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Message: 2900 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 21:11:27

Subject: Re: Some 7-limit superparticular pentatonics

From: clumma


>> >ene, are you allowing 9-limit edges?
> 


> No, but it would be easy enough to do so.



Oh, it wasn't a request.  I just wanted to make
sure we were on the same page, as I seem to remember
you using prime limits in the past.

-Carl
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Message: 2901 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 03:23:57

Subject: Re: Some 10 note 22 et scales

From: genewardsmith

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:


> We can also use the assoicated graph to analyze scales other than RI scales; here is the connectivity ...



The 7-limit edge-connectivity ...
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Message: 2902 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 22:48:27

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:


> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 
> There were the usual repetions (meantone, 1/2 fifth meantone, 1/2 


fourth meantone, etc) 

To a mathematician focussing on approximation of ratios for harmony 
these may be repetitions, but to a musician they are quite distinct 
and it is quite wrong to call them "meantones". But it is important to 
point out their relationship to meantone.



>as well as a lot of systems which I ranked 


pretty low on this list. 

Me too.



>The worst badness measure belonged to this 
one:
> 


>> 144.5 [ 5  11] 
> 
> Comma: 200000/177147
> 
> Map:
> 


> [ 0  1]
> [ 5  1]
> [11  1]
> 
> Generators: a = 3.0066/25 = 144.315 cents; b = 1
> 
> badness: 7358
> rms: 15.57
> g: 7.89
> errors: [19.62, 1.15, -18.48]

Clearly junk.



But what about those that were on my earlier list (as better than 
pelogic), but not on yours. Have you figured out why that is?
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Message: 2903 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 03:29:29

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:


> Dave, I'll do the arithmetic for you and give you the RMS optima for 
> the ones with RMS less than 20 cents and g<8. Is there anything 
> _missing_ that is as good as one of these?
> 
> Generator 522.86¢, Period 1 oct.
> Generator 505.87¢, Period 1/4 oct.
> Generator 163.00¢, Period 1 oct.
> Generator 491.20¢, Period 1/3 oct.
> Generator 379.97¢, Period 1 oct.
> Generator 503.83¢, Period 1 oct. 
> Generator 494.55¢, Period 1/2 oct.
> Generator 442.98¢, Period 1 oct.
> Generator 387.82¢, Period 1 oct.
> Generator 271.59¢, Period 1 oct.
> Generator 317.08¢, Period 1 oct.
> Generator 498.27¢, Period 1 oct.
> 
> Did I miss any, Gene?
> Thanks for checking, Dave.



Ok Paul, here are all those I have found with a whole octave period 
where the rms error is no worse than the worst of these [which is 
pelogic (18.2 c)] and the log-odd-limit-weighted rms gens is no worse 
than the worst of these [which is orwell (6.3 gens)]. They are listed 
in order of generator size.

Gen     Gens per
(cents) 3   5
-----------------
78.0  [ 9   5]
81.5  [-6  10]
98.3  [-5   4]
102.0 [-5  -8]
126.2 [-4   3]
137.7 [ 5  -6]
144.5 [ 5  11]
163.0 [-3  -5]
176.3 [ 4   9]
226.3 [ 3   7]
251.9 [-2  -8]
271.6 [ 7  -3]
317.1 [ 6   5]
336.9 [-5  -6]
348.1 [ 2   8]
356.3 [ 2  -9]
380.0 [ 5   1]
387.8 [ 8   1]
414.5 [-7  -2]
443.0 [ 7   9]
471.2 [ 4  11]
490.0 [-1  -9]
498.3 [-1   8]
503.8 [-1  -4]
518.5 [ 6  10]
522.9 [-1   3]
561.0 [-3 -10]
568.6 [-3   7]

Note that this includes those I gave in the earlier list based on my 
own badness measure, except for 339.5c [-5 -13] and 351.0c [2 1]. I 
still think that earlier list is more relevant.
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Message: 2904 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 22:56:08

Subject: Re: coordinates from unison-vectors

From: dkeenanuqnetau

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:


> Hi Dave (Keenan),
> 
> 
> Aren't you the current resident Excel-meister?
> (suggestive evidence: your tumbling dekany)
> 
> I've posted my Periodicity-Block Calculator spreadsheet
> in the Files section, and have some comments below.
> Please take a look.



Sorry Monz, I'm preparing to go away with my family in a few days to 
camp for two weeks on an coral island. (Lady Musgrave Island, southern 
end of the Great Barrier Reef.)



> (And if anyone else feels up to the challenge to dethrone
> Dave, that's OK with me... all I'm after is good solid code.)



