Tuning-Math Digests messages 11050 - 11074

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Message: 11050

Date: Sat, 12 Jun 2004 02:12:40

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:
> 
> > 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}.

Oh, so it's not considered part of the diaschismic family because its 
generator is only half the diaschismic one (yes?) . . .


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Message: 11052

Date: Sat, 12 Jun 2004 02:32:10

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:
> > --- 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!! 
> 
> And I got the val from the commas.

But if you go directly from the commas, it's unequivocally torsion 
and not contorsion. Remember, soon after your joining, that Fokker 
periodicity block, that you initially said represented 24-equal, but 
then you retracted that and said it was really 12-equal with torsion?


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Message: 11061

Date: Mon, 14 Jun 2004 18:00:24

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:
> 
> 
> > > > > >  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.
> 
> Only at the cost of ever-increasing complexity. If we cap badness to
> something reasonable, we get only a few possibilities, or sometimes
> only one. However, here both "augie" and "tripletone" are live
> possibilities. It can be even worse than you might think in a sense,
> since more than one temperament can have the same TOP generators.
> Consider the following three versions of meantone:
> 
> <<1 4 10 4 13 12|| [<1 2 4 7|, <0 -1 -4 -10|]
> 
> <<1 4 41 4 62 84|| [<1 2 4 20|, <0 -1 -4 -41|]
> 
> <<1 4 -164 4 -262 -292|| [<1 2 4 -66|, <0 -1 -4 164|]
> 
> All have the same TOP tuning, but it is clear which one we prefer.

In some cases the choice might not be so clear.

> 
> > 
> > > > > 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.
> 
> It's from the name for the porcupine family, Erethizontidae. Don't
> know if it is better.

How about something related to a porcupine, like say "hedgehog"?


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Message: 11063

Date: Mon, 14 Jun 2004 18:02:52

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:
> 
> > But if you go directly from the commas, it's unequivocally 
torsion 
> > and not contorsion. Remember, soon after your joining, that 
Fokker 
> > periodicity block, that you initially said represented 24-equal, 
but 
> > then you retracted that and said it was really 12-equal with 
torsion?
> 
> Which is why I picked this example. If we are thinking Fokker 
blocks,
> it is torsion; however why do we have to be thinking Fokker blocks?

Who said anything about Fokker blocks? If you start with JI (as true 
temperaments do), and then temper out the two commas in question, you 
end up with 12-equal, not 24-equal. It's that simple.

> Admittedly, the contorsion interpretation may be a little 
articifical,
> but there's nothing else for it to be, basically.

??


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Message: 11064

Date: Mon, 14 Jun 2004 22:02:55

Subject: Re: Paul's nifty fifty

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 a porcupine is a member of the erethezontic family, wouldn't 
that 
> > falsely suggest that one temperament is more general than the 
other?
> 
> "Hedgehog" seems good.
> 
> > > Minorsemi from the approximate 21/20 generator, and
> > > duodecal because like waage it's got 12 on the brain.
> > 
> > What happened to "catler"?
> 
> I forgot about it. Is the attribution of this to Catler securely 
>based?

Well, his tuning is very different from TOP, but the addition of 7-
limit notes at a fixed offset from a basic 12-equal 5-limit system is 
clearly the basis behind the FreeNote 12-tone-plus guitars.

> > > [2, 8, 1, 8, -4, -20] hemifourths
> > semifourths, I think.
> 
> OK.
> 
> 
> > > 135/128 pelogic
> > mavila.
> 
> Is that one "l" or two?

One.

> > > 6561/6250 ragisemi
> > A little too wild.
> 
> "Ragisemitonic" seems worse; what about "ragitonic"? See message 
8331
> for this comma.

I remember you mentioned it was a ragisma off from 21/20. That's what 
I see as a little too wild -- bringing in all these 7-limit intervals 
when we're still in 5-limit land.

> > > 262144/253125 tertiatonic
> > meaning?
> 
> It's another 12-et comma, which I couldn't find a name for when I 
was
> considering 12-note Fokker blocks. The name comes from the fact that
> it is a third of a tone--see 
> 
> Yahoo groups: /tuning-math/message/8338 *

This seems to go against the grain -- if the comma vanishes, it's not 
a third of any nonzero interval.

> > > 393216/390625 wuerschmidt
> > > 531441/524288 aristoxenan
> > compton.
> 
> This is the second time you've wanted to change this one. :)

As I've mentioned before, Carl pointed out the Compton reference for 
this, and it seems the earliest we have.


