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Message: 11026 - Contents - Hide Contents
Date: Thu, 10 Jun 2004 22:13:03
Subject: Re: The meantone family
From: Herman Miller
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.
Message: 11027 - Contents - Hide Contents
Date: Thu, 10 Jun 2004 22:39:32
Subject: Re: The diaschismic family
From: Herman Miller
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]], wouldn't 11-limit
Shrutar be something starting with <<4, -8, 14...?
<<4, -8, 14, -2, -22, 11, -17, 55, 23, -54]] would work.
Map: [[2, 3, 5, 5, 7], [0, 2, -4, 7, -1]]; it matches the description of
Shrutar on Graham Breed's catalog of linear temperaments.
It also shows up on my collection of 11-limit temperaments that have
consistent mappings of the intervals between 1/1 and 2/1 with numerators
of 11 or less (which I found by a brute force approach of running a
program to check all possible periods and generators to an accuracy of
one cent, up to 31 steps per octave).
<<6, 10, 3, 8, 2, -12, -8, -21, -16, 12]] #54
<<2, 8, 8, 12, 8, 7, 12, -4, 0, 6]] Injera
<<3, 5, -6, 4, 1, -18, -4, -28, -8, 32]] Porcupine
<<1, -13, -2, -6, -23, -6, -13, 32, 31, -10]] #109
<<4, 16, 9, 10, 16, 3, 2, -24, -32, -3]] Squares
<<8, 6, 6, -4, -9, -13, -34, -3, -30, -32]] #92
<<3, 5, -13, 4, 1, -29, -4, -44, -8, 56]] unnamed?
<<3, 5, 16, 4, 1, 17, -4, 23, -8, -44]] unnamed?
<<7, -3, 8, 2, -21, -7, -21, 27, 15, -22]] Orwell
<<5, 1, 12, -8, -10, 5, -30, 25, -22, -64]] Magic
<<4, -8, 14, -2, -22, 11, -17, 55, 23, -54]] Shrutar
<<7, 2, -11, -10, -13, -37, -40, -31, -30, 10]] unnamed?
<<5, -11, -12, -3, -29, -33, -22, 3, 31, 33]] Tritonic
<<10, 9, 7, 25, -9, -17, 5, -9, 27, 46]] Nonkleismic
<<1, 21, 15, 11, 31, 21, 14, -24, -47, -21]] unnamed?
<<6, -7, -2, 15, -25, -20, 3, 15, 59, 49]] Miracle
<<9, 5, -3, 7, -13, -30, -20, -21, -1, 30]] Quartaminorthirds (Valentine)
Message: 11028 - Contents - Hide Contents
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.
Message: 11029 - Contents - Hide Contents
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!
Message: 11030 - Contents - Hide Contents
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.
Message: 11031 - Contents - Hide Contents
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?
Message: 11032 - Contents - Hide Contents
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]
Message: 11033 - Contents - Hide Contents
Date: Fri, 11 Jun 2004 21:50:36
Subject: Re: The meantone family
From: Gene Ward Smith
--- 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.
Message: 11034 - Contents - Hide Contents
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)
Message: 11035 - Contents - Hide Contents
Date: Fri, 11 Jun 2004 22:21:25
Subject: Re: The diaschismic family
From: Gene Ward Smith
--- 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.
TM basis: {245/243, 2048/2025}
Mapping: [<2 3 5 5|, <0 2 -4 7|]
TOP generators: [599.46, 52.64]
TOP error: 1.079
L1 TOP complexity 8.438
L1 TOP badness: 76.826
There's not much of an 11-limit family tree, shrutar-22 and shrutar+46
are marginal and the rest worse. The TOP generators for 11-limit
shrutar are [599.82, 52.37], pretty close. This is a pretty clear-cut
case, it seems to me.
Naturally, this still works for higher limits; to project a 13-limit
wedgie to the 11-limit, we take
<<a1 a2 a3 a4 a6 a7 a8 a10 a11 a13||
We are simply ignoring the basis elements associated with the 13adic
valuation.
Message: 11036 - Contents - Hide Contents
Date: Fri, 11 Jun 2004 22:25:33
Subject: Re: Nexials
From: Gene Ward Smith
--- 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.
Message: 11037 - Contents - Hide Contents
Date: Fri, 11 Jun 2004 22:48:33
Subject: Re: Family commas
From: Gene Ward Smith
--- 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. Otherwise you are looking at a rank
two group [Z, Z/2Z] as the image, which doesn't make a lot of sense
musically.
>> 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.
Message: 11038 - Contents - Hide Contents
Date: Fri, 11 Jun 2004 08:06:26
Subject: Re: The diaschismic family
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx Herman Miller <hmiller@I...> wrote:
> <<3, 5, -13, 4, 1, -29, -4, -44, -8, 56]] unnamed?
> <<3, 5, 16, 4, 1, 17, -4, 23, -8, -44]] unnamed?
