Tuning-Math Digests messages 2500 - 2524

This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).

Contents Hide Contents S 3

Previous Next

2000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500 2550 2600 2650 2700 2750 2800 2850 2900 2950

2500 - 2525 -



top of page bottom of page down


Message: 2500

Date: Sun, 16 Dec 2001 05:05:17

Subject: formula for meantone implications?

From: monz

Hello all,


Please take a look at
Yahoo groups: /tuning-math/files/monz/formula1-6-cmt.gif *

This is an x-y plot of the numeric relationship between the
pitches of 1/6-comma meantone their their acoustically closest
implied 5-limit JI ratios, as illustrated on my lattice at
lattices comparing various Meantone Cycles,  (c)2001 by Joseph L. Monzo *


I suppose calculus is need to derive this numerically,
since some values of x have two values for y and z, yes?

In cases where there are two values for both y and z
(i.e., -3 and +3 on my graph, and -3-6x and 3+6x if
it were to be continued), the lower z goes with the
top y and vice versa.

If someone is interested in this, I would also appreciate formulas
for the other meantone systems on my webpage, as well as a
generalized formula if one can be derived.



love / peace / harmony ...

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


 



_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 2501

Date: Sun, 16 Dec 2001 22:42:03

Subject: Vitale 19 (was: Re: Temperament calculations online)

From: dkeenanuqnetau

--- In tuning-math@y..., "clumma" <carl@l...> wrote:
> You said, "that doesn't quite do it"....
> 
> Anyway, without the givens, one could read... "all linear
> temperaments have the same number of o- and u-tonal chords",
> as Paul seems to have done.

You seem to have missed the important change I made. I put "same" in 
between "o" and "u", instead of after them.

Instead of 

the number of o and u are the same in any ...

I made it 

the number of o is the same as the number of u in any ...


top of page bottom of page up down


Message: 2503

Date: Sun, 16 Dec 2001 22:50:51

Subject: Re: Badness with gentle rolloff

From: dkeenanuqnetau

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > It has to do with Diophantine approximation theory. Have you 
read 
> > > Dave Benson's course notes?
>
> Well, he does mention the Diophantine approximation exponent for
> N-term ratios.

Could you tell me what section this is in? I have searched all 8 pdf 
files for the word "diophantine" with no success.


top of page bottom of page up down


Message: 2505

Date: Sun, 16 Dec 2001 22:59:44

Subject: Vitale 19 (was: Re: Temperament calculations online)

From: dkeenanuqnetau

--- In tuning-math@y..., graham@m... wrote:
> This is a useful thing to know for a temperament finder.  When 
considering 
> unison vectors, you can check which ones can never produce a 
temperament 
> as accurate as the one you want.  Do people have other rules of 
thumb for 
> filtering unison vectors, ETs or wedgies according to the simplest 
or most 
> accurate temperaments they can give rise to?  It would make the 
search 
> less arbitrary.

See A method for optimally distributing any comma *


top of page bottom of page up down


Message: 2507

Date: Sun, 16 Dec 2001 17:18 +0

Subject: Re: Vitale 19 (was: Re: Temperament calculations online)

From: graham@xxxxxxxxxx.xx.xx

Dave Keenan:
> > It has 5 otonal and 5 utonal 7-limit tetrads with max error of 2.7 
> c.

Paul Erlich
> Now -- if you think of this as a planar temperament where _only_ 
> 224:225 is tempered out, I bet you can reduce that error even further.

224:225 comes from 14:15 and 15:16 being equivalent.  These are both 
second-order 7-limit intervals.  So, the error has to be shared amongst 4 
7-limit intervals.  224:225 is 7.7 cents, so any scale tempering it out 
can't be closer than 7.7/4=1.9 cents to 7-limit JI.  So that's what the 
minimax for the planar temperament will be.

This is a useful thing to know for a temperament finder.  When considering 
unison vectors, you can check which ones can never produce a temperament 
as accurate as the one you want.  Do people have other rules of thumb for 
filtering unison vectors, ETs or wedgies according to the simplest or most 
accurate temperaments they can give rise to?  It would make the search 
less arbitrary.


                 Graham


top of page bottom of page up down


Message: 2509

Date: Mon, 17 Dec 2001 19:37:22

Subject: Re: Badness with gentle rolloff

From: clumma

> You can search .pdf files for a particular word?

Absolutely.

-Carl


top of page bottom of page up down


Message: 2510

Date: Mon, 17 Dec 2001 20:18:31

Subject: Vitale 19 (was: Re: Temperament calculations online)

From: paulerlich

--- In tuning-math@y..., graham@m... wrote:

> 224:225 comes from 14:15 and 15:16 being equivalent.  These are 
both 
> second-order 7-limit intervals.  So, the error has to be shared 
amongst 4 
> 7-limit intervals.

