Re: Rational Proportional Help Nec.

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> John Gilbert wrote:
> > Hi I have a problem here. What I am attempting to do is take a set of
> > ratios for 1 octave (1/1-2/1) and translate them over a wider range of 8
> > octaves. The original octave contains highly specified ratios which are
> > all
> > decimals. It seems to be moderately difficult. One idea is to map it as
a
> > triangle where the top smaller area has the line for the 1 octave range
> > while the base of the triangle is the 8 octave range. I am using scala
and
> > will want the results to be in hz ( or possibly cents would work) for a
> non
> > linear scale. If anyone would like to try to solve this problem please
> > don't hesitate to help out.
>
> I can't see the problem. What do you mean by non linear?
> Why do you need these triangles? To move ratios up or down
> an octave you multiply or divide by 2.
>
>
>
> Hi Graham also thank you for your interest in the solution to this. As I
> have explained above, by non linear mapping I mean no octave repeats.
What
> it really means I believe is complete freedom in tuning, no certain
> temperament all the way through the sonic range. If you wanted an
arbitrary
> random pattern as your tuning you can do so with a pattern actually I
> really meant to say is "LINEAR" so thats my bad. Anyhow...LINEAR MAPPING
is
> the correct term.
>
>
> If you are still with me after that, ok, let me explain. I want to take a
> one octave set of ratios, apply them over an 8 octave range with no
regards
> to repetitive octaves in the tuning. This will appear as a 27 note piano
> with unique frequencies and specific ones.

Ah, I think I'm beginning to understand what you want to get. I'm only not
sure if you wanted to preserve frequency ratios or length ratios and
therefore if you meant linear frequency shifting or linear length shifting.
So I'll assume you meant frequency shifting now. Well, a standard piano has
a range of approx. 7 octaves, not 8, so let me stay with this. Okay, if
you're using Scala, then load your set of intervals, type
"Move/frequency -126/127", and see the result. If you want to preserve
length proportions instead, type "Move/length -126/254". As to my view, I
don't think these very unequal sets of tones are an easy choice to tune on a
piano indeed. Is this really what you're looking for?

Petr

I wrote:

> Ah, I think I'm beginning to understand what you want to get. I'm only not
> sure if you wanted to preserve frequency ratios or length ratios and
> therefore if you meant linear frequency shifting or linear length
shifting.
> So I'll assume you meant frequency shifting now. Well, a standard piano
has
> a range of approx. 7 octaves, not 8, so let me stay with this. Okay, if
> you're using Scala, then load your set of intervals, type
> "Move/frequency -126/127", and see the result. If you want to preserve
> length proportions instead, type "Move/length -126/254". As to my view, I
> don't think these very unequal sets of tones are an easy choice to tune on
a
> piano indeed. Is this really what you're looking for?

If you actually wanted to preserve the exponential interval size proportions
instead of the linear frequency or length proportions, you do it with
"Multiply 7", provided you're going to stretch a one-octave scale into a
seven-octave one.
By the way, you didn't say a word about stretching, as far as I've read. If
you said you wanted to stretch the interval sizes in such a way you got
seven octaves instead of one, I think you had been understood sooner. Sorry
I'm saying 7 octaves instead of 8. If you want to tune this on a piano, 8 is
not the best way to start.

Petr

Hi John,

--- In tuning@yahoogroups.com, "John Gilbert" <preciousatonement@...>
wrote:
>
> On 4/7/07, John Gilbert
> <preciousatonement@...<preciousatonement%40gmail.com>>
> wrote:
> > Hi I have a problem here. What I am attempting to do is
> > take a set of ratios for 1 octave (1/1-2/1) and translate
> > them over a wider range of 8 octaves.
>
> Why can't you just multiply by powers of two? I must be
> missing something.
>
> Keenan
>
>
> Keenan, thanks for your interest. The more that I have looked
> into solving this problem the grander it seems to reveal itself
> as being moderately difficult. The reason I don't think you
> can multiply by powers of 2 or any simple multiplication is
> that I need to resolve the ratios in a larger system and then
> attain exact frequencies. Let me restate the problem: I
> have a 1 octave series of ratios in decimals for instance
> the 15th note is 1/.64641 and the octave would be 1/.5 or 2/1.
> This is a one octave set of 27 ratios all in decimal format
> which you can resolve to simply a number and not a fraction.
> I am looking to use the same proportional ratios over a
> wider range, in my case, the piano's range approx. Which
> we'll say 8 octaves to make it simple. So I'm looking to use
> the same proportions only this time over a wider area without
> any repeats.

