Tuning-Math Digests messages 1175 - 1199

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Message: 1175

Date: Sun, 29 Jul 2001 11:56 +0

Subject: Re: Magic lattices

From: graham@xxxxxxxxxx.xx.xx

I wrote:

> Now Dave Keenan's found an alternative simplified Miracle lattice, 
> let's see if he can make anything of this.

I came up with something overnight:

                                          Eb----Bb----F-----C
                                       B#/ \ Gb/ \ Db/ \ Ab/   Ev
                                    G^  / D#\ / A#\ / E#\ / Cb
                                       B-----F#----C#----G#
                                    G / \ D / \ A / \ E /
                                 Eb  / Bb\ / F \ / C \ /
                              B#----Gb----Db----Ab----Ev
                           G^/ \ D#/ \ A#/ \ E#/   Cb
                            / B \ / F#\ / C#\ / G#
                           G-----D-----A-----E
                        Eb/ \ Bb/ \ F / \ C /
                     B#  / Gb\ / Db\ / Ab\ /
                  G^----D#----A#----E#----Cb
                 / \ B / \ F#/ \ C#/   G#
                / G \ / D \ / E \ / E
               Eb----Bb----F-----C
            B#/ \ Gb/ \ Db/ \ Ab/   Ev
         G^  / D#\ / A#\ / E#\ / Cb
            B-----F#----C#----G#
         G / \ D / \ A / \ E /
      Eb  / Bb\ / F \ / C \ /
   B#----Gb----Db----Ab----Ev
G^/ \ D#/ \ A#/ \ E#/   Cb
 / B \ / F#\ / C#\ / G#
G-----D-----A-----E


The template is

   5
1-----3-----9
       \   /
        \ /
         7

Like Dave's new septimal-kleismic lattice, but unlike a normal 7-limit 
lattice, pitch increases left-right for a 4:5:6:7:9 chord.  That'd make it 
good as a mapping for a hexagonal keyboard.



I also found this:


G#--B+--D#--F#+-A#
   / \     / \
G+/ B \ D+/ F#\ A+
 /     \ /     \
G---Bt--D---F+--A
 \     / \     /
Gt\ Bb/ Dt\ Ft/ At
   \ /     \ /
Gb--Bbt-Db--Fbt-Ab

With the template

    5

1-------3-------9---11
         \     /
          \   /
           \ /
            7


It works with 31-equal, but I haven't found any other consistent 
temperaments for it.  Although it does work with the meantone-like 
neutral-third family 7, 24, 31, 38, ...  with the mapping (2 8 -11 5), a 
complexity measure of 20 and an approximation of the 11-limit to within 
10.8 cents (not much of an improvement on 31-equal).  It looks like a good 
mapping for a rectangular keyboard, and might work with adaptive tuning.

Unison vectors are 176:175 or (4 0 -2 -1 1)H and 243:242 or (-1 5 0 0 
-2)H.  These combine to give 31104:30625 or (7 5 -4 -2)H.  Dieses are 
36:35, 45:44 and 55:54.  Two dieses of 36:35 make a 25:24.


And a unified neutral-second/neutral-third lattice


B--D+-F#-A+-C#
|\    |\    |
| \   | \   |
A  C+ E  G+ B
|   \ |   \ |
|    \|    \|
G--Bt-D--F+-A
|\    |\    |
| \   | \   |
F  At C  Et G
|   \ |   \ |
|    \|    \|
Eb-Gt-Bb-Dt-F


It's like Dave Keenan's new Miracle lattice, but with extra rows.  
Template

            7
            |
            |
            |
            |
            |
5           |
|           |
|           |
| 11        |
|           |
|           |
1-----3-----9--11


I don't know if there's a simpler position for the 7 ...


                  Graham


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Message: 1177

Date: Mon, 30 Jul 2001 18:48:46

Subject: Re: BP linear temperament

From: Paul Erlich

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
> 
> Paul wrote:
> 
> >> Probably, but I don't have time. A suggested approach: Give the
> >> simplest frequency ratio you can for each scale degree.
> 
> >Hmm . . . this would be the Kees van Prooijen definition of a
> >periodicity block. Have you studied this page? Manuel, can we do 
this
> >in SCALA?
> 
> Yes, with APPROXIMATE while using one of the attribute functions
> for your definition of "simple".
> 
> >> Then list the
> >> step sizes between these, classify them into two sizes, L and s,
> >> then figure out what the generator is from that.
> 
> > The Hypothesis in action!
> 
> In case of the Triple BP scale I don't see a generator for it.

