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Message: 5275 - Contents - Hide Contents

Date: Fri, 04 Oct 2002 17:07:58

Subject: Re: A common notation for JI and ETs

From: gdsecor

(This is a continuation of my message #4664.)

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote (#4662):
> At 06:19 PM 17/09/2002 -0700, George Secor wrote:
>> From: George Secor (9/17/02, #4626) >> Subject: A common notation for JI and ETs >> >> --- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote: >>> ...
>>> Here are some others for your consideration: >>> 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >> 15
>>> 282: )| ~| ~)| |~ /| |) )|) (| (|( //| /|) (|~ /|\ (|) >>> |( ~|( /|~ ~|\ |~) >>> >>> 11deg282 is the difficult one. /|) is only correct as the 5- comma + >>> 7-comma, not the 13-comma, and |~) is a two-flags-on-the-same- side symbol >>> I'm proposing to stand for the 13:19-comma (and possibly the 5:13-comma). >>> But if you'd rather, I'll just accept that 282-ET and 294-ET
are not notatable. I find 282 a little difficult, but still notatable. If we don't use |\, then we can't have both matching symbols and ||\ as RC of /|. With that constraint I would do 282 this way with rational complementation: 282a: |( ~| ~)| |~ /| |) )|) (| (|( //| |~) (|~ /|\ (|) ||( )||( ~|| ~||( )||~ )/|| ||) ||\ ~||) ~||\ //|| /||) /||\ (RC) The )/|| symbol is the double-shaft version of the one that I am proposing below for 306 and 494; here it is the proposed rational complement of )|). But if we use |\ with matching symbols, then I get this: 282b: |( ~| ~)| |~ /| |) )|) |\ (|( //| |~) (|~ /|\ (|) ||( ~|| ~)|| ||~ /|| ||) )||) ||\ (||( //|| ||~) (||~ /||\ (MS) But this shifts symbols such as ||) into the wrong positions and makes them almost meaningless, besides not having ||\. So I prefer 282a.
>> Yes, I think that there are too many problems. >>>
>>> However, 306-ET _is_ notatable without using any two-flags-on- the-same-side >>> symbols. Alternatives for some degrees are given on the line below. >>> >>> 306: )| |( )|( ~|( /| ~|~ |) (| |\ //| ~|\ /|)
(|~ /|\ (|)
>>> ~| ~)| |~ )|) ~|) (|( |~) >>
>> (|( is a better choice than //| for the comma roles it fulfills. >
> I guess so. Since //| only works as 5+5 comma and (|( works in all its > possible roles. > >> (|~
>> and ~|~ look like they may be a little shaky in the flag arithmetic for >> |~. (A wavy flag becomes a shaky flag?) >
> I hadn't noticed that, thanks. But in cases like this (where the only > alternative is incomplete notation, I don't think we should let flag > arithmetic stop us.
>>> 318 is notatable if you accept (/| (the 31' comma) for 15 steps. >>
>> Neither 306 nor 318 are 7-limit consistent, so I don't see much point >> in doing these, other than they may have presented an interesting >> challenge. >
> Good point. Forget 318-ET, but 306-ET is of interest for being strictly > Pythagorean. The fifth is so close to 2:3 that even god can barely tell the > difference. ;-)
What's making me hesitate about 306 is a 5 factor 49 percent of a degree false. But I tried it anyway without looking at what you have and came up with the following, which surprised me with how well it works. It eliminates the shaky flag with a new symbol )/|, which I will explain below when I discuss 494: 306: )| |( )|( ~|( /| )/| |) )|) |\ (|( |~) /|) (|~ /|\ (|) )|| (|\ )||( ~||( /|| )/| ||) )||) ||\ (||( ||~) /||) (||~ /||\ (RC & MS)
> If we can accept fuzzy arithmetic with the right wavy flag, and the > addition of the 13:19 comma symbol |~) then the 31-limit-consistent 388-ET > can be notated (but surprisingly, not 311-ET). > > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 > 388: )| |( ~| ~)| ~|( |~ /| ~|~ |) |\ (| ~|) ~|\ //| > > 15 16 17 18 19 20 21 22 > |~) /|) /|\ (/| |\) (|) (|\ ||( ... (MS) > > The symbols (/| and |\) are of course the 31-comma symbols we agreed on > long ago.
Yes, and they work quite well here, as well as in 494, below. Rational complementation doesn't work very well when /| and |\ are 3 degrees apart, so I will go along with the matching symbols, even if they don't really mean much of anything; 388 is therefore agreed! I was wondering why you said that we can't do 311. Is it because (/| is not the proper number of degrees for the 31 comma? But neither is |~ as 6deg388, the 23 comma, nor is )|~ as 8deg494 valid as the 19' comma, but you have proposed these here. And I agree with your decision, because there is no alternative. So I would do 311 thus: 311: |( )|( ~)| ~|( |~ /| |) |\ (| (|( ~|\ //| /|) /|\ (/| (|) (|\ ~|| ~)|| ~|( )|~ /|| ||) ||\ ~||) (||( ~||\ ||~) /||) /||\ (RC) I have selected the best single-shaft symbols and used their rational complements. The symbols are not matched in the half-apotomes.
> Here's another one I think should be on the list, 494-ET, if only because > of the fineness of the division, and because it shows all our rational > complements*. It is 17-limit consistent. Somewhat surprisingly, it is fully > notatable with the addition of the 13:19 comma symbol |~). It has the same > problem as 306 and 388, with right-wavy being fuzzy, taking on
values 6, 7
