-----------------------------------------------------------
Four Chromatic Tetrachords with 13/11, 168/143, or 11/9
An addendum to _Divisions of the Tetrachord_
-----------------------------------------------------------
This is an addendum to that landmark publication of John Chalmers,
_Divisions of the Tetrachord_ (DOTT), and more specifically to its catalogue
of tetrachords in Chapter 9, pp. 180-181 (C19), listing tetrachords with a
characteristic interval of 13/11 (289.210 cents). Additionally, I address a
tetrachord with a characteristic interval not included in the catalogue,
168/143 (278.935 cents), which would have appeared on p. 181 between 20/17
(281.358 cents) and 27/23 (277.591 cents), and another which does not appear
in the list for its characteristic interval of 11/9, p. 175 (C8).
What I should explain is that the tetrachords I'm about to document in JI
form are what I use in a tempered form, also given in what follows, in the
MET-24 tuning system. In MET-24, a near-just 13:11 (288.867 cents in a
2048-ed2 version) is the regular minor third from three fifths at 703.711
cents (or more precisely three fourths at 496.289 cents). There are two
chains of these fifths (at 1.756 cents wide of 3:2), spaced at 57.422
cents. This "spacing" at 57.4 cents is a featured interval in the first
division I describe, where it can represent a just 121:117 (58.198 cents). It
often represents the familiar ratios of 28:27 at 62.961 cents (e.g. 9/8-7/6,
a tempered 207.422-264.844 cents), and 33:32 at 53.273 cents (e.g. 4/3-11/8,
a tempered 496.289-553.711 cents); and also 91:88 at 58.036 cents
(e.g. 22/13-7/4, a tempered 911.133-968.555 cents).
First, a quick listing of the four tetrachord divisions of interest, two of
which use 13:11 as a characteristic interval, plus a division of its 4/3's
complement, the pyknon at 44:39 (208.835 cents), involving a superparticular
step.
1. 121/117 x 12/11 x 13/11................58.198 + 150.637 + 289.210 cents
2. 176/169 x 13/12 x 13/11................70.263 + 138.573 + 289.210 cents
The third division features a characteristic interval not listed in the DOTT
catalogue: 168/143 (278.935 cents), with the pyknon at 143:126 (219.110
cents) divided into two superparticular steps, 22:21 and 13:12.
3. 22/21 x 13/12 x 168/143................80.537 + 138.573 + 278.935 cents
The fourth division, of interest as a possible realization of a Byzantine
Hard Chromatic or Palace Mode, has 11/9 as the characteristic interval and
12:11 as the pyknon (see DOTT, p. 175, C8), and like the last division
features a superparticular step of 22:21.
4. 22/21 x 11/9 x 126/121.................80.537 + 347.408 + 70.100 cents
In MET-24, I sometimes use the first two divisions as variations on the
Archytas Chromatic (28:27-243:224-32:27). The second and third divisions in
neochromatic or Hijaz permutations with the large interval as the middle step
play a vital role for me in the context of Near Eastern styles, as explained
below. The fourth division is relevant to Byzantine music, where the Palace
Mode favors a middle step at around 333 cents or 20 steps of 72-ed2 (a size
recommended by the Patriarchal Committee on Music in 1881), and accounting to
some comments on the World Wide Web may be around 11:9 (347.408 cents) in
certain schools of practice.
-----------------------------------
1. The 121:117-12:11-13:11 division
-----------------------------------
After using the following tetrachord as one not very accurate approximation
in MET-24 of the Archytas Chromatic (28:27-243:224-32:27), I sought out a
possible JI interpretation and arrived at this, then finding that it was not
listed in DOTT. This diagram shows JI ratios in the upper portion, and
tempered intervals in the lower portion:
....58.198.......150.637.............289.210..............
...121:117........12:11...............13:11...............
.0........58.198........208.835.......................498.045
1/1------121/117---------44/39--------------------------4/3
117........121............132...........................156
484........468............429...........................363
.0........57.422........207.422.......................496.289
...57.422........150.000.............288.867..............
This tetrachord follows the harmonic ratios 117:121:132:156, or the string
ratios 484:468:429:363.
A permutation of theoretical interest in Near Eastern music, although Amine
Beyhom notes it as unidiomatic in practice, is Baron d'Erlanger's tuning
for the tetrachord of `Awj Ara at 3-6-1 steps of 24-ed2. or 150-300-50
cents (0-150-450-500 cents). Beyhom suggests that this type of Hijaz or
neochromatic tetrachord (Gevaert 1875; DOTT, p. 15) might be tuned in
practice at or close to Qutb al-Din al-Shirazi's Hijaz from around 1300,
12:11-7:6-22:21 or 150.637-266.871-80.537 cents, a tuning which seems
closely approximated in some measurements from 20th-century Iranian
performances of the tetrachord called Chahargah, equivalent to Arab Hijaz.
