Yahoo Tuning Groups Ultimate Backup tuning [tuning] theory on possible JI construction of harmony, melody and music

I found it more apropriate to turn this into a seperate thread

2009/2/19 hstraub64 <straub@...>

> > Amongst other things I have discovered / created a huge theory /
> > mathematical framework which is so beautifull and perfect I still
> > can't beleive it.
> > It is the most beautifull and perfect thing I've ever seen in my
> > life. And it looks like music. Can't fully explain it yet, too much
> > research to still do, I will post it later.
>
> Please do post about this. But be prepared for critisism :-)

Ok I will post a small part of it :)
I'm not afraid of critisism, if I can't agree with the critisism I have no
problem with it, if I can agree with the ciritisism I will learn from it.
Similar to how critisism on my mess of Lasso made me realise something about
retuning sounding notes in JI in a score written in 12tet (which I'll
revisit later)

Ok so here's part of the thing I discovered.

I think that music is allways about harmony (the tonal part) and can't
escape from harmony unless it's noise.
Not even monophonic melody can escape harmony since every note is judged
against its preceding note even though they don't sound at thesame time.

Now harmony is infinite in it's possibilities of construction.
But to try to understand harmony I tried to rate harmony from the most
simple contruction to more complex contructions.
For this I looked at the harmonic overtone series as it is the most perfect
harmonic construction possible.
It is also rateable from simple to more complex. seen from 1/1 you get 1/1
2/1 simplest, 1/1 2/1 3/1 next simplest, 1/1 2/1 3/1 4/1 next simplest, etc.
When looking at the harmonic series I can see different parts of it that
make it perfect.
First the consecutive parts of 2/1 3/2 4/3 etc, and the order from big to
small.
There are more parts and reasons i can see that make it perfect but i'll
keep this mail small.

Now if one takes one part that makes the harmonic overtone series
construction perfect, namely that it is constructed from ever smaller
intervals, then you can construct a different series by reshifting the
order.
This retains the ever smaller interval size perfection and simplest to more
complex ordering that comes with it. It loses the order perfection.
So this new series goes like this: simplest 1/1 2/1, next simplest 1/1 2/1
3/1 and 1/1 3/2 3/1. both having thesame source of 1/1.
first one is 1/1 + 2/1 + 3/2, second one is 1/1 + 3/2 + 2/1.
put one on top of the other and you get 1/1 3/2 2/1 3/1
next simplest harmonic construction is 1/1 2/1 3/1 4/1 (1/1 + 2/1 + 3/2 +
4/3) which also becomes 1/1 3/2 2/1 4/1 (1/1 + 3/2 + 4/3 + 2/1), and 1/1 2/1
8/3 4/1 (1/1 + 2/1 + 4/3 + 3/2), and 1/1 4/3 2/1 4/1 (1/1 + 4/3 + 3/2 +
2/1).
These are all the possible orderings when going till the 4th harmonic.
All on top of eachother they give 1/1 4/3 3/2 2/1 8/3 3/1 4/1
Please note that I'm not bringing everything down into 1 octave.

If you go untill the 5th harmonic you get this:

1/1 2/1 3/1 4/1 5/1 1-2-3-4-5
1/1 3/2 3/1 4/1 5/1 1-3-2-4-5
1/1 3/2 2/1 4/1 5/1 1-3-4-2-5
1/1 3/2 2/1 5/2 5/1 1-3-4-5-2
1/1 2/1 8/3 4/1 5/1 1-2-4-3-5
1/1 2/1 8/3 10/3 5/1 1-2-4-5-3
1/1 4/3 8/3 4/1 5/1 1-4-2-3-5
1/1 4/3 8/3 10/3 5/1 1-4-2-5-3
1/1 4/3 5/3 10/3 5/1 1-4-5-2-3
1/1 4/3 2/1 4/1 5/1 1-4-3-2-5
1/1 4/3 2/1 5/2 5/1 1-4-3-5-2
1/1 4/3 5/3 5/2 5/1 1-4-5-3-2
1/1 2/1 3/1 15/4 5/1 1-2-3-5-4
1/1 2/1 5/2 15/4 5/1 1-2-5-3-4
1/1 5/4 5/2 15/4 5/1 1-5-2-3-4
1/1 3/2 3/1 15/4 5/1 1-3-2-5-4
1/1 3/2 15/8 15/4 5/1 1-3-5-2-4
1/1 3/2 15/8 5/2 5/1 1-3-5-4-2
1/1 5/4 15/8 15/4 5/1 1-5-3-2-4
1/1 5/4 15/8 5/2 5/1 1-5-3-4-2
1/1 2/1 5/2 10/3 5/1 1-2-5-4-3
1/1 5/4 5/2 10/3 5/1 1-5-2-4-3
1/1 5/4 5/3 10/3 5/1 1-5-4-2-3
1/1 5/4 5/3 5/2 5/1 1-5-4-3-2

With all harmonic constructions put on top of eachother you get this: 1/1
5/4 4/3 3/2 5/3 15/8 2/1 5/2 8/3 3/1 10/3 15/4 4/1 5/1

Now this is harmonic structure simplicity in the way i described it.
It does not mean that every one of those harmonies is the most simple or
consonant harmony, since we abandoned the perfect ordering of the harmonic
series.
Of the above only 1/1 2/1 3/1 4/1 5/1 has the perfect ordering. Nonetheless
all the above harmonic structures are simplest and most perfect in their
interval sizes construction.

Btw the number of different base harmonic structures for the above 5th
harmonic limit structure is 1*2*3*4 = 24
If one counts all the number of pitches (non unique) making up those
harmonic structures the number is 1*2*3*4*5 = 120

Now if one takes this structure to the 6th harmonic you get:

1*2*3*4*5 = 120 different harmonic structures.
Comprised of 1*2*3*4*5*6 = 720 non unique notes.

I'm not going to paste this list of 120 harmonic constructions here now for
the readability of this email, I'll paste it in the next message.

When you put all the harmonic structures on top of eachother you get:
1/1 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1 9/4 12/5 5/2 8/3 3/1 16/5 10/3 18/5
15/4 4/1 9/2 24/5 5/1 6/1

When you look at for instance the octave from 2/1 till 4/1 you see:
1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 2/1

Now these are the most simple harmonic structures based on interval sizes.
But without any ordering, infact with every possible ordering.

