Tuning-Math Digests messages 11028 - 11052

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Message: 11028

Date: Fri, 11 Jun 2004 01:07:20

Subject: Re: Family commas

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> One way to sort out the family relationships is to use a comma 
scheme
> which I have intended for some time to discuss, but which I may have
> neglected to do. This is defining a linear temperament in terms of a
> sequence of commas, each at a succesively higher prime limit, and 
each
> with a minimal Tenney height given that all the previous commas are
> fixed. This sort of whatzit reduction, for meantone, would say
> meantone is the 81/80-temperament, dominant sevenths the [81/80,
> 36/35] temperament, septimal meantone the [81/80,
> 126/125]-temperament, flattone the [81/80, 525/512]-temperament. 
Then
> 11-limit meantone is the [81/80, 126/125, 385/384]-temperament and
> huygens the [81/80, 126/125, 99/98]-temperament. And so forth.
> 
> It should be noted that while this keeps track of the familial
> relationships, we don't necessarily get corresponding generators in
> these family trees, nor do we necessarily get rid of contorsion.

How is it possible to get contorsion from commas? I don't think it is.

> 7-limit ennealimmal is the [ennealimma, breedsma]-temperament, but 
the
> wedge product of this has a common factor of 4.

But this is torsion, not contorsion, right?

> Hemiennealimmal is
> then the [ennealimma, breedsma, lehmerisma]-temperament, again with 
a
> common factor of 4, but now with non-corresponding generators.

Don't know what "non-corresponding" means.


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Message: 11029

Date: Fri, 11 Jun 2004 19:54:09

Subject: Re: The meantone family

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:
> Paul Erlich wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
> >>Hmm, I must have the brackets backward on my web page; I have 
> > 
> > meantone 
> > 
> >>as [1, 4, 10, 4, 13, 12>.
> > 
> > 
> > These can't both be right, because taking the complement changes 
some 
> > of the signs. And in fact, your web page is incorrect. The bival 
> > bracket points to the left but has the numbers you cite. The 
bimonzo 
> > bracket points to the right but the sign on the second and fifth 
> > elements would have to be reversed. You should compute these 
yourself 
> > to see it. Try computing the wedge product of the monzos for 
81/80 
> > and 126/125, for example.
> 
> I've since updated the web page (I was intending the bival bracket 
but 
> had the brackets pointing the wrong way). Although it seems that I 
never 
> got around to uploading the corrected version with double brackets. 
> Well, it's up now.

Your webpage still seems wrong. You say:

"Take for instance the commas 81/80 and 126/125: you can wedge them 
to get <<1, 4, 10, 4, 13, 12]]."

But this is not correct. Putting aside the direction of the brackets, 
the numbers are simply not right. Either two or four of the signs 
must be reversed, depending on the order in which 81/80 and 126/125 
are wedged together. Try it!


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Message: 11030

Date: Fri, 11 Jun 2004 01:14:57

Subject: Re: The meantone family

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:
> Gene Ward Smith wrote:
> > This gives part of the family tree of the meantone family; I 
don't go
> > into cousins such as the various 11-limit versions of flattone or
> > dominant sevenths. In each limit, I propose giving the 
name "meantone"
> > to the temperament with the same TOP generators as 5-limit 
meantone.
> > The numbers before the colon give the nexial val with the base
> > meantone, plus or minus depending on whether adding or 
subtracting is
> > required. Also, the TM basis and the TOP octave and fourth.
> > 
> > meantone 81/80 comma 
> > [1201.70, 504.13]
> > 
> > 7-limit meantone family
> > 
> > 0: <<1 4 10 4 13 12|| meantone {81/80, 126/125}
> > [1201.70, 504.13]
> 
> Hmm, I must have the brackets backward on my web page; I have 
meantone 
> as [1, 4, 10, 4, 13, 12>.

These can't both be right, because taking the complement changes some 
of the signs. And in fact, your web page is incorrect. The bival 
bracket points to the left but has the numbers you cite. The bimonzo 
bracket points to the right but the sign on the second and fifth 
elements would have to be reversed. You should compute these yourself 
to see it. Try computing the wedge product of the monzos for 81/80 
and 126/125, for example.

