difference tones - how computer can calculate it?

hello all.

I need something, maybe you can help me...

I try to find some software that can calculate out difference tones
from a chord (@ nontepmered system!).

For istance, we have the C maj chord and we will get:
C-E=C
C-G=C
E-G=C

What I was thinking is following:

1. we know that there is acoustical phenomena called difference tone.
That we can hear when two tones are sounding at the same time. That
was discovered by Tartini (maybe even earlier).
2. If we have tone A4 of 440 Hz and A5 that is 880 Hz, difference tone
would be 440 hz, well, the same bottom (ground) note.
3. If we have note C4 and G4 played together in nontempered system
difference tone will be exactly bottom note C but (C3), if we play
perfect fourth (C4-F4) the difference tone will be F2 (upper note!).

4. If we play a chord, we have to calculate all the distances between
all notes to get their rellevant difference tones.

5. The program would not display tones and chords more than in one, or
two octaves. It is just practical, since the minor second (C4-Db4)
gives very low note (sub-contra), and it can be transposed to the
octave no 4 (treble cleff) for easy reading.

Well, if we have chord C-Eb-G (minor) it will calculate distances:

C Eb = Ab

C G = C

Eb G = Eb

and we have got Ab, C and Eb as difference tones, and will continue:

Ab C = Ab

Ab Eb = Ab

C Eb = Ab

That is the end: difference tones of C-maj are Ab-maj, and difference
of that chord is just tone Ab.

Since the relations between notes can be given in hertz, I consider
that it can be possible to give instruction to a program as MIDI or as
some numbers, not as wave. Actually, point is to get results as MIDI,
or at least as tone names.
For instance, all octave intervals give the ground note as difference,
as well as major second, major third, perfect fifth and perfect
octave, and so on.

That program I need would be able to calculate even more complex
(four, five, six tone) chords (let us say: C#,D,Eb,Fb,G,Ab) and get
difference tones and than calculate again that difference tones again
and again up to the end (the end can be: one tone, one chord, several
repeating chords, 12-chord). One can do it manually, but I have to
research some harmonic structures of hundreds of chords and it can
take a lot of time (and mistakes too).
Do you know if there is anything like that, some software or plugin?
I use PC (win and linux), but at school we have Mac. I am not good in
computing, my skills are almost at zero.

I study composition.
Thank you very much in advance,

My best regards,
Djuro

Hi Djoro !
I have done this with quite a few simple just chords and there inversions. in fact all the tetrads within the 1-3-5-7-9-11 matrix.
I tried doing these with hexanies which are 6 tone structures and these took days, yet i learned allot about what is happening.
while it is good to have answers , it often is good to know exactly what is going on, at least to start.
one of the problems with ETs on general is how out of tune the difference tone is, on others it can be quite surprising and gives us one more clue why some sound better than other factors tells us.

Message: 7 Date: Mon, 07 Nov 2005 21:59:45 -0000
From: "djuro_z" <djuro_z@yahoo.com>
Subject: difference tones - how computer can calculate it?

hello all.

I need something, maybe you can help me...

I try to find some software that can calculate out difference tones
from a chord (@ nontepmered system!). For istance, we have the C maj chord and we will get:
C-E=C
C-G=C
E-G=C

What I was thinking is following:

1. we know that there is acoustical phenomena called difference tone.
That we can hear when two tones are sounding at the same time. That
was discovered by Tartini (maybe even earlier).
2. If we have tone A4 of 440 Hz and A5 that is 880 Hz, difference tone
would be 440 hz, well, the same bottom (ground) note.
3. If we have note C4 and G4 played together in nontempered system
difference tone will be exactly bottom note C but (C3), if we play
perfect fourth (C4-F4) the difference tone will be F2 (upper note!).

4. If we play a chord, we have to calculate all the distances between
all notes to get their rellevant difference tones.

5. The program would not display tones and chords more than in one, or
two octaves. It is just practical, since the minor second (C4-Db4)
gives very low note (sub-contra), and it can be transposed to the
octave no 4 (treble cleff) for easy reading. Well, if we have chord C-Eb-G (minor) it will calculate distances:

C Eb = Ab

C G = C

Eb G = Eb

and we have got Ab, C and Eb as difference tones, and will continue:

Ab C = Ab

Ab Eb = Ab

C Eb = Ab

That is the end: difference tones of C-maj are Ab-maj, and difference
of that chord is just tone Ab.