Are you including the full 39 pitches per tritave?


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Message: 1179

Date: Mon, 30 Jul 2001 00:44:14

Subject: Re: Hey Carl

From: Paul Erlich

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Paul,
> 
> Lots of stuff to chew on there, but here's some quick thoughts...
> 
> <<Still can't see what a 2D generator could mean.>>
> 
> When I say it I mean it in a very simple way -- as I see it it's
> exactly the generalized difference between the 7-tone Pythagorean
> single 1D plane built from 2:3s, and the classic JI, 5-limit 
diatonic
> 2D plane built from 2:3s and 4:5s.

So wouldn't any connected, convex chunk of an N-dimensional JI 
lattice be "generatable" by this definition?
> 
> 
> <<Here's why the hypothesis should work [SNIP] QED -- right?>>
> 
> Well it might make sense right off to some folks but it's a lot for 
me
> to follow right now without some examples... do you think you could
> flesh out the steps with some?

I need to draw some diagrams for you . . .
> 
> 
> <<another thought is that trivalency is a very special case>>
> 
> Certainly when compared to "bivalency" it's not as omnipresent (as
> I've said before all M-out-of-N sets are bivalent within a period),
> but I could come up with several trivalent examples pretty easily, 
so
> I don't think it's exactly "very special".

By "very special" here I didn't mean to imply that it would be hard 
for you to find examples . . . just that in general, a 2D hyper-MOS 
would have up to four specific sizes for each generic interval.


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Message: 1180

Date: Mon, 30 Jul 2001 22:45:05

Subject: Another BP linear temperament? (was: Magic lattices)

From: Dave Keenan

--- In tuning-math@y..., "D.Stearns" <STEARNS@C...> wrote:
> Hi Graham,
> 
> <<I've also updated the Bohlen-Pierce entry.>>
> 
> The BP linear temperament was so obvious that I could hardly call it
> much of a contribution... it's an observation at best!
> 
> Personally I'm much more satisfied with the 2D and especially the 
1/6
> comma generalized meantone contributions, as I think they're more 
than
> simple observations of the obvious.
> 
> The 1/6 comma generalization would be accomplished by using 1/6 of 
the
> 118098/117649 2D comma to make the 177147/117649 in a chain of 9/7s
> pure:
> 
> 1468   434
> 1034   868
>  600  1302
>  166  1736
> 1634   268
> 1200   702
>  766  1136
>  332  1570
> 1800   102
> 1366   536
>  932   970
>  498  1404
> 
> Here's the generalized or remapped 1/6 comma Bohlen-Pierce meantone
> rotations:
> 
> 0 268 434 600 868 1034 1302 1468 1736 1902
> 0 166 332 600 766 1034 1200 1468 1634 1902
> 0 166 434 600 868 1034 1302 1468 1736 1902
> 0 268 434 702 868 1136 1302 1570 1736 1902
> 0 166 434 600 868 1034 1302 1468 1634 1902
> 0 268 434 702 868 1136 1302 1468 1736 1902
> 0 166 434 600 868 1034 1200 1468 1634 1902
> 0 268 434 702 868 1034 1302 1468 1736 1902
> 0 166 434 600 766 1034 1200 1468 1634 1902
> 
> Note the pure 1:2s, 2:3s and the perfect 1:2^(1/2)s.
> 
> (Come to think of it, even the n*n figurate number lattice based on
> the 3:5:7 seems like more of a contribution to me...)

Dan,

Can you give us the mapping from primes to numbers of generators for 
this temperament?

Also the mapping between primes (or whatever) that takes us from 
meantone to this temperament?

