Yahoo Tuning Groups Ultimate Backup tuning A temperament for Maqam

Hi folks,
These are my additions to a recent post by Gene intitled "Bleu", and its comments from Keenan Pepper (message 100439, digest #7274)
My excuses for not being very attentive to the TL these days (I am involved in the process of creating an ecovillage and music is sacrified !), but as it happened I noticed Gene was speaking french and that's how I found that what he was talking about was not far from a model I have been working on for years, based on the just intonation qanun tunings of my friend and arabic music master Julien Jalaleddin Weiss.

Julien Jalaleddine Weiss is the Director of the world-famous arabian classical music ensemble Al-Kindi, and has been playing qanun with some of the greatest singers and musicians of the Arab world. I am sure you can find some infos on internet about him as he recorded 20 CD with Al-Kindi, along with such famous singers as Sheikh Hamza Shakkur, Sabri Mudallal, Omar Sarmini, Adib Dayir, Husseyn Al Azami, Lotfi Bouchnak, Sheikh Habboush, Dogan Dikmen or Bekir Buyukbas.
What is of the highest interest to us, if we want to know more about the classical arabian Maqam intonation, is that he has been designing the tuning luthery of 9 qanun prototypes based on JI variations, all in a microtonal system of his own, that ables him to be in tune with the diversity of the whole span of schools and Maqam traditions of Middle-East.
In order to achieve this he has extended the usual number of Mandal-Orabs (tuning stops) of all his qanuns to 15 positions for each string, which he controls with an outstanding virtuosity. I have seen Julien in concert many times and I am always amazed by the richness of modulations he uses, in order to follow the most subtle inflexions of the music, of the singers and the other instruments.

But before I present more precisely Julien's system, let me relate a story :
On the 2nd of january 2011, after they gave a concert for the new year's eve in my village, I was invited by the German "ethnic jazz" group Embryo on their way to Marocco, for a jam with them. Embryo are pioneer musicians in their domain, as they join with local musicians wherever they go in the world, and have a certain experience with oriental music. I knew they were fond of quartertones, and since they had mostly western instruments (trombone, sax, trumpet) except a oud, a retuned marimba and the santoor, played by Christian Burchard, I decided to make it simple, by bringing with me a banjo refretted to a JI version of Mohajira.
The only problem was that my banjo was tuned around the Bayati mode (with low-type neutral tones such as 13/12, 39/32 etc. above the open strings), while their quartertones instruments were tuned on the Rast mode, with higher types of neutral thirds (128/117, 16/13, etc. above the tonic). Of course I could have simply pulled my strings up each time I would play a quartertone, but instead I prefered to take the risk to raise my open strings to the only 3 quartertones they had (half-flat Re, Mi, Si), that I choosed (mischieviously) to emphasize as the tonic notes...
The musicians were a bit surprised at the beginning but they heard it was sounding good and soon we were having the most delightful time together, passing from Indonesian pelogs to Rast or Bayati in various keys, with occasional touches of Mauritanian and Malagasian music ; up till the end nobody could tell what the tonic was, and we left it unresolved.
If I tell this story it is to remind us of one essential feature of Arabian music, that it has no one single sort of neutral tones, but in essence at least two distinct ones.
For example if you play a traditional Bayati pentachord in D :
3 3 4 4 (= quartertones)
and change your tonic to C :
4 3 3 4 4
you may get roughly the same quartertones structure as a Rast mode, but it will not sound Rast, because the neutral third would be some kind of a 39/32 and that's simply too low for Rast's Segah.
This is a basic example of why the neutral seconds, the neutral thirds etc. in the Arabic Maqam are (at least) of two types (or, if the two are'nt needed together, at least of one dissymetric type).

The same necessity applies to differential coherence, as I mentioned on this list last year in several exchanges with Margo Schulter. For an optimal -c, the neutral thirds in a chain need to be of two distinct types again : one of a "11 - 9 = 2" type, that generates a wholetone (here a 9/8) under the tonic, and one of the "59 - 48 = 11" type, that generates a neutral tone (here a 12/11) under the tonic : two types.

Some time ago Cameron explained this wonderful way to fret a 16/13 on a string, that is as old as Maqam :
> Drop a finger halfway between 4:3 and the nut, you've got 8:7,
> drop a finger halfway between there and 4:3 and you've got 16:13.

Likewise, if you take the aliquot point between 9/8 and 4/3 (other very basic proportions) then you have :
32 + 27 / 72 of the string, or a 72/59 interval, precisely the complement in a fifth of the -c neutral third 59/48 I just mentionned : high-limit ratios, but very old luthery practices (a division suggested by Ibn-Sina...).
And between these two classical aliquot neutral thirds 72/59 and 16/13 is a 118/117 comma, that we may not want to temper in just intoned Maqam.
Likewise Mohajira rational sequences, such as : 26 : 32 : 39 : 48 : 59 : 72 : 88 propose also several neutral thirds in their beginning, but will tend towards an average unique neutral third, dividing the fifth into two equal parts, such as 7 + 7 steps in a 24-edo, or 9 + 9 steps in 31-edo.
These do not follow the original patterns of the classical Arabic Maqam.
Mohajira also dissolves the 81/80 comma by tempering the fifth, this is not the essence of the classical Maqam either (neither Arabic nor Turkish), that is typically based on tetrachords, and pure fifths all the way (it's also pure fifths you get naturally from a ney flute, that passes from h.2 to h.3, to reach the higher tetrachord).
Furthermore, the simplest harmonic neutral thirds we know, such as 11/9 or 16/13 divide a 3/2 fifth in two unequal parts.

Among other features, this distinction between at least two types of neutral thirds is a characteristic of Julien's system.
Those are found inside the span of the division of an apotome (16/15 or rather 2187/2048) in 7 unequal commas, under and above each note of a Pythagorean diatonic scale : 1/1, 9/8, 81/64, 4/3, 3/2, 27/16, 243/128 (here 1/1 or the note "DO" of Julien's diatonic open tuning actually refers to B flat (-10 of my symetric horagram) ;
in most of his tunings, these apotomes are boundered with syntonic commas and each remnant lagu (25/24) is divided in 5 smaller commas of unequal but more or less even sizes.
These are Julien's qanun tuning criterias :
- pure octaves
- pure fifths (as a major criteria)
- pure major thirds
- these three leading to a frame equivalent to the 22 shrutis, extensible to 29 notes of a sequence of fifths
- those are completed by a variety of JI intervals of all possible limits, in which the factor 13 is always present.

For years Julien and I have shared discussions without end about the ratios he wanted to tune his qanuns in, and about the coherence these should have together. Though he does not want to hear about temperament, I came to see that he makes very precise distinctions between the commas he wants to keep in control while playing, and the less relevant smaller variations between them. And since the basic frame of his tunings remains stable I always thought he was turning around some kind of a hidden temperament, explainable by the common structure of all his tunings.
After trying several potential generators, the lower neutral second revealed to be the best one.
For one thing, the "3/4 tone" or neutral second is an omnipresent interval in Maqam music : whether in Rast or Bayati, each have 2 of those in each tetrachord, and a whole tone.
If we focus closer on the 32/27 area between 9/8 and 4/3 as it gets divided by the two traditional aliquot divisions I described before, the lower 3/4 tone between 9/8 and 72/59 is 64/59, while the higher 3/4 tone is a 13/12 and in the middle is a 118/117 comma.
The difference between 13/12 and 64/59 is only 768/767, and this small dissymmetry certainly can be tempered.
By chance, if you divide a pure fifth in 5 equal parts you will get an interval of 1.084471771 or 140.391 c., that falls in between 13/12 and 64/59 : I think you now get the idea of my temperament...

You may have a better vision of it by opening the file "Tsaharuk.jpg" in my TL folder.
There are two ways to arrive to this structure of 12 lagus (each divided in five small commas), and separated by 17 syntonic commas evenly distributed.
A convenient way is to go through a planar temperament using such a generator of one 1/5 of a fifth, and a complementary generator of a pure or quasi-pure 5/4.
Or, the same structure can be attained by a linear temperament using the first generator only, knowing that 40 generators (or 8 fifths just like in any schismatic temperament) lead to the equivalent of a 128/5 ratio : when using 8 fourths minus 3 octaves as a major third in the planar version, both are equivalent.
In both cases the fifth acts in practice as an extra-periodic interval.
So Tsaharuk in its linear version tempers 32805/32768, and knowing this feature it suggests that the 1/5 of a POTE L-5 schismatic temperament could be a solution for a generator (approximated by 244/225) in favor of harmonic 5.
However, between the best values that can be found for h. 3 and 5, Tsaharuk proposes also a very precise resolution of harmonic 7, after 24 generators, that raises the optimal solutions for h 3, 5 , 7 altogether a little higher, and closer to Julien's criterias.
Using exactly one 1/24th of a (7:1) as a generator is another solution, itself very close to the algebraic solution of x^24 = 2x^5 + 4 (x = 1.084455776530908 = 140.3654663448 c., giving a fifth of 701.827331724 c.) that offers more rigorous -c 7/4 intervals, and triple equal-beating properties.
Not far from this area is a found a Meta-temperament version of Tsaharuk (showing perfectly self-similar interval patterns) when the generator size is 2/ (12 + sqrt(26)) of one octave, or 140.3589251 c.

