The Cawapu Comma

The de facto pitch bend range is +-200 cents, leading to a de facto
pitch resolution of a cawapu, which is 1/4096 of a cent (as opposed to
the midipu, which does not seem important in practice but which would
give you 1/4 cawapu resolution in case you thought you needed it.)

This lead me to wonder what tuning properties, in theory, the 12 *
4096 = 49152 et would have. It turns out to have a 7-limit comma which
stands out from the pack: (4375/4374)/(250047/250000) =
(4375/4374)^2/(2401/2400) = 78125000/78121827. I think a good name for
this comma would the the cawapu comma.

There's another noteworthy comma, this one actually less than a cawapu
in size, for the 11-limit version of the 49152 et. This one is
(2401/2400)^3/(5632/5625) = 7^24 / 2^24 3 5^2 11.

Gene Ward Smith wrote:

<<The de facto pitch bend range is +-200 cents, leading to a de facto
pitch resolution of a cawapu, which is 1/4096 of a cent (as opposed to
the midipu, which does not seem important in practice but which would
give you 1/4 cawapu resolution in case you thought you needed it.)
>>

Be advised that Joe Monzo's definitions of 'midipu' and 'cawapu' are incorrect. I have already informed him of this. Midipu results in 12 x 2^13 = 98,304 units per octave, not 196,608. Cawapu results in 12 x 2^11 = 24576, not 49152. The results in each case are half of the values Joe gives, because pitch bend works in both directions from a given midi note (+/-) which means 'semitone' bends overlap at the quartertone. Half of the units need to be thrown out of the 'EDO' result based on the semitone pitch bend subdivision.

Aaron

--- In tuning-math@yahoogroups.com, pitchcolor@a... wrote:
> Gene Ward Smith wrote:
>
> <<The de facto pitch bend range is +-200 cents, leading to a de
facto
> pitch resolution of a cawapu, which is 1/4096 of a cent (as opposed
to
> the midipu, which does not seem important in practice but which
would
> give you 1/4 cawapu resolution in case you thought you needed it.)
> >>
>
> Be advised that Joe Monzo's definitions of 'midipu' and 'cawapu'
are incorrect. I have already informed him of this. Midipu results in
12 x 2^13 = 98,304 units per octave, not 196,608. Cawapu results in
12 x 2^11 = 24576, not 49152.

The de facto standard, as I mentioned, is 49152 pitches per octave,
so we need a word for it. I presume Cakewalk follows this standard;
do you have evidence it does not?

hi Aaron and Gene,

> From: "Gene Ward Smith" <gwsmith@svpal.org>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, June 29, 2003 2:55 PM
> Subject: [tuning-math] Re: The Cawapu Comma
>
>
> --- In tuning-math@yahoogroups.com, pitchcolor@a... wrote:
> > Gene Ward Smith wrote:
> >
> > > The de facto pitch bend range is +-200 cents, leading
> > > to a de facto pitch resolution of a cawapu, which is
> > > 1/4096 of a cent (as opposed to the midipu, which does
> > > not seem important in practice but which would
> > > give you 1/4 cawapu resolution in case you thought
> > > you needed it.)
> >
> >
> > Be advised that Joe Monzo's definitions of 'midipu'
> > and 'cawapu' are incorrect. I have already informed
> > him of this. Midipu results in 12 x 2^13 = 98,304 units
> > per octave, not 196,608. Cawapu results in 12 x 2^11 =
> > 24576, not 49152.
>
> The de facto standard, as I mentioned, is 49152 pitches
> per octave, so we need a word for it. I presume Cakewalk
> follows this standard; do you have evidence it does not?

lately there's been quite a bit of discussion of this
subject on both this list and the main tuning list.
i've already mentioned that Aaron informed me of the
error and that i haven't yet fixed my webpages or
definitions. the main reason for the delay is that
i simply haven't had any spare time to check up on it.

also, when i first created my "MIDI tuning spec" page
i was informed of something else there that was not
correct, having to do with two different MIDI tuning
standards, one for sys-ex and the other for pitch-bend
commands. but i didn't fully understand that comment
and just ignored it, hoping to get back to it later.

can some of you do some digging and establish once and
for all what it is that should be correctly stated in
my webpages? i'd appreciate that a lot. thanks.

