Tuning-Math Digests messages 7951 - 7975

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Message: 7951

Date: Mon, 03 Nov 2003 16:54:14

Subject: Re: Some 11-limit TM reduced et bases

From: Carl Lumma

>This gives the basis for the corresponding standard val, and then the
>characteristic temperament (the linear temperament obtained from the
>first three of the four basis commas.)

It seems you order these largest-to-smallest.  Why do we want to leave
out the smallest comma in the basis -- my guess was we'd want to omit
the largest.

-Carl


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Message: 7952

Date: Mon, 03 Nov 2003 19:25:58

Subject: Re: Some 11-limit TM reduced et bases

From: Carl Lumma

>> It seems you order these largest-to-smallest.  Why do we want to 
>>leave out the smallest comma in the basis -- my guess was we'd want
>>to omit the largest.
>
>No, I ordered them by Tenney height, but why in the world would we 
>ditch the largest comma?

Why in the world would we ditch the highest comma?  Once again,
isn't it a combination of these two factors that decides a comma's
'goodness'?

Then again, I'm not sure of the relative benefit of ditching vs.
keeping the best commas.

-Carl


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Message: 7953

Date: Tue, 04 Nov 2003 15:35:00

Subject: Re: Some 11-limit TM reduced et bases

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >This gives the basis for the corresponding standard val, and then 
the
> >characteristic temperament (the linear temperament obtained from 
the
> >first three of the four basis commas.)
> 
> It seems you order these largest-to-smallest.

no, simplest to most complex.

> Why do we want to leave
> out the smallest comma in the basis -- my guess was we'd want to 
omit
> the largest.

he leaves out the most complex, which is intuitive. the simplest will 
have the most effect on harmonic progressions in the tuning.


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Message: 7954

Date: Tue, 04 Nov 2003 15:55:14

Subject: Re: reduced basis for 24-ET??

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> hi Gene,
> 
> 
> i tried deriving a periodicity-block for
> 24-ET from a <3,5,11>-prime-space by using
> the following unison-vectors:
> 
> 
> [2, 3, 5, 11]-monzo  ratio
> 
> 
> [ -4, 4, -1, 0]      81:80
> 
> [  7, 0, -3, 0]     128:125
> 
> [-17, 2,  0, 4]  131769:131072
> 
> 
> 
> but instead of getting a 24-tone periodicity-block,
> i got a 48-tone torsional-block.
> 
> 
> 24-ET represents ratios-of-11 so well that there
> has to be a periodicity-block hiding in here somewhere.
> can you help?
> 
> 
> 
> -monz

i'm not sure what you're asking gene. you'd like to remove the 
torsion? simple -- note that the sum of the three rows in the matrix 
above is

[-14 6 -4 4]

which is the square of

[-7 3 -2 2] 3267:3200

using this for the third row of the matrix, you get

 [ -4, 4, -1, 0]      81:80
 
 [  7, 0, -3, 0]     128:125
 
 [-7, 3,  -2, 2]    3267:3200

and the torsion is gone.


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Message: 7955

Date: Tue, 04 Nov 2003 15:59:17

Subject: Re: hey Paul

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> I'm interested in these scales...
> 
> >> [4, -3, 2, 13, 8, -14] [[1, 2, 2, 3], [0, 4, -3, 2]]
> >> complexity 14.729697 rms 12.188571 badness 2644.480844
> >> generators [1200., -125.4687958]
> >
> >25:24 chroma = -6 - 4 = -10 generators -> 10 note scale
> >graham complexity = 7 -> 6 tetrads
> 
> Not sure of the significance of the - in -125.  I realize
> that might have been Gene.

yup. anyway, this is a negri scale, yes?