Any of those generators, close enough to one 1/5 of a pure fifth leads to a main DE scale of 77 notes, that can be extended further to reach Julien JW's full qanun tuning (105 notes or a little less in pratice). Here are some JI approximations of the first intervals generated :

(0 = 1/1)
1 ~ 13/12, 1024/945, 64/59, 243/224
2 ~ 20/17, 153/130, 207/176
3 ~ 14/11, 51/40, 88/69
4 ~ 18/13, 112/81, 177/128
5 ~ 3/2
6 ~ 13/8, 512/315, 96/59, etc.

As the mapping for the first harmonics (3 5 7 11 13 17 19 23 29) is : (5 -40 24 21 6 -42 -15 13 33),
the larger neutral 2th is attained by -16 generators, which considers these ratios as variations :
12/11, 130/119, 153/140, 59/54, 176/161, 35/32, 128/117, etc.
Which means that Tsaharuk also tempers 4096/4095, 3520/3519, 2601/2600, 2185/2184, 1521/1520, 945/944, 833/832, 768/767, 729/728, 561/560, 484/483, etc., along with 385/384, 378/377, 352/351.
The harmonic suite 116:117:118:119:120 is equal in Tsaharuk and lets you figurate some more of the commas it dissolves.

A few other approximations :
-17 ~ 81/80
10 ~ 9/8
-15 ~ 32/27
11 ~ 39/32
-6 ~ 16/13
-23 ~ 119/96 (a useful and fairly good Turkish low major 3rd)
20 ~ 81/64, 224/177
-5 ~ 4/3
21 ~ 351/256
-13 ~ 39/28
-4 ~ 13/9
-25 ~ 256/243, 59/56, etc.

And a basic mapping for a Bayati in C = 0 for those who want to test it :
0 1 -15 -5 5 6 -10
along with variations for Rast in Bb (= -10) : -16 -11

The "parametric" horagram representation I used for my octave wheel shows the developpement of the temperament in a 77 notes DE, but along with a radial micromodulation of the generator, resulting into a continuous variation of structure.
The outer rings pass by several edos but for more clarity I reported only three of them, in the most pertinent zone close to pure fifths : 77-edo, 171-edo, 94-edo, with respectiveley 9, 20, 11 steps for the generator.
The 94-edo is the only one with fifths larger than 3/2 ; a recurrent sequence such as x^29 = 4x^6 + 4 provides such a cycle, with x = 1.0844858376952085 = 140.4134555676 c., and a fifth of 702.067277838 c.
One interesting circle between 171 and 94 would be a 359-edo (with 42 steps for the generator) as it's near to it we find the purest fifths, while the best h.13 is found beyond the 77-edo ring (that would reduce to a 60-edo at some point) ; the 60-edo ring is not represented here but we can figure easily where it would be situated, passing by the crossing of the lines extended after 17 and -17 (BTW 4x^6 = x^5 + 5 (x = 1.0842367808 = 140.01582452796 c.), or x^19 = x^15 + 1, (x = 1.084180083774 = 139.925292225 c.) are correct algebraic solutions for "Bleu", and the last one should be a fun sequence maker).

As a conclusion, while Tsaharuk as a temperament does not offer the simplicity of a Mohajira or a Buzurg (a temperament for Persian music I described some time ago), as a 7-limit schismatic micro-temperament integrating realistic neutral intervals it proposes a range of tunings for Arabic Maqam (as well as for Turkish Maqam to a certain extent), leading to scale structures that have been demonstrated by Julien Jalaleddine Weiss as completely relevant to Maqam music.
I hope this humble contribution will motivate some of you to discover new oriental flavors, to experiment with Maqam scales and tunings and perhaps to learn more about one of the world's richest musical treasures of all times.
- - - - - - -
Jacques

🔗Margo Schulter <mschulter@...>

[This is the first part of a two-part reply, with the second
part to be posted immediately after this one. My greetings
also to Mike Battaglia, with many thanks for his courtesy
and graciousness -- M.S.]

> Hi folks,These are my additions to a recent post by
> Geneï¿œintitled "Bleu", and its comments fromï¿œKeenan
> Pepperï¿œ(message 100439,ï¿œdigest #7274) My excuses for not
> being very attentive to the TL these days (I am involved in the
> process of creating an ecovillage and music is sacrified !),
> but as it happened I noticed Gene was speaking french and
> that's how I found that what he was talking about was not far
> from a model I have been working on for years, based on
> theï¿œjust intonationï¿œqanun tunings of my friend and arabic
> music master Julien Jalaleddin Weiss.

Dearest Jacques,

Thank you for such a wonderful and elevating letter at once
bringing us, as it were, into the presence of Julien Jalaleddin
Weiss and his inspired ensemble, and inviting us to contemplate
more closely the fine points of Maqam music and tuning whether in
just or in tempered approaches.

And I send greetings also to Keenan Pepper, who provided the
foundation for what I still consider one of the most excellent
approaches to a more modest maqam temperament for a fixed-pitch
instrument, of which I have recently devised a variation which
may be slightly more accurate in representing the ratio of 11:10
which seems often useful in Turkish music.

> Julien Jalaleddine Weiss is the Director of the world-famous
> arabian classical music ensemble Al-Kindi,ï¿œand has been
> playing qanun with some of the greatest singers and musicians
> of theï¿œArabï¿œworld. I am sure you can find some infos on
> internet about him as heï¿œrecorded 20 CD withï¿œAl-Kindi, along
> with such famous singers asï¿œSheikh Hamza Shakkur,ï¿œSabri
> Mudallal, Omar Sarmini,ï¿œAdib Dayir,ï¿œHusseyn Al Azami,ï¿œLotfi
> Bouchnak, Sheikh Habboush, Dogan Dikmen or Bekir Buyukbas.ï¿œ

Indeed I would imagine that to hear such an ensemble would be
like hearing the music at St. Mark's Cathedral in Venice in the
time of Giovanni Gabrieli, an experience related by a visitor
from England, Fynes Moryson.

And this should be an occasion at once of joy for you, to have
the benefit of sharing in and learning from such consummate
mastery, and of humility for me, to remember that my concepts of
maqam music must quite literally be "outlandish," since I have
not had the benefit of such sound and able tuition.

> What is of the highest interest to us, if we want to know more
> about the classical arabian Maqam intonation, is that he has
> been designing the tuning luthery of 9 qanun prototypes based
> on JI variations, all in a microtonal system of his own, that
> ables him to be in tune with the diversity of the whole span of
> schools and Maqam traditions of Middle-East.

Indeed this seems like the "universal Maqam instrument" to which
some musicians of the Middle East have aspired from time to
time. And for those of us who may design and use less perfect and
universal tuning systems also, your account is an invaluable
guide as to what is possible, and may be realized in some small
part in more modest systems also.

> In order to achieve this he has extended the usual number of
> Mandal-Orabs (tuning stops) of all his qanuns to 15 positions
> for each string, which he controls with an outstanding
> virtuosity. I have seen Julien in concert many times and I am
> always amazed by the richness of modulations he uses, in order
> to follow the most subtle inflexions of the music, of the
> singers and the other instruments.

Here I might comment that the modest problem to which I have
given almost my attention is the tuning of a solo keyboard
instrument, an electronic synthesizer in 1024-EDO as it happens,
where no other instruments or performers are involved, and so
many of fine points and subtleties of practical ensemble playing
which have shaped Julien's system do not come into the question.
Yet the desire for intonational variety, flexibility, and choice
which is central to the highest level of _tarab_ ("ecstasy" or
"enchantment") in maqam performance, may be realized to some
degree even in such modest systems.

At this point I should duly apologize for the fact that, reading
such a magnificent letter as yours, I find that any fitting reply
attempting a similar scope would at once become longer than even
I might care to post <grin>, and on many points would go beyond
my knowledge and experience.

Yet there are a few points on which I might comment. The first is
your delightful discussion of Rast and Bayyati, which invites a
due recognition of the most noble Mustaqim of Ibn Sina, which
among the Arabs is sometimes styled Rast Jadid or "New Rast."
It might also be of interest here to show how the fine nuance in
the placement of the step Segah (Arabic Sikah) for these maqamat
may sometimes be addressed, albeit imperfectly, on a 24-note
keyboard.

The second concerns the comma between the smaller and larger size
of neutral or Zalzalian second in certain divisions -- for
example, 64/59 and 128/117 in your example with a difference or
comma at 118/117. This raises some fine questions which might
apply to either just or tempered tuning.

