Subgroup scales

Does anyone know of a listing of musical scales such as ancient Greek, Alexandrian, or medieval Islamic which clearly use a subgroup of a prime limit to construct the scale intervals? I mean for instance constructed from 2, 3 and 7, or 2, 3 and 11, amd so forth.

I'm not sure about a complete listing, but I believe that George Secor
touched on this in his 17-tet paper. One of the Greek tunings for the
various tetrachords was 2.3.7-limit I think, but I can't remember which one.

-Mike

On Sun, May 16, 2010 at 12:12 AM, genewardsmith <genewardsmith@sbcglobal.net
> wrote:

>
>
> Does anyone know of a listing of musical scales such as ancient Greek,
> Alexandrian, or medieval Islamic which clearly use a subgroup of a prime
> limit to construct the scale intervals? I mean for instance constructed from
> 2, 3 and 7, or 2, 3 and 11, amd so forth.
>
>
>

http://raven.dartmouth.edu/~larry/published_articles/divisions_of_the_tetrachord/index.html

"scale" isn't the right concept for ancient tuning.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> Does anyone know of a listing of musical scales such as ancient Greek, Alexandrian, or medieval Islamic which clearly use a subgroup of a prime limit to construct the scale intervals? I mean for instance constructed from 2, 3 and 7, or 2, 3 and 11, amd so forth.
>

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
>
> http://raven.dartmouth.edu/~larry/published_articles/divisions_of_the_tetrachord/index.html
>
> "scale" isn't the right concept for ancient tuning.

Not very informative. Are you claiming a division of the tetrachord doesn't count as a scale? That Avidenna or Averroes never discussed scales in their work on music theory? Genuine questions, since I'm no expert, but so far as I know you aren't either, so convince me otherwise.

You don't find John Chalmer's Divisions of the Tetrachord very informative? I believe it has just what you're looking for.

And no, a tetrachord is not a scale- even in modern western usage where scalar thinking dominates it is considered a scale fragment, in ancient and eastern usage clearly a structural unit of greater strength and identity than a "fragement". It doesn't matter a bit whether Avicenna or Aristoxenos or Sorbo talked about scales, the structural unit, and even more importantly for your question, the tuning unit, is the tetrachord.

-Cameron Bobro

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> >
> > http://raven.dartmouth.edu/~larry/published_articles/divisions_of_the_tetrachord/index.html
> >
> > "scale" isn't the right concept for ancient tuning.
>
> Not very informative. Are you claiming a division of the tetrachord doesn't count as a scale? That Avidenna or Averroes never discussed scales in their work on music theory? Genuine questions, since I'm no expert, but so far as I know you aren't either, so convince me otherwise.
>

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:

> And no, a tetrachord is not a scale- even in modern western usage where scalar thinking dominates it is considered a scale fragment, in ancient and eastern usage clearly a structural unit of greater strength and identity than a "fragement".

And this matters because...?

It doesn't matter a bit whether Avicenna or Aristoxenos or Sorbo talked about scales, the structural unit, and even more importantly for your question, the tuning unit, is the tetrachord.

This has no relevance to my question, which was about prime factoriztion and which could perfectly well be answered by citing tetrachords if there is some way of reconstructing the entire musical gamut from them. Certainly, if all you are playing are the tetrachords themselves that would suffice, "fragment" or not.

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> You don't find John Chalmer's Divisions of the Tetrachord very informative? I believe it has just what you're looking for.

I was hoping someone actually knew the answer. That's a lot of tetrachords; I'm interested primarily in where they can be factored by {2,3,p}, where p>5 is prime.

Dear Gene (and all),

Thank you for asking a question where I may have a special
interest in these "subgroup scales," quite the norm for me
since some of my favorite systems, just or tempered, are
based on or approximate 2-3-7-11-13, which if I'm correct
would be a "subgroup" in your meaning.

Anyway, let's go through a few ancient Greek or medieval
Islamic tetrachords, pentachords, or modes of this kind,
duly noting that John Chalmers discusses a number of them
in _Divisions of the Tetrachord_, as has already been
pointed out, and that the Scala scale archive is another
good source:

Classic Greek
-------------

Archytas Diatonic (2-3-7)

1/1 8/7 32/27 4/3 3/2 12/7 16/9 2/1
8:7 28:27 9:8 9:8 8:7 28:27 9:8

Ptolemy Intense Chromatic (2-3-7-11)

1/1 22/21 8/7 4/3 3/2 11/7 12/7 2/1
22:21 12:11 7:6 9:8 22:21 12:11 7:6

Medieval Islamic
----------------

Zalzal, al-Farabi's version (2-3-11)

1/1 9/8 27/22 4/3 3/2 18/11 16/9 2/1
9:8 12:11 88:81 9:8 12:11 88:81 9:8

Mustaqim mode, Ibn Sina (2-3-13)

1/1 9/8 39/32 4/3 3/2 13/8 16/9 2/1
9:8 13:12 128:117 9:8 13:12 128:117 9:8

Septimal tuning, Ibn Sina (2-3-7-13)

1/1 14/13 7/6 4/3 3/2 21/13 7/4 2:1
14:13 13:12 8:7 9:8 14:13 13:12 8:7

Septimal tuning, Safi al-Din (2-3-7-13)

1/1 8/7 16/13 4/3 32/21 64/39 16/9 2/1
8:7 14:13 13:12 8:7 14:13 13:12 9:8

Septimal tuning, Safi al-Din (2-3-7)

1/1 8/7 9/7 4/3 32/21 12/7 16/9 2/1
8:7 9:8 28:27 8:7 9:8 28:27 9:8

Buzurg, Safi al-Din and Qutb al-Din (2-3-7-13)

1/1 14/13 16/13 4/3 56/39 3/2
14:13 8:7 13:12 14:13 117:112

Hijaz, Qutb al-Din (2-3-7-11)

1/1 12/11 14/11 4/3
12:11 7:6 22:21

There are a number of interesting connections
between these tunings, for example Qutb al-Din
al-Shirazi's Hijaz tetrachord as a permutation
of Ptolemy's Intense Chromatic; Safi al-Din's
1/1-8/7-9/7-4/2 as a tuning using the same steps
as the Archyas diatonic (8:7, 9:7, 28:27); and
three different permutations of (14:13, 13:12, 8:7).

Note that tetrachords may be conjunct or disjunct
in medieval and modern Near Eastern modes, with
both arrangements illustrated here.

You'll understand why tunings of this kind are
very attractive to me, and I hope this quick
survey gives some answer to your question.

Best,

Margo
mschulter@...

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Dear Gene (and all),

Thanks, Margo for a very informative answer. Would you object to my mining this for information for a Xenharmonic Wiki article? I want to put up something on subgroups, and the fact that there is all of this history made it something I would wish to include.

Note that not all divisions of the tetrachord are or were conceived of in ratios. Modern divisions can be concieved in cents, and historical divisions are/were sometimes concieved in terms of divisions of string lengths (or other measurement, eg. "parts").