> I can never remember which way they go. But is 
> there any significance to the double brackets?

To remind you that these are bivectors rather than vectors.


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Message: 11031

Date: Fri, 11 Jun 2004 19:55:25

Subject: Re: The diaschismic family

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> 
wrote:
> Paul Erlich wrote:
> 
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> > wrote:
> >>11-limit diaschismic family
> >>
> >>0: <<2 -4 -16 -24 -11 -31 -45 -26 -42 -12|| diaschismic {126/125,
> >>176/175, 896/891}
> >>[599.37, 103.79]
> > 
> > 
> > Shrutar?
> 
> If 7-limit Shrutar is <<4, -8, 14, -22, 11, 55]],

Hmm? There's no such thing as 7-limit Shrutar. Where are you getting 
this wedgie from?


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Message: 11032

Date: Fri, 11 Jun 2004 01:17:00

Subject: Re: The diaschismic family

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> diaschismic 2048/2025 comma
> [599.56, 104.70]
> 
> 7-limit diaschismic family
> 
> 0: <<2 -4 -16 -11 -31 -26|| diaschismic {126/125, 2048/2025}
> [599.37, 103.79]
> 
> 12: <<2 -4 -4 -11 -12 2|| pajara {50/49, 64/63}
> [598.45, 106.57]
> 
> 22: <<2 -4 6 -11 4 25|| {28/27, 525/512}
> [599.74, 115.71]
> 
> 24: <<2 -4 8 -11 7 30||  {36/35, 2048/2025}
> [598.71, 97.51]
> 
> 34: <<2 -4 18 -11 23 53|| {875/864, 2048/2025}
> [599.27, 107.35]
> 
> 11-limit diaschismic family
> 
> 0: <<2 -4 -16 -24 -11 -31 -45 -26 -42 -12|| diaschismic {126/125,
> 176/175, 896/891}
> [599.37, 103.79]

Shrutar?


> 12: <<2 -4 -16 -12 -11 -31 -26 -26 -14 22|| {56/55, 100/99, 
2048/2025}
> [598.18, 103.58]
> 
> 46: <<2 -4 -16 22 -11 -31 28 -26 65 117|| {126/125, 385/384, 
1232/1215}
> [599.74, 104.26]
> 
> 58: <<2 -4 -16 34 -11 -31 47 -26 93 151|| {126/125, 540/539, 
2048/2025}
> [599.30, 103.47]
> 
> 11-limit pajara family
> 
> 0: <<2 -4 -4 -12 -11 -12 -26 2 -14 -20|| pajara {50/49, 64/63, 
99/98}
> [598.45, 106.57]
> 
> 12: <<2 -4 -4 0 -11 -12 -7 2 14 14|| {45/44, 50/49, 56/55}
> [596.50, 101.36]
> 
> 22: <<2 -4 -4 10 -11 -12 9 2 37 42|| {50/49, 55/54, 64/63}
> [599.27, 108.86]


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Message: 11034

Date: Fri, 11 Jun 2004 01:23:38

Subject: Re: Nexials

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" 
<paul.hjelmstad@m...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
> wrote:
> > Suppose p>5 is prime and q is the next largest prime. For any p-
> limit
> > linear temperament wedgie, we can project down to a q-limit 
wedgie 
> by
> > for instance taking the wedge product of the truncated mapping. 
Its
> > elements will be a subset of certain specific p-limit elements, 
and 
> in
> > the case which will interest us, we suppose these are all 
relatively
> > prime and we have a well-behaved temperament.
> > 
> > If w is a p-limit val belonging to this temperament, and 
> > u = <0 0 ... 1| is p-limit, then both truncated u and tuncated w
> > belong to the ttemperament, and therefore u^w belongs to the 
> projected
> > temperament. Adding or subtracting multiples of it will lead to
> > wedgies which define different p-limit temperaments, but the same
> > q-limit temperament. We can say they are the same, mod the 
> truncated w.
> 
> How is u raised to w? I am confused about w. I know how to raise
> u to a bra-ket value, how do you raise a value to a value?

I believe Gene was using the symbol "^" to represent a wedge product, 
not exponentiation. Better would be to write u/\w, since /\ looks a 
lot more like the standard wedge product symbol.