-monz

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

> can some of you do some digging and establish once and
> for all what it is that should be correctly stated in
> my webpages? i'd appreciate that a lot. thanks.

I imagine Manuel knows the answer without digging. My concern is
really not with the official standard, but the default standard.

hi Gene,

> From: "Gene Ward Smith" <gwsmith@svpal.org>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, June 29, 2003 7:22 PM
> Subject: [tuning-math] Re: The Cawapu Comma
>
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > can some of you do some digging and establish once and
> > for all what it is that should be correctly stated in
> > my webpages? i'd appreciate that a lot. thanks.
>
> I imagine Manuel knows the answer without digging. My concern is
> really not with the official standard, but the default standard.

understood ... but as the guy to whose work everyone
refers in trying to understand the terminology, i'd
better make it *all* correct!

(in fact, i think it was Manuel who wrote to me about
the "two different specs" to which i referred.)

-monz

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

> understood ... but as the guy to whose work everyone
> refers in trying to understand the terminology, i'd
> better make it *all* correct!

I've just been looking at a web page Manuel gave on MMM:

http://www.midi.org/about-midi/tuning.shtml

According to this, the "de facto standard" I've been talking about is
*not* the same as MTS. MTS expresses a pitch by means of a three digit
number in base 128, leaving off the last number, which means a number
from 0 to 2097150. This number gives the pitch in terms of the
196608-et, where 0 is a pitch of 8.1758 Hz, five octaves below middle
C. Middle C itself of course is 5*196608 = 983040, or 3C 00 00 when
written in base 128 with hex digits as the computers prefer.

A midipu therefore should be 2^(1/196608), which I think is what you
already have.

I think Joe's numbers are correct, although the standard pitch
bend range isn't something special to Cakewalk.

The cawapu value range is wrong though, should be -8192 .. 8191.

Also the Turkish cent is 10 times too big, should be 10600 per
octave.

Manuel

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:

> I think Joe's numbers are correct, although the standard pitch
> bend range isn't something special to Cakewalk.

I've noticed. But it seems MTS is another kettle of fish, and I think
it might be nice to have an entry explaining it. You could start with
the post I just made which explains it in terms (equal temperaments
and base 128 numbers) which make more sense to me, at least, than
what the midi spec has. Add in the crap in the beginning (including
that what looks like a bunch of numbers near the start when you run
it through mf2t is actually ascii characters which say something or
other, though it doesn't much matter what.) Once you know what the
seeming gibberish at the beginning is all about and where the actual
pitch information starts, it's not too bad.

The figures in each case are:

'midipu' = 12 x 2^13 = 98304

'cawapu' = 12 x 2^11 = 24576

I'm not sure why there is any confusion about this. The values above are correct. I do not use cakewalk, but the same logic holds for all pitch bend messages. The 'middle' value is a midi note number with no bend. The bend range is thus a semitone below for half the values and a halfstep above for the other half. For 7 bit pitch bend we have 0-63 below a midi note, 64 as the note with no bend, and 65-127 for the halfstep above. That is a range of a whole step, not a halfstep. This is the smallest pitch bend range available. In 14 bit precision, each of the 7 bit values has 128 values of its own, where 0-127 is a range from lowest to highest within the 7 bit step. Any way you look at it, we are talking about a wholestep range which is a halfstep above and below a given midi note. So when you try to make an equal division of the octave out of it, you have to throw out half of the values. If this still doesn't make sense, I will write up a better explanation. I know for a fact that what I am saying is correct, because I use pitchbend in the microtonal instruments I build:

http://www.members.aol.com/pitchcolor/music.html

Aaron

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:

> Also the Turkish cent is 10 times too big, should be 10600 per
> octave.

huh? 53*20=1060.

