Two Gr-20 Improvisations

A couple of semi-improvisations in two (new to me) tunings - that is I found some chords and then improvised around them. The technique uses Fractal Tune Simthy as a relay tuner to my Roland GR-20.

The first uses Gene Ward Smith's
1/4-comma meantone with a period of 5^(1/4) and a tone generator - over a period of 5 octaves I believe - this is supposed to be a non-octave tuning. Gene is a nice guy and put it together for me - thanks Gene!! I had some issues with the Fractal Tune Smithy relay - so only the top 4 strings could be used for this piece. :-/

Title: A Quart of Genes

http://www.notonlymusic.com/board/download/file.php?id=222

The second uses Carl Lumma's 10 out of 13-tET MOS, TL 21-12-1999 => Actually done earlier and the relay + Gr-20 were acting nicer.

You will hear a bit of similarity in the motives of the two improvisations - a danger of improvising is getting in a rut and I sort of do here.

Title: 10 Lummas for the Price of 13

http://www.notonlymusic.com/board/download/file.php?id=221

--- In tuning@yahoogroups.com, "christopherv" <chrisvaisvil@...> wrote:
>
> A couple of semi-improvisations in two (new to me) tunings - that is I found some chords and then improvised around them. The technique uses Fractal Tune Simthy as a relay tuner to my Roland GR-20.
>
> The first uses Gene Ward Smith's
> 1/4-comma meantone with a period of 5^(1/4) and a tone generator - over a period of 5 octaves I believe - this is supposed to be a non-octave tuning. Gene is a nice guy and put it together for me - thanks Gene!!

You are welcome. Of course, the scale has lots of octaves but they aren't periods or generators.

ohh...

I thought this was a tuning that passed the octaves. I don't always trust my
ears, especially with alternate tunings, as to what intervals I'm using. I
go by the overall sound but w/o names.

Well, I do want to try something along the lines of say a JI tuning that has
no compromises - which means of course it includes the errors.
I'm thinking it might be possible to live without octave equivalence where
C5 is not octave equivalent to C4 - so if you want to return to tonic you'd
have to return to the original register.

Chris

On Fri, May 14, 2010 at 12:15 AM, genewardsmith <genewardsmith@...
> wrote:

>
>
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "christopherv"
> <chrisvaisvil@...> wrote:
> >
> > A couple of semi-improvisations in two (new to me) tunings - that is I
> found some chords and then improvised around them. The technique uses
> Fractal Tune Simthy as a relay tuner to my Roland GR-20.
> >
> > The first uses Gene Ward Smith's
> > 1/4-comma meantone with a period of 5^(1/4) and a tone generator - over a
> period of 5 octaves I believe - this is supposed to be a non-octave tuning.
> Gene is a nice guy and put it together for me - thanks Gene!!
>
> You are welcome. Of course, the scale has lots of octaves but they aren't
> periods or generators.
>
>
>

I ran across one of Jacky Ligon's scales on
http://xenharmonic.wikispaces.com/Superparticular-Nonoctave-MOS

It's an MOS of
9:8, 12:11, 9:8, 12:11, 9:8, 12:11, 12:11
...that has all perfect fifths except for one interval (1.44927..which still sounds quite consonant to my ears) and tons of other good intervals...the only one of which I didn't particularly like nears 17/10. Plus the 9:8, 12:11 combinations allow for lots of cool clustered "neutral second" chords along with "alternative 7th" chords. I also found if you replace the last 12:11 interval in his above scale with a 13:12 you essentially nail the 2/1 octave.

I jumped at the chance to try that scale as I've heard Ligon's music on Sevish's Split Notes label and consider it the best micro-tonal music I've heard so far. It's both incredibly consonant and with vast tonal color and expression...not to mention fantastic drum work. I had a funny feeling, even though he's a great composer, he also did so with the help of a great scale system.

Does anyone here know more about Jacky Ligon's scales?

On 14 May 2010 09:56, Michael <djtrancendance@...> wrote:
>
> I ran across one of Jacky Ligon's scales on
>   http://xenharmonic.wikispaces.com/Superparticular-Nonoctave-MOS
>
> It's an MOS of
>    9:8, 12:11, 9:8, 12:11, 9:8, 12:11, 12:11
> ...that has all perfect fifths except for one interval (1.44927..which still sounds quite consonant to my ears) and tons of other good intervals...the only one of which I didn't
<snip>

It looks like Mohajira to me. Stretched a bit. Why didn't you like
Mohajira before?

There should be two imperfect fifths.

Graham

Graham>"It looks like Mohajira to me. Stretched a bit. Why didn't you like Mohajira before?"
I've only had time to try the 24tet version so far...which I know is not the ideal temperament for it.

[This message is best read, if you are on the Yahoo
site, using the "Fixed Font Width" option.]

Dear Michael and Graham,

Please let me clarify, as a longtime friend and admirer
of Jacky Ligon's artistry as a tuning designer and
performer, that his superparticular JI scheme we're
discussing is quite distinct from "Mohajira" either
as the name of a Near Eastern mode evidently like
the medieval `Iraq (J T J - T - J T J), or from a
type of generator for a generally tempered kind of
tuning based on a neutral third at or near precisely
half the size of a fifth, as in 17-EDO, 24-EDO,
31-EDO, 1/4-comma meantone, etc.

One use of "Mohajira" on this group, for example by
Jacques Dudon, relates to a mode J T J - T - J T J
where T is a whole tone (typically around 9:8), and
J a type of interval usually assumed to be a neutral
second step in this context, although in medieval
Near Eastern theory it might also be interpreted
as a Pythagorean apotome or diminished third very
close respectively to 16:15 or 10:9. In either
reading, this would be the same as the medieval
Maqam `Iraq, which the Lebanese composer and scholar
Nidaa Abou Mrad gives in one likely interpretation as
follows, based on al-Farabi's neutral second steps
of 12:11 and 88:81.