> and 8 here. > > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 > 494: )| |( )|( ~| ~)| ~|( |~ )|~ /| ~|~ |) )|) |\ (| ~|) > > 16 17 18 19 20 21 22 23 24 25 26 27 28 > (|( ~|\ //| |~) /|) (|~ /|\ (/| |\) (|) )|| (|\ )||( ... > (RC* & MS) > > * It agrees with all our rational complements so far, except that we'd need > to accept > ~|~ <---> )|) [where the |~ flag corresponds to 6 steps of 494] > instead of > ~|~ <---> /|( > which might become an alternative complement. > > and we'd need to add > > )|( <---> |~) [where the |~ flag corresponds to 8 steps of 494] > > In all other symbols above, the |~ flag corresponds to 7 steps of 494. > > My interpretations are > ~|~ 5:19 comma > )|) 7:19 comma > )|( 19 comma + 5:7 comma > |~) 13:19 comma > > Obviously these symbols should be the last to be chosen for any purpose. > > So we see that the addition of that one new symbol |~) for the 13:19 comma > and the acceptance of a fuzzy right wavy flag, lets the maximum notatable > ET leap from 217 to 494, more than double! > > So who cares about notating 282, 388 and 494? I dunno, but here's a funny > thing: The difference between them is 106. 176 is the next one down.
And (surprise!) 600 is the next one up (but 7 and 17 are really bad). All I can say about 106 is that it's twice 53. I first found 494 in the 1970s when I was looking for a division with a low-error 17 limit. I noticed that two excellent 7-limit divisions, 99 and 171, have their 11 errors in opposite directions, so in their sum, 270, they cancel out (reckoned as fractions of a degree). For the 13 limit both 224 and 270 are good, but their 17 errors are in opposite directions, so in their sum, 494, they also cancel out. (Also note their difference of 46, which is also quite good for the 17 limit.) But I digress. I have a problem changing ~|~ to represent 10deg494 in that it must be given a different complement to make this work. The proposed complement, )||), has an offset of -2.64 cents, large enough that it would be invalid in most other larger divisions. This would also make the complementation we previously had for ~|~ <--> /||( and /|( <--> ~||~ (offset of 0.49 cents) unavailable for other divisions such as 342 and 388 (except as an alternate complement). Instead of ~|~ I propose )/| for 10deg494 (and 6deg306 above), which is the correct number of degrees and has the actual flags for the 5:19 comma (hence is easy to remember; besides, the symbol that I made for this looks pretty good). This also makes a consistent complement to )||) in 282, 306 and 494 (the three places where I have found a use for it); the offset of -2.25 cents is still rather large, but not as much as before. (It makes a nice alternate complement with /|( with an offset of 0.88 cents.) It also restricts the fuzzy arithmetic to only one symbol, |~), which has its two flags on the same side. This would put the total number of single-shaft symbols at 30, and the only symbols that would be left without rational complements are )|\ and /|~. I don't object to the fuzzy |~) arithmetic for 19deg494, because this makes it consistent with its proposed complement )||(, which has an offset of only 0.09 cents (and would probably be valid a lot of other places). The symbol does somewhat resemble |\), but I believe that the two are sufficiently different in size that this shouldn't cause any problem. So I get: 494: )| |( )|( ~| ~)| ~|( |~ )|~ /| )/| |) )|) |\ (| ~|) (|( ~|\ //| |~) /|) (|~ /|\ (/| |\) (|) )|| (|\ )||( ~|| ~)|| ~||( ||~ )||~ /|| )/|| ||) )||) ||\ (|| ~||) (||( ~||\ //|| ||~) /||) (||~ /||\ (RC & MS) The only irregularities with this are the fuzzy symbol arithmetic with |~) and ||~) and the fact that )|~ is not valid as the 19' comma. Considering that 19 is not well represented in 494 and that the 19' comma will be the much less used of the two 19 commas, I think that this is inconsequential. I tried messing around with some 3-flag symbols as alternatives to |~), which would eliminate the remaining fuzzy symbol arithmetic. Since )/| looked so good, I tried ~|\( for the 37 comma for 19deg494, which seems pretty easy to distinguish from everything else. As a u- d complement to )|( it has an offset of -2.60 cents, rather large, so it's not valid in a lot of other places. I eventually decided that it wasn't worth it, especially since the symbol would have 3 flags, so I would stick with your proposal for |~). However, I am intrigued by the idea of )|)), the 19+7^2 diesis, as being very close to half an apotome (and thus its own rational complement); this would be very useful in a lot of places, e.g., 270, 311, and 400. We may have to explore this a bit more, or at least leave open the possibility of future expansion, i.e., more flag combinations. I figure that the more bells and whistles we have, the less likely it is that anybody is ever going to use all of them.
> Here's another big one we can notate this way. Only 11-limit consistent, > but its relative accuracy at that limit is extremely good. 342 = 2*3*3*19. > > 342: > )| |( )|( ~|( )|~ /| ~|~ |) |\ ~|) (|( //|
|~) /|) /|\ (/| (|) (|\ Agreed! I spoke about 224 and 270 above, but we don't have a notation for them. How about this: 224: |( )|( ~|( /| |) |\ (|( //| /|) /|\ (|) (|\ ~|| ~||( /|| ||) ||\ (||( ~||\ /||) /||\ (RC) 270: |( ~| ~)| )|~ /| |) |\ (| (|( //| /|) /|\ (/| (|) (|\ ~|| ~||( )||~ /|| ||) ||\ (|| ~||\ //|| /||) /||\ (RC) --George
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Message: 5276 - Contents - Hide Contents