Paul wrote:
>huh? 53*20=1060.

You corrected me once before on this and then I thought I was
wrong, but not. It's 53*200=10600. I've seen it in Karadeniz.

Manuel

--- In tuning-math@yahoogroups.com, pitchcolor@a... wrote:

I know for a fact that what I am saying is correct, because I use
pitchbend in the microtonal instruments I build:
>
> http://www.members.aol.com/pitchcolor/music.html

What happens if you play a standard midi file which uses pitch bends
on the the instruments you build?

<<
What happens if you play a standard midi file which uses pitch bends
on the the instruments you build?
>>

I'm only building controllers, not tone modules. But your question suggests that I'm using a nonstandard approach to pitchbend, which is not the case. My controllers work with all standard MIDI gear having pitch bend capabilities. I'll defer to a cakewalk user on 'cawapu', but the rest of what I posted is correct.
Aaron

--- In tuning-math@yahoogroups.com, pitchcolor@a... wrote:
> <<
> What happens if you play a standard midi file which uses pitch
bends
> on the the instruments you build?
> >>
>
> I'm only building controllers, not tone modules. But your question
suggests that I'm using a nonstandard approach to pitchbend, which is
not the case.

There's a lot of evidence in what you say that you are. I suggest
using a fairly extreme test case--something with pitch bends giving
more or less than 12 notes to the octave--and seeing if the result
makes any sense.

>My controllers work with all standard MIDI gear having pitch bend
>capabilities.

I'm talking about playing a midi file. Do you do that?

hi Gene and Aaron,

> From: "Gene Ward Smith" <gwsmith@svpal.org>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, June 30, 2003 6:44 PM
> Subject: [tuning-math] Re: The Cawapu Comma
>
>
> --- In tuning-math@yahoogroups.com, pitchcolor@a... wrote:
> >
> > > What happens if you play a standard midi file which
> > > uses pitch bends on the the instruments you build?
> > >
> >
> > I'm only building controllers, not tone modules. But
> > your question suggests that I'm using a nonstandard
> > approach to pitchbend, which is not the case.
>
> There's a lot of evidence in what you say that you are. I suggest
> using a fairly extreme test case--something with pitch bends giving
> more or less than 12 notes to the octave--and seeing if the result
> makes any sense.
>
> > My controllers work with all standard MIDI gear having pitch bend
> > capabilities.
>
> I'm talking about playing a midi file. Do you do that?

no, Gene, Aaron's instruments are MIDI *controllers*.
they *create* MIDI-files, not play them.

-monz

hi Manuel,

> From: "Manuel Op de Coul" <manuel.op.de.coul@eon-benelux.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, June 30, 2003 2:57 AM
> Subject: Re: [tuning-math] Re: The Cawapu Comma
>
>
> I think Joe's numbers are correct, although the
> standard pitch bend range isn't something special
> to Cakewalk.

thanks for pointing that out. i said as much
when i originally uploaded the Tuning Dictionary
"cawapu" entry, and solicited suggestions for
other names, but no-one responded.

i really am not fond of either "cawapu" or
"midipu" as tuning terms, but i felt that they
were tuning units of measurement which had
enough importance to merit names.

so, suggestions for other names are still welcome.
use of my two terms hasn't become widespread
enough to be a concern.

> The cawapu value range is wrong though, should be
> -8192 .. 8191.
>
> Also the Turkish cent is 10 times too big, should
> be 10600 per octave.

hmmm ... can you give the full reference to Karadeniz?

so then is my "T�rk cent" Dictionary webpage
completely wrong?

-monz

Joe wrote:
>hmmm ... can you give the full reference to Karadeniz?

Karadeniz, M. Ekrem. _Türk Mûsikîsinin Nazariye ve Esaslari_ (Theory and
principles of Turkish music). Türkiye IS Bankasi Kültür Yayinlari,
Publ. no. 237/238, Ankara, 1965, 1983.

>so then is my "Türk cent" Dictionary webpage
>completely wrong?