Since the relations between notes can be given in hertz, I consider
that it can be possible to give instruction to a program as MIDI or as
some numbers, not as wave. Actually, point is to get results as MIDI,
or at least as tone names. For instance, all octave intervals give the ground note as difference,
as well as major second, major third, perfect fifth and perfect
octave, and so on.

That program I need would be able to calculate even more complex
(four, five, six tone) chords (let us say: C#,D,Eb,Fb,G,Ab) and get
difference tones and than calculate again that difference tones again
and again up to the end (the end can be: one tone, one chord, several
repeating chords, 12-chord). One can do it manually, but I have to
research some harmonic structures of hundreds of chords and it can
take a lot of time (and mistakes too).
Do you know if there is anything like that, some software or plugin?
I use PC (win and linux), but at school we have Mac. I am not good in
computing, my skills are almost at zero. I study composition.
Thank you very much in advance,

My best regards,
Djuro

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, "djuro_z" <djuro_z@y...> wrote:
>
> hello all.

Hi!

> I need something, maybe you can help me...

You got it.

> I try to find some software that can calculate out difference tones
> from a chord (@ nontepmered system!).

So you're using JI? Perhaps you should give the notes in a format
other than simple note names?

> For istance, we have the C maj chord and we will get:
> C-E=C
> C-G=C
> E-G=C

This is true for first-order difference tones but it's important to
specify the octave.

> What I was thinking is following:
>
> 1. we know that there is acoustical phenomena called difference
tone.
> That we can hear when two tones are sounding at the same time. That
> was discovered by Tartini (maybe even earlier).

We now know that Tartini was actually hearning second-order
difference tones . . .

> 2. If we have tone A4 of 440 Hz and A5 that is 880 Hz, difference
tone
> would be 440 hz, well, the same bottom (ground) note.
> 3. If we have note C4 and G4 played together in nontempered system
> difference tone will be exactly bottom note C but (C3), if we play
> perfect fourth (C4-F4) the difference tone will be F2 (upper note!).
>
> 4. If we play a chord, we have to calculate all the distances
between
> all notes to get their rellevant difference tones.
>
> 5. The program would not display tones and chords more than in one,
or
> two octaves. It is just practical, since the minor second (C4-Db4)
> gives very low note (sub-contra), and it can be transposed to the
> octave no 4 (treble cleff) for easy reading.

Yes, but is that transposition a musically fair representation of
what is heard?

> Well, if we have chord C-Eb-G (minor) it will calculate distances:
>
> C Eb = Ab
>
> C G = C
>
> Eb G = Eb
>
> and we have got Ab, C and Eb as difference tones, and will continue:
>
> Ab C = Ab
>
> Ab Eb = Ab
>
> C Eb = Ab
>
> That is the end: difference tones of C-maj are Ab-maj, and
difference
> of that chord is just tone Ab.

Yes; this can be easily understood in that the fundamental of the
minor triad C Eb G or 10:12:15 is 1 or Ab. You can't get any smaller
difference than 1 from a set of integers no matter how many
differences you take!

> Since the relations between notes can be given in hertz, I consider
> that it can be possible to give instruction to a program as MIDI or
as
> some numbers, not as wave. Actually, point is to get results as
MIDI,
> or at least as tone names.
> For instance, all octave intervals give the ground note as
difference,
> as well as major second, major third, perfect fifth and perfect
> octave, and so on.
>
> That program I need would be able to calculate even more complex
> (four, five, six tone) chords (let us say: C#,D,Eb,Fb,G,Ab) and get
> difference tones and than calculate again that difference tones
again
> and again up to the end (the end can be: one tone, one chord,
several
> repeating chords, 12-chord). One can do it manually, but I have to
> research some harmonic structures of hundreds of chords and it can
> take a lot of time (and mistakes too).

Hmm . . . I'm not sure what the difficulty is. But since you say
you're only dealing with nontempered, JI chords, let me ask you: how
do you tune the chord C#,D,Eb,Fb,G,Ab?

> Do you know if there is anything like that, some software or plugin?
> I use PC (win and linux), but at school we have Mac. I am not good
in
> computing, my skills are almost at zero.

