Re:Sad symmetric septimal minor tune

Hi Paul

(continued from MakeMicroMusic, see end of:
/makemicromusic/topicId_6638.html#6700
)

> > BTW I have added a new option to FTS
> > to make any formula you like into
> > a wave to use e.g. for example
> > chords in harmonic timbres
> > - but also they can be made into
> > sound fonts e.g. for the SB Live! or
> > for Giga. After I have finished the
> > upload I'll try making a few fonts
> > for both of those, and make
> > them available. (FTS can't play the
> > waves directly at present, so has to make them
> > into wave files then play those,
> > or sound fonts for a wavetable midi synth).
> >
> > It includes Bessel functions
> > which make an interesting rounded sawtooth
> > and I tried x*sin(1/x) and variants
> > such as x^3*sin(1/x^3) which have
> > infinitely many waves getting smaller
> > and smaller as you approach the origin.
> >
> > It also lets you build the functions up into
> > a harmonic series and you can quantise the
> > series to the current scale so I can experiment
> > with Setherising, I was surprised to find that
> > sine waves didn't work so well there
> > as more complex functions - you got beats
> > within a single note if you use
> > sine waves for e.g. equal tempered partials
> > in a harmonic series but when you replaced
> > them with more complex curves then the beats
> > went away. So there is clearly a bit of an art
> > to this.
>
> This is one of the most fascinating posts I've seen in a long time.
>
> Would you mind getting a bit deeper on all this on one of the more
> analytical lists, such as harmonic_entropy, tuning, or tuning-math?
>
> I'd enjoy that, anyway.

The curves are maybe best explained
with screen shots of them - I'll
do some to follow this up.

The basic mathematical idea I used
is to let the user enter any mathematical formula
to set a curve for either the
quarter, half or complete wave.

If it is intended for a quarter wave, FTS adds a constant
term to ensure that it starts at 0.
Then it does the other three quarters
so that they all join up seamlessly
- continuous lines anyway though they
may not be smooth where the various
quarters meet.

If it is intended for a half curve, then it
does two rotated copies, so again
they meet up and indeed this time
they will meet up smoothly too if the
original curve is smooth at origin.
This time it adds a linear term to
ensure that the start and end points of the
curve are both 0.

If it is a complete wave, then FTS
will add a quadratic term to ensure
that the two endpoints of the curve
are both zero, and that in between
it has as much of the curve above
as below the zero line (so that
the areas above and below are equal).
This won't necessarily always be a smooth
curve where they endpoints meet up
from one wave to the next however.

It then normalises the result so that
whatever your formula, the maximum
height of the wave remains the same.

So you can use any formula for any
of those, e.g. you could use say, x^3
for all three and it would still
work and make continuous
repeating sound waves you can play
and listen to.

You also have a choice of a few
presets too - sine waves,
triangles, a sine triangle, and
repeated sines which make a kind
of a rounded square wave,
and you can morph any of your
curves with a sine wave to
make a curve that is between
the two (e.g. 80 % your curve
and 20 % sine or whatever).

Then it adds together any desred number
of these waves to make
a harmonic series. The series
can be quantised to the current
scale e.g.to 7-et or 12-et or whatever.

Then the Setherised thing is a matter
partly of personal preference I think
- maybe the Setherised sine curve harmonic timbres
are in a sense smoother but they also have more obvious beats
(i.e. for each single note on its own)
while more complex curves end up with
a kind of uniform texture or roughness
so that you don't hear the beats within
that texture so clearly, so it
may sound more harmonious in some sense
if not really smoother.

The x^n*sin(1/x^n) etc family of curves
which I tried out is quite interesting here - when you set
n high you get something more like a series
of fast clicks, so a buzz, rather than a
wave, and the more it is like that then
the less you notice any beats, but the
note gets very kind of rough (an interesting
texture and still with a distinctive
pitch in it but a bit more towards a kind of
coloured noise rather than a note).

Anyway when I have some time after the upload
I'll do a web page about it with graphics
and sound clips, and some sound fonts
and Gigs to download for users of the
SB soundcards or others that can use
SF2 fonts, and for users of GigaSampler
/ GigaStudio. They consist of just
single waves so are tiny and the new

FTS has an option to make wave clips
for all the octaves for a sound font in one go.

However for best pitch accuracy you want
the high pitch waves to be an integral
number of samples. At present it
uses 1/100 of the sample rate
for the 1/1 of the sound font samples as it were,
and then goes up and down in octaves
from that point. So that pitches
them at A = 441, then if you want
A = 440 you can adjust the
result down by 4 cents when you use
the sound font.

