---------------------------------------------- Evolution of Goya-Supergoya Subsets of Zest-24 Some scenarios for musical expansion ---------------------------------------------- Various theories and scenarios seek to account for changing musical systems: for example, Joseph Yasser's theory of evolving tonality which posits a kind of quasi-Fibonacci sequence of systems based on respectively 5, 7, 12, 19, and 31 notes per octave, etc. Here I would like to look at the practical considerations which can influence the development of subsets of a tuning system such as Zest-24, which consists of two 12-note modified meantone circles based on Zarlino's 2/7-comma temperament. Here I address the slightly modified version of this system on a synthesizer in 1024-EDO. ! zest24n.scl ! 1024-tET/EDO nanotemperament of Zest-24 24 ! 50.39062 70.31250 120.70312 191.01562 241.40625 287.10938 337.50000 383.20313 433.59375 503.90625 554.29687 574.21875 624.60938 696.09375 746.48437 778.12500 828.51562 887.10938 937.50000 996.09375 1046.48438 1079.29688 1129.68750 2/1 Specifically, this article focuses on the development of the subset called "Supergoya," a 20-note set which grows out of a 17-note system called "Goya." Both the 17-note set and its 20-note superset are named in honor of a giant herbivorous rodent, informally known as "Goya," inhabiting South America about 8 million years ago. ------------------------------------------ 1. Foundations of Goya: From 8 notes to 12 ------------------------------------------ The foundations of Goya may be sought in the system of 8 notes (Gb-G) in either 12-note circle of Zest-24 where major and minor thirds have a range of roughly from Pythagorean to septimal in sizes and colors. Thus within this 8-note system, major thirds have a range of 408-434 cents, and minor thirds of 274-300 cents. This palette of colors is very attractive for neomedieval music, the main caution being that, as in a tuning such as 22-EDO, the fifths and fourths are notably more impure than in a milder temperament such as Peppermint, let alone Pythagorean just intonation. We begin with this 8-note system on the lower keyboard, showing both the actual keyboard names of the notes (Gb-G) and equivalents in an untransposed medieval modal system, where these notes could correspond to the standard or _musica recta_ gamut of Bb-B. Here we take Db as the "1/1" of what will become a larger system. (Bb) 504 Gb Db Eb F G Ab Bb C Db 0 217 434 626 708 926 1130 1200 (F) (G) (A) (B) (C) (D) (E) (F) What we have here is a kind of neomedieval Lydian mode on Db, equivalent to an untransposed medieval range of F-F, with a fluid fourth degree which may be realized as either G or Gb -- much like the fluid B/Bb of the untransposed medieval range. Taking Db as the 1/1, our chain of seven fifths provides more or less accurate approximations for the major septimal intervals (second, third, sixth, and seventh) and the augmented fourth. Thus we have: 4/3 Gb 504 +6 1/1 8/7 9/7 81/56 3/2 12/7 27/14 2/1 Db Eb F G Ab Bb C Db 0 217 434 626 708 926 1130 1200 just -14 -2 -13 +6 -7 -7 just The next step is to derive a larger 12-note set by adding four notes from the upper keyboard which provide septimal approximations of the minor second, third, sixth, and seventh. Here is the arrangement as a system derived from two chains of fifths: 28/27 14/9 7/6 7/4 50 758 267 976 Db* Ab* Eb* Bb* | | | | | | | | 4/3 1/1 3/2 8/7 12/7 Gb Db Ab Eb Bb F C G 504 0 708 217 925 434 1130 626 Because of the factor of temperament, the approximate equivalents of just septimal intervals sometimes involve melodic steps of different sizes than would obtain in a JI system. Thus the 50-cent diesis between the two keyboards might be seen as a kind of tempered compromise about midway between the septimal ratios of 49:48 at 36 cents and 28:27 at 63 cents. The actual step is very close to 35:34, which represents for example the difference between a just 17:15 and 7:6, ratios which occur in virtually just form at Db-Eb and Db-Eb* respectively. ! zest24-subgoya12_Db.scl ! Zest-24 subset (12) of Goya-17, near-septimal major/minor intervals over Db 12 ! 50.39062 216.79688 267.18750 433.59375 503.90625 