> >> [4, 2, 2, -1, 8, -6] [[2, 0, 3, 4], [0, 2, 1, 1]]
> >> complexity 10.574200 rms 23.945252 badness 2677.407574
> >> generators [600.0000000, 950.9775006]
> >
> >//
> >
> >> [2, 6, 6, -3, -4, 5] [[2, 0, -5, -4], [0, 1, 3, 3]]
> >> complexity 11.925109 rms 18.863889 badness 2682.600333
> >> generators [600.0000000, 1928.512337]
> >
> >25:24 chroma = 6 - 1 = 5 generators -> 10 note scale
> >graham complexity = 3*2 = 6 -> 8 tetrads
> 
> I've never noticed "generators" being expressed as larger
> than "periods".  Why?  Can't we just reduce by the periods
> here, getting
> 
> 350.9775006
> 
> and
> 
> 600., 128.512337
> 
> resp.?
> 
> Again, sorry if this is more of a question for the poster
> of the >>'d text (Gene?).

it is. but the answer is yes, you can so reduce by the periods. gene 
was just trying to give a "hermite-reduced basis" or some such 
abstractly interesting form for the generators.

> >> [6, -2, -2, 1, 20, -17] [[2, 2, 5, 6], [0, 3, -1, -1]]
> >> complexity 19.126831 rms 11.798337 badness 4316.252447
> >> generators [600.0000000, 231.2978354]
> >
> >25:24 chroma = -2 - 3 = 5 generators -> 10 note scale
> >graham complexity = 8 -> 4 tetrads
> 
> By the way, do these temperaments have names?

many of them do, thanks to gene . . .


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Message: 7956

Date: Tue, 04 Nov 2003 16:51:45

Subject: Re: More TM base postings

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:

> ... of course, noting that the chromatic semitone disappears
> because 2^(7/24) is 350 cents, and is thus a neutral-3rd
> which represents both the major-3rd and minor-3rd on this
> particular bingo-card, which in turn means that it's not
> really doing much in the way of representing 5-limit JI
> in the first place.

well, it's still doing so at least in theory, just like all the ETs 
on the 'dicot' line on the zoom-1 and zoom-10 charts:

Definitions of tuning terms: equal temperament, (c) 1998 by Joe Monzo *

(this mapping of 24-equal would appear at the intersection of the 
aristoxenean and dicot lines; i didn't label it because 24-equal is 
consistent in the 5-limit with the same approximations as 12-equal, 
so you'll see 24 on the same point as 12 instead.)

and like those with a bingo card where the chromatic semitone 
vanishes, for example

Yahoo groups: /tuning/files/perlich/10.gif *
Yahoo groups: /tuning-math/files/Paul/7p.gif *


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Message: 7957

Date: Tue, 04 Nov 2003 08:52:57

Subject: Re: Some 11-limit TM reduced et bases

From: Carl Lumma

>he leaves out the most complex, which is intuitive. the simplest will 
>have the most effect on harmonic progressions in the tuning.

But isn't this also true for chromatic vectors?

-Carl


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Message: 7958

Date: Tue, 04 Nov 2003 09:02:43

Subject: Re: hey Paul

From: Carl Lumma

>> >> [4, -3, 2, 13, 8, -14] [[1, 2, 2, 3], [0, 4, -3, 2]]
>> >> complexity 14.729697 rms 12.188571 badness 2644.480844
>> >> generators [1200., -125.4687958]
>> >
>> >25:24 chroma = -6 - 4 = -10 generators -> 10 note scale
>> >graham complexity = 7 -> 6 tetrads
>> 
>> Not sure of the significance of the - in -125.  I realize
>> that might have been Gene.
>
>yup. anyway, this is a negri scale, yes?

Yep.

>> >> [4, 2, 2, -1, 8, -6] [[2, 0, 3, 4], [0, 2, 1, 1]]
>> >> complexity 10.574200 rms 23.945252 badness 2677.407574
>> >> generators [600.0000000, 950.9775006]
>> >
>> >//
>> >
>> >> [2, 6, 6, -3, -4, 5] [[2, 0, -5, -4], [0, 1, 3, 3]]
>> >> complexity 11.925109 rms 18.863889 badness 2682.600333
>> >> generators [600.0000000, 1928.512337]
>> >
>> >25:24 chroma = 6 - 1 = 5 generators -> 10 note scale
>> >graham complexity = 3*2 = 6 -> 8 tetrads
>> 
>> I've never noticed "generators" being expressed as larger
>> than "periods".  Why?  Can't we just reduce by the periods
>> here, getting
>> 
>> 350.9775006
>> 
>> and
>> 
>> 600., 128.512337
>> 
>> resp.?
>> 
>> Again, sorry if this is more of a question for the poster
>> of the >>'d text (Gene?).
>
>it is. but the answer is yes, you can so reduce by the periods. gene 
>was just trying to give a "hermite-reduced basis" or some such 
>abstractly interesting form for the generators.