That's OK with me too. :-)
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Message: 2905 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 23:03:31

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: dkeenanuqnetau

I'm preparing to go away with my family in a few days for two weeks on 
a coral island, so it doesn't look like I'm going to get to check 
those 1/2 octave and 1/3 octave temperaments. Sorry Paul.

By the way, I had some misplaced parentheses in my formulae for rms 
error and log-odd-limit-weighted rms gens. The square root operation 
should of course be performed last, i.e. _after_ diving by the sum of 
the weights.
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Message: 2906 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 23:05:38

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:



>> There were the usual repetions (meantone, 1/2 fifth meantone, 1/2 


> fourth meantone, etc) 
> 
> To a mathematician focussing on approximation of ratios for harmony 
> these may be repetitions, but to a musician they are quite distinct 
> and it is quite wrong to call them "meantones". But it is important to 
> point out their relationship to meantone.



You
have an even and odd set of pitches, meaning an even or odd number of
generators to the pitch. You can't get from even to odd by way of
consonant 7-limit intervals, so basically we have two unrelated
meantone systems a half-fifth or half-fourth apart. You can always
glue together two unrelated systems and call it a temperament, and
this differs only because it does have a single generator.



> But what about those that were on my earlier list (as better than 
> pelogic), but not on yours. Have you figured out why that is?



They weren't junk, but they were below my cutoff; if I raised it from
badness 500 to badness 1000 they would have been on it. Should they
be?
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Message: 2907 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 15:35:35

Subject: Re: coordinates from unison-vectors

From: monz

Hey Dave,




> From: dkeenanuqnetau <d.keenan@xx.xxx.xx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Monday, December 31, 2001 2:56 PM
> Subject: [tuning-math] Re: coordinates from unison-vectors
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:


>> Hi Dave (Keenan),
>> 
>> 
>> Aren't you the current resident Excel-meister?
>> (suggestive evidence: your tumbling dekany)
>> 
>> I've posted my Periodicity-Block Calculator spreadsheet
>> in the Files section, and have some comments below.
>> Please take a look.
> 


> Sorry Monz, I'm preparing to go away with my family in a few days to 
> camp for two weeks on an coral island. (Lady Musgrave Island, southern 
> end of the Great Barrier Reef.)




OK, well, then have a good holiday!  (sounds pretty cool)

Anyway, I'm working on a Dictionary entry for "Transformation",
so I've finally gotten the hang of it.  I think I'll be able
to clean up my own code... but if anyone else wants to take
a shot at it, please do!



-monz




 



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Message: 2908 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 08:48:51

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: genewardsmith

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

There
were the usual repetions (meantone, 1/2 fifth meantone, 1/2 fourth
meantone, etc) as well as a lot of systems which I ranked pretty low
on this list. The worst badness measure belonged to this one:



> 144.5 [ 5  11] 

Comma: 200000/177147

Map:



[ 0  1]
[ 5  1]
[11  1]

Generators: a = 3.0066/25 = 144.315 cents; b = 1

badness: 7358
rms: 15.57
g: 7.89
errors: [19.62, 1.15, -18.48]
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Message: 2909 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 23:34:39

Subject: Re: OPTIMAL 5-LIMIT GENERATORS FOR DAVE

From: dkeenanuqnetau

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:


> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> 


>>> There were the usual repetions (meantone, 1/2 fifth meantone, 
1/2 


>> fourth meantone, etc) 
>> 
>> To a mathematician focussing on approximation of ratios for 
harmony 
>> these may be repetitions, but to a musician they are quite 
distinct 
>> and it is quite wrong to call them "meantones". But it is 
important to 
>> point out their relationship to meantone.
> 


> You have an even and odd set of pitches, meaning an even or odd 


number of generators to the pitch.

You mean even or odd number of generators to the intervals between the 
pitches. "Generators making up a pitch" doesn't make sense to me. 
Which reminds me: I think it would be a help to readers of your posts 
if you adopted the long-standing convention on this list of giving 
intervals as m:n (or n:m) and pitches as n/m, and when referring to a 
rational part of an octave, writing "n/m oct".



> You can't get from even to odd by 


way of consonant 7-limit intervals, so basically we have two unrelated 
meantone systems a half-fifth or half-fourth apart. You can always 
glue together two unrelated systems and call it a temperament, and 
this differs only because it does have a single generator.


> 



I see your point now, and it's a very good one. However, they _are_ 
linear temperaments by all the definitions I am aware of, and they 
_are_ very different from meantone melodically, and despite the 
doubling of the gens measure relative to meantone they _are_ better 
than some others on your list (at least according to me).



>> But what about those that were on my earlier list (as better than 
>> pelogic), but not on yours. Have you figured out why that is?
> 


> They weren't junk, but they were below my cutoff; if I raised it 


from badness 500 to badness 1000 they would have been on it. Should 
they be?