-- Dave Keenan


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Message: 1181

Date: Mon, 30 Jul 2001 00:46:22

Subject: Re: TGs, Unison Vectors, and Octaves

From: Paul Erlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> > From: Paul Erlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Saturday, July 28, 2001 10:35 AM
> > Subject: [tuning-math] Re: TGs, Unison Vectors, and Octaves
> >
> >
> > 
> > [My?] thoughts on tonics are in a much different realm.
> > I see periodicity blocks, etc. as fundamentally "pre-tonal",
> > as the appearance of diatonic and chromatic scales in Pythagorean 
> > and meantone tuning preceded the appearance [of] tonality
> > in Western music.
> 
> 
> Hey Paul,
> 
> This is a cool idea, and I agree with it.
> 
> It seems to me like you're alluding to my idea of "finity",
> in that the composers make use of unison-vectors and the
> listeners pick that up, without anyone really being very
> conscious of it all.  Am on I the right track?

Sure -- I've been implying something to that effect specifically on 
the Tuning List, rather than here. And of course, your inquiries into 
finity are what led me to study PBs in the first place.


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Message: 1182

Date: Mon, 30 Jul 2001 11:08:06

Subject: Re: BP linear temperament

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Paul wrote:

>> Probably, but I don't have time. A suggested approach: Give the
>> simplest frequency ratio you can for each scale degree.

>Hmm . . . this would be the Kees van Prooijen definition of a
>periodicity block. Have you studied this page? Manuel, can we do this
>in SCALA?

Yes, with APPROXIMATE while using one of the attribute functions
for your definition of "simple".

>> Then list the
>> step sizes between these, classify them into two sizes, L and s,
>> then figure out what the generator is from that.

> The Hypothesis in action!

In case of the Triple BP scale I don't see a generator for it.
If we omit one of 13/11 and 25/21 and do FIT/MODE, we see there
are two approximations that have two step sizes, a mode of 26th
root of 3 and one of 32nd root of 3, but neither is Myhill.
If there would be one, finding the generator is easy by doing
QUANTIZE for that ET and then SHOW DATA.

Manuel


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Message: 1183

Date: Mon, 30 Jul 2001 11:23 +0

Subject: Re: Magic lattices

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <3.0.6.32.20010729153925.00a89630@xx.xxx.xx>
Dave Keenan wrote:

> > I've discovered that "Multiple Approximations Generated Iteratively 
> and 
> > Consistently" is an acronym for "MAGIC".  What a coincidence!
> 
> Tee hee! Yes I _had_ noticed that.
> 
> By the way, you can delete the second ocurrence of it in your catalog. 
> The
> 5-limit one. That was my fault.

Oops.

I've also updated the Bohlen-Pierce entry.

> > Now Dave Keenan's found an alternative simplified Miracle lattice, 
> let's 
> > see if he can make anything of this.
> 
> The lattice I gave for Miracle works just as well for this temperament
> (without the 11s of course), because the 224:225 is distributed in this
> temperament too.

Yes, it does!  I also ended up with my own lattice that removes the 
224:225.


                  Graham


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Message: 1184

Date: Tue, 31 Jul 2001 18:37:33

Subject: Re: BP linear temperament

From: Paul Erlich

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
> 
> Paul wrote:
> >Are you including the full 39 pitches per tritave?
> 
> No, I wasn't. But trying that, after filling the gaps I got the 
scale:
> 1/1 65/63 35/33 27/25 55/49 15/13 25/21 11/9 49/39 9/7 65/49 15/11 
7/5
> 13/9  49/33 75/49 11/7 21/13 5/3 77/45 135/77 9/5 13/7 21/11 49/25
> 99/49 27/13  15/7 11/5 147/65 7/3 117/49 27/11 63/25 13/5 147/55 
25/9
> 99/35 189/65 3/1
> I saw there's no L+S approximation to it that is Myhill.
> If you give me the set of unison vectors

You can find the unison vectors by calculating the ratios of 
(differences between) various pairs of generic seconds, pairs of 
generic thirds, etc.

> I can try to find some other
> versions using the periodicity block shaping of Scala and see if any
> of those can be approximated by a MOS, but the chance looks small 
that
> it can be found like this by hand.

Well at least you're not trying to do it entirely by hand!


