Solving a tuning-math-like task -- looking for help

Hi tuners.

I hope someone here can give me a clue for answering this question. I'll first explain the topic using 2D temperaments and then go on to the question for the 3D example I'm trying to solve. For the purpose of this explanation, I'll use "cent(x)" as a replacement for the ratio to cent conversion.

In the case of meantone, for example, let's have A=cent(10/9), B=cent(16/15), C=cent(81/80). If we know, for example, that 5/2 is then mapped to "7A 2B 4C" and that 6/1 is "13A 5B 8C", we may get a pure 2:5:12 in our temperament by multiplying our numbers by an appropriate common coefficient so that subtracting the two results gives 0 at the second or the first position, respectively. This is "9, 0, 4" and "0, 9, 4", which means that both A and B should be widened by 4/9 C and C is tempered out.

But now I'm getting stuck. I have four intervals:
A=cent(27/25)
B=cent(15/14)
C=cent(28/27)
D=cent(4375/4374)

The mapping for the lowest 4 primes in terms of A|B|C|D counts is:
4 4 3 1
7 6 4 2
10 9 6 3
13 10 7 4

Now if I want to temper out D and have a pure 1:2:3:5, I'd like to multiply the numbers by appropriate common coefficients and subtract the results to get 4 + 4 + 4 numbers, where there are non-zero value only at the first and fourth, second and fourth, or third and fourth position, respectively. And I don't know how to do that. Could anyone help me?

Thanks in advance.

Petr

Petr,

The basic idea is to subtract appropriate multiples of the first row from
the second and third rows so they have a zero in their first entry. Then you
can use the second row to zero out the second entry from the first and third
rows and then finally use the third row to zero out the third entry of the
first and second rows. If you need more details, look for Gauss elimination
on some basic math web sites and you should find some hits.

David Bowen

2009/9/21 Petr Pařízek <p.parizek@...>

>
>
> Hi tuners.
>
> I hope someone here can give me a clue for answering this question. I'll
> first explain the topic using 2D temperaments and then go on to the
> question
> for the 3D example I'm trying to solve. For the purpose of this
> explanation,
> I'll use "cent(x)" as a replacement for the ratio to cent conversion.
>
> In the case of meantone, for example, let's have A=cent(10/9),
> B=cent(16/15), C=cent(81/80). If we know, for example, that 5/2 is then
> mapped to "7A 2B 4C" and that 6/1 is "13A 5B 8C", we may get a pure 2:5:12
> in our temperament by multiplying our numbers by an appropriate common
> coefficient so that subtracting the two results gives 0 at the second or
> the
> first position, respectively. This is "9, 0, 4" and "0, 9, 4", which means
> that both A and B should be widened by 4/9 C and C is tempered out.
>
> But now I'm getting stuck. I have four intervals:
> A=cent(27/25)
> B=cent(15/14)
> C=cent(28/27)
> D=cent(4375/4374)
>
> The mapping for the lowest 4 primes in terms of A|B|C|D counts is:
> 4 4 3 1
> 7 6 4 2
> 10 9 6 3
> 13 10 7 4
>
> Now if I want to temper out D and have a pure 1:2:3:5, I'd like to multiply
>
> the numbers by appropriate common coefficients and subtract the results to
> get 4 + 4 + 4 numbers, where there are non-zero value only at the first and
>
> fourth, second and fourth, or third and fourth position, respectively. And
> I
> don't know how to do that. Could anyone help me?
>
> Thanks in advance.
>
> Petr
>
>
>

--- In tuning@yahoogroups.com, Petr Pařízek <p.parizek@...> wrote:
> Hi tuners.
> I hope someone here can give me a clue for answering this question. > But now I'm getting stuck. I have four intervals:
> A=cent(27/25)
> B=cent(15/14)
> C=cent(28/27)
> D=cent(4375/4374)
> The mapping for the lowest 4 primes in terms of A|B|C|D counts is:
> 4 4 3 1
> 7 6 4 2
> 10 9 6 3
> 13 10 7 4

Petr,
I don't have an answer yet, but I am interested in this, and am working on it. How did you obtain the "mappings" by the way? (I know how I'd do it, but am interested in how you did it)

Steve.

Steve wrote:

> How did you obtain the "mappings" by the way? (I know how I'd do it,
> but am interested in how you did it)

Well, it depends on the method I decide to use. For this particular case, I started with a chord of 2:3:5:7:9 and I was subtracting smaller interval sizes from the larger ones until I got to the four I mentioned. So if, for example, 5/3 is the largest one and you "destroy it" with 3/2 (i.e. you subtract the size of 3/2 from that), then there are two 3/2s, one 7/5, one 9/7, and one 10/9. And if you go on doing this further and further, you can eventually get the small interval of 4375/4374, which is what I did.

