Temperament seaches using wedgies only

I did a 7-limit temperament search using only wedgie methods, and it went very quickly, so I am going to push it farther. First, however, I'd like some feedback.

I started from <<i j k ...||, with |i|<=57, |j|<=83, |k|<=100. I then extended this to a wedgie by wedging <0 i j K| with <1 p3 p5 p7|, where the pi are log2(prime), and rounding to the nearest integer, getting <<a1 a2 a3 a4 a5 a6||. I tested if this was an actual bival by checking if a1*a6-a2*a5+a3*a4 = 0. If it was, I reduced the thing to a standard wedgie, and then computed a logflat wedgie badness, defined as

rank two badness = (||w^J||/sqrt(n)) * ||w||^(2/(n-2))

Here n is the number of primes in the prime limit, so n=4 in this case and we get

badness = (||w^J||/2) * ||w||

Here w is the weighted wedgie, and J = <1 1 1 1|.

After eliminating everything with a badness score over 1/10, I got the following list, giving name, wedgie, and badness. It seems to me some of the unnamed temperaments are worthy of consideration.

ennealimmal [18 27 18 1 -22 -34] .0088435418089696885579
supermajor [37 46 75 -13 15 45] .026543157974387598427
enneadecal [19 19 57 -14 37 79] .026832031985490745136
sesquiquartififths [4 -32 -15 -60 -35 55] .027542717576045142214
tertiaseptal [22 -5 3 -59 -57 21] .031831551710259372284
meantone [1 4 10 4 13 12] .033576042872227173750
no name [33 78 90 47 50 -10] .034504309118488562018
pontiac [1 -8 39 -15 59 113] .034617762502821380063
no name [55 73 93 -12 -7 11] .036925968737179370934
no name [14 59 33 61 13 -89] .040995164725513287308
miracle [6 -7 -2 -25 -20 15] .041009960000702024453
no name [15 51 72 46 72 24] .041185711149189375697
beep [2 3 1 0 -4 -6] .045653799875247522090
magic [5 1 12 -10 5 25] .046339704004986845808
no name [0 2 2 3 3 -1] .047453786308905762605
dicot [2 1 3 -3 -1 4] .048829872997826245170
term [3 -24 -54 -45 -94 -58] .048868070382821418532
pajara [2 -4 -4 -11 -12 2] .049071374294939127432
hemiwuerschmidt [16 2 5 -34 -37 6] .049741937737284037808
dominant [1 4 -2 4 -6 -16] .050679669572590145120
orwell [7 -3 8 -21 -7 27] .050791239759919002996
father [1 -1 3 -4 2 10] .052203479931224961019
catakleismic [6 5 22 -6 18 37] .052667158516598435633
garibaldi [1 -8 -14 -15 -25 -10] .053017597436298734989
hemififths [2 25 13 35 15 -40] .054483592578059287625
diminished [4 4 4 -3 -5 -2] .054871417886598491093
no name [1 2 1 1 -1 -3] .054950842151695402173
no name [1 0 1 -2 -1 2] .055417554000827271634
no name [1 1 1 -1 -1 0] .056617360390913000797
neptune [40 22 21 -58 -79 -13] .057383951044240165788
no name [0 0 1 0 2 2] .057712803594492961710
amity [5 13 -17 9 -41 -76] .057928962599295443222
no name [1 -1 0 -4 -3 3] .058596599872625268258
no name [1 2 3 1 2 1] .058695303078569718055
mother [1 -1 -2 -4 -6 -2] .059160037375258796686
no name [0 1 0 2 0 -3] .059606131422749986582
augene [3 0 -6 -7 -18 -14] .060786201399615867500
no name [1 1 0 -1 -3 -3] .060852731968265522580
sharptone [1 4 3 4 2 -4] .060863829795678222166
mitonic [17 35 -21 16 -81 -147] .061688229146787075789
no name [2 1 -1 -3 -7 -5] .062170143550530937255
sensi [7 9 13 -2 1 5] .062760062697460327001
blacksmith [0 5 0 8 0 -14] .062805016195047016959