The 7-limit reductions of these are, in terms of nexials,
porcupine-7 and porcupine+22 respectively.
> <<7, 2, -11, -10, -13, -37, -40, -31, -30, 10]] unnamed?
TM basis {100/99, 245/242, 525/512}; 26&29.
> <<1, 21, 15, 11, 31, 21, 14, -24, -47, -21]] unnamed?
TM basis {121/120, 441/440, 686/675}. The 7-limit TM basis is
{686/675, 5120/5103}, and the 5-limit comma |31 -21 1>. This is the
29&46 system.
Message: 11040 - Contents - Hide Contents
Date: Fri, 11 Jun 2004 23:05:07
Subject: Re: The meantone family
From: Herman Miller
Paul Erlich wrote:
> --- 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.
Well, it's a tricky situation. I need to explain just enough about
wedgies so that visitors to the page have at least a rough idea of what
those strange numbers by the temperament names are and why they're of
any use, without introducing all sorts of other concepts that aren't
relevant to the page. After all, I'm using them more or less as unique
"index numbers" for temperaments, for cross-reference purposes. But I
can see how the description might be misleading. I do remark that
wedging commas isn't the same as wedging maps (i.e., monzos vs. vals); a
more in-depth explanation would reveal that the difference involves
taking the complement, but I didn't want to get into that much detail.
How about changing "you can wedge them to get <<1, 4, 10, 4, 13, 12]]"
to something like "you can wedge them and ultimately end up with <<1, 4,
10, 4, 13, 12]]"? Or would it be better to add more detail and go
through a process like "You can represent 81/80 as [-4 4 -1 0> and
126/125 as [1 2 -3 1>; taking the wedge product gives a result of [[12,
-13, 4, 10, -4, 1>> which after some further manipulation gives a
normalized result of <<1, 4, 10, 4, 13, 12]]"?
I guess really the only way around it is to write a page on linear
temperament theory and link to that. But that's another project for a
later date.
Message: 11041 - Contents - Hide Contents
Date: Fri, 11 Jun 2004 23:35:31
Subject: Re: The diaschismic family
From: Herman Miller
Paul Erlich wrote:
> 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.
I was just going from Gene's Jan. 21 list of 114 7-limit temperaments,
which identified <<4, -8, 14, -22, 11, 55]] as "Shrutar". The map of
[[2, 3, 5, 5, 7], [0, 2, -4, 7, -1]] is consistent with 896/891 and
2048/2025 vanishing; this temperament has a 52 cent generator and
half-octave period, consistent with the description of (the original
11-limit) Shrutar at Catalogue of linear temperaments * [with cont.] (Wayb.).
Message: 11042 - Contents - Hide Contents
Date: Sat, 12 Jun 2004 00:50:33
Subject: 41 "Hermanic" 7-limit linear temperaments (was: Re: 114 7-limit temperaments)
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> Now updating to include last 3 additions:
Is this now the full list?
>>> 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.
>> IN
>>> Number 11 Schismic
>>>
>>> [1, -8, -14, -15, -25, -10] [[1, 2, -1, -3], [0, -1, 8, 14]]
>>> TOP tuning [1200.760625, 1903.401919, 2784.194017, 3371.388750]
>>> TOP generators [1200.760624, 498.1193303]
>>> bad: 8.5260 comp: 5.618543 err: .912904
This isn't too close to 5-limit schismic; I'm suggesting septishis or
something.
>> IN
>>> Number 14 Augmented
>>>
>>> [3, 0, 6, -7, 1, 14] [[3, 5, 7, 9], [0, -1, 0, -2]]
>>> TOP tuning [1199.976630, 1892.649878, 2799.945472, 3385.307546]
>>> TOP generators [399.9922103, 107.3111730]
>>> bad: 8.7811 comp: 2.147741 err: 5.870879
Augie?
>> 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?
>> 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?
>> 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.
> That's all, folks! All 28 I'm including were in the top 30 of
> the "Hermanic" list I posted 3 and a half months ago . . .
Could we have the final lists posted?
Message: 11043 - Contents - Hide Contents
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.
Message: 11044 - Contents - Hide Contents
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.
Message: 11045 - Contents - Hide Contents
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.
Message: 11046 - Contents - Hide Contents
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?
Message: 11047 - Contents - Hide Contents
Date: Sat, 12 Jun 2004 02:02:28
Subject: Re: The diaschismic family
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> 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.
It's h46/\h22 =<<4 -8 14 -2 -22 11 -17 55 23 -54||, and 896/891 and
2048/2025 do indeed vanish, though the TM basis is actually
{121/120, 176/175, 245/243}.
Message: 11048 - Contents - Hide Contents
Date: Sat, 12 Jun 2004 02:03:41
Subject: Re: Nexials
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
>>
>
>>> 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?
OK, no. :)
Message: 11049 - Contents - Hide Contents
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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