Even better would be if Gene would answer your questions himself. 
(and mine)


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Message: 11043

Date: Sat, 12 Jun 2004 01:29:42

Subject: Re: The meantone family

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > Your webpage still seems wrong. You say:
> > 
> > "Take for instance the commas 81/80 and 126/125: you can wedge 
them 
> > to get <<1, 4, 10, 4, 13, 12]]."
> > 
> > But this is not correct.
> 
> It's correct shorthand; given that we see a wedgie (normalized 
bival)
> above, it means we wedge the monzos for 81/80 and 126/125, take the
> complement, and normalize to a wedgie. In other words, you can wedge
> the two intervals and get the wedgie.

Look at the sentence Herman had on his webpage immediately before 
this one, which includes the term "wedge product". In the context, 
the statement is simply not correct. Herman would have to add "and 
take the complement", at a minimum.


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Message: 11044

Date: Sat, 12 Jun 2004 01:36:25

Subject: Re: The diaschismic family

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > > If 7-limit Shrutar is <<4, -8, 14, -22, 11, 55]],
> > 
> > Hmm? There's no such thing as 7-limit Shrutar. Where are you 
getting 
> > this wedgie from?
> 
> No such thing as 7-limit Shrutar?? That's news to me.
> 
> If <<a1 a2 a3 a4 a5 a6 a7 a8 a9 a10|| is an 11-limit wedgie, then 
> <<a1 a2 a3 a5 a6 a8||, so long as gcd(a1,a2,a3,a5,a6,a8)=1, is a
> 7-limit wedgie. While we can debate the merits of names going 
uplimit,
> obviously if no other name takes precedence this deserves to be 
called
> by the 11-limit name. Hence the wedgie above is 7-limit Shrutar, as 
it
> has been called by some of us for some time now. 

Well enough, but please take a step back, Gene. Herman was starting 
with 7-limit shrutar, and trying to construct 11-limit shrutar from 
it. But the tuning was originally defined as an 11-limit one, and 
that's what I was asking about. Instead of an answer, we're going 
around in circles. All I remember for sure is that 896/891 and 
2048/2025 vanish.


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Message: 11045

Date: Sat, 12 Jun 2004 01:39:17

Subject: Re: Nexials

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
> <paul.hjelmstad@m...> wrote:
> 
> > Thanx. I can't believe I missed something this obvious. Regarding 
my 
> > second question: Is the normalization of a wedgie something that 
> > would yield a single value (number?) like 53?
> 
> Normalization of a bival to a wedgie is accomplished by dividing by
> the GCD of the coefficients and making the first nonzero one 
positive;
> this is the convention Graham and I have agreed on.

So the anwser is no, right? Gene, I think it would be really helpful 
if you tried to answer yes/no questions with a "yes" or a "no", 
because often I can't tell which it is from your answer, and others 
probably have even less idea. I'm sure it always seems obvious to you 
but don't forget that you're dealing with a bunch of blockheads like 
me.


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Message: 11046

Date: Sat, 12 Jun 2004 01:43:29

Subject: Re: Family commas

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> 
> > How is it possible to get contorsion from commas? I don't think 
it is.
> 
> Interpreting the GCD > 1 to mean contorsion rather than torsion 
seems
> to make the most sense if you are considering it as representing a
> temperament. You get torsion out of 648/625 and 2048/2025, but you
> could also count it as contorsion by taking the <24 38 56| val
> literally, as 24-equal, being contorted 12-equal, so that the 
mapping
> of the 5-limit is not surjective.

Yes, exactly! But this is not coming from commas, it's coming from a 
val!! You seem to have mentally ignored that we were talking about 
commas and somehow mentally substituted wedgies or something without 
saying so. Or . . . (??)

> > > Hemiennealimmal is
> > > then the [ennealimma, breedsma, lehmerisma]-temperament, again 
with 
> > a
> > > common factor of 4, but now with non-corresponding generators.
> > 
> > Don't know what "non-corresponding" means.
> 
> They are not approxmiately the same, nor are they mapped from the 
same
> JI intervals.

Hmm . . . can you elaborate on this with more detail, please?