Or, the Hahn length of 224:225 in the 7-limit is 4. (Scala is 
supposedly able to compute this)


top of page bottom of page up down


Message: 2511

Date: Mon, 17 Dec 2001 12:41:01

Subject: Re: formula for meantone implications?

From: monz

Hi J,


> From: unidala <JGill99@xxxxxx.xxx>
> To: <tuning-math@xxxxxxxxxxx.xxx>
> Sent: Sunday, December 16, 2001 7:56 PM
> Subject: [tuning-math] Re: formula for meantone implications?
>
>
> J Gill: Monz, it sounds like you want to build a machine
> than can "think" (like people do)! I guess if you can
> define a set of JI ratios (which you like, or which meet 
> some "man-made" criteria for the numerical size of the
> numerator/denominator involved, etc.),  you could write
> a program to "decide" which of those ratios your meantone
> pitch value is "closest" to [by some pre-determined measure
> such as RMS error in deviation from a function such as
> 2^(pitch/reference)].


Not at all!  It's much simpler than that.  I'm just looking
for an elegant mathematical formula to explain what I'm showing
on my lattices.

The only measure I'm using is simple closeness in pitch-height.
The only reason it gets complicated and requires two solutions
sometimes is because some meantone pitches are exactly midway
between the two closest implied ratios.



-monz

 



_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 2512

Date: Mon, 17 Dec 2001 21:01:20

Subject: Re: formula for meantone implications?

From: paulerlich

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

> The only measure I'm using is simple closeness in pitch-height.
> The only reason it gets complicated and requires two solutions
> sometimes is because some meantone pitches are exactly midway
> between the two closest implied ratios.

Each meantone pitch implies an infinite number of ratios on the just 
5-limit lattice. Restricting yourself to the two closest would be 
severely insufficient to describe a piece by, say, Mozart, where the 
tonic alone would have to imply several different 81:80 
transpositions of itself over the course of the piece.


top of page bottom of page up down


Message: 2513

Date: Mon, 17 Dec 2001 22:24:41

Subject: Re: inverse of matrix --> for what?

From: genewardsmith

--- In tuning-math@y..., graham@m... wrote:

> Each column is a generator mapping.  The left hand one corresponds to the 
> top row of the original, 50:49, being the chromatic unison vector.  That 
> gives a 710 cent generator that approximates 3;2, with 9 octave reduced 
> fifths approximating 5:4 and 2 octave reduced fourths approximating 7:4.

It can be done as 9/22, better as 11/27, and best of all as 20/49, where it is the 27+22 system.

> The next column is for 64:63 being the chromatic unison vector.  As it has 
> a common factor of 2, you know the octave is divided into 2 equal parts.  
> You could set the generator as 434 cents.  Then, 3 generators are a 3:2, 
> and 5 could be either 5:4 or 7:4 (with tritone reduction).  Because 7:4 
> and 5:4 are the same tritone-reduced, 7:5 must be a tritone.  So 7:5 and 
> 10:7 are the same, and 50:49 is tempered out, as expected.  I think this 
> one is Paultone.

This is the chain-of-supermajor-thirds system, an interesting system with a unique association to the 22-et.

> The last column is for 245:243 tempered out.  I get a 109.4 cent 
> generator, with a 7-limit error of 17.5 cents.

This is twintone, aka Paultone.
 
> According to Gene, this:
> 
> ( 1 -6 -2 )
> ( 9 -10 4 )
> (-2 -10 4 )
> 
> is the adjoint of the original matrix, and each column is the wedge 
> product
of the relevant commatic unison vectors.

It's the adjoint matrix; the columns are wedge products in a 3D space
of octave equivalence classes, where the wedge product becomes a
cross-product. I don't recommend this point of view, which throws away
some valuable information.

From the full 7-limit point of view, we can do something equivalent by
taking the odd part of the commas; we then have

25/49^1/63 = [-2,4,4,0,0,0]
25/49^245/243 = [6,10,10,0,0,0]
1/63^245/243 = [1,9,-2,0,0,0]

Looking at this, we might think we have torsion in the first two
examples; however

50/49^64/63 = [-2,4,4,-2,-12,11]
50/49^245/243 = [6,10,10,-5,1,2]
64/63^245/243 = [1,9,-2,-30,6,12]

The above shows we do not have torsion, and tells us other things,
such as how to calculate the second column of the period matrix.

> > Do these integers tell us something about 22-EDO?
> > Or about 22-EDO's representation of the prime-factors?
> > 
> > ????
> 
> You should have left the factors of 2 in for that.  Add the octave
to the 
> matrix:

Another way is to add a top row of basis vectors to the matrix; this
is is the same as taking the triple wedge product: 
50/49^64/63^245/243 = 22 i + 35 j + 51 k + 62 l; we see that the
triple wedge product gives us the 22-et from the three commas.


top of page bottom of page up down


Message: 2515

Date: Mon, 17 Dec 2001 12:49 +0

Subject: Re: inverse of matrix --> for what?