>



Ask a musician, e.g Paul. I don't think I've ever seen them before. I 
wouldn't miss them. But I do think they look better than pelogic. If 
you raised your badness cutoff to 1000 you'd probably end up including 
a lot more that I'd consider junk, either because of too many gens or 
too big errors.
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Message: 2910 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 01:09:49

Subject: Re: coordinates from unison-vectors

From: monz


> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, December 30, 2001 8:38 PM
> Subject: Re: [tuning-math] coordinates from unison-vectors
>
>
> The first part of the "LOOP" treats the unison-vectors
> as boundaries of a unit-cube, and calculates values p,q for
> the coordinates within that unit-cube, on the transformed lattice.
> I think this is working OK... but if anyone wants to check...





Oops!... of course, my work on this is all 2-dimensional,
so that's a "unit square" and not a "unit cube".

(but *you* all knew that! ...)



-monz





 



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Message: 2911 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 02:16:26

Subject: Re: coordinates from unison-vectors

From: monz

Hi Dave (Keenan),


Aren't you the current resident Excel-meister?
(suggestive evidence: your tumbling dekany)

I've posted my Periodicity-Block Calculator spreadsheet
in the Files section, and have some comments below.
Please take a look.

(And if anyone else feels up to the challenge to dethrone
Dave, that's OK with me... all I'm after is good solid code.)




> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, December 30, 2001 8:07 PM
> Subject: Re: [tuning-math] coordinates from unison-vectors
> <Yahoo groups: /tuning-math/message/2332 * [with cont.] >
>
>
> A plea to all who understand matrix math:
>
>
> A week ago, I posted the pseudo-code for the formulas
> in my Excel spreadsheet which calculates the coordinates
> of a 2-dimensional periodicity-block from a given pair
> of unison-vectors.  <etc.>




I've simplified and uncluttered my spreadsheet, and uploaded
it again to the Files section:

Yahoo groups: /tuning-math/files/monz/5-limit ... * [with cont.] 
UVs.xls


Rather than even bothering to deal with that broken link,
it's much better to simply go to the Files section and
download it, because there's a description of it there
that explains how to use it.

Yahoo groups: /tuning-math/files/monz/ * [with cont.] 


Look for the file named "5-limit PBs from UVs.xls".


Here are my detailed comments:


The unison-vector pair which I put in the spreadsheet is the
exponent matrix:

  [ 5 -6]
  [-4  1]

This is an example of one of the problems I mention in that
quoted post: in order to get all the coordinates within the
calculated boundaries which are centered on [0 0], the second
unison-vector has to have the signs reversed -- it's supposed
to be the usual old syntonic comma [4 -1].

Generally, when my spreadsheet produces garbage, it can be
corrected by changing the signs of either one or both of the
unison-vector exponents.

That post also quotes all the pseudo-code for the formulas in
my spreadsheet.



The formulas for p,q, in spreadsheet cells A21..B75, seem to
work correctly to find all the coordinates which fall within
the parallelogram bounded by the unison-vectors... but that
should be checked anyway.

I simply start at the origin [0 0] and add one of the
unison-vectors continously until the results exceed the
boundary +/- 1/2 of the other unison-vector, then divide
it modulo that unison-vector.



Then that lattice must be transformed back into the original
ratio-space.  This is where I'm having problems.

The formulas for x,y, in spreadsheet cells D21..E75, don't
always work.  Sometimes they produce exactly the correct
coordinates, other times the shape is right but not centered
within the parallelogram.

I've tried changing the formulas for x,y according to what
I've seen in Paul's _Gentle Introduction, part 3_ and in
textbooks on matrix math, but none of those coordinates
worked at all.  So I'm really confused.


The formulas in G4..J5 find the corners of the parallelogram
when it is centered on the origin coordinates [0 0].
An adjustment for either axis or both axes may be entered
in cells I6 and J6, for prime-factors 3 and 5 respectively.
If it's necessary at all (which it often isn't), a value
of 0.5 generally works to put all the coordinates within
the unison-vector boundaries.


The formula in cells C76..C134 finds the bounding notes of
the meantone chain specified by the integer fraction-of-a-comma
numerator and denominator entered into cells D12 and E12.
Cells D76..E134 calculate the coordinates for that, according
to the meantone's fractional exponents of 3 and 5.  This also
doesn't always work, and again, is usually corrected when
the signs in the original unison-vector matrix are changed.

(And according to Paul's views it's entirely unnecessary anyway
... but this is my spreadsheet and I like this theory...)



