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