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <002301c18678$e1e58f00$af48620c@xxx.xxx.xxx>
monz wrote:

> Excel's "mdeterm" function gives 22 as the determinant of
> the original matrix.  Multiplying the inverse of the matrix
> by the determinant gives the inverse as fractional parts of 22:
>   
> fractional inverse   
> | 1 -6 -2 | *  1
> | 9 -10 4 |   --
> |-2 -10 4 |   22

You shouldn't use | for the brackets.  They're for determinants.

> My questions: what does this inverse explain?
> What purpose does it serve?

Each column is a generator mapping.  The left hand one corresponds to the 
top row of the original, 50:49, being the chromatic unison vector.  That 
gives a 710 cent generator that approximates 3;2, with 9 octave reduced 
fifths approximating 5:4 and 2 octave reduced fourths approximating 7:4.

The next column is for 64:63 being the chromatic unison vector.  As it has 
a common factor of 2, you know the octave is divided into 2 equal parts.  
You could set the generator as 434 cents.  Then, 3 generators are a 3:2, 
and 5 could be either 5:4 or 7:4 (with tritone reduction).  Because 7:4 
and 5:4 are the same tritone-reduced, 7:5 must be a tritone.  So 7:5 and 
10:7 are the same, and 50:49 is tempered out, as expected.  I think this 
one is Paultone.

The last column is for 245:243 tempered out.  I get a 109.4 cent 
generator, with a 7-limit error of 17.5 cents.

According to Gene, this:

( 1 -6 -2 )
( 9 -10 4 )
(-2 -10 4 )

is the adjoint of the original matrix, and each column is the wedge 
product of the relevant commatic unison vectors.

> Do these integers tell us something about 22-EDO?
> Or about 22-EDO's representation of the prime-factors?
> 
> ????

You should have left the factors of 2 in for that.  Add the octave to the 
matrix:

( 1  0  0  0)
( 1  0  2 -2)
( 6 -2  0 -1)
( 0 -5  1  2)

then the adjoint is

(22  0   0  0)
(35  1  -6 -1)
(51  9 -10  4)
(62 -2 -10  4)

so you now have an extra column that tells you the number of steps to each 
prime interval.  It's also the wedge product of the three unison vectors.


                    Graham


top of page bottom of page up down


Message: 2516

Date: Mon, 17 Dec 2001 08:34:56

Subject: Re: inverse of matrix --> for what?

From: monz

----- Original Message -----
From: <graham@xxxxxxxxxx.xx.xx>
To: <tuning-math@xxxxxxxxxxx.xxx>
Sent: Monday, December 17, 2001 4:49 AM
Subject: [tuning-math] Re: inverse of matrix --> for what?



> You shouldn't use | for the brackets.  They're for determinants.

Wow -- thanks for clearing that up!


>
> > My questions: what does this inverse explain?
> > What purpose does it serve?
>
> Each column is a generator mapping.  The left hand one corresponds to the
> top row of the original, <snip...>
>
> According to Gene, this:
>
> ( 1 -6 -2 )
> ( 9 -10 4 )
> (-2 -10 4 )
>
> is the adjoint of the original matrix, and each column is the wedge
> product of the relevant commatic unison vectors.


Thanks very much for explaining this, Graham.  Now I'm at least
beginning to hope that someday I'll understand Gene's work.

Shouldn't I have Tuning Dictionary definitions for "wedge product"
and "adjoint"?  Please help.  ... Gene?  Paul?


> then the adjoint is
>
> (22  0   0  0)
> (35  1  -6 -1)
> (51  9 -10  4)
> (62 -2 -10  4)


Looks like a typo... shouldn't the second row be  (35 1 -6 -2) ?



love / peace / harmony ...

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






_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 2517

Date: Mon, 17 Dec 2001 16:50 +0

Subject: Re: inverse of matrix --> for what?

From: graham@xxxxxxxxxx.xx.xx

In-Reply-To: <00c901c18718$c3a638a0$af48620c@xxx.xxx.xxx>
monz wrote:

> Shouldn't I have Tuning Dictionary definitions for "wedge product"
> and "adjoint"?  Please help.  ... Gene?  Paul?

I don't know.  It depends on how bloated you want it to get.  They're both 
linear algebra terms that have a specialist application to tuning theory.  
And wedge products conceptually make the adjoint obsolete anyway.  The 
adjoint's only useful because it can sometimes be calculated more 
efficiently if you already have a library that does inverses (or solves 
systems of linear equations, which comes to the same thing).  Even then, 
it'll probably mean taking the inverse, multiplying by the determinant, 
and rounding off to integers.  So knowing it's called an "adjoint" isn't 
much help.