-monz







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Message: 2912 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 10:40:38

Subject: Some 7-limit superparticular pentatonics

From: genewardsmith

These are the ones which employ the two most proper possibilities,
(15/14)(8/7)(7/6)^2(6/5) and (15/14)(10/9)(7/6)(6/5)^2; both with a Blackwood index of 2.64 (largest over smallest scale step ratio.)

1--6/5--7/5--3/2--7/4
[6/5 7/6 15/14 7/6 8/7] c = 3

1--7/6--4/3--10/7--12/7
[7/6 8/7 15/14 6/5 7/6] c = 2

1--7/6--7/5--3/2--12/7
[7/6 6/5 15/14 8/7 7/6] c = 2

1--6/5--7/5--3/2--12/7
[6/5 7/6 15/14 8/7 7/6] c = 2

1--8/7--4/3--8/5--12/7
[8/7 7/6 6/5 15/14 7/6] c = 2

1--7/6--5/4--10/7--12/7
[7/6 15/14 8/7 6/5 7/6]

1--7/6--4/3--8/5--12/7
[7/6 8/7 6/5 15/14 7/6] c = 1



1--6/5--4/3--8/5--12/7
[6/5 10/9 6/5 15/14 7/6] c = 2

1--7/6--7/5--3/2--9/5
[7/6 6/5 15/14 6/5 10/9] c = 1

1--7/6--7/5--3/2--5/3
[7/6 6/5 15/14 10/9 6/5] c = 1

1--6/5--7/5--3/2--5/3
[6/5 7/6 15/14 10/9 6/5] c = 1

1--6/5--9/7--3/2--5/3
[6/5 15/14 7/6 10/9 6/5] c = 1
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Message: 2913 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 02:44:39

Subject: Re: coordinates from unison-vectors

From: monz


> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>; Dave Keenan <d.keenan@xx.xxx.xx>
> Sent: Monday, December 31, 2001 2:16 AM
> Subject: Re: [tuning-math] coordinates from unison-vectors
>
>
> Yahoo groups: /tuning-math/files/monz/ * [with cont.] 
> 
> 
> Look for the file named "5-limit PBs from UVs.xls".
> 
> 
> Here are my detailed comments:
> 
> 
> The unison-vector pair which I put in the spreadsheet is the
> exponent matrix:
> 
>   [ 5 -6]
>   [-4  1]
>
> ...
>
> The formula in cells C76..C134 finds the bounding notes of
> the meantone chain specified by the integer fraction-of-a-comma
> numerator and denominator entered into cells D12 and E12.





The meantone I entered in this example is -5/16-comma meantone.
I chose that value because it visually splits the periodicity-block
in half right down the center.


For an another example of meantone which is a pretty good fit
(according to my discredited-by-Paul theory) with this
periodicity-block, try entering the value -1 into cell D12
and the value 3 into cell E12, for -1/3-comma meantone.



Here's another, totally different, example which I 
found to be very interesting, according to my 
"meantone-rational-implications" theory:

Enter this unison-vector matrix into cells A7..B8:

   [-8 -1]
   [-4  1]

These unison-vectors, the reversed equivalents of the
skhisma and the syntonic comma, define a typical 12-tone
periodicity-block.


Enter 0.5 into cell I6 to adjust the boundaries of the
parallelogram slightly to the right, so that coordinate (6,0)
is included within it (it previously fell right on a corner).

Now, enter the value -1 into cell D12 and the value 11 into
cell E12, for -1/11-comma meantone, which we all know is
nearly identical to our familiar old 12-EDO tuning.  Notice
how it visually splits *this* parallelogram in half almost
exactly down the center.  Intriguing...  

So according to my "meantone-rational-implications" theory,
use of 12-EDO strongly implies the application of this JI
periodicity-block -- among others, of course... but the fact
that it goes right down the middle of this one, averaging as
perfectly as possible the pitch-height distance from this
particular set of JI pitch-classes, suggests to me that this
periodicity-block would be the one most likely to be 
interpreted by the listener.

Or, looked at another way, I would say that if a composer
wanted to find a "best fit" meantone or EDO for this particular
periodicity-block, it would be 12-EDO ~= -1/11-comma meantone.




