It should be

 [2 3 5 7 11][ 31   41   72]
             [ 49   65  114]
             [ 72   95  167]
             [ 87  115  202]
             [107  142  249]

And then, you shouldn't be multiplying frequency integers like that, so 
change it to

 [log(2) log(3) log(5) log(7) log(11)][ 31   41   72]
                                      [ 49   65  114]
                                      [ 72   95  167]
                                      [ 87  115  202]
                                      [107  142  249]

And you still aren't saying anything about nearest approximations or how 
they add up.

> and can you explain what you mean by 'a "val" ... is in
> a sense the dual concept to interval' ?  i don't get it.

It helps if you understand wedge products.  But yes, it's something like 
that.  I think the definition is that the wedge product of a val and an 
integer will always be a scalar.


                       Graham


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Message: 3866

Date: Tue, 12 Feb 2002 10:29:31

Subject: Re: h72 = h31 + h41

From: monz

> From: <graham@xxxxxxxxxx.xx.xx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Tuesday, February 12, 2002 4:49 AM
> Subject: [tuning-math] Re: h72 = h31 + h41
>
>
> In-Reply-To: <00cb01c1b332$dcdd1e60$af48620c@xxx.xxx.xxx>
> monz wrote:
> 
> > so, then does this express what you're saying? :
> > 
> > [ 31   41   72] [2 3 5 7 11]
> > [ 49   65  114]
> > [ 72   95  167]
> > [ 87  115  202]
> > [107  142  249]
> 
> Not really.  The multiplication's the wrong way round.  It should be
> 
>  [2 3 5 7 11][ 31   41   72]
>              [ 49   65  114]
>              [ 72   95  167]
>              [ 87  115  202]
>              [107  142  249]


hmmm . . . i had a hunch that that was the case.
i only wrote it that way so that the Yahoo interface
would put the bigger matrix in proper columns.
but why does it make a difference?

 
> And then, you shouldn't be multiplying frequency integers like that, so 
> change it to
> 
>  [log(2) log(3) log(5) log(7) log(11)][ 31   41   72]
>                                       [ 49   65  114]
>                                       [ 72   95  167]
>                                       [ 87  115  202]
>                                       [107  142  249]


i had a hunch about that too.  can't that be written
more easily as:

log([2 3 5 7 11]) [ 31   41   72]
                  [ 49   65  114]
                  [ 72   95  167]
                  [ 87  115  202]
                  [107  142  249]


 
> And you still aren't saying anything about nearest approximations or how 
> they add up.


why not?  please clarify, because that's exactly what i'm
trying to understand here.

 
> > and can you explain what you mean by 'a "val" ... is in
> > a sense the dual concept to interval' ?  i don't get it.
> 
> It helps if you understand wedge products.  But yes, it's something like 
> that.  I think the definition is that the wedge product of a val and an 
> integer will always be a scalar.


ok, thanks . . . i still haven't learned what a wedgie is.
still studying.


-monz


 



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Message: 3867

Date: Tue, 12 Feb 2002 21:20 +0

Subject: Re: h72 = h31 + h41

From: graham@xxxxxxxxxx.xx.xx

Me:
> > Not really.  The multiplication's the wrong way round.  It should be
> > 
> >  [2 3 5 7 11][ 31   41   72]
> >              [ 49   65  114]
> >              [ 72   95  167]
> >              [ 87  115  202]
> >              [107  142  249]
> 

Monz:
> hmmm . . . i had a hunch that that was the case.
> i only wrote it that way so that the Yahoo interface
> would put the bigger matrix in proper columns.
> but why does it make a difference?

Try multiplying it out in Excel, using MMULT.  You'll get different 
results.


> i had a hunch about that too.  can't that be written
> more easily as:
> 
> log([2 3 5 7 11]) [ 31   41   72]
>                   [ 49   65  114]
>                   [ 72   95  167]
>                   [ 87  115  202]
>                   [107  142  249]

There is a definition of the logarithm of a matrix, but I don't think this 
is it.

Me:
> > And you still aren't saying anything about nearest approximations or 
> > how they add up.

Monz:
> why not?  please clarify, because that's exactly what i'm
> trying to understand here.

There's nothing in the formula to say "take the nearest approximation" or 
"this column plus this other one makes that column".

You can expend h31+h41=h72 to

[ 31]   [ 41]   [ 72]
[ 49]   [ 65]   [114]
[ 72] + [ 95] = [167]
[ 87]   [115]   [202]
[107]   [142]   [249]

If you want to understand it, try setting up a spreadsheet that calculates 
the last column from the first two.  Then get it to generate those columns 
from their top entries, and the list of prime numbers.


> > It helps if you understand wedge products.  But yes, it's something 
> > like that.  I think the definition is that the wedge product of a val 
> > and an integer will always be a scalar.
> 
> 
> ok, thanks . . . i still haven't learned what a wedgie is.
> still studying.

The thing you have here as h72 is a val:

[ 72]
[114]
[167]
[202]
[249]

for an interval, take the comma 81:80.  That's 2**-4 * 3**4 * 5**-1 or [-4 
4 -1 0 0].  You get the representation of comma in h72 from the wedge 
product comma^h72.  That expands to give the matrix product

`[-4 4 1 0 0][ 72] = [1]
`            [114]
`            [167]
`            [202]
`            [249]

If you want to play with wedge products, you'll have to get my Python 
library from <Automatically generated temperaments *>.  Then you can do 
things like

>>> import temper
>>> comma = temper.WedgableRatio(81,80)
>>> h72 = temper.PrimeET(72, temper.primes[:4])
>>> comma^temper.Wedgable(h72).complement()
{(0, 1, 2, 3, 4): 1}

I think my complement() method is doing what Gene calls the "dual" so I'll 
rename it sometime.