no name [23 -13 42 -74 2 134] .063295104220269201032
august [3 0 6 -7 1 14] .064812271333133398656
negri [4 -3 2 -14 -8 13] .064869954663840959102
no name [1 1 2 -1 0 2] .065449280653394808127
semaphore [2 8 1 8 -4 -20] .065517440821311467590
myna [10 9 7 -9 -17 -9] .066244067682785170638
keemun [6 5 3 -6 -12 -7] .067136241448043576591
parakleismic [13 14 35 -8 19 42] .067193154495232664698
decimal [4 2 2 -6 -8 -1] .069403796983723031308
mutt [21 3 -36 -44 -116 -92] .069580916598702692105
no name [1 2 0 1 -3 -6] .069697278831743886129
no name [0 0 3 0 5 7] .070234938128662398287
sharp [2 1 6 -3 4 11] .070893754890569808508
no name [0 0 2 0 3 5] .071150102239865996936
no name [21 56 -77 40 -181 -336] .071333903167321860511
no name [41 14 60 -73 -20 100] .071688215165355985601
no name [5 -40 24 -75 24 168] .075191224258516782502
no name [3 29 -95 39 -159 -302] .075571757638310008357
valentine [9 5 -3 -13 -30 -21] .076071683287586520518
injera [2 8 8 8 7 -4] .076252833373444862989
superpyth [1 9 -2 12 -6 -30] .079163522175890294473
octacot [8 18 11 10 -5 -25] .082902877087567710674
no name [20 -30 -10 -94 -72 61] .083041752202331554639
harry [12 34 20 26 -2 -49] .083470470889225375035
no name [0 1 1 2 2 0] .085124620115314044806
no name [39 83 -59 41 -203 -370] .086793510978608975313
compton [0 12 24 19 38 22] .087411648537062961210
no name [1 0 0 -2 -3 0] .087419450960145542180
quasiorwell [38 -3 8 -93 -94 27] .087769697809951034545
no name [1 -1 1 -4 -1 5] .089207181830935273535
octokaidecal [2 6 6 5 4 -3] .090012152300061693196
misty [3 -12 -30 -26 -56 -36] .090145511047380827593
no name [4 21 -56 24 -100 -189] .090593294938592443471
no name [22 48 -38 25 -122 -223] .090630234506419815348
rodan [3 17 -1 20 -10 -50] .090904936486659899125
mothra [3 12 -1 12 -10 -36] .090988220918553563877
no name [2 -1 1 -6 -4 5] .091138598189929348973
no name [34 29 23 -33 -59 -28] .091419854858945765224
no name [0 2 -2 3 -3 -10] .091946908896390849259
no name [2 1 -4 -3 -12 -12] .092014646882220038289
gamera [23 40 1 10 -63 -110] .092218248573179013168
diaschismic [2 -4 -16 -11 -31 -26] .092870774684982262522
no name [40 75 -20 26 -144 -257] .093315036173585053875
no name [1 0 -1 -2 -4 -2] .093352084694613673393
unidec [12 22 -4 7 -40 -71] .094042546293449117203
flattone [1 4 -9 4 -17 -32] .094434671390882865736
mavila [1 -3 -4 -7 -9 -1] .094699273102126715892
no name [24 20 16 -24 -42 -19] .096094993650177715510
no name [0 2 0 3 0 -6] .097028899702290217408
opossum [3 5 9 1 6 7] .099572528622990912999

Gene wrote:

>I did a 7-limit temperament search using only wedgie methods, and it
>went very quickly, so I am going to push it farther. First, however,
>I'd like some feedback.
>
>I started from <<i j k ...||, with |i|<=57, |j|<=83, |k|<=100.
>I then extended this to a wedgie

It isn't already a wedgie?

>by wedging <0 i j K| with <1 p3 p5 p7|,
>where the pi are log2(prime), and rounding to the nearest integer,
>getting <<a1 a2 a3 a4 a5 a6||.

Ah, right, the back bit. 100^6 I suppose is a large number.
Or so I was told when I suggested working directly from wedgies
5+ years ago. Though a cutoff far below 100 should catch
everything (I'm interested in, anyway).

