Tuning-Math Digests messages 5278 - 5302

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Message: 5278

Date: Thu, 10 Oct 2002 01:34:32

Subject: Re: mathematical model of torsion-block symmetry?

From: monz

hi Hans,

thanks very much for your replies to this, but
i'm afraid some of the math language is over my head.
i defer to Gene, paul, Graham, et al. for comment.


-monz
"all roads lead to n^0"



----- Original Message -----
From: "Hans Straub" <straub@xxxxxxxx.xx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Wednesday, October 09, 2002 2:47 PM
Subject: [tuning-math] Re: mathematical model of torsion-block symmetry?


> From:  "monz" <monz@a...>:
> >
> >Is there some way to mathematically model
> >the symmetry in a torsion-block?
> >
> >see the graphic and its related text in my
> >Tuning Dictionary definition of "torsion"
> >-- i've uploaded it to here:
> >Yahoo groups: /monz/files/dict/torsion.htm *
> >
>
> Well, they are translation symmetries in the quotient group of the full
lattice
> and the subgroup generated by the unison vectors. The symmetries in the
> example are pairs because the element has order 2 in the quotient group,
> but there are other elements such as (0,1) with order 6 or (0,2), (1,1)
with
> order 3. Something like this?
>
>
> BTW, I think the definition of torsion can be made simpler. You do not
need
> the condition that some power of the interval is in the unison vector
group,
> because this is always the case (at least when the periodicity block is
finite).
> Do I see this correctly?
>
> Hans Straub


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Message: 5279

Date: Thu, 10 Oct 2002 17:20:27

Subject: Re: Piano tuning and "BODE'S LAW EXPLAINED" II

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

You can add another note to your solar system scale now.
Perhaps it's also an escaped moon from Neptune?

Manuel


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Message: 5280

Date: Thu, 10 Oct 2002 15:10:26

Subject: EDO superset containing approximation of Werckmeister III?

From: monz

could someone please explain how to find an EDO superset
that gives a good approximation of the 12 pitches in
Werckmeister III, with the scale data given here?

Yahoo groups: /monz/files/dict/werckmeister.htm *




-monz


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Message: 5281

Date: Thu, 10 Oct 2002 14:43:36

Subject: Re: Piano tuning and "BODE'S LAW EXPLAINED" II

From: monz

----- Original Message ----- 
From: <manuel.op.de.coul@xxxxxxxxxxx.xxx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Thursday, October 10, 2002 8:20 AM
Subject: Re: [tuning-math] Re: Piano tuning and "BODE'S LAW EXPLAINED" II


> You can add another note to your solar system scale now.
> Perhaps it's also an escaped moon from Neptune?


thanks -- john chalmers and david beardsley wrote me
about this already a few days ago.

unofortunately, even Pluto is already beyond the audible
range in my sonic mapping, and so since it's more distant
than Pluto, this planet won't sound like much either!  ;-)



-monz


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Message: 5282

Date: Thu, 10 Oct 2002 16:05:18

Subject: 7-limit signatures

From: Gene W Smith

Recall that cubic lattice coordinates for 7-limit tetrads associate the 
3-tuple of integers [a,b,c] with the major triad with root

3^((-a+b+c)/2) 5^((a-b+c)/2) 7^((a+b-c)/2)

if a+b+c is even, and the minor tetrad with root

3^((-a+b+c-1)/2) 5^((a-b+c+1)/2) 7^((a+b-c+1)/2)

if a+b+c is odd.  This means that [2,0,0], [0,2,0], [0,0,2] represent the

major tetrads with roots 5*7/3, 3*7/5, 3*5/7 respectively; when octave 
reduced these are 35/24, 21/20, and 15/14.