`Iraq tone `Iraq
|---------------------| |------------------|
1/1 12/11 27/22 4/3 3/2 18/11 81/44 2/1
0 151 355 498 702 853 1057 1200
12:11 9:8 88:81 9:8 12:11 9:8 88:81
151 204 143 204 151 204 143

Note that this form of medieval `Iraq or Mohajira has
three sizes of intervals: a tone (here 9:8), a larger
neutral second at 12:11, and a smaller one at 81:44.
The two neutral seconds add up to a regular minor third,
which in this context where Pythagorean tuning is assumed
is 32:27 or 294 cents. I should clarify that in this
just system based on ratios of 2-3-11, the Pythagorean
chain of fifth does _not_ generate the neutral intervals,
which are introduced as independent factors.

In a medieval context, Mrad and I agree that the relevant
source appears to indicate that the larger neutral second
step comes first in a tetrachord. In a modern context,
however, `Iraq is a general term for a tetrachord on this
pattern of J T J. The generic system of modern Arab theory
often expresses this in terms of a 24-step system as 3 4 3,
not necessarily implying equal steps but simply identifying
the types of intervals as a tone "4" and neutral second "3".
Using the 53-comma system often favored in Turkey and Syria,
we could write 7 9 6 or 6 9 7 to indicate whether a larger
or smaller neutral second step comes first -- or in Ozan
Yarman's 79MOS-of-159 notation, 10 14 9 or 9 14 10.

To understand Jacky Ligon's ingenious superparticular scale,
we must consider a related form discussed by Erv Wilson
in his _Rast-Bayyati Matrix_ paper. Suppose we generate
a tuning by chaining two _unequal_ neutral thirds, the
sizes above of 12:11 and 27:22 given by al-Farabi as an
interpretation of Mansur Zalzal's famous neutral third
fretting for the `oud or lute in the 8th century. We get
this tuning, starting from the 1/1 of the chain:

"Zalzalian Penchgah" Rast
|--------------------------------|--------------------|
1/1 9/8 27/22 243/176 3/2 27/16 81/44 2/1
0 204 355 558 702 906 1057 1200
9:8 12:11 9:8 88:81 9:8 12:11 88:81
204 151 204 143 204 151 143

Here the pattern we get is T J T J - T J J, which borrowing
a Turkish usage might be called a Zalzalian Maqam Penchgah,
with a lower pentachord alternating tones and neutral seconds,
and an upper Rast tetrachord with a tone followed by a larger
12:11 and smaller 88:81 neutral second. Note, again, that
there are _three_ interval sizes, with 12:11 plus 88:81 equal
to 32:27.

(Note that the modern Turkish Penchgah as reported by Ozan
Yarman is different, with "J" equal to 10:9 or 16:15, as also
happens in a common form of Turkish Rast, so that the maqam
is taken as ~1/1-9/8-5/4-45/32-3/2-27/16-15/8-2/1, with a
suggested nuancing of the 45/32 step to a nearby 7/5. For
our present purposes, however, it is the neutral third
version that is relevant in relation to Jacky's tuning.)

In Scala terms, this could also be called a tritriadic tuning
of the 22:27:33 sonority: it's a JI scheme distinct from any
tempered tuning using a single neutral third generator.

The key to Jacky's ingenious variation is that the comma or
kleisma between al-Farabi's two neutral second steps at
12:11 (151 cents) and 88:81 (143 cents), and likewise between
his neutral thirds at 27:22 (355 cents) and 11:9 (347 cents),
is rather small: 243/242 or 7.139 cents.

Thus, a natural given Jacky's love of superparticular JI steps
and stretched or compressed octave tunings (the latter also
arising with TOP temperaments, of course), he hit on his
clever takeoff on al-Farabi, whether or not he conceived of
it in those historical terms:

"Ligonian Penchgah" "Ligonian Rast"
|--------------------------------|-----------------------|
1/1 9/8 27/22 243/176 729/484 6561/3872 19683/ 59040/
10648 29282
0 204 355 558 709 913 1064 1214
9:8 12:11 9:8 12:11 9:8 12:11 12:11
204 151 204 151 204 151 151

Here we have only _two_ step sizes, the 9:8 tone and 12:11 neutral
second -- so that the fifth gets stretched by 7.139 cents, almost
identical to 13/22 octave! -- and the octave by twice this amount.

While the concept here is JI, a curious aside would note that
a 27/22 generator produces a tuning virtually identical to 44-EDO,
which unlike 22-EDO has a complement of neutral intervals. While
all neutral thirds will be precisely or virtually equal to 27/22,
we will have two _unequal_ neutral seconds together adding up to
a minor third of precisely 300 cents (11/44 octave), the larger
at 164 cents or 6/44 octave (an interval found in 22-EDO), and
the smaller at 136 cents or 5/44 octave.

In short, JI tunings and TOP temperaments or the like are distinct
approaches, although they may sometimes yield curiously similar
results. Anyway, when I look at Jacky's stretched octave tuning
with its all-superparticular steps, I would think back to Zalzal
and al-Farabi, with some influence from stretched octave styles
such as gamelan a not unlikely factor.

Most appreciatively,

Margo Schulter
mschulter@...

Margo>"While all neutral thirds will be precisely or virtually equal to 27/22,
we will have two _unequal_ neutral seconds together adding up to
a minor third of precisely 300 cents (11/44 octave)"

This is where I think the ingenuity really hits...good smaller interval "adding up" to good mid and large sized ones that are pretty much uniform across the scale. And while one or two off the fifth-ish interval are off (around 1.4444 or 13/9)...they aren't "wolf" by any means and most of the other intervals sound incredibly confident and expressive.