So, strictly speaking, we could disingeneously describe a number of ancient tetrachords in terms of whatever primes we wish. We could even speculate, tongue in cheek (but "accurately"! in terms of numbers and even physically possible monochord implementation) that Aristoxenos' soft chromatic is based on the Golden section.

And so Gene, in my opinion it would be best to limit your search even further than "primes". Limited exponents, preference for superparticular divisions, etc.

-Cameron Bobro

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Dear Gene (and all),
>
> Thank you for asking a question where I may have a special
> interest in these "subgroup scales," quite the norm for me
> since some of my favorite systems, just or tempered, are
> based on or approximate 2-3-7-11-13, which if I'm correct
> would be a "subgroup" in your meaning.
>
> Anyway, let's go through a few ancient Greek or medieval
> Islamic tetrachords, pentachords, or modes of this kind,
> duly noting that John Chalmers discusses a number of them
> in _Divisions of the Tetrachord_, as has already been
> pointed out, and that the Scala scale archive is another
> good source:
>
> Classic Greek
> -------------
>
> Archytas Diatonic (2-3-7)
>
> 1/1 8/7 32/27 4/3 3/2 12/7 16/9 2/1
> 8:7 28:27 9:8 9:8 8:7 28:27 9:8
>
> Ptolemy Intense Chromatic (2-3-7-11)
>
> 1/1 22/21 8/7 4/3 3/2 11/7 12/7 2/1
> 22:21 12:11 7:6 9:8 22:21 12:11 7:6
>
>
> Medieval Islamic
> ----------------
>
> Zalzal, al-Farabi's version (2-3-11)
>
> 1/1 9/8 27/22 4/3 3/2 18/11 16/9 2/1
> 9:8 12:11 88:81 9:8 12:11 88:81 9:8
>
>
> Mustaqim mode, Ibn Sina (2-3-13)
>
> 1/1 9/8 39/32 4/3 3/2 13/8 16/9 2/1
> 9:8 13:12 128:117 9:8 13:12 128:117 9:8
>
> Septimal tuning, Ibn Sina (2-3-7-13)
>
> 1/1 14/13 7/6 4/3 3/2 21/13 7/4 2:1
> 14:13 13:12 8:7 9:8 14:13 13:12 8:7
>
> Septimal tuning, Safi al-Din (2-3-7-13)
>
> 1/1 8/7 16/13 4/3 32/21 64/39 16/9 2/1
> 8:7 14:13 13:12 8:7 14:13 13:12 9:8
>
> Septimal tuning, Safi al-Din (2-3-7)
>
> 1/1 8/7 9/7 4/3 32/21 12/7 16/9 2/1
> 8:7 9:8 28:27 8:7 9:8 28:27 9:8
>
> Buzurg, Safi al-Din and Qutb al-Din (2-3-7-13)
>
> 1/1 14/13 16/13 4/3 56/39 3/2
> 14:13 8:7 13:12 14:13 117:112
>
> Hijaz, Qutb al-Din (2-3-7-11)
>
> 1/1 12/11 14/11 4/3
> 12:11 7:6 22:21
>
>
> There are a number of interesting connections
> between these tunings, for example Qutb al-Din
> al-Shirazi's Hijaz tetrachord as a permutation
> of Ptolemy's Intense Chromatic; Safi al-Din's
> 1/1-8/7-9/7-4/2 as a tuning using the same steps
> as the Archyas diatonic (8:7, 9:7, 28:27); and
> three different permutations of (14:13, 13:12, 8:7).
>
> Note that tetrachords may be conjunct or disjunct
> in medieval and modern Near Eastern modes, with
> both arrangements illustrated here.
>
> You'll understand why tunings of this kind are
> very attractive to me, and I hope this quick
> survey gives some answer to your question.
>
> Best,
>
> Margo
> mschulter@...
>

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:

> And so Gene, in my opinion it would be best to limit your search even further than "primes". Limited exponents, preference for superparticular divisions, etc.

I'm a mathematician and I think like one. To me, first looking at groups, which takes the issue to a high and conceptually clean level of abstraction, makes far more sense than trying to add flounces and furbelows in the form of seemingly extraneous assumptions. If someone else wants to expound using another point of view on these scales, I'm interested.

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> Hijaz, Qutb al-Din (2-3-7-11)
>
> 1/1 12/11 14/11 4/3
> 12:11 7:6 22:21

If this is completed to an octave--conjunct or disjunct tetrachords, it doesn't matter--then it generates the [2,3,7,11] group. But just by itself, it only generates [4/3, 8/7, 16/11]. Should we assume it completes to an octave some way or another?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> I'm a mathematician and I think like one. To me, first looking at >groups, which takes the issue to a high and conceptually clean level >of abstraction, makes far more sense than trying to add flounces and >furbelows in the form of seemingly extraneous assumptions. If someone >else wants to expound using another point of view on these scales, I'm >interested.
>

Well, you were looking into ancient scales, so a preference for superparticular intervals would not be flouncy at all. Rather, it would be historical as well as practical from an acoustic viewpoint (easier to tune by ear, more realistically implemented with physical measuring instruments).

And, 5573 is a prime- were you really considering 2,3,5573? Or 3-limit figures like 4782969? Such ludicrosities aren't ruled out by your original quest, are they? They would be ruled out if ancient tuning theory were restricted to reasonable harmonic ratios, but it isn't- it contains such potentially controversial divisions such as "parts". I'd call putting prime and exponential limits, and ruling such historical theories out of your quest, "elegant" rather than "frambubulous" or whatever.

And understanding tetrachords is of deep and vital importance in the study of ancient and eastern music, that's just a plain old brute fact (as far as plain old brute facts can exist in music theory).

-Cameron Bobro

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:

> Well, you were looking into ancient scales, so a preference for superparticular intervals would not be flouncy at all.

I am interested in the groups the scales generate, and to that question this would be irrelevant, and hence flouncy.

> And, 5573 is a prime- were you really considering 2,3,5573? Or 3-limit figures like 4782969? Such ludicrosities aren't ruled out by your original quest, are they?

Obviously, I am only interested in small primes. OF COURSE they are of no interest. Is there a point to bringing up such absurdities?

> And understanding tetrachords is of deep and vital importance in the study of ancient and eastern music, that's just a plain old brute fact (as far as plain old brute facts can exist in music theory).

Argument for this alleged "fact"? If identical (or even inverted, for that matter) tetrachords are always to be stuck together with a 9/8 in between, whether conjointly or disjointly, they are completely unimportant as such so far as the issue of the groups which scales generate goes, and of mainly historical interest otherwise so far as I can see. You can't tell the difference between someone playing on a diatonic scale composed of two tetrachords and someone who got it from a MOS if they are the exact same set of notes, can you?

You've several times now asserted the cosmic importance of tetrachords in understanding certain kinds of music, but you've given NO argument. It remains a bare assertion of dogma until you can manage one. That people found it helpful to construct scales (sorry, but the word is appropriate) via tetrachords by no means settles the matter, or even starts in on settling it.