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Message: 1185

Date: Tue, 31 Jul 2001 18:39:30

Subject: Re: BP linear temperament

From: Paul Erlich

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
> 
> 
> If you give me the set of unison vectors I can try to find some 
other
> versions using the periodicity block shaping of Scala and see if any
> of those can be approximated by a MOS, but the chance looks small 
that
> it can be found like this by hand.

P.S. if you temper out all but one of the unison vectors, you're 
guaranteed to get an MOS, by my Hypothesis . . . right? I'd look for 
a set of unison vectors that are all super-super-particular (n=d+2), 
then temper out all but the largest one (the one with the smallest 
numbers) . . .


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Message: 1186

Date: Tue, 31 Jul 2001 19:47:03

Subject: Re: BP linear temperament

From: Paul Erlich

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> 
> You can find the unison vectors by calculating the ratios of 
> (differences between) various pairs of generic seconds, pairs of 
> generic thirds, etc.

Here are some I found, just by looking at the generic seconds:

243:245
273:275
1323:1331
1617:1625

That should be enough . . . in 4-D, (5,7,11,13) space, with 3 as the 
interval of equivalence, these UVs become

(-1 -2 0 0)
(-2 1 -1 1)
( 0 2 -3 0)
(-3 2 1 -1)

and the determinant of this matrix is -39. Whew!

Now what's the generator of the MOS we get by tempering out all of 
the above UVs, except for, say, 243:245?


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Message: 1187

Date: Tue, 31 Jul 2001 11:46:26

Subject: Re: BP linear temperament

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Hi Dan,

>I suppose I really should just get off my duff and install your
>wonderful program Scala (though I can do the above without it easy
>enough as well, and old habits sometimes die hard)!

Yes, I know. Well you could still decide not to use it after having
it tried.

>Does Scala have a function that can take and arbitrary set of notes
>and quantize it so that it can be rendered as a 2D lattice (i.e., in
>"triads" where the 1D plane is also the two stepsize MOS generator) --
>preferably in its simplest (nearest to the 1/1) configuration?

If I understand what you mean, i.e. can Scala calculate a multilinear
temperament that approximates an arbitrary scale in some optimal way,
no. It can however calculate a linear temperament, least squares and
minimax optimal. It can also approximate a scale by a JI scale for a
given set of primes.

Manuel


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Message: 1188

Date: Tue, 31 Jul 2001 11:50:09

Subject: Re: BP linear temperament

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Paul wrote:
>Are you including the full 39 pitches per tritave?

No, I wasn't. But trying that, after filling the gaps I got the scale:
1/1 65/63 35/33 27/25 55/49 15/13 25/21 11/9 49/39 9/7 65/49 15/11 7/5
13/9  49/33 75/49 11/7 21/13 5/3 77/45 135/77 9/5 13/7 21/11 49/25
99/49 27/13  15/7 11/5 147/65 7/3 117/49 27/11 63/25 13/5 147/55 25/9
99/35 189/65 3/1
I saw there's no L+S approximation to it that is Myhill.
If you give me the set of unison vectors I can try to find some other
versions using the periodicity block shaping of Scala and see if any
of those can be approximated by a MOS, but the chance looks small that
it can be found like this by hand.

Manuel


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Message: 1189

Date: Tue, 31 Jul 2001 15:31 +0

Subject: Re: Another BP linear temperament?

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9k4o1h+jqn@xxxxxxx.xxx>
Dan Stearns wrote:

> > The BP linear temperament was so obvious that I could hardly call it
> > much of a contribution... it's an observation at best!

Well, that's all I'm cataloguing.

I've written a little script to go with my temperament finder for this.  
Your BP mapping is the best in the "7-limit" with a tritave as the 
interval of equivalence.

Here's the printout:

7/30, 279.0 cent generator

basis:
(1.0, 0.23246075055125442)

mapping by period and generator:
[(1, 0), (-1, 7), (1, 2), (2, -1)]

mapping by steps:
[(17, 13), (11, 8), (25, 19), (30, 23)]

unison vectors:
[[9, 2, -7, 0], [-13, 1, 0, 7]]

highest interval width: 8
complexity measure: 8  (9 for smallest MOS)
highest error: 0.003704  (4.445 cents)
unique

The top 10 list should be at <2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 8 *> and 
the script at <import temper, string *>.