-monz




 



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Message: 2914 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 02:55:23

Subject: Re: coordinates from unison-vectors

From: monz


> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Monday, December 31, 2001 2:44 AM
> Subject: Re: [tuning-math] coordinates from unison-vectors
>
>


>> From: monz <joemonz@xxxxx.xxx>
>> To: <tuning-math@xxxxxxxxxxx.xxx>; Dave Keenan <d.keenan@xx.xxx.xx>
>> Sent: Monday, December 31, 2001 2:16 AM
>> Subject: Re: [tuning-math] coordinates from unison-vectors
>> 
>> 
>> Yahoo groups: /tuning-math/files/monz/ * [with cont.] 
>> 
>> 
>> Look for the file named "5-limit PBs from UVs.xls".
>
> ...
> 


> Enter this unison-vector matrix into cells A7..B8:
> 
>    [-8 -1]
>    [-4  1]
> 
> These unison-vectors, the reversed equivalents of the
> skhisma and the syntonic comma, define a typical 12-tone
> periodicity-block.
> 
> 
> Enter 0.5 into cell I6 to adjust the boundaries of the
> parallelogram slightly to the right, so that coordinate (6,0)
> is included within it (it previously fell right on a corner).
> 
> Now, enter the value -1 into cell D12 and the value 11 into
> cell E12, for -1/11-comma meantone, which we all know is
> nearly identical to our familiar old 12-EDO tuning.  Notice
> how it visually splits *this* parallelogram in half almost
> exactly down the center.  Intriguing...  





Oops... the meantone doesn't follow the boundary adjustment
entered into cell I6.  That needs to be fixed.  So until it is...

If you try this example, leave the boundary adjustment blank
and simply realize that either JI coordinate (6,0) or (-6,0) 
may be included.  That way the -1/11-comma meantone will be 
more nearly centered within the parallelogram, and will actually
reflect its true relationship with the unison-vectors.
(According to my potential invalid theory, that is...)




-monz


 



 




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Message: 2915 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 03:23:13

Subject: Re: coordinates from unison-vectors

From: monz


> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Monday, December 31, 2001 2:55 AM
> Subject: Re: [tuning-math] coordinates from unison-vectors
>
>
>>> Yahoo groups: /tuning-math/files/monz/ * [with cont.] 
>>> 
>>> 


>>> Look for the file named "5-limit PBs from UVs.xls".
>> 




Here's another interesting example:

Enter this unison-vector exponent pair into the matrix
in cells A7..B8:

  [-4 -2]
  [-4  1]

The top one is the "diaschisma", the bottom one is the
"8ve"-complement (i.e., sign-reverse) of the syntonic comma.

Enter -1/6-comma as the value of the meantone tempering fraction
in cells D12 and E12.


The (1,-1), (3,0), and (2,1) coordinate pairs all fall on edges
of the bounding parallelogram, thus they each have an alternate
a unison-vector away which falls on the opposite edge.  The
first two may lowered be a comma and the third by a diaschisma:

(1,-1) + [-4  1]  =  (-3, 0)
(3, 0) + [-4  1]  =  (-1, 1)
(2, 1) + [-4 -2]  =  (-2,-1)

Since the parallelogram is centered on (0,0), it should be
easy to see that these are all equivalent pairs.


The meantone chain produces 13 notes, of which the first (-2,-1)
and last (2,1) both fall on edges of the parallegram and are
separated by the [-4 -2] unison-vector, and thus are alternates.
It visually splits the parallelogram in half running nearly
down the center.

This suggests to me that a 12-tone chain of -1/6-comma meantone
is a "best fit" meantone for this particular JI periodicity-block.

(disclaimer: Paul Erlich has discredited this "best-fit" theory of mine)



-monz


 



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Message: 2916 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 03:31:58

Subject: Re: coordinates from unison-vectors

From: monz


> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Monday, December 31, 2001 2:55 AM
> Subject: Re: [tuning-math] coordinates from unison-vectors
>
>
> Yahoo groups: /tuning-math/files/monz/ * [with cont.] 
> 
 > Look for the file named "5-limit PBs from UVs.xls".




I forget to mention this:

The blue lines, which connect the coordinates within the
periodicity-block, show the postions of the JI "wolves" as
one traces the chain of 3:2s from the bottom to the top.



-monz



 




 



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Message: 2917 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 13:34:47

Subject: Re: more tetrachordality results

From: clumma

I wrote...



> Returns the minimum mean deviation (in cents, and as
> a percentage of the smallest interval in the scale),
> of the pitches in any order, of a pitch set
> representing the given scale, and its transposition
> at 702 cents, for all modes of the given scale.



I found the bug that was given different results for
different modes (should have known that couldn't be
right).  Fortunately, I was taking the minimum across
modes, and the bug inflates the score, and at least
one mode usually comes out right.  Therefore, most of
the results I posted were right.