One common mistake in ethnomusicology -- and doubly in ancient
ethnomusicology -- is trusting the theorists. We are currently
living in the first and only era where musicians are capable
of precisely intoning arbitrary scales prescribed by theory
(otherwise we would not be so excited on this list). The notion
that any significant number of musicians of the Islamic golden
age -- or any era past -- tuned their instruments with 17-limit
ratios is laughable. Even in the Islamic world today, most
instruments are tuned to rough quartertone scales. I'm sure
Ozan and others will now chop my head off, but listening to
maqam music over the years, I have yet to hear any intonation not
adequately captured by 24-ET. This does not mean I could capture
any single performance with 24-ET, but rather that the differences
between performances are not systematic and when boiled down,
24-ET is clearly what we get.

-Carl

Off with his head!

LOL

Oz.

✩ ✩ ✩
www.ozanyarman.com

On May 16, 2010, at 11:47 PM, Carl Lumma wrote:

> One common mistake in ethnomusicology -- and doubly in ancient
> ethnomusicology -- is trusting the theorists. We are currently
> living in the first and only era where musicians are capable
> of precisely intoning arbitrary scales prescribed by theory
> (otherwise we would not be so excited on this list). The notion
> that any significant number of musicians of the Islamic golden
> age -- or any era past -- tuned their instruments with 17-limit
> ratios is laughable. Even in the Islamic world today, most
> instruments are tuned to rough quartertone scales. I'm sure
> Ozan and others will now chop my head off, but listening to
> maqam music over the years, I have yet to hear any intonation not
> adequately captured by 24-ET. This does not mean I could capture
> any single performance with 24-ET, but rather that the differences
> between performances are not systematic and when boiled down,
> 24-ET is clearly what we get.
>
> -Carl
>

Gene wrote:

> You've several times now asserted the cosmic importance of
> tetrachords in understanding certain kinds of music, but you've
> given NO argument. It remains a bare assertion of dogma until
> you can manage one. That people found it helpful to construct
> scales (sorry, but the word is appropriate) via tetrachords by
> no means settles the matter, or even starts in on settling it.

I don't pretend to understand the first thing about maqam music
theory (talking about their practicing theory here, not the
ethno-intonation stuff) but when I have looked into it, it has
seemed that the proper lego block is the tetrachord. Take that
for what it's worth. -Carl

Dear Gene,

Please feel welcome to use the little compilation
of tetrachords, pentachords, and modes that I gave
in a Wiki article or the like. They come from various
sources I've read, not least Chalmers on _Divisions
of the Tetrachord_ and the Scala scale archives.

Here I should confess my mathematical naivete: I
simply understand "subgroup" to mean a set of
prime factors where not all possible primes smaller
than the largest used are represented: e.g.
2-3-7 out of 2-3-5-7. Since I take this kind of
thing for granted in lots of JI systems or
temperaments, I didn't have any special word for
it, but your usage understood in this simple way
is certainly logical and intuitive to me.

Thus to me the question is simply, "Taking a given
tetrachord or pentachord or octave-species or whatever,
what prime factors are represented?" For example:

1/1 12/11 14/11 4/3
12:11 7:6 22:21

Here I'd say that we have ratios formed by
prime factors of 2, 3, 7, and 11. All I need is
the tetrachord to see that we can form all of
its ratios with these four primes, and need all
four of these primes to form its ratios.

There are various ways this tetrachord might be
used as part of a mode, and I should add the
caution that Near Eastern modal systems are
indeed built from tetrachords, not always
resulting in strict octave reduplication, as
one may note in a modern Maqam Bayyati or
Shur Dastgah, both of which often feature
a neutral third below the final but a minor
sixth above it in the "textbook" form of
the modal type.

For example, let's look at some possibilities
for the above Hijaz tetrachord. One of these
given in by Qutb al-Din would be to use it
as an upper tetrachord above a Buzurg
pentachord:

Buzurg Hijaz
|-------------------------------|-----------------|
1/1 14/13 16/13 4/3 56/39 3/2 18/11 21/11 2/1
14:13 8:7 13:12 14:13 117:112 12:11 7:6 22:21

Combined, these tetrachords give us ratios based on
primes 2-3-7-11-13. Here's another possibility which
is rather like Qutb al-Din's Zankule mode:

Rast Hijaz tone
|-----------------|--------------------| | 1/1 9/8 27/22 4/3 16/11 56/33 16/9 2/1
9:8 12:11 88:81 12:11 7:6 22:21 9:8

Here the factors for the whole octave, as for
Hijaz alone, are 2-3-7-11. However, if one follows
the 5-based ratios for Rast as given by Qutb al-Din
(and still used in some Near Eastern traditions,
notably the Ottoman), then we would have:

Rast Hijaz 9:8 tone
|-----------------|--------------------| | 1/1 9/8 5/4 4/3 16/11 56/33 16/9 40/21 2/1
9:8 10:9 16:15 12:11 7:6 22:21 15:14 21:20

Qutb al-Din's Zankule is thus not a subgroup but
what you would call simply an "11-limit" tuning,
if I'm correct, i.e. 2-3-7-11.

Here are modern uses of a Hijaz tetrachord which do
fit a subgroup concept as I understand it. First,
as you observed, we could use the tetrachord
disjunctly for one form of what is called
Maqam Hijaz-Kar, here 2-3-7-11:

|-----------------------| |--------------------|
1/1 12/11 14/11 4/3 3/2 18/11 21/11 2/1
12:11 7:6 22:21 9:8 12:11 7:6 22:21

Another possibility is a form of Maqam Hijaz with a
conjunct upper tetrachord of Rast:

Hijaz Rast
|-----------------------|--------------------| tone |
1/1 12/11 14/11 4/3 3/2 18/11 16/9 2/1
12:11 7:6 22:21 9:8 12:11 88:81 9:8

This, again, would be 2-3-7-11.

A very helpful bit of advice in the maqam tradition is
to "follow the ajnas" -- ajnas being the Arabic plural or
jins or "genus" -- that is, to be aware of and be guided
by one's progressions through the ajnas or genera of a
given maqam, often trichords, tetrachords, or pentachords.
These ajnas provide one guide in negotiating the _sayr_
(Turkish _seyir_) or "path" of a given maqam, which often
may involve expected tetrachord mutations or modulations.
Thus I would say that for this music, jins-based concepts
are vital; and writing about the Persian dastgah system,
Hormoz Farhat warns that thinking in terms of octave scales
can often be misleading as to the actual structure of
the music.

With many thanks,

Margo
mschulter@...

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> I don't pretend to understand the first thing about maqam music
> theory (talking about their practicing theory here, not the
> ethno-intonation stuff) but when I have looked into it, it has
> seemed that the proper lego block is the tetrachord. Take that
> for what it's worth. -Carl

Not much until you define what you mean; this is the same problem once again. Do awful things happen if you listen to Rast and think of it in octave form?

Dear Carl and Ozan and all,

Please let me add to this dialogue a view that
all theorists, ancient or medieval or modern,
should be read carefully and also critically,
exercising one's reason and musical discernment
while seeking to improve these.