Tx.

>> >> [6, -2, -2, 1, 20, -17] [[2, 2, 5, 6], [0, 3, -1, -1]]
>> >> complexity 19.126831 rms 11.798337 badness 4316.252447
>> >> generators [600.0000000, 231.2978354]
>> >
>> >25:24 chroma = -2 - 3 = 5 generators -> 10 note scale
>> >graham complexity = 8 -> 4 tetrads
>> 
>> By the way, do these temperaments have names?
>
>many of them do, thanks to gene . . .

Is there a way for people to look them up?

-Carl


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Message: 7959

Date: Tue, 04 Nov 2003 17:07:58

Subject: Re: Some 11-limit TM reduced et bases

From: Paul Erlich

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >he leaves out the most complex, which is intuitive. the simplest 
will 
> >have the most effect on harmonic progressions in the tuning.
> 
> But isn't this also true for chromatic vectors?
> 
> -Carl

not really, chromatic vectors only determine how far the temperament 
is carried out to form a scale, and can look the same for unrelated 
temperaments and scales, but the temperament itself is characterized 
by the commatic vectors. if you take one of the simplest commas which 
vanishes in the equal temperament and re-interpret it as a chromatic 
vector, you'll end up with a system that differs more strongly from 
the 'native harmony' of the equal temperament than when you do this 
with a more complex comma. for example, 31 in the 5-limit is [81/80, 
393216/390625], and making the 393216/390625 chromatic maintains the 
meantone character that dominates 31-equal's 5-limit behavior, while 
making 81/80 chromatic yields the more tenuous würschmidt system . . .


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Message: 7960

Date: Tue, 04 Nov 2003 17:18:58

Subject: Re: Some 11-limit TM reduced et bases

From: Paul Erlich

moreover, when there are three or more commatic vectors, the 
reduction definition is more arbitrary -- the simplest (or shortest 
in the lattice) comma is uniquely and unambiguously defined, but the 
rest depend on the precise reduction definition -- for example 
minkowski reduction may lead to a very simple second comma and a more 
complex third comma, while another basis may sacrifice the simplicity 
of the second comma so that the third comma comes out less 
complex . . .

--- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> 
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> > >he leaves out the most complex, which is intuitive. the simplest 
> will 
> > >have the most effect on harmonic progressions in the tuning.
> > 
> > But isn't this also true for chromatic vectors?
> > 
> > -Carl
> 
> not really, chromatic vectors only determine how far the 
temperament 
> is carried out to form a scale, and can look the same for unrelated 
> temperaments and scales, but the temperament itself is 
characterized 
> by the commatic vectors. if you take one of the simplest commas 
which 
> vanishes in the equal temperament and re-interpret it as a 
chromatic 
> vector, you'll end up with a system that differs more strongly from 
> the 'native harmony' of the equal temperament than when you do this 
> with a more complex comma. for example, 31 in the 5-limit is 
[81/80, 
> 393216/390625], and making the 393216/390625 chromatic maintains 
the 
> meantone character that dominates 31-equal's 5-limit behavior, 
while 
> making 81/80 chromatic yields the more tenuous würschmidt 
system . . .


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Message: 7961

Date: Tue, 04 Nov 2003 09:19:20

Subject: Re: Some 11-limit TM reduced et bases

From: Carl Lumma

>>>he leaves out the most complex, which is intuitive. the simplest 
>>>will have the most effect on harmonic progressions in the tuning.
>> 
>> But isn't this also true for chromatic vectors?
>> 
>> -Carl
>
>not really, chromatic vectors only determine how far the temperament 
>is carried out to form a scale, and can look the same for unrelated 
>temperaments and scales,

But still seems important, in light of the current "hey paul" thread,
and Gene's T[n] thread.