To work with these one still has take into account ordering.
Now we make a simple ordering division into this list of harmonic
structures.
We devide the list into 2 lists first.

1 list is all the different orderings of interval sizes except the 5/4 then
6/5 order we find in the harmonic series.
So the below list is allways 4:5:6, the rest of the intervals still have all
possible orders.
You get:
1/1 5/4 4/3 3/2 5/3 15/8 2/1 9/4 5/2 8/3 3/1 10/3 15/4 4/1 9/2 5/1 6/1
Seen in 1 octave (or from 2/1) you get:
1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

Now this is the major scale!

When you now do the oposite, namely order it with first 6/5 then 5/4 then
you get:
1/1 6/5 4/3 3/2 8/5 9/5 2/1 9/4 12/5 8/3 3/1 16/5 18/5 4/1 9/2 24/5 6/1
Seen in 1 octave (or from 2/1) you get:
1/1 9/8 6/5 4/3 3/2 8/5 9/5 2/1

This is the minor scale!

And not only did we get the major and minor scale, we got them including
every single on of their most simple interval based harmonic structures.
(we did not get more complex harmonic structures which can also underly
these scales, the list of which is infinite, though we can list the "next
simplest")

So we can see these scales as coming from the most simple harmonic
structures in which you can reverse the ordering of the harmonic overtone
series, but not reverse the order of 5/4 + 6/5 in the scale (or while
playing).
Simplest reversion would be the octave, next simplest the 3rd harmonic, next
simplest 4th harmonic next simplest 5th harmonic, next simplest 6th
harmonic.
The reversion of the 6th harmonic we don't seem to like a lot.
Not only seen from the scales above, but play 1/1 5/4 3/2 then 1/1 6/5 3/2.
It's not pretty.
The reversion of the 5th harmonic is allready a bit strong, play 1/1 4/3 5/3
then 1/1 5/4 5/3.
(please note these are "reversions" or whatever they should be called, not
inversions)

Ok I'm tired of typing for right now, bussy day but I'll add a lot to this
later.
Below I'll paste the diatonic major and minor scales seen from all their
different degrees (which can all function as a tonal center, though none as
good as the 1/1 of the harmonic structure offcourse)
Also below are diatonic like scales based on thesame harmonic structure of
the 6th harmonic but with slightly different ordering rules.
Also soon i'll explain harmonic interval structures of higher harmonics,
this is where it gets even more interesting :)

Base Modes:

mode 1 (major)
1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
9/8 : 10/9 32/27 4/3 40/27 5/3 16/9 2/1
5/4 : 16/15 6/5 4/3 3/2 8/5 9/5 2/1
4/3 : 9/8 5/4 45/32 3/2 27/16 15/8 2/1
3/2 : 10/9 5/4 4/3 3/2 5/3 16/9 2/1
5/3 : 9/8 6/5 27/20 3/2 8/5 9/5 2/1
15/8: 16/15 6/5 4/3 64/45 8/5 16/9 2/1

mode 2 (minor)
1/1 9/8 6/5 4/3 3/2 8/5 9/5 2/1
9/8: 16/15 32/27 4/3 64/45 8/5 16/9 2/1
6/5: 10/9 5/4 4/3 3/2 5/3 15/8 2/1
4/3: 9/8 6/5 27/20 3/2 27/16 9/5 2/1
3/2: 16/15 6/5 4/3 3/2 8/5 16/9 2/1
8/5: 9/8 5/4 45/32 3/2 5/3 15/8 2/1
9/5: 10/9 5/4 4/3 40/27 5/3 16/9 2/1

mode 3
1/1 9/8 5/4 4/3 3/2 5/3 9/5 2/1
9/8: 10/9 32/27 4/3 40/27 8/5 16/9 2/1
5/4: 16/15 6/5 4/3 36/25 8/5 9/5 2/1
4/3: 9/8 5/4 27/20 3/2 27/16 15/8 2/1
3/2: 10/9 6/5 4/3 3/2 5/3 16/9 2/1
5/3: 27/25 6/5 27/20 3/2 8/5 9/5 2/1
9/5: 10/9 5/4 25/18 40/27 5/3 50/27 2/1

mode 4
1/1 9/8 6/5 4/3 3/2 5/3 9/5 2/1
9/8: 16/15 32/27 4/3 40/27 8/5 16/9 2/1
6/5: 10/9 5/4 25/18 3/2 5/3 15/8 2/1
4/3: 9/8 5/4 27/20 3/2 27/16 9/5 2/1
3/2: 10/9 6/5 4/3 3/2 8/5 16/9 2/1
5/3: 27/25 6/5 27/20 36/25 8/5 9/5 2/1
9/5: 10/9 5/4 4/3 40/27 5/3 50/27 2/1

mode 5
1/1 9/8 5/4 4/3 3/2 8/5 15/8 2/1
9/8 : 10/9 32/27 4/3 64/45 5/3 16/9 2/1
5/4 : 16/15 6/5 32/25 3/2 8/5 9/5 2/1
4/3 : 9/8 6/5 45/32 3/2 27/16 15/8 2/1
3/2 : 16/15 5/4 4/3 3/2 5/3 16/9 2/1
8/5 : 75/64 5/4 45/32 25/16 5/3 15/8 2/1
15/8: 16/15 6/5 4/3 64/45 8/5 128/75 2/1

mode 6
1/1 9/8 6/5 4/3 3/2 8/5 15/8 2/1
9/8 : 16/15 32/27 4/3 64/45 5/3 16/9 2/1
6/5 : 10/9 5/4 4/3 25/16 5/3 15/8 2/1
4/3 : 9/8 6/5 45/32 3/2 27/16 9/5 2/1
3/2 : 16/15 5/4 4/3 3/2 8/5 16/9 2/1
8/5 : 75/64 5/4 45/32 3/2 5/3 15/8 2/1
15/8: 16/15 6/5 32/25 64/45 8/5 128/75 2/1

mode 7
1/1 9/8 5/4 4/3 3/2 8/5 9/5 2/1
9/8: 10/9 32/27 4/3 64/45 8/5 16/9 2/1
5/4: 16/15 6/5 32/25 36/25 8/5 9/5 2/1
4/3: 9/8 6/5 27/20 3/2 27/16 15/8 2/1
3/2: 16/15 6/5 4/3 3/2 5/3 16/9 2/1
8/5: 9/8 5/4 45/32 25/16 5/3 15/8 2/1
9/5: 10/9 5/4 25/18 40/27 5/3 16/9 2/1

mode 8
1/1 9/8 6/5 4/3 3/2 5/3 15/8 2/1
9/8 : 16/15 32/27 4/3 40/27 5/3 16/9 2/1
6/5 : 10/9 5/4 25/18 25/16 5/3 15/8 2/1
4/3 : 9/8 5/4 45/32 3/2 27/16 9/5 2/1
3/2 : 10/9 5/4 4/3 3/2 8/5 16/9 2/1
5/3 : 9/8 6/5 27/20 36/25 8/5 9/5 2/1
15/8: 16/15 6/5 32/25 64/45 8/5 16/9 2/1