>I tested if this was an actual bival by
>checking if a1*a6-a2*a5+a3*a4 = 0.

Somehow this restricts us to rank 2 temperaments. Do you ever
reveal how you arrive at these crazy polynomials?

>If it was, I reduced the thing to a
>standard wedgie, and then computed a logflat wedgie badness, defined as
>
>rank two badness = (||w^J||/sqrt(n)) * ||w||^(2/(n-2))
>
>Here n is the number of primes in the prime limit, so n=4 in this case
>and we get
>
>badness = (||w^J||/2) * ||w||
>
>Here w is the weighted wedgie, and J = <1 1 1 1|.

So J is the JIP. I recognize the 2nd term as the complexity,
so the 1st must be the error. Is this w^J only valid with a
Euclidean norm? Looks like the thing Graham supplied. Is there
a xenwiki on it?

>After eliminating everything with a badness score over 1/10,

Why not just combinate the entire wedgie with a smaller cutoff?
It looks like by using <0 i j K| with <1 p3 p5 p7| back at step
one you're only getting true linear temperaments. But nope, huh,
you've got pajara in your results...?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> >I tested if this was an actual bival by
> >checking if a1*a6-a2*a5+a3*a4 = 0.
>
> Somehow this restricts us to rank 2 temperaments. Do you ever
> reveal how you arrive at these crazy polynomials?

Right. It's in the literature in terminology involving things called Grassmann varieties and the Plucker embedding. Not to mention Pfaffians, Plucker ideals, and complete intersections of quadric varieties. So if I was to do something on it, it would involve translation, but the actual ideas, as is so often is the case, isn't nearly as nasty as the terminology or the highbrow math lurking in the background.

> >badness = (||w^J||/2) * ||w||
> >
> >Here w is the weighted wedgie, and J = <1 1 1 1|.
>
> So J is the JIP. I recognize the 2nd term as the complexity,
> so the 1st must be the error.

Graham calls it badness, but with what qualifier I don't know. It isn't logflat badness, which is what I used. Error would be

(||w^J||/2)/||w||

Is this w^J only valid with a
> Euclidean norm?

It can be used with other norms; the arguments favoring its relevance may not be as clear, but using L-infinity (max) norms makes sense.

Looks like the thing Graham supplied. Is there
> a xenwiki on it?

Not yet, but I guess I've gotten everything out of Graham I'm going to get, so I should write one.

> >After eliminating everything with a badness score over 1/10,
>
> Why not just combinate the entire wedgie with a smaller cutoff?

If I knew what combinating a wedgie was I might do that.

> It looks like by using <0 i j k| with <1 p3 p5 p7| back at step
> one you're only getting true linear temperaments. But nope, huh,
> you've got pajara in your results...?

Just stick <0 2 -4 -4| in the extension machine and there it is.

At 08:05 PM 6/12/2010, you wrote:

>>> badness = (||w^J||/2) * ||w||
>>>
>>> Here w is the weighted wedgie, and J = <1 1 1 1|.
>>
>> So J is the JIP. I recognize the 2nd term as the complexity,
>> so the 1st must be the error.
>
> Graham calls it badness, but with what qualifier I don't know. It
> isn't logflat badness, which is what I used.

You simplified the 2nd term to ||w|| from

||w||^(2/(n-2))

based on the usual pi - rank formalism for logflat and ||w||
certainly looks like complexity. So I think it's logflat.
And the 1st term

||w^J||/sqrt(n)

must be an error.

> Error would be
>
> (||w^J||/2)/||w||

I don't get where the denominator is coming from here. Maybe
you can decompose it fresh so we aren't lost in quotes.

>Not yet, but I guess I've gotten everything out of Graham I'm going
>to get, so I should write one.

Sweet.

>>> After eliminating everything with a badness score over 1/10,
>>
>> Why not just combinate the entire wedgie with a smaller cutoff?
>
> If I knew what combinating a wedgie was I might do that.