In fact, looking at modern practice where, as
you have noted, Carl, quite accurate measurements
_are_ possible, we find lots of maqam intonations
very distinct from 24-EDO. Ozan, doubtless, might
cite the very common interpretation of Maqam Rast
in the Ottoman tradition as based on a tetrachord
of 1/1 9/8 5/4 4/3, with Safi al-Din and Qutb al-Din
giving these ratios in the epoch 1250-1300.

In the modern Persian and Turkish traditions, and
likely also in some Arab styles, Maqam Hijaz or
the corresponding Dastgah-e Chahargah typically
uses a middle step around 7:6, rather different
from 24-EDO, although the 250-cent step of that
tuning is a possible variation.

Consider also this measured Hijaz intonation of
the Turkish performer Kudsi Erguner, as reported
by the Lebanese composer and scholar Amine Beyhom:

0 131 368 501
131 237 133

Actually this is quite close to the lower tetrachord
of Buzurg as recorded by Qutb al-Din and Safi al-Din,
two theorists whom I _would_ trust, albeit with an
appreciation of the ambiguities and questions which
do arise, as with most theorists:

1/1 14/13 16/13 4/3
0 128 359 498
14:13 8:7 13:12
128 231 139

Yes, Erguner's version of 131-237-133 -- and this
fluctuates a bit, not so surprisingly, during his
performance -- is a bit different than 128-231-139,
but a lot closer to that than to 150-200-150, for
example. If we want for some reason a 12n-EDO
approximation, how about 36-EDO: 133-233-133?

While we don't have recordings from the 8th-13th
centuries to test what the theorists of that era
report, a lot of it seems intuitively very credible,
especially in view of what is being advocated,
documented, and measured among Near Eastern musicians
today.

Thus al-Farabi's version of Zalzal's `oud or lute
tuning seems to me a quite reasonable approximation
of the kind of Rast which might be played in Egypt;
as you have noted, Ozan, the third step Segah
(or Sikah in Arabic), which al-Farabi places at 27/22,
will typically be rather higher as one hears the maqam
in parts more to the east such as Palestine, Syria, and
Turkey: maybe around 21/17 or 26/21, and in the
Ottoman tradition often around 5/4.

Likewise, Ibn Sina's placement of Zalzal's wusta or
neutral third fret considerably lower at 39/32 may
reflect a preference still present in much Persian
music today for a small neutral third, for example
in Dastgah-e Segah, where Hormoz Farhat, averaging
some tar tunings, places this step at about 340 cents
above the note representing the final of a mode
like Ibn Sina's (in Dastgah-e Segah, the third step
is now regarded as the final).

Carl, while a ratio like 81:68 (the wusta Fers or
"Persian middle finger" of medieval theory) might
seem rather complex, on a fretboard a medieval
or modern musician can make an arithmetic division
of 9:8 into 18:17:16, so we have a 9:8 step plus
this division of another 9:8 tone, here shown in
string ratios:

81 72 68 64
1/1 9/8 81/68 4/3
9:8 18:17 17:16

In fact, nuances such as distinctions in the placement
of smaller and larger neutral seconds are a concern
of traditional performers which tend to get lost in
the teaching of some Egyptian conservatories, for
example, that 24-EDO is an adequate model.

Best,

Margo

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> One common mistake in ethnomusicology -- and doubly in ancient
> ethnomusicology -- is trusting the theorists.

Let's go by performance practice in ancient Greece and the medieval Islamic world instead. Except, does anyone *know* anything about that?

Even in the Islamic world today, most
> instruments are tuned to rough quartertone scales.

Which could possibly represent a falling away from a more sophisticated approach. That seems to happen in music, or at least it can be seen to be happening now.

I'm sure
> Ozan and others will now chop my head off, but listening to
> maqam music over the years, I have yet to hear any intonation not
> adequately captured by 24-ET.

If you look at 24-et, one of the most striking things about it is what a good {2,3,11} system it is. If you look at 159-et, one of the most striking things about it is what an even better {2,3,11} system it is. I doubt if either of us is enough of an expert to pronounce on whether that is an accident or not, but if someone wants to give the {2,3,11} 24-note MOS of 24&159, which tempers out two cents worth of 1771561/1769472, has a period of 1/3 octave and a generator of 12/11, I'm all for it. Retune those 24-et instruments and play some maqams, sez I!

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> > One common mistake in ethnomusicology -- and doubly in ancient
> > ethnomusicology -- is trusting the theorists.
>
> Let's go by performance practice in ancient Greece and the
> medieval Islamic world instead. Except, does anyone *know*
> anything about that?

Not really, no. People have tried to reconstruct instruments,
but what they start with never specifies the intonation in any
meaningful way. The alternative is Ptolemy and so on, who
clearly made stuff up and wrote it down.

> > Even in the Islamic world today, most
> > instruments are tuned to rough quartertone scales.
>
> Which could possibly represent a falling away from a more
> sophisticated approach. That seems to happen in music, or at
> least it can be seen to be happening now.

It's possible, yes.

> > I'm sure
> > Ozan and others will now chop my head off, but listening to
> > maqam music over the years, I have yet to hear any intonation
> > not adequately captured by 24-ET.
>
> If you look at 24-et, one of the most striking things about it
> is what a good {2,3,11} system it is.

Yup. But to make those 11s meaningful as 11s, one needs the
2s and the 3s (at least) at the same time. The quartertones
of maqam music are melodically motivated. And I think it's
likely to be a straightforward case of aiming for the middle.
It's funny this came up, because I was just thinking about it
as I was listening to a Munir Bashir album yesterday.

> If you look at 159-et, one of the most striking things about
> it is what an even better {2,3,11} system it is. I doubt if
> either of us is enough of an expert to pronounce on whether
> that is an accident or not,

Ozan actively considered approximations to extended JI when
he selected 159.

-Carl

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> > I don't pretend to understand the first thing about maqam music
> > theory (talking about their practicing theory here, not the
> > ethno-intonation stuff) but when I have looked into it, it has
> > seemed that the proper lego block is the tetrachord. Take that
> > for what it's worth. -Carl
>
> Not much until you define what you mean; this is the same
> problem once again. Do awful things happen if you listen to Rast
> and think of it in octave form?

There are fewer tetrachords than scales, for one thing.
If you get into translating the terminology of their practical
theory, you'll... hopefully have better luck than I did at
putting it all together. But one thing that seemed evident is
they do not have octave equivalence to the extent Western
music does. They have scales defined over a 9th compass, for
one. For another, they apparently 'modulate' between scales
through common tetrachords.