                    Graham


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Message: 3868

Date: Tue, 12 Feb 2002 00:13:29

Subject: Re: secondary generator definition

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>  
> there was a need to supplement the definition of "generator",
> which my Dictionary had as a generator of a scale, with a
> secondary definition describing the generator of a kernel.
> 
> Gene, if you already posted your definition of "generator",
> i'm sorry i can't recall it now.
> 
> this is what i added:
> 
> Internet Express - Quality, Affordable Dial Up, DSL, T-1, Domain Hosting, Dedicated Servers and Colocation *
> 
> >> In periodicity-block theory, there are small intervals
> > called unison-vectors, a select few of which are able to
> > generate a kernel, which in JI is the periodicity-block
> > enclosing a finite set of ratios on the lattice.

what does this have to do with the definition of 'generator'??


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Message: 3869

Date: Tue, 12 Feb 2002 00:37:58

Subject: Re: secondary generator definition

From: genewardsmith

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

> what does this have to do with the definition of 'generator'??

I've been using "generator" in
its meaning from the theory of finitely-generated abelian groups, in
which sense both the generators of a kernel or the generators of a
temperament are generators.


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Message: 3870

Date: Wed, 13 Feb 2002 14:13:56

Subject: Re: h72 = h31 + h41

From: monz

hi Graham,


> From: <graham@xxxxxxxxxx.xx.xx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Tuesday, February 12, 2002 1:20 PM
> Subject: [tuning-math] Re: h72 = h31 + h41
>
>
> The thing you have here as h72 is a val:
>
> [ 72]
> [114]
> [167]
> [202]
> [249]
>
> for an interval, take the comma 81:80.  That's 2**-4 * 3**4 * 5**-1 or [-4
> 4 -1 0 0].  You get the representation of comma in h72 from the wedge
> product comma^h72.


ahh . . . so now we have to use ** instead of ^ for
"raise to the power of", because now ^ is the wedge product.
yes?


> That expands to give the matrix product
>
> `[-4 4 1 0 0][ 72] = [1]
> `            [114]
> `            [167]
> `            [202]
> `            [249]


there's a typo in the row matrix on the left.  the exponent of 5
should be -1 not 1, so that matrix should read `[-4 4 -1 0 0].

so then if ^ is the wedge product, is (81/80)^h72 = 1
the proper way to notate this?



-monz






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Message: 3871

Date: Wed, 13 Feb 2002 22:46:12

Subject: Re: h72 = h31 + h41

From: genewardsmith

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

> so then if ^ is the wedge product, is (81/80)^h72 = 1
> the
proper way to notate this?

It's certainly *possible* to interpret this in a way which makes
sense, and notate it thusly, but I think it would be much preferable
to write this as h72(81/80) = 1.


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Message: 3872

Date: Wed, 13 Feb 2002 21:20:59

Subject: Re: h72 = h31 + h41

From: monz

> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Wednesday, February 13, 2002 2:46 PM
> Subject: [tuning-math] Re: h72 = h31 + h41
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > so then if ^ is the wedge product, is (81/80)^h72 = 1
> > the proper way to notate this?
> 
> It's certainly *possible* to interpret this in a way which
> makes sense, and notate it thusly, but I think it would be
> much preferable to write this as h72(81/80) = 1.


well, ok . . . now  t h a t  notation looks familiar, and
i understand it.

but i've seen ^ used in connection with wedgies, so my
question is still not really answered: do we have to use **
now to represent "raise to the power of"?  apparently,
whatever ^ is being used for, it's something else other
than that.


-monz


 



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Message: 3873

Date: Wed, 13 Feb 2002 23:44:18

Subject: ^ and ** (was: h72 = h31 + h41)

From: monz

> From: genewardsmith <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Wednesday, February 13, 2002 11:03 PM
> Subject: [tuning-math] Re: h72 = h31 + h41
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > but i've seen ^ used in connection with wedgies, so my
> > question is still not really answered: do we have to use **
> > now to represent "raise to the power of"?  apparently,
> > whatever ^ is being used for, it's something else other
> > than that.
> 
> The "^" symbol is well-established as a notation both
> for exponentiation


aha!  i   k n e w   there was a one-word term for that,
but i couldn't think of it.  thanks.


> and wedge product; I would use it for either myself so
> long as there seemed no potential for confusion. Fortran
> gave us "**" for exponentiation also, which is fine,
> and not used for anything else to my knowledge.


well, i've certainly used ^ a heck of a lot for exponentiation,
and i sure don't want to change now.  but in the interests
of standardization and consistency (in the general sense, not
the technical tuning sense), since ** is unique for exponentiation,
then maybe we should adopt this separation.

^ for wedge product
** for exponentiation


should we create a poll?  any other opinions on it?



-monz


 





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Message: 3874

Date: Thu, 14 Feb 2002 07:57:41

Subject: Re: ^ and ** (was: h72 = h31 + h41)

From: paulerlich

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

> ^ for wedge product
> ** for exponentiation
> 
> 
> should we create a poll?  any other opinions on it?

let's keep ^ for exponentiation and use
/\ for wedge product


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