The third is a fascinating passage in your letter mentioning the
division of a "remnant lagu" (25/24) into "5 smaller commas of
unequal but more or less even sizes." Here I shall propose one
interpretation of these five commas related to the four ratios
for superparticular neutral seconds -- 14:13, 13:12, 12:11, and
11:10 -- and show how, in some tempered systems used for maqam or
dastgah music, the commas between these ratios may be represented
imperfectly but still more or less accurately by aliquot
logarithmic parts. And such divisions, of course, should not be
confused with the most noble aliquot divisions one make on a
monochord or fretboard and to which you rightly give place of
honor in your discourse, as 16:14:13:12 or 72:64:59:54.

As it happens, I find that the exposition of even this ground
requires a reply broken into parts, of which this is the first.

* * *

Before delving into these points, I would take care to say that
the majesty and excellence of the music and tuning you describe
go far beyond such points, which at most may perhaps illustrate a
freedom, accuracy, and flexibility for which we all may strive.

Your brief summary of Julien's criteria for tuning the qanun
might itself inspire many voluminous treatises, and while some
might quibble over details of regional taste here and there, I
must conclude that he and his colleagues have set a standard for
those of us pursuing more modest or limited goals at once to
admire and to emulate at least in part.

Let me briefly add something about the role of ratios, which you
and I alike find a natural language for discussing not only just
intonation but also tempered systems. What I would like to
emphasize is that some of the best Middle Eastern musicians and
theorists such as Amine Beyhom, Ali Jihad Racy, Hormoz Farhat,
and Dariush Tala`i do not dwell much on ratios, but are very
concerned indeed with artful maqam intonation, including, of
course, the use of unequal neutral second steps. And scholars
from other parts of the world such as Scott Marcus, likewise, use
values in cents (or sometimes, as with Racy, in 53-commas) to
document the "understanding" of traditional performers in Egypt,
for example, as to the higher or lower placement of certain
neutral steps and intervals.

Myself, I am an unapologetic lover of ratios, but must admit also
to resorting to temperament, with your wise discussion as to some
commas perhaps being more essential than others in a given style
as one fine starting point for a fuller discussion. Here,
however, I shall seek to focus mainly on the aforementioned three
points.

----------------------
1. Of Rast and Bayyati
----------------------

There could be no better beginning on this point than to quote
from your most delightful story of your encounter with a German
"ethnic jazz" group called Embryo in your village two days after
they have given a New Years' Eve concert:

> The only problem was that my banjo was tuned around the Bayati
> mode (with low-type neutral tones such as 13/12, 39/32
> etc. above the open strings), while their quartertones
> instruments were tuned on the Rast mode, with higher types of
> neutral thirds (128/117, 16/13, etc. above the tonic). Of
> course I could have simply pulled my strings up each time I
> would play a quartertone, but instead I prefered to take the
> risk to raise my open strings to the only 3 quartertones they
> had (half-flat Re, Mi, Si), that I choosed (mischieviously) to
> emphasize as the tonic notes...

One small detail is that, for Rast on Ut, I might have also
desired a half-flat La for the neutral sixth step of conjunct
Rast, placed by al-Farabi (not yet under the name of Rast) at
18/11; or at 64/39 to match a 16/13 third at rast-segah, for
example (assuming pure fourths). However, it may be that this
neutral sixth is not so commonly used by all performers, some of
them possibly keeping La unchanged (27/16 or whatever) while
placing Si half-flat (e.g. 81/44 or 24/13) when ascending and
Si-b (e.g. 16/9) when descending, for the most part.

> The musicians were a bit surprised at the beginning but they
> heard it was sounding good and soon we were having the most
> delightful time together, passing from Indonesian pelogs to
> Rast or Bayati in various keys, withï¿œoccasionalï¿œtouches of
> Mauritanian and Malagasian music ; up till the end nobody could
> tell what the tonic was, and we left it unresolved.

This must have been a joy to hear, but a yet greater joy to
play.

> If I tell this story it is to remind us of one essential
> feature of Arabian music, that it has no one single sort of
> neutral tones, but in essence at least two distinct ones.ï¿œ

A vital point which you well make is that while a tuning system
as impressive as Julien's follows this rule, so does the more
modest system of a 17-note Persian tar or setar, for example, or
even the more limited gamut of the santur which generally has
fewer than 12 notes per octave.

> For example if you play a traditional Bayati pentachord in D :
> 3 ï¿œ3 ï¿œ4 ï¿œ4 (= quartertones)
> and change your tonic to C :
> 4 ï¿œ3 ï¿œ3 ï¿œ4 ï¿œ4

> you may get roughly the same quartertones structure as a
> Rastï¿œmode, but it will not sound Rast, because the neutral
> third would be some kind of a 39/32 and that's simply too low
> for Rast's Segah.

One way sometimes used by Arab theorists to illustrate this point
is by counting the intervals in approximate commas -- not as an
exact measure, but to show which neutral steps are understood as
larger or smaller. And it may be done by ratios or cents also.

Here there are actually four possibilities, each of which has
musical charm and value, but two of which indeed define the
favored tunings for Rast and Bayyati. Thus let us begin, as in
your example, with Bayyati, and then add a step at an exact or
approximate 9:8 tone below, producing a maqam somewhat different
from a modern Arab or Turkish Rast, although beautiful:

Here the ASCII accidental "d" signifies a half-flat, and "+" a
half-sharp:

Bayyati:

dugah segah chargah neva huseyni
D E-d F G A
Re Mi-d Fa Sol La
Quarters: 0 3 6 10 14
3 3 4 4

53-commas: 0 6 13 22 31
6 7 9 9

ratios: 1/1 13/12 32/27 4/3 3/2
cents: 0 139 294 498 702
13:12 128:117 9:8 9:8
139 155 204 204

Here, while 24-step notation tells us that there are two neutral
seconds followed by two usual tones, 3-3-4-4, any of the other
notations shows that the first neutral second step is smaller
than the second in a stylish Bayyati. I have chosen the ratios
mentioned in your story, but rounded cents of the kind used by
Farhat or Beyhom would make the same point: say, for example,
that the steps are 140-155-205-200 cents.

> and change your tonic to C :
> 4 ï¿œ3 ï¿œ3 ï¿œ4 ï¿œ4

Mustaqim:

rast dugah segah chargah neva
C D E-d F G
Ut Re Mi-d Fa Sol
Quarters: 0 4 7 10 14
4 3 3 4

53-commas: 0 9 15 22 31
9 6 7 9

ratios: 1/1 9/8 39/32 4/3 3/2
cents: 0 204 342 498 702
9:8 13:12 128:117 9:8
204 139 155 204

Here we have a usual 9:8 tone from rast to dugah, plus the 13:12
step at dugah-segah so excellent for Bayyati, arriving at a
rast-segah interval of 39/32 -- exactly what Ibn Sina called for
in his description of a mode called Mustaqim, Arabic for the
"true, correct, or regular" mode, a meaning likewise signified by
the Persian name Rast.

This beautiful mode may be heard today in some of the Persian
gushe-ha, or modal themes which make up the dastgah-ha or modal
families, although it does not seem to be recognized as a dastgah
or avaz (satellite dastgah) in itself. The gushe of Mahur Dastgah
called Shekaste may give one of the best examples. And the
beautiful Mustaqim tetrachord may also be found, for example, on
the fourth step of your Ibina.

However, as you emphasize, 39/32 is simply not high enough for a
stylish Arab or Turkish Rast, with al-Farabi's 27/22 (355 cents)
or Beyhom's 200-155-145 cents for a typical modern Lebanese Rast
(with rast-segah at around 355 cents, very close to al-Farabi) as
a guide for a low or moderate tuning.

Thus, as Scott Marcus sums things up, segah must be higher for
Rast than for Bayyati; and as your story suggests, as well as a
detail of Beyhom's suggestions as to modern Lebanese tunings,
musicians not able or inclined to make this distinction will tend
to favor the steps of Rast.

Rast:

rast dugah segah chargah neva
C D E-d F G
Ut Re Mi-d Fa Sol
Quarters: 0 4 7 10 14
4 3 3 4

53-commas: 0 9 16 22 31
9 7 6 9

ratios: 1/1 9/8 16/13 4/3 3/2
cents: 0 204 359 498 702
9:8 128:117 13:12 9:8
204 155 204 204

An advantage of rast-segah at 16/13 is that it should be pleasing
in many Arab localities, and at the same time high enough to
qualify as a correct if "low" Turkish Rast, as Ozan Yarman has
said.

Now let us assume that a fixed-pitch instrument with only one
step to represent segah has been tuned for Rast, and is now used
to play in Bayyati. We thus have:

High Bayyati (or Huseyni):

dugah segah chargah neva huseyni
D E-d F G A
Re Mi-d Fa Sol La
Quarters: 0 3 6 10 14
3 3 4 4

53-commas: 0 7 13 22 31
7 6 9 9

ratios: 1/1 128/117 32/27 4/3 3/2
cents: 0 155 294 498 702
128:117 13:12 9:8 9:8
155 139 204 204

Indeed some distinguished Arab musicians have accepted, or at
least resigned themselves to this high neutral second at
dugah-segah in Maqam Bayyati, so that the Syrian musician Tawfiq
al-Sabbagh, for example, gives this tetrachord as 7-6-9 commas,
although another Syrian source as well as the experience of
Beyhom and Marcus in other parts of the Arab world would favor a
notation of 6-7-9 commas. And the intent of such 53-comma
notation is often not to specify any exact tuning (read
literally, around 158-136-204 cents or 136-158-204 cents), but to
indicate the ordering of the unequal neutral seconds, with the
smallest placed below (6-7-9 commas) -- in contrast to Rast
(9-7-6 commas) where the larger neutral second precedes the
smaller.