> > then the adjoint is
> >
> > (22  0   0  0)
> > (35  1  -6 -1)
> > (51  9 -10  4)
> > (62 -2 -10  4)
> 
> 
> Looks like a typo... shouldn't the second row be  (35 1 -6 -2) ?

Yes, looks like it, although I've lost the original calculation.


                      Graham


top of page bottom of page up down


Message: 2518

Date: Mon, 17 Dec 2001 19:00:20

Subject: Re: formula for meantone implications?

From: paulerlich

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> 
> I suppose calculus is need to derive this numerically,
> since some values of x have two values for y and z, yes?

Calculus???


top of page bottom of page up down


Message: 2519

Date: Mon, 17 Dec 2001 19:03:04

Subject: Vitale 19 (was: Re: Temperament calculations online)

From: paulerlich

--- In tuning-math@y..., graham@m... wrote:

> This is a useful thing to know for a temperament finder.  When 
considering 
> unison vectors, you can check which ones can never produce a 
temperament 
> as accurate as the one you want.  Do people have other rules of 
thumb for 
> filtering unison vectors, ETs or wedgies according to the simplest 
or most 
> accurate temperaments they can give rise to?

Yes, I've talked about this before, but my version does not 
correspond to the minimax view of things.


top of page bottom of page up down


Message: 2520

Date: Mon, 17 Dec 2001 19:04:56

Subject: Re: Badness with gentle rolloff

From: paulerlich

--- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> > > > It has to do with Diophantine approximation theory. Have you 
> read 
> > > > Dave Benson's course notes?
> >
> > Well, he does mention the Diophantine approximation exponent for
> > N-term ratios.
> 
> Could you tell me what section this is in?

I don't remember.

> I have searched all 8 pdf 
> files for the word "diophantine" with no success.

You can search .pdf files for a particular word? I've never heard of 
this ability. Try searching for "the".


top of page bottom of page up down


Message: 2521

Date: Mon, 17 Dec 2001 19:05:51

Subject: Vitale 19 (was: Re: Temperament calculations online)

From: paulerlich

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

> See A method for optimally distributing any comma *

Seems like a dead end. Time to redo this page with linear programming?


top of page bottom of page up down


Message: 2522

Date: Mon, 17 Dec 2001 19:10:18

Subject: Re: inverse of matrix --> for what?

From: paulerlich

--- In tuning-math@y..., graham@m... wrote:

> The next column is for 64:63 being the chromatic unison vector.
> As it has 
> a common factor of 2, you know the octave is divided into 2 equal 
parts.  
> You could set the generator as 434 cents.  Then, 3 generators are a 
3:2, 
> and 5 could be either 5:4 or 7:4 (with tritone reduction).  Because 
7:4 
> and 5:4 are the same tritone-reduced, 7:5 must be a tritone.  So 
7:5 and 
> 10:7 are the same, and 50:49 is tempered out, as expected.  I think 
this 
> one is Paultone.

Generator of 434 cents? I don't think so!
> 
> The last column is for 245:243 tempered out.

You mean _not_ tempered out.

> I get a 109.4 cent 
> generator, with a 7-limit error of 17.5 cents.

_That's_ paultone!


top of page bottom of page up down


Message: 2523

Date: Tue, 18 Dec 2001 19:08:50

Subject: Re: 55-tET (was: Re: inverse of matrix --> for what?)

From: monz

> So, rewritten in a form that I'm more familiar with, that's:
> 
> where unison-vector = 2^x * 3^y * 5^z,
> 
>   x   y   z
> 
> ( 90 -26 -21 )
> ( 82 -18 -23 )
> (  7  25 -20 )
> ( 31   1 -14 )
> ( 27   5 -15 )  <etc. -- snip>


And of course, Yahoo's new space-removing "feature" ruined
the careful formatting I put into that matrix, on the web-based
version of the list.



-monz


 



_________________________________________________________

Do You Yahoo!?

Get your free @yahoo.com address at Yahoo! Mail Setup *


top of page bottom of page up down


Message: 2524

Date: Tue, 18 Dec 2001 19:49:42

Subject: Vitale 19 (was: Re: Temperament calculations online)

From: paulerlich

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

> Actually, what would be the point. The point of my attempt on that 
> page, is that you can do it with nothing more than pen and paper 
and 
> you can follow what and why.

But it doesn't work right -- though of course if you could find a 
general fix, I'd be all for it . . . Linear programming can usually 
be done with pen and paper too.

> If you just want an algorithm for computer, then numerical methods 
> (sucessive approximations) work just fine.

You'd be surprised what a black-box minimization program can do with 
absolute value functions.


top of page bottom of page up

Previous Next

2000 2050 2100 2150 2200 2250 2300 2350 2400 2450 2500 2550 2600 2650 2700 2750 2800 2850 2900 2950

2500 - 2525 -

top of page