Ocarena Tuning

I have a feeling this has been explored by someone, somewhere, but I
don't know enough to know what to look for.

I was thinking about building an Ocarena, and the best way to tune it.
I thought that if I could finger 8 holes, I should try to maximize my
use of the resulting 256 fingerings. I'm still in the planning phase,
but the mathematical construct that came out of my investigation,
which I'll call the Ocarena Product Set, was interesting enough to
pursue on its own.

The fingering I settled on is as follows:
Right Hand (9/8) (10/9) (11/10) (12/11)
Left Hand (13/12) (14/13) (15/14) (16/15)

This way, a tin-whistle-fingering (straight up the right hand then
straight up the left) gives a harmonic (8-16)/8 scale and the opposite
fingering gives the subharmonic 16/(16-8) scale. Other fingerings fill
out some, but I think not all of the 16th harmonic diamond, as well as
some more distant pitches from transposing a harmonic pitch by a
superparticular.

The truth is, I know almost nothing about tuning, so any really simple
explanation of what I have wrought would be most appreciated. If this
has been written up, I would like to see what has been discovered
about it. I'm more interested in the practical applications, I've
worked out some common modes that are contained in the set. And if any
of you want to think about tempering it, I'll just sit over here and
fidget until you're done.

--TRISTAN
Dreaming of Eden is a Comic with no Pictures
http://dreamingofeden.smackjeeves.com

Hi Tristan,

> I'm still in the planning phase,
> but the mathematical construct that came out of my investigation,
> which I'll call the Ocarena Product Set, was interesting enough to
> pursue on its own.
>
> The fingering I settled on is as follows:
> Right Hand (9/8) (10/9) (11/10) (12/11)
> Left Hand (13/12) (14/13) (15/14) (16/15)
>
> This way, a tin-whistle-fingering (straight up the right hand
> then straight up the left) gives a harmonic (8-16)/8 scale and
> the opposite fingering gives the subharmonic 16/(16-8) scale.
> Other fingerings fill out some, but I think not all of the 16th
> harmonic diamond, as well as some more distant pitches from
> transposing a harmonic pitch by a superparticular.
>
> The truth is, I know almost nothing about tuning, so any really
> simple explanation of what I have wrought would be most
> appreciated. If this has been written up, I would like to see
> what has been discovered about it. I'm more interested in the
> practical applications, I've worked out some common modes that
> are contained in the set.

I don't know anything about the ocarina (I assume the spellings
are interchangeable?). But if you really have produced a
15-limit diamond, you should be a happy camper. Lots has been
written about diamonds, of course, but nothing earth-shattering.
They give you a lot of chords, and the 8-16 harmonics scale
is a good one. Prent Rodgers' music is a great place to start
with the diamond.

According to Wikipedia, the ocarina's timbre is nearly
sinusoidal. But it sounds similar to a pan flute, which,
like the clarinet, has mostly odd harmonics. If you're
building your own, I guess it depends on how you build it.

Good luck!

-Carl

Wow. I just now noticed that I completely didn't say what I was
actually doing. Thank you for the ideas, though.

Since each finger-hole is independent of position, only its size
matters, opening two holes is essentially the same as adding two
ratios. But since my fingers are limited in size, the amount added by
each hole has to be small.

I can take any sized subset out of the set of 8 superparticulars and
add them together by opening those specific holes. All of them closed
makes 1/1, and all open makes 2/1. There are some notes that clearly
lie on a diamond, like the ones I mentioned, but there are others that
seem more like a CPS, for instance if I open 9/8 and 11/10 I get
99/80, and there are a lot of these distant-key notes that I'm not
really sure how they relate to each other. I mean, I know how they
relate to the fundamental and how they relate to their component
finger-holes, but do they lie in a (incomplete) transposed diamond, or
is it more complicated?

I know that there are overlaps in this set, I've found them by
accident, but I don't know what their pattern is, so I wonder if there
is any way to enumerate the entire set without calculating all 256
fingerings.