>"The key to Jacky's ingenious variation is that the comma or
kleisma between al-Farabi's two neutral second steps at
12:11 (151 cents) and 88:81 (143 cents), and likewise between
his neutral thirds at 27:22 (355 cents) and 11:9 (347 cents),
is rather small: 243/242 or 7.139 cents."
Right so, as I understand it, he manages to reap both a lot of the purity of a JI scale while keeping a lot of the constant interval size-like nature of TET scales.

To my ears, Jacky's scale there is a bit more useful than Ptolemy's Homalon scale...which in turn I believe is a fair bit more useful to Pythagorus' scale.
In both Jacky and Ptolemy's scale I find the neutral seconds and new 7th interval's ability to make tons of new chords well worth the loss of a mere two perfect fifths...and in fact a wider variety of total chords possible (or so I've found by composing with them).

If you're not obsessed with getting near-perfect standard (IE using near-perfect 5ths) triads almost everywhere I highly recommend both Ptolemy's and Jacky's scales.
And Margo I agree, Jacky's scale is an original and not any sort of deliberate Arab scale knock-off or "tweak". It's a shame that when a few characteristics of a scale are similar to a historical one people say "all you're doing a slight variation of the 'real' historic version of the scale"...which IMVHO may explain why so few people seem to be bothering with creating new scales that don't deliberately model themselves off older ones. Needless to say, I highly respect those who even at least try to make new scales and again say I think Jacky's skill with scales has a profound effect on adding a great deal of originality on top of his/?her? already superb composition skills.

Again I ask though...is that scale his only major invented/new scale? Because if he has others I'm dying to try them out. :-)

_,_._,___

On 15 May 2010 02:22, Margo Schulter <mschulter@...> wrote:

> Please let me clarify, as a longtime friend and admirer
> of Jacky Ligon's artistry as a tuning designer and
> performer, that his superparticular JI scheme we're
> discussing is quite distinct from "Mohajira" either
> as the name of a Near Eastern mode evidently like
> the medieval `Iraq (J T J - T - J T J), or from a
> type of generator for a generally tempered kind of
> tuning based on a neutral third at or near precisely
> half the size of a fifth, as in 17-EDO, 24-EDO,
> 31-EDO, 1/4-comma meantone, etc.

Right, so how is it different? It's a different mode. It starts on a
different note. But it is generated by a neutral third (of exactly
half the size of a fifth) and an octave, both of which happen to be
impure (but just). That's what we've been calling Mohajira for all
those years.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> Right, so how is it different? It's a different mode. It starts on a
> different note. But it is generated by a neutral third (of exactly
> half the size of a fifth) and an octave, both of which happen to be
> impure (but just). That's what we've been calling Mohajira for all
> those years.

To start out with, we have a scale with two generators, 12/11 and 9/8, and octaves tuned sharp by *two* 243/242, fifths sharp by one. So we could just call it rank two and leave it at that, but if we take the above approximations seriously, then we have two generators in {2,3,11} and one comma, and hence a rank one {2,3,11} temperament. Taking a look at it, we discover that in terms of octave-and-generator, it's got a generator of 11/9. So by the regular mapping paradigm, at any rate, it's a strangely tuned mohajira scale. But that paradigm wields a pretty blunt scalpel when it comes to analyzing tuning issues, since it isn't actually about tuning at all.

Graham wrote:

> Right, so how is it different? It's a
> different mode. It starts on a different
> note. But it is generated by a neutral
> third (of exactly half the size of the
> fifth) and an octave, both of which happen
> to be impure (but just). That's what we've
> been calling Mohajira for all those years.

Dear Graham,

Please let me begin by agreeing that you
certainly _can_ stretch your concept of
"Mohajira" under your regular mapping
model or whatever so as to arrive at a
result similar or identical to Jacky's
superparticular JI stretched-octave
tuning.

However, it seems to me that do so and
create your very interesting replication,
you must adopt his motivations and then
reframe them to fit your approach. Thus
you are emulating his concept, rather than
vice versa -- and in a very interesting way!

First of all, I see the use of the term
"Mohajira" for a neutral third generator,
as opposed to a kind of Near Eastern tetrachord
or maqam, as an idiom specific to a group of
people, of whom you are a very noteworthy member,
exploring these "regular mappings." I might speak
simply of a "hemififth" generator or the like.

Secondly, terminology quite aside, let's consider
what happens if we build a tuning based on a
regular generator of 27/22, repeating it, say,
to form a 7-note tuning. We won't get either
of Jacky's steps, 9:8 or 12:11, but rather
steps of around 218 and 136 cents, or rather
to close to 17:15 and 13:12, for example.

Yes, you can then stretch the octave by two
243:242 commas, take that as the periodicity,
and arrive at Jacky's result. But would you
normally do so under your paradigm, even a
TOP version? Looking at Paul Erlich's article
on "A Middle Path" (_Xenharmonikon_ 18), I
see that TOP adjustments are generally rather
smaller than 14 cents. Of course, you _can_
do it, and it's a very instructive exercise;
but no substitute, I would, say, for Jacky's
original JI conception.

My own reaction on seeing Jacky's tuning is
rather like one of the models offered by Gene:
we have in effect two basic generators, 9:8
and 12:11. Those two suffice eloquently to
explain the tuning.

More generally, I would say that any person
or group exploring the interval spectrum will
be likely to take different views of things,
and to use different language to describe some
of these things.

Thus Jacky's tuning immediately reminded me
of Zalzal, or more specifically of al-Farabi's
version of his lute tuning, rather than of some
regular temperament derived from a 27/22 neutral
third plus a stretched octave. They are both
valid worldviews, and mutual recognition of the
validity of each seems to me the best conclusion.

With many thanks,

Margo Schulter
mschulter@...

