Using just-intonation mixed with the harmonic series to form a new scale

http://en.wikipedia.org/wiki/Just_intonation
This shows two ways to derive a 7-note scale favoring certain
chords/intervals.
The two scales they derive are

SCALE ONE SCALE TWO
1/1 (root) 1/1 (root)
9/8 = 1.125 9/8 = 1.125
5/4 = 1.25 6/5 = 1.2
4/3 = 1.333333 4/3 = 1.333333
3/2 = 1.5 8/5 = 1.6
5/3 = 1.6666666 9/5 = 1.8
15/8 = 1.875 15/8 = 1.875
____________________________________________________
I made my own scale most by "mirroring" the harmonic series around
an octave causing

MY HYBRID JI/HARMONIC-SERIES SCALE

1
12/11 = 1.090909 NEW FREQUENCY
12/10 = 1.2 SAME AS IN SECOND JI example
12/9 = 1.3333 SAME AS IN BOTH JI EXAMPLES
12/8 = 1.5 SAME AS IN FIRST JI EXAMPLE
5/3(12/8*10/9) = 1.66666 SAME AS IN FIRST JI EXAMPLE
11/6(5/3*11/10)= 1.83333 NEW FREQUENCY
2 (11/6*12/11) OCTAVE

It just seems profound to me...that my new scale seems to be
mixing the mathematics of different forms of JI with the harmonic
series (note the harmonic series fractions 12/11*11/10*10/9*9/8 used
to create the first 4 notes of my scale...and the fractions in
reverse order IE 10/9*11/10*12/11 to create the last 3).

Any idea how or why this works (or doesn't)?...far as special
matrices or other properties that are beyond my knowledge?
________________________________________________________________

All I know is...when I play notes using the new scale it, at
least to my ears, seems to sound a good deal more natural to my ears
than 12TET regardless of which chords are played. I can even jam
out all 7 notes within a single octave...and it seems quite obvious
where the "tone center" is.

Hopefully gaining more knowledge will help me fine-tune this
scale...and ultimately help musicians find good use for it.

-Michael

Michael wrote:

> http://en.wikipedia.org/wiki/Just_intonation
> This shows two ways to derive a 7-note scale favoring certain
> chords/intervals.
> The two scales they derive are
>
> SCALE ONE SCALE TWO
> 1/1 (root) 1/1 (root)
> 9/8 = 1.125 9/8 = 1.125
> 5/4 = 1.25 6/5 = 1.2
> 4/3 = 1.333333 4/3 = 1.333333
> 3/2 = 1.5 8/5 = 1.6
> 5/3 = 1.6666666 9/5 = 1.8
> 15/8 = 1.875 15/8 = 1.875
> ____________________________________________________
> I made my own scale most by "mirroring" the harmonic series around
> an octave causing
>
> MY HYBRID JI/HARMONIC-SERIES SCALE
>
> 1
> 12/11 = 1.090909 NEW FREQUENCY
> 12/10 = 1.2 SAME AS IN SECOND JI example
> 12/9 = 1.3333 SAME AS IN BOTH JI EXAMPLES
> 12/8 = 1.5 SAME AS IN FIRST JI EXAMPLE
> 5/3(12/8*10/9) = 1.66666 SAME AS IN FIRST JI EXAMPLE
> 11/6(5/3*11/10)= 1.83333 NEW FREQUENCY
> 2 (11/6*12/11) OCTAVE
//
> Any idea how or why this works (or doesn't)?...

What you've got there is a lower tetrachord consisting
of subharmonics 12-8, and an upper tetrachord consisting
of harmonics 9-12. Tetrachordal construction is a great
way to make scales, and taking one's tetrachords from
the series is a cracking good idea too. I don't think
you'll find it contains 6 triads as the diatonic scale
in meantone does, but you will be able to play a lot of
things that sound good. I've tried similar things in the
past but I don't see this scale in my archive, so I'll
add it.

-Carl

Interesting, so "that's all it is".  Then again it seems increasingly apparent to me...the more you can ultimately simplify a scales formulaic composition (no matter how many steps that simplification takes), the more natural it sounds.
 
    So if I have it right
1) The note 9/8 is the only note in my scale not part of a tetrachord
2) The two tetrachords are on each side, each occupying the ratio 4/3 IE a perfect 4th.  Since 4/3 * 4/3 = 16/9 and 16/9 times 9/8 = 2 it is able to fill an octave.
 
   The other thing I have blatantly notices is the sense of where the "center" of the scale is. 
Even played with just sine wave...scales like 10TET sound "warped" IE sound either slightly lower than their centers.  The ideal location of where the center of the sound should be, of course, seems to be the perfect 5th (which my scale both has and seems to "point" to with its other notes).
 
  All of this seems to go back to your discussion earlier of why periodicity matters so much and is needed to define "concordance" just as much as the absense of beating is. 
 
   Indeed, the ear does seem to gravitate toward combinations of tones that point to (IE the series itself) or intersect with (IE diatonic just intonation) the harmonic series.  The harmonic series also seems to lend all its harmonics to pointing to the sense of a single tone/center making it "tonal".  Thus, by nature, it appears nothing can provide a stronger sense of location of a single note than JI itself.
 
