MAQAMAT: Cameron's charming "soldierly/desertly" tetrachord

Hello, Cameron and all.

Please let me thank you for a delightful post about deriving what I
would call a septimal flavor of Hijaz tetrachord, or in Iranian
terms Chahargah, from four "soldierly" 3:2's up (i.e. an 81:64
ditone) and a "desertly" 7:6 down.

As you note, this basic superparticular concept produces a
tetrachord of approximately 1/1-13/12-81/64-4/3 -- or, as you
note, something practically synonymous in musical terms with
this, in theory 1/1-243/224-81/64-4/3 (141-267-90 cents) if
we take the 7:6 step to be precisely that.

While I'm not sure if Ibn Sina actually describes such a
tetrachord, it's certainly available in his `oud tuning from the
early 11th century as given in the Scala archives under the file
name avicenna_17.scl (according to Ahmed Mahmud Hifni, Cairo,
1977). The basic tetrachord would be:

1/1 13/12 81/64 4/3
0 139 408 498

Here the middle interval is actually 243/208, or around 269
cents, or greater than 7/6 by 729/728, the same amount by which
243/224 is greater than 13/12. From a musical perspective, we
have something just about identical to what you described trying
with your sons!

According to Hormoz Farhat, a modern Iranian Chahargah would be
tuned like this on the tar tuning he recommends, based on the
average values for some instruments:

0 135 410 500
G Ap B C

Obviously we have a very similar concept, although the middle
step is slightly wider than 7/6 at around 275 cents.

Around 1300, Qutb al-Din al-Shirazi suggests a tuning for Hijaz,
very much the same type of tetrachord, at 1/1-12/11-14/11-4/3 or
12:11-7:6-22:21 (150-267-81 cents). Thus we would have, in
Persian notation:

0 151 418 498
G Ap B C

Interestingly, in a tempered system like O3, there are rough
analogues of both your "soldierly/desertly" form and the close
equivalents in the Ibn Sina and Farhat tunings, with a middle
second at around 13/12; and the Qutb al-Din tuning, with a middle
second of around 12/11 (or the "150 cents" or so you mentioned).

Given the Pythagorean framework for the regular major and minor
intervals which you and also Scott Marcus, for example, have
noted in much maqam music, O3 might be described as a subtle and
(hopefully) artful distortion of this basic framework.

By extending each regular fifth by one or two binary millioctaves
(1.17 or 2.34 cents), the most notable alteration is that
augmented and diminished intervals shift from near-5-based to
near-13-based (e.g. apotome ~14/13; diminished third ~11/10;
augmented second ~40/33; diminished fourth ~26/21).

Another, more subtle, effect is to stretch the ditone from 81/64
to somewhere between 33/26 and 14/11, the latter used by Qutb
al-Din.

Let us first consider the nearest approximation of your
"soldierly and desertly" Hijaz or Chahargah. With this tempered
system, as with superparticular JI, a simple chain of 12 fifths
will not supply intervals such as 13/12 or 12/11. In O3, two
12-note chains are spaced so that a regular major sixth plus the
distance between the chains produces a pure 7/4 (or actually one
of the best two 1024-EDO approximations thereof), a procedure
which also produces versions of 13/12 and 12/11 always within a
cent of just.

Thus for your "soldierly/desertly" version with 13/12, or
Farhat's Chahargah also available in a rational version on Ibn
Sina's `oud tuning, we have:

Persian notation: G Ap B C
O3 keyboard: E F* G# A
0 138 416 497

This tetrachord of 0-138-416-497 cents or 138-278-81 cents has a
central interval close to Farhat's 275 cents, although a tad
larger, and a notably narrower limma than Pythagorean at 81 cents,
almost identtcal to Qutb al-Din's just 22:21 step.

For Qutb al-Din's tuning with its 12/11 step, we must shift to a
note in O3 which is part of the upper chain of fifths:

Persian notation: G Ap B C
O3 keyboard: G* A B* C*
0 151 416 497

This time we have 151-265-81 cents, very close to Qutb al-Din's
12:11-7:6-22:21 (151-267-81 cents).