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Message: 11065

Date: Mon, 14 Jun 2004 18:09:58

Subject: Re: Paul's nifty fifty

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:
> Here is what I am suggesting for names for Paul's list of 50
> temperaments.

Thanks.

> Septischismic in place of schismic, augie in the place
> of augmented, and erethezontic in the place of biporky have been
> discussed.

If a porcupine is a member of the erethezontic family, wouldn't that 
falsely suggest that one temperament is more general than the other?

> Minorsemi from the approximate 21/20 generator, and
> duodecal because like waage it's got 12 on the brain.

What happened to "catler"?

> 
> In the 5-limit, the names minorsemi and tertiatonic come from the
> discussion of 12-note 5-limit Fokker blocks around tuning-math 8300 
to
> 8400. Superpythagorean corresponds to the 7-limit name, and 
corrects a
> typo for the comma.
> 
> [1, 4, 10, 4, 13, 12] meantone
> [5, 1, 12, -10, 5, 25] magic
> [2, -4, -4, -11, -12, 2] pajara
> [7, 9, 13, -2, 1, 5] semisixths
> [1, 4, -2, 4, -6, -16] dominant sevenths
> [2, 8, 8, 8, 7, -4] injera
> [6, 5, 3, -6, -12, -7] kleismic
> [2, 8, 1, 8, -4, -20] hemifourths
semifourths, I think.

> [4, -3, 2, -14, -8, 13] negri
> [3, 0, -6, -7, -18, -14] augmented
> [1, -8, -14, -15, -25, -10] septischismic?
> [1, 9, -2, 12, -6, -30] superpythagorean
> [7, -3, 8, -21, -7, 27] orwell
> [3, 0, 6, -7, 1, 14] augie
> [3, 5, -6, 1, -18, -28] porcupine
> [6, 10, 10, 2, -1, -5] erethezontic?
> [3, 12, -1, 12, -10, -36] supermajor seconds
Anyone have another suggestion?

> [1, 4, -9, 4, -17, -32] flattone
> [4, 4, 4, -3, -5, -2] diminished
> [6, 10, 3, 2, -12, -21] minorsemi?
> [0, 0, 12, 0, 19, 28] duodecal?
> [3, 12, 11, 12, 9, -8] gawel
> [10, 9, 7, -9, -17, -9] nonkleismic
> [6, -7, -2, -25, -20, 15] miracle
> [2, -9, -4, -19, -12, 16] beatles
> [6, -2, -2, -17, -20, 1] lemba
> [8, 6, 6, -9, -13, -3] doublewide
> [0, 5, 0, 8, 0, -14] blackwood
> [18, 27, 18, 1, -22, -34] ennealimmal
> 
> 
> 25/24 dicot
> 81/80 meantone
> 128/125 augmented
> 135/128 pelogic
mavila.

> 250/243 porcupine
> 256/243 blackwood
> 648/625 diminished
> 2048/2025 diaschismic
> 3125/3072 magic
> 6561/6250 ragisemi
A little too wild.

> 15625/15552 hanson
> 16875/16384 negri
> 20000/19683 tetracot
> 20480/19683 superpythagorean
> 32805/32768 schismic
> 78732/78125 semisixths
> 262144/253125 tertiatonic
meaning?

> 393216/390625 wuerschmidt
> 531441/524288 aristoxenan
compton.

> 1600000/1594323 amity
> 2109375/2097152 orwell


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Message: 11073

Date: Tue, 15 Jun 2004 19:38:15

Subject: Re: Paul's nifty fifty

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:
> >>Septischismic in place of schismic, augie in the place
> >>of augmented, and erethezontic in the place of biporky have been
> >>discussed.
> > 
> > 
> > If a porcupine is a member of the erethezontic family, wouldn't 
that 
> > falsely suggest that one temperament is more general than the 
other?
> 
> Actually, "porcupine" is more general; there are New World 
porcupines 
> (Erethizontidae) (note the "i": not "Erethezontidae") and Old World 
> porcupines (Hystricidae). So if <<3, 5, 1, 1, -7, -12]] is 
tentatively 
> "Hystrix", maybe <<6, 10, 10, 2, -1, -5]] could be "Erethizon".
> 
> >>[3, 12, -1, 12, -10, -36] supermajor seconds
> > 
> > Anyone have another suggestion?
> 
> Trimeantone? (3 generators = a meantone fifth)
> Or just plain supermajor?

If you ignore the 5 axis, this temperament has been referred to 
as "wonder" and "slendric" -- correct?



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