Posted:



> Wholetone scale
> (0 2 4 6 8 10) -> ((102 $) (51 %))

Should be:

Wholetone scale


(0 2 4 6 8 10) -> ((98 $) (49 %))

Correct as posted:



> Pentatonic Scale
> (0 2 5 7 9) -> ((21 $) (10 %))
> Diatonic Scale
> (0 2 4 5 7 9 11) -> ((16 $) (16 %))
> Diminished chord
> (0 3 6 9) -> ((102 $) (34 %))
> Diminished scale
> (0 2 3 5 6 8 9 11) -> ((50 $) (50 %))
> Minor scales w/'gypsy' tetrachord
> (0 1 4 5 7 8 11) -> ((44 $) (44 %))
> (0 1 4 5 7 8 10) -> ((44 $) (44 %))
> (0 1 4 5 7 9 10) -> ((44 $) (44 %))

Posted:

> 1--6/5--5/4--3/2--5/3
> [6/5, 25/24, 6/5, 10/9, 6/5]
> (0 9 11 20 25) -> ((73 $) (103 %))
>
> 1--25/24--5/4--3/2--9/5
> [25/24, 6/5, 6/5, 6/5, 10/9]
> (0 2 11 20 29) -> ((128 $) (181 %))

Should be:

1--6/5--5/4--3/2--5/3


[6/5, 25/24, 6/5, 10/9, 6/5]
(0 9 11 20 25) -> ((73 $) (104 %))

1--25/24--5/4--3/2--9/5
[25/24, 6/5, 6/5, 6/5, 10/9]
(0 2 11 20 29) -> ((128 $) (180 %))

Correct as posted:



> 1--6/5--4/3--3/2--5/3
> [6/5, 10/9, 9/8, 10/9, 6/5]
> (0 9 14 20 25) -> ((73 $) (41 %))
> 
> 1--5/4--4/3--3/2--8/5
> [5/4, 16/15, 9/8, 16/15, 5/4]
> (0 11 14 20 23) -> ((129 $) (122 %))

-Carl
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Message: 2918 - Contents - Hide Contents

Date: Mon, 31 Dec 2001 13:56:11

Subject: Re: Some 7-limit superparticular pentatonics

From: clumma

et__7-(odd)limit dyadic rms (cents)
22| 11
27| 8
37| 7
31| 4

Gene, are you allowing 9-limit edges?



> 1--6/5--7/5--3/2--7/4
> [6/5 7/6 15/14 7/6 8/7] c = 3

et__steps__________tetrachordality_


22| 0 6 11 13 18 _| 78$, 72% _____|
27| 0 7 13 16 22 _| 64$, 48% _____|
37| 0 10 18 22 30 | 74$, 57% _____|
31| 0 8 15 18 25 _| 71$, 61% _____|



> 1--6/5--4/3--8/5--12/7
> [6/5 10/9 6/5 15/14 7/6] c = 2

et__steps__________tetrachordality_


22| 0 6 9 15 17 __| 59$, 54% _____|
27| 0 7 11 18 21 _| 50$, 38% _____|
37| 0 10 15 25 29 | 52$, 40% _____|
31| 0 8 13 21 24 _| 55$, 47% _____|

Tomorrow, I'm going to try and tune these
up and see what they sound like.

I'm guessing a 10-cent difference between
every note any its 3:2 transposition isn't
as bad as a 50-cent difference between one
note and its transposition.  Thus, the
next version of this software will offer
rms in addition to mad.

-Carl
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Message: 2920 - Contents - Hide Contents

Date: Tue, 01 Jan 2002 23:42:27

Subject: Some 10-tone, 72-et scales

From: genewardsmith

I started out looking at these as 7-limit 225/224 planar temperament
scales, but decided it made more sense to check the 5 and 11 limits
also, and to take them as 72-et scales; if they are ever used that is
probably how they will be used. I think anyone interested in the 
72-et should take a look at the top three, which are all 5-connected,
and the top scale in particular, which is a clear winner. The "edges"
number counts edges (consonant intervals) in the 5, 7, and 11 limits,
and the connectivity is the edge-connectivity in the 5, 7 and 11
limits.

[0, 5, 12, 19, 28, 35, 42, 49, 58, 65]
[5, 7, 7, 9, 7, 7, 7, 9, 7, 7]
edges   15   27   35   connectivity   2   5   6

[0, 5, 12, 19, 28, 35, 42, 51, 58, 65]
[5, 7, 7, 9, 7, 7, 9, 7, 7, 7]
edges   14   25   35   connectivity   1   3   6

[0, 5, 12, 21, 28, 35, 42, 51, 58, 65]
[5, 7, 9, 7, 7, 7, 9, 7, 7, 7]
edges   13   25   35   connectivity   1   3   6

[0, 5, 12, 19, 26, 35, 42, 51, 58, 65]
[5, 7, 7, 7, 9, 7, 9, 7, 7, 7]
edges   11   21   35   connectivity   0   2   6

[0, 5, 12, 21, 28, 35, 42, 49, 58, 65]
[5, 7, 9, 7, 7, 7, 7, 9, 7, 7]
edges   12   25   33   connectivity   0   3   6