If you try A = 440 at 44.1 Khz
that doesn't work well. It is important
to get the numbers right there as
the higher notes have very few samples,
e.g.at A = 441, then a uses
100 samples for a single wave, a' uses 50, and a'' uses
only 25. Clearly something like
49.5 approximated by 50 there would throw
the pitch well out. Best solution would
be if you could use the lowest pitch wave
for everything but in the SF2 fonts
anyway it seems it won't let you raise the
pitch more than about an octave or two.
(will be interested to know if anyone
has any tips about this).

The other solution is to record
a train of e.g. 100 waves
instead of a single wave,
which is an alternative - though the single
wave is more elegant.

Another idea is to set A=440 and
then save the sound clips at
44 Khz which you can do in FTS
and I think the SF2 format anyway
can handle data in that format,
and play it at the appropriate pitch, will see.

More later when I do the web page,
and will be interested to know if
anyone has any tips on the sound fonts.
Also has anyone come across any other program
that lets you play any mathematical equation
you like to enter as a sound wave?

If anyone has any particular questions
be sure to ask too,

Best,

Robert

Hi Robert!

> (continued from MakeMicroMusic, see end of:
> /makemicromusic/topicId_6638.html#6700
> )
>
> > > BTW I have added a new option to FTS
> > > to make any formula you like into
> > > a wave to use e.g. for example
> > > chords in harmonic timbres
> > > - but also they can be made into
> > > sound fonts e.g. for the SB Live! or
> > > for Giga. After I have finished the
> > > upload I'll try making a few fonts
> > > for both of those, and make
> > > them available. (FTS can't play the
> > > waves directly at present, so has to make them
> > > into wave files then play those,
> > > or sound fonts for a wavetable midi synth).
> > >
> > > It includes Bessel functions
> > > which make an interesting rounded sawtooth
> > > and I tried x*sin(1/x) and variants
> > > such as x^3*sin(1/x^3) which have
> > > infinitely many waves getting smaller
> > > and smaller as you approach the origin.
> > >
> > > It also lets you build the functions up into
> > > a harmonic series and you can quantise the
> > > series to the current scale so I can experiment
> > > with Setherising, I was surprised to find that
> > > sine waves didn't work so well there
> > > as more complex functions - you got beats
> > > within a single note if you use
> > > sine waves for e.g. equal tempered partials
> > > in a harmonic series but when you replaced
> > > them with more complex curves then the beats
> > > went away. So there is clearly a bit of an art
> > > to this.
> >
> > This is one of the most fascinating posts I've seen in a long
time.
> >
> > Would you mind getting a bit deeper on all this on one of the more
> > analytical lists, such as harmonic_entropy, tuning, or tuning-
math?
> >
> > I'd enjoy that, anyway.

> Then the Setherised thing is a matter
> partly of personal preference I think
> - maybe the Setherised sine curve harmonic timbres
> are in a sense smoother but they also have more obvious beats
> (i.e. for each single note on its own)

Yes. These beats result from inharmonicity, which is pleasant to a
degree (e.g., for piano strings). But beyond a certain threshold
string inharmonicity becomes associated with "poor tone quality", and
the same thing is true for synthesizers. Sonic artistry can
compensate for this to some degree, as Carlos, Sethares, Ligon, and
others' music shows.

> while more complex curves end up with
> a kind of uniform texture or roughness
> so that you don't hear the beats within
> that texture so clearly, so it
> may sound more harmonious in some sense
> if not really smoother.

These are periodic curves and thus have harmonic partials, correct?

I'm eagerly looking forward to your graphs and sound examples.

> Also has anyone come across any other program
> that lets you play any mathematical equation
> you like to enter as a sound wave?

Yes, I do that in Matlab, and have posted a number of sound examples
that way.

Later,
Paul

Hi Paul,

> I'm assuming you mean Sethares-style inharmonic timbres, which are
> neither sine curves nor harmonic timbres. Am I understanding you
> correctly?

Yes, I think so. Made by stacking e.g. sine curves, or also
other curves such as rounded saw tooth etc into a harmonic
series of pitches with the pitches quantised to whatever
scale you want e.g. 12-et or 17-et or whatever.