However, while it may be less than ideal for Bayyati, 7-6-9
commas or the like is a beautiful intonation for Maqam Huseyni,
which might call for a more bright or outgoing quality, or for
one shading of Turkish Ushshaq.

Now it is not necessarily a flaw of the 24-step notation that it
does not specify the intonational details of which neutral
seconds are larger or smaller -- as long as we understand that it
is best used as merely a guide to the general types of intervals
(e.g. major, minor, and neutral seconds), as likewise with a
notation such as that of Safi al-Din al-Urmawi in the 13th
century based on 17 steps to the octave, which Beyhom has come to
favor as preferable to 24.

As Beyhom emphasizes, having used both the 24-step and 17-step
notations, they merely define moveable steps or "locations," and
certainly need not imply an _equal_ spacing of these steps!

What I urge is that while the proper patterns of modern Rast and
Bayyati (approximately 9-7-6 and 6-7-9 commas) may be most
familiar, Mustaqim or Rast Jadid (9-6-7) and a high Arab Huseyni
(7-6-9) should also be deliberately cultivated rather than fallen
into mainly by mistake or inertia.

If any may desire a "ratio-unromantic" accounting of these four
tunings in a nutshell, and using the same step sizes in rounded
cents for each, with a 500-cent fourth and a 155-145 division of
a 300-cent third (as in Beyhom's estimate for a Lebanese Rast),
then we have:

Arrangement for D Bayyati or C Mustaqim

0 200 345 500 700
ut re mi-d fa sol
(C) D Ed F G
(200) 145 155 200

Arrangement for D Huseyni or C Rast
(sometimes used for high D Bayyati)

0 200 355 500 700
ut re mi-d fa sol
(C) D Ed F G
(200) 155 145 200

Let it be added that the reality is much more subtle, with
Beyhom's estimated position for Ed in a "popular" Lebanese
Bayyati being around 130 cents above D rather than the 145 cents
we would get simply by reversing the order of steps in his Rast.
His account of an "academic" or "learned" (_musique savante_)
Bayyati, if I correctly understand the French, reports a
placement at 155 cents, from which we might infer that persons of
this school leave the accustomed position of segah in Rast
unchanged for Bayyati. Your own merry tale of playing with the
group Embryo might suggest this possibility.

> This is a basic example of why the neutral seconds, the neutral
> thirds etc. in the Arabic Maqam are (at least) of two types
> (or, if the two are'nt needed together, at least of one
> dissymetric type).

Ideally, of course, on a qanon for example, one might simply have
two locations for segah, say at 39/32 above rast for Bayyati, and
16/13 above it for Rast. However, on a keyboard in the O3 tuning,
there are some interesting situations revealing both the
possibilities and manifest imperfections of this tuning.

Suppose, for example, we wish to have both a higher segah for
Rast and a lower one for Bayyati -- by reference to the same 1/1
or rast step, that is! One possibility would be to choose F* on
the upper keyboard as our rast or 1/1:

F* G* A Bb* C* D* E F*
0 207.4 358.6 496.9 704.3 911.7 1061.7 1200
207,4 151.2 138.2 207.4 207.4 150.0 138.2

Here each Rast tetrachord is very close to 39:44:48:52, with F*-A
at not quite a cent narrow of 16/13. For Bayyati on our dugah
step, G*, we use G#* rather than A as segah:

Eb*
785.2
G* G#* Bb* C* D* E F* G*
0 127.7 289.5 496.9 704.3 854.3 992.6 1200
127,7 161.7 207.4 207.4 150.0 138.2 207.4

Now we have dugah-segah at 127.7 cents, very close to 14/13,
which may a bit low for Egyptian tastes (Marcus suggests 135-145
cents or so), but more congenial to popular Lebanese tastes,
favoring according to Beyhom a step of around 130 cents.

A fine point that is that while the minor sixth is the usual step
above the final of Beyhom, here at 785.2 cents, or between 11/7
and 52/33, the neutral sixth at E is at 854.3 cents, very close
to 18/11, and much higher than a 3:2 fifth above our low segah
would produce! Marcus reports an "understanding" among
traditional Egyptian performers that this neutral sixth degree
should be somewhat higher than 850 cents, just as the neutral
second step should be somewhat lower than 150 cents. And it is
interesting that the mostly corresponding Turkish Maqam Ushshaq,
as measured from recorded performances of esteemed artists by
Baris Bozkurt and others including Can Akkoc and Ozan Yarman,
should show a low neutral second peaking in the region of around
130-142 cents, but a very high neutral sixth (or small major
sixth) peaking at around 872 cents!

For a Turkish Rast with a high neutral or submajor third around
26/21, one favorite location is B on the lower keyboard, and here
we may obtain a moderate Arab Bayyati with steps very close to
13:12 and 12:11.

B C# Eb E F# G# Bb B
0 207.4 369.1 495.7 703.1 911.7 1073.4 1200
207.4 161.7 126.6 207.4 208.6 161.7 126.6

This Rast, quite high by most Arab standards although moderate in
a Turkish context where segah may often approach a ratio of 5/4,
includes large and small neutral seconds of around 11:10 and
14:13 which also nicely serve a Turkish Huseyni. For an Arab
Bayyati (or Turkish Ushshaq or Arazbar), however, a lower segah
at D* is used:

784.0
A
C# D* E F# G# Bb B C#
0 138.3 288.3 495.7 704.3 866.0 992.6 1200
138.3 150.0 207.4 208.6 161.7 126.6 207.4

The Bayyati tetrachord at 138.3-150.0-207.4 cents is very close
to a division of 52:48:44:39, the lower minor third receiving an
aliquot arithmetic division on monochord of 13:12:11.

One question of taste is whether, in an Arab Bayyati, to use the
high neutral sixth step Bb at 864.8 cents (between 28/17 and
104/63, for example), or the lower neutral sixth A* at 842.6
cents. The first choice might be a bit bright for an ideal
Bayyati in some parts of the Arab world, although fine for a
Turkish Ushshaq or Huseyni; while the second would disregard the
understanding reported by Marcus that the neutral sixth should be
rather more than a perfect fifth above segah.

In these examples, the steps rast and dugah have remained fixed,
while segah takes a higher position for Rast and a lower one for
Bayyati. A different solution offers itself if we place Rast at
G* on the upper keyboard, where a usual Arab Rast is to be
obtained:

G* A* B C* D* E* F G*
0 207.4 358.6 496.9 704.3 911.7 1061.7 1200
208.6 150.0 138.2 207.4 207.4 150.0 138.2

To obtain a low Bayyati, we keep B as segah, but shift the second
step dugah of Rast from A* at 207.4 cents to Bb at 230.9 cents, a
comma higher (a virtually just 8/7), and use this as the tonic or
final of Bayyati!

F*
760.5
Bb B C* Eb F F# G* Bb
0 126.6 264.8 495.7 703.1 829.7 968.0 1200
126.6 138.3 230.9 207.4 126.6 138.3 232.0

This septimal flavor of Bayyati might fit the observation made to
Marcus by Racy, with a demonstration on a fretted instrument,
that the minor third is lower than that produced by strings tuned
in pure 4:3 fourths at 32/27. One consequence here is a
commitment to an impressively low regular minor sixth step at
760.5 cents (near 273/176, by comparison to 14/9 at 764.9 cents),
and to a low neutral sixth at 829.7 cents near 21/13, the higher
position favored in much Arab and Turkish practice not being
available at this location. Here the emphasis, of course, is on
an approximation of the 14:13:12 division of the 7:6 minor third,
on monochord 28:26:24:21.

One might say that the motto here is that it is better to strive
for variety imperfectly, and sometimes with curious if not
anomalous results, than not to strive for variety at all.

And just as a polygon with an increasing number of sides may more
and more perfectly approach a circle, so Julien's tuning may have
approached if not attained the point where the flexibility
becomes almost infinite and the potential anomalies virtually
imperceptible to most listeners.

At this spot I close this first portion of my reply, with the
next question that of the division of 25:24 into five more or
less even commas, as it may be applied to tempered as well as
just systems, for in this teaching of Julien and your report of
it I discern a high wisdom.

Most appreciatively,

Margo Schulter
mschulter@...

🔗Margo Schulter <mschulter@...>

[This is the second part of a two-part reply to Jacques
Dudon, offered with the hope that any ponderousness in my
mathematics will not distract from his lively account
of musicmaking and fascinating sketch of the philosophy
of a master in the art of Maqam. To celebrate such an
worthy epistle by a gloss or commentary such as this
may be a curious comment on my own tastes, but the
radiant spirit of musical joy which Jacques transmits
is the ultimate message which I hope may be amplified
rather than obscured by my reflections here. -- M.S.]