[0, 5, 14, 21, 28, 35, 42, 51, 58, 65]
[5, 9, 7, 7, 7, 7, 9, 7, 7, 7]
edges   10   24   33   connectivity   0   3   6

[0, 5, 14, 21, 28, 35, 44, 51, 58, 65]
[5, 9, 7, 7, 7, 9, 7, 7, 7, 7]
edges   10   23   33   connectivity   0   3   5

[0, 5, 12, 21, 28, 35, 44, 51, 58, 65]
[5, 7, 9, 7, 7, 9, 7, 7, 7, 7]
edges   11   22   33   connectivity   0   2   5

[0, 5, 12, 19, 28, 35, 44, 51, 58, 65]
[5, 7, 7, 9, 7, 9, 7, 7, 7, 7]
edges   10   20   33   connectivity   0   2   5

[0, 5, 12, 19, 26, 35, 44, 51, 58, 65]
[5, 7, 7, 7, 9, 9, 7, 7, 7, 7]
edges   7   15   32   connectivity   0   1   5

[0, 5, 14, 21, 28, 35, 42, 49, 58, 65]
[5, 9, 7, 7, 7, 7, 7, 9, 7, 7]
edges   9   23   31   connectivity   0   3   5

[0, 5, 12, 21, 28, 35, 42, 49, 56, 65]
[5, 7, 9, 7, 7, 7, 7, 7, 9, 7]
edges   9   22   31   connectivity   0   3   5

[0, 5, 14, 21, 28, 35, 42, 49, 56, 63]
[5, 9, 7, 7, 7, 7, 7, 7, 7, 9]
edges   7   22   31   connectivity   0   4   5

[0, 5, 14, 21, 28, 37, 44, 51, 58, 65]
[5, 9, 7, 7, 9, 7, 7, 7, 7, 7]
edges   8   21   31   connectivity   0   2   5

[0, 5, 14, 21, 28, 35, 42, 49, 56, 65]
[5, 9, 7, 7, 7, 7, 7, 7, 9, 7]
edges   7   21   31   connectivity   0   3   5

[0, 5, 14, 21, 30, 37, 44, 51, 58, 65]
[5, 9, 7, 9, 7, 7, 7, 7, 7, 7]
edges   6   19   31   connectivity   0   2   5

[0, 5, 12, 21, 28, 37, 44, 51, 58, 65]
[5, 7, 9, 7, 9, 7, 7, 7, 7, 7]
edges   7   18   31   connectivity   0   2   5

[0, 5, 14, 23, 30, 37, 44, 51, 58, 65]
[5, 9, 9, 7, 7, 7, 7, 7, 7, 7]
edges   6   18   30   connectivity   0   1   5

[0, 5, 12, 19, 28, 37, 44, 51, 58, 65]
[5, 7, 7, 9, 9, 7, 7, 7, 7, 7]
edges   6   15   30   connectivity   0   2   5

[0, 5, 12, 21, 30, 37, 44, 51, 58, 65]
[5, 7, 9, 9, 7, 7, 7, 7, 7, 7]
edges   5   15   30   connectivity   0   1   5
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Message: 2921 - Contents - Hide Contents

Date: Tue, 1 Jan 2002 23:40:48

Subject: yesterday: December 31, 2002?

From: monz

I read the tuning lists in my Microsoft Outlook
email program, and I see that all of yesterday's
posts have the wrong date: "2002" for the year
instead of "2001".

Did this happen to anyone else?  Posts may be
easily missed this way, by those who don't use
the web interface.


-monz



 



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Message: 2922 - Contents - Hide Contents

Date: Tue, 01 Jan 2002 05:32:35

Subject: Some 12-tone meantone scales/temperaments

From: genewardsmith

Here are up to isomorphism by mode and inversion all of the meantone
scales of twelve tones which have a 7-limit edge-connectivity greater
than two. While the usual meantone scale (with a connectivity of six)
wins, it does not dominate, and the other scales/temperaments are
worth considering. While the results are given in terms of the 31-et,
they do not depend on the precise tuning, and are generic meantone
results.

I am not aware if this sort of thing has ever been investigated, but
it certainly seems worth pursuing.

