Hi All,

I was looking at oscil3 and table3 and cubic interpolation in general.
Could someone tell me what kind of interpolation this is?  (This is in
OOps/ugens2.c) I'm wondering too if we couldn't make an inline
function reuse in the various places to make it a little easier to
understand this bit of code.

Thanks!
steven

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I was looking at this paper "Polynomial Interpolators for High-Quality
Resampling of Oversampled Audio" by Olli Niemitalo [1] and found a
number of approaches. For example, it looks like he's contributed a
cubic formula to the musicdsp wiki[2] (looks to correspond to the
"4-point, 3rd-order Hermite (x-form)" in the paper).  I wasn't sure
what interpolation scheme was used in oscil3 (plus, my maths is not
very good. :P).

[1] - http://yehar.com/blog/?p=197
[2] - http://musicdsp.org/archive.php?classid=5#49

On Mon, Jul 14, 2014 at 1:49 PM,   wrote:
> ??  Cubic interpolation fits a cubic curve (Ax^3 + Bx^2 + Cx +D)
> through four points, two below point in question and 2 above.  Not
> sure what you are asking
>
> Quoting Steven Yi :
>
>> Hi All,
>>
>> I was looking at oscil3 and table3 and cubic interpolation in general.
>> Could someone tell me what kind of interpolation this is?  (This is in
>> OOps/ugens2.c) I'm wondering too if we couldn't make an inline
>> function reuse in the various places to make it a little easier to
>> understand this bit of code.
>>
>> Thanks!
>> steven
>>
>> ------------------------------------------------------------------------------
>> Want fast and easy access to all the code in your enterprise? Index and
>> search up to 200,000 lines of code with a free copy of Black Duck®
>> Code Sight™ - the same software that powers the world's largest code
>> search on Ohloh, the Black Duck Open Hub! Try it now.
>> http://p.sf.net/sfu/bds
>> _______________________________________________
>> Csound-devel mailing list
>> Csound-devel@lists.sourceforge.net
>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>
>
>
>
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Have a look at the Audio Programming book, “time-domain audio programming”, I explain 
the cubic interpolation method there.

========================
Dr Victor Lazzarini
Senior Lecturer
NUI Maynooth, Ireland
victor dot lazzarini at nuim dot ie




On 14 Jul 2014, at 19:09, Steven Yi  wrote:

> I was looking at this paper "Polynomial Interpolators for High-Quality
> Resampling of Oversampled Audio" by Olli Niemitalo [1] and found a
> number of approaches. For example, it looks like he's contributed a
> cubic formula to the musicdsp wiki[2] (looks to correspond to the
> "4-point, 3rd-order Hermite (x-form)" in the paper).  I wasn't sure
> what interpolation scheme was used in oscil3 (plus, my maths is not
> very good. :P).
> 
> [1] - http://yehar.com/blog/?p=197
> [2] - http://musicdsp.org/archive.php?classid=5#49
> 
> On Mon, Jul 14, 2014 at 1:49 PM,   wrote:
>> ??  Cubic interpolation fits a cubic curve (Ax^3 + Bx^2 + Cx +D)
>> through four points, two below point in question and 2 above.  Not
>> sure what you are asking
>> 
>> Quoting Steven Yi :
>> 
>>> Hi All,
>>> 
>>> I was looking at oscil3 and table3 and cubic interpolation in general.
>>> Could someone tell me what kind of interpolation this is?  (This is in
>>> OOps/ugens2.c) I'm wondering too if we couldn't make an inline
>>> function reuse in the various places to make it a little easier to
>>> understand this bit of code.
>>> 
>>> Thanks!
>>> steven
>>> 
>>> ------------------------------------------------------------------------------
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>>> _______________________________________________
>>> Csound-devel mailing list
>>> Csound-devel@lists.sourceforge.net
>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>> 
>> 
>> 
>> 
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> 
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Hi Victor,

Thanks for mentioning that. I read through the section but I'm afraid
I'm still a little bit in the dark as to how these coefficients were
chosen.  Perhaps I'll just leave it for now and come back to this
another time.