Trying all combinations of wedgie terms < cutoff, just as you did,
but for all terms, not just the front half. However, your results
had wedgies with numbers like -336 in the back half, so maybe you
don't want to do that. Such temperaments have complexity far too
high for my taste, I suspect.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:

> > Error would be
> >
> > (||w^J||/2)/||w||
>
> I don't get where the denominator is coming from here. Maybe
> you can decompose it fresh so we aren't lost in quotes.

If I scale everything in w by 2, and use 2*w instead of w, the error should be the same. The denominator knocks out any dependence on w; another way of writing it would be ||w/||w|| ^ J/||J|| ||, so that it's a wedge of two unit vectors.

> >> Why not just combinate the entire wedgie with a smaller cutoff?
> >
> > If I knew what combinating a wedgie was I might do that.
>
> Trying all combinations of wedgie terms < cutoff, just as you did,
> but for all terms, not just the front half.

Since that would make everything far, far harder an get me exactly zero more wedgies than what I did, there's no point to it.

genewardsmith wrote:

> After eliminating everything with a badness score over 1/10, I got the following list, giving name, wedgie, and badness. It seems to me some of the unnamed temperaments are worthy of consideration.

Most of these are showing up in my recent list, so I've got ET equivalents and generator mappings for them. You might want to try adding error and complexity cutoffs. Many of these are high on the complexity range:

> no name [33 78 90 47 50 -10] .034504309118488562018
171&210 [<3, 9, 17, 20], <0, -11, -26, -30]>

> no name [55 73 93 -12 -7 11] .036925968737179370934
171&179 [<1, -19, -25, -32], <0, 55, 73, 93]>

> no name [14 59 33 61 13 -89] .040995164725513287308
58&113 [<1, -3, -17, -8], <0, 14, 59, 33]>

> no name [15 51 72 46 72 24] .041185711149189375697
60&111 [<3, 3, 1, 0], <0, 5, 17, 24]>

> no name [23 -13 42 -74 2 134] .063295104220269201032
22&149 [<1, 11, -3, 20], <0, -23, 13, -42]>

> no name [21 56 -77 40 -181 -336] .071333903167321860511
7&217 [<7, 11, 16, 20], <0, 3, 8, -11]>

> no name [41 14 60 -73 -20 100] .071688215165355985601
50&121 [<1, -14, -3, -20], <0, 41, 14, 60]>

> no name [5 -40 24 -75 24 168] .075191224258516782502
17&77 [<1, 1, 7, 0], <0, 5, -40, 24]>

> no name [3 29 -95 39 -159 -302] .075571757638310008357
53&335 [<1, 3, 16, -42], <0, -3, -29, 95]>

> no name [20 -30 -10 -94 -72 61] .083041752202331554639
10&120 [<10, 16, 23, 28], <0, -2, 3, 1]>

Others have too much error to be useful as temperaments, e.g.:

> no name [0 2 2 3 3 -1] .047453786308905762605
2&4 [<2, 3, 5, 6], <0, 0, -1, -1]>

> no name [1 2 1 1 -1 -3] .054950842151695402173
[<1, 2, 3, 3], <0, -1, -2, -1]>

> no name [1 0 1 -2 -1 2] .055417554000827271634
[<1, 2, 2, 3], <0, -1, 0, -1]>

> no name [1 1 1 -1 -1 0] .056617360390913000797
[<1, 2, 3, 3], <0, -1, -1, -1]>

> no name [0 0 1 0 2 2] .057712803594492961710

> no name [1 -1 0 -4 -3 3] .058596599872625268258
1&2 [<1, 2, 2, 3], <0, -1, 1, 0]>

> no name [1 2 3 1 2 1] .058695303078569718055
[<1, 2, 3, 4], <0, -1, -2, -3]>

> no name [0 1 0 2 0 -3] .059606131422749986582
[<1, 2, 3, 3], <0, 0, -1, 0]>

> no name [1 1 0 -1 -3 -3] .060852731968265522580
[<1, 2, 3, 3], <0, -1, -1, 0]>

> no name [2 1 -1 -3 -7 -5] .062170143550530937255
3&4 [<1, 1, 2, 3], <0, 2, 1, -1]>

At least one of them looks reasonably good and has come up in lists before, but I'm not aware of a name for it.