Marcel

🔗Marcel de Velde <m.develde@...>

Major:

1/1 2/1 3/1 4/1 5/1 6/1 1-2-3-4-5-6
1/1 3/2 3/1 4/1 5/1 6/1 1-3-2-4-5-6
1/1 3/2 2/1 4/1 5/1 6/1 1-3-4-2-5-6
1/1 3/2 2/1 5/2 5/1 6/1 1-3-4-5-2-6
1/1 3/2 2/1 5/2 3/1 6/1 1-3-4-5-6-2

1/1 2/1 8/3 4/1 5/1 6/1 1-2-4-3-5-6
1/1 4/3 8/3 4/1 5/1 6/1 1-4-2-3-5-6
1/1 2/1 8/3 10/3 5/1 6/1 1-2-4-5-3-6
1/1 4/3 8/3 10/3 5/1 6/1 1-4-2-5-3-6
1/1 4/3 5/3 10/3 5/1 6/1 1-4-5-2-3-6
1/1 2/1 8/3 10/3 4/1 6/1 1-2-4-5-6-3
1/1 4/3 8/3 10/3 4/1 6/1 1-4-2-5-6-3
1/1 4/3 5/3 10/3 4/1 6/1 1-4-5-2-6-3
1/1 4/3 5/3 2/1 4/1 6/1 1-4-5-6-2-3
1/1 4/3 2/1 4/1 5/1 6/1 1-4-3-2-5-6
1/1 4/3 2/1 5/2 5/1 6/1 1-4-3-5-2-6
1/1 4/3 5/3 5/2 5/1 6/1 1-4-5-3-2-6
1/1 4/3 2/1 5/2 3/1 6/1 1-4-3-5-6-2
1/1 4/3 5/3 5/2 3/1 6/1 1-4-5-3-6-2
1/1 4/3 5/3 2/1 3/1 6/1 1-4-5-6-3-2

1/1 2/1 5/2 3/1 9/2 6/1 1-2-5-6-3-4
1/1 2/1 3/1 15/4 5/1 6/1 1-2-3-5-4-6
1/1 2/1 5/2 15/4 5/1 6/1 1-2-5-3-4-6
1/1 5/4 5/2 15/4 5/1 6/1 1-5-2-3-4-6
1/1 2/1 3/1 15/4 9/2 6/1 1-2-3-5-6-4
1/1 2/1 5/2 15/4 9/2 6/1 1-2-5-3-6-4
1/1 5/4 5/2 15/4 9/2 6/1 1-5-2-3-6-4
1/1 5/4 5/2 3/1 9/2 6/1 1-5-2-6-3-4
1/1 5/4 3/2 3/1 9/2 6/1 1-5-6-2-3-4
1/1 3/2 3/1 15/4 5/1 6/1 1-3-2-5-4-6
1/1 3/2 3/1 15/4 9/2 6/1 1-3-2-5-6-4
1/1 5/4 3/2 9/4 9/2 6/1 1-5-6-3-2-4
1/1 5/4 3/2 9/4 3/1 6/1 1-5-6-3-4-2
1/1 3/2 15/8 15/4 5/1 6/1 1-3-5-2-4-6
1/1 5/4 15/8 15/4 5/1 6/1 1-5-3-2-4-6
1/1 3/2 15/8 5/2 5/1 6/1 1-3-5-4-2-6
1/1 5/4 15/8 5/2 5/1 6/1 1-5-3-4-2-6
1/1 3/2 15/8 5/2 3/1 6/1 1-3-5-4-6-2
1/1 5/4 15/8 5/2 3/1 6/1 1-5-3-4-6-2
1/1 3/2 15/8 15/4 9/2 6/1 1-3-5-2-6-4
1/1 3/2 15/8 9/4 9/2 6/1 1-3-5-6-2-4
1/1 3/2 15/8 9/4 3/1 6/1 1-3-5-6-4-2
1/1 5/4 15/8 15/4 9/2 6/1 1-5-3-2-6-4
1/1 5/4 15/8 9/4 9/2 6/1 1-5-3-6-2-4
1/1 5/4 15/8 9/4 3/1 6/1 1-5-3-6-4-2

1/1 2/1 5/2 3/1 4/1 6/1 1-2-5-6-4-3
1/1 5/4 5/2 3/1 4/1 6/1 1-5-2-6-4-3
1/1 5/4 3/2 3/1 4/1 6/1 1-5-6-2-4-3
1/1 5/4 3/2 2/1 4/1 6/1 1-5-6-4-2-3
1/1 2/1 5/2 10/3 4/1 6/1 1-2-5-4-6-3
1/1 2/1 5/2 10/3 5/1 6/1 1-2-5-4-3-6
1/1 5/4 5/2 10/3 5/1 6/1 1-5-2-4-3-6
1/1 5/4 5/3 10/3 5/1 6/1 1-5-4-2-3-6
1/1 5/4 5/2 10/3 4/1 6/1 1-5-2-4-6-3
1/1 5/4 5/3 10/3 4/1 6/1 1-5-4-2-6-3
1/1 5/4 5/3 2/1 4/1 6/1 1-5-4-6-2-3
1/1 5/4 3/2 2/1 3/1 6/1 1-5-6-4-3-2
1/1 5/4 5/3 5/2 5/1 6/1 1-5-4-3-2-6
1/1 5/4 5/3 5/2 3/1 6/1 1-5-4-3-6-2
1/1 5/4 5/3 2/1 3/1 6/1 1-5-4-6-3-2

Minor (exact mirror of major)