Thanks!
steven

On Mon, Jul 14, 2014 at 2:48 PM, Victor Lazzarini
 wrote:
> Have a look at the Audio Programming book, “time-domain audio programming”, I explain
> the cubic interpolation method there.
>
> ========================
> Dr Victor Lazzarini
> Senior Lecturer
> NUI Maynooth, Ireland
> victor dot lazzarini at nuim dot ie
>
>
>
>
> On 14 Jul 2014, at 19:09, Steven Yi  wrote:
>
>> I was looking at this paper "Polynomial Interpolators for High-Quality
>> Resampling of Oversampled Audio" by Olli Niemitalo [1] and found a
>> number of approaches. For example, it looks like he's contributed a
>> cubic formula to the musicdsp wiki[2] (looks to correspond to the
>> "4-point, 3rd-order Hermite (x-form)" in the paper).  I wasn't sure
>> what interpolation scheme was used in oscil3 (plus, my maths is not
>> very good. :P).
>>
>> [1] - http://yehar.com/blog/?p=197
>> [2] - http://musicdsp.org/archive.php?classid=5#49
>>
>> On Mon, Jul 14, 2014 at 1:49 PM,   wrote:
>>> ??  Cubic interpolation fits a cubic curve (Ax^3 + Bx^2 + Cx +D)
>>> through four points, two below point in question and 2 above.  Not
>>> sure what you are asking
>>>
>>> Quoting Steven Yi :
>>>
>>>> Hi All,
>>>>
>>>> I was looking at oscil3 and table3 and cubic interpolation in general.
>>>> Could someone tell me what kind of interpolation this is?  (This is in
>>>> OOps/ugens2.c) I'm wondering too if we couldn't make an inline
>>>> function reuse in the various places to make it a little easier to
>>>> understand this bit of code.
>>>>
>>>> Thanks!
>>>> steven
>>>>
>>>> ------------------------------------------------------------------------------
>>>> Want fast and easy access to all the code in your enterprise? Index and
>>>> search up to 200,000 lines of code with a free copy of Black Duck®
>>>> Code Sight™ - the same software that powers the world's largest code
>>>> search on Ohloh, the Black Duck Open Hub! Try it now.
>>>> http://p.sf.net/sfu/bds
>>>> _______________________________________________
>>>> Csound-devel mailing list
>>>> Csound-devel@lists.sourceforge.net
>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>>
>>>
>>>
>>>
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>>
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>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>
>
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The coefficients are chosen by considering this:

(1) We need to fit the polynomial f(x) = ax^3+bx^2+cx+d so it passes through 
the 4 points (p0,p1,p2,p3) we have sampled.

(2) The value we want is between p1 and p2, so we can start by considering what
the polynomial values are for p1=0 and p2=1, f(0) and f(1)

f(0) = d
f(1) = a+b+c+d

(3) Continuing in this way, we look for the slope of the function at 0 and 1, to 
determine the rest of the polynomial shape. This will help us bring p0 and p3 
into the expression. The slope is the derivative, ie f’(x) = 3ax^2 + 2bx + c,

f’(0) = c
f’(1) = 3a + 2b + c

(4) To relate the above to the sampled points p0…p3, we can do:

f(0) = p1 and f(1) = p2   (the two points to the left and right of the region we want)

and the slope at f(0) and f(1) can be determined by the simple (first-order) 
relations:

f’(0) = (p2 - p0)/2
f’(1) = (p3 - p1)/2

(4) Now we have a system of equations which we can use to determine
a, b, c and d. 

p1 =  d
p2 =  a+b+c+d
(p2 - p0)/2 =  c
(p3 - p1)/2 = 3a + 2b + c 

and from this we can work out our four coefficients.

========================
Dr Victor Lazzarini
Senior Lecturer
NUI Maynooth, Ireland
victor dot lazzarini at nuim dot ie




On 15 Jul 2014, at 00:44, Steven Yi  wrote:

> Hi Victor,
> 
> Thanks for mentioning that. I read through the section but I'm afraid
> I'm still a little bit in the dark as to how these coefficients were
> chosen.  Perhaps I'll just leave it for now and come back to this
> another time.
> 
> Thanks!
> steven
> 
> On Mon, Jul 14, 2014 at 2:48 PM, Victor Lazzarini
>  wrote:
>> Have a look at the Audio Programming book, “time-domain audio programming”, I explain
>> the cubic interpolation method there.
>> 
>> ========================
>> Dr Victor Lazzarini
>> Senior Lecturer
>> NUI Maynooth, Ireland
>> victor dot lazzarini at nuim dot ie
>> 
>> 
>> 
>> 
>> On 14 Jul 2014, at 19:09, Steven Yi  wrote:
>> 
>>> I was looking at this paper "Polynomial Interpolators for High-Quality
>>> Resampling of Oversampled Audio" by Olli Niemitalo [1] and found a
>>> number of approaches. For example, it looks like he's contributed a
>>> cubic formula to the musicdsp wiki[2] (looks to correspond to the
>>> "4-point, 3rd-order Hermite (x-form)" in the paper).  I wasn't sure
>>> what interpolation scheme was used in oscil3 (plus, my maths is not
>>> very good. :P).
>>> 
>>> [1] - http://yehar.com/blog/?p=197
>>> [2] - http://musicdsp.org/archive.php?classid=5#49
>>> 
>>> On Mon, Jul 14, 2014 at 1:49 PM,   wrote:
>>>> ??  Cubic interpolation fits a cubic curve (Ax^3 + Bx^2 + Cx +D)
>>>> through four points, two below point in question and 2 above.  Not
>>>> sure what you are asking
>>>> 
>>>> Quoting Steven Yi :
>>>> 
>>>>> Hi All,
>>>>> 
>>>>> I was looking at oscil3 and table3 and cubic interpolation in general.
>>>>> Could someone tell me what kind of interpolation this is?  (This is in
>>>>> OOps/ugens2.c) I'm wondering too if we couldn't make an inline
>>>>> function reuse in the various places to make it a little easier to
>>>>> understand this bit of code.
>>>>> 
>>>>> Thanks!
>>>>> steven
>>>>> 
>>>>> ------------------------------------------------------------------------------
>>>>> Want fast and easy access to all the code in your enterprise? Index and
>>>>> search up to 200,000 lines of code with a free copy of Black Duck®
>>>>> Code Sight™ - the same software that powers the world's largest code
>>>>> search on Ohloh, the Black Duck Open Hub! Try it now.
>>>>> http://p.sf.net/sfu/bds
>>>>> _______________________________________________
>>>>> Csound-devel mailing list
>>>>> Csound-devel@lists.sourceforge.net
>>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>>> 
>>>> 
>>>> 
>>>> 
>>>> ------------------------------------------------------------------------------
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>>>> Csound-devel mailing list
>>>> Csound-devel@lists.sourceforge.net
>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>> 
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>> 
>> 
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> 
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> (2) The value we want is between p1 and p2, so we can start by considering what
> the polynomial values are for p1=0 and p2=1, f(0) and f(1)


Quick correction, p1 and p2 are not 0 and 1, but the values of f(x) at x=0 and x=1,
sorry for the confusion.

========================
Dr Victor Lazzarini
Senior Lecturer
NUI Maynooth, Ireland
victor dot lazzarini at nuim dot ie




On 15 Jul 2014, at 11:34, Victor Lazzarini  wrote:

> The coefficients are chosen by considering this:
> 
> (1) We need to fit the polynomial f(x) = ax^3+bx^2+cx+d so it passes through 
> the 4 points (p0,p1,p2,p3) we have sampled.
> 
> (2) The value we want is between p1 and p2, so we can start by considering what
> the polynomial values are for p1=0 and p2=1, f(0) and f(1)
> 
> f(0) = d
> f(1) = a+b+c+d
> 
> (3) Continuing in this way, we look for the slope of the function at 0 and 1, to 
> determine the rest of the polynomial shape. This will help us bring p0 and p3 
> into the expression. The slope is the derivative, ie f’(x) = 3ax^2 + 2bx + c,
> 
> f’(0) = c
> f’(1) = 3a + 2b + c
> 
> (4) To relate the above to the sampled points p0…p3, we can do:
> 
> f(0) = p1 and f(1) = p2   (the two points to the left and right of the region we want)
> 
> and the slope at f(0) and f(1) can be determined by the simple (first-order) 
> relations:
> 
> f’(0) = (p2 - p0)/2
> f’(1) = (p3 - p1)/2
> 
> (4) Now we have a system of equations which we can use to determine
> a, b, c and d. 
> 
> p1 =  d
> p2 =  a+b+c+d
> (p2 - p0)/2 =  c
> (p3 - p1)/2 = 3a + 2b + c 
> 
> and from this we can work out our four coefficients.
> 
> ========================
> Dr Victor Lazzarini
> Senior Lecturer
> NUI Maynooth, Ireland
> victor dot lazzarini at nuim dot ie
> 
> 
> 
> 
> On 15 Jul 2014, at 00:44, Steven Yi  wrote:
> 
>> Hi Victor,
>> 
>> Thanks for mentioning that. I read through the section but I'm afraid
>> I'm still a little bit in the dark as to how these coefficients were
>> chosen.  Perhaps I'll just leave it for now and come back to this
>> another time.
>> 
>> Thanks!
>> steven
>> 
>> On Mon, Jul 14, 2014 at 2:48 PM, Victor Lazzarini
>>  wrote:
>>> Have a look at the Audio Programming book, “time-domain audio programming”, I explain
>>> the cubic interpolation method there.
>>> 
>>> ========================
>>> Dr Victor Lazzarini
>>> Senior Lecturer
>>> NUI Maynooth, Ireland
>>> victor dot lazzarini at nuim dot ie
>>> 
>>> 
>>> 
>>> 
>>> On 14 Jul 2014, at 19:09, Steven Yi  wrote:
>>> 
>>>> I was looking at this paper "Polynomial Interpolators for High-Quality
>>>> Resampling of Oversampled Audio" by Olli Niemitalo [1] and found a
>>>> number of approaches. For example, it looks like he's contributed a
>>>> cubic formula to the musicdsp wiki[2] (looks to correspond to the
>>>> "4-point, 3rd-order Hermite (x-form)" in the paper).  I wasn't sure
>>>> what interpolation scheme was used in oscil3 (plus, my maths is not
>>>> very good. :P).
>>>> 
>>>> [1] - http://yehar.com/blog/?p=197
>>>> [2] - http://musicdsp.org/archive.php?classid=5#49
>>>> 
>>>> On Mon, Jul 14, 2014 at 1:49 PM,   wrote:
>>>>> ??  Cubic interpolation fits a cubic curve (Ax^3 + Bx^2 + Cx +D)
>>>>> through four points, two below point in question and 2 above.  Not
>>>>> sure what you are asking
>>>>> 
>>>>> Quoting Steven Yi :
>>>>> 
>>>>>> Hi All,
>>>>>> 
>>>>>> I was looking at oscil3 and table3 and cubic interpolation in general.
>>>>>> Could someone tell me what kind of interpolation this is?  (This is in
>>>>>> OOps/ugens2.c) I'm wondering too if we couldn't make an inline
>>>>>> function reuse in the various places to make it a little easier to
>>>>>> understand this bit of code.
>>>>>> 
>>>>>> Thanks!
>>>>>> steven
>>>>>> 
>>>>>> ------------------------------------------------------------------------------
>>>>>> Want fast and easy access to all the code in your enterprise? Index and
>>>>>> search up to 200,000 lines of code with a free copy of Black Duck®
>>>>>> Code Sight™ - the same software that powers the world's largest code
>>>>>> search on Ohloh, the Black Duck Open Hub! Try it now.
>>>>>> http://p.sf.net/sfu/bds
>>>>>> _______________________________________________
>>>>>> Csound-devel mailing list
>>>>>> Csound-devel@lists.sourceforge.net
>>>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>>>> 
>>>>> 
>>>>> 
>>>>> 
>>>>> ------------------------------------------------------------------------------
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>>>>> search up to 200,000 lines of code with a free copy of Black Duck®
>>>>> Code Sight™ - the same software that powers the world's largest code
>>>>> search on Ohloh, the Black Duck Open Hub! Try it now.
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>>>>> _______________________________________________
>>>>> Csound-devel mailing list
>>>>> Csound-devel@lists.sourceforge.net
>>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>>> 
>>>> ------------------------------------------------------------------------------
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>>>> Csound-devel@lists.sourceforge.net
>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>> 
>>> 
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>> 
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> 
> 
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Actually, scrap that. The cubic interpolation used by Csound does not
use the coefficients found by this method. It’s actually much simpler:

(1) Set  f(-1) = p0, f(0) = p1, f(1) = p2 and  f(2) = p3, with
f(x) = ax^3 + bx^2 + cx + d

(2) Solve the system
p0 = -a + b - c + d 
p1 = d
p2 = a + b + c + d
p3 = 8a + 4b + 2c + d