If L is a wedgie for a 7-limit linear temperament, we may define the 
*signature* of L as S = [-L[1]+L[2]+L[3], L[1]-L[2]+L[3],
L[1]+L[2]-L[3]].  
This is a 3-tuple representing the number of generator steps in the
octave 
plus generator formulation of the temperament for 35/24, 21/20, 15/14 
respectively, weighted by the number of periods to the octave.  In the
case 
where the octave is the period, it uniquely defines the tetrad in terms
of 
steps by sending the tetrad [a,b,c] to S[1]*a + S[2]*b + S[3]*c steps. 
For 
example, taking the meantone wedgie of [1,4,10,12,-13,4] gives us a 
signature of [13,-7,5], so the minor tonic tetrad [-1,0,0] is sent to -13
steps,
the dominant major tetrad [0,1,1] to -2 steps, and so forth; for major 
tetrads these steps are twice the number of generator steps for the root
of 
the tetrad, while the minor tetrads fill in the gaps in ways which depend

on the temperament--for instance, here we get [-1,1,-1] ~ [1,-2,0] at -1 
step, equivalent under 126/125.

Just as temperaments with a generator which is a consonant interval are
of 
particular interest, temperaments where one of the signature values is
+-1 
are of interest, with miracle, whose signature is [-15,11,1] an example. 

In this case the ordering of tetrads by steps corresponds to a chain of 
adjacent tetrads in the lattice, so the step ordering is of particular 
interest.  Miracle now relates [-1,0,0] not just to 15 steps, but to the 
tetrad [0,0,15], and [0,1,1] to [0,0,12], and so forth.  This helps to
keep 
track of the connectivity of the tetrads when using miracle.  Moreover,
we 
may define miracle MOS in terms of tetrads--Blackjack for instance can be

described as a chain of sixteen consecutive [0,0,n] tetrads, where n
starts 
from an even number (representing a major tetrad) and runs up to an odd 
number (minor tetrad.)  For example, the chain from [0,0,0] (major tonic)

to [0,0,15] (minor tonic.)

Here is a list of temperaments with this unital signature property:


[[1, 1, 3, 3], [0, 6, -7, -2]]   [6, -7, -2, 15, 20, -25] Miracle

generators   [1200., 116.5729472]   signatures   [-15, 11, 1]

rms   1.637405196   comp   24.92662917   bad   1017.380173

ets   [10, 21, 31, 41, 72, 103]



[[1, 0, -4, 6], [0, 1, 4, -2]]   [1, 4, -2, -16, 6, 4] Dominant seventh

generators   [1200., 1902.225977]   signatures   [1, -5, 7]

rms   20.16328150   comp   9.836559603   bad   1950.956872

ets   [5, 7, 12]



[[1, 1, 2, 3], [0, 9, 5, -3]]   [9, 5, -3, -21, 30, -13]
Quartaminorthirds

generators   [1200., 77.70708739]   signatures   [-7, 1, 17]

rms   3.065961726   comp   27.04575317   bad   2242.667500

ets   [15, 16, 31, 46]



[[1, 1, 1, 2], [0, 8, 18, 11]]   [8, 18, 11, -25, 5, 10] Octafifths

generators   [1200., 88.14540671]   signatures   [21, 1, 15]

rms   2.064339812   comp   34.23414357   bad   2419.357925

ets   [27, 41, 68]



[[1, 2, 2, 3], [0, 4, -3, 2]]   [4, -3, 2, 13, 8, -14] Tertiathirds

generators   [1200., -125.4687958]   signatures   [-5, 9, -1]

rms   12.18857055   comp   14.72969740   bad   2644.480844

ets   [1, 9, 10, 19, 29]



[[1, 0, 7, -5], [0, 1, -3, 5]]   [1, -3, 5, 20, -5, -7] Hexadecimal

generators   [1200., 1873.109081]   signatures   [1, 9, -7]

rms   18.58450012   comp   12.33750942   bad   2828.823679

ets   [7, 9, 16]



[[1, 25, -31, -8], [0, 26, -37, -12]]   [26, -37, -12, 76, 92, -119]

generators   [1200., -1080.705187]   signatures   [-75, 51, 1]

rms   .2219838332   comp   118.1864167   bad   3100.676640

ets   [10, 171, 513]



[[1, 3, 6, 5], [0, 20, 52, 31]]   [20, 52, 31, -74, 7, 36]

generators   [1200., -84.87642563]   signatures   [63, -1, 41]

rms   .3454637898   comp   96.52895120   bad   3218.975773

ets   [99, 212, 311, 410]