[<1, 1, 2, 4], <0, 2, 1, -4]> [[2 1 -4 -3 -12 -12>>

Notable for their absence:

[<1, 2, 2, 3], <0, -2, 2, -1]> [[2 -2 1 -8 -4 8>>
[<1, 2, 1, 2], <0, -1, 3, 2]> [[1 -3 -2 -7 -6 4>>
[<1, 2, 3, 2], <0, -3, -5, 6]> [[3 5 -6 1 -18 -28>> porcupine
[<2, 5, 6, 7], <0, -4, -3, -3]> [[8 6 6 -9 -13 -3>> doublewide
[<1, 0, 1, 4], <0, 12, 10, -9]> [[12 10 -9 -12 -48 -49>> hemikleismic
[<2, 1, 5, 2], <0, 6, -1, 10]> [[12 -2 20 -31 -2 52>> wizard

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> At least one of them looks reasonably good and has come up in lists
> before, but I'm not aware of a name for it.
>
> [<1, 1, 2, 4], <0, 2, 1, -4]> [[2 1 -4 -3 -12 -12>>

I was just discussing the dicot family on rhe Xenwili, and this isn't in there, and should be, I suppose. Adds 64/63 to 25/24.

> Notable for their absence:
>
> [<1, 2, 2, 3], <0, -2, 2, -1]> [[2 -2 1 -8 -4 8>>

Badness 0.10856

> [<1, 2, 1, 2], <0, -1, 3, 2]> [[1 -3 -2 -7 -6 4>>

Badness 0.10462

> [<1, 2, 3, 2], <0, -3, -5, 6]> [[3 5 -6 1 -18 -28>> porcupine

Badness 0.10057

> [<2, 5, 6, 7], <0, -4, -3, -3]> [[8 6 6 -9 -13 -3>> doublewide

Badness 0.10646

> [<1, 0, 1, 4], <0, 12, 10, -9]> [[12 10 -9 -12 -48 -49>> hemikleismic

Badness 0.12751

> [<2, 1, 5, 2], <0, 6, -1, 10]> [[12 -2 20 -31 -2 52>> wizard

Badness 0.10005

A lot of these are very close to the line, especially wizard. One thing to recall about some of these, and wizard is a good example, is that they aren't really notable as 7-limit temperaments. Wizard is more an 11-limit temperament, and not bad in the 13-limit either.

The bigger news is that none of the things not on the list fell off because of an error in the method of generating wedgies, which I told Carl would get all of them, and I am fairly sure that is true. The intermediate region, between low and high complexity, is as usual the only palce anyone would want to push badness scores up a bit. Perhaps next time I'll revert to the method of listing by complexity, that way you can ignore both ends.

What's the name for that {25/24, 64/63} temperament?

On 12 June 2010 22:29, genewardsmith <genewardsmith@sbcglobal.net> wrote:
> I did a 7-limit temperament search using only wedgie methods, and it went very quickly, so I am going to push it farther. First, however, I'd like some feedback.
>
> I started from <<i j k ...||, with |i|<=57, |j|<=83, |k|<=100. I then
> extended this to a wedgie by wedging <0 i j K| with
> <1 p3 p5 p7|, where the pi are log2(prime), and rounding to the
> nearest integer, getting <<a1 a2 a3 a4 a5 a6||. I tested if this
> was an actual bival by checking if a1*a6-a2*a5+a3*a4 = 0. If it
> was, I reduced the thing to a standard wedgie, and then
> computed a logflat wedgie badness, defined as

So you're choosing all values for the octave-equivalent part of the
wedgie? You can make that more efficient if you enforce a complexity
cutoff. Then, you know exactly which values of j will work with your
chosen i, and which values of k will work with your i and j.

> rank two badness = (||w^J||/sqrt(n)) * ||w||^(2/(n-2))
>
> Here n is the number of primes in the prime limit, so n=4 in this case and we get
>
> badness = (||w^J||/2) * ||w||
>
> Here w is the weighted wedgie, and J = <1 1 1 1|.