                    Graham


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Message: 1190

Date: Wed, 01 Aug 2001 20:03:41

Subject: Re: BP linear temperament

From: Paul Erlich

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
> 
> Paul wrote:
> 
> > Here are some I found, just by looking at the generic seconds:
> >
> > 243:245
> > 273:275
> > 1323:1331
> > 1617:1625
> 
> Creating a periodicity block with these unison intervals was an
> immediate result! This is the scale:
> 1/1 77/75 35/33 27/25 55/49 63/55 25/21 11/9 1701/1375 9/7 33/25
> 15/11 7/5 275/189 81/55 75/49 11/7 441/275 5/3 77/45 135/77 9/5
> 275/147 21/11 49/25 99/49 567/275 15/7 11/5 25/11 7/3 825/343 27/11
> 63/25 55/21 147/55 135/49 99/35 225/77 3/1
> It is 11-limit. The 56th root of 3 quantisation is Myhill and the
> generator is 11/7 (782.49 c.) or 21/11 (1119.46 c.).
> The least squares best generator for the above scale is 780.8095
> cents, s=39.924 cents (22) and L=60.213 cents (17).

Your least squares optimization uses what mapping from generators to 
primes? And it considers the errors for intervals 3:5, 3:7, 3:11, 
3:13, 5:7, 5:11, 5:13, 7:11, 7:13, 11:13 . . . right?

> So it's very
> close to the 95th root of 3.
> 1/1 39.9241 100.1373 140.0614 200.2747 240.1987 300.4120 340.3361
> 380.2601 440.4734 480.3975 540.6107 580.5348 640.7481 680.6721
> 740.8854 780.8095 820.7335 880.9468 920.8709 981.0841 1021.0082
> 1081.2215 1121.1455 1161.0696 1221.2829 1261.2069 1321.4202
> 1361.3443 1421.5575 1461.4816 1521.6949 1561.6189 1601.5430
> 1661.7563 1701.6803 1761.8936 1801.8177 1862.0309 3/1
> 
> Manuel


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Message: 1191

Date: Wed, 1 Aug 2001 03:10:28

Subject: Re: finding n-limit ratios

From: Robert Walker

Hi there,

I've been working recently on section of FTS that finds close
approximations to n-limit ratios

You can find it in current beta preview, but it is very slow
for large quotients. It does it just by testing all quotients,
and working out what the denominator would be for that quotient,
then if denom / denum is within the desired tolerance,
checks if they are both n-limit. If denom is n-limit, and denum isn't,
does an increment / decrement of denum until it finds one that is okay,
or until it goes outside the desired tolerance.

However, even that fairly crude technique is useful for this
work, where small values of denom. and denum. are interesting to find.
It's not too bad for max quotient of 1000.

I'm interested to extend the range.

I've been trying a very simple technique:

suppose r = ratio, and suppose you want it 5 limit,
then write

r~= 2^a*3^b*5^c

as
log r ~= a log 2 + b log 3 + c log 5

Now set a max quotient to find and tolerance in cents
in advance.

From that, one can calculate a max possible power for each
of the exponents. E.g. b < log(N)/log 3 where N is the max quotient.

Now just loop over all the possible values for b and c.
Leave a free, since as a power of 2, it will be the largest.

For each value of b, c, you can then work out a value for a
as
a = (log r - b log 3 - c log 5)/ log 2

and since you want an integer, use the floor and ceil of a
i.e. nearest integers above and below.

This is very fast for 5 limit ratios.

E.g. for max quotient of 1000000, you have 
integer part of log(1000000)/log(5)= 10
and of log(1000000)/log(3)= 12

So, taking negative numbers into account you have
24*16 = 384 loops to do for each ratio. That's tiny,
even if you are testing maybe 20 numbers or more,
and doing a fair bit of processing in each loop.

Even up to max quotient of 1000000000000 works fine.

Doubling the number of digits will only double the
range for each of the powers, and since there are
two involved, that quadruples the time.

Add in fourth factor though
2 3 5 7
and you begin to get into seriously large numbers.
Each time you double the number of digits you
multiply the time needed by 8. That's a siginificant
difference.