   How much of what I have said, far as others' past research, has proven true?
______________________________________________________________________
 
    I am just trying to piece various bits of the puzzle together...and figure out how they can be re-arranged to open up more compositional possibilities than just pure JI or the pure harmonic series.
   I also wonder how Wilson's horogram scales work relative to these concepts and what they add...as, so far, they are by far the most "concordant" scales I have heard, more so than the MOS scales or virtually any of the others.
 
-Michael
 

--- On Thu, 12/25/08, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@lumma.org>
Subject: [tuning] Re: Using just-intonation mixed with the harmonic series to form a new scale
To: tuning@yahoogroups.com
Date: Thursday, December 25, 2008, 2:11 AM

Michael wrote:

> http://en.wikipedia .org/wiki/ Just_intonation
> This shows two ways to derive a 7-note scale favoring certain
> chords/intervals.
> The two scales they derive are
>
> SCALE ONE SCALE TWO
> 1/1 (root) 1/1 (root)
> 9/8 = 1.125 9/8 = 1.125
> 5/4 = 1.25 6/5 = 1.2
> 4/3 = 1.333333 4/3 = 1.333333
> 3/2 = 1.5 8/5 = 1.6
> 5/3 = 1.6666666 9/5 = 1.8
> 15/8 = 1.875 15/8 = 1.875
> ____________ _________ _________ _________ _________ ____
> I made my own scale most by "mirroring" the harmonic series around
> an octave causing
>
> MY HYBRID JI/HARMONIC- SERIES SCALE
>
> 1
> 12/11 = 1.090909 NEW FREQUENCY
> 12/10 = 1.2 SAME AS IN SECOND JI example
> 12/9 = 1.3333 SAME AS IN BOTH JI EXAMPLES
> 12/8 = 1.5 SAME AS IN FIRST JI EXAMPLE
> 5/3(12/8*10/ 9) = 1.66666 SAME AS IN FIRST JI EXAMPLE
> 11/6(5/3*11/ 10)= 1.83333 NEW FREQUENCY
> 2 (11/6*12/11) OCTAVE
//
> Any idea how or why this works (or doesn't)?...

What you've got there is a lower tetrachord consisting
of subharmonics 12-8, and an upper tetrachord consisting
of harmonics 9-12. Tetrachordal construction is a great
way to make scales, and taking one's tetrachords from
the series is a cracking good idea too. I don't think
you'll find it contains 6 triads as the diatonic scale
in meantone does, but you will be able to play a lot of
things that sound good. I've tried similar things in the
past but I don't see this scale in my archive, so I'll
add it.

-Carl

--- In tuning@yahoogroups.com, djtrancendance@... wrote:

>     So if I have it right
> 1) The note 9/8 is the only note in my scale not part of a
> tetrachord
> 2) The two tetrachords are on each side, each occupying the
> ratio 4/3 IE a perfect 4th.  Since 4/3 * 4/3 = 16/9 and 16/9
> times 9/8 = 2 it is able to fill an octave.

That's right. Most commonly, two tetrachords are put together
to make a 7-tone scale, with one rooted on 1/1 and the other
rooted on 3/2. Though sometimes the upper tetrachord is rooted
on 4/3 and the 9/8 gap is left on top as you suggest. And all
other manner of things are sometimes done.

> Interesting, so "that's all it is". Then again it seems
> increasingly apparent to me...the more you can ultimately
> simplify a scales formulaic composition (no matter how many
> steps that simplification takes), the more natural it sounds.

I think there are two primary kinds of simplicity worth
mentioning when it comes to scales:

There's something about the number of total number of intervals
in a scale. Usually one can just look at all the 2nds of the
scale (e.g. 9/8 10/9 etc.) and get a good idea of the total
number of unique intervals. A scale with only one kind of 2nd
can only have one kind of anything else (is an ET). With two
kinds of 2nd the order matters; LsLsLs... gives only 2 kinds
of 3rd, 4th, 5th, etc. but LLssLLss gives three kinds of 3rd
(2L, L+s, and 2s).

Scales with a lower number of intervals seem to me to have a
more "coherent" sound, for lack of a better word. Others have
noticed something similar... they seem to be more singable.
On the other hand, having a high number of different intervals
(like many JI scales) also gives a unique sound that is
certainly valid material for music-making.

The second kind of simplicity has to do with symmetry at the
3/2. Octave-equivalent scales are fully "closed" at the
octave. This means that if you sing a note an octave away
from a pitch in the scale you'll wind up on another pitch in
the scale. There's some evidence that a similar effect may
be important for the next most powerful consonance after
the octave, the 3/2. But no scale can be fully "closed" with
respect to both the octave and the 3/2 (this is equivalent
to some number of stacked fifths equally some number of
octaves, which we know is impossible). So octaves get
preference, and then we try to do as best we can with 3/2s.
A Pythagorean chain will be optimal here. After that, using
tetrachordal construction will give the best results.

>    I also wonder how Wilson's horogram scales work relative
> to these concepts and what they add...as, so far, they are by
> far the most "concordant" scales I have heard, more so than
> the MOS scales or virtually any of the others.

Er, well the horograms show MOS scales. :) It's just a way
of visualizing MOS.

-Carl