> Yes. Inharmonic timbres will generally have combinational tones which
> beat against the partials and against one another, and according to
> John Chalmers, even virtual pitches can beat against combinational
> tones. I replied to your message on the tuning list, and I'm hoping I
> got your meaning right above, since that's what I assumed in my
> tuning list reply.

Ah right that may be what it is, beating of combinational tones
with each other or the partials. I wondered how it happened.
I was using a harmonic series of quite a few terms, e.g.
5 terms or 13. I wonder if missing out some of the partials
may be a way to reduce those beats then. Well tht will
probably be for another upload after this - I plan to
add options to let the user set the volumes individually
for each partial (and the pitches too if they like).
Right now it just does them with the volumes
decreasing as you go up the series.

Meanwhile what I can do is to add radios to do
only the odd harmonics or only odd prime
harmonics - to reduce the number of partials
so of combination tones - maybe that will help make
the Setherised chords smoother.

Just as a taster, here is a screen shot:

http://www.robertinventor.com/xpn_sin_1oxpn.gif

Shows x^n sin(1/x^n) though can't remember what n was
- it's just a screen shot I made for debugging.

Normally all the ripples would be about the zero line,
but it is curved like this because of the quadratic term
that gets added to make the areas above and below
the zero line the same size, as they are in normal
sound waves.

As you vary the parameters in another window,
(including the n) then the curve
changes instantly in the oscilloscope and you can click
the Play this button to hear it.

I've cc'd this to tuning, and posted another reply
to your post there.

Thanks,

Robert

Hi Paul,

> Yes. These beats result from inharmonicity, which is pleasant to a
> degree (e.g., for piano strings). But beyond a certain threshold
> string inharmonicity becomes associated with "poor tone quality", and
> the same thing is true for synthesizers. Sonic artistry can
> compensate for this to some degree, as Carlos, Sethares, Ligon, and
> others' music shows.

Yes indeed.

> > while more complex curves end up with
> > a kind of uniform texture or roughness
> > so that you don't hear the beats within
> > that texture so clearly, so it
> > may sound more harmonious in some sense
> > if not really smoother.

> These are periodic curves and thus have harmonic partials, correct?

I meant periodic curves of any shape, then stacked in a harmonic series
with the series retuned to the scale, with all of those curves added
together, i.e. played simultaneously to make the inharmonic timbre.

e.g. a sequence of rounded saw tooth curves tuned to the nearest pitches to
the harmonic series in 12-et or 17-et then played together
as a series of say 5 or 13 pitches or whatever, with the volumes
reduced as you go up the series.

> I'm eagerly looking forward to your graphs and sound examples.

Will be fun to make them too. You can also try them out in
FTS once it is uploaded.

> > Also has anyone come across any other program
> > that lets you play any mathematical equation
> > you like to enter as a sound wave?

> Yes, I do that in Matlab, and have posted a number of sound examples
> that way.

Ah yes I remember now that you did some examples of chords tuned
in various ways made in Matlab for us to listen to and compare.

Thanks,

Robert

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
wrote:
> Hi Paul,
>
> > I'm assuming you mean Sethares-style inharmonic timbres, which are
> > neither sine curves nor harmonic timbres. Am I understanding you
> > correctly?
>
> Yes, I think so. Made by stacking e.g. sine curves, or also
> other curves such as rounded saw tooth etc into a harmonic
> series of pitches with the pitches quantised to whatever
> scale you want e.g. 12-et or 17-et or whatever.
>
> > Yes. Inharmonic timbres will generally have combinational tones
which
> > beat against the partials and against one another, and according
to
> > John Chalmers, even virtual pitches can beat against combinational
> > tones. I replied to your message on the tuning list, and I'm
hoping I
> > got your meaning right above, since that's what I assumed in my
> > tuning list reply.
>
> Ah right that may be what it is, beating of combinational tones
> with each other or the partials. I wondered how it happened.

It's pretty hard to avoid when constructing inharmonic timbres.
Besides the combinational tones and virtual pitch, there's also a
phenomenon called "second-order beats" which can affect sine waves in
out-of-tune octaves, but is demonstrably not due to nonlinearity in
the ear. Roederer's book is a good reference for stuff like this.

> Meanwhile what I can do is to add radios to do
> only the odd harmonics

This might alleviate the second-order beating, but would make the
timbre clarinet-like in a way you might not want . . . also it might
defeat the purpose if some of the intervals in the tuning you were
trying to "Setharize" for had become intervals involving an even
partial in the Setharized timbre.