Add a couple more factors and it is getting on for as
slow as the method of checking all the quotients in
succession, for small quotients (of course
it will always be faster for large quotients
if you are willing to wait for the answer).

I can refine this a little by just doing things
more efficiently.

However, I wondered if there is any way to go more
directly to the solution rather than search all
the points of the lattice. I've been puzzling it
over but haven't come up with anything yet.

I've heard of the LLL and PLSQ algorithms.
However, seems to me if you look for an integer
relation between (r, log 2, log 3, log 5, log 7)
then you are as likely to find a relation with 0
coeff. for r as one involving r and the remaining
ones. Also if you find a relation involving r,
then the coeff. for r could be greater than 1,
and that would be no good either. Am I missing
something?

I've had a look for free source code
for PLSQ but only found gpld code which I can't
use in FTS. However, if it is likely to solve this
I'll be interested to look more thoroughly, or
if nec, try and see if I can code it myself,
which should be poss, if I can find a good
clear description somewhere of how it works.

Any thoughts? Or hints of an idea even that might
lead to something?

Robert


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Message: 1192

Date: Wed, 1 Aug 2001 03:21:15

Subject: Re: finding n-limit ratios

From: Robert Walker

Hi there,

> However, seems to me if you look for an integer
> relation between (r, log 2, log 3, log 5, log 7)
that of course should be

> relation between (log r, log 2, log 3, log 5, log 7)

so if coeff of log r > 1 you end up with a nice expression
of, say, r^3 as an n-limit ratio.

Robert


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Message: 1194

Date: Wed, 1 Aug 2001 00:07:52

Subject: Sumerian 12-EDO? (was: [tuning] fractional exponents of prime-factors)

From: monz

I wrote:

> From: monz <joemonz@xxxxx.xxx>
> To: <tuning@xxxxxxxxxxx.xxx>
> Sent: Wednesday, July 18, 2001 3:09 AM
> Subject: [tuning] fractional exponents of prime-factors
>
>
> For a very simple example, if we use vector subtraction to
> calculate what note results from tempering the Pythagorean
> "perfect 5th" by 1/12 of a Pythagorean comma, we get:
> 
>     2^x   3^y
> 
>   |-12/12 12/12|  =  2^-1 * 3^1   =   3/2    =  "perfect 5th"
> - |-19/12 12/12|  =  ((2^-19)*(3^12))^(1/12) =  1/12 Pythagorean comma
> -----------------
>   | 7/12   0   |  =  2^(7/12)     =             12-EDO "5th"



Upon re-reading this, it just struck me that perhaps this
was the reasoning which may have led the Sumerians to 12-EDO
(if I'm correct that they did "go there").


I'm not suggesting that the Sumerians conceived of numbers
in exponential terms, especially not numbers which describe
ratios of *frequencies* (well, OK, maybe I am), but if they
did, this calculation would look like this in base-60:

  |-1       1|
- |-1,35    1|
  ------------
  | 35      0|


So the "cycle of 5ths" in 12-EDO would be:

          exponent
         in base-60

2^(6/12)    30
2^(11/12)   55
2^(4/12)    20
2^(9/12)    45
2^(2/12)    10
2^(7/12)    35
2^(0/12)     0
2^(5/12)    25
2^(10/12)   50
2^(3/12)    15
2^(8/12)    40
2^(1/12)     5
2^(6/12)    30

which form an exponent-cycle of +35 (or -25) mod 60.


They certainly knew about the so-called "Pythagorean" cycle,
and my research leads me to believe that they knew how to
calculate 12-EDO.  If they realized that the 12-EDO "5th"
is tempered by 1/12 of a Pythagorean comma, as is shown by
my vector subtraction, then there ought to be some evidence
of their reasoning.


On firmer ground, let's take a look at string-length calculations.
The string-length which produces 2^(7/12), which is quite
close to 1772/2655, can be described extremely accurately
in base-60 as 40,2,42,42,15.


Do any of you know of any "ancient mathematics" groups
whose subscribers might know about ancient references
to these numbers, if there are any?
(either groups or references, that is)



love / peace / harmony ...