I'd be happy to write a program for you. Let me know what you'd like
it to accomplish.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "djuro_z" <djuro_z@y...> wrote:
> >
> > hello all.
>
> Hi!
>
> > I need something, maybe you can help me...
>
> You got it.
>
> > I try to find some software that can calculate out difference tones
> > from a chord (@ nontepmered system!).
>
> So you're using JI? Perhaps you should give the notes in a format
> other than simple note names?

IT CAN BE LIKE THAT. BUT I DON'T WANT TO MAKE IT COMPLICATED.

> > For istance, we have the C maj chord and we will get:
> > C-E=C
> > C-G=C
> > E-G=C
>
> This is true for first-order difference tones but it's important to
> specify the octave.
>
> > What I was thinking is following:
> >
> > 1. we know that there is acoustical phenomena called difference
> tone.
> > That we can hear when two tones are sounding at the same time. That
> > was discovered by Tartini (maybe even earlier).
>
> We now know that Tartini was actually hearning second-order
> difference tones . . .
>
> > 2. If we have tone A4 of 440 Hz and A5 that is 880 Hz, difference
> tone
> > would be 440 hz, well, the same bottom (ground) note.
> > 3. If we have note C4 and G4 played together in nontempered system
> > difference tone will be exactly bottom note C but (C3), if we play
> > perfect fourth (C4-F4) the difference tone will be F2 (upper note!).
> >
> > 4. If we play a chord, we have to calculate all the distances
> between
> > all notes to get their rellevant difference tones.
> >
> > 5. The program would not display tones and chords more than in one,
> or
> > two octaves. It is just practical, since the minor second (C4-Db4)
> > gives very low note (sub-contra), and it can be transposed to the
> > octave no 4 (treble cleff) for easy reading.
>
> Yes, but is that transposition a musically fair representation of
> what is heard?

AT THE MOMENT, I USE IT AS A COMPOSITION PROCESS, NOT AS AN ACOUSTIC
PHENOMEN. ANYWAY, I FIND IT VERY INTERESTING IN A WAY I DO FOR MY
WORK. I CANNOT EXPLAIN HOW INTERESTING FOR ME IS THAT MAJOR CHORD HAS
ITS BOTTOM NOTE AS DIFFERENCE NOTE 8THAT IS OF COURSE WELL-KNOWN).
EVEN THAT TONE(S) IS VERY DEEP I CONSIDER IT AS VERY IMPORTANT EVEN IF
I TRANSPOSE IT TWO OR MORE OCTAVES UP.
THE COMPOSING PROCESS DOESN'T SAY IF I AM GOING TO USE THAT NOTE IN
PRECISE OCTAVE.

>
> > Well, if we have chord C-Eb-G (minor) it will calculate distances:
> > C Eb = Ab
> > C G = C
> > Eb G = Eb
> > and we have got Ab, C and Eb as difference tones, and will continue:
> > Ab C = Ab
> > Ab Eb = Ab
> > C Eb = Ab
> > That is the end: difference tones of C-maj are Ab-maj, and
> difference
> > of that chord is just tone Ab.
>
> Yes; this can be easily understood in that the fundamental of the
> minor triad C Eb G or 10:12:15 is 1 or Ab. You can't get any smaller
> difference than 1 from a set of integers no matter how many
> differences you take!

NO, I DON'T SPEAK ABOUT FUNDAMENTAL TONE. I AM INTERESTED IN
DIFFERENCE TONE. DIFFERENCE TONE AND FUNDAMENTAL SOMETIMES CAN BE THE
SAME (INTERVALS UP TO FIFTH). TRY TO FIND DIFFERENCE TONE OF MINOR
SIXTH OR SEVENTH. IN THAT CASE DIFFERENCE TONE AND FUNDAMENTAL ARE
DIFFERENT.