> Just as a taster, here is a screen shot:
>
> http://www.robertinventor.com/xpn_sin_1oxpn.gif
>
> Shows x^n sin(1/x^n) though can't remember what n was
> - it's just a screen shot I made for debugging.
>
> Normally all the ripples would be about the zero line,
> but it is curved like this because of the quadratic term
> that gets added to make the areas above and below
> the zero line the same size, as they are in normal
> sound waves.

So this is a harmonic, i.e., periodic timbre. Have you looked at the
amplitudes in Fourier transform (i.e., the amplitudes of each of the
partials)? It's particularly efficient to do when you have a periodic
timbre like this. It also tells you a lot more about the audible
parts of the sound than the shape of the waveform does -- some
totally different waveshapes sound virtually identical since their
partials have the same amplitudes.

P.S. You don't really need to add a quadratic term or even a linear
term to get the areas above and below zero to be the same, a constant
would suffice. If you don't add a linear term, you might end up with
a discontinuous waveform but plenty of waveforms are discontinuous,
for example sawtooth, square, and pulse waves.

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
wrote:

> > > while more complex curves end up with
> > > a kind of uniform texture or roughness
> > > so that you don't hear the beats within
> > > that texture so clearly, so it
> > > may sound more harmonious in some sense
> > > if not really smoother.
>
> > These are periodic curves and thus have harmonic partials,
correct?
>
> I meant periodic curves of any shape,

Right . . . periodic curves all have harmonic partials, and no
inharmonic partials.

> then stacked in a harmonic series

Whoa! So you're stacking "timbres" into "macrotimbres"
or "metatimbres" in a sense?

> with the series retuned to the scale, with all of those curves added
> together, i.e. played simultaneously to make the inharmonic timbre.

If the answer is yes, I didn't get this message before, and one would
certainly expect these "metatimbres" to be very rich in beating --
good old, first-order beating between near-coincident partials.

> e.g. a sequence of rounded saw tooth curves tuned to the nearest
pitches to
> the harmonic series in 12-et or 17-et then played together
> as a series of say 5 or 13 pitches or whatever, with the volumes
> reduced as you go up the series.

OK. I'd love to see a specific example you've worked with so I can
produce the sound using Matlab and maybe we could even compare our
results.

Best Wishes,
Paul

Hi Paul,

> > Meanwhile what I can do is to add radios to do
> > only the odd harmonics

> This might alleviate the second-order beating, but would make the
> timbre clarinet-like in a way you might not want . . . also it might
> defeat the purpose if some of the intervals in the tuning you were
> trying to "Setharize" for had become intervals involving an even
> partial in the Setharized timbre.

Ok I'll try even as well (+ 1) i.e.
1 2 4 6 8 10 ... shifting the odd harmonics up the
series.
Also do primes rather than odd primes.

> Whoa! So you're stacking "timbres" into "macrotimbres"
> or "metatimbres" in a sense?

Yes, for instance that curve I just showed might
be the curve used for all the "partial"s in the metatimbre.

> If the answer is yes, I didn't get this message before, and one would
> certainly expect these "metatimbres" to be very rich in beating --
> good old, first-order beating between near-coincident partials.

Yes. I think what happens is that you have so very many in the way of
simultaneous beats that they sound more like a kind of a texture,
depending on the waveform. But they don't go away completely yet
with the ones I tried so far apart from the near noise fuzzy buzzy
timbres I tried. With those, I didn't pick out any beats, but can
nevertheless hear all the chord pitches in the noise.
Somewhere between the two with some art one can probably
get something that sounds smooth and pleasant and hasn't
got any beats, and I suppose that is what Setharising is
all about. I hadn't appreciated the art involved in it
before.

Robert

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
wrote:

> Ok I'll try even as well (+ 1) i.e.
> 1 2 4 6 8 10 ...

I liked your odd idea -- seemed quite clever. But this one would have
a lot of coinciding and thus potentially beating first-order
difference tones. 4-2=2, 6-4=2, 8-6=2, 10-8=2. Second-order
difference tones are often even louder, and a lot of these would be
potentially beating too: 2*4-6=2, 2*6-10=2, a lot of 4s, etc . . .