1/1 2/1 3/1 4/1 24/5 6/1 1-2-3-4-6-5
1/1 2/1 3/1 18/5 24/5 6/1 1-2-3-6-4-5
1/1 2/1 12/5 18/5 24/5 6/1 1-2-6-3-4-5
1/1 6/5 12/5 18/5 24/5 6/1 1-6-2-3-4-5
1/1 6/5 9/5 18/5 24/5 6/1 1-6-3-2-4-5
1/1 6/5 9/5 12/5 24/5 6/1 1-6-3-4-2-5
1/1 6/5 9/5 12/5 3/1 6/1 1-6-3-4-5-2
1/1 3/2 3/1 4/1 24/5 6/1 1-3-2-4-6-5
1/1 3/2 2/1 4/1 24/5 6/1 1-3-4-2-6-5
1/1 3/2 2/1 12/5 24/5 6/1 1-3-4-6-2-5
1/1 3/2 2/1 12/5 3/1 6/1 1-3-4-6-5-2
1/1 3/2 3/1 18/5 24/5 6/1 1-3-2-6-4-5
1/1 3/2 9/5 18/5 24/5 6/1 1-3-6-2-4-5
1/1 3/2 9/5 12/5 24/5 6/1 1-3-6-4-2-5
1/1 3/2 9/5 12/5 3/1 6/1 1-3-6-4-5-2

1/1 2/1 12/5 16/5 24/5 6/1 1-2-6-4-3-5
1/1 2/1 12/5 16/5 4/1 6/1 1-2-6-4-5-3
1/1 6/5 12/5 16/5 24/5 6/1 1-6-2-4-3-5
1/1 6/5 12/5 16/5 4/1 6/1 1-6-2-4-5-3
1/1 6/5 8/5 16/5 24/5 6/1 1-6-4-2-3-5
1/1 6/5 8/5 16/5 4/1 6/1 1-6-4-2-5-3
1/1 6/5 8/5 2/1 4/1 6/1 1-6-4-5-2-3
1/1 2/1 8/3 4/1 24/5 6/1 1-2-4-3-6-5
1/1 4/3 8/3 4/1 24/5 6/1 1-4-2-3-6-5
1/1 2/1 8/3 16/5 24/5 6/1 1-2-4-6-3-5
1/1 4/3 8/3 16/5 24/5 6/1 1-4-2-6-3-5
1/1 4/3 8/5 16/5 24/5 6/1 1-4-6-2-3-5
1/1 2/1 8/3 16/5 4/1 6/1 1-2-4-6-5-3
1/1 4/3 8/3 16/5 4/1 6/1 1-4-2-6-5-3
1/1 4/3 8/5 16/5 4/1 6/1 1-4-6-2-5-3
1/1 4/3 8/5 2/1 4/1 6/1 1-4-6-5-2-3
1/1 6/5 8/5 12/5 24/5 6/1 1-6-4-3-2-5
1/1 6/5 8/5 12/5 3/1 6/1 1-6-4-3-5-2
1/1 6/5 8/5 2/1 3/1 6/1 1-6-4-5-3-2
1/1 4/3 2/1 4/1 24/5 6/1 1-4-3-2-6-5
1/1 4/3 2/1 12/5 24/5 6/1 1-4-3-6-2-5
1/1 4/3 2/1 12/5 3/1 6/1 1-4-3-6-5-2
1/1 4/3 8/5 12/5 24/5 6/1 1-4-6-3-2-5
1/1 4/3 8/5 12/5 3/1 6/1 1-4-6-3-5-2
1/1 4/3 8/5 2/1 3/1 6/1 1-4-6-5-3-2

1/1 2/1 3/1 18/5 9/2 6/1 1-2-3-6-5-4
1/1 2/1 12/5 18/5 9/2 6/1 1-2-6-3-5-4
1/1 2/1 12/5 3/1 9/2 6/1 1-2-6-5-3-4
1/1 6/5 12/5 18/5 9/2 6/1 1-6-2-3-5-4
1/1 6/5 12/5 3/1 9/2 6/1 1-6-2-5-3-4
1/1 6/5 3/2 3/1 9/2 6/1 1-6-5-2-3-4
1/1 3/2 3/1 18/5 9/2 6/1 1-3-2-6-5-4
1/1 3/2 9/5 18/5 9/2 6/1 1-3-6-2-5-4
1/1 6/5 9/5 18/5 9/2 6/1 1-6-3-2-5-4
1/1 3/2 9/5 9/4 9/2 6/1 1-3-6-5-2-4
1/1 3/2 9/5 9/4 3/1 6/1 1-3-6-5-4-2
1/1 6/5 9/5 9/4 9/2 6/1 1-6-3-5-2-4
1/1 6/5 9/5 9/4 3/1 6/1 1-6-3-5-4-2
1/1 6/5 3/2 9/4 9/2 6/1 1-6-5-3-2-4
1/1 6/5 3/2 9/4 3/1 6/1 1-6-5-3-4-2

1/1 2/1 12/5 3/1 4/1 6/1 1-2-6-5-4-3
1/1 6/5 12/5 3/1 4/1 6/1 1-6-2-5-4-3
1/1 6/5 3/2 3/1 4/1 6/1 1-6-5-2-4-3
1/1 6/5 3/2 2/1 4/1 6/1 1-6-5-4-2-3
1/1 6/5 3/2 2/1 3/1 6/1 1-6-5-4-3-2

🔗Claudio Di Veroli <dvc@...>

🔗Marcel de Velde <m.develde@...>

🔗hstraub64 <straub@...>

🔗Cameron Bobro <misterbobro@...>

🔗Marcel de Velde <m.develde@...>

🔗Michael Sheiman <djtrancendance@...>

--The best way to go about music theory is to
--make music.
Agreed! Actually most of my learning about scales under tunings has come from having a general mathematical idea and pushing it into place by trying to compose with it. Doing so forces me to consider, for example, ability for melody and not just harmony...and things like ability to stay in a mood and how many moods are possible with chords in a new scale within a tuning.
I used to the the harmonic series was restricted to the consistent 1/1, 2/1, 3/1 pattern and started sounding bad after about the 19th harmonic until I started trying to compose with it and found that eliminating certain notes and wrapping around the octave IE 7/7 8/7 9/7 10/7 11/7 12/7 13/7 2/1 2/1 * 8/7....allowed me to avoid the "too many or too few notes" problem and avoid the intense beating that happens over the 19th harmonic.
I also found, for example, that doing things imperfect to the mathematical series formula ALA 9/8 * 10/9 * 11/10 * 12/11 * 10/9 (instead of 13/12) * 11/10 * 12/11....actually increased the emotionality of the scale (kind of more of a "major/minor" contrast) far as composition even though it is not as mathematically formal.