[[1, 2, 2, 2], [0, 5, -4, -10]]   [5, -4, -10, -12, 30, -18]

generators   [1200., -97.68344522]   signatures   [-19, -1, 11]

rms   6.041345016   comp   24.27272426   bad   3559.349900

ets   [12, 37]



[[1, 3, 0, 2], [0, 14, -23, -8]]   [14, -23, -8, 46, 52, -69]

generators   [1200., -121.1940013]   signatures   [-45, 29, -1]

rms   .8353054234   comp   68.53846955   bad   3923.865443

ets   [10, 99]



[[1, 12, 15, 1], [0, 23, 28, -4]]   [23, 28, -4, -88, 71, -9]

generators   [1200., -543.2692838]   signatures   [1, -9, 55]

rms   .7218691130   comp   78.22290415   bad   4416.989140

ets   [53]



[[1, 2, 3, 4], [0, 5, 8, 14]]   [5, 8, 14, 10, -8, 1]

generators   [1200., -102.3994286]   signatures   [17, 11, -1]

rms   8.609470174   comp   22.70605087   bad   4438.739304

ets   [12]



[[1, 2, 1, 1], [0, 6, -19, -26]]   [6, -19, -26, -7, 58, -44]

generators   [1200., -83.37933102]   signatures   [-51, -1, 13]

rms   1.487254275   comp   55.50097036   bad   4581.275174

ets   [29, 72]



[[1, 43, -58, -17], [0, 46, -67, -22]]   [46, -67, -22, 137, 164, -213]

generators   [1200., -1080.392876]   signatures   [-135, 91, 1]

rms   .1267147296   comp   211.5126443   bad   5668.912722

ets   [10, 301, 311, 612]



[[1, 2, 3, 3], [0, 6, 10, 3]]   [6, 10, 3, -21, 12, 2]

generators   [1200., -82.00647655]   signatures   [7, -1, 13]

rms   12.62928610   comp   21.39334917   bad   5780.113425

ets   [15, 29]



[[1, 2, 1, 2], [0, 4, -13, -8]]   [4, -13, -8, 18, 24, -30]

generators   [1200., -122.3321832]   signatures   [-25, 9, -1]

rms   6.403982242   comp   31.21994593   bad   6241.865585

ets   [10]



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

generators   [1200., 187.6316444]   signatures   [3, 7, 1]

rms   47.68000484   comp   11.69073209   bad   6516.579639

ets   [6]



[[1, 0, -3, 6], [0, 3, 10, -6]]   [3, 10, -6, -42, 18, 9]

generators   [1200., 638.4642643]   signatures   [1, -13, 19]

rms   9.885351494   comp   25.98120378   bad   6672.839126

ets   [15]



[[1, 2, 3, 3], [0, 5, 8, 2]]   [5, 8, 2, -18, 11, 1]

generators   [1200., -100.0317906]   signatures   [5, -1, 11]

rms   21.64417648   comp   17.58481613   bad   6692.936885

ets   [12]



[[1, 2, 5, 6], [0, 4, 26, 31]]   [4, 26, 31, -1, -38, 32]

generators   [1200., -123.5352658]   signatures   [53, 9, -1]

rms   2.267858844   comp   56.46645397   bad   7230.978171

ets   [29, 68]



[[1, 2, 2, 3], [0, 5, -4, 2]]   [5, -4, 2, 16, 11, -18]

generators   [1200., -99.19646785]   signatures   [-7, 11, -1]

rms   21.21541236   comp   18.58251802   bad   7325.893533

ets   [1, 12]



[[1, 3, 2, 4], [0, 13, -3, 11]]   [13, -3, 11, 34, 19, -35]

generators   [1200., -130.2049690]   signatures   [-5, 27, -1]

rms   4.481233722   comp   41.46170034   bad   7703.566083

ets   [9, 37, 46]



[[1, 12, 10, 5], [0, 19, 14, 4]]   [19, 14, 4, -30, 47, -22]

generators   [1200., -657.8863907]   signatures   [-1, 9, 29]

rms   3.032624788   comp   52.44877824   bad   8342.369709

ets   [31]