"Türk sent". Not if you multiply/divide the values by 10.
Karadeniz invented it, but it's unclear why he wanted such a
fine division, maybe only to surpass Ellis's cent.

Manuel

hi Manuel,

> From: "Manuel Op de Coul" <manuel.op.de.coul@eon-benelux.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Tuesday, July 01, 2003 9:04 AM
> Subject: Re: [tuning-math] Re: The Cawapu Comma
>

> Joe wrote:
> > hmmm ... can you give the full reference to Karadeniz?
>
> Karadeniz, M. Ekrem.
> _T�rk M�sik�sinin Nazariye ve Esaslari_
> (Theory and principles of Turkish music).
> T�rkiye IS Bankasi K�lt�r Yayinlari,
> Publ. no. 237/238, Ankara, 1965, 1983.
>
> > so then is my "T�rk cent" Dictionary webpage
> > completely wrong?
>
> "T�rk sent". Not if you multiply/divide the values by 10.
> Karadeniz invented it, but it's unclear why he wanted such a
> fine division, maybe only to surpass Ellis's cent.

OK, thanks for that. but i'm still not clear:
does the division of 1060 per 8ve have any
historical relevance for Turkish music, or for
any other for that matter?

-monz

hi Aaron,

> From: <pitchcolor@aol.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Monday, June 30, 2003 9:34 AM
> Subject: [tuning-math] Re: The Cawapu Comma
>
>
> The figures in each case are:
>
> 'midipu' = 12 x 2^13 = 98304
>
> 'cawapu' = 12 x 2^11 = 24576
>
> I'm not sure why there is any confusion about this.
> The values above are correct. I do not use cakewalk,
> but the same logic holds for all pitch bend messages.
> The 'middle' value is a midi note number with no bend.
> The bend range is thus a semitone below for half the
> values and a halfstep above for the other half. For
> 7 bit pitch bend we have 0-63 below a midi note,
> 64 as the note with no bend, and 65-127 for the
> halfstep above. That is a range of a whole step,
> not a halfstep. This is the smallest pitch bend range
> available. In 14 bit precision, each of the 7 bit
> values has 128 values of its own, where 0-127 is a
> range from lowest to highest within the 7 bit step.
> Any way you look at it, we are talking about a wholestep
> range which is a halfstep above and below a given midi
> note. So when you try to make an equal division of the
> octave out of it, you have to throw out half of the
> values. If this still doesn't make sense, I will write
> up a better explanation. I know for a fact that what
> I am saying is correct, because I use pitchbend in the
> microtonal instruments I build:
>
> http://www.members.aol.com/pitchcolor/music.html

i appreciate your pursuing this topic ... but i'm still
having trouble seeing it.

i'm sorry that i have to speak in terms of Cakewalk,
since you don't use it, but i do and i know this works:

each 12edo semitone is divided in 4096 cawapus.
thus, each cent is divided into exactly 40.96 cawapus.
12 * 4096 = 49152.

i know that the standard pitch-bend range is a
whole-step, one half-step on either side of the
MIDI-note. but Cakewalk uses 8192 as the size of
a whole-step. 6 * 8192 = 49152.

please keep this dialog going, because i really want
to clear this up.

-monz

>but i'm still not clear:
>does the division of 1060 per 8ve have any
>historical relevance for Turkish music, or for
>any other for that matter?

No, but 106-tET does. Karadeniz lists some makams
as modes of 106 in his book. I've put them in the
mode list.