---------------------------------------------
2. Of aliquot frettings and the 118/117 comma
---------------------------------------------

Discussing such neutral thirds as the 11/9 type (with a
difference tone of 11 - 9 = 2) or the 59/48 type (with a
difference tone of 59 - 48 = 11), respectively at a 9:8 or a
12:11 below the lower note of the interval, both of which arise
in a Mohajira sequence, you turn to a topic of special delight as
well presented by Cameron.

Indeed in Ibn Sina we find the division of a fourth into
16:14:13:12, with 4:3 divided by equal or aliquot string lengths
first into 16:14:12 (8:7:6), and then the upper 7:6 minor third
into steps of 14:13:12. And this he calls a jins or genus most
noble, as Cris Forster explains also in his book on _Musical
Mathematics_.

16:13
|---------------|
16 : 14 : 13 : 12
1/1 8/7 16/13 4/3
8:7 14:13 13:12

While favoring the division of a 32/27 third into a lower
interval of 13:12 and a higher of 128:117, he also notes an
approximation of this later favored by Safi al-Din: an aliquot
arithmetic division of the fourth 72:64:59:54, with a lower tone
at 72:64 (9:8), and the upper third at 64:54 (or 32:27) divided
into a lower step of 64:59 and an upper step of 59:54.

72 : 64 : 59 : 54
1/1 9/8 72/59 4/3
9:8 64:59 59:54

If we expand this tetrachord to a pentachord, then we shall
obtain two sizes of neutral thirds, at 72/59 and a larger 59/48
which you have mentioned:

59:48
|----------------|
72 : 64 : 59 : 54 : 48
1/1 9/8 72/59 4/3 3/2
9:8 64:59 59:54 9:8

Here it is interesting that while 59/48 has a difference tone of
11, 72/59 has one of 13, which I suppose would produce this tone
at an interval below the lower note of the interval at 59:52, a
ratio also featured in at least one of your Mohajira tunings.

Now from Ibn Sina's "very noble" 16:14:13:12, we arrive at a
neutral third of 16:13; and this arises also if we take his ideal
tetrachord of Mustaqim and expand it to a pentachord, thus

16:13
|------------|
468 : 416 : 384 : 351 : 312
1/1 9/8 39/32 4/3 3/2
9:8 13:12 128:117 9:8

As you have well noted, the Mustaqim resulting from the aliquot
fretting of 72:64:59:54 which he mentions and Safi al-Din
espouses as one of his principal ajnas or genera has neutral
steps and intervals differing from those of his 1/1-9/8-39/32-4/3
tuning by only a comma of 768:767 (2.256 cents). And now to your
comma of 118/117:

> And between these two classical aliquot neutral thirds 72/59
> and 16/13 is a 118/117 comma, that we may not want toï¿œtemper
> in just intoned Maqam.

Here I might imagine, for example, a qanun tuned so as to support
some steps like this -- not a complete tuning, but merely an
example of how this comma might arise:

59:48
|-----------------------|
9/8 8/7 7/6 32/27 72/59
|--------------------------------------|--------------------|
1/1 16/13 4/3 3/2
|--------------------|
39:32

Here there is one neutral third at 72/59, and another at 16/13,
both in relation to the 1/1 or rast. As you observe, between
these notes there is a comma of 118:117 or 14.734 cents, let us
say about two-thirds of a comma.

Now these two thirds with respect to rast give rise to two others
of slightly different size. For example, suppose that we begin on
the 16/13 location of segah apt for an Arab or Turkish Rast, and
play in Maqam Segah (or Maqam Sikah as it is known in Arabic),
with 16/13-4/3-3/2 as our first trichord. Now between 16/13 and
4/3 we have a neutral third at 39:32, a favorite of Ibn Sina.

And suppose that playing in a Persian manner, we use the lower
position of segah at 72/59, and play Dastgah-e Segah, with a
tetrachord above this final of 72/59-4/3-3/2-96/59. Between 72/59
and 3/2 we have a third at 59:48.

If we use pure fifths and fourths, as I understand that Julien
does, then while 118:117 or not quite 15 cents will measure the
space between the 72/59 and 16/13 steps, we shall have some
additional commas expressing the difference in the sizes of a
pair of neutral thirds (e.g. 39:32 and 16:13, or 72:59 and 59:48)
together making up a 3:2 fifth.

Thus with Ibn Sina's 39:32 and 16:13, we have a difference of
512:507, or 16.990 cents, close to 3/4 of a comma. And these
sizes might be taken roughly to mark the lower and upper borders
of the realm of "central neutral thirds," since neighboring
ratios substantially smaller or larger might instead be called
supraminor or submajor thirds (e.g. 40/33 at 333 cents, or 26/21
at 370 cents).

Now with the 72:59 and 59:48 thirds brought about by the
arithmetic division which he mentions and Safi al-Din recommends,
the difference is 3481:3456 or 12.478 cents, slightly more than
half of a Pythagorean comma (23.460 cents) or equal Holderian
53-comma (22.642 cents).

With al-Farabi's neutral thirds at 11:9 and 27:22, the difference
is a smaller 243:242 or 7.139 cents, or roughly 1/3 comma. Amine
Beyhom's rounded approximation for a modern Lebanese Rast with
steps of 200-155-145 cents, and thus thirds at 345 and 355 cents,
has a comparable difference of about 10 cents.

On a justly tuned instrument such as I picture schematically
above, there might be both a 72/59 step and 16/13 step. In a
tuning system such as O3, however, we find some important commas
or differences which may not occur on the keyboard as direct
steps -- unlike the 118:117 step as imagined above.

For example, in O3 we find thirds at 345.7 cents (e.g. C-Eb*) and
357.4 cents (e.g. D*-F#), quite close to 72:59 and 59:48 at 344.7
and 357.2 cents. Here the comma is about 11.719 cents -- or almost
exactly half a Pythagorean comma, as compared to 12.478 cents for
the just 72:59 and 59:48.

In O3, however, there is no direct step on the keyboard
representing this comma -- indeed, the smallest is a full comma
of 20 tuning steps in 1024-EDO, or 23.4375 cents. Thus it is not
possible to obtain thirds at both 345.7 and 357.4 cents above the
same note -- for example, the step we choose as rast, where these
values might nicely serve respectively an Arab Bayyati and an
Arab Rast. Rather, at best, we may find 345.7 cents and 369.1
cents (an Arab Bayyati but more of a bright Turkish Rast), or
334.0 cents and 357.4 cents (a low Bayyati by at least some Arab
tastes, although a characteristic Rast).

In this kind of limited system, one makes intonational choices in
good part by moving around the keyboard and deciding where one
will notionally place rast or the gamut as a whole for a given
piece or improvisation. There is variety, but not necessarily a
high degree of consistency or congruity!

However, whatever the scope of the system, I believe that the
comma or difference in sizes between two neutral thirds together
forming a fifth is one important parameter of style and taste.
Thus we have the quite subtle contrast of al-Farabi's 27/22 and
11/9, the somewhat greater contrast of Ibn Sina's 39/32 and
16/13, representing shadings still much relished in Arab music,
and sometimes in Persian and Turkish styles as well.

Other traditional tetrachords, such as Safi al-Din's "Medium
Sundered" at 9:8-11:10-320:297 (204-165-129 cents), represent a
more pronounced degree of contrast where we may speak of submajor
and supraminor thirds, a shading which may occur in parts of the
Arab world but seems most characteristic, perhaps, of Turkish and
Persian music. Thus in Safi al-Din's tetrachord, extended to a
pentachord 1/1-9/8-99/80-4/3-3/2, we have a large 99:80 or
submajor third at 368.9 cents, and a smaller 40:33 or supraminor
third at 333.0 cents -- a difference of 3267/3200 or 35.873
cents, about 1.5 commas (Pythagorean or Holderian), and more like
a small diesis (compare the smaller diesis step on an
archicembalo in 1/4-comma meantone at 34.99 cents, which together
with the more familiar diesis at 128:125 or 41.06 cents makes up
a minor semitone at 76.05 cents).

From this it also follows that when a Middle Eastern theorist
speaks of a tetrachord of Rast being 9-7-6 commas, or of Bayyati
as 6-7-9 commas, typically no exact measurement, much less
prescription, is intended. Rather the main purpose is to show the
general types of steps and intervals, and the ordering of larger
and smaller neutral second steps.

Thus a "9-7-6" Rast in many parts of the Arab world might have
neutral second steps differing by much less than a Holderian
comma. as with al-Farabi's steps of 12:11 and 88:81; or in Turkey
by rather more, as with Safi al-Din's steps of 11:10 and 320:297.
Indeed, in Turkey, the contrast may become so great that the
notation changes to "9-8-5," with a small whole tone and a large
semitone step close to 10:9 and 16:15, and the difference of
about three commas close to the pental or 5-limit chromatic
semitone at 25:24 (70.7 cents).

