Can this equation be simplified?

(I've added brackets above the section whose log is taken,

and above the entire power of 10, to make them easier to see.)

|----------------------------------------------------|

|-----------------------|

v = 10 ^ ( LOG( 1 / (2 ^ (9r - 1/r) ) ) / ( -15r + 2/r - 1) )

Thanks.

-monz

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

>

> v = 10 ^ ( LOG( 1 / (2 ^ (9r - 1/r) ) ) / ( -15r + 2/r - 1) )

I don't know what the base of the log is, presuming it is e, we get

b = ln(2)(3r+1)/(5r+1) for the exponent, and so v = 10^b.

Hi Gene,

> From: genewardsmith <genewardsmith@juno.com>

> : <tuning-math@yahoogroups.com>

> Sent: Sunday, January 06, 2002 3:12 AM

> Subject: [tuning-math] Re: please simplify equation

>

>

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

>

> >

> > v = 10 ^ ( LOG( 1 / (2 ^ (9r - 1/r) ) ) / ( -15r + 2/r - 1) )

>

> I don't know what the base of the log is, presuming it is e, we get

>

> b = ln(2)(3r+1)/(5r+1) for the exponent, and so v = 10^b.

Thanks for doing that. Paul and I had a long online chat last

night in which I showed him what I had derived and he simplified

things for me.

But I tried plugging your equation into my spreadsheet, and

got the wrong results. The base of the log is 10, but that's

irrelevant now anyway, because I see now how I'm really looking

for an exponent that goes with base 2.

This formula expresses the golden meantone "5th",

where "r" is PHI = [1 + 5^(1/2)] / 2 .

By plugging in (1/r) = (r-1), my equation reduces to:

2^[ (8r+1) / (13r+3) ]

And Paul gave me these equivalent simplifications of it:

= 2^[ (2r-1) / (3r-1) ]

= 2^[ (3-r) / (4-r) ]

I plotted the numbers of all three of the above formulas

into a graph, and can see how they're all related linearly.

Can you explain algebraically what's going on? Please

be as detailed as possible. Thanks.

-monz

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

> 2^[ (8r+1) / (13r+3) ]

>

> And Paul gave me these equivalent simplifications of it:

>

> = 2^[ (2r-1) / (3r-1) ]

>

> = 2^[ (3-r) / (4-r) ]

>

>

> I plotted the numbers of all three of the above formulas

> into a graph, and can see how they're all related linearly.

> Can you explain algebraically what's going on? Please

> be as detailed as possible. Thanks.

Not really. My (3r+1)/(5r+1) is (r+9)/19, your (8r+1)/(13r+3) is

(r+18)/31, and Paul's (2r-1)/(3r-1) = (3-r)/(4-r) = (8-r)/11, so these are not the same. If you tell me what recurrence you are seeking the limit of, I'll tell you the answer.

Hi Gene,

> From: genewardsmith <genewardsmith@juno.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Sunday, January 06, 2002 4:00 PM

> Subject: [tuning-math] Re: please simplify equation

>

>

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

>

> > 2^[ (8r+1) / (13r+3) ]

> >

> > And Paul gave me these equivalent simplifications of it:

> >

> > = 2^[ (2r-1) / (3r-1) ]

> >

> > = 2^[ (3-r) / (4-r) ]

> >

> >

> > I plotted the numbers of all three of the above formulas

> > into a graph, and can see how they're all related linearly.

> > Can you explain algebraically what's going on? Please

> > be as detailed as possible. Thanks.

>

> Not really. My (3r+1)/(5r+1) is (r+9)/19,

> your (8r+1)/(13r+3) is (r+18)/31, and

> Paul's (2r-1)/(3r-1) = (3-r)/(4-r) = (8-r)/11,

> so these are not the same.

Can you show me how you work this magic?

Here are my comments:

First, your (3r+1)/(5r+1) definitely isn't right anyway.

The exponent of 2 has to be ~0.580178728. If r is PHI,

(3r+1)/(5r+1) = ~0.644003578 =/= ~0.580178728.

However, for r = PHI = [1 + 5^(1/2)] / 2 ,

(8r+1)/(13r+3)

= (2r-1)/(3r-1)

= (3-r)/(4-r)

= (8-r)/11

= ~0.580178728

but

(3r+1)/(5r+1) =/= (r+9)/19 and

(8r+1)/(13r+3) =/= (r+18)/31

So how do you get (8-r)/11 from (2r-1)/(3r-1) and

(3-r)/(4-r), and why are the other solutions incorrect?

> If you tell me what recurrence you are seeking the limit of,

> I'll tell you the answer.

Thanks for the offer, but... umm... I don't know what that means.

But these are the two things I'm looking for:

1)

Where r = PHI = [1 + 5^(1/2)] / 2 ,

my spreadsheet is calculating all three equations

2^[ (8r+1) / (13r+3) ]

= 2^[ (2r-1) / (3r-1) ]

= 2^[ (3-r) / (4-r) ]

to be the same to 9 decimal places.

I can see that they follow the general formula 2^x,

x = (ar+b)/(cr+d), where r = PHI = [1 + 5^(1/2)] / 2 .

I'm looking for the function which calculates a,b,c,d.

2)

I want to be able to describe some basic intervals of

golden meantone mathmatically, in terms of nothing but

PHI and numbers, as "ratios":

v = 5th

t = tone = major 2nd

s = diatonic semitone = minor 2nd

t^2 = major 3rd

t*s = minor 3rd

I already have several equivalent expressions for v :

2^[(8r+1)/(13r+3)]

= 2^[(2r-1)/(3r-1)]

= 2^[(3-r)/(4-r)]

= 2^[(8-r)/11]

I'd like to have something like that final form

for t, s, t^2, and t*s as well.