Manuel

Hi Joe,

You wrote:

<<i appreciate your pursuing this topic ... but i'm still
having trouble seeing it.
i'm sorry that i have to speak in terms of Cakewalk,
since you don't use it, but i do and i know this works:
each 12edo semitone is divided in 4096 cawapus.
thus, each cent is divided into exactly 40.96 cawapus.
12 * 4096 = 49152.
i know that the standard pitch-bend range is a
whole-step, one half-step on either side of the
MIDI-note. �but Cakewalk uses 8192 as the size of
a whole-step. �6 * 8192 = 49152.
please keep this dialog going, because i really want
to clear this up.>>

From Manuel's input and what you wrote there, it looks like you've got it figured out. Looks like the cawapu ET (or ED2) should be what you thought it was all along:

6 * 8192 = 49152

The only concern I would raise is that since cakewalk uses both data bytes then it is likely that the second data byte is ignored by the midi module. I defer to Manuel if there is something I am missing, but it looks like if the second data byte is ignored, then 7 bit cawapu should be the same as standard pitch bend:

6 x 2^7 = 768 ET

According to your webpage giving specifics about the data format for cakewalk, this would be correct. I raise this point because if the second byte is simply ignored, then there are bend messages being sent out which are incorrect in 7-bit precision. They will be incorrect because they need to be rounded from 12-bits to 7-bits. In other words, the MSB needs to be rounded up or down according to the value of the LSB. Truncated 7-bit values will sometimes be wrong.

Aaron

hi Aaron,

> From: <pitchcolor@aol.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, July 02, 2003 11:05 AM
> Subject: [tuning-math] Re: The Cawapu Comma
>
>
> Hi Joe,
>
> You wrote:
>
> <<i appreciate your pursuing this topic ... but i'm still
> having trouble seeing it.
> i'm sorry that i have to speak in terms of Cakewalk,
> since you don't use it, but i do and i know this works:
> each 12edo semitone is divided in 4096 cawapus.
> thus, each cent is divided into exactly 40.96 cawapus.
> 12 * 4096 = 49152.
> i know that the standard pitch-bend range is a
> whole-step, one half-step on either side of the
> MIDI-note. but Cakewalk uses 8192 as the size of
> a whole-step. 6 * 8192 = 49152.
> please keep this dialog going, because i really want
> to clear this up.>>
>
>
> From Manuel's input and what you wrote there,
> it looks like you've got it figured out. Looks like
> the cawapu ET (or ED2) should be what you thought it
> was all along:
>
> 6 * 8192 = 49152
>
> The only concern I would raise is that since cakewalk
> uses both data bytes then it is likely that the second
> data byte is ignored by the midi module. I defer to Manuel
> if there is something I am missing, but it looks like if
> the second data byte is ignored, then 7 bit cawapu should
> be the same as standard pitch bend:
>
> 6 x 2^7 = 768 ET
>
> According to your webpage giving specifics about the
> data format for cakewalk, this would be correct. I
> raise this point because if the second byte is simply
> ignored, then there are bend messages being sent out
> which are incorrect in 7-bit precision. They will be
> incorrect because they need to be rounded from 12-bits
> to 7-bits. In other words, the MSB needs to be rounded
> up or down according to the value of the LSB. Truncated
> 7-bit values will sometimes be wrong.

hmmm ... well, i use Cakewalk to play MIDI-files thru
my computer's soundcard, and the "cawapu" pitch-bend
data i use gives me the correct tuning.

i have no idea what happens when you play use Cakewalk
to play MIDI-files thru another instrument.

i coined the term "cawapu" to have a handy word to
represent the 4096-pitch-bend-units-per-semitone
unit of interval measurement, which is one degree
of 49152edo.

therefore, it would be incorrect to use the term
"cawapu" when referring to an instrument that only
has 7-bit precision. as you point out, even if the
MIDI-file uses 49152edo as its tuning basis, the
result is truncated to 768edo resolution.

in view of the fact that so many instruments use
7-bit precision, i suppose we really need another
new term to represent one degree of 768edo.
any suggestions?

PS -- "midipu" covers the full 14-bit resolution
that is possible in MIDI, but "cawapu" seems to
be the resolution which is much more frequently
implemented in practice.

-monz

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

> PS -- "midipu" covers the full 14-bit resolution
> that is possible in MIDI, but "cawapu" seems to
> be the resolution which is much more frequently
> implemented in practice.

In practice, pitch bends use cawapus; on the other hand, MTS uses
midipus. Both terms seem useful.