-monz
Yahoo! GeoCities *
"All roads lead to n^0"


 



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Message: 1195

Date: Wed, 1 Aug 2001 00:31:16

Subject: Re: Sumerian 12-EDO? (was: [tuning] fractional exponents of prime-factors)

From: monz

> From: monz <joemonz@xxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>; <celestial-tuning@xxxxxxxxxxx.xxx>
> Sent: Wednesday, August 01, 2001 12:07 AM
> Subject: [tuning-math] Sumerian 12-EDO? (was: [tuning] fractional
exponents of prime-factors)
>

>
>
> I wrote:
>
> > ... if we use vector subtraction to
> > calculate what note results from tempering the Pythagorean
> > "perfect 5th" by 1/12 of a Pythagorean comma, we get:
> >
> >     2^x   3^y
> >
> >   |-12/12 12/12|  =  2^-1 * 3^1   =   3/2    =  "perfect 5th"
> > - |-19/12 12/12|  =  ((2^-19)*(3^12))^(1/12) =  1/12 Pythagorean comma
> > -----------------
> >   | 7/12   0   |  =  2^(7/12)     =             12-EDO "5th"
>
>
>
> Upon re-reading this, it just struck me that perhaps this
> was the reasoning which may have led the Sumerians to 12-EDO
> (if I'm correct that they did "go there").



A few more quite accurate base-60 string-length ratios
which are important in pursuing this idea:


Pythagorean comma:  59,11,32,43,11

1/12 Pythagorean comma:  59,55,56,13,9





love / peace / harmony ...

-monz
Yahoo! GeoCities *
"All roads lead to n^0"






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Message: 1196

Date: Wed, 1 Aug 2001 10:11 +01

Subject: Re: BP linear temperament

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <9k71vn+ll5m@xxxxxxx.xxx>
Paul wrote:

> > You can find the unison vectors by calculating the ratios of 
> > (differences between) various pairs of generic seconds, pairs of 
> > generic thirds, etc.
> 
> Here are some I found, just by looking at the generic seconds:
> 
> 243:245
> 273:275
> 1323:1331
> 1617:1625
> 
> That should be enough . . . in 4-D, (5,7,11,13) space, with 3 as the 
> interval of equivalence, these UVs become
> 
> (-1 -2 0 0)
> (-2 1 -1 1)
> ( 0 2 -3 0)
> (-3 2 1 -1)
> 
> and the determinant of this matrix is -39. Whew!

Oh, you want 13-limit now, do you?  My program gives


13/56, 279.0 cent generator

basis:
(1.0, 0.23248676035896343)

mapping by period and generator:
[(1, 0), (-1, 7), (1, 2), (2, -1), (-2, 18), (0, 10)]

mapping by steps:
[(43, 13), (27, 8), (63, 19), (76, 23), (94, 28), (100, 30)]

unison vectors:
[[-2, -1, -1, 1, 0, 1], [9, 2, -7, 0, 0, 0], [-13, 1, 0, 7, 0, 0], [4, 18, 
0, 0, -7, 0]]

highest interval width: 19
complexity measure: 19  (30 for smallest MOS)
highest error: 0.009850  (11.820 cents)

(Caveat: cents are not cents, the consonance limit is arbitrary)

> Now what's the generator of the MOS we get by tempering out all of 
> the above UVs, except for, say, 243:245?

(-1 -2 0 0)(2)     (0)
(-2 1 -1 1)(-1)  = (-13)
( 0 2 -3 0)(18)    (-56)
(-3 2 1 -1)(10)    (0)


So only two of your UVs work with my MOS.


Taking my UVs

[[-2, -1, -1, 1, 0, 1], [9, 2, -7, 0, 0, 0], [-13, 1, 0, 7, 0, 0], [4, 18, 
0, 0, -7, 0]]

These are in (3,2,5,7,11,13) space.  So they are, um, 91/90, 78732/78125, 
1647086/1594323 and 21233664/19487171.  I expect they can be simplified.

The real generator of my MOS is 
0.23248676035896343*log2(3)*1200=442.2 cents.  Not considering octaves, I 
don't get a scale that looks right in the top 10.  See 
<import temper, string *> and 
<2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 8 *> as well as 
<import temper, string *> and 
<2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 8 *>.