[[1, 23, -56, 83], [0, 47, -128, 176]]   [47, -128, 176, 768, -147, -312]

generators   [1200., -546.7680257]   signatures   [1, 351, -257]

rms   .3610890892e-1   comp   481.2637469   bad   8363.357505

ets   [1578]



[[1, 13, 17, -1], [0, 21, 27, -7]]   [21, 27, -7, -92, 70, -6]

generators   [1200., -652.3887024]   signatures   [-1, -13, 55]

rms   1.469925034   comp   75.92946624   bad   8474.535049

ets   [46, 57, 103]



[[1, 2, 4, 5], [0, 4, 16, 21]]   [4, 16, 21, 4, -22, 16]

generators   [1200., -125.5372720]   signatures   [33, 9, -1]

rms   6.562501740   comp   35.99263747   bad   8501.523814

ets   [19]



[[1, 1, 3, 4], [0, 7, -8, -14]]   [7, -8, -14, -10, 42, -29]

generators   [1200., 101.5775171]   signatures   [-29, 1, 13]

rms   7.012328960   comp   35.52454740   bad   8849.513343

ets   [12]



[[1, 2, -1, -1], [0, 6, -48, -55]]   [6, -48, -55, 7, 104, -90]

generators   [1200., -83.05774075]   signatures   [-109, -1, 13]

rms   .6644554968   comp   115.7156146   bad   8897.127847

ets   [29, 130]



[[1, 2, 3, 3], [0, 7, 11, 3]]   [7, 11, 3, -24, 15, 1]

generators   [1200., -73.16557361]   signatures   [7, -1, 15]

rms   16.40779159   comp   24.26315309   bad   9659.276719

ets   [16]



[[1, 2, 3, 3], [0, 8, 13, 4]]   [8, 13, 4, -27, 16, 2]

generators   [1200., -63.00613990]   signatures   [9, -1, 17]

rms   12.64637740   comp   28.07029990   bad   9964.608569

ets   [19]


It might be remarked that the signatures with the middle-sized (in 
absolute value) components relatively small are an interesting subclass
of 
these unital signature temperaments; they are associated with certain 
planar temperaments of a kind not usually considered. Examples are
[-5,27,-1], 
covered by 46, [-109,-1,13], covered by 130, and [1,-9,55], covered
(though 
not very well) by 53.


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Message: 5283

Date: Fri, 11 Oct 2002 08:13:06

Subject: Werckmeister as subset of 612edo

From: monz

hi Gene,


i've just done a comprehensive analysis of
Werckmeister III as a subset of 612edo:

Yahoo groups: /monz/files/dict/werckmeister.htm *


i've put an entry for this into the EDO historical table:
Yahoo groups: /monz/files/dict/eqtemp.htm *

have you analyzed Werckmeister III like this before?
has anyone else?  

the only reference i've found to 612edo besides your posts
is a mention by Bosanquet in his book, referring to
Captain Herschel's advocacy of this tuning.



-monz


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Message: 5286

Date: Fri, 11 Oct 2002 14:23:25

Subject: Re: EDO superset containing approximation of Werckmeister III?

From: monz

> From: "Gene Ward Smith" <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Friday, October 11, 2002 12:37 PM
> Subject: [tuning-math] Re: EDO superset containing approximation of
Werckmeister III?
>
>
> --- In tuning-math@y..., "monz" <monz@a...> wrote:
>
> > awesome!!  i was hoping you'd give some details as to how
> > you found out that 612edo was the best approximation.
>
> I ran a search and 612 came out the best,


well, OK, but ... AARRRGGH! -- *how* did you do that search?

since i'm math-challenged, the only way i know how to do it
is to set up an Excel spreadsheet with the EDO-cardinality
as a variable, but then i have to manually enter each cardinality
and look at the graphs of deviation to see which EDOs are best.