As for timbre, the one I use right now is made by clamping my hands
together, and I think that a softer material adds significantly to the
harmonics. For this one I will probably use modeling clay, and add a
soft resonator like on a kazoo.

On 1/5/07, Carl Lumma <clumma@yahoo.com> wrote:
>
> I don't know anything about the ocarina (I assume the spellings
> are interchangeable?). But if you really have produced a
> 15-limit diamond, you should be a happy camper. Lots has been
> written about diamonds, of course, but nothing earth-shattering.
> They give you a lot of chords, and the 8-16 harmonics scale
> is a good one. Prent Rodgers' music is a great place to start
> with the diamond.
>
> According to Wikipedia, the ocarina's timbre is nearly
> sinusoidal. But it sounds similar to a pan flute, which,
> like the clarinet, has mostly odd harmonics. If you're
> building your own, I guess it depends on how you build it.
>
> Good luck!
>
> -Carl

--TRISTAN
Dreaming of Eden is a Comic with no Pictures
http://dreamingofeden.smackjeeves.com

Hi tristan and happy new year!

you know that you are using two scales based on harmonic series. these two different scales have divisions which are superparticular ratios.
(9/8) (10/9) (11/10) (12/11)(13/12) (14/13) (15/14) (16/15) shows 8-ADO (Arithmetic divisions of octave) ((instead of (8-16)/8)) and the opposit fingering shows 16-EDL( Equal divisions of length)((instead of 16/(16-8))
ADO and EDL systems are complementary systems and cover 15-limit . you can have a look at :
http://240edo.googlepages.com/ADO-EDL.XLS <http://240edo.googlepages.com/ADO-EDL.XLS>
to see how to construct these 2 and their relation with harmonic series.
for more about EDL you can look at :
http://240edo.googlepages.com/equaldivisionsoflength(edl <http://240edo.googlepages.com/equaldivisionsoflength(edl> )
There are some resources about historic background of EDL systems (based on cordophones and aerophones) and also , you can look at john chalmer's book :
http://eamusic.dartmouth.edu/~larry/published_articles/divisions_of_the_tetrachord/chapter7.pdf <http://eamusic.dartmouth.edu/~larry/published_articles/divisions_of_the_tetrachord/chapter7.pdf>

Shaahin Mohajeri

Tombak Player & Researcher , Microtonal Composer

My web site?? ???? ????? ?????? <http://240edo.googlepages.com/>

My farsi page in Harmonytalk ???? ??????? ?? ??????? ??? <http://www.harmonytalk.com/mohajeri>

Shaahin Mohajeri in Wikipedia ????? ?????? ??????? ??????? ???? ???? <http://en.wikipedia.org/wiki/Shaahin_mohajeri>

________________________________

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of Rozencrantz the Sane
Sent: Saturday, January 06, 2007 3:42 AM
To: Tuning List
Subject: [tuning] Ocarena Tuning

I have a feeling this has been explored by someone, somewhere, but I
don't know enough to know what to look for.

I was thinking about building an Ocarena, and the best way to tune it.
I thought that if I could finger 8 holes, I should try to maximize my
use of the resulting 256 fingerings. I'm still in the planning phase,
but the mathematical construct that came out of my investigation,
which I'll call the Ocarena Product Set, was interesting enough to
pursue on its own.

The fingering I settled on is as follows:
Right Hand (9/8) (10/9) (11/10) (12/11)
Left Hand (13/12) (14/13) (15/14) (16/15)

This way, a tin-whistle-fingering (straight up the right hand then
straight up the left) gives a harmonic (8-16)/8 scale and the opposite
fingering gives the subharmonic 16/(16-8) scale. Other fingerings fill
out some, but I think not all of the 16th harmonic diamond, as well as
some more distant pitches from transposing a harmonic pitch by a
superparticular.

The truth is, I know almost nothing about tuning, so any really simple
explanation of what I have wrought would be most appreciated. If this
has been written up, I would like to see what has been discovered
about it. I'm more interested in the practical applications, I've
worked out some common modes that are contained in the set. And if any
of you want to think about tempering it, I'll just sit over here and
fidget until you're done.