We have these basic relationships, for r=PHI:

v = (t^3)*s

t = (v^2)/2 = s*r

s = (2^3)/(v^5) = t^(1/r)

I derived my 2^[(8r+1)/(13r+3)] by plugging the values

for t and s involving v, into the v equation.

-monz

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

> > Not really. My (3r+1)/(5r+1) is (r+9)/19,

> > your (8r+1)/(13r+3) is (r+18)/31, and

> > Paul's (2r-1)/(3r-1) = (3-r)/(4-r) = (8-r)/11,

> > so these are not the same.

>

>

> Can you show me how you work this magic?

Ratios of the sort (a+br)/(c+dr) define an algebraic number field, which can always be put into the form of a sum of rational numbers times powers of a single algebraic number r. In this case, that results in

(a+br)/(c+dr) = (ac+ad-bd + (bc-ad)r)/(c^2+cd-d^2)

This form of the algebraic numbers in the field Q(r) is unique, since

{1, r} are a basis for a vector space over the rationals Q; hence we can determine if two elements of Q(r) are the same by putting them both into this form.

> First, your (3r+1)/(5r+1) definitely isn't right anyway.

> The exponent of 2 has to be ~0.580178728.

By taking the continued fraction for .580178728 I get something

very close to 1/1+1/1+1/(1+r), which simplifies to (r+7)/11, which is presumably the meantone fifth you are looking for--Kornerup's, I imagine. (8-r)/11 is the conjugate of (7+r)/11, and will give the right answer only if you replace r with r` = -1-r.

> So how do you get (8-r)/11 from (2r-1)/(3r-1) and

> (3-r)/(4-r), and why are the other solutions incorrect?

The division formula I gave should work--one way to derive it is to multiply numerator and denomiantor by the conjugate of the denominator. I think it should do for the rest of your quest also.

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

> By taking the continued fraction for .580178728 I get something

> very close to 1/1+1/1+1/(1+r), which simplifies to (r+7)/11, which is presumably the meantone fifth you are looking for--Kornerup's, I imagine. (8-r)/11 is the conjugate of (7+r)/11, and will give the right answer only if you replace r with r` = -1-r.

Whups, I goofed--I was trying to use r, but ended up using r`; so it seems (8-r)/11 is correct.

> From: genewardsmith <genewardsmith@juno.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Monday, January 07, 2002 4:31 PM

> Subject: [tuning-math] Re: please simplify equation

>

>

> Ratios of the sort (a+br)/(c+dr) define an algebraic number

> field, which can always be put into the form of a sum of

> rational numbers times powers of a single algebraic number r.

> In this case, that results in

>

> (a+br)/(c+dr) = (ac+ad-bd + (bc-ad)r)/(c^2+cd-d^2)

Thanks, Gene! That was the final piece of the puzzle

which I needed to complete my new Dictionary entry

for "golden meantone":

http://www.ixpres.com/interval/dict/golden.htm

love / peace / harmony ...

-monz

http://www.monz.org

"All roads lead to n^0"

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

> > Ratios of the sort (a+br)/(c+dr) define an algebraic number

> > field, which can always be put into the form of a sum of

> > rational numbers times powers of a single algebraic number r.

> > In this case, that results in

> >

> > (a+br)/(c+dr) = (ac+ad-bd + (bc-ad)r)/(c^2+cd-d^2)

>

>

> Thanks, Gene! That was the final piece of the puzzle

> which I needed to complete my new Dictionary entry

> for "golden meantone":

Hmmm...if it's in your dictionary, it might be well to be more precise and point out that for a quadratic number field, like the

golden ratio field Q(r), that (a+br)/(c+dr) gives all of the elements, but for a field of degree greater than three that would not be so. In general, however, for a field of degee d, elements of the form a_0 + a_1 r + ... + a_{n-1} r^{d-1}

with a_i a rational number and r an algebraic number define the number field Q(r) of degree d, where d is the degree of the irrreducible polynomial satisfied by r.

Maybe you should just skip the generalities and say that every number of the form (a+br)/(c+dr) with a,b,c,d rational can be reduced by the formula I gave to a unique form A + Br, with A and B rational, and that Q(r) defined by this is an algebraic number field, analogous to the ordinary rational numbers.

Hi Gene,

> From: genewardsmith <genewardsmith@juno.com>

> To: <tuning-math@yahoogroups.com>

> Sent: Tuesday, January 08, 2002 11:55 AM

> Subject: [tuning-math] Re: please simplify equation

>

>

> Hmmm...if it's in your dictionary, it might be well to

> be more precise and point out that for a quadratic number

> field, like the golden ratio field Q(r), that (a+br)/(c+dr)

> gives all of the elements, but for a field of degree greater

> than three that would not be so. In general, however, for

> a field of degee d, elements of the form

> a_0 + a_1 r + ... + a_{n-1} r^{d-1} with a_i a rational number

> and r an algebraic number define the number field Q(r) of

> degree d, where d is the degree of the irrreducible

> polynomial satisfied by r.

>

> Maybe you should just skip the generalities and say that

> every number of the form (a+br)/(c+dr) with a,b,c,d rational

> can be reduced by the formula I gave to a unique form A + Br,

> with A and B rational, and that Q(r) defined by this is an

> algebraic number field, analogous to the ordinary rational

> numbers.

Thanks for the further elaboration! But I hesitate to put all

of this into the "golden meantone" definition. Wouldn't it be

better as part of the "algebraic number" definition? If the

latter, then please suggest how the information specific to

each Dictionary entry should be placed and how they should

be linked. I'm understanding you as saying that golden

meantone is an example of a specific type of algebriac number

field, and so the Dictionary entries should be written to

reflect that.

-monz

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