fifteen plus nine

on the 18th of april wallyesterpaulrus wrote :

> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> > Hello Jacques!
> > i am a bit comfused as to where the term Mohajira series comes
> >from. Is it your own or is it Persian term.
> > Erv use of these terms comes from a performance of Nabil
> >Assani's at the hollywood bowl museum in 1992 which happen to be a
> >show shared by all three of us.
> > Your own scale tackles a similar problem as allows has its own
> >properties that cycle at coincides at
> > 7 and 24 with Erv's. Page 2 also shows other places this interval
> >forms cycles (look at the denominator). Erv sees it cycling at 41
> >and 65 and beyond. not to mention the 10 and 17 tone scales which is
> >isolated as possibily important and useful. He also shows
> > how all these scales can be mapped onto a bosanquet keyboard.
>
> more information can be found here:
>
> http://x31eq.com/7plus3.htm

Thank you for the link. I found this page very interesting. Who is the author ? I don't see his
name.
One general remark :
There is not *one* spiral of neutral thirds, because there are several concepts of that interval,
and several philosophies about how to link them together.
We often see neutral thirds associated by pairs, producing pure fifths. That's the strong
spectral attractor with these intervals when chained. But neutral thirds can go also their own life

independantly of the fifths. This allows for more adaptive and consonant systems.
For instance,
"xy = always 3/2" systems may work fine with 10 and 17 tones scales (Erv's are a good example),
but I think not as good with 24 tones scales (and of course 31). I found more equality,
melodic coherence and consonance in 11-limit systems for 24 tones, when you allow for more
11/9 than 27/22 to fit in the cycle one way or another, such as fifteen for nine. It will end up
with
fifteen slightly larger quartertones (33/32) for nine slightly smaller (4096/3993 and 1331/1296)
as well, and in several orders will show remarquable differential coherence.
So this is like meantone, with a little bit of microtonality different spirals will find their way
for each purpose.

jacques dudon wrote:

> Thank you for the link. I found this page very interesting. Who is the author ? I don't see his
> name.

That's me. It does say in the meta tags.

> One general remark :
> There is not *one* spiral of neutral thirds, because there are several concepts of that interval,
> and several philosophies about how to link them together.
> We often see neutral thirds associated by pairs, producing pure fifths. That's the strong
> spectral attractor with these intervals when chained. But neutral thirds can go also their own life
> independantly of the fifths. This allows for more adaptive and consonant systems.

There's only one spiral the way I do them, at least when I wrote that page. If you look at a link Kraig gave recently, Erv Wilson has looked at those scales generated by alternating sizes of neutral thirds. These could also be looked at as two spirals of fifths.

> For instance,
> "xy = always 3/2" systems may work fine with 10 and 17 tones scales (Erv's are a good example),
> but I think not as good with 24 tones scales (and of course 31). I found more equality,
> melodic coherence and consonance in 11-limit systems for 24 tones, when you allow for more
> 11/9 than 27/22 to fit in the cycle one way or another, such as fifteen for nine. It will end up
> with
> fifteen slightly larger quartertones (33/32) for nine slightly smaller (4096/3993 and 1331/1296)
> as well, and in several orders will show remarquable differential coherence.
> So this is like meantone, with a little bit of microtonality different spirals will find their way
> for each purpose.

24 notes per octave gives a better 3/2 than 17 notes.

I have been looking at a guitar mapping for the "mystery" temperment, where the neutral thirds alternate between approximations to 11:9 and 16:13. Both are 17 steps of 58-equal, but the tuning is a pair of scales of 29-equal somewhere between 58- and 87-equal. One way, Mohajira comes easily to the fingers, as do 8:11:13 chords.

Graham

Graham Breed wrote :

> jacques dudon wrote:
>
> > One general remark :
> > There is not *one* spiral of neutral thirds, because there are several concepts of that interval,
> > and several philosophies about how to link them together.
> > We often see neutral thirds associated by pairs, producing pure fifths. That's the strong
> > spectral attractor with these intervals when chained. But neutral thirds can go also their own life
> > independantly of the fifths. This allows for more adaptive and consonant systems.
>
> There's only one spiral the way I do them, at least when I wrote that
> page. If you look at a link Kraig gave recently, Erv Wilson has looked
> at those scales generated by alternating sizes of neutral thirds. These
> could also be looked at as two spirals of fifths.
>
> > For instance,
> > "xy = always 3/2" systems may work fine with 10 and 17 tones scales (Erv's are a good example),
> > but I think not as good with 24 tones scales (and of course 31). I found more equality,
> > melodic coherence and consonance in 11-limit systems for 24 tones, when you allow for more
> > 11/9 than 27/22 to fit in the cycle one way or another, such as fifteen for nine. It will end up
> > with
> > fifteen slightly larger quartertones (33/32) for nine slightly smaller (4096/3993 and 1331/1296)
> > as well, and in several orders will show remarquable differential coherence.
> > So this is like meantone, with a little bit of microtonality different spirals will find their way
> > for each purpose.
>
> 24 notes per octave gives a better 3/2 than 17 notes.