Here is the relevant permutation with JI ratios and tempered interval sizes
in MET-24:
......150.637...................289.210..................58.198
.......12:11.....................13:11..................121:117
.0..............150.636..........................439.847.....498.045
1/1--------------12/11---------------------------156/132-------4/3
363...............396..............................468.........484
156...............143..............................121.........117
.0..............150.000..........................438.867.....496.289
......150.000...................288.867..................57.422
The third of the tempered version at 438.867 cents is MET-24's
not-so-accurate approximation of 9/7 (435.084 cents), but quite close to a
just 156/132 (12/11 x 13/12), since 12:11, 13:12, and 13:11 are all
near-just. In JI, this tetrachord has harmonic ratios of 363:396:468:484,
and string ratios of 156:143:121:117.
MET-24 also has a close approximation of Qutb al-Din's 1/1-12/11-14/11-4/3
(0-150.637-417.508-498.045 cents) at 0-150.000-414.844-496.289 cents; but
this variation is what I consider an "exuberant" variation that can be used
now and then in certain context.
This 121:117-12:11-13:11 division in its various permutations may be notable
among tetrachords with a characteristic interval of 13:11 because of its
superparticular step at 12:11.
-----------------------------------
2. The 176:169-13:12-13:11 division
-----------------------------------
The other division of interest as an addendum to DOTT involving 13:11 also
features a superparticular step at 13:12. This is 176:169-13:12-13:11 as
presented in the form of a classic Greek chromatic with the smallest step
first.
....70.263.......138.573..................289.210
...176:169........13:12....................13:11...............4/3
.0........70.263........208.835..............................498.045
1/1------176/169---------44/39---------------------------------4/3
507........528............572..................................676
176........169............156..................................132
.0........68.555........207.422..............................496.289
....68.655.......138.867..................288.867
The harmonic ratios for the JI version are 507:528:572:676, and the string
ratios 176:169:156:132. The JI division of the lower 44:39 tone, and even
more so the tempered 2048-ed2 division in MET-24, are very close to a
geometric division into a literal 1/3-tone and 2/3-tone. In the 1024-ed2
version of MET-24, this precise division of the tempered 9:8 or 44:39 at
207.422 cents (177 tuning steps) actually occurs at certain locations into
59 and 118 tuning steps, or 69.141 cents and 138.282 cents.
This division is most often used in a neochromatic or Hijaz permutation,
where the third step is 357.422 cents, an apt tuning for a moderate Arab
Rast, not too far from al-Farabi's middle third fret or wusta of Zalzal at
27/22 (354.547 cents), closer to 16/13 (359.472 cents) that occurs in some
tetrachords and permutations of Ibn Sina, and a virtually just 59/48
(357.217 cents), a ratio that occurs in permutations of Safi al-Din
al-Urmawi's 72:64:59:54, and recently has been featured in some of the
Mohajira tunings of Jacques Dudon. Here the idea is to have a lower semitone
(or here really thirdtone) step, a third step identical to that of Rast, and
a 4/3 fourth.
....70.263...............289.210...................138.573
...176:169................13:11.....................13:12.......
1/1------176/169---------------------------16/13--------------4/3
.0........70.263..........................359.472...........498.045
507........528..............................624...............676
176........169..............................143...............132
.0........68.555..........................357.422...........496.289
....68.555...............288.867...................138.867
Here the harmonic ratios are 507:528:624:676, and the string ratios
176:169:143:132. This is not necessarily an ideal tuning of this form of
Hijaz tetrachord, which Beyhom terms Zirkula (also a name for the step in
the Arab gamut at a limma or regular diatonic semitone above the step rast,
the usual 1/1 for the untransposed Maqam Rast). Such an ideal form might be
something like 1/1-256/243-16/13-4/3, or 0-90.225-359.472-498.045 cents,
with the middle step a "plus-tone" notably larger than a regular tone at
around 9:8 but smaller than a regular minor third (e.g. the Pythagorean
32/27 at 294.135 cents, or here 13/11 or its tempered equivalent). One
preferred region for tuning this plus-tone step for many Near Eastern
musicians is somewhere around 7:6.