At times, of course, Holderian commas are used in decimal form as
one unit for measuring actual performances, as with Baris Bozkurt
and his colleagues. Also, because 31/53 octave is so close to a
pure 3/2, a tuning based on 53-EDO or its subdivisions of 106-EDO
and 159-EDO is sometimes advocated or indeed implemented by
Middle Eastern musicians, with Ozan Yarman's 79-note or 80-note
MOS subset of 159-EDO as a famous example. However, it is good to
explain that often the 53-comma notation is used simply as a
generic device to show the types and positions of steps and
intervals, not as a precise measurement or prescription.

---------------------------------------------------------
3. The "five commas" within a 25:24 lagu, and temperament
---------------------------------------------------------

One the most fascinating comments in your letter, whose wonder I
hope will not be eclipsed by my pedestrian commentary, is the
following:

> in most of his tunings, these apotomes are boundered with
> syntonic commas and each remnant lagu (25/24) is divided in 5
> smaller commas of unequal but more or less even sizes.

Here I begin by picturing a framework for this the 25:24 space as
follows, for example between the regular 256:243 limma and the
regular 9:8 tone, a difference of an apotome or 2187:2048.

2187:2048
|-------------------------------------------------------|
256/243 16/15 10/9 9/8
|---------|...................................|---------|
81:80 25:24 81:80

Now this space between a large minor second or semitone at 16/15
and a small tone at 10/9, indeed a territory measured by the
25:24 ratio, is the realm known in Middle Eastern theory as the
"mujannab zone," the rather wide territory within which a
mujannab interval (shown in the notation used by Safi al-Din and
others by an Arabic letter "J" and in the modern Turkish alphabet
as "C") may fall. For Safi al-Din and modern Turkish theorists
also, it may include small mujannab steps around 16/15 and large
ones around 10/9, as well as middle or Zalzalian steps in the
myriad shades represented, for example, by the intervening
superparticular ratios of 14/13, 13/12, 12/11, and 11/10.

Let us for the moment assume that a transition from a large
5-limit variety of minor second to a small neutral or supraminor
second occurs somewhere between 15:14 (119.4 cents) and 14:13
(128.3 cents); Hormoz Farhat suggests that neutral seconds may
typically range upward in Persian music from about 125 cents.

The next transition might be from a supraminor or small neutral
second to a "smaller central" neutral second, perhaps at around
135 cents, with Ibn Sina's 13:12, his alternative of 64:59
adopted by Safi al-Din, and al-Farabi's 88:81 all exemplifying
this range.

Then comes the "larger central" region presumably starting, if we
assume pure fifths, fourths, and 32:27 minor thirds, at somewhere
roughly between 147 cents (half the size of such a third at 294.1
cents) and al-Farabi's step of 12/11 (150.6 cents). This zone
also includes the 59:54 step of Ibn Sina and Safi al-Din (153.3
cents), and the 128:117 (155.6 cents) of Ibn Sina.

Perhaps somewhere around 160 cents. we enter a zone of "submajor
seconds" with 11:10 (165.0 cents) well exemplifying this
category, as it occurs in Safi al-Din's 9:8-11:10-320:297
tetrachord which makes a fine Ottoman Rast.

Much beyond 11:10, we enter first an interesting region around
170 cents or 1/7 octave where it may be difficult to discern the
exact distinction between a very large neutral or very small
major second, and then the precincts of 10/9 (182.4 cents), or
the slightly smaller Pythagorean diminished third at 65536/59049
(180.45 cents).

One simplification of this intricate situation might be to
conceive of four regions of Zalzalian or neutral intervals, plus
a lower region still partaking of pental or 5-limit minor (near
16:15 or 15:14), and an upper region approaching or partaking of
pental major.(near 10:9). While superparticular ratios often
provide helpful landmarks, as well as some of the other
traditional ratios going back in the literature the better part
of a millennium or more, the Turkish and Syrian system of 53
commas also offers some helpful guideposts. By around 6 Holderian
commas or Hc for short (135.8 cents), neutral second steps are
making a transition from "supraminor" to more of a "central
neutral" quality; while at around 7 Hc (158.5 cents), we are
moving from the "large central" zone to the realm of the
"submajor" second.

Such a scheme with six conceptual zones between 16/15 and 10/9,
and thus five intervals or "commas" often very fuzzily delimiting
these zones, might go about as follows, with the transitions
impressionistic and highly debatable, not to speak of the role of
musical context:

~125c? ~135c? ~147c? ~158c?
112c 119c . 128c . 139c 141c 143c . 151c 153c 156c .
|--------...-------..,-----------------...--------------------...
16/15 15/14 . 14/13 . 13/12 64/59 88/81 . 12/11 59/54 128/117 .
5 Hc 5.5 Hc 6 Hc 6.5 Hc 7 Hc
113c 125c 136c 147c 158c

pental minor supraminor small central large central

~168c?
162c 165c . 170c 182c
..--------------..........-----------|
56/51 11/10 . 32/29 10/9
7.5 Hc 8 Hc
170c 181c
submajor pental major

The superparticular Zalzalian or neutral ratios are helpful in
exemplifying each of the four middle zones. Thus 14/13 is a fine
supraminor or small neutral second; 13/12 likewise a small
central neutral second; 12/11 a large central neutral second; and
11/10 a submajor second. Further it seems that the first three of
these are toward the lower portions of their respective zones,
while 11/10 may be not too far from the upper limits of its zone.

Comparing these ratios will yield commas in the traditional sense
which give some idea of the spacing of these regions. From 16/15
(clearly pental minor) to 14/13 (clearly supraminor or small
neutral) is 105:104, or 16.6 cents. From 14/13 to 13/12 (clearly
small central neutral) is 169:168, or 10.3 cents. From 13/12 to
12/11 (large central neutral) is 144:143, or 12.1 cents. From
12/11 to 11/10 (clearly submajor or large neutral) is 121:120, or
14.4 cents. And from 11/10 to the clearly pental major 10/9 is
100:99, or 17.4 cents.

Since the transitions between zones are fuzzy, to seek exact
"commas" defining their metes and bounds would be
unwise. However, if we adopt Farhat's view of neutral seconds as
generally ranging from about 125 to 170 cents, then a region
encompassing the realm from around 5.5 to 7.5 Hc would nicely
convey this impressionistic estimate.

Curiously, each of the six zones may be crudely conceived as
occupying the space of about 1/2-Hc, thus:

Pental minor (5 - 5.5 Hc, or 112-124 cents)
Supraminor (5.5 - 6 Hc, or 124-135 cents)
Smaller central (6 - 6.5 Hc, or 135-147 cents)
Larger central (6.5 - 7 Hc, or 147-158 cents)
Submajor (7 - 7.5 Hc, or 158-170 cents)
Pental major (7.5 - 8 Hc, or 170-182 cents)

Here it must be added that the pental minor region may begin with
steps rather smaller than 16/15 or 5 Hc, with the 104-cent step
in O3 sometimes serving this purpose although not as a main
aspect of the tuning design, and the 106-cent step in Ozan
Yarman's 79/80-MOS doing so as an essential feature of the
design. As to the "pental major" region around 10/9, we must take
caution that the lower part of this region around 1/7 octave may
have more of an "equable heptatonic" or uncertain character,
while the realm of small major seconds extends considerably
beyond a just 10/9, or 8 Hc, to include for example the steps of
a characteristic meantone at around 1/3-comma (189.6 cents) or
the true mean-tone of 1/4-comma (193.2 cents) at midway between
10/9 and 9/8.

Focusing on the four Zalzalian categories between pental minor
and pental major, we may take the ratios of 14/13, 13/12, 12/11,
and 11/10 as exemplary. Thus a JI system or temperament striving
for variety might seek to approximate these ratios, or others,
representing each of the four zones.

Now while the commas between these ratios are necessarily
unequal, yet they may be reasonably although not perfectly
represented by logarithmic aliquot steps. Let us first consider
the situation with the 24-note O3 temperament in 1024-EDO:

Ratio JI Cents JI Hc TU Cents Hc
------------------------------------------------------------
14/13 128.298 5.67 108/109 126.6/127.7 5.59/5.64
------------------------------------------------------------
13/12 138.573 6.12 118 138.3 6.11
------------------------------------------------------------
12/11 150.637 6.65 128/129 150.0/151.2 6.625/6.68
------------------------------------------------------------
11/10 165.004 7.29 138 161.7 7.14
------------------------------------------------------------

Here 13/12 and 12/11 are the most accurately represented steps,
each being always within a cent of just; 14/13 is usually about
1.7 cents narrow of just at 126.6 cents; and 11/10 is least
accurate, 3.3 cents narrow at 161.7 cents (close to 56/51).

If we consider the spacings or commas between the JI ratios, and
compare those between the tempered O3 sizes (taking the more
common smaller sizes for 14/13 and 12/11), then we find that the
latter are equally spaced at 10 tuning units or 11.72 cents,
almost exactly half of a Pythagorean comma at 531441/524288 or
23.46 cents.