> > Whoa! So you're stacking "timbres" into "macrotimbres"
> > or "metatimbres" in a sense?
>
> Yes, for instance that curve I just showed might
> be the curve used for all the "partial"s in the metatimbre.
>
> > If the answer is yes, I didn't get this message before, and one
would
> > certainly expect these "metatimbres" to be very rich in beating --
> > good old, first-order beating between near-coincident partials.
>
> Yes. I think what happens is that you have so very many in the way
of
> simultaneous beats that they sound more like a kind of a texture,
> depending on the waveform. But they don't go away completely yet
> with the ones I tried so far apart from the near noise fuzzy buzzy
> timbres I tried. With those, I didn't pick out any beats, but can
> nevertheless hear all the chord pitches in the noise.

Cool! I look forward to hearing examples . . .

> Somewhere between the two with some art one can probably
> get something that sounds smooth and pleasant and hasn't
> got any beats, and I suppose that is what Setharising is
> all about. I hadn't appreciated the art involved in it
> before.

I'm not aware of anyone combining harmonic timbres inharmonically
into "metatimbres" before. It's certainly not what Sethares has
suggested in his book or articles. So you may be creating a whole new
art form here ;)

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
wrote:

> Also has anyone come across any other program
> that lets you play any mathematical equation
> you like to enter as a sound wave?

Goldwave has an expression evaluator. With some imagination it can do
FM synthesis, Karplus-Strong, additive, waveshaping etc...

Kalle

Hi Kalle,

> Goldwave has an expression evaluator. With some imagination it can do
> FM synthesis, Karplus-Strong, additive, waveshaping etc...

Great, found it, thanks! I use Goldwave a lot, but this window was
new to me :-).

Thanks,

Robert

Hi Kalle

> Goldwave has an expression evaluator. With some imagination it can do
> FM synthesis, Karplus-Strong, additive, waveshaping etc...

Just looked up Karplus-Strong and that is a nice easy thing to implement
to get a plucked string effect so I've added that in as an option,
thanks, getting some nice effects - a saw tooth + Karplus Strong
sounds pretty much like a plucked string, and then I've
added a couple of ways of warping the decay algorithrm
to get interesting sounds.

Here is a fun example, Keplus- Strong
decay of saw tooth + a couple of warps of the algorithm
and a swell and decay of each note superimposed on top
of that all (so you don't actually hear the initial pluck
at all).

http://www.robertinventor.com/21o16_3o2_saw_tooth_warped_swell_decay.mp3
:-)

Playing 1/1 21/16 3/2 2/1

Thanks,

Robert

Hi Paul,

> I think the harshness may have to do with the higher partials having
> amplitude that's inversely proportional to partial number, while
> higher-partial amplitudes inversely proportional to *squared* partial
> number are a lot less harsh on the ear. Can you verify this (or help
> me verify it) for the discontinuous waveforms you had in mind? It
> might be interesting to construct some continuous waveforms which
> however maintain the same amplitude spectrum, and see how the sounds
> compare.

Yes, I expect that's it. FTS has an option to set the exponent
for the harmonic series, and when set high it sounds smoother.
Sometims I've tried it really high e.g. 100 which just
adds a touch of the higher partials but with quite a few of those
it does change the wave shape in interesting ways.

Thanks,

Robert

Hi Kurt,

Thanks for all your suggestions.

I've added in the Heavyside functions, which let you
do anything you like in the way of continuous
functions or stretching a function over part of
its range linearly,

They are easy to use once one gets the idea
so I think it will do for now,
though later I could add some interface
to let user stretch and move the curve
interactively and maybe make it so
FTS itself generates the functions to
make that all work.

I've defined an extension of
it to make it easier to set ranges.

The Heavyside function is

H(x) = 0 if x<0 and 1 if x >1.

I've made a step function

K(a, x, b) is 1 if x is in the range a to b
and 0 otherwise.

So for example, sin(x*2*pi) * K(0,x,0.5)
is sin(x*2*pi) between 0 and 0.5 and 0
elsewhere.

To stretch one half of the sine wave you could
use e.g.

sin(x*3*pi) * K(0,x,1/3) - sin( (x-1/3)*(4/3)*pi) * K(1/3,x,1)

sin(x*6*pi) * K(0,x,1/6) - sin( (x-1/6)*(7/6)*pi) * K(1/6,x,1)

which I have just tested in FTS, and it works.

To have piece of various functions depending
on the range, just do the likes of:
K(0,x,0.5) *sin(x*pi*2) + K(0.5,x.1) *(0.5 - x)

That one is sin(2*pi*x) up to 0.5 then (0.5 - x) after that.

You could do them all using H instead
but you can't so easily read off the
ranges of values, especially if there
are sevearl steps, not just 2.