But a completely different way is if the starting point is the interval which is tempered out rather than a chord. Then you can easily get the mapping from the prime exponents of this small interval .

Petr

--- In tuning@yahoogroups.com, Petr Pařízek <p.parizek@...> wrote:
> ... I have four intervals:
> A=cent(27/25)
> B=cent(15/14)
> C=cent(28/27)
> D=cent(4375/4374)
> ... if I want to temper out D and have a pure 1:2:3:5 ...

Petr,
I have an answer, but my method I think is different from yours; and I am not 100% sure that what I did addresses the problem. i.e. I posed myself a mathematical problem and then solved it, but was it the right problem?!

My Answer:
A=27/25 (unchanged)
B=3125/2916 (raised by 4375/4374)
C=648/625 (lowered by 4375/4374)

Does that seem plausible? I see that there are no fractional powers, and no factors of 7 at all. Did you mean 1:2:3:5 pure, or something else?

Steve.

I wrote:

> (47/46 narrowed by the ragisma).

I meant 36/35, of course.

Petr

Steve wrote:

> A=27/25 (unchanged)
> B=3125/2916 (raised by 4375/4374)
> C=648/625 (lowered by 4375/4374)

Thanks an awful lot, Steve, I'm not sure how you did that but I've eventually discovered the same. Unfortunately, anyway, I later learned that I would need to split the scale into even smaller intervals to be able to get a tempered 7/4 in a more or less regular fashion (i.e. if the octave is 4A+4B+3C, then two As in a row is not the best choice). So I completely remade the scale by taking A=27/25, B=3125/2916 (i.e. 15/14 widened by the "ragisma"), and C=250/243 (47/46 narrowed by the ragisma). Then the octave is mapped to 7+1+3, which makes two As in a row perfectly okay. So I've eventually used the octave of AABAACAACAC, which resulted in a 5-limit untempered scale where the 7/4 was mistuned by less than half a cent.

I'm currently playing around with it, I'll see if I can make some usable music out of that.

Petr

So where do the Bs go?

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of Petr Parízek
Sent: Tuesday, September 22, 2009 12:58 PM
To: tuning@yahoogroups.com
Subject: Re: [tuning] Re: Solving a tuning-math-like task -- looking for help

My previous answer apparently did not get through, so I'll try to send it once more.

Steve wrote:
> A=27/25 (unchanged)
> B=3125/2916 (raised by 4375/4374)
> C=648/625 (lowered by 4375/4374)
Thanks an awful lot, Steve, I'm not sure how you did that but I've eventually discovered the same. Unfortunately, anyway, I later learned that I would need to split the scale into even smaller intervals to be able to get a tempered 7/4 in a more or less regular fashion (i.e. if the octave is 4A+4B+3C, then two As in a row is not the best choice). So I completely remade the scale by taking A=27/25, B=3125/2916 (i.e. 15/14 widened by the "ragisma"), and C=250/243 (36/35 narrowed by the ragisma). Then the octave is mapped to 7+1+3, which makes two As in a row perfectly okay. So I've eventually used the octave of AABAACAACAC, which resulted in a 5-limit untempered scale where the 7/4 was mistuned by less than half a cent.
I'm currently playing around with it, I'll see if I can make some usable music out of that.
Petr

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
> ... I'm not sure how you did that but I've eventually discovered the same. ...
I did it by inverting a matrix or two; I think I can now explain it in terms of Gaussian elimination (as David Bowen pointed out).

For your first example, I assume you took the "mapping" coefficients and the "target ratios" coefficients and combined them into a two-row matrix:

7 2 4 -1 0 1
13 5 8 1 1 0

and then used Gaussian elimination to change it to:

1 0 4/9 -7/9 -2/9 5/9
0 1 4/9 20/9 7/9 -13/9

The result is found in the last three numbers in each row. I have done it this way, and "my" way, and got the same answer.

For your second example, there are three rows:
4 4 3 1 1 0 0
7 6 4 2 0 1 0
10 9 6 3 0 0 1

which you need to change to:
1 0 0 x x x x
0 1 0 x x x x
0 0 1 x x x x

I have been too lazy to do it this way. The final three numbers in each row give the answer. It looks like the 4th column is not actually needed in this example, I guess because there is no power of 7 in the target ratios.

S.