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

[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

> 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.

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@...

[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@...

Dear Margo,

Thanks for your enthusiastic response, fully documented as always with historical references and complements.

I know Julien Jalaleddine Weiss is very busy, preparing and giving concerts between Beyrouth, Damas et Alep for a huge ensemble these days, but he should be able to read your posts soon, for on many musicological points he should be the one to reply to you !

I told him the interest you showed to his system and I am certain he will be delighted to exchange his knowledge of Arabic Maqam with you.

Just to complete my post, and for all our friends curious of Maqam, here is a close-up of Julien and the way he switches his Mandal-Orabs on one of his very microtonal qanuns :

http://www.youtube.com/watch?v=lFIQMM8bZQk&NR=1

And an interesting sufi-style interpretation of I suppose maqam Bayyati, with a Saba variation in the end, by Julien's ensemble in the Fes sacred music festival 2008 :

http://www.youtube.com/watch?v=2-nI7QVuUDQ

Something very microtonal I particularly enjoyed here is when the singer Sheikh Hamza Shakkur grabs the tonic at the upper octave at 2'58" shortly but impressively, before he passes the torch to the ney player - the way he pitches this note incredibly higher of God knows how many cents, while of course he could sing a 2/1 perfectly in tune if he wanted, makes me feel he is tuning to something beyond this world.

I wonder if Julien can explain the interval he sings in this moment, I would say perhaps a 65/32, which would make sense as a "Picardy third" above 13/8, unless if in the context he forces a 8/7 over 16/9, or 128/63, which in both cases amounts to more or less the same thing. The needle of my temperament indicate -34 generators for both, I am glad it handles it... ;)

It will take me some time to read thorougly such a complete answer you did to my post with the limited knowledge I have, in comparison to yours, of Maqam theory.

Sorry to be completly invested somewhere else these days, but hopefully I am realizing some of my dreams, along with a new place for even more beautiful music feasts in the next years, if I am not dead !)

Cheers,

- - - - - - -

Jacques

Margo Schulter wrote :

> [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.]

>

> 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.

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.

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.

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.

The neutral intervals of Rast will be the tonic of Bayyati or Mustaqim, and reversely.

This has been concocted simply with a sequence of 11 schismatic fifths, transposed by 28/27.

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.

Nine quasi-pure 7/4 appear between the two fifth sequences, which can be surprising for a Maqam tuning.

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.

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.

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.

You will notice there are neither 11 nor 13 primes in this tuning, but only 2, 3, 5 , 7 and 59.

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).

I am only giving a rational version here, the more useful for fretting applications ; but it expresses a temperament.

! tsaharuk24.scl

!

Rational version of Tsaharuk linear temperament

24

!

28/27

59/56

35/32

9/8

7/6

32/27

59/48

5/4

35/27

4/3

112/81

59/42

35/24

3/2

14/9

128/81

59/36

27/16

7/4

16/9

59/32

15/8

35/18

2/1

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Jacques

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