--TRISTAN
Dreaming of Eden is a Comic with no Pictures
http://dreamingofeden.smackjeeves.com <http://dreamingofeden.smackjeeves.com>

--- In tuning@yahoogroups.com, "Rozencrantz the Sane"
<rozencrantz@...> wrote:
>
> I have a feeling this has been explored by someone, somewhere, but I
> don't know enough to know what to look for.
>
> I was thinking about building an Ocarena, and the best way to tune it.
>

I'm afraid things are probably much more complicated than you imagine.

The problem is that wind instruments with finite size holes do not
react in a simple and predictable way to the opening and closing of
the holes. There is a complex nonlinear feedback from the resonating
cavity, which is coupled both to the outside air and the source of
vibrations (here the knife-edge). It is very unlikely that the change
in pitch can be predicted by simple mathematical formulae.

I believe concert flutes, for example, are designed mainly by
painstaking trial and error rather than any simple formula. And you'd
expect them to be relatively simple (a single vibrating column).

Even the simplest 'acoustic' instrument, the monochord, cannot be
controlled with absolute accuracy - there are end effects, bridge
effects, etc etc...

~~~T~~~

This reminds me of an interesting thing an aerospace engineer (with
patents and all) told me. According to him, it's actually the trial
and error of a hundred years and plain old craftsmanship that's most
responsible for making things fly.

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> I'm afraid things are probably much more complicated than you
imagine.
>
> The problem is that wind instruments with finite size holes do not
> react in a simple and predictable way to the opening and closing of
> the holes. There is a complex nonlinear feedback from the
resonating
> cavity, which is coupled both to the outside air and the source of
> vibrations (here the knife-edge). It is very unlikely that the
change
> in pitch can be predicted by simple mathematical formulae.
>
> I believe concert flutes, for example, are designed mainly by
> painstaking trial and error rather than any simple formula. And
you'd
> expect them to be relatively simple (a single vibrating column).
>
> Even the simplest 'acoustic' instrument, the monochord, cannot be
> controlled with absolute accuracy - there are end effects, bridge
> effects, etc etc...
>
> ~~~T~~~
>

Hi folks.

Nice to see you again, I was away for some time.

Tom wrote:

> The problem is that wind instruments with finite size holes do not
> react in a simple and predictable way to the opening and closing of
> the holes. There is a complex nonlinear feedback from the resonating
> cavity, which is coupled both to the outside air and the source of
> vibrations (here the knife-edge). It is very unlikely that the change
> in pitch can be predicted by simple mathematical formulae.

It depends a lot on the shape of the instrument. The formulae are not simple indeed, that's true, but I don't think the pitches are that much unpredictable as you seem to find.

> I believe concert flutes, for example, are designed mainly by
> painstaking trial and error rather than any simple formula. And you'd
> expect them to be relatively simple (a single vibrating column).

I do not.

> Even the simplest 'acoustic' instrument, the monochord, cannot be
> controlled with absolute accuracy - there are end effects, bridge
> effects, etc etc...

Even though the measures cannot be predicted with absolute accuracy, they can be predicted with enough accuracy to make as small corrections afterwards as possible. This can also include end correction and other things. Maybe you should read this:

www.phys.unsw.edu.au/jw/musFAQ.html

Petr

--- In tuning@yahoogroups.com, Petr Par�zek <p.parizek@...> wrote:
>
> Hi folks.
>
> Nice to see you again, I was away for some time.
>
>
> It depends a lot on the shape of the instrument. The formulae are
not simple indeed, that's true, but I don't think the pitches are that
> much unpredictable as you seem to find.

I'm not saying totally unpredictable, but certainly enough to make
thinking about small (below quartertone) pitch difference on a
prototype ocarina design pointless.

> Maybe you should read this:
>
> www.phys.unsw.edu.au/jw/musFAQ.html
>

All very well, Petr, but that website seems to have about a hundred
pages! Which ones would you recommend for the question under discussion?

There are indeed bits about clarinet, oboe, flute etc., but their
quirks seem to be idiosyncratic. In other words you need one formula
for the clarinet bell, another for the oboe reed, etc. in an empirical
fashion. So I would think the formulae simply encode the results of
experiment, i.e. sophisticated trial and error. The ocarina formula
doesn't exist yet and I don't think you can work one out from scratch.
~~~T~~~