In general...I have found it's not a game of math dominating composition or composition dominating math...but one where it helps immensely to have a foot in both.

-Michael

--- On Thu, 2/19/09, Cameron Bobro <misterbobro@...> wrote:

From: Cameron Bobro <misterbobro@...>
Subject: [tuning] Re: theory on possible JI construction of harmony, melody and music
To: tuning@yahoogroups.com
Date: Thursday, February 19, 2009, 5:43 AM

--- In tuning@yahoogroups. com, Marcel de Velde <m.develde@. ..> wrote:

> > - has not been already done by others

> > - has not been alredy proven impossible

> > - will serve a practical purpose

> >

> Though I can never be sure it has not allready been done by someone

else, I

> have not encountered it.

> Many theories abound but all that I've found work in a different

way, also

> all I've read that comes near brings all intervals back to the

octave which

> I don't)

> It's allready serving a practical purpose for me.

> The scales I and musical I get out of my theory make sense to me

and help me

> understand music.

> Marcel

Have you read "Genesis of a Music"?

I'd like to give you some advice, obviously noone is under any

obligation to take it.

The best way to go about music theory is to

make music. If your theories work, other people will come to you

and ask you what you're doing, they may even pay you to tell them.

Well that's it. The chances of you paying attention to what I'm

saying are probably nil, whatever! Just don't say in 30 years that I

didn't warn you. :-)

-Cameron Bobro

🔗hstraub64 <straub@...>

🔗Marcel de Velde <m.develde@...>

🔗justin_tone52 <kleisma7@...>

> Have you done this process for the 7th harmonic, too? I would be
> interested what the results are.

Dear Hans and Marcel,

I have been working on extending this theory to the 7-limit and higher
limits, and I have found a very interesting and quick way to construct
these scales (much quicker than listing all permutations, and gives
the identical result). I am going to illustrate this algorithm using
the example of finding the scale of all permutations of 2/1 3/2 4/3
5/4 6/5 7/6 8/7. It is a very simple algorithm:

STEP 1: Start with 1/1.

STEP 2: Transpose this by the first interval in the list (namely,
2/1), and combine with the original. Since 2/1 adds no new pitches to
the list, one stays at 1/1.

STEP 3: Transpose this scale by the next interval in the list (namely,
3/2), and combine with the original. In this case, one arrives at 1/1 3/2.

STEP 4: Transpose this scale by the next interval in the list (namely,
4/3), and combine with the original (1/1 3/2).
So one arrives at 1/1 4/3 3/2.

STEP 5: Transpose this scale by the next interval in the list (namely,
5/4), and combine with the original (1/1 4/3 3/2).
So one arrives at 1/1 5/4 4/3 3/2 5/3 15/8.

STEP 6: Transpose this scale by the next interval in the list (namely,
6/5), and combine with the original (1/1 5/4 4/3 3/2 5/3 15/8).
So one arrives at 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 (this is
just the combination of the major and minor scales).

STEP 7: Transpose this scale by the next interval in the list (namely,
7/6), and combine with the original (1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3
9/5 15/8).
So one arrives at 1/1 21/20 35/32 9/8 7/6 6/5 5/4 21/16 4/3 7/5 35/24
3/2 14/9 8/5 5/3 7/4 9/5 28/15 15/8 35/18

STEP 8: Transpose this scale by the next (and last) interval in the
list (namely, 8/7), and combine with the original (1/1 21/20 35/32 9/8
7/6 6/5 5/4 21/16 4/3 7/5 35/24 3/2 14/9 8/5 5/3 7/4 9/5 28/15 15/8
35/18).
One finally arrives at the following 33-tone scale:

1/1 36/35 21/20 16/15 15/14 35/32 10/9 9/8 8/7 7/6 6/5 5/4 9/7 21/16
4/3 48/35 7/5 10/7 35/24 3/2 32/21 14/9 8/5 5/3 12/7 7/4 16/9 9/5
64/35 28/15 15/8 40/21 35/18 2/1

This is the scale that one obtains when combining all permutations of
2/1 3/2 4/3 5/4 6/5 7/6 8/7.

I have used this algorithm to create an equivalent scale for the 11
limit, using harmonics 1-12 (essentially I added four more steps,
transposing at the 9/8, 10/9, 11/10 and 12/11). I got this 250-tone
(!) scale:

1/1 385/384 176/175 100/99 81/80 64/63 56/55 55/54 45/44 77/75 36/35
33/32 28/27 80/77 25/24 22/21 21/20 81/77 135/128 200/189 35/33
297/280 16/15 77/72 15/14 189/176 320/297 27/25 175/162 693/640 88/81
12/11 35/32 192/175 11/10 243/220 256/231 10/9 891/800 352/315 28/25
432/385 9/8 112/99 198/175 25/22 154/135 8/7 63/55 55/48 405/352
800/693 231/200 81/70 297/256 64/55 220/189 7/6 90/77 75/64 88/75
350/297 33/28 189/160 32/27 385/324 25/21 105/88 6/5 77/64 135/112
40/33 243/200 175/144 128/105 11/9 27/22 315/256 216/175 100/81 99/80
56/45 96/77 5/4 44/35 63/50 486/385 81/64 80/63 14/11 891/700 225/176
32/25 77/60 9/7 567/440 165/128 128/99 35/27 100/77 2079/1600 176/135
72/55 55/42 21/16 405/308 33/25 175/132 297/224 4/3 385/288 75/56
945/704 400/297 27/20 693/512 110/81 15/11 175/128 48/35 11/8 243/176
320/231 243/175 25/18 891/640 88/63 7/5 108/77 45/32 140/99 99/70
567/400 64/45 77/54 10/7 63/44 36/25 231/160 81/56 16/11 35/24 225/154
22/15 81/55 165/112 189/128 40/27 297/200 525/352 576/385 3/2 385/256
264/175 50/33 243/160 32/21 84/55 55/36 135/88 77/50 54/35 99/64 14/9
120/77 25/16 11/7 63/40 243/154 100/63 35/22 891/560 8/5 77/48 45/28
567/352 160/99 81/50 175/108 2079/1280 44/27 18/11 105/64 288/175
33/20 128/77 5/3 176/105 42/25 648/385 27/16 320/189 56/33 297/175
75/44 77/45 12/7 189/110 55/32 140/81 400/231 693/400 243/140 96/55
110/63 7/4 135/77 44/25 175/99 99/56 567/320 16/9 385/216 25/14
315/176 9/5 231/128 20/11 175/96 64/35 11/6 81/44 1280/693 324/175
50/27 297/160 352/189 28/15 144/77 15/8 560/297 66/35 189/100 154/81
40/21 21/11 48/25 77/40 27/14 64/33 35/18 150/77 88/45 108/55 55/28
63/32 160/81 99/50 175/88 768/385 2/1

which shows clearly that the number of tones increases rapidly with
the prime limit, so that the scales quickly get impractical.