That one is quite simple using H though, since H(0.5-x) is
1 for x < 0.5, and H( x - 0.5) is 1 if x > 0.5, giving:

H(0.5- x) *sin(x*pi*2) + H(x -0.5)*(0.5 - x)

In the case of the derivatives, I've add an option to make the pieces join
together smoothly in the sense in which the word is sometimes
used in Maths, as meaning that the curve needs to have infinitely many derivatives.

I've made a smoothness parameter which sets how much of the adjoining part of the
curve you permit to be changed to make it smooth at the end points.

E.g. if you smooth sin(1/x) which has infiniely many ripples getting
cloer and cloer together at the origin, and set the parameter high then you
smooth out all the ripples altogether so they are no
longer large enough to be visible, even though they
are of height 1. Set it low and you mainly smooth out the last few of them
near the origin.

> It also occurs to me it might be useful to specify what fraction of the
> waveform period th entered waveform should corresond to, with the option of
> "padding" the rest either with:
>
> * a mirror of the original (probably just like what you do when a "half
> wave" is entered)

I did the half wave as a rotation (or mirror twice) so it automtically
meets upr at the end points, and also does so smoothly if the
original curves are smooth at those points.

The single mirror (i.e. reflect first then rotate) could be another
option if you want the curve to joni up with discontinous derivatives.

Later if I spread this feature out over several
windows in FTS, or make a new app for it, then
it could be nice to add in an option to transform the
curve with a click and drag on the
curve to stretch it or compress it in the x direction
in some way.

Meanwhile just adding in the Heavyside functions
is a way to let more advanced users do
anything they like in the way of discontinuous
or continuous and piece-wise smooth curves
without adding any more to the interface for
first time visitors to the window.

Thanks,

Robert

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
wrote:
> Hi Kurt,
>
> Thanks for all your suggestions.
>
> I've added in the Heavyside functions, which let you
> do anything you like in the way of continuous
> functions or stretching a function over part of
> its range linearly,
>
> They are easy to use once one gets the idea
> so I think it will do for now,
> though later I could add some interface
> to let user stretch and move the curve
> interactively and maybe make it so
> FTS itself generates the functions to
> make that all work.
>
> I've defined an extension of
> it to make it easier to set ranges.
>
> The Heavyside function is
>
> H(x) = 0 if x<0 and 1 if x >1.

So what is H(x) if 0<=x<=1?

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
wrote:
> The Heavyside function is
>
> H(x) = 0 if x<0 and 1 if x >1.

Ah. So what's the Lightside? :)

--- In tuning@yahoogroups.com, "wallyesterpaulrus" <paul@s...> wrote:

> > The Heavyside function is
> >
> > H(x) = 0 if x<0 and 1 if x >1.
>
> So what is H(x) if 0<=x<=1?

The Heaviside function is 1 for x>0, 0 for x<0, and most usefully 1/2
when x=0, though some people use 0 or 1.

on 5/20/04 7:59 PM, Robert Walker <robertwalker@ntlworld.com> wrote:

> Also has anyone come across any other program
> that lets you play any mathematical equation
> you like to enter as a sound wave?

This is probably not the kind of thing you meant, but you can play *any*
function as sound in mathematica, just as easily as you can plot it.

-Kurt

on 5/23/04 8:53 PM, Robert Walker <robertwalker@ntlworld.com> wrote:

> Hi Kurt,
>
> Thanks for all your suggestions.
>
> I've added in the Heavyside functions, which let you
> do anything you like in the way of continuous
> functions or stretching a function over part of
> its range linearly,

That was fast! Too bad I don't have a PC (that I use).

> In the case of the derivatives, I've add an option to make the pieces join
> together smoothly in the sense in which the word is sometimes
> used in Maths, as meaning that the curve needs to have infinitely many
> derivatives.

To have infinitely many derivatives defined over the whole domain? That's a
tall order.

> I've made a smoothness parameter which sets how much of the adjoining part of
> the
> curve you permit to be changed to make it smooth at the end points.s

Ah, I see, you are doing it by smoothing the end result, not by adjusting
the spliced-in section to achieve the desired continuity?

Maybe we should talk about getting FTS ported to the Mac? How much of the
code is user-interface and how much of it is sythesis and other
implementation, and how well are these aspects separated out? If there was
a quick-and-dirty way to do it it might be worth it. Just a thought. Not
that I have any time for it at the moment or in the next year! ;)

-Kurt

>
> Thanks,
>
> Robert