[0, 2, 5, 8, 10, 13, 15, 18, 20, 23, 26, 28]
[2, 3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 3]   6

[0, 3, 6, 8, 11, 13, 16, 19, 21, 23, 26, 29]
[3, 3, 2, 3, 2, 3, 3, 2, 2, 3, 3, 2]   5

[0, 3, 6, 8, 11, 13, 16, 18, 21, 23, 26, 29]
[3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 3, 2]   5

[0, 3, 6, 8, 10, 13, 16, 19, 21, 23, 26, 29]
[3, 3, 2, 2, 3, 3, 3, 2, 2, 3, 3, 2]   5

[0, 3, 6, 8, 11, 13, 16, 18, 21, 23, 26, 28]
[3, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3]   4

[0, 3, 6, 8, 11, 13, 16, 18, 21, 24, 26, 28]
[3, 3, 2, 3, 2, 3, 2, 3, 3, 2, 2, 3]   4

[0, 2, 5, 8, 10, 13, 15, 17, 20, 23, 25, 28]
[2, 3, 3, 2, 3, 2, 2, 3, 3, 2, 3, 3]   4

[0, 3, 6, 8, 11, 13, 15, 18, 21, 23, 26, 28]
[3, 3, 2, 3, 2, 2, 3, 3, 2, 3, 2, 3]   3

[0, 3, 6, 8, 11, 13, 16, 18, 20, 23, 26, 28]
[3, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3]   3

[0, 3, 6, 9, 11, 13, 15, 18, 21, 24, 26, 28]
[3, 3, 3, 2, 2, 2, 3, 3, 3, 2, 2, 3]   3

[0, 3, 6, 9, 11, 13, 15, 17, 19, 22, 25, 28]
[3, 3, 3, 2, 2, 2, 2, 2, 3, 3, 3, 3]   3

[0, 2, 5, 8, 10, 12, 15, 18, 20, 22, 25, 28]
[2, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3]   3

[0, 2, 5, 8, 11, 13, 15, 17, 20, 23, 26, 28]
[2, 3, 3, 3, 2, 2, 2, 3, 3, 3, 2, 3]   3

[0, 3, 5, 7, 10, 13, 15, 18, 20, 22, 25, 28]
[3, 2, 2, 3, 3, 2, 3, 2, 2, 3, 3, 3]   3
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Message: 2923 - Contents - Hide Contents

Date: Tue, 01 Jan 2002 07:58:51

Subject: 7-limit hexatonic scales

From: genewardsmith

Here are the two most proper classes of superparticular scales

Blackwood = 2.825

[1, 16/15, 8/7, 4/3, 8/5, 28/15]
[16/15, 15/14, 7/6, 6/5, 7/6, 15/14]   3

[1, 16/15, 8/7, 4/3, 14/9, 28/15]
[16/15, 15/14, 7/6, 7/6, 6/5, 15/14]   2

[1, 16/15, 8/7, 4/3, 8/5, 12/7]
[16/15, 15/14, 7/6, 6/5, 15/14, 7/6]   2

[1, 16/15, 56/45, 4/3, 14/9, 5/3]
[16/15, 7/6, 15/14, 7/6, 15/14, 6/5]   2

[1, 16/15, 8/7, 4/3, 10/7, 5/3]
[16/15, 15/14, 7/6, 15/14, 7/6, 6/5]   1

[1, 16/15, 8/7, 4/3, 10/7, 12/7]
[16/15, 15/14, 7/6, 15/14, 6/5, 7/6]   1

[1, 16/15, 8/7, 4/3, 14/9, 5/3]
[16/15, 15/14, 7/6, 7/6, 15/14, 6/5]   1

[1, 16/15, 8/7, 48/35, 8/5, 12/7]
[16/15, 15/14, 6/5, 7/6, 15/14, 7/6]   1

[1, 16/15, 56/45, 4/3, 10/7, 5/3]
[16/15, 7/6, 15/14, 15/14, 7/6, 6/5]   1

[1, 16/15, 56/45, 4/3, 10/7, 12/7]
[16/15, 7/6, 15/14, 15/14, 6/5, 7/6]   1

[1, 16/15, 56/45, 4/3, 8/5, 12/7]
[16/15, 7/6, 15/14, 6/5, 15/14, 7/6]   1


Blackwood = 3.159

[1, 21/20, 6/5, 7/5, 3/2, 7/4]
[21/20, 8/7, 7/6, 15/14, 7/6, 8/7]   4

[1, 21/20, 6/5, 7/5, 3/2, 12/7]
[21/20, 8/7, 7/6, 15/14, 8/7, 7/6]   3

[1, 21/20, 9/8, 9/7, 3/2, 7/4]
[21/20, 15/14, 8/7, 7/6, 7/6, 8/7]   2

[1, 21/20, 9/8, 21/16, 3/2, 12/7]
[21/20, 15/14, 7/6, 8/7, 8/7, 7/6]   2
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Message: 2924 - Contents - Hide Contents

Date: Tue, 01 Jan 2002 08:13:32

Subject: The 7-limit connectivity of 7-tone meantone scales

From: genewardsmith

I might as well give this:

[0 5 10 15 18 23 28]
5553553 c = 4

[0 5 10 15 18 23 26]
5553535 c = 3

[0 5 10 15 18 21 26]
5553355 c = 3
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