While a middle step in Hijaz at or near the size of a regular minor third,
often around 32:27 (294.135 cents), or here 13:11, is common for example in
Turkish music despite a widespread preference for the plus-tone at or
slightly larger than 7:6, I am not sure how common it is in the context of
Beyhom's Zirkula to use a lower step smaller than a regular semitone (the
Pythagorean 256/243, or here a tempered 22/21, see below). However, I often
use this form at locations apt for Rast where the regular semitone is not
available but the "middle thirdtone" step, as I call it, at 68.6 cents is,
since Maqam Hijazkar (featuring two disjunct Hijaz tetrachords) is a common
modulation from Rast sharing the same 1/1 or final. Thus my liking for this
tetrachord as a variation on Beyhom's Zirkula may be a case of making a
virtue of necessity -- or of necessity as the mother of an addendum to DOTT,
at any rate.
-----------------------------------
3. The 22:21-13:12-168:143 division
-----------------------------------
A MET-24 tuning of this Zirkula variety of Hijaz closer to the ideal is
81.455-275.977-138.867 cents, with the same Rast third near 59/48 or 16/13,
and a regular limma at a tempered 22/21 (80.537 cents). The plus-tone step
here might be interpreted, for example, as 168:143, as in 176:168:143:132 or
22:21-168:143-13:12 (80.537-278.935-138.573 cents). This is one of my
favorite Hijaz tetrachords to use in such modulations from a moderate Arab
Rast (with the third at 357.4 cents) as Hijazkar, Nahawand, and Nakriz.
......80.537..............278.935..................138.573
......22:21...............168:143...................13:12
.0.............80.537...................359.472.............498.045
1/1------------22/21---------------------16/13----------------4/3
273.............286.......................336.................364
176.............168.......................143.................132
.0.............81.445...................357.422.............496.289
......81.455..............275.977..................138.867
Here the harmonic ratios re 273:286:336:364, while the string ratios are
176:168:143:132. With just ratios, the lower step at 22/21 is larger than
176/169 in the previous example of Zirkula by 169:168, and the middle
interval accordingly smaller by this ratio, with the 16/13 step common to
both tunings.
This division also has another permutation which I use for one possible
interpretation of Maqam Sazkar, which is related to Rast but in its
ascending form replaces the usual 9/8 step of the lower Rast tetrachord with
an interval sometimes understood as a kind of plus-tone, perhaps around 7/6,
or even in some accounts a regular minor third. Thus the Baron d'Erlanger
specifies 6-1-3 quartertones, or literally taken, 300-50-150 cents.
Following a plus-tone interpretation, I use this tuning:
............278.935...............80.537.........138.573
............168:143...............22:21...........13:12
.0........................278.935.......359.472.............498.045
1/1-----------------------168/143--------16/13----------------4/3
429.........................504...........528.................572
336.........................286...........273.................252
.0........................275.977.......357.422.............496.289
............275.997...............81.445.........138.867
The harmonic ratios are 429:504:528:572, and the string ratios
336:286:273:252. This is only one possible interpretation of Sazkar, and I
am told by an Arab musician that sometimes the second step may be taken at a
comma or so higher than in usual Rast, or around 8/7. With a Rast third at
around 16/13, this would be at or close to Ibn Sina's "very noble genus" in
his permutation featuring an elegant monochord division of 16:14:13:12
(8:7-14:13-13:12, 231.174-128.298-138.573 cents). Ibn Sina also gives
another permutation with 8:7 as the lower step and 13:12 preceding 14:13,
resulting in a higher third at 26/21 (369.747 cents). This interpretation is
also available at certain locations in MET-24 (231.445-125.977-138.867
cents), a tuning system where the various permutations of this genus of Ibn
Sina are one focus of the design.
Incidentally, a corresponding permutation of the preceding division of
176:169-13:12-13:11 (Section 2) can be used if one wants to experiment with
a version of Maqam Sazkar not too far from d'Erlanger's description of 6-1-3
quartertone steps also noted by Scott Marcus in some other 19th-20th sources
in Arab music theory, a "kinder, gentler" variation on the general theme of
a lower step synonymous with a regular minor third, a middle step smaller
than a regular semitone, and an upper step as in usual Rast.
..............289.210..............70.263........138.573
...............13:11..............176:169.........13:12
.0..........................289.210.....359.472.............498.045
1/1--------------------------13/11-------16/13----------------4/3
429...........................507.........528.................572
208...........................176.........169.................156
.0..........................288.867.....357.422.............496.289
..............288.867..............68.555........138.867
Here the middle thirdtone step from the regular 13/11 third to the 16/13
Rast third is 68.6 cents, as compared to the literal 50 cents of the
theoretical quartertone system (which Arab theorists such as Mikhail Mashaqa
in the 19th century and Amine Beyhom caution is not the most accurate
representation of intonation, with Beyhom explaining that the intention is
a flexible concept of 24 positions, not 24 equal quartertones.