On 17 May 2010 03:07, Margo Schulter <mschulter@...> wrote:

> Please let me begin by agreeing that you
> certainly _can_ stretch your concept of
> "Mohajira" under your regular mapping
> model or whatever so as to arrive at a
> result similar or identical to Jacky's
> superparticular JI stretched-octave
> tuning.

My concept is of a neutral third MOS, which seems to match Jacques
Dudon's concept of Mohajira. It doesn't need regular mappings. It
could give you exactly Jacky's tuning.

> However, it seems to me that do so and
> create your very interesting replication,
> you must adopt his motivations and then
> reframe them to fit your approach. Thus
> you are emulating his concept, rather than
> vice versa -- and in a very interesting way!

Well, what were his motivations? He's deleted the original message
from the archives. But here I read it and say he was talking about a
neutral third MOS:

/makemicromusic/topicId_1827.html#1902

It looks like we have essentially the same concept.

The message we still have from Jacky is here:

http://www.nonoctave.com/forum/messages/318.html?n=18

He calls his scale a "variation on a theme" where the theme is neutral
third MOS scales.

> Yes, you can then stretch the octave by two
> 243:242 commas, take that as the periodicity,
> and arrive at Jacky's result. But would you
> normally do so under your paradigm, even a
> TOP version? Looking at Paul Erlich's article
> on "A Middle Path" (_Xenharmonikon_ 18), I
> see that TOP adjustments are generally rather
> smaller than 14 cents. Of course, you _can_
> do it, and it's a very instructive exercise;
> but no substitute, I would, say, for Jacky's
> original JI conception.

Whether I'd normally do it or not doesn't stop it belonging to a
particular type of scales. It's clearly a tuning of Mohajira with a
stretched octave. A 14 cent stretch is consistent with observed
scales in the Scala archive. This particular tuning is clearly
Jacky's.

Note that the piece it comes from uses bell timbres, apparently chosen
to fit this scale. So TOP is irrelevant.

What I originally gave Michael, and that he claims to not like, was
the tuning to 24-equal. The steps Jacky chose are both close to that.
Of course the error accumulates as you build up an octave.

> My own reaction on seeing Jacky's tuning is
> rather like one of the models offered by Gene:
> we have in effect two basic generators, 9:8
> and 12:11. Those two suffice eloquently to
> explain the tuning.

Yes, you can use different pairs of generators to define an MOS.

> More generally, I would say that any person
> or group exploring the interval spectrum will
> be likely to take different views of things,
> and to use different language to describe some
> of these things.

Yes, the language may be different, but the content will be the same.
This is an MOS of the type we call Mohajira with a stretched octave
and rational scale steps.

> Thus Jacky's tuning immediately reminded me
> of Zalzal, or more specifically of al-Farabi's
> version of his lute tuning, rather than of some
> regular temperament derived from a 27/22 neutral
> third plus a stretched octave. They are both
> valid worldviews, and mutual recognition of the
> validity of each seems to me the best conclusion.

The old page where I talk about these things does cover tetrachords:

http://x31eq.com/7plus3.htm#tetra

Regardless of the MOS concept, there are very few ways of constructing
tetrachordal scales with only two step sizes. They are the usual
(meantone) diatonics, Mohajira, a certain interpretation of Rast with
different tonics, and an equable diatonic where you divide the fourth
into equal steps. So there are, indeed, different worldviews in which
these scales look important.

Jacky happened to choose a mode that emphasizes neutral thirds, not tetrachords.

Graham

Dear Margo,

I never heard about thoses scales from Jacky Ligon and I am off topic
here,
but there are a few things I must clarify about "Mohajira", since I
have my part in the confusion !
When I created my first photosonic disks based on chains of neutral
thirds (that was in 1985),
I felt these scales had to be traditionally from somewhere. But I
couldn't find them in usage neither in Arabic nor in Persian music
then. The closest I could find was with the Persian Dastgah-e Segâh,
where the seventh degree was quite lower, and listening even more
closely to several records lately I still think Dastgah Segah follows
a much more complex model.
But perhaps I missed it in Arabic music, did you mentionned "Irak" to
have the same tetrachordal structure ?
(then this would be a beautiful coïncidence because my piece "Sumer",
started in 1991, that uses intensively the heptatonic form of
Mohajira, was inspired by visions I received during the first Gulf
war...)
Then I tried to look for more ancient sources and I asked John
Chalmers, among others, about it.
John pointed to me a tetrachord (#458) from Ibn Sina :
1/1 13/12 39/32 4/3
(13/12 - 9/8 - 128/117)
= the "s L s" tetrachord I was looking for ! (the only one that can
generate the heptatonic form of Mohajira).
But still I had no name for it and Mohajira (= "migratory" in
Arabian) was clearly what the heptatonic versions expressed for me,
so this is how I described it, close to the Persian "Segah", of the
form [s L s L s L s] and generated by a succession of six WustaZalzal.
Years later when I joined the Tuning list I found that people were
using that name and thought it was traditional... I thought it was
funny, but I had to rectify, because Mohajira was only my own naming.
Anyway, neither Graham nor others are responsible of this, and if I
can have the possiblity myself to hear what I mean by a Mohajira
heptaphone in a Maqam where it has a traditional name, first I will
be glad to hear it and then I would not use the name "Mohajira" again
for such heptatonic scales. So may be you, Ozan, or others can help
(I think you mentionned "Mustaqim" for this same Ibn Sina's
tetrachord lately ? and would "Irak" be the same ?).
Also, you mentionned an article you wrote on the subject, but I
haven't read it. Can we find it somewhere ?