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Message: 11049

Date: Sat, 12 Jun 2004 02:09:02

Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments)

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> > Now updating to include last 3 additions:
> 
> Is this now the full list?

There were 28 "INs" in this list:

[1, 4, 10, 4, 13, 12]
[5, 1, 12, -10, 5, 25]
[2, -4, -4, -11, -12, 2]
[7, 9, 13, -2, 1, 5]
[1, 4, -2, 4, -6, -16]
[2, 8, 8, 8, 7, -4]
[6, 5, 3, -6, -12, -7]
[2, 8, 1, 8, -4, -20] 
[4, -3, 2, -14, -8, 13]
[3, 0, -6, -7, -18, -14]
[1, -8, -14, -15, -25, -10]
[1, 9, -2, 12, -6, -30]
[7, -3, 8, -21, -7, 27]
[3, 0, 6, -7, 1, 14]
[3, 5, -6, 1, -18, -28] 
[6, 10, 10, 2, -1, -5]
[3, 12, -1, 12, -10, -36]
[1, 4, -9, 4, -17, -32]
[4, 4, 4, -3, -5, -2] 
[6, 10, 3, 2, -12, -21]
[0, 0, 12, 0, 19, 28]
[3, 12, 11, 12, 9, -8]
[10, 9, 7, -9, -17, -9]
[6, -7, -2, -25, -20, 15]
[2, -9, -4, -19, -12, 16]
[6, -2, -2, -17, -20, 1]>
[8, 6, 6, -9, -13, -3]
[0, 5, 0, 8, 0, -14]

Add ennealimmal and you've got the 29 7-limit ones.

There are also 21 5-limit ones:

>25/24 
>81/80 
>128/125 
>135/128 
>250/243 
>256/243 
>648/625 
>2048/2025
>3125/3072 
>6561/6250 
>15625/15552 
>16875/16384 
>20000/19683 
>20480/19638 
>32805/32768
>78732/78125
>262144/253125
>393216/390625
>531441/524288
>1600000/1594323
>2109375/2097152




> > > >  Number 10 Tripletone
> > > >  
> > > >  [3, 0, -6, -7, -18, -14] [[3, 5, 7, 8], [0, -1, 0, 2]]
> > > >  TOP tuning [1197.060039, 1902.640406, 2793.140092, 
3377.079420]
> > > >  TOP generators [399.0200131, 92.45965769]
> > > >  bad: 8.4214 comp: 4.045351 err: 2.939961
> 
> I'm suggesting calling this "augmented", since the TOP generators 
are
> close to 5-limit augmented.

Makes some sense, but somewhere out there there's something even 
closer -- an infinite number of somethings, I think.


> > > IN
> > > >  Number 16
> > > >  
> > > >  [6, 10, 10, 2, -1, -5] [[2, 4, 6, 7], [0, -3, -5, -5]]
> > > >  TOP tuning [1196.893422, 1906.838962, 2779.100462, 
3377.547174]
> > > >  TOP generators [598.4467109, 162.3159606]
> > > >  bad: 8.9422 comp: 4.306766 err: 3.106578
> 
> This one needs a name. The TM basis is {50/49, 245/243}. 
Erethezontic?

Where does that come from? We were calling this "Biporky", but of 
course we'd like something better.

> > > IN
> > > >  Number 27 -- formerly Number 82
> > > >  
> > > >  [6, -2, -2, -17, -20, 1] [[2, 2, 5, 6], [0, 3, -1, -1]]
> > > >  TOP tuning [1203.400986, 1896.025764, 2777.627538, 
3379.328030]
> > > >  TOP generators [601.7004928, 230.8749260]
> > > >  bad: 10.0002 comp: 4.619353 err: 3.740932
> 
> Didn't we decide to call this one Lemba?

Yup.

> > > IN
> > > >  Number 29
> > > >  
> > > >  [8, 6, 6, -9, -13, -3] [[2, 5, 6, 7], [0, -4, -3, -3]]
> > > >  TOP tuning [1198.553882, 1907.135354, 2778.724633, 
3378.001574]
> > > >  TOP generators [599.2769413, 272.3123381]
> > > >  bad: 10.1077 comp: 5.047438 err: 3.268439
> 
> Doublewide.

Sure -- but why? Wide what?


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