-Carl

On Sun, May 16, 2010 at 8:16 PM, Margo Schulter <mschulter@...> wrote:
>
> Carl, while a ratio like 81:68 (the wusta Fers or
> "Persian middle finger" of medieval theory) might
> seem rather complex, on a fretboard a medieval
> or modern musician can make an arithmetic division
> of 9:8 into 18:17:16, so we have a 9:8 step plus
> this division of another 9:8 tone, here shown in
> string ratios:

Margo makes an excellent point here that I think bears repeating, and
one I have never really thought of. If on a fretboard, one places a
fret at a perfect 9/8 from the open string -- and then simply places
another fret linearly equidistant to the nut and this 9/8 fret -- the
3-note "scale" resulting will be:

1/1 18/17 9/8

For those playing at home, and wanting a quick and simple mathematical
pseudo-"proof" of this:

If "d" represents the distance along the string's length from the nut
a fret is placed as a value going from 0-1, where 0 is the nut and 1
is the bridge, the formula to get from "d" to the relative frequency
produced is

f = 1/(1-d)

So that a fret placed halfway down the string (d=1/2) yields f=2, a
fret placed 3/4 of the way down the string (d=3/4) yields f=4, etc.

Turns out that a fret placed 1/9 of the way down the string yields
f=9/8. And if you place another fret in between this and the nut, so d
= 1/18, you get f = 18/17.

So it's possible that medieval musicians might have been using
17-limit ratios because of clever little tricks like this (possibly
without knowing what they were doing). The limiting factor as to how
relevant that is, in this case, wouldn't be a lack of mathematical
tools in antiquity, but rather whether the ear really perceives that
scale as involving ratios of 17 or so.

-Mike

On Sun, May 16, 2010 at 8:16 PM, Margo Schulter <mschulter@...> wrote:
>
> Carl, while a ratio like 81:68 (the wusta Fers or
> "Persian middle finger" of medieval theory) might
> seem rather complex, on a fretboard a medieval
> or modern musician can make an arithmetic division
> of 9:8 into 18:17:16, so we have a 9:8 step plus
> this division of another 9:8 tone, here shown in
> string ratios:
>
> 81 72 68 64
> 1/1 9/8 81/68 4/3
> 9:8 18:17 17:16

Although let me add to this that I'm not seeing how 81/68 is generated
here - if you arithmetically divide the space between the 9/8 and
4/3, you end up getting 72/59, which is a 59-limit ratio. Unless I've
screwed up the math here somehow.

-Mike

> On Sun, May 16, 2010 at 12:12 AM, genewardsmith <genewardsmith@...
> > wrote:
> >
> > Does anyone know of a listing of musical scales such as ancient Greek,
> > Alexandrian, or medieval Islamic which clearly use a subgroup of a prime
> > limit to construct the scale intervals? I mean for instance constructed from
> > 2, 3 and 7, or 2, 3 and 11, amd so forth.
>
--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I'm not sure about a complete listing, but I believe that George Secor
> touched on this in his 17-tet paper. One of the Greek tunings for the
> various tetrachords was 2.3.7-limit I think, but I can't remember which one.
>
> -Mike

Archytas' tunings for all three ancient Greek genera of are 7 prime limit.

There are basically 3 different ancient Greek scales (or genera): enharmonic, chromatic, & diatonic genus, and various theorists had one or more tunings for each genus.

My source or ancient Greek tunings is the chapter on ancient Greek scales from J. Murray Barbour's _Tuning and Temperament_. Rather than list all of these here, I've uploaded an Excel spreadsheet that I made some years ago, which lists all of these, in the order in which they appear in the book.

/tuning/files/secor/Greek.xls

--George

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> So it's possible that medieval musicians might have been using
> 17-limit ratios because of clever little tricks like this (possibly
> without knowing what they were doing). The limiting factor as to how
> relevant that is, in this case, wouldn't be a lack of mathematical
> tools in antiquity, but rather whether the ear really perceives that
> scale as involving ratios of 17 or so.

Of course it's possible. It's also not significant, whether
a semitone is tuned exactly to 18/17 or not, especially on the
quickly-decaying timbres produced by the plucked strings of
the day.

Galileo Sr published a method for getting viols frets near 12-ET
by using 18/17. Does that make early string ensemble music
17-limit (in the regular mapping sense Gene is asking about)?
No. -Carl

Of course modern theorists aren't much more trustworthy. Claudio Di Veroli (among others) convinced me that Barbour was a modern-day Ptolemy, as it were. -Carl

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
>
> > I don't pretend to understand the first thing about maqam music
> > theory (talking about their practicing theory here, not the
> > ethno-intonation stuff) but when I have looked into it, it has
> > seemed that the proper lego block is the tetrachord. Take that
> > for what it's worth. -Carl
>
> Not much until you define what you mean; this is the same problem >once again. Do awful things happen if you listen to Rast and think >of it in octave form?
>

Curious you should pick Rast, for if I recall correctly (Ozan can correct me if I'm wrong), Rast is an excellent example of how important tetrachordal thinking is in maqam music. For it is quite typical that it goes up in two disjunct Rast tetrachords, but descends with the upper tetrachord swapped out for another (it's got a darker "third" in the tetrachord, don't know what it's called), while retaining the Rast tetrachord on the bottom.

Well you could say that's the same as going down with a lower seventh in an octave species scale, a la minor/harmonic-minor movement in Western music. And who would know the difference? But that would be wrong, for there are other standard shared-tetrachord (or trichord or pentachord) movements in Rast which are NOT equivalent to simply flatting a descending diatonic tone. The interlocking units are embedded in the scale that results from the original disjunct Rast tetrachords, and lead to heavier modulations.

And, if you play an acoustic instrument with strings tuned to open fifths or fourths, it is immediately obvious just how incredibly handy (sorry for the pun) this method of construction is, in a practical way as well.

-Cameron Bobro

> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > One common mistake in ethnomusicology -- and doubly in ancient
> > ethnomusicology -- is trusting the theorists.

That's a good point. For example, you'll find writings which simply quote the figures given by ancient authors without much in the way of either arguing for or against the realism of the figures, and you'll find writings with comments dismissing the likelihood of the figures as being realistic. But if you take a monochord and tune the things up yourself, you may find that what looked like a bit of ancient sci-fi or numerology is in fact almost trivially simple to accurately implement in "real life", and, far from being worthy of smug dismissal, is more deserving of a stamp of approval by Occam's razor (to unwisely mix metaphors, ouch).

Take, for example, the 28:27 bottom interval found in all of Archytus' tetrachords. Now, this probably seems a far-fetched and numerological at first glance, but it is actually a very easily implemented interval. For it is simply the difference between a 7:6 and a 9:8. After tuning it up on an acoustic instrument yourself, it is obvious that this is a completely plausible real-life historical interval, tunable "in a couple of easy steps" which you can pass down to your child via oral tradition, even without reference to any math at all. My six-year-old can already recognize the intervals needed here, 3:2 and 7:6, he even comes up with his own names ("soldier-like" and "desert-like" in this case).

Add to this ease of practical implementation the eerie similarity of such tunings to tunings found in the Middle East and Balkans to this day and just plain common sense has moved our ancient writing from implausible to very likely indeed. All the theoretical writings can be tested this way, it's fun and easy and you can form your own informed opinion as to what was likely or not.

Of course, my mild and reasonable comments on this topic don't exist in a vacuum- rather, they're set afloat in the same dark and smelly virtual sea that also homes Joe Monzo's 12-tET ancient Sumerians, Neandrathal bone flutes in C Major, and other frightening vondanikenisms.