Now, as I said not long ago, "Mohajira" means many things for me, not
to forget a generator of the half of a meantone fifth, or of the
triple of a Secor, and certainly that the notion of "a linear
temperament that leads to 24 > 31 > 55 tones per octave mapped <0, 2,
8, -11, 5, -1, -10, -6], of which one interesting generator is
1.2232849566 or 348.91261178844 c., the solution of the recurrent
sequence x^5 - x^4 = 1/2"... should not interfere too much with any
traditional knowledge at least !
- - - - - -
Jacques

Margo wrote :

> Graham wrote:
>
> > Right, so how is it different? It's a
> > different mode. It starts on a different
> > note. But it is generated by a neutral
> > third (of exactly half the size of the
> > fifth) and an octave, both of which happen
> > to be impure (but just). That's what we've
> > been calling Mohajira for all those years.
>
> Dear Graham,
>
> Please let me begin by agreeing that you
> certainly _can_ stretch your concept of
> "Mohajira" under your regular mapping
> model or whatever so as to arrive at a
> result similar or identical to Jacky's
> superparticular JI stretched-octave
> tuning.
>
> However, it seems to me that do so and
> create your very interesting replication,
> you must adopt his motivations and then
> reframe them to fit your approach. Thus
> you are emulating his concept, rather than
> vice versa -- and in a very interesting way!
>
> First of all, I see the use of the term
> "Mohajira" for a neutral third generator,
> as opposed to a kind of Near Eastern tetrachord
> or maqam, as an idiom specific to a group of
> people, of whom you are a very noteworthy member,
> exploring these "regular mappings." I might speak
> simply of a "hemififth" generator or the like.

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> John pointed to me a tetrachord (#458) from Ibn Sina :
> 1/1 13/12 39/32 4/3
> (13/12 - 9/8 - 128/117)
> = the "s L s" tetrachord I was looking for ! (the only one that can
> generate the heptatonic form of Mohajira).

Looking at it from the point of groups, it generates the entire {2,3,13} group from this tetrachord alone, without the need of anything more. I'm less happy about it generating the name "mohajira", as dividing the fifth into 39/32 and 16/13 parts isn't quite what I have in mind with the term myself. I'd like to get a consensus about what "mohajira" should mean to the tuning community.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@> wrote:
>
> > John pointed to me a tetrachord (#458) from Ibn Sina :
> > 1/1 13/12 39/32 4/3
> > (13/12 - 9/8 - 128/117)
> > = the "s L s" tetrachord I was looking for ! (the only one that can
> > generate the heptatonic form of Mohajira).
>
> Looking at it from the point of groups, it generates the entire {2,3,13} group from this tetrachord alone, without the need of anything more.

Interesting, you have to explain more about that.

> I'm less happy about it generating the name "mohajira", as dividing the fifth into 39/32 and 16/13 parts isn't quite what I have in mind with the term myself. I'd like to get a consensus about what "mohajira" should mean to the tuning community.

No idea what "Mohajira" means to the tuning community, perhaps rather a chain of neutral thirds ?
But one thing I should say about 13 and a "Mohajira sequence" :
13, it's true, is perhaps the prime that's approached the worse way, in a "Mohajira temperament" (by -1 generator, but as you said, with some approximation). But on the other hand, it is a necessary basic ingredient to start a differentially-coherent sequence such as :
[18 22 26 32 39 48 59 72 88 >>...
where x^5 - x^4 = 1/2, that will much later on converge to closer and closer semififths.
From my own musical experience, I played many of those scales and the chords based on "not so symmetrical" factors like 26 : 32 : 39 sound better than more symmetrical ones like 40 : 49 : 60, and 18 : 22 : 27 is in between. So I'm not surprised that many traditional Middle-East scales, such as the ones Margo reported, make use of factor 13.

--- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@...> wrote:

> But one thing I should say about 13 and a "Mohajira sequence" :
> 13, it's true, is perhaps the prime that's approached the worse way, in a "Mohajira temperament" (by -1 generator, but as you said, with some approximation). But on the other hand, it is a necessary basic ingredient to start a differentially-coherent sequence such as :
> [18 22 26 32 39 48 59 72 88 >>...
> where x^5 - x^4 = 1/2, that will much later on converge to closer and >closer semififths.

I don't know how to calculate this sequence further- what are the figures when taken out to 17 tones?

-Cameron Bobro

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "jacques.dudon" <fotosonix@> wrote:
>
> > But one thing I should say about 13 and a "Mohajira sequence" :
> > 13, it's true, is perhaps the prime that's approached the worse way, in a "Mohajira temperament" (by -1 generator, but as you said, with some approximation). But on the other hand, it is a necessary basic ingredient to start a differentially-coherent sequence such as :
> > [18 22 26 32 39 48 59 72 88 >>...
> > where x^5 - x^4 = 1/2, that will much later on converge to closer and >closer semififths.
>
> I don't know how to calculate this sequence further- what are the figures when taken out to 17 tones?

The polynomial x^5 - x^4 - 1/2 is the key. It is a "characteristic polynomial" for the linear homogenous recurrence relationship with constant coefficients given above, meaning Jacques gave the first few terms of it, and the polynomial tells us that if A[n] is the nth term, then A[n] = A[n-1] + A[n-5]/2. The sequence thus continues
215/2, 263/2, 161, 197, 241, 1179/4 ... All of the roots except for the unique real root are less than one in absolute value, which means that the recurrence is dominated by this root (like the Fibonacci sequence) and the ratios between successive terms quickly approach the real root. That root is the square root of a meantone generator, and hence is a mohajira generator--and, as it happens, a good one.

Dear Jacques and all,

Thank you for your moving explanation of how you came
to use the term Mohajira, a usage I warmly embrace and
hope that everyone will respect and honor. It is often
the nature of musical terms, of course, to be take on
a range of meanings, so my purpose is simply to thank
you for a most illuminating explanation and to clarify
a few points to my best knowledge. To answer some of
the questions you raise, we need to turn to the
relevant text of Ibn Sina, or to a reliable translation,
with d'Erlanger as one candidate.