Thus in usual Arab practice, if the "6-1-3" prescription were applied, the
result, assuming a Pythagorean intonation of regular diatonic steps and
intervals (as Marcus suggests), might be something like 1/1-32/27-16/13-4/3,
or 0-294.135-359.472-498.045 cents, with steps at around 32:27-27:26-13:12,
294.135-65.337-138.573 cents. This is a permutation of DOTT tetrachord 247
(p. 179, C17, characteristic interval 32/27, pyknon 9/8). MET-24 varies from
this mainly in having a slightly smaller regular minor third at 13/11,
narrower by 352:351 (4.925 cents).
How often Sazkar in its ascending form is played with a lower step at a
regular minor third somewhere around 32/27, a plus-tone step around 7/6, or
a step around 8/7 as in Ibn Sina's indeed "very noble" genus of 16:14:13:12,
remains an open question. But MET-24 has versions of all three options,
although all three are not available at any single location. The first
two options, with a lower step at a tempered 13/11 or 168/143, approximate
just divisions that may supplement the compendious catalogue in DOTT, while
the third simply affirms the beauty of a division, both aural and
intellectual, which Ibn Sina (c. 980-1037) documented about a millennium
ago.
----------------------------------
4. The 22:21-11:9-126:121 division
----------------------------------
As mentioned in the introduction, this division is a possible interpretation
for what I might term an accentuated Byzantine Palace Mode. It should be
noted that both Chrysanthos of Madytos in 1832 and the Committee on Music in
1881 associated this genus with a narrow upper step: a thirdtone at 66.667
cents for the Committee (4 steps of 72-ed2); and for Chrysanthos, an even
narrower 3 steps of 68-ed2 at 52.941 cents, which he associated also with
the Greek enharmonic genus. While the Committee's division is 6-20-4 steps
of 72-ed2 (100.000-333.333-66.667 cents), Chrysanthos specifies 7-18-3 steps
of 68-ed2 (123.529-317.647-52.941 cents), with a larger lower step but a
smaller middle step close to a just 6:5 (315.641 cents). A possible JI
interpretation for the latter is DOTT, tetrachord 218 (p. 177, C14), in the
permutation 14:13-6:5-65:63 (128.298-315.641-54.105). For the Committee's
tuning, a possible interpretation is a permutation of DOTT, tetrachord 198
(p. 176, C11) at 18:17-17:14-28:27 (98.955-336.129-62.961 cents).
The division I present here has a smaller lower step at 22:21 than either of
the above interpretations, a larger middle step than either at 11:9, and an
upper step at 126:121 (70.100 cents), a thirdtone comparable to the
Committee's 66.7 cents and rather larger than the 52.9 cents of Chrysanthos,
a quartertone close to 33:32 (53.273 cents).
.....80.537......................347.408.....................70.100
.....22:21........................11:9......................126:121
.0...........80.537...................................427.945.....498.045
1/1----------22/21------------------------------------242/189-------4/3
6237.........6534......................................7986........8316
484..........462........................................378.........363
The harmonic ratios are 6237:6534:7986:8316, while the string ratios are a
bit simpler at 484:462:378:363.
This division may have some kinship to DOTT, tetrachord 173 (p. 175, C7)
with a characteristic interval of 27:22 (354.547 cents), al-Farabi's ratio
for the wusta of Zalzal also noted by Ibn Sina, in a permutation at
22:21-27:22-28:27 or 21:22:27:28 (80.537-354.537-62.961 cents). This
permutation was famously used in Rod Poole's JI guitar tunings, and is
available in MET-24 at a tempered 81.445-357.422-57.422 cents
(0-81.455-438.867-496.289 cents).
While the division with a middle step of 11:9 was suggested to me by some
accounts of certain forms of practice in Byzantine music for the Palace
Mode, I am not sure of any sources of precedents upon which Rod Poole or Erv
Wilson, with whom he often collaborated, may have drawn upon for the
division with 27:22. The superparticular steps at 22:21 and 28:27 make it
noteworthy, whatever its possible connections with Near Eastern or other
musical traditions.
Also, while the permutations on which I have focused and also Rod Poole's
21:22:27:28 are neochromatic, with the characteristic interval as the middle
step, divisions like these as DOTT reminds us belong also to the spectrum of
Greek chromatic genera as surveyed, for example, by Aristoxenos and his
commentator Cleonides.
If this supplementary report on four tetrachords evidently not listed in
DOTT can help to make the database offered by that compendious source yet
more complete, and also to show how the scholarly landmark of John Chalmers
continues to inspire the study of the fine nuances of such divisions both in
theory and in musical practice, it will have served its purpose.
Margo Schulter
24 January 2019