Date: Wed, 9 Oct 2002 23:47:34

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

From: monz" :

>
>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 * [with cont.] >
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: 5278 - Contents - Hide Contents

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 * [with cont.] >> >
> 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 - Contents - Hide Contents

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 - Contents - Hide Contents

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 * [with cont.] 




-monz


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Message: 5281 - Contents - Hide Contents

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 - Contents - Hide Contents

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 - Contents - Hide Contents

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 * [with cont.] 


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

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: 5284 - Contents - Hide Contents

Date: Fri, 11 Oct 2002 19:37:16

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

From: Gene Ward Smith

--- 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, but other strange-looking possibilities are out there, such as 200 and 412 (200+412=612, of course.)
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Message: 5286 - Contents - Hide Contents

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: 5287 - Contents - Hide Contents

Date: Fri, 11 Oct 2002 22:46:56

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

From: Gene Ward Smith

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

> well, OK, but ... AARRRGGH! -- *how* did you do that search?
Brute force, much like a search for good ets. I totaled up the relative error for each n from 1 to 1000 by running a simple Maple routine, and insisted they be at least as good as 12-et.
> 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.
Mathematica is very similar to Maple, but you need to learn how to use it.
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Message: 5288 - Contents - Hide Contents

Date: Fri, 11 Oct 2002 05:08:28

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

From: Gene Ward Smith

--- 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 * [with cont.]
I used Manual's scale data rather than trying to figure out where the data was on your page. 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
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Message: 5289 - Contents - Hide Contents

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 * [with cont.] >
> 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: 5291 - Contents - Hide Contents

Date: Fri, 11 Oct 2002 05:08:28

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

From: Gene Ward Smith

--- 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? >=20 > Yahoo groups: /monz/files/dict/werckmeister.htm * [with cont.]
I used Manual's scale data rather than trying to figure out where the data = was on your page. It turns out that Werckmeister III can be expressed with = extreme accuracy in terms of what I call "schismas", steps of the 612 et. I= n 612-et terms, it is 0, 46, 98, 150, 199, 254, 300, 355, 404, 453, 508, 557
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Message: 5293 - Contents - Hide Contents

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 * [with cont.] >
> 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 - Contents - Hide Contents

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 - Contents - Hide Contents

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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Message: 5298 - Contents - Hide Contents

Date: Sat, 12 Oct 2002 22:25:30

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

From: Gene Ward Smith

--- In tuning-math@y..., manuel.op.de.coul@e... wrote:
> > 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
Nice! Is there a way to go beyond the stop point, and *not* show all divisions? I notice 612 popping up a lot with temperaments I hadn't looked at yet, but the stop point is set too low to easily see it.
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Message: 5299 - Contents - Hide Contents

Date: Sat, 12 Oct 2002 22:34:00

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

From: Gene Ward Smith

--- In tuning-math@y..., "hs" <straub@d...> wrote:

> The base lattice (Z^2) is a Z-module (like a vector space but only integers as > coefficients and for scalar multiplication), and so is the quotient (the > elements of which are simply equivalence classes of intervals with respect to > the unison vectors). A periodicity block, BTW, is nothing else but a set of > adjacent representants of the quotient module (one representant for each > equivalence class).
I've mentioned this before, but readers used to the "abelian group" terminology should keep in mind that abelian group and Z-module mean the same thing.
> Now, the quotient module being finite...
Whups--you are sticking "2" into the mix when you conclude this. The math is more straightforward if you treat 2 as just another prime number.
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