> but other strange-looking possibilities are out there,
> such as 200 and 412 (200+412=612, of course.)


ah, now that's useful!  i was hoping to find something smaller
than 612edo which could describe Werckmeister III, and 200
does the trick nicely.

unfortunately, however, neither 200 nor 412 give
integer-divisions for 12edo, so they're not as useful
for comparing Werckmeister III to 12edo as 612edo is.


please, Gene, more info on how your search method works.
do you know how to set it up in an Excel spreadsheet?
if not, then do you have some code that i could run on
my PC?  i have Mathematica -- just don't know a lot
about how to use it.


-monz
"all roads lead to n^0"


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Message: 5289

Date: Fri, 11 Oct 2002 00:57:35

Subject: Re: EDO superset containing approximation of Werckmeister III?

From: monz

hi Gene,


> From: "Gene Ward Smith" <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Thursday, October 10, 2002 10:08 PM
> Subject: [tuning-math] Re: EDO superset containing approximation of
Werckmeister III?
>
>
> --- In tuning-math@y..., "monz" <monz@a...> wrote:
>
> > could someone please explain how to find an EDO superset
> > that gives a good approximation of the 12 pitches in
> > Werckmeister III, with the scale data given here?
> >
> > Yahoo groups: /monz/files/dict/werckmeister.htm *
>
> I used Manual's scale data rather than trying to figure out
> where the data was on your page.


there's a table showing the tunings as a chain of generators.
anyway, i tried it and came up with the same results you did.


> It turns out that Werckmeister III can be expressed with
> extreme accuracy in terms of what I call "schismas", steps
> of the 612 et. In 612-et terms, it is
>
> 0, 46, 98, 150, 199, 254, 300, 355, 404, 453, 508, 557



awesome!!  i was hoping you'd give some details as to how
you found out that 612edo was the best approximation.




-monz
"all roads lead to n^0"


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Message: 5293

Date: Fri, 11 Oct 2002 00:57:35

Subject: Re: EDO superset containing approximation of Werckmeister III?

From: monz

hi Gene,


> From: "Gene Ward Smith" <genewardsmith@xxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Thursday, October 10, 2002 10:08 PM
> Subject: [tuning-math] Re: EDO superset containing approximation of
Werckmeister III?
>
>
> --- In tuning-math@y..., "monz" <monz@a...> wrote:
>
> > could someone please explain how to find an EDO superset
> > that gives a good approximation of the 12 pitches in
> > Werckmeister III, with the scale data given here?
> >
> > Yahoo groups: /monz/files/dict/werckmeister.htm *
>
> I used Manual's scale data rather than trying to figure out
> where the data was on your page.


there's a table showing the tunings as a chain of generators.
anyway, i tried it and came up with the same results you did.


> It turns out that Werckmeister III can be expressed with
> extreme accuracy in terms of what I call "schismas", steps
> of the 612 et. In 612-et terms, it is
>
> 0, 46, 98, 150, 199, 254, 300, 355, 404, 453, 508, 557



awesome!!  i was hoping you'd give some details as to how
you found out that 612edo was the best approximation.




-monz
"all roads lead to n^0"


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Message: 5295

Date: Fri, 11 Oct 2002 22:52:04

Subject: Re: Historical well-temeraments, 612, and 412

From: monz

----- Original Message ----- 
From: "Gene Ward Smith" <genewardsmith@xxxx.xxx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Friday, October 11, 2002 5:31 PM
Subject: [tuning-math] Historical well-temeraments, 612, and 412


> It seems that Werckmeister III is not the only well-temperament
> to be nailed by 612. Here are some others, using data taken from
> Manual's list of scales:
> <snip>


wow, Gene, thanks for these!!!
they'll eventually all become Tuning Dictionary webpages.

my guess is that the reason 612 works so well has something
to do with the fact that these temperaments temper out the
Pythagorean comma.  wanna look into that more?


-monz


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Message: 5297

Date: Sat, 12 Oct 2002 13:37:39

Subject: Re: EDO superset containing approximation of Werckmeister III?

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Joe and Gene,

I must have told this before but in Scala it's very easy
to do too:

load werck3
fit/mode

This show successively better approximations and stops at
some point. To go beyond that, and show all divisions,
use a negative number:

fit/mode -612

With a positive parameter it only shows that division.

Manuel


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