So- LOL.

-Cameron Bobro

Dear Margo,

✩ ✩ ✩
www.ozanyarman.com

On May 17, 2010, at 3:16 AM, Margo Schulter wrote:

> Dear Carl and Ozan and all,
>
> Please let me add to this dialogue a view that
> all theorists, ancient or medieval or modern,
> should be read carefully and also critically,
> exercising one's reason and musical discernment
> while seeking to improve these.
>

Agreed!

There is a quotidian tendency to brush aside "antique/moderntheoretical pursuits" as not reflecting the musical situation of the
particular period they were formulated in. While there is indeed some
concern as to the validity of some 20th Century theoretical models
which appear to be influenced by ideological, supremacist or
pretentious worries, it would be an utter folly to disregard their
contributions to the explication of music-making altogether.

Yes, trust not the theorists, but heed them! Especially the older
ones! I know not of any Art Music in the world bereft of at least the
semblance of a theoretical model. As such, Folk and Ethnic musics are
picking up the pace in the theoretical arena. Jazz and Pop are already
in the lead.

> In fact, looking at modern practice where, as
> you have noted, Carl, quite accurate measurements
> _are_ possible, we find lots of maqam intonations
> very distinct from 24-EDO. Ozan, doubtless, might
> cite the very common interpretation of Maqam Rast
> in the Ottoman tradition as based on a tetrachord
> of 1/1 9/8 5/4 4/3, with Safi al-Din and Qutb al-Din
> giving these ratios in the epoch 1250-1300.
>

This tetrachord, a likely remnant from Ptolemy, is mentioned in the
passing, but does not seem to be the component of any particular
famous "maqam" of that era. Doubtless, if we had the means of time-
travel and witnessed the city-scapes of medieval Baghdad or Basra, we
would surely be dumbstruck by the exotic music-making of that age, and
if we fast-forwarded the scenes, we might perhaps observe the
flickering of perde segah up and down the mujannab continuum, finally
splitting into two or three fragments between the Islamic Trichotomy:
Turkey, Persia, Arabia.

> In the modern Persian and Turkish traditions, and
> likely also in some Arab styles, Maqam Hijaz or
> the corresponding Dastgah-e Chahargah typically
> uses a middle step around 7:6, rather different
> from 24-EDO, although the 250-cent step of that
> tuning is a possible variation.
>

My ears indicate that the whereabouts of 250 cents is a desirable
concordant augmented second interval in the Hijaz tetrachord. It is
curious, because 144/125, 125/108 or 196/169 are not easy JI ratios.

Arguably, since JI is meant to justify Western tonality, not any other
thing!

> Consider also this measured Hijaz intonation of
> the Turkish performer Kudsi Erguner, as reported
> by the Lebanese composer and scholar Amine Beyhom:
>
> 0 131 368 501
> 131 237 133
>
> Actually this is quite close to the lower tetrachord
> of Buzurg as recorded by Qutb al-Din and Safi al-Din,
> two theorists whom I _would_ trust, albeit with an
> appreciation of the ambiguities and questions which
> do arise, as with most theorists:
>
> 1/1 14/13 16/13 4/3
> 0 128 359 498
> 14:13 8:7 13:12
> 128 231 139
>
> Yes, Erguner's version of 131-237-133 -- and this
> fluctuates a bit, not so surprisingly, during his
> performance -- is a bit different than 128-231-139,
> but a lot closer to that than to 150-200-150, for
> example. If we want for some reason a 12n-EDO
> approximation, how about 36-EDO: 133-233-133?
>

I used to toy with 36-EDO during my master's studies 9 years ago. My
master's dissertation is a microtonal theory trial based on it.
Looking back, I see how at certain moments I came close to grasping
some hidden features of Maqam music, and how equally those same have
eluded me!

> While we don't have recordings from the 8th-13th
> centuries to test what the theorists of that era
> report, a lot of it seems intuitively very credible,
> especially in view of what is being advocated,
> documented, and measured among Near Eastern musicians
> today.
>

And what if theoretical models were not at all formulated in those
times? What insights would we have as to the music-making of a
millenium ago? Shudder at the thought! One could even have assumed
that 12-equal or multiples thereof was the norm for all times
everywhere.

Oh wait, Carl has already assumed that.

Hah hah!

> Thus al-Farabi's version of Zalzal's `oud or lute
> tuning seems to me a quite reasonable approximation
> of the kind of Rast which might be played in Egypt;
> as you have noted, Ozan, the third step Segah
> (or Sikah in Arabic), which al-Farabi places at 27/22,
> will typically be rather higher as one hears the maqam
> in parts more to the east such as Palestine, Syria, and
> Turkey: maybe around 21/17 or 26/21, and in the
> Ottoman tradition often around 5/4.
>
> Likewise, Ibn Sina's placement of Zalzal's wusta or
> neutral third fret considerably lower at 39/32 may
> reflect a preference still present in much Persian
> music today for a small neutral third, for example
> in Dastgah-e Segah, where Hormoz Farhat, averaging
> some tar tunings, places this step at about 340 cents
> above the note representing the final of a mode
> like Ibn Sina's (in Dastgah-e Segah, the third step
> is now regarded as the final).
>
> Carl, while a ratio like 81:68 (the wusta Fers or
> "Persian middle finger" of medieval theory) might
> seem rather complex, on a fretboard a medieval
> or modern musician can make an arithmetic division
> of 9:8 into 18:17:16, so we have a 9:8 step plus
> this division of another 9:8 tone, here shown in
> string ratios:
>
> 81 72 68 64
> 1/1 9/8 81/68 4/3
> 9:8 18:17 17:16
>
> In fact, nuances such as distinctions in the placement
> of smaller and larger neutral seconds are a concern
> of traditional performers which tend to get lost in
> the teaching of some Egyptian conservatories, for
> example, that 24-EDO is an adequate model.
>

Hence the ages old dilemma: More power to simplicity and regularity of
pitches? Or more power to the subtleties of nuance at the expense of
the easy-learning curve?

The oscillation between these two poles never cease.

> Best,
>
> Margo
>

Cordially,
Oz.

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:

> My ears indicate that the whereabouts of 250 cents is a desirable
> concordant augmented second interval in the Hijaz tetrachord. It >is
> curious, because 144/125, 125/108 or 196/169 are not easy JI ratios.
>
> Arguably, since JI is meant to justify Western tonality, not any other
> thing!

15/13 would be the "justification" here, I'm quite sure. I believe this:

1/1
13/12
5/4
4/3
3/2
13/8
7/4
2/1

might be the simplest Hijaz with conjunct Rast for maqam Hijaz, (8/5 instead of 13/8 for descending). This puts the nice 15/13 in the middle of the Hijaz tetrachord and surely would be the Just "explanation", were one required. Correct me if this is wrong.