(3) The coefficients are
a = (p3 - 3p2 + 3p1 - p0)/6
b = (p2 - 2p1 + p0)/2 
c = -p3/6 + p2 - p1/2 - p0/3
d = p1

This actually ensures that the polynomial fits the points correctly.
Sorry again for any confusion caused ;)

========================
Dr Victor Lazzarini
Senior Lecturer
NUI Maynooth, Ireland
victor dot lazzarini at nuim dot ie

On 15 Jul 2014, at 11:39, Victor Lazzarini  wrote:

>> (2) The value we want is between p1 and p2, so we can start by considering what
>> the polynomial values are for p1=0 and p2=1, f(0) and f(1)
> 
> 
> Quick correction, p1 and p2 are not 0 and 1, but the values of f(x) at x=0 and x=1,
> sorry for the confusion.
> 
> ========================
> Dr Victor Lazzarini
> Senior Lecturer
> NUI Maynooth, Ireland
> victor dot lazzarini at nuim dot ie
> 
> 
> 
> 
> On 15 Jul 2014, at 11:34, Victor Lazzarini  wrote:
> 
>> The coefficients are chosen by considering this:
>> 
>> (1) We need to fit the polynomial f(x) = ax^3+bx^2+cx+d so it passes through 
>> the 4 points (p0,p1,p2,p3) we have sampled.
>> 
>> (2) The value we want is between p1 and p2, so we can start by considering what
>> the polynomial values are for p1=0 and p2=1, f(0) and f(1)
>> 
>> f(0) = d
>> f(1) = a+b+c+d
>> 
>> (3) Continuing in this way, we look for the slope of the function at 0 and 1, to 
>> determine the rest of the polynomial shape. This will help us bring p0 and p3 
>> into the expression. The slope is the derivative, ie f’(x) = 3ax^2 + 2bx + c,
>> 
>> f’(0) = c
>> f’(1) = 3a + 2b + c
>> 
>> (4) To relate the above to the sampled points p0…p3, we can do:
>> 
>> f(0) = p1 and f(1) = p2   (the two points to the left and right of the region we want)
>> 
>> and the slope at f(0) and f(1) can be determined by the simple (first-order) 
>> relations:
>> 
>> f’(0) = (p2 - p0)/2
>> f’(1) = (p3 - p1)/2
>> 
>> (4) Now we have a system of equations which we can use to determine
>> a, b, c and d. 
>> 
>> p1 =  d
>> p2 =  a+b+c+d
>> (p2 - p0)/2 =  c
>> (p3 - p1)/2 = 3a + 2b + c 
>> 
>> and from this we can work out our four coefficients.
>> 
>> ========================
>> Dr Victor Lazzarini
>> Senior Lecturer
>> NUI Maynooth, Ireland
>> victor dot lazzarini at nuim dot ie
>> 
>> 
>> 
>> 
>> On 15 Jul 2014, at 00:44, Steven Yi  wrote:
>> 
>>> Hi Victor,
>>> 
>>> Thanks for mentioning that. I read through the section but I'm afraid
>>> I'm still a little bit in the dark as to how these coefficients were
>>> chosen.  Perhaps I'll just leave it for now and come back to this
>>> another time.
>>> 
>>> Thanks!
>>> steven
>>> 
>>> On Mon, Jul 14, 2014 at 2:48 PM, Victor Lazzarini
>>>  wrote:
>>>> Have a look at the Audio Programming book, “time-domain audio programming”, I explain
>>>> the cubic interpolation method there.
>>>> 
>>>> ========================
>>>> Dr Victor Lazzarini
>>>> Senior Lecturer
>>>> NUI Maynooth, Ireland
>>>> victor dot lazzarini at nuim dot ie
>>>> 
>>>> 
>>>> 
>>>> 
>>>> On 14 Jul 2014, at 19:09, Steven Yi  wrote:
>>>> 
>>>>> I was looking at this paper "Polynomial Interpolators for High-Quality
>>>>> Resampling of Oversampled Audio" by Olli Niemitalo [1] and found a
>>>>> number of approaches. For example, it looks like he's contributed a
>>>>> cubic formula to the musicdsp wiki[2] (looks to correspond to the
>>>>> "4-point, 3rd-order Hermite (x-form)" in the paper).  I wasn't sure
>>>>> what interpolation scheme was used in oscil3 (plus, my maths is not
>>>>> very good. :P).
>>>>> 
>>>>> [1] - http://yehar.com/blog/?p=197
>>>>> [2] - http://musicdsp.org/archive.php?classid=5#49
>>>>> 
>>>>> On Mon, Jul 14, 2014 at 1:49 PM,   wrote:
>>>>>> ??  Cubic interpolation fits a cubic curve (Ax^3 + Bx^2 + Cx +D)
>>>>>> through four points, two below point in question and 2 above.  Not
>>>>>> sure what you are asking
>>>>>> 
>>>>>> Quoting Steven Yi :
>>>>>> 
>>>>>>> Hi All,
>>>>>>> 