I think you might be having some trouble expressing exactly
what you want, because maybe you're not familiar enough with
the mathematics. Here's my go at it:

It seems to me that what you want is to take the same
proportional relationships between the notes as they are
currently expressed within one octave (2/1 ratio), and
stretch all of them so that the range which formerly
covered one octave covers 8 octaves in the new tuning.
Yes?

To do that, you have to use logarithms. For tuning math
calculations, it almost always makes sense to use logs
with base 2, since we generally treat the 2/1 as an
equivalence interval. In your case, you're not doing
that, so you could use any base you like, but i'll stick
with base 2. Using the two example pitches you provided:

1/.5 = 2.000000
log_2(2.0) = 1.0

Now just multiply that by 8 and you get your new exponent
of 2: 8.0. And 2^8 = 256, so that's your new ratio.

1/.64641 = 1.547006
log_2(1.547006) = ~0.629479

Multiply that by 8 and you get ~5.035829.
And 2^5.035829 = ~32.804663, your new ratio for that note.

Understand how to do that?

If the method i've explained is what you were looking for,
and you still don't understand how to do the math, i'll
be happy to do it for you if you provide the whole list
of original ratios. It also simple to convert to Hz as well.

-monz
http://tonalsoft.com
Tonescape microtonal music software

Monz wrote:

It seems to me that what you want is to take the same
proportional relationships between the notes as they are
currently expressed within one octave (2/1 ratio), and
stretch all of them so that the range which formerly
covered one octave covers 8 octaves in the new tuning.
Yes?

I have understood it exactly the same, Monz. Unfortunately, I'm not sure if
he wanted to preserve the interval size proportions or the linear frequency
proportions or the linear length proportions. Fortunately, he sais he uses
Scala so I showed him how to get any of these (see my last letter). And I
hope it helps him at least a little.

Petr

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> It seems to me that what you want is to take the same
> proportional relationships between the notes as they are
> currently expressed within one octave (2/1 ratio), and
> stretch all of them so that the range which formerly
> covered one octave covers 8 octaves in the new tuning.
> Yes?
>
> To do that, you have to use logarithms.

You don't have to use logarithms; in this case,
you can merely cube the numbers. Replace
3/2 with (3/2)^3 = 27/8 and so forth.

Gene wrote:

You don't have to use logarithms; in this case,
you can merely cube the numbers. Replace
3/2 with (3/2)^3 = 27/8 and so forth.

First of all, this would change one octave into three octaves. Perhaps you
meant to say "power of 8"? Then, I'm still not sure if he didn't actually
mean to preserve linear proportions of frequencies or possibly of lengths
(instead of interval sizes), for which I have given explanation earlier.

Petr

--- In tuning@yahoogroups.com, Petr Paríº¥k <p.parizek@...> wrote:
>
> Gene wrote:
>
> You don't have to use logarithms; in this case,
> you can merely cube the numbers. Replace
> 3/2 with (3/2)^3 = 27/8 and so forth.
>
> First of all, this would change one octave into three octaves.
Perhaps you
> meant to say "power of 8"?

I thought he *wanted* to change one octave to three,
by changing 2 to 8 and leaving 1 as 1.

Then, I'm still not sure if he didn't actually
> mean to preserve linear proportions of frequencies or possibly of
lengths
> (instead of interval sizes), for which I have given explanation
earlier.

I'm responding to Monz, really. U don't know what
the OP meant, only what people think he might
have meant.

--- In tuning@yahoogroups.com, "John Gilbert" <preciousatonement@...>
wrote:
>
> Anyhow...LINEAR MAPPING is
> the correct term.
>

Linear in pitch, or in frequency?

To be very simple: suppose you had a tone in the exact middle of the
original octave pitch-wise

i.e. sqrt(2)=1.414...

Where would you want this to come out in the resulting 8-octave range?

You could use a linear frequency map between one octave 2:1 and eight
octaves 256:1 which would take 1.414 to 1 + (255*0.414). Or you could
use a linear pitch map which would take the tritone to the quadruple
octave 16.

Clearly if you use a linear frequency map between 2:1 and 256:1 on a
set of pitches which are more or less uniformly spaced with an octave,
you will end up with a new set of pitches which are strongly clustered
towards the top of the range.

I would therefore recommend a linear pitch map, which means taking the
eighth power of the ratios you started with (as someone already pointed
out) or stretching the cent values.

> If you are still with me after that, ok, let me explain. I want to
take a
> one octave set of ratios, apply them over an 8 octave range with no
regards
> to repetitive octaves in the tuning. This will appear as a 27 note
piano
> with unique frequencies and specific ones.
>

PS I believe the Boesendorfer Imperial has 8 octaves. However it is
inadvisable to write specifically for this piano as it is rare and
extremely expensive.

~~~T~~~