It looks right. Why is that factor of 2 (or sqrt(n)) in there? It
never makes a difference comparing like with like.

Graham

On 13 June 2010 07:05, genewardsmith <genewardsmith@sbcglobal.net> wrote:

>> >badness = (||w^J||/2) * ||w||
>> >
>> >Here w is the weighted wedgie, and J = <1 1 1 1|.
>>
>> So J is the JIP.  I recognize the 2nd term as the complexity,
>> so the 1st must be the error.
>
> Graham calls it badness, but with what qualifier I don't know. It
> isn't logflat badness, which is what I used.

It's scalar badness. I've also called it simple badness and 0-badness.

Graham

Graham wrote:

>>> >badness = (||w^J||/2) * ||w||
>>> >
>>> >Here w is the weighted wedgie, and J = <1 1 1 1|.
>>>
>>> So J is the JIP. I recognize the 2nd term as the complexity,
>>> so the 1st must be the error.
>>
>> Graham calls it badness, but with what qualifier I don't know. It
>> isn't logflat badness, which is what I used.
>
>It's scalar badness. I've also called it simple badness and 0-badness.

It isn't clear which of the two terms Gene was talking about.
I assumed the 1st since that's where he clipped it. But neither
of the terms can be a badness or you wouldn't need the other one.
So I can only conclude what I wrote was completely ignored and
people have just gone on talking about the whole expression.

-Carl

On 13 June 2010 10:10, Carl Lumma <carl@lumma.org> wrote:

> It isn't clear which of the two terms Gene was talking about.
> I assumed the 1st since that's where he clipped it.  But neither
> of the terms can be a badness or you wouldn't need the other one.
> So I can only conclude what I wrote was completely ignored and
> people have just gone on talking about the whole expression.

||w^J|| is (proportional to) scalar badness. You still need a
complexity term because it isn't a very useful badness, and Gene wants
logflat badness. In general, badness is a function of error and
complexity. In this case, scalar badness is easier to calculate than
the error, so the useful badness is a function of the simple badness
and complexity.

Graham

Graham wrote:

>> It isn't clear which of the two terms Gene was talking about.
>> I assumed the 1st since that's where he clipped it. But neither
>> of the terms can be a badness or you wouldn't need the other one.
>> So I can only conclude what I wrote was completely ignored and
>> people have just gone on talking about the whole expression.
>
>||w^J|| is (proportional to) scalar badness. You still need a
>complexity term because it isn't a very useful badness, and Gene wants
>logflat badness. In general, badness is a function of error and
>complexity. In this case, scalar badness is easier to calculate than
>the error, so the useful badness is a function of the simple badness
>and complexity.

Thank you! -Carl

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> So you're choosing all values for the octave-equivalent part of the
> wedgie? You can make that more efficient if you enforce a complexity
> cutoff.

I thought restricting i, j, and k *was* a complexity cutoff. Why do I need another one?

> > badness = (||w^J||/2) * ||w||
> >
> > Here w is the weighted wedgie, and J = <1 1 1 1|.
>
> It looks right. Why is that factor of 2 (or sqrt(n)) in there? It
> never makes a difference comparing like with like.

The point is, it should help when comparing unlike to unlike. The numbers you get for one prime limit should be more like what you get for another.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> > Graham calls it badness, but with what qualifier I don't know. It
> > isn't logflat badness, which is what I used.
>
> It's scalar badness. I've also called it simple badness and 0-badness.

Simple badness sounds good. Let's not call it "scalar badness" unless you can justify the word "scalar".

--- In tuning-math@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> > > badness = (||w^J||/2) * ||w||
> > >
> > > Here w is the weighted wedgie, and J = <1 1 1 1|.
> >
> > It looks right. Why is that factor of 2 (or sqrt(n)) in there? It
> > never makes a difference comparing like with like.
>
> The point is, it should help when comparing unlike to unlike. The numbers you get for one prime limit should be more like what you get for another.
>

To get that idea to actually work, though, I also need to rescale complexity by using ||w||/(n(n-1)/2) to make it an average size of component, it strikes me. Better check things out, I suppose.