>>>>>>> I was looking at oscil3 and table3 and cubic interpolation in general.
>>>>>>> Could someone tell me what kind of interpolation this is?  (This is in
>>>>>>> OOps/ugens2.c) I'm wondering too if we couldn't make an inline
>>>>>>> function reuse in the various places to make it a little easier to
>>>>>>> understand this bit of code.
>>>>>>> 
>>>>>>> Thanks!
>>>>>>> steven
>>>>>>> 
>>>>>>> ------------------------------------------------------------------------------
>>>>>>> Want fast and easy access to all the code in your enterprise? Index and
>>>>>>> search up to 200,000 lines of code with a free copy of Black Duck®
>>>>>>> Code Sight™ - the same software that powers the world's largest code
>>>>>>> search on Ohloh, the Black Duck Open Hub! Try it now.
>>>>>>> http://p.sf.net/sfu/bds
>>>>>>> _______________________________________________
>>>>>>> Csound-devel mailing list
>>>>>>> Csound-devel@lists.sourceforge.net
>>>>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>>>>> 
>>>>>> 
>>>>>> 
>>>>>> 
>>>>>> ------------------------------------------------------------------------------
>>>>>> Want fast and easy access to all the code in your enterprise? Index and
>>>>>> search up to 200,000 lines of code with a free copy of Black Duck®
>>>>>> Code Sight™ - the same software that powers the world's largest code
>>>>>> search on Ohloh, the Black Duck Open Hub! Try it now.
>>>>>> http://p.sf.net/sfu/bds
>>>>>> _______________________________________________
>>>>>> Csound-devel mailing list
>>>>>> Csound-devel@lists.sourceforge.net
>>>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>>>> 
>>>>> ------------------------------------------------------------------------------
>>>>> Want fast and easy access to all the code in your enterprise? Index and
>>>>> search up to 200,000 lines of code with a free copy of Black Duck®
>>>>> Code Sight™ - the same software that powers the world's largest code
>>>>> search on Ohloh, the Black Duck Open Hub! Try it now.
>>>>> http://p.sf.net/sfu/bds
>>>>> _______________________________________________
>>>>> Csound-devel mailing list
>>>>> Csound-devel@lists.sourceforge.net
>>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>>> 
>>>> 
>>>> ------------------------------------------------------------------------------
>>>> Want fast and easy access to all the code in your enterprise? Index and
>>>> search up to 200,000 lines of code with a free copy of Black Duck®
>>>> Code Sight™ - the same software that powers the world's largest code
>>>> search on Ohloh, the Black Duck Open Hub! Try it now.
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>>>> _______________________________________________
>>>> Csound-devel mailing list
>>>> Csound-devel@lists.sourceforge.net
>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>> 
>>> ------------------------------------------------------------------------------
>>> Want fast and easy access to all the code in your enterprise? Index and
>>> search up to 200,000 lines of code with a free copy of Black Duck
>>> Code Sight - the same software that powers the world's largest code
>>> search on Ohloh, the Black Duck Open Hub! Try it now.
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>>> _______________________________________________
>>> Csound-devel mailing list
>>> Csound-devel@lists.sourceforge.net
>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>> 
>> 
>> ------------------------------------------------------------------------------
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>> Csound-devel mailing list
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>> https://lists.sourceforge.net/lists/listinfo/csound-devel
> 
> 
> ------------------------------------------------------------------------------
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Hi Victor,

Thanks very much for this!  I'll look forward to studying it. :)