Normally, but it depends if they follow cycles of 3/2 or not. My point was rather to say that concerning
chains of neutral thirds, there are different possible attractors :
1.224 744 871 - which I call "Ozir" is naturally the spectral-coherent one (perfect for overtone 3)
1.223 284 956 6 which I call "Mohajira" ... is better for differential coherence
17 (unequal) tones systems may sound well with Ozir,
24 (unequal) tones systems may sound even better, with a balance of Mohajira & Ozir
To explain briefly how Mohajira's differential coherence works, here is an example in a 11-limit type of
scale as the one I suggested :
36 44 54 66 81 99 121 ...(two consecutive 11/9 in the end) make 121 - 99 = 11 coherent, while
36 44 54 66 81 99 243 ...(always alternances of 11/9 and 27/22) keep the last 3/2 harmonic
For a guitar tuning, I would choose 121, of course.

--- In tuning@yahoogroups.com, jacques dudon <aeh@f...> wrote:
> Graham Breed wrote :
>
> > jacques dudon wrote:

> > 24 notes per octave gives a better 3/2 than 17 notes.
>
> Normally, but it depends if they follow cycles of 3/2 or not. My
point was rather to say that concerning
> chains of neutral thirds, there are different possible attractors :
> 1.224 744 871 - which I call "Ozir" is naturally the spectral-
coherent one (perfect for overtone 3)

ok, that's just the square root of 3/2 . . .

> 1.223 284 956 6 which I call "Mohajira" ... is better for
differential coherence
> 17 (unequal) tones systems may sound well with Ozir,
> 24 (unequal) tones systems may sound even better, with a balance of
Mohajira & Ozir

hmm . . . you should educate us some more on these entities . . .
graham however is no stranger to different "attractors" for the
neutral thirds generator -- for example on graham's page
http://x31eq.com/limit9.txt he finds that the third-best
linear temperament for the 9-limit has a 351.5 cent generator,
0.292907877459 octave or, as a ratio, 1.2251071, where prime 5 is
represented by 25 generators and prime 7 by 13 generators . . .
meanwhile, at http://x31eq.com/limit5.txt, there is a
different linear system, generator 348.1 cents = 0.290068685822
octave = ratio 1.2227, where prime 5 is represented by 8
generators . . .

wallyesterpaulrus a �crit :

> --- In tuning@yahoogroups.com, jacques dudon <aeh@f...> wrote:
> > concerning chains of neutral thirds, there are different possible attractors :
> > 1.224 744 871 - which I call "Ozir" is naturally the spectral-
> coherent one (perfect for overtone 3)
>
> ok, that's just the square root of 3/2 . . .
>
> > 1.223 284 956 6 which I call "Mohajira" ... is better for
> differential coherence
> > 17 (unequal) tones systems may sound well with Ozir,
> > 24 (unequal) tones systems may sound even better, with a balance of
> Mohajira & Ozir
>
> hmm . . . you should educate us some more on these entities . . .
> graham however is no stranger to different "attractors" for the
> neutral thirds generator -- for example on graham's page
> http://x31eq.com/limit9.txt he finds that the third-best
> linear temperament for the 9-limit has a 351.5 cent generator,
> 0.292907877459 octave or, as a ratio, 1.2251071, where prime 5 is
> represented by 25 generators and prime 7 by 13 generators . . .
> meanwhile, at http://x31eq.com/limit5.txt, there is a
> different linear system, generator 348.1 cents = 0.290068685822
> octave = ratio 1.2227, where prime 5 is represented by 8
> generators . . .

I have only been interested in attractors useful for differential coherence,
except like with Ozir, when it is difficult to escape close harmonicity.
But I am totally ignorant about the subject of these pages from Graham,
I am lacking many notions here.
On the other side,
1.226 884 113 3, which I call "Zal" appears with the fractal series
9 11 27 265 2601... that generates a second similar series -
5 49 481 ... so this creates a "Zal+Zal" system
that could be experienced in 17, 61, 139, 200 tones per octave systems.
It does not seem to have anything to see. Perhaps parallel worlds.