> > Since the relations between notes can be given in hertz, I consider
> > that it can be possible to give instruction to a program as MIDI or
> as
> > some numbers, not as wave. Actually, point is to get results as
> MIDI,
> > or at least as tone names.
> > For instance, all octave intervals give the ground note as
> difference,
> > as well as major second, major third, perfect fifth and perfect
> > octave, and so on.
> >
> > That program I need would be able to calculate even more complex
> > (four, five, six tone) chords (let us say: C#,D,Eb,Fb,G,Ab) and get
> > difference tones and than calculate again that difference tones
> again
> > and again up to the end (the end can be: one tone, one chord,
> several
> > repeating chords, 12-chord). One can do it manually, but I have to
> > research some harmonic structures of hundreds of chords and it can
> > take a lot of time (and mistakes too).
>
> Hmm . . . I'm not sure what the difficulty is. But since you say
> you're only dealing with nontempered, JI chords, let me ask you: how
> do you tune the chord C#,D,Eb,Fb,G,Ab?

I TAKE THE INTERVALS "AS THEY ARE" IN THEIR RELLEVANT RATIOS.

>
> > Do you know if there is anything like that, some software or plugin?
> > I use PC (win and linux), but at school we have Mac. I am not good
> in
> > computing, my skills are almost at zero.
>
> I'd be happy to write a program for you. Let me know what you'd like
> it to accomplish.
>
FIRST, LOOK AT THE PICTURE HERE:
www.adagio.se/differencetones.TIF
THERE ARE DIFFERENCE TONES, SIMPLIFIED.

HERE IS "WHAT I NEED":
www.adagio.se/difference-calculation.TIF

HERE IS THE COMPLETE EXPLAINE OF THE PICTURES:
-----------------------------------------------------------------
DIFFERENCE TONES

(Prime gives the bottom tone as difference tone)
Minor second gives the upper note of intervall as difference tone
Major second gives the bottom note of intervall as difference tone
Minor third gives minor sixth over the bottom note of intervall as difference tone
Major third gives the bottom note of intervall as difference tone
Perfect fourth gives the upper note of intervall as difference tone
Diminushed fifth gives minor sixth over the bottom note of intervall as difference tone
Perfect fifth gives the bottom note of intervall as difference tone
Minor sixth gives minor third over the bottom note of intervall as difference tone
Major sixth gives perfect fourth over the bottom note of intervall as difference tone
Minor septime gives minor sixth over the the bottom note of intervall as difference tone
Major septime gives minor seventh over the bottom note of intervall as difference tone
**********

HOW TO DO IT?
I enter notes D Eb F# G A Bb (ex. of some chord):

1 - D Eb F# G A Bb <enter>
2 - D Eb F Gb Abb Bb Cb <enter>
3 - Db D Eb F Gb G Bb Cb <enter>
4 - C# D Eb F F# G A Bb B <enter>
...etc

What happens here?
1 - Notes are entered as pitch not in octaves.

I press ENTER to calculate.

2 - Calculation no 2 are difference tones.
All duplicate tones are anullated.
All tones are put in the most narrow position (distance between D and
Cb is major sixth!). There is also one distance of major sixth Bb Abb,
in other position but root D is closest to the root of the chord No1 D
(the same note).

I press ENTER to calculate.

3 - These notes are difference tones of the chord 2. This is the
second level of the chord no 1. Everything is done as with
manipulation between chord 1 and 2.

I press ENTER to calculate.

4 - These notes are difference tones of the chord 3. This is the third
level of the chord no 1. Everything is done as with manipulation
between chord 1 and 2.

I press ENTER to calculate.

I get more and more tones. The end is (probably) in 12 tone chord.
************

MORE INFO:

all notes have got numbers in the program:
C-1 C#-2 D-3 D#-4 etc... untill B-24 (octave + major seventh over the
C-1, that would be enough space to "play" with chords, - maybe even
more??)
If the distance between two numbers are 3 the difference will be "8th
note from bottom note":
if we have C and Eb
we will get numbers 1 and 4
and the difference note is 8 - Ab (or G#)

or if we have notes no 2 and 3 difference is 3 (program says: if
distance is 1 take number that is top: in our case 3!)
or if distance between notes is 2 take bottom note as difference
(C#-13 and D#-14 = C#-13)

During calculations there will be a lot of multiplies. Tell to the
program to delete multiplies even they appear in other octaves
for instance we have got from a chord several notes that are the same:
C-1, C-1, C-1, C-13, C-13,
program says: All letters C will be annulated, only one will survive.

well, we get C-1

Because during the calculations we have got many notes, program has to
organise them, to gather them.
Program thinks: C-1 can jump to C-13.
Or: all letters can jump over or under to the same letter to force the
chord (notes in the chord) to be in closest position.
and program says:
gather all tones that is distance between top and bottom at least 1,
if not possible
gather all tones that is distance between top and bottom at least 2,
if not possible
gather all tones that is distance between top and bottom at least 3,
if not possible
gather all tones that is distance between top and bottom at least 4,
if not possible
gather all tones that is distance between top and bottom at least 5,
....etc
untill program gathers all notes in the closest, most narrow position.