As Ozan might have me observe, Turkish theory and practice often
favor a middle step around 7:6 or 12 commas of 53 (either in
Pythagorean or 53-EDO, an interval slightly larger than 7:6,
around 271 or 272 cents). However, in Turkish practice, the major
third step of Hijaz is often smaller than 81/64 or 14/11. Amine
Beyhom, a Lebanese composer and theorist, measured one
interpretation at around 0-130-395-485 cents (130-265-90 cents).
While the near-7:6 is in keeping with the soldierly/desertly or
Farhat version also available in Ibn Sina's `oud tuning, as well
as Qutb al-Din's tuning, the narrower major third is not too far
from 5/4 -- with the upper step at 90 cents, a usual Pythagorean
limma, here leading to a narrow fourth at around 485 cents.

Anyway, I'm one of those people who takes medieval
superparticular ratios for maqam music quite seriously, and seeks
to realize reasonably accurate approximations on a modern
keyboard.

This doesn't mean that performances approximating 24-EDO aren't
also possible; and I have seen at least one Iranian theorist
propose 43-EDO. However, where a superparticular model fits well,
I see no reason not to celebrate this millennium-long tradition
which is, as you say, ultimately a matter of musical feeling
rather than abstract mathematics alone.

A final note: Hijaz, often (although not always) tending toward
to a middle step close to 7:6, does indeed carry the association
of the Hijaz region of desert for maqam musicians, so that your
"desertly" description of this step seems to fit the Near Eastern
tradition. While I'm not sure if a 3:2 carries a "soldierly" or
martial connotation in the world of maqam, I recall that it does
have this association in William Byrd, for example, around 1600,
who uses lots of fifths in his suite for harpsichord about a
battle. Maybe I might say, speaking only for myself, that it
might more generally carry the character which is in Arabic
called _sumud_ or "steadfastness," a virtue in many situations.

Best,

Margo Schulter
mschulter@...

Hi Margo, thanks for yet another detailed and fascinating post!

The terms "soldierly" and "desert-ly" ("voja¹ko" and "pu¹èavkso" respectively) are my 6-year old son's descriptions, of his own device.
Of course these are associative- he recognizes them as characteristic intervals from films, cartoons and documentaries, as my wife immediately recognizes 5-limit (and higher) JI as "church music". I think it's also interesting that a musician I know who grew up singing in choirs but had no technical knowledge whatsoever about tuning immediately recognized 11:8 as an "augmented fourth, perfectly in tune".

I rediscovered the "6/7 from Pythagorean ditone" when trying tune along with recordings. Even if the theoretical frame of reference really were 24-tET, on paper so to speak, consistently a little flat on the "150" and sharp on the "400" gives us these ancient, and tunable by ear! versions. Occam's razor, I say. :-)