My point of uncertainty, not having such a translation
at hand, is whether Ibn Sina himself discussed the
specific permutation 13:12-9:8-128:117, or someone
else derived this from his Mustaqim, which I have read
is 9:8-13:12-128:117. John Chalmers quotes the first
form, and I am not sure if Ibn Sina himself states this
permutation or if Chalmers or someone else derives it
from what Ibn Sina terms Mustaqim, perhaps following
the custom of tending to put the smallest interval of a
tetrachord first, for example.

Indeed I agree that if we take Segah Dastgah from the
step now considered a neutral third below the usual final,
but sometimes taken as the 1/1 in reckoning the intervals
of this dastgah (e.g. segah2.scl in the Scala archives),
we have Ibn Sina's Mustaqim!

Mustaqim Mustaqim tone
|------------------|-------------------| |
1/1 9/8 39/32 4/3 3/2 13/8 16/9 2/1
0 204 342 498 702 841 996 1200
9:8 13:12 128:117 9:8 13:12 128:117 9:8
Farhat: 0 200 335 495 700 835 995 1200
segah2: 0 200 340 500 700 840 1000 1200

Now let us compare this to Zalzal's tuning as described
somewhat earlier by al-Farabi, a form which might now be
called a conjunct Maqam Rast:

Rast Rast tone
|------------------|-------------------| |
1/1 9/8 27/22 4/3 3/2 18/11 16/9 2/1
0 204 355 498 702 853 996 1200
9:8 12:11 88:81 9:8 12:11 88:81 9:8

It is curious that Ibn Sina is the first to use the term
Mustaqim (Arabic for the "correct" or "standard" form)
for a tetrachord with lower tone and two neutral seconds,
while Arab usage applies the equivalent Persian term Rast!

As you observe, Ibn Sina's steps of 13:12 and 128:117 differ
more decidedly at 139 and 155 cents than al-Farabi's 12:11
and 88:81 at 151 and 143 cents. The other distinction is
that in al-Farabi's tuning of a type later called Rast,
the larger neutral second step is placed before the smaller
(151-143), while the converse applies in Mustaqim (139-155).

Another aspect of this latter distinction is that even
today, a larger neutral third above the final is
characteristic of Arab Rast, and a smaller neutral third
of Persian modal systems such as Afshari (often F-Ap), or
indeed some gusheh-ha of Mahur where the final of C may
have an inflected Ep above it.

From the viewpoint of permutations, it then would be natural
to associate Ibn Sina's Mustaqim with a tetrachord like
1/1-13/12-39/32-4/3 in Chalmers, or 13:12-9:8-128:117, where
again we have the smaller neutral second placed first. However,
if we derive a tetrachord of the J-T-J type ("J" here meaning
a neutral second) by rotation, then the larger 128:117 actually
comes first, as may be seen by looking at the tetrachord above
the third step, typically the final of a modern Segah Dastgah:

1/1 9/8 39/32 4/3 3/2 13/8 16/9 2/1
0 204 342 498 702 841 996 1200
|--------------------|
1/1 128/117 16/13 4/3
0 155 359 498
128:117 9:8 13:12
155 204 139
segah2.scl 160 200 140
Farhat 160 205 135

The same kind of thing happens with al-Farabi's tuning
later called Rast. It would be natural, thinking in terms
of permutation, to associate 9:8-12:11-88:81 with a
tetrachord with 9:8 as the central step where the larger
12:11 precedes the smaller 88:81, thus 1/1-12/11-27/22-4/3
(0-151-355-498 cents or 151-204-143 cents).

Indeed, in the later 13th century, roughly three centuries
after al-Farabi and some two after Ibn Sina, Safi al-Din
al-Urmawi describes a tetrachord `Iraq in which we have
J T J, and a mode `Iraq with two such tetrachords. However,
there are some questions of interpretation. Taken literally
as transcribed by Dr. Fazli Arslan, we have two _conjunct_
tetrachords of a 5-based structure 1/1-10/9-5/4-4/3, with
"J" representing 10:9 or 16:15 rather than a neutral second
step of the kind used by al-Farabi or Ibn Sina. Here I ignore
the 1.95-cent schisma (32805:32768) in Safi al-Din's 17-note
Pythagorean scheme used for some of his tunings:

`Iraq `Iraq tone
|------------------|-------------------|
1/1 10/9 5/4 4/3 40/27 5/3 16/9 2/1
0 182 386 498 680 884 996 1200
10:9 9:8 16:15 10:9 9:8 16:15 9:8

The Lebanese scholar Nidaa Abou Mrad offers a very
appealing Zalzalian tuning, taking two `Iraq tetrachords
based on a permutation of al-Farabi's tuning,
i.e. 12:11-9:8-88:81, and deeming these to genres to
be "separated by a whole step," i.e. disjunct, or
J T J - T - J T J, the basic form of your Mohajira:

`Iraq tone `Iraq
|------------------| |---------------------|
1/1 12/11 27/22 4/3 3/2 18/11 81/44 2/1
0 151 355 498 702 853 1057 1200
12:11 9:8 88:81 9:8 12:11 9:8 88:81
151 204 143 204 151 204 143

If we adopt this reading of the tetrachord but a
conjunct arrangement of the kind often favored by
medieval Islamic sources -- as in al-Farabi's tuning
later called Rast, or Ibn Sina's Mustaqim -- then we
get:

`Iraq `Iraq tone
|------------------|--------------------| |
1/1 12/11 27/22 4/3 16/11 18/11 16/9 2/1
0 151 355 498 649 853 996 1200
12:11 9:8 88:81 12:11 9:8 88:81 9:8
151 204 143 151 204 143 204

It is interesting that in modern Arab theory, an
`Iraq tetrachord means the generic arrangement "3 4 3"
in the 24-step notation. However, a caution is
necessary: while this continues to be a theoretical
construct, the musically most salient concept focuses
on the lower trichord of "3 4" known as Sikah, and
common to the Sikah family of maqamat (the same term
as Persian Segah, i.e. the "third" step).