0.45 Hc 0.53 Hc 0.63 Hc
10.3c 12.1c 14.4c
169:168 144:143 121:120
JI: 14/13 -------- 13/12 --------- 12/11 ----------- 11/10
128.3c 138.6c 150.6c 165.0c
5.67 Hc 6.12 Hc 6.65 Hc 7.29 Hc

0.52 Hc 0.52 Hc 0.52 Hc
11.7c 11.7c 11.7c
10 TU 10 TU 10 TU
O3: 126.6c ------- 138.3c -------- 150.0c -------- 161.7c
5.59Hc 6.11 Hc 6.625 Hc 7.14 Hc
108 TU 118 TU 128 TU 138 TU

Here the tempered comma of 10 TU or 11.7 cents approximates the
just 144:143 (13/12 to 12/11) at 12.1 cents, but is a bit large
for the 169:168 (14/13 to 13/12) at 10.3 cents, and considerably
smaller than the 121:120 (12/11 to 11/10) at 14.4 cents.

It must be borne in mind that although all these steps are
represented in O3, only two at most will be available above any
one step; and where such pairs occur, they will be at a 20-TU
comma apart (23.4375 cents). Thus C*-C#* (126.6 cents) or C*-D
(150.0 cents); and C#-D* (138.3 cents) or C#-Eb (161.7 cents).

A consideration of just and tempered versions of the beautiful
Mohajira mode Ibina (72:78:88:96:108:117:128:144) may illustrate
some fine points about these systems:

Mohajira Bayyati
|-----------------------| |-----------------------|
72 78 88 96 108 117 128 144
JI: 1/1 13/12 11/9 4/3 3/2 13/8 16/9 2/1
0 138.6 347.4 498.0 702.0 840.5 996.1 1200
13:12 44:39 12:11 9:8 13:12 128:117 9:8
138.6 208.8 150.6 203.9 138.6 155.6 203.9
|----------------------|
Mustaqim

Mohajira Bayyati
|------------------------| |------------------------|
A Bb* C* D E F* G A
O3: 0 138.3 345.7 495.7 703.1 841.4 991.4 1200
138.3 207.4 150.0 207.4 138.3 150.0 208.6
|----------------------|
Mustaqim

The lower Mohajira tetrachord at 36:39:44:48 or 13:12-44:39-12:11
seems almost custom-designed for near-just representation in O3,
although it is above all, of course, custom-designed for JI!

The middle Mustaqim and upper Bayyati tetrachords, however, both
in themselves and as compared to the lower Mohajira tetrachord,
reveal some notable melodic divergences between the just and
tempered versions. In the just tuning, both the Mustaqim and
Bayyati tetrachords use Ibn Sina's steps of 9:8, 13:12, and
128:117. The latter two steps at 138.6 and 155.6 cents differ by
almost 17 cents; but the O3 values of 138.3 and 150.0 cents
subtract over 5 cents from the 128:117 step, and reduce the
difference between the two steps to only 11.7 cents, missing the
subtle variations from step sizes in the lower tetrachord.

Generally O3 tends to level out the variations in the just
version between the pure fifths or fourths at four locations and
the "justly tempered" fourth at 117:88 (493.120 cents) found at
11/9-13/8, at a 352:351 or 4.925 cents wide. In 03, where this is
one of the main commas tempered out, all fifths are impure by
1.170 or 2.342 cents (1 or 2 TU).

In a polyphonic setting, the just Ibina with its 4:3 and 117:80
fourths might somewhat resemble a well-temperament, while O3
presents a near-regular "gentle temperament." However, one of the
more notable inaccuracies involves the beautiful Ibina sonority
of 1/1-4/3-16/9 or 9:12:16 much favored by European composers
around 1200 such as Perotin, and described by Jacobus of Liege in
the early 14th century as a 16:9 "split" into two equal 4:3
fourths. A classic resolution has the 4/3 and 16/9 steps each
ascend by a tone above the stationary 1/1, arriving at
1/1-3/2-2/1 or 2:3:4, the complete or perfect stable harmony of
the era.

In O3, however, the 16/9 of Ibina is compressed by almost a full
352:351 to 991.4 cents, or about 4.7 cents narrow. This minor
seventh, or the slightly wider shading at 992.6 cents found for
example at C*-Bb* (11/9-13/6), can nicely represent the just
39/22 (991.165 cents) occurring at the latter location in Ibina.
There is, however, no way of obtaining a just 4:3, or 16:9, or
the pure form of the exquisite 9:12:16 sonority combining these
ratios.

Also, from a melodic perspective, the subtle JI distinctions
between steps of 9:8 and 44:39, as well as 12:11 and 128:117, are
lost in the temperament process.

In O3 a regular minor third is at 288.3 cents (here E-G) or 289.5
cents, both near 13:11 (289.210 cents), nicely representing a
52:48:44:39 interpretation of the upper Bayyati tetrachord
(E-F*-G-A) using much the same steps as those actually called for
by the lower Mohajira tetrachord (13:12-44:39-12:11) in a
different permutation. The Ibina sequence, however, calls for a
Bayyati tetrachord at 108:117:128:144, with the slightly wider
minor third at 32:27 (294.135 cents) which shapes many classic
neutral third divisions including that of Ibn Sina (into 13:12
and 128:117) featured in this sequence.

Of course, this is not to exclude O3 and other systems with
extended fifths, but only to say that these systems are
variations on the theme of JI which should leave room for that
theme itself.

Again I am not sure how close I have come to your meaning in the
discussion of Julien's five commas within a 25:24. Yet I regard
the idea of four Zalzalian zones or regions, exemplified by the
steps of 14/13, 13/12, 12/11, and 11/10, as a guide for
temperaments as well as just tunings which strive for subtlety in
maqam and dastgah music. And I would emphasize that such classic
ratios as al-Farabi's 88/81, Ibn Sina's 128/117, or his 64/59 and
59/54 variations adopted by Safi al-Din, are also basic landmarks
often making a welcome appearance in your tunings.

With deepest thanks,

Margo Schulter
mschulter@...

🔗Jacques Dudon <fotosonix@...>

Dear Margo,

Yes indeed it is incredible, astonishing, and wonderful, how your Bamm24b and my Tsaharuk24 share common features, structure, and ratios !
But I swear I never saw your tuning, nowhere, when I conceived mine.
And I am amazed, because in fact I would have never imagined that anyone else could have supported the idea of such an UNEQUAL and bizarre (?) quasi-7-limit 24 notes tuning for Maqam !!!!
"Les grands esprits se rencontrent" as people say in french...
And you perhaps didn't know either, when you composed "Bamm24b", that it could be seen as a subgroup of a linear temperament !

I have not look in detail in the structures differences (5/4 instead of 81/64 , but this seems to be only the result of a transposition), but what seem to me the most important differences between our scales only lie in schismatic considerations.
The ratio similarities come from the presence of factor 59, omnipresent in your tuning (in fact a planar 2 - 3 - 59 temperament), while you do not use its septimal approximations as I do in Tsaharuk24, nor fives, in order to follow (roughly), a schismatic sequence.
Here it is in one of the two sequences, going up by fifths :
[35/27 35/18 35/24 35/32] (7*5*3^3/59*2^4) [59/36 59/48 59/32] (59*3^5:7*2^11) [112/81 28/27 14/9 7/6 7/4]

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> > I am pleased to share this very specific 24 notes/octave unequal
> > tuning, based on a linear version of Tsaharuk temperament,
> > especially designed for Maqam music.
>
> Dear Jacques,
>
> How curious that your 24-note Tsaharuk temperament is very similar to
> a tuning I described last December in another forum -- but the
> differences show the special logic of Tsaharuk in meeting Julien
> Jalaleddine Weiss's criteria for Maqam music, some of which do not
> apply to my tuning despite the often identical or similar interval
> sizes!
>
> </justintonation/topicId_unknown.html#1005>
> <http://www.bestII.com/~mschulter/bamm24b_C.scl>
>
> > It contains five classical Rast heptaphones transpositions, plus 3
> > Turkish Rast versions, one in between, and one folk style Rast,
> > seven Bayyati heptaphones transpositions, two perfect Syntonon
> > Diatonon (quartertone related), six Mustaqim heptaphones (from Ibn
> > Sina and Safi al-Din tetrachords mentionned by Margo Schulter), and
> > probably many other modes.
>
> Here I should ask the ratios for the "folk style Rast."

What I call "folk style Rast" here, without irrefutable musicological claims, are just Rast heptaphones using 5/3 (or meantones versions such as in mohajira) instead of 27/16, which belongs to the classical version.

> The other
> categories are generally clear to me, and nicely sum up some of the
> modes in this tuning. Note that my tuning mentioned above, Bamm24b,
> does not have any just 5-limit ratios or Syntonon Diatonon tetrachords
> or modes.
>
> > It provides two sizes of each of the neutral intervals : seconds,
> > thirds, etc., except that within this limited number of notes
> > selected from the complete set of 77 notes, they can't be
> > experienced with the same tonic.
>
> This raises an important point you discussed in your earlier long post
> on Maqam tunings. The vital thing is to have at least two sizes of
> neutral intervals, even if only one is available from a given step.