-Cameron Bobro

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Hello, Cameron and all.
>
> Please let me thank you for a delightful post about deriving what I
> would call a septimal flavor of Hijaz tetrachord, or in Iranian
> terms Chahargah, from four "soldierly" 3:2's up (i.e. an 81:64
> ditone) and a "desertly" 7:6 down.
>
> As you note, this basic superparticular concept produces a
> tetrachord of approximately 1/1-13/12-81/64-4/3 -- or, as you
> note, something practically synonymous in musical terms with
> this, in theory 1/1-243/224-81/64-4/3 (141-267-90 cents) if
> we take the 7:6 step to be precisely that.
>
> While I'm not sure if Ibn Sina actually describes such a
> tetrachord, it's certainly available in his `oud tuning from the
> early 11th century as given in the Scala archives under the file
> name avicenna_17.scl (according to Ahmed Mahmud Hifni, Cairo,
> 1977). The basic tetrachord would be:
>
> 1/1 13/12 81/64 4/3
> 0 139 408 498
>
> Here the middle interval is actually 243/208, or around 269
> cents, or greater than 7/6 by 729/728, the same amount by which
> 243/224 is greater than 13/12. From a musical perspective, we
> have something just about identical to what you described trying
> with your sons!
>
> According to Hormoz Farhat, a modern Iranian Chahargah would be
> tuned like this on the tar tuning he recommends, based on the
> average values for some instruments:
>
> 0 135 410 500
> G Ap B C
>
> Obviously we have a very similar concept, although the middle
> step is slightly wider than 7/6 at around 275 cents.
>
> Around 1300, Qutb al-Din al-Shirazi suggests a tuning for Hijaz,
> very much the same type of tetrachord, at 1/1-12/11-14/11-4/3 or
> 12:11-7:6-22:21 (150-267-81 cents). Thus we would have, in
> Persian notation:
>
> 0 151 418 498
> G Ap B C
>
> Interestingly, in a tempered system like O3, there are rough
> analogues of both your "soldierly/desertly" form and the close
> equivalents in the Ibn Sina and Farhat tunings, with a middle
> second at around 13/12; and the Qutb al-Din tuning, with a middle
> second of around 12/11 (or the "150 cents" or so you mentioned).
>
> Given the Pythagorean framework for the regular major and minor
> intervals which you and also Scott Marcus, for example, have
> noted in much maqam music, O3 might be described as a subtle and
> (hopefully) artful distortion of this basic framework.
>
> By extending each regular fifth by one or two binary millioctaves
> (1.17 or 2.34 cents), the most notable alteration is that
> augmented and diminished intervals shift from near-5-based to
> near-13-based (e.g. apotome ~14/13; diminished third ~11/10;
> augmented second ~40/33; diminished fourth ~26/21).
>
> Another, more subtle, effect is to stretch the ditone from 81/64
> to somewhere between 33/26 and 14/11, the latter used by Qutb
> al-Din.
>
> Let us first consider the nearest approximation of your
> "soldierly and desertly" Hijaz or Chahargah. With this tempered
> system, as with superparticular JI, a simple chain of 12 fifths
> will not supply intervals such as 13/12 or 12/11. In O3, two
> 12-note chains are spaced so that a regular major sixth plus the
> distance between the chains produces a pure 7/4 (or actually one
> of the best two 1024-EDO approximations thereof), a procedure
> which also produces versions of 13/12 and 12/11 always within a
> cent of just.
>
> Thus for your "soldierly/desertly" version with 13/12, or
> Farhat's Chahargah also available in a rational version on Ibn
> Sina's `oud tuning, we have:
>
> Persian notation: G Ap B C
> O3 keyboard: E F* G# A
> 0 138 416 497
>
> This tetrachord of 0-138-416-497 cents or 138-278-81 cents has a
> central interval close to Farhat's 275 cents, although a tad
> larger, and a notably narrower limma than Pythagorean at 81 cents,
> almost identtcal to Qutb al-Din's just 22:21 step.
>
> For Qutb al-Din's tuning with its 12/11 step, we must shift to a
> note in O3 which is part of the upper chain of fifths:
>
> Persian notation: G Ap B C
> O3 keyboard: G* A B* C*
> 0 151 416 497
>
> This time we have 151-265-81 cents, very close to Qutb al-Din's
> 12:11-7:6-22:21 (151-267-81 cents).
>
> As Ozan might have me observe, Turkish theory and practice often
> favor a middle step around 7:6 or 12 commas of 53 (either in
> Pythagorean or 53-EDO, an interval slightly larger than 7:6,
> around 271 or 272 cents). However, in Turkish practice, the major
> third step of Hijaz is often smaller than 81/64 or 14/11. Amine
> Beyhom, a Lebanese composer and theorist, measured one
> interpretation at around 0-130-395-485 cents (130-265-90 cents).
> While the near-7:6 is in keeping with the soldierly/desertly or
> Farhat version also available in Ibn Sina's `oud tuning, as well
> as Qutb al-Din's tuning, the narrower major third is not too far
> from 5/4 -- with the upper step at 90 cents, a usual Pythagorean
> limma, here leading to a narrow fourth at around 485 cents.
>
> Anyway, I'm one of those people who takes medieval
> superparticular ratios for maqam music quite seriously, and seeks
> to realize reasonably accurate approximations on a modern
> keyboard.
>
> This doesn't mean that performances approximating 24-EDO aren't
> also possible; and I have seen at least one Iranian theorist
> propose 43-EDO. However, where a superparticular model fits well,
> I see no reason not to celebrate this millennium-long tradition
> which is, as you say, ultimately a matter of musical feeling
> rather than abstract mathematics alone.
>
> A final note: Hijaz, often (although not always) tending toward
> to a middle step close to 7:6, does indeed carry the association
> of the Hijaz region of desert for maqam musicians, so that your
> "desertly" description of this step seems to fit the Near Eastern
> tradition. While I'm not sure if a 3:2 carries a "soldierly" or
> martial connotation in the world of maqam, I recall that it does
> have this association in William Byrd, for example, around 1600,
> who uses lots of fifths in his suite for harpsichord about a
> battle. Maybe I might say, speaking only for myself, that it
> might more generally carry the character which is in Arabic
> called _sumud_ or "steadfastness," a virtue in many situations.
>
> Best,
>
> Margo Schulter
> mschulter@...
>