Looking at the modern Sikah family sharing the theme
of the trichord derived by rotation, i.e. by starting
on the third step of Rast, we find that in fact it is
the _smaller_ neutral second that precedes the larger,
more as in a permutation of Ibn Sina's Mustaqim, although
with varying degrees of contrast between the two neutral
steps depending on local intonation practices in different
parts of the Arab world. Let us first consider al-Farabi's
tuning:

Rast Rast tone
|------------------|-------------------| |
1/1 9/8 27/22 4/3 3/2 18/11 16/9 2/1
0 204 355 498 702 853 996 1200
9:8 12:11 88:81 9:8 12:11 88:81 9:8
`Iraq
|--------------------|
1/1 88/81 11/9 4/3
0 143 347 498
88:81 9:8 12:11
143 204 151

Here the smaller neutral third and tone make up a Sikah
trichord, with the larger neutral third completing the
theoretical tetrachord of `Iraq, which may indeed be
found in modern Maqam `Iraq. However, the usual modern
interpretation of this Maqam divides the genera or ajnas
(Arabic plural of jins or "genus") otherwise:

Sikah Bayyati Rast
|--------------|-----------------------|------------|
1/1 88/81 11/9 4/3 352/243 44/27 11/6 2/1
0 143 347 498 642 845 1049 1200
88:81 9:8 12:11 88:81 9:8 9:8 12:11
|---------------------|
`Iraq

In a usual account, we have a Sikah trichord, Bayyati
tetrachord, and Rast trichord. However, the `Iraq tetrachord
is certainly recognized in current theory, and is there
if we wish to celebrate it and explore it, following the
adage: "Follow the ajnas." If we do so, then raising the
fifth step of Maqam `Iraq to a 3/2 would produce a disjunct
Mohajira. And in a more "contrasty" tuning of Maqam `Iraq,
we would get a shade of intonation more like the Chalmers
permutation of Ibn Sina's Mustaqim (whatever its original
source), as in my regular temperament at 704.607 cents:

Sikah Bayyati Rast
|--------------|-----------------------|------------|
0 132 341 495 628 837 1046 1200
132 209 154 132 209 209 154
|---------------------|
`Iraq

and by raising the fifth degree:

Sikah Mrad's `Iraq Rast
|--------------|-----------------------|------------|
0 132 341 495 705 837 1046 1200
132 209 154 209 132 209 154
|---------------------| |-------------------|
`Iraq tone `Iraq

Here the Maqam I suspect we may both be delighted to call
Mohajira with its disjunct `Iraq (or Mohajira) tetrachords
is shown below the degrees and adjacent steps in cents, while a trichord-tetrachord-trichord division fitting
the modern pattern of a lower Sikah and upper Rast is
shown above. Because the name "Maqam `Iraq" is now so
widely understood to mean the previous form with a
middle Bayyati tetrachord, Maqam Mohajira seems to
a fitting name, with an explanation that this may be
synonymous with a 13th-century interpretation of
Maqam `Iraq as suggested by Mrad. A neat feature of
our Maqam Mohajira is that it includes in the
trichord-tetrachord-trichord interpretation a middle
tetrachord like that of Mrad, with with smaller neutral
second, tone, and larger neutral second (here 154-209-132)!

Apart from Mrad's medieval Maqam `Iraq, is there any strong
precedent for our `Iraq or Mohajira tetrachord as a focal
unit? Curiously, an example that stands out to me is an
intonation of Dastgah-e Bayat-e Esfahan sometimes known as
the "Old Esfahan," with small neutral second, 9:8 tone, and
large neutral second leading up to the final. As I tune
this interpretation:

|--------------------|
F# G* A* B
0 132 341 495
132 209 154

Indeed I would be very comfortable calling this Mohajira,
by contrast with the Buzurg which would result if I
should raise A* by a comma to Bb at 363 cents.

Of course, seeing exactly what Ibn Sina himself said
about Mustaqim and any of its permutations he may have
mentioned would permit a more informed discussion on
the question of the earliest mention of our Mohajira;
in my view, Safi al-Din's `Iraq tetrachord of J T J
qualifies the moment anyone reads this in a Zalzalian
intonation, as Mrad suggests likely happened at the
time or soon after. Would John Chalmers, Ozan, or
anyone here have convenient access to Ibn Sina's
text or a translation?

And whether or not Ibn Sina himself mentioned this
permutation of his Mustaqim or not, associating
permutations of a given tetrachord with the person
describing or advocating that tetrachord is a very
common practice, so your beautiful narrative can
stand, Jacques, regardless of this detail.

If you asked me for a definition of Mohajira, I might
propose the following:

The term Mohajira, literally "migratory," can refer
to either a type of tetrachord or mode; or to a method
of scale generation chaining either a single neutral
third or two unequal neutral thirds, with a 7-note
chain producing a Mohajira mode. The term was
originated by Jacques Dudon.

1. A tetrachord with neutral second, tone around 9:8,
and neutral second, sometimes specifically applied
to a shade of intonation like 1/1-13/12-39/32-4/3
(0-139-342-498 cents), a permutation of Ibn Sina's
Mustaqim at 1/1-9/8-39/32-4/3, where the two
neutral second steps differ considerably in size
and the smaller precedes the larger. From Jacques
Dudon, who explored a modality with two disjunct
tetrachords of this type in a piece called "Sumer"
inspired by visions received during the first
U.S.-Iraq Gulf War of 1991, and then was informed
by John Chalmers of Ibn Sina's tetrachord (whether
presented as 13:12-9:8-128:117 by Ibna Sina himself
or derived later as a permutation of Mustaqim).