--- In tuning@yahoogroups.com, jacques dudon <aeh@f...> wrote:
> wallyesterpaulrus a écrit :
>
> > --- In tuning@yahoogroups.com, jacques dudon <aeh@f...> wrote:
> > > concerning chains of neutral thirds, there are different
possible attractors :
> > > 1.224 744 871 - which I call "Ozir" is naturally the spectral-
> > coherent one (perfect for overtone 3)
> >
> > ok, that's just the square root of 3/2 . . .
> >
> > > 1.223 284 956 6 which I call "Mohajira" ... is better for
> > differential coherence
> > > 17 (unequal) tones systems may sound well with Ozir,
> > > 24 (unequal) tones systems may sound even better, with a
balance of
> > Mohajira & Ozir
> >
> > hmm . . . you should educate us some more on these entities . . .
> > graham however is no stranger to different "attractors" for the
> > neutral thirds generator -- for example on graham's page
> > http://x31eq.com/limit9.txt he finds that the third-
best
> > linear temperament for the 9-limit has a 351.5 cent generator,
> > 0.292907877459 octave or, as a ratio, 1.2251071, where prime 5 is
> > represented by 25 generators and prime 7 by 13 generators . . .
> > meanwhile, at http://x31eq.com/limit5.txt, there is a
> > different linear system, generator 348.1 cents = 0.290068685822
> > octave = ratio 1.2227, where prime 5 is represented by 8
> > generators . . .
>
> I have only been interested in attractors useful for differential
coherence,
> except like with Ozir, when it is difficult to escape close
harmonicity.
> But I am totally ignorant about the subject of these pages from
Graham,
> I am lacking many notions here.

the 5-limit temperaments seek to approximate all ratios of odd
numbers through 5 (with arbitrary factors of 2 allowed too, of
course) . . . 9-limit, odd numbers through 9 . . . this is pretty
basic stuff, so we can discuss it here, though many of the more
mathematical discussions between graham, myself, and others occur on
the tuning-math list instead . . .

jacques dudon wrote :

> wallyesterpaulrus wrote :
>
> > --- In tuning@yahoogroups.com, jacques dudon <aeh@f...> wrote:
> > > concerning chains of neutral thirds, there are different possible attractors :
> > > 1.224 744 871 - which I call "Ozir" is naturally the spectral-
> > coherent one (perfect for overtone 3)
> >
> > ok, that's just the square root of 3/2 . . .
> >
> > > 1.223 284 956 6 which I call "Mohajira" ... is better for
> > differential coherence
> > > 17 (unequal) tones systems may sound well with Ozir,
> > > 24 (unequal) tones systems may sound even better, with a balance of
> > Mohajira & Ozir
> >
> > hmm . . . you should educate us some more on these entities . . .
> > graham however is no stranger to different "attractors" for the
> > neutral thirds generator -- for example on graham's page
> > http://x31eq.com/limit9.txt he finds that the third-best
> > linear temperament for the 9-limit has a 351.5 cent generator,
> > 0.292907877459 octave or, as a ratio, 1.2251071, where prime 5 is
> > represented by 25 generators and prime 7 by 13 generators . . .
> > meanwhile, at http://x31eq.com/limit5.txt, there is a
> > different linear system, generator 348.1 cents = 0.290068685822
> > octave = ratio 1.2227, where prime 5 is represented by 8
> > generators . . .
>
> I have only been interested in attractors useful for differential coherence,
> except like with Ozir, when it is difficult to escape close harmonicity.
> But I am totally ignorant about the subject of these pages from Graham,
> I am lacking many notions here.
> On the other side,
> 1.226 884 113 3, which I call "Zal" appears with the fractal series
> 9 11 27 265 2601... that generates a second similar series -
> 5 49 481 ... so this creates a "Zal+Zal" system
> that could be experienced in 17, 61, 139, 200 tones per octave systems.
> It does not seem to have anything to see. Perhaps parallel worlds.

Sorry, I forgot about
1.225802981476, between Ozir and Zal (let's call it Olzal...), interesting variation
because the differentials of the neutral thirds of the differentials of the neutral thirds
of the series come back to the original series :
series : 36 40 44 49 54 60 265 294 325...
differentials : 9 10 11 49 54 60 265 294 325...
(49 - 40 = 9 and so on) ; more fractal and much simpler also.
Could be well implemented in 7, 34, 143 tones per octave systems.
Still not certain it has something to see with Graham's.

jacques dudon wrote:

> But I am totally ignorant about the subject of these pages from Graham,
> I am lacking many notions here.

There's an explanation at

http://x31eq.com/temper/method.html

Graham