On 1/8/07, Tom Dent <stringph@gmail.com> wrote:

> I'm afraid things are probably much more complicated than you imagine.

I don't doubt it. Still, I'm as interested in the pure mathematical
object as in the practical realization.

> The problem is that wind instruments with finite size holes do not
> react in a simple and predictable way to the opening and closing of
> the holes. There is a complex nonlinear feedback from the resonating
> cavity, which is coupled both to the outside air and the source of
> vibrations (here the knife-edge). It is very unlikely that the change
> in pitch can be predicted by simple mathematical formulae.

So if I drill hole A and test it so that I know it gives me 9/8 above
all-holes-closed, and I drill hole B and test it so that I know it
gives 10/9 over all-holes-closed, Opening A and B together won't give
me 5/4, is this what you're saying? Of course every time I drill a new
hole I'll change the total volume slightly, so after I drill the 10/9
hole I'll have to fine tune the 9/8 hole, which would throw the 10/9
hole out again.

But once I got them both in tune against the fundamental, they still
wouldn't add up to 5/4?

--TRISTAN
Dreaming of Eden is a Comic with no Pictures
http://dreamingofeden.smackjeeves.com

--- In tuning@yahoogroups.com, "Rozencrantz the Sane"
<rozencrantz@...> wrote:
> if I drill hole A and test it so that I know it gives me 9/8 above
> all-holes-closed, and I drill hole B and test it so that I know it
> gives 10/9 over all-holes-closed, Opening A and B together won't give
> me 5/4, is this what you're saying? Of course every time I drill a new
> hole I'll change the total volume slightly, so after I drill the 10/9
> hole I'll have to fine tune the 9/8 hole, which would throw the 10/9
> hole out again.
>
> But once I got them both in tune against the fundamental, they still
> wouldn't add up to 5/4?
>

That would only be true in the limit of infinitely small holes... I
would bet money against it.

For one thing, can it really be true that the (relative) position of
the holes has zero effect?
~~~T~~~

Tom Dent schrieb:
> --- In tuning@yahoogroups.com, "Rozencrantz the Sane"
> <rozencrantz@...> wrote:
> >> if I drill hole A and test it so that I know it gives me 9/8 above
>> all-holes-closed, and I drill hole B and test it so that I know it
>> gives 10/9 over all-holes-closed, Opening A and B together won't give
>> me 5/4, is this what you're saying? Of course every time I drill a new
>> hole I'll change the total volume slightly, so after I drill the 10/9
>> hole I'll have to fine tune the 9/8 hole, which would throw the 10/9
>> hole out again.
>>
>> But once I got them both in tune against the fundamental, they still
>> wouldn't add up to 5/4?
>>
>> >
> That would only be true in the limit of infinitely small holes... I
> would bet money against it.
>
> For one thing, can it really be true that the (relative) position of
> the holes has zero effect?
> Position and ordering of the tone holes should change exactly nothing. To make sure, it is probaby wise to start aus with an as globular design as possible. The perfect string and the perfect ocarina (probably realizable with beatehn ballons minus the high-pitched sound of the tightened skin) work on two very different principles with the cylindricality (I HAD to write this) of real-life wind instruments as compromised strings with volume-dependent and correction.

But as far as I can remember, it (pitch being dependent on volume and area of the opening[s]) is true for infinitely thin walls. So forget about clay, try origami!

I guess a design of any shape with the holes all being in one plain could also work (I'm thinking of recorders with all finger holes covered and the end hole manipulated on the knee).

klaus