I have also tried this method for the 15 limit, using harmonics 1-16,
and I obtained a 1775-tone scale. No doubt this method is not ideal
for composers who wish to use tonal materials based on higher
harmonics (this method is pretty much unique to the 5 and 7 limits).

Justin Tone

(P.S.) The seven-limit scale is a very interesting scale to work with,
as it can also be derived in a Tonality-Diamond fashion using the
identities 1-3-5-7-9-15-21-35. (This is not true in general for scales
created by permuting intervals in a harmonic series, and is definitely
not true for the 1775-tone scale I obtained in the 15 limit.)

🔗justin_tone52 <kleisma7@...>

As I examine this algorithm closer, I can find equivalents to the major
scale and minor scale, using higher harmonics. For example, the familiar
major and minor scale were created in Step 6, essentially by taking the
scale obtained in Step 6:
1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8
and tempering out the comma 25:24 to obtain a 7 note scale:
1/1 9/8 (6/5,5/4) 4/3 3/2 (8/5,5/3) (9/5,15/8)
Selecting the higher member of each equivalence class gives rise to the
major scale, and selecting the lower member of each equivalence class
gives rise to the major scale.
I can now do the same thing with, for example, Step 8:

1/1 36/35 21/20 16/15 15/14 35/32 10/9 9/8 8/7 7/6 6/5 5/4 9/7 21/16 4/3
48/35 7/5 10/7 35/24 3/2 32/21 14/9 8/5 5/3 12/7 7/4 16/9 9/5 64/35
28/15 15/8 40/21 35/18 2/1

Instead of tempering out 25/24, I temper out 49/48 and select the
pitches accordingly, giving:

1/1 21/20 16/15 35/32 10/9 9/8 7/6 6/5 5/4 21/16 4/3 7/5 35/24 3/2 14/9
8/5 5/3 7/4 16/9 9/5 28/15 15/8 35/18 2/1

This is a 7-limit equivalent of the "major scale" (however it is not
really useful as a melodic construct because of its large number, 23, of
tones). The minor scale is obtained by replacing every 7 in the ratios
with 48/7, or alternately, inverting the whole scale around the 1/1-2/1
axis.

This works with any step number, so at Step n, I temper out the comma
(n-1)^2 / n(n - 2). However, there is a challenge when the step number
(n) is odd, namely, there is a note in the tuning that is a comma lower
than unison. So the "minor scale" would end up omitting the unison. This
is not a big issue, however, because the whole scale can be transposed
up a comma without problem.

By the way, just as Erv Wilson's hexany and its analogs are called
"combination-product sets" (CPS), can these types of scales be called
"permutation-product sets" (PPS)? This is just a suggestion. Then I
could describe Marcel's theory of JI scales in one succinct sentence,
along these lines: "The 6-step PPS consists of the major and minor
scales on a common tonic."

Justin Tone

-P.S. As a side note, it must be possible to create new PPS's by
skipping some harmonics in the overtone series. For example, 2/1 3/2 4/3
7/6 8/7 9/8 would give the following 16-tone PPS:

1/1 9/8 8/7 7/6 9/7 21/16 4/3 3/2 32/21 14/9 27/16 12/7 7/4 16/9 27/14
63/32 2/1

This may create new and useful resources in microtonal composition.

------------------------------------------------------------------------\
-------------

> I have been working on extending this theory to the 7-limit and higher
> limits, and I have found a very interesting and quick way to construct
> these scales (much quicker than listing all permutations, and gives
> the identical result). I am going to illustrate this algorithm using
> the example of finding the scale of all permutations of 2/1 3/2 4/3
> 5/4 6/5 7/6 8/7. It is a very simple algorithm:
>
> STEP 1: Start with 1/1.
>
> STEP 2: Transpose this by the first interval in the list (namely,
> 2/1), and combine with the original. Since 2/1 adds no new pitches to
> the list, one stays at 1/1.
>
> STEP 3: Transpose this scale by the next interval in the list (namely,
> 3/2), and combine with the original. In this case, one arrives at 1/1
3/2.
>
> STEP 4: Transpose this scale by the next interval in the list (namely,
> 4/3), and combine with the original (1/1 3/2).
> So one arrives at 1/1 4/3 3/2.
>
> STEP 5: Transpose this scale by the next interval in the list (namely,
> 5/4), and combine with the original (1/1 4/3 3/2).
> So one arrives at 1/1 5/4 4/3 3/2 5/3 15/8.
>
> STEP 6: Transpose this scale by the next interval in the list (namely,
> 6/5), and combine with the original (1/1 5/4 4/3 3/2 5/3 15/8).
> So one arrives at 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8 (this is
> just the combination of the major and minor scales).
>
> STEP 7: Transpose this scale by the next interval in the list (namely,
> 7/6), and combine with the original (1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3
> 9/5 15/8).
> So one arrives at 1/1 21/20 35/32 9/8 7/6 6/5 5/4 21/16 4/3 7/5 35/24
> 3/2 14/9 8/5 5/3 7/4 9/5 28/15 15/8 35/18
>
> STEP 8: Transpose this scale by the next (and last) interval in the
> list (namely, 8/7), and combine with the original (1/1 21/20 35/32 9/8
> 7/6 6/5 5/4 21/16 4/3 7/5 35/24 3/2 14/9 8/5 5/3 7/4 9/5 28/15 15/8
> 35/18).
> One finally arrives at the following 33-tone scale:
>
> 1/1 36/35 21/20 16/15 15/14 35/32 10/9 9/8 8/7 7/6 6/5 5/4 9/7 21/16
> 4/3 48/35 7/5 10/7 35/24 3/2 32/21 14/9 8/5 5/3 12/7 7/4 16/9 9/5
> 64/35 28/15 15/8 40/21 35/18 2/1
>
> This is the scale that one obtains when combining all permutations of
> 2/1 3/2 4/3 5/4 6/5 7/6 8/7.
>
> I have used this algorithm to create an equivalent scale for the 11
> limit, using harmonics 1-12 (essentially I added four more steps,
> transposing at the 9/8, 10/9, 11/10 and 12/11). I got this 250-tone
> (!) scale:
>
> (scale)
>
> which shows clearly that the number of tones increases rapidly with
> the prime limit, so that the scales quickly get impractical.
>
> I have also tried this method for the 15 limit, using harmonics 1-16,
> and I obtained a 1775-tone scale. No doubt this method is not ideal
> for composers who wish to use tonal materials based on higher
> harmonics (this method is pretty much unique to the 5 and 7 limits).
>
> Justin Tone
>