Cheers!
steven

On Tue, Jul 15, 2014 at 3:50 PM, Victor Lazzarini
 wrote:
> Actually, scrap that. The cubic interpolation used by Csound does not
> use the coefficients found by this method. It’s actually much simpler:
>
> (1) Set  f(-1) = p0, f(0) = p1, f(1) = p2 and  f(2) = p3, with
> f(x) = ax^3 + bx^2 + cx + d
>
> (2) Solve the system
> p0 = -a + b - c + d
> p1 = d
> p2 = a + b + c + d
> p3 = 8a + 4b + 2c + d
>
> (3) The coefficients are
> a = (p3 - 3p2 + 3p1 - p0)/6
> b = (p2 - 2p1 + p0)/2
> c = -p3/6 + p2 - p1/2 - p0/3
> d = p1
>
> This actually ensures that the polynomial fits the points correctly.
> Sorry again for any confusion caused ;)
>
> ========================
> Dr Victor Lazzarini
> Senior Lecturer
> NUI Maynooth, Ireland
> victor dot lazzarini at nuim dot ie
>
> On 15 Jul 2014, at 11:39, Victor Lazzarini  wrote:
>
>>> (2) The value we want is between p1 and p2, so we can start by considering what
>>> the polynomial values are for p1=0 and p2=1, f(0) and f(1)
>>
>>
>> Quick correction, p1 and p2 are not 0 and 1, but the values of f(x) at x=0 and x=1,
>> sorry for the confusion.
>>
>> ========================
>> Dr Victor Lazzarini
>> Senior Lecturer
>> NUI Maynooth, Ireland
>> victor dot lazzarini at nuim dot ie
>>
>>
>>
>>
>> On 15 Jul 2014, at 11:34, Victor Lazzarini  wrote:
>>
>>> The coefficients are chosen by considering this:
>>>
>>> (1) We need to fit the polynomial f(x) = ax^3+bx^2+cx+d so it passes through
>>> the 4 points (p0,p1,p2,p3) we have sampled.
>>>
>>> (2) The value we want is between p1 and p2, so we can start by considering what
>>> the polynomial values are for p1=0 and p2=1, f(0) and f(1)
>>>
>>> f(0) = d
>>> f(1) = a+b+c+d
>>>
>>> (3) Continuing in this way, we look for the slope of the function at 0 and 1, to
>>> determine the rest of the polynomial shape. This will help us bring p0 and p3
>>> into the expression. The slope is the derivative, ie f’(x) = 3ax^2 + 2bx + c,
>>>
>>> f’(0) = c
>>> f’(1) = 3a + 2b + c
>>>
>>> (4) To relate the above to the sampled points p0…p3, we can do:
>>>
>>> f(0) = p1 and f(1) = p2   (the two points to the left and right of the region we want)
>>>
>>> and the slope at f(0) and f(1) can be determined by the simple (first-order)
>>> relations:
>>>
>>> f’(0) = (p2 - p0)/2
>>> f’(1) = (p3 - p1)/2
>>>
>>> (4) Now we have a system of equations which we can use to determine
>>> a, b, c and d.
>>>
>>> p1 =  d
>>> p2 =  a+b+c+d
>>> (p2 - p0)/2 =  c
>>> (p3 - p1)/2 = 3a + 2b + c
>>>
>>> and from this we can work out our four coefficients.
>>>
>>> ========================
>>> Dr Victor Lazzarini
>>> Senior Lecturer
>>> NUI Maynooth, Ireland
>>> victor dot lazzarini at nuim dot ie
>>>
>>>
>>>
>>>
>>> On 15 Jul 2014, at 00:44, Steven Yi  wrote:
>>>
>>>> Hi Victor,
>>>>
>>>> Thanks for mentioning that. I read through the section but I'm afraid
>>>> I'm still a little bit in the dark as to how these coefficients were
>>>> chosen.  Perhaps I'll just leave it for now and come back to this
>>>> another time.
>>>>
>>>> Thanks!
>>>> steven
>>>>
>>>> On Mon, Jul 14, 2014 at 2:48 PM, Victor Lazzarini
>>>>  wrote:
>>>>> Have a look at the Audio Programming book, “time-domain audio programming”, I explain
>>>>> the cubic interpolation method there.
>>>>>
>>>>> ========================
>>>>> Dr Victor Lazzarini
>>>>> Senior Lecturer
>>>>> NUI Maynooth, Ireland
>>>>> victor dot lazzarini at nuim dot ie
>>>>>
>>>>>
>>>>>
>>>>>
>>>>> On 14 Jul 2014, at 19:09, Steven Yi  wrote:
>>>>>
>>>>>> I was looking at this paper "Polynomial Interpolators for High-Quality
>>>>>> Resampling of Oversampled Audio" by Olli Niemitalo [1] and found a
>>>>>> number of approaches. For example, it looks like he's contributed a
>>>>>> cubic formula to the musicdsp wiki[2] (looks to correspond to the
>>>>>> "4-point, 3rd-order Hermite (x-form)" in the paper).  I wasn't sure
>>>>>> what interpolation scheme was used in oscil3 (plus, my maths is not