-Cameron Bobro

Oh, Ozan- I was meaning to ask you about this:

1/1
13/12
26/21
4/3
3/2
13/8
7/4
2/1

which I really like but don't know where to compare in a maqam system. (with a 14/9 instead of 13/8 in descent, pretty slick :-) )Isn't the 8/7 in the lower tetrachord too narrow for this to be Hijaz though?

-Cameron Bobro

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
>
> > My ears indicate that the whereabouts of 250 cents is a desirable
> > concordant augmented second interval in the Hijaz tetrachord. It >is
> > curious, because 144/125, 125/108 or 196/169 are not easy JI ratios.
> >
> > Arguably, since JI is meant to justify Western tonality, not any other
> > thing!
>
> 15/13 would be the "justification" here, I'm quite sure. I believe this:
>
> 1/1
> 13/12
> 5/4
> 4/3
> 3/2
> 13/8
> 7/4
> 2/1
>
> might be the simplest Hijaz with conjunct Rast for maqam Hijaz, (8/5 instead of 13/8 for descending). This puts the nice 15/13 in the middle of the Hijaz tetrachord and surely would be the Just "explanation", were one required. Correct me if this is wrong.
>
> -Cameron Bobro
>

Oh, forgot to say that I realize that should be a 16/9 (it's conjunct Rast), but I was thinking in terms of conjunct pentachord, probably not authentic (but sounds good).

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
>
> > My ears indicate that the whereabouts of 250 cents is a desirable
> > concordant augmented second interval in the Hijaz tetrachord. It >is
> > curious, because 144/125, 125/108 or 196/169 are not easy JI ratios.
> >
> > Arguably, since JI is meant to justify Western tonality, not any other
> > thing!
>
> 15/13 would be the "justification" here, I'm quite sure. I believe this:
>
> 1/1
> 13/12
> 5/4
> 4/3
> 3/2
> 13/8
> 7/4
> 2/1
>
> might be the simplest Hijaz with conjunct Rast for maqam Hijaz, (8/5 instead of 13/8 for descending). This puts the nice 15/13 in the middle of the Hijaz tetrachord and surely would be the Just "explanation", were one required. Correct me if this is wrong.
>
> -Cameron Bobro
>

Indeed, Barbour is not reliable.  The entire field was in a primitive state when he wrote his book, and his comments about just intonation near the end are demonstrably false.  There's been a ton of high-quality work done in historical tunings since his time.
However, the theorists of the Renaissance in Italy were often also composers and practical musicians, and there was an quasi-scientific interchange of opinions that led them to revise their opinions based on criticism and other research.  When one reads about the tuning systems of theorists such as Zarlino and Vincentino, one can their systems seriously.
Franklin

--- On Mon, 5/17/10, Carl Lumma <carl@...> wrote:

From: Carl Lumma <carl@...>
Subject: [tuning] Re: Subgroup scales
To: tuning@yahoogroups.com
Date: Monday, May 17, 2010, 3:59 AM

Of course modern theorists aren't much more trustworthy. Claudio Di Veroli (among others) convinced me that Barbour was a modern-day Ptolemy, as it were. -Carl

Dear Cameron,

24:26:30:32:36:39:42:48 sounds most concordant to my ears, even when
all of them are played together.

Another surprising thing is that, 12:16:21 also sounds delicious
although 21:16 is a 471 cent grave fourth that should sound
discordant, but does not do so in this maqam.

Why? The maqam is, as you noted, nothing other than the principal
scale of Hijaz with 3:2 as perde huseyni, which is the asma-karar (sub-
dominant-like degree) of this maqam. There is a nifty 7:6 there
between huseyni and ajem, which is 7:4. Good show!

Then again, the dominant 4:3 (perde neva) is more important toward the
cadence, so one can alterate ajem to 16/9 to form a pure fourth in
between them.

But these are all besides the point. The main reason was to illustrate
the presence of 15:13 amidst the Hijaz tetrachord. It works very well
if indeed some JI explanation was saught. But so high a limit? Why
not! This type of Hijaz is one of those tetrachords in 13-limit that I
had formulated two years ago. There is another 13-limit Hijaz
tetrachord which goes:

39:42:48:52

where the augmented second is now a 8:7.

An AEU-compliant JI version in 7-limit is thus:

45:48:56:60

where the augmented second jumps all the way up to 7:6.

And here is a 48-tET compliant version in 67-limit:

54:58:67:72

where the 250 cent interval is preserved, albeit in 67-limit, with the
edge intervals equalized.

Oz.

✩ ✩ ✩
www.ozanyarman.com

On May 19, 2010, at 8:54 AM, cameron wrote:

> Oh, forgot to say that I realize that should be a 16/9 (it's
> conjunct Rast), but I was thinking in terms of conjunct pentachord,
> probably not authentic (but sounds good).
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>>
>>
>>
>> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
>>
>>> My ears indicate that the whereabouts of 250 cents is a desirable
>>> concordant augmented second interval in the Hijaz tetrachord. It >is
>>> curious, because 144/125, 125/108 or 196/169 are not easy JI ratios.
>>>
>>> Arguably, since JI is meant to justify Western tonality, not any
>>> other
>>> thing!
>>
>> 15/13 would be the "justification" here, I'm quite sure. I believe
>> this:
>>
>> 1/1
>> 13/12
>> 5/4
>> 4/3
>> 3/2
>> 13/8
>> 7/4
>> 2/1
>>
>> might be the simplest Hijaz with conjunct Rast for maqam Hijaz,
>> (8/5 instead of 13/8 for descending). This puts the nice 15/13 in
>> the middle of the Hijaz tetrachord and surely would be the Just
>> "explanation", were one required. Correct me if this is wrong.
>>
>> -Cameron Bobro
>>

Dear Cameron,

Please let me comment quickly on your most
interesting tuning:

1/1
13/12
26/21
4/3
3/2
13/8
7/4
2/1

Your lower tetrachord is a fine example of
a medieval type called Buzurg: a smallish
neutral second, a central 8:7, and another
smallish neutral second.

Safi al-Din al-Urmawi and Qutb al-Din
al-Shirazi, writing around 1250-1300, give
one form as 14/13-16/13-4/3, but with some
variations: your 13/12-26/21-4/3 fits right
in, and uses the same steps (13:12, 8:7, 14:13)
with the larger neutral second placed below.

While Ozan is the best authority on Turkish
theory and practice, I do know, as was discussed
in a recent thread, that this kind of tetrachord
is used by Turkish performers as a kind of
Hijaz. For example, one performer was measured at
131-368-501 cents.

Going with your very characteristic scheme for Maqam
Hijaz of Hijaz plus a conjunct Rast, and also
following your suggestion to change 7/4 to a 16/9,
we have

4/3 3/2 13/8 16/9
498 702 841 996
9:8 13:12 128:117
204 139 155

Being a lover of Buzurg, whether considered as
a separate genus or a variety of Hijaz, I find this
a natural choice, and would suggest a fine point
of naming. The 13/8, or in this tetrachord 39:32,
nicely concords with the 13/12 of your Buzurg,
but would be considerably lower than the term
Rast would suggest in a typical Arab or Turkish
usage. However, Ibn Sina, writing about a millennium
ago, described _precisely_ this tetrachord, a
beautiful companion to your Buzurg.