Exactly.

> Even in O3, a 24-note system which does have some steps with two sizes
> of neutral intervals available at a comma apart, we can't get, for
> example, a moderate Rast (357.4 or 358.6 cents) and a moderate
> Mustaqim (345.7 or 346.9 cents) above the same note. That would
> require a lower and higher segah only 12 cents or so apart. What we
> can sometimes get, for example, is a high segah (369.1 cents) for a
> Turkish Rast or moderately low segah (345.7 cents) for a typical Arab
> Bayyati, a difference of a comma at 23.4 cents.
>
> However, of course, a complete Tsaharuk does provide choices like a
> virtually just 72/59 or 59/48 above the same note, which I found in a
> 94-note version with five generators equal to a pure 3/2.
>
> > The neutral intervals of Rast will be the tonic of Bayyati or
> > Mustaqim, and reversely.
>
> This is also a pattern we see in Mohajira or Wilson's Rast-Bayyati
> matrix, with a Rast heptatone appearing in a version extended to 10 or
> more notes. And likewise an Arab Makam Sikah has the smaller neutral
> intervals in contrast to Rast; while Persian Dastgah-e Segah has the
> larger neutral intervals, in contrast to Mustaqim (or a modern gushe
> such as Shekaste like the old Mustaqim).

Would you make special distinctions between Sikah and Mustaqim, besides being related to different cultures ?

> > This has been concocted simply with a sequence of 11 schismatic
> > fifths, transposed by 28/27.
>
> Curiously, Bamm24b results from extending a variation on a 17-note
> tuning for the `oud explained by Cris Forster in his book _Musical
> Mathematics_, but may be concocted as two chains of 11 pure fifths at
> a 531/512 apart (63.1 cents),

If this is the case then I confirm that both scales have quasi-identical structures, and differ by only minute variations. The identity of structure is not totally surprising since both express the best DE (distributionnally even) solution (if we make abstraction of the schismatic variations, more diverse in my own rational approximations)

> or 14337:14336 more than 28/27!

You must know more than me about this extraordinary schisma (...I mean extrordinary for prime 59, and maqam lovers, as we both are !).
It was present from the beginning in my rational approximations of Tsaharuk schismatic fifths sequences, from the "4 generators" = 112/81 (~ 177/128), then all the way to 7/4. But you were actually the first one to reveal it to me this winter in its impressive arithmetical expression, and whoever the "inventor" of it is, I wish to give credit, otherwise I will call it the Margo schisma !

> Thus the only primes are 2, 3, and 59 -- while your Tsaharuk also has
> 5 and 7! The biggest difference may be that Tsaharuk 24 has pure
> 5-limit thirds, in line with Julien's standard, while in Bamm24b all
> usual fifths are pure.

I am not sure if pure 5-limit thirds would be the standard for the Julien, as for his qanuns he would rather have pure fifths everywhere.

> > So 28/27 appears 12 times as the larger step, alternatively with
> > five 36/35 and seven septimal commas 64/63, all evenly
> > distributed. Their relative size in 171-edo is 9, 7, and 4 steps.
>
> In Bamm24b we have 12 larger steps at 531/512 (63.1 cents), five at
> 243/236 (50.6 cents), and seven at 131072/129033, smaller than 64/63
> by 14337:14336.
>
> > Nine quasi-pure 7/4 appear between the two fifth sequences, which
> > can be surprising for a Maqam tuning.
>
> On this point Tsaharuk-24 and Bamm24b are similar, and I agree that a
> lot of modern Maqam theory doesn't emphasize septimal intervals. However, it
> said that in Maqam Ushshaq Masri (a variation on Nahawand
> associated with Egypt, somewhat like Persian Nava, with an upper
> Bayyati, e.g. re mi fa sol la si-d ut re, with "si-d" a half-flat, it
> is sad that the third is a comma lower than usual, which could mean
> around 7/6, with the seventh step maybe near 7/4. For a low Nahawand
> or Ushshaq Masri, Tsaharuk-24 might be very useful!
>
> Also, I've heard that Maqam Buselik in Turkey may favor a low third
> around 7/6, and the 5-limit intervals of Tsaharuk-24 should fit a
> Turkish style also.

Thanks for these interesting informations !
And have you any clue about what major third that would be used in Saba (5/4 or 81/64 ?)
The first 3 notes of the Bayyatis found in both our tunings are followed by 81/64s.

> > But as I said in my first post, an optimal generator would arrive at
> > 7:1 in 24 reiterations, while it gives also almost pure fifths,
> > which is the first criteria Julien Jalaleddine Weiss and other
> > musicians want for Maqam tunings.
>
> Indeed it will give a pure 7/4 and an almost pure 3/2 (very slightly
> narrow) and 5/4 (likewise, with -40 generators at 385.290 cents).
>
> > In fact, a pure 7:1 attained in 24 generators creates a slightly
> > schismatic tuning of the fifths, that produces also quasi-pure 5/4
> > thirds, and this is the idea of this tuning, in this central zone
> > near 171-edo.
>
> For this generator, Scala shows 140.368 cents, yielding a fifth around
> 701.839 cents. What I note is that 24 generators give a pure 7/4 and
> -40 a near-pure 5/4. With a fifth generator around 3/2, getting 7/4
> pure (-14 fifths) would require slightly extending the fifth, which
> would make 5/4 (-8 fifths) less pure.
>
> > The average Tsaharuk generator in the tuning below, five of them
> > giving a fifth, is 1.08444952 ~20/171 of octave, itself very close
> > to the meta-temperament version of Tsaharuk, 1.084451679092 ; it
> > often appears as 64/59 in this rational version of the temperament.
>
> An interesting point is that Bamm24b takes a pure 64/59 and 59/54 as
> one of its starting points (using generators of 3/2 and 531/512), but
> your Tsaharuk has generators of varying sizes around 3/2^1/5: for
> example 64/59 and 243/224, differing by 14337:14336, as well as two
> others a bit smaller helping to produce the pure 5-limit thirds.

Very well observed !

> > You will notice there are neither 11 nor 13 primes in this tuning,
> > but only 2, 3, 5 , 7 and 59.
>
> It's curious that in Bamm24b, we have only 2, 3, and 59 -- but with
> the same types of approximations you mention next.
>
> > One of the reasons is that Tsaharuk tempers 352/351, and 59 finds
> > its place precisely in between (59/48 for example is in the middle
> > of 27/22 and 16/13).
>
> Yes, 59 is very close to exactly between 11 and 13. A 59/48, for
> example, is larger than 27/22 by 649:648, and smaller than 16/13 by
> 768:767, respectively 2.670 and 2.256 cents. Early this year I did a
> lattice for Bamm24b showing some of these near-equivalences:
>
> <http://www.bestII.com/~mschulter/hexapentadic17.txt>

Very nice matrix - you shoud definitely show this to Julien some day !

> > I am only giving a rational version here, the more useful for
> > fretting applications ; but it expresses a temperament.
>
> The pure thirds, fitting Julien's criteria, do make the tempering
> clear.

I am not certain of what is Julien Jalaleddine Weiss's criteria about major thirds actually, between pythagorean or 5-limit, or between a consonant ideal and actual frettings - or if he just admits certain schismatic variations, in the indian shrutis sense.

> With Bamm24b, we have basically two chains of 11 pure fifths at a
> 531/512 apart, almost identical to a 28/27, which could be tuned by
> ear,

I never thought of it but it's true - perhaps with the help of intermediaries like 16/9 and 32/27 ?
While for 27/26 it would perhaps be more difficult -

so the only "tempering" is the 14337:14336 difference (0.121
> cents). In 1024-ED2, the generators are 599 steps (703.953 cents) and
> 54 steps (63.281 cents) -- a virtually pure 3/2 I suspect Julien would
> accept, and an approximation of 531/512 about 0.200 cents wide.
>
>
> However, while we have lots of 72/59 and 59/54 thirds for rast-segah
> in Bayyati or Mustaqim and Rast (but, as with Tsaharuk-24, not both positions
> above a single step), and nine virtually pure 7/4's, Julien
> would notice that there are no pure ratios of 5. They are not a part
> of the design, although some schismatic thirds will occur -- in
> contrast to the pure 5/4 and 6/5 thirds of tsaharuk-24, or near-pure
> thirds with complete Tsaharuk tunings using 7^1/24, for example.
>
> I also notice that a Tsaharuk-94 with the 7^1/24 generator has some
> other near-just intervals like 13/11 and 17/14

Interesting - JJW actually uses more than 77 notes in his qanuns and could very well use those.

> -- and lots of places
> supporting both Rast with segah near 59/54 and Mustaqim or Bayyati
> with segah near 72/59!
>
> Best,
>
> Margo

Thanks for this detailed analyse and comparison. More than ever, I am convinced you and Julien, as soon as he can, should exchange your respective knowledge of Maqam intonation !

Harmonically yours,
- - - - -
Jacques