>but the temperament itself is characterized 
>by the commatic vectors. if you take one of the simplest commas which 
>vanishes in the equal temperament and re-interpret it as a chromatic 
>vector, you'll end up with a system that differs more strongly from 
>the 'native harmony' of the equal temperament than when you do this 
>with a more complex comma. for example, 31 in the 5-limit is [81/80, 
>393216/390625], and making the 393216/390625 chromatic maintains the 
>meantone character that dominates 31-equal's 5-limit behavior, while 
>making 81/80 chromatic yields the more tenuous würschmidt system . . .

Ok, I'll buy that.

-Carl


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Message: 7970

Date: Tue, 04 Nov 2003 06:46:54

Subject: Re: More TM base postings

From: monz

--- 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:
> > 
> > > the first one assumes the standard val (or something)
> > > for inconsistent ets; i'd at least show the results
> > > for other vals.
> > 
> > The 5-limit needs to be what it is in order to include 
> > 81/80, and for some of these, the 7-limit needs to be
> > what it is to avoid torsion.
> 
> yes, that process should be made explicit though -- i'd
> rather explain that 24-equal sometimes doesn't have a
> reasonable basis due to torsion, and in those cases is
> best understood as an equal halving of each 12-equal step,
> instead of using hidden rules to provide a nice-looking 
> answer.



yes, i agree totally with paul.

in fact, the precise thing that prompted me to write my
original post requesting these TM-reduced bases was that
i tried to create 41-ET with our software, and while it
did give 41edo as the temperament, it gave an 82-tone
periodicity-block for the JI scale.

as soon as i saw that, i suspected that it was due to
torsion, and sure enough, that turned out to be the case.


in the specific case of 24-ET in a [3,5]-prime-space,
it's nice to be able to see how choosing a val of h(5)=8
results in a "double 12-ET", whereas h(5)=7 results in
a true 24-tone periodicity-block ...

... of course, noting that the chromatic semitone disappears
because 2^(7/24) is 350 cents, and is thus a neutral-3rd
which represents both the major-3rd and minor-3rd on this
particular bingo-card, which in turn means that it's not
really doing much in the way of representing 5-limit JI
in the first place.



-monz


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Message: 7971

Date: Tue, 04 Nov 2003 13:16:53

Subject: Re: hey Paul

From: Carl Lumma

>> Is there a way for people to look them up?
>
>Should I put up a web page? Dave, do you have an objection?

Graham's catalog is neither complete or up to date, last I
checked.

The existence of a single resource is a lot to ask, I know...

-Carl


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Message: 7972

Date: Tue, 04 Nov 2003 07:44:29

Subject: reduced basis for 24-ET?

From: monz

hi Gene,


i tried deriving a periodicity-block for
24-ET from a <3,5,11>-prime-space by using
the following unison-vectors:


[2, 3, 5, 11]-monzo  ratio


[ -4, 4, -1, 0]      81:80

[  7, 0, -3, 0]     128:125

[-17, 2,  0, 4]  131769:131072



but instead of getting a 24-tone periodicity-block,
i got a 48-tone torsional-block.


24-ET represents ratios-of-11 so well that there
has to be a periodicity-block hiding in here somewhere.
can you help?



-monz


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Message: 7974

Date: Tue, 04 Nov 2003 07:53:43

Subject: Re: reduced basis for 24-ET??

From: monz

--- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> hi Gene,
> 
> 
> i tried deriving a periodicity-block for
> 24-ET from a <3,5,11>-prime-space by using
> the following unison-vectors:
> 
> 
> [2, 3, 5, 11]-monzo  ratio
> 
> 
> [ -4, 4, -1, 0]      81:80
> 
> [  7, 0, -3, 0]     128:125
> 
> [-17, 2,  0, 4]  131769:131072
> 
> 
> 
> but instead of getting a 24-tone periodicity-block,
> i got a 48-tone torsional-block.
> 
> 
> 24-ET represents ratios-of-11 so well that there
> has to be a periodicity-block hiding in here somewhere.
> can you help?



i tried using [14 -3 1 2] = 16384:16335 for the third
unison-vector, along with 81:80 and 125:128, and it
worked beautifully.

i got one 12-tone PB in the 11^0 [3,5]-plane, and
another very much like it in the 11^1 plane.




-monz


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