He called it Mustaqim: 1/1-9/8-39/32-4/3, as has been
discussed in another thread about Jacques Dudon's
Mohajira. Some Arab theorists call it Rast Jadid,
or "New Rast," meaning that the neutral third is
about a comma lower than in a usual Rast, where it
might be somewhere around 27/22, 16/13, 21/17, or
26/21, for example. In Turkey, it's often as high
as 5/4, or as Ozan has said, sometimes a tad lower,
say 380-384 cents.

With this arrangement, you also have a fine disjunct
tetrachord to the lower Buzurg which the Arabs call
Bayati, the Persians Shur, and the Turks might call
Ushshaq or the like, with Ozan as our guide.

3/2 13/8 16/9 2/1
702 841 996 1200
13:12 128:117 9:8
139 155 204

This is another permutation of Ibn Sina's Mustaqim,
and still quite useful after roughly 1000 years!

Let me not neglect your other tuning of this
tetrachord, which also happens to have been
described by -- guess who! -- Ibn Sina.

3/2 13/8 7/4 2/1
702 841 969 1200
13:12 14:13 8:7
139 128 231

I'm not sure if it would fit with the lower
Buzurg, but if you have a 7/6 available, it
could make a nice modulation: your lower
tetrachord would be 1/1-13/12-7/6-4/3, and
you'd have a version of Ibn Sina's octave
scale for this septimal genus. People have
differed as to whether he has the 14:13
or the 13:12 come first, but your version
of 12:13:14:16 is a fine choice.

As you noted, when descending in Maqam
Hijaz, one often leans to a minor sixth
above the final: I tend to favor something
around 11/7 for this, although a Pythagorean
128/81 would be a common choice, and 14/9
might be very expressive, as I've found in
some other contexts.

Anyway, I sometimes rely on the Turkish
practice of using a Buzurg intonation
as a kind of Hijaz, and would add that
the highest form of praise for the
classic medieval Islamic tunings is to
use them well, and indeed from time to
time to rediscover them in our quest
for musical beauty.

With many thanks,

Margo Schulter
mschulter@...

Margo and Ozan, thank you for your enlightening posts. I'm replying in great detail to your observations, but by way of working on a piece of music, in which the tetrachords are subjected to the physical limitations of specific instruments as well as theoretical concerns.

For example, I must use the Ibn Sina voicing
14/13-8/7-13/12 because on my Bb Boehm-system clarinet, I cannot get a satisfactory 13:12 and 26:21 above the 1/1 I am using, while 14:13 and 16:13 are strong and easy (with cross-fingering and klezmer-style embouchure of course). Although the story will probably be different once I get a Turkish clarinet (Albert on G, very open mouthpiece), on the Boehm the fingerings and embouchure which can obtain the 13:12 and 26:21 keep wanting to leap to 12:11 and 5:4.

And so on, hopefully I'll have the tune for you asap, very busy as always...

-Cameron Bobro

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Dear Cameron,
>
> Please let me comment quickly on your most
> interesting tuning:
>
> 1/1
> 13/12
> 26/21
> 4/3
> 3/2
> 13/8
> 7/4
> 2/1
>
> Your lower tetrachord is a fine example of
> a medieval type called Buzurg: a smallish
> neutral second, a central 8:7, and another
> smallish neutral second.
>
> Safi al-Din al-Urmawi and Qutb al-Din
> al-Shirazi, writing around 1250-1300, give
> one form as 14/13-16/13-4/3, but with some
> variations: your 13/12-26/21-4/3 fits right
> in, and uses the same steps (13:12, 8:7, 14:13)
> with the larger neutral second placed below.
>
> While Ozan is the best authority on Turkish
> theory and practice, I do know, as was discussed
> in a recent thread, that this kind of tetrachord
> is used by Turkish performers as a kind of
> Hijaz. For example, one performer was measured at
> 131-368-501 cents.
>
> Going with your very characteristic scheme for Maqam
> Hijaz of Hijaz plus a conjunct Rast, and also
> following your suggestion to change 7/4 to a 16/9,
> we have
>
> 4/3 3/2 13/8 16/9
> 498 702 841 996
> 9:8 13:12 128:117
> 204 139 155
>
> Being a lover of Buzurg, whether considered as
> a separate genus or a variety of Hijaz, I find this
> a natural choice, and would suggest a fine point
> of naming. The 13/8, or in this tetrachord 39:32,
> nicely concords with the 13/12 of your Buzurg,
> but would be considerably lower than the term
> Rast would suggest in a typical Arab or Turkish
> usage. However, Ibn Sina, writing about a millennium
> ago, described _precisely_ this tetrachord, a
> beautiful companion to your Buzurg.
>
> He called it Mustaqim: 1/1-9/8-39/32-4/3, as has been
> discussed in another thread about Jacques Dudon's
> Mohajira. Some Arab theorists call it Rast Jadid,
> or "New Rast," meaning that the neutral third is
> about a comma lower than in a usual Rast, where it
> might be somewhere around 27/22, 16/13, 21/17, or
> 26/21, for example. In Turkey, it's often as high
> as 5/4, or as Ozan has said, sometimes a tad lower,
> say 380-384 cents.
>
> With this arrangement, you also have a fine disjunct
> tetrachord to the lower Buzurg which the Arabs call
> Bayati, the Persians Shur, and the Turks might call
> Ushshaq or the like, with Ozan as our guide.
>
> 3/2 13/8 16/9 2/1
> 702 841 996 1200
> 13:12 128:117 9:8
> 139 155 204
>
> This is another permutation of Ibn Sina's Mustaqim,
> and still quite useful after roughly 1000 years!
>
> Let me not neglect your other tuning of this
> tetrachord, which also happens to have been
> described by -- guess who! -- Ibn Sina.
>
> 3/2 13/8 7/4 2/1
> 702 841 969 1200
> 13:12 14:13 8:7
> 139 128 231
>
> I'm not sure if it would fit with the lower
> Buzurg, but if you have a 7/6 available, it
> could make a nice modulation: your lower
> tetrachord would be 1/1-13/12-7/6-4/3, and
> you'd have a version of Ibn Sina's octave
> scale for this septimal genus. People have
> differed as to whether he has the 14:13
> or the 13:12 come first, but your version
> of 12:13:14:16 is a fine choice.
>
> As you noted, when descending in Maqam
> Hijaz, one often leans to a minor sixth
> above the final: I tend to favor something
> around 11/7 for this, although a Pythagorean
> 128/81 would be a common choice, and 14/9
> might be very expressive, as I've found in
> some other contexts.
>
> Anyway, I sometimes rely on the Turkish
> practice of using a Buzurg intonation
> as a kind of Hijaz, and would add that
> the highest form of praise for the
> classic medieval Islamic tunings is to
> use them well, and indeed from time to
> time to rediscover them in our quest
> for musical beauty.
>
> With many thanks,
>
> Margo Schulter
> mschulter@...
>