IS THERE POSSIBLE TO MAKE SUCH PROGRAM, OR THERE IS ONE SIMILAR?

ALL THE BEST
DJURO

--- In tuning@yahoogroups.com, "djuro_z" <djuro_z@y...> wrote:
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> >
> > --- In tuning@yahoogroups.com, "djuro_z" <djuro_z@y...> wrote:
> > >
> > > hello all.
> >
> > Hi!
> >
> > > I need something, maybe you can help me...
> >
> > You got it.
> >
> > > I try to find some software that can calculate out difference
tones
> > > from a chord (@ nontepmered system!).
> >
> > So you're using JI? Perhaps you should give the notes in a format
> > other than simple note names?
>
> IT CAN BE LIKE THAT. BUT I DON'T WANT TO MAKE IT COMPLICATED.

Unfortunately, you won't be able to make correct inferences about
difference tones otherwise.

> > > For istance, we have the C maj chord and we will get:
> > > C-E=C
> > > C-G=C
> > > E-G=C
> >
> > This is true for first-order difference tones but it's important
to
> > specify the octave.
> >
> > > What I was thinking is following:
> > >
> > > 1. we know that there is acoustical phenomena called difference
> > tone.
> > > That we can hear when two tones are sounding at the same time.
That
> > > was discovered by Tartini (maybe even earlier).
> >
> > We now know that Tartini was actually hearning second-order
> > difference tones . . .
> >
> > > 2. If we have tone A4 of 440 Hz and A5 that is 880 Hz,
difference
> > tone
> > > would be 440 hz, well, the same bottom (ground) note.
> > > 3. If we have note C4 and G4 played together in nontempered
system
> > > difference tone will be exactly bottom note C but (C3), if we
play
> > > perfect fourth (C4-F4) the difference tone will be F2 (upper
note!).
> > >
> > > 4. If we play a chord, we have to calculate all the distances
> > between
> > > all notes to get their rellevant difference tones.
> > >
> > > 5. The program would not display tones and chords more than in
one,
> > or
> > > two octaves. It is just practical, since the minor second (C4-
Db4)
> > > gives very low note (sub-contra), and it can be transposed to
the
> > > octave no 4 (treble cleff) for easy reading.
> >
> > Yes, but is that transposition a musically fair representation of
> > what is heard?
>
> AT THE MOMENT, I USE IT AS A COMPOSITION PROCESS, NOT AS AN ACOUSTIC
> PHENOMEN. ANYWAY, I FIND IT VERY INTERESTING IN A WAY I DO FOR MY
> WORK. I CANNOT EXPLAIN HOW INTERESTING FOR ME IS THAT MAJOR CHORD
HAS
> ITS BOTTOM NOTE AS DIFFERENCE NOTE 8THAT IS OF COURSE WELL-KNOWN).
> EVEN THAT TONE(S) IS VERY DEEP I CONSIDER IT AS VERY IMPORTANT
EVEN IF
> I TRANSPOSE IT TWO OR MORE OCTAVES UP.
> THE COMPOSING PROCESS DOESN'T SAY IF I AM GOING TO USE THAT NOTE IN
> PRECISE OCTAVE.
>
>
> >
> > > Well, if we have chord C-Eb-G (minor) it will calculate
distances:
> > > C Eb = Ab
> > > C G = C
> > > Eb G = Eb
> > > and we have got Ab, C and Eb as difference tones, and will
continue:
> > > Ab C = Ab
> > > Ab Eb = Ab
> > > C Eb = Ab
> > > That is the end: difference tones of C-maj are Ab-maj, and
> > difference
> > > of that chord is just tone Ab.
> >
> > Yes; this can be easily understood in that the fundamental of the
> > minor triad C Eb G or 10:12:15 is 1 or Ab. You can't get any
smaller
> > difference than 1 from a set of integers no matter how many
> > differences you take!
>
> NO, I DON'T SPEAK ABOUT FUNDAMENTAL TONE. I AM INTERESTED IN
> DIFFERENCE TONE. DIFFERENCE TONE AND FUNDAMENTAL SOMETIMES CAN BE
THE
> SAME (INTERVALS UP TO FIFTH). TRY TO FIND DIFFERENCE TONE OF MINOR
> SIXTH OR SEVENTH. IN THAT CASE DIFFERENCE TONE AND FUNDAMENTAL ARE
> DIFFERENT.