>>>>>> very good. :P).
>>>>>>
>>>>>> [1] - http://yehar.com/blog/?p=197
>>>>>> [2] - http://musicdsp.org/archive.php?classid=5#49
>>>>>>
>>>>>> On Mon, Jul 14, 2014 at 1:49 PM,   wrote:
>>>>>>> ??  Cubic interpolation fits a cubic curve (Ax^3 + Bx^2 + Cx +D)
>>>>>>> through four points, two below point in question and 2 above.  Not
>>>>>>> sure what you are asking
>>>>>>>
>>>>>>> Quoting Steven Yi :
>>>>>>>
>>>>>>>> Hi All,
>>>>>>>>
>>>>>>>> I was looking at oscil3 and table3 and cubic interpolation in general.
>>>>>>>> Could someone tell me what kind of interpolation this is?  (This is in
>>>>>>>> OOps/ugens2.c) I'm wondering too if we couldn't make an inline
>>>>>>>> function reuse in the various places to make it a little easier to
>>>>>>>> understand this bit of code.
>>>>>>>>
>>>>>>>> Thanks!
>>>>>>>> steven
>>>>>>>>
>>>>>>>> ------------------------------------------------------------------------------
>>>>>>>> Want fast and easy access to all the code in your enterprise? Index and
>>>>>>>> search up to 200,000 lines of code with a free copy of Black Duck®
>>>>>>>> Code Sight™ - the same software that powers the world's largest code
>>>>>>>> search on Ohloh, the Black Duck Open Hub! Try it now.
>>>>>>>> http://p.sf.net/sfu/bds
>>>>>>>> _______________________________________________
>>>>>>>> Csound-devel mailing list
>>>>>>>> Csound-devel@lists.sourceforge.net
>>>>>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>>
>>>>>>> ------------------------------------------------------------------------------
>>>>>>> Want fast and easy access to all the code in your enterprise? Index and
>>>>>>> search up to 200,000 lines of code with a free copy of Black Duck®
>>>>>>> Code Sight™ - the same software that powers the world's largest code
>>>>>>> search on Ohloh, the Black Duck Open Hub! Try it now.
>>>>>>> http://p.sf.net/sfu/bds
>>>>>>> _______________________________________________
>>>>>>> Csound-devel mailing list
>>>>>>> Csound-devel@lists.sourceforge.net
>>>>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>>>>>
>>>>>> ------------------------------------------------------------------------------
>>>>>> Want fast and easy access to all the code in your enterprise? Index and
>>>>>> search up to 200,000 lines of code with a free copy of Black Duck®
>>>>>> Code Sight™ - the same software that powers the world's largest code
>>>>>> search on Ohloh, the Black Duck Open Hub! Try it now.
>>>>>> http://p.sf.net/sfu/bds
>>>>>> _______________________________________________
>>>>>> Csound-devel mailing list
>>>>>> Csound-devel@lists.sourceforge.net
>>>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>>>>
>>>>>
>>>>> ------------------------------------------------------------------------------
>>>>> Want fast and easy access to all the code in your enterprise? Index and
>>>>> search up to 200,000 lines of code with a free copy of Black Duck®
>>>>> Code Sight™ - the same software that powers the world's largest code
>>>>> search on Ohloh, the Black Duck Open Hub! Try it now.
>>>>> http://p.sf.net/sfu/bds
>>>>> _______________________________________________
>>>>> Csound-devel mailing list
>>>>> Csound-devel@lists.sourceforge.net
>>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>>>
>>>> ------------------------------------------------------------------------------
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>>>> Csound-devel mailing list
>>>> Csound-devel@lists.sourceforge.net
>>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>>
>>>
>>> ------------------------------------------------------------------------------
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>>> https://lists.sourceforge.net/lists/listinfo/csound-devel
>>
>>
>> ------------------------------------------------------------------------------
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>
>
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