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Re: [tuning] theory on possible JI construction of harmony, melody and music

🔗djtrancendance@...

2/19/2009 9:21:03 AM

   This does look like a good theory, at least on paper: I can see how it chains the harmonic series in such a way that transposition is possible without destroying interval consistency (an advantage my "old" JI-based scales lack).
 
   Your major scale was stated as:
1/1     9/8     5/4       4/3            3/2          5/3       15/8         2/1 =
 1   1.125 1.25    1.3333    1.5     1.6666 
1.875     2

--------------------------------------------------------
     What I wonder is how this would compare to my old "chained harmonic series" scale to the human ear:
1/1 * 9/8 * 10/9 * 11/10 * 12/11 * 10/9 * 11/10  * 12/10  =
1/1   9/8    5/4     11/8        3/2     5/3      33/18    2/1 =
1     1.125  1.25   1.375     1.5     1.6666  1.83333  2

   Note that 2/1 (of course) but also 9/8, 5/4, 3/2, and 5/3 are exactly the same as in your scale...in fact just two notes (15/8 and 4/3 from your scale) are different.

  The one pattern I see in your scale and not mine is the use of those "reversed harmonic series" IE from 5/4 to 4/3.  In fact, if you
consider "reversed" in your scale a "minor", you seem to end up with the more-or-less the exact order of major-minor intervals in the major scale (perhaps this is why 12TET style transposition and assigning of keys within the tuning is mathematically possible within your scale). :-)

  However I don't quite understand how 5/3  and 15/8 (in your scale) connect.
********************************************************
  I also wonder...if any of you, just by listening, can compare the JI scales Marcel and I created.  It's not exactly fair for me to rate them. :-D

-Michael

                         

                               

--- On Thu, 2/19/09, Marcel de Velde <m.develde@...> wrote:

From: Marcel de Velde <m.develde@...>
Subject: [tuning] theory on possible JI construction of harmony, melody and music
To: tuning@yahoogroups.com
Date: Thursday, February 19, 2009, 3:04 AM

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

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

> 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 3/1next 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 4/1Please 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  1-2-3-4-51/1 3/2 3/1  4/1  5/1  1-3-2-4-51/1 3/2 2/1  4/1  5/1  1-3-4-2-5
1/1 3/2 2/1  5/2  5/1  1-3-4-5-21/1 2/1 8/3  4/1  5/1  1-2-4-3-51/1 2/1 8/3  10/3 5/1  1-2-4-5-31/1 4/3 8/3  4/1  5/1  1-4-2-3-51/1 4/3 8/3  10/3 5/1  1-4-2-5-31/1 4/3 5/3  10/3 5/1  1-4-5-2-3
1/1 4/3 2/1  4/1  5/1  1-4-3-2-51/1 4/3 2/1  5/2  5/1  1-4-3-5-21/1 4/3 5/3  5/2  5/1  1-4-5-3-21/1 2/1 3/1  15/4 5/1  1-2-3-5-41/1 2/1 5/2  15/4 5/1  1-2-5-3-41/1 5/4 5/2  15/4 5/1  1-5-2-3-4
1/1 3/2 3/1  15/4 5/1  1-3-2-5-41/1 3/2 15/8 15/4 5/1  1-3-5-2-41/1 3/2 15/8 5/2  5/1  1-3-5-4-21/1 5/4 15/8 15/4 5/1  1-5-3-2-41/1 5/4 15/8 5/2  5/1  1-5-3-4-21/1 2/1 5/2  10/3 5/1  1-2-5-4-3
1/1 5/4 5/2  10/3 5/1  1-5-2-4-31/1 5/4 5/3  10/3 5/1  1-5-4-2-31/1 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 6/1Seen 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 31/1 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 41/1 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 51/1 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 61/1 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 71/1 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 81/1 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