🔗Marcel de Velde <m.develde@...>

You know what.My theory is right!!

I've been doubting it because of the lasso en beethoven pieces and the
trouble they gave.
(Not that I was able to use my theory in a practical way to analyse these
pieces but I did expect them to be perfectly harmonious)
But it's not the problem of my theory. I've been thinking it over deeply
lately and my theory is perfect.
It gives the perfect harmonious music and describes perfect melody in
perfect harmony.
It's that many of the music written till today just isn't very harmonious!
The commen practice music is based on wrong theories about harmony and
melody, and they're kinda getting away with it because of 12tet.
Besides this, in perfectly harmonious music you get wolf fourths, fifths and
"wolf octaves" and they sound perfectly ok.
But in commen practice music I suspect these (and other much less harmonious
harmonic structures) are abused because of how they sound in 12tet.
This probably means that much music written in 12tet will not sound very
pretty in just intonation, and that it's incredibly hard to analyse.

Take for instance the 5-limit dominant 7th chord.
According to my theory it's harmonious construction is 1/1 9/8 45/32 27/16
9/4
The 7th is below the major chord, it may not be above or it's another
structure much less harmonious.
You cannot invert this chord in the octave as it's a chord that does not
exist within 2 notes an octave appart.
You can invert this chord in the 9/4 for instance (among many other options)
This will give chords like 1/1 5/4 3/2 2/1 9/4
1/1 6/5 8/5 9/5 9/4
1/1 4/3 3/2 15/8 9/4

But you can also see it as 1/1 9/8 45/32 27/16 and invert it between 1/1 and
27/16
1/1 5/4 3/2 27/16
1/1 6/5 27/20 27/16

You can also reposition the 9/8 in 1/1 9/8 45/32 27/16 9/4 like this
1/1 5/4 3/2 2/1 9/4
1/1 5/4 3/2 2716 9/4
1/1 5/4 45/32 27/16 9/4

Now the above chord had a few 27/20
But thesame way of seeing harmony gives intervals like 81/40.
I had trouble with accepting these before but after more thinking and
experimenting I now see them as correct and nessecary.
Take for instance this chord 1/1 5/4 3/2 9/4 which exists in 6harmonic limit
harmonic structure.
And make it a 7th chord like the chords above like this.
1/1 9/8 45/32 27/16 81/32
Inverting this in the 81/32 will give the wide octave.
1/1 5/4 3/2 9/4 81/32
see the 81/40 between 5/4 and 81/32
Now offcourse you can't tranpose things by an octave to create 81/80 which
is done a lot in commen practice music it seems.
Also I'm not saying you can invert chords like this in actual music since
you have to take melody (permutations) into account, but you can get to
these chords in many ways and they'll be the only correct way to tune them
in many cases.

Now I allready think that due to my bad explenations and radical things I'm
saying the list will largely not agree.
But I'm going to write a program that makes music according to the rules of
my theory.
Will have it finished enough to let you hear things in about a month.
Then there will be no denying anymore :)

Marcel

🔗Marcel de Velde <m.develde@...>

I know it's right now.My theory just solved Beethoven's Drei Equali :)
All I had to accept was that not all chords are to be seen as their most
harmonious translation from 12tet.
And that in common practice music everything can get inverted in octaves and
notes transposed by octaves.
When taking this into account my theory gives the complete construction of
all harmony and of the movement of the melodies that make up the harmonies.

Marcel

2009/3/29 Marcel de Velde <m.develde@...>

> You know what.My theory is right!!
>
> I've been doubting it because of the lasso en beethoven pieces and the
> trouble they gave.
> (Not that I was able to use my theory in a practical way to analyse these
> pieces but I did expect them to be perfectly harmonious)
> But it's not the problem of my theory. I've been thinking it over deeply
> lately and my theory is perfect.
> It gives the perfect harmonious music and describes perfect melody in
> perfect harmony.
> It's that many of the music written till today just isn't very harmonious!
> The commen practice music is based on wrong theories about harmony and
> melody, and they're kinda getting away with it because of 12tet.
> Besides this, in perfectly harmonious music you get wolf fourths, fifths
> and "wolf octaves" and they sound perfectly ok.
> But in commen practice music I suspect these (and other much less
> harmonious harmonic structures) are abused because of how they sound in
> 12tet.
> This probably means that much music written in 12tet will not sound very
> pretty in just intonation, and that it's incredibly hard to analyse.
>
> Take for instance the 5-limit dominant 7th chord.
> According to my theory it's harmonious construction is 1/1 9/8 45/32 27/16
> 9/4
> The 7th is below the major chord, it may not be above or it's another
> structure much less harmonious.
> You cannot invert this chord in the octave as it's a chord that does not
> exist within 2 notes an octave appart.
> You can invert this chord in the 9/4 for instance (among many other
> options)
> This will give chords like 1/1 5/4 3/2 2/1 9/4
> 1/1 6/5 8/5 9/5 9/4
> 1/1 4/3 3/2 15/8 9/4
>
> But you can also see it as 1/1 9/8 45/32 27/16 and invert it between 1/1
> and 27/16
> 1/1 5/4 3/2 27/16
> 1/1 6/5 27/20 27/16
>
> You can also reposition the 9/8 in 1/1 9/8 45/32 27/16 9/4 like this
> 1/1 5/4 3/2 2/1 9/4
> 1/1 5/4 3/2 2716 9/4
> 1/1 5/4 45/32 27/16 9/4
>
> Now the above chord had a few 27/20
> But thesame way of seeing harmony gives intervals like 81/40.
> I had trouble with accepting these before but after more thinking and
> experimenting I now see them as correct and nessecary.
> Take for instance this chord 1/1 5/4 3/2 9/4 which exists in 6harmonic
> limit harmonic structure.
> And make it a 7th chord like the chords above like this.
> 1/1 9/8 45/32 27/16 81/32
> Inverting this in the 81/32 will give the wide octave.
> 1/1 5/4 3/2 9/4 81/32
> see the 81/40 between 5/4 and 81/32
> Now offcourse you can't tranpose things by an octave to create 81/80 which
> is done a lot in commen practice music it seems.
> Also I'm not saying you can invert chords like this in actual music since
> you have to take melody (permutations) into account, but you can get to
> these chords in many ways and they'll be the only correct way to tune them
> in many cases.
>
> Now I allready think that due to my bad explenations and radical things I'm
> saying the list will largely not agree.
> But I'm going to write a program that makes music according to the rules of
> my theory.
> Will have it finished enough to let you hear things in about a month.
> Then there will be no denying anymore :)
>
> Marcel
>