                    Graham


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Message: 1197

Date: Wed, 1 Aug 2001 11:44:30

Subject: Re: BP linear temperament

From: manuel.op.de.coul@xxxxxxxxxxx.xxx

Paul wrote:

> Here are some I found, just by looking at the generic seconds:
>
> 243:245
> 273:275
> 1323:1331
> 1617:1625

Creating a periodicity block with these unison intervals was an
immediate result! This is the scale:
1/1 77/75 35/33 27/25 55/49 63/55 25/21 11/9 1701/1375 9/7 33/25
15/11 7/5 275/189 81/55 75/49 11/7 441/275 5/3 77/45 135/77 9/5
275/147 21/11 49/25 99/49 567/275 15/7 11/5 25/11 7/3 825/343 27/11
63/25 55/21 147/55 135/49 99/35 225/77 3/1
It is 11-limit. The 56th root of 3 quantisation is Myhill and the
generator is 11/7 (782.49 c.) or 21/11 (1119.46 c.).
The least squares best generator for the above scale is 780.8095
cents, s=39.924 cents (22) and L=60.213 cents (17). So it's very
close to the 95th root of 3.
1/1 39.9241 100.1373 140.0614 200.2747 240.1987 300.4120 340.3361
380.2601 440.4734 480.3975 540.6107 580.5348 640.7481 680.6721
740.8854 780.8095 820.7335 880.9468 920.8709 981.0841 1021.0082
1081.2215 1121.1455 1161.0696 1221.2829 1261.2069 1321.4202
1361.3443 1421.5575 1461.4816 1521.6949 1561.6189 1601.5430
1661.7563 1701.6803 1761.8936 1801.8177 1862.0309 3/1

Manuel


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Message: 1198

Date: Thu, 02 Aug 2001 19:39:09

Subject: Re: BP linear temperament

From: Paul Erlich

--- In tuning-math@y..., <manuel.op.de.coul@e...> wrote:
> 
> Paul wrote:
> >Your least squares optimization uses what mapping from generators 
to
> >primes? And it considers the errors for intervals 3:5, 3:7, 3:11,
> >3:13, 5:7, 5:11, 5:13, 7:11, 7:13, 11:13 . . . right?
> 
> Not exactly, it was an approximation to all pitches of the "11-
limit"
> scale given.

Now why would anyone want that? It's the consonant intervals, not the 
pitches, that we care about approximating well. (At least that's my 
philosophy.)

> A bit more work, but can do that too. Then the result
> is a virtually equal scale

You mean virtually the 39th root of 3? That's more what I would have 
expected.

> with a LS generator of 780.2702 cents.
> Differences are
> 3:5    6.647 /  5.906
> 3:7    3.772 /  4.513
> 3:11   5.993 /  6.734
> 3:13   2.880 /  2.139
> 5:7   -2.876 / -2.135
> 5:11  -0.654 /  0.087
> 5:13  -3.768 / -4.509
> 7:11   2.222 /  1.481
> 7:13  -0.892 / -1.633
> 11:13 -3.114 / -3.855
> The first one is the optimised value, the second one is the one
> occurring fewer times in the scale.

Can you explain what the latter means?


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Message: 1199

Date: Thu, 02 Aug 2001 01:51:52

Subject: Re: Another BP linear temperament?

From: Dave Keenan

--- In tuning-math@y..., graham@m... wrote:
> Here's the printout:
> 
> 7/30, 279.0 cent generator

Oops! Your generator should be 442.1 cents. You've multiplied by cents 
per octave when it should have been cents per tritave.

I get 442.6 cents as the 7-limit optimum. I notice you put "7 limit" 
in scare quotes, and rightly so. I'd call what you've given "7-limit 
without 2s".

By analogy: 
For an octave repeating scale, "7 limit" means ratios of all (whole 
number) non-multiples of 2 up to 7, i.e {1,3,5,7} (and their octave 
equivalents). 

So for a tritave repeating scale I think "7 limit" should mean ratios 
of all non-multiples of 3 up to 7, i.e. {1,2,4,5,6,7}.

-- Dave Keenan


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