You aren't paying enough attention to what I wrote. I know that
difference tones and fundamental tones are different in general! What
I was saying was that, if you take difference tones of difference
tones and so on ad infinitum, you'll eventually arrive at the
fundamental tone and never anything lower.

> > > Since the relations between notes can be given in hertz, I
consider
> > > that it can be possible to give instruction to a program as
MIDI or
> > as
> > > some numbers, not as wave. Actually, point is to get results as
> > MIDI,
> > > or at least as tone names.
> > > For instance, all octave intervals give the ground note as
> > difference,
> > > as well as major second, major third, perfect fifth and perfect
> > > octave, and so on.
> > >
> > > That program I need would be able to calculate even more complex
> > > (four, five, six tone) chords (let us say: C#,D,Eb,Fb,G,Ab) and
get
> > > difference tones and than calculate again that difference tones
> > again
> > > and again up to the end (the end can be: one tone, one chord,
> > several
> > > repeating chords, 12-chord). One can do it manually, but I have
to
> > > research some harmonic structures of hundreds of chords and it
can
> > > take a lot of time (and mistakes too).
> >
> > Hmm . . . I'm not sure what the difficulty is. But since you say
> > you're only dealing with nontempered, JI chords, let me ask you:
how
> > do you tune the chord C#,D,Eb,Fb,G,Ab?
>
> I TAKE THE INTERVALS "AS THEY ARE" IN THEIR RELLEVANT RATIOS.

Which are?

> > > Do you know if there is anything like that, some software or
plugin?
> > > I use PC (win and linux), but at school we have Mac. I am not
good
> > in
> > > computing, my skills are almost at zero.
> >
> > I'd be happy to write a program for you. Let me know what you'd
like
> > it to accomplish.
> >
> FIRST, LOOK AT THE PICTURE HERE:
> www.adagio.se/differencetones.TIF
> THERE ARE DIFFERENCE TONES, SIMPLIFIED.

It seems simplified for one thing because, whether you realize it or
not, you're allowing the tuning of each note to be "fudged" depending
on context to get the difference tone you want.

> HERE IS "WHAT I NEED":
> www.adagio.se/difference-calculation.TIF

I don't think this is valid because in general you'll be assuming
ratio-interpretations for the intervals which are not even consistent
with the possibility of them coexisting in the chord.

IMO, you'd be far better served actually specifying the notes you're
using if you're using JI. If not, you should calculate the actual
difference tones generated in the tuning you are using.

Hi Djuro.

> > > > I try to find some software that can calculate out difference
> tones
> > > > from a chord (@ nontepmered system!).
> > >
> > > So you're using JI? Perhaps you should give the notes in a format
> > > other than simple note names?
> >
> > IT CAN BE LIKE THAT. BUT I DON'T WANT TO MAKE IT COMPLICATED.
>
> Unfortunately, you won't be able to make correct inferences about
> difference tones otherwise.

At this point, I can nothing but agree with Paul that you seem to want the
program to distinguish many things somehow "automatically" (without actually
telling it what these things are). First of all, as far as I had been
reading this till your last message, I thought you were supposing 5-limit
chords in every case. But now, when you have said that the diff. tone of
C-F# should come out as Ab, I'm realizing you're also taking 7-limit
intervals into account. Now I'm asking: How do you explain this to me that
all the other intervals (except the diminished fourth) are 5-limit ones? And
then, if you say that the diff. tone of C-Bb should come out as Ab, I see
you mean a minor seventh of 9/5 (and not 16/9 as I'd expect). Again, how do
you explain this to the computer? If this is the case, why have you chosen a
major second of 9/8 and not 10/9 then?
Please be more specific about the commatic or kleismatic variants of the
chosen intervals if you wish to work with them.

Petr