As Dudon has remarked, it is thus notable that a
tetrachord of this general type is known in modern
Arab theory as `Iraq, and makes up the lowest four
notes above the final of modern Maqam `Iraq. The
Lebanese scholar Nidaa Abou Mrad has deduced a
likely medieval interpretation of Safi al-Din's
tetrachord and octave species of `Iraq which would
result in a maqam like Dudon's with two disjunct
tetrachords each with neutral second, tone, neutral
second, the larger neutral step here being placed
first.

2. A method of scale generation based on chaining
either a single neutral third generator often
at precisely half the size of a fifth at or
relatively close to 3/2; or two such generators
typically adding up to such a fifth. The first
method is often applied to tunings such as
17-EDO, 24-EDO, 31-EDO, or 44-EDO where the usual
fifth is divisible into two equal neutral thirds;
while Erv Wilson and others have used the second
to generate a variety of just or tempered scales
of a tritriadic variety (e.g. 22:27:33 by chaining
al-Farabi's neutral thirds at 27:22 and 11:9; or
32:39:48 from Ibn Sina's 39:32 and 16:13).

A 7-note chain using the first method produces a
Moment of Symmetry (MOS) scale with only two
adjacent step sizes, while such a chain using the
second method produces a "trivalent" tuning with
three such step sizes.

Using either of these methods, and starting within
the resulting octave species from the scale degree a
neutral seventh above the note which served as the
origin of the chain of neutral thirds, we find a
Mohajira mode in the sense of definition (1). Thus
the two definitions are connected.

It should be noted as a property of the second
method chaining two unequal neutral thirds that
if the smaller precedes the larger (e.g. 32:39:48),
then the resulting Mohajira mode on the seventh
step of the resulting octave species will feature
_larger_ neutral intervals above this step; while
a chain where the larger neutral third precedes
the smaller (e.g. 22:27:33) will feature a Mohajira
mode with smaller neutral intervals above this step.
The resulting tetrachords of the Mohajira mode in
these two examples would be 1/1-128/117-16/13-4/3
(0-155-359-498 cents), or 128:117-9:8-13:12
(155-204-139 cents); and 1/1-88/81-11/9-3/2
(0-143-347-498 cents), or 88:81-9:8-12:11
(143-204-151 cents).

My purpose is to suggest a definition of Mohajira that will
embrace both the tetrachord or mode and the method of scale
generation, and to honor the composition and experiences
which inspired this worthy term.

It is curious and enlightening how claims of either complete
originality or definitive precedent can often be elusive.
Perhaps that is one precious lesson of a thread which has
illuminated some of the origins of a most beautiful term
which I am delighted to embrace.

With deepest thanks,

Margo Schulter
mschulter@...

On Tue, 18 May 2010, Margo Schulter wrote:

> and by raising the fifth degree:
>
> Sikah Mrad's `Iraq Rast
> |--------------|-----------------------|------------|
> 0 132 341 495 705 837 1046 1200
> 132 209 154 209 132 209 154
> |---------------------| |-------------------|
> `Iraq tone `Iraq
>

> shown above. Because the name "Maqam `Iraq" is now so
> widely understood to mean the previous form with a
> middle Bayyati tetrachord, Maqam Mohajira seems to
> a fitting name, with an explanation that this may be
> synonymous with a 13th-century interpretation of
> Maqam `Iraq as suggested by Mrad. A neat feature of
> our Maqam Mohajira is that it includes in the
> trichord-tetrachord-trichord interpretation a middle
> tetrachord like that of Mrad, with with smaller neutral
> second, tone, and larger neutral second (here 154-209-132)!

Please let me note that obviously I meant to say in the
last quoted sentence "a middle tetrachord like that of
Mrad, with _larger_ neutral second, tone, and _smaller_
neutral second (here 154-209-132)!"

Now that I've made that correction, I should also explain
to Jacques and others that my article on Mohajira isn't
yet written, but should be soon, analyzing some types
I've used or seen in the Scala archives, and telling
the curious story of how I arrived at a result quite
similar to dudon_a.scl in the Scala archives.

Also, if we are looking for examples of Mohajira or
the `Iraq tetrachord used as an important genre in
Near Eastern music, one area of investigation stands
out to me: the maqam called Sikah Gharib or Sikah
Baladi (a "foreign" or "estranged" Sikah; or a
"popular" Sikah -- I'm almost tempted to say a
"down home" Sikah).

The accounts seem to agree that in Sikah Baladi, a
usual Hijaz tetrachord in the Arab tradition with
a central step around 12 or 13 commas has this
step "condensed" to something closer to Maqam
Sikah with its lower trichord of neutral second
and tone around 9:8.

Here the question is: "Just how much is that
step typically condensed?" If it's to an 8:7
or the like, then we would have Buzurg. If it's
really to around a 9:8, then we would have
our Mohajira or `Iraq tetrachord; and since
this Maqam seems typically to use two disjunct
tetrachords of this "compressed Hijaz" type,
we might have precisely our Maqam Mohajira.

One approach would be to see if anyone has
done measurement studies on how this maqam
(often favored as a modulation from Rast
or Bayyati, if reports are accurate) is
intoned in practice.

Another might be to seek out videos of
performances identified as "Sikah Gharib"
or "Sikah Baladi," and have someone
either make measurements (if possible)
or give an opinion on listening as to
whether a given performer evokes our
Mohajira.

With or without such precedents, Jacques,
it's a beautiful maqam, and could serve
as a worthy variation on Sikah Baladi
or the like if it isn't already being
used as a manifestation of that style.

Best,

Margo