🔗Marcel de Velde <m.develde@...>

Hi Justin,

Bit of a late in depth reply but I had to get things more clear to myself
first :)

As I examine this algorithm closer, I can find equivalents to the major
> scale and minor scale, using higher harmonics. For example, the familiar
> major and minor scale were created in Step 6, essentially by taking the
> scale obtained in Step 6:
>

> 1/1 9/8 6/5 5/4 4/3 3/2 8/5 5/3 9/5 15/8
> and tempering out the comma 25:24 to obtain a 7 note scale:
> 1/1 9/8 (6/5,5/4) 4/3 3/2 (8/5,5/3) (9/5,15/8)
> Selecting the higher member of each equivalence class gives rise to the
> major scale, and selecting the lower member of each equivalence class gives
> rise to the major scale.

Yes this seems to be the easyest way to do it practically but it doesn't
make enough sense to me in a more abstract logical way.
I still prefer to do it by setting a rule of order in which the harmonic
intervals 5/4 and 6/5 have to follow eachother in the underlying harmonies
of the scales. 5/4 before 6/5 is major, 6/5 before 5/4 is minor.
You get the 25/24 stepsize in melodies only when switching the order of 5/4
and 6/5 in the underlying harmonies.
This only concerns melodies.
I do think 25/24 is a perfectly valid stepsize in melodies, as is for
instance 27/25. They're just one step more "complex" than 16/15.

> I can now do the same thing with, for example, Step 8:
>
> 1/1 36/35 21/20 16/15 15/14 35/32 10/9 9/8 8/7 7/6 6/5 5/4 9/7 21/16 4/3
> 48/35 7/5 10/7 35/24 3/2 32/21 14/9 8/5 5/3 12/7 7/4 16/9 9/5 64/35 28/15
> 15/8 40/21 35/18 2/1
>
> Instead of tempering out 25/24, I temper out 49/48 and select the pitches
> accordingly, giving:
>
> 1/1 21/20 16/15 35/32 10/9 9/8 7/6 6/5 5/4 21/16 4/3 7/5 35/24 3/2 14/9 8/5
> 5/3 7/4 16/9 9/5 28/15 15/8 35/18 2/1
>
> This is a 7-limit equivalent of the "major scale" (however it is not really
> useful as a melodic construct because of its large number, 23, of tones).
> The minor scale is obtained by replacing every 7 in the ratios with 48/7, or
> alternately, inverting the whole scale around the 1/1-2/1 axis.
>

This should be thesame or almost thesame as setting a rule of order in the
underlying harmonies of either 8/7 before 7/6 or the other way around.
Btw I beleive the size of the scale and the small stepsizes you're getting
is correct.
After long thinking about how to use the 7th melodically I now think
strongly that the 7th harmony is for very small stepsizes.
The 5th and 6th harmonic have their limit, they ultimately build 12 tone
scales.
The 7th - 8th harmonics together with the lower harmonics seem to divide the
12 tone 5-limit scales further into 24? tone scales.

>
> This works with any step number, so at Step *n*, I temper out the comma (*
> n*-1)^2 / *n*(*n* - 2). However, there is a challenge when the step number
> (*n*) is odd, namely, there is a note in the tuning that is a comma lower
> than unison. So the "minor scale" would end up omitting the unison. This is
> not a big issue, however, because the whole scale can be transposed up a
> comma without problem.

This problem doesn't occur when doing this by limiting the order of the
intervals in the underlying harmonies.
However there does to me seem to be a bit of trouble with some chords giving
for instance 81/40.
Does this mean that we're supposed to build scales that are non octave
repeating? Or does it mean this chord can only be played in large scales
that have very narrow stepsizes? I think the 2nd one, and the chords that
give 81/40 only start at the 9th harmonic.

By the way, just as Erv Wilson's hexany and its analogs are called
> "combination-product sets" (CPS), can these types of scales be called
> "permutation-product sets" (PPS)? This is just a suggestion. Then I could
> describe Marcel's theory of JI scales in one succinct sentence, along these
> lines: "The 6-step PPS consists of the major and minor scales on a common
> tonic."
>

I do like the name :)
However it may not make clear enough we're talking about permuations of the
harmonic series. I don't want confusion and people starting to use thesame
name for non harmonic permutation sets.
How about HPPS :)

> -P.S. As a side note, it must be possible to create new PPS's by skipping
> some harmonics in the overtone series. For example, 2/1 3/2 4/3 7/6 8/7 9/8
> would give the following 16-tone PPS:
>
> 1/1 9/8 8/7 7/6 9/7 21/16 4/3 3/2 32/21 14/9 27/16 12/7 7/4 16/9 27/14
> 63/32 2/1
>
> This may create new and useful resources in microtonal composition.

I've looked at this too, but now I use a different way of skipping certain
steps.
By not allowing for comma shifts in a composition I let the composition
select the ratios from larger scales and make 12tone subset scales this way
(based on all harmonies in the composition following the underlying harmonic
structure that underlies this whole system)
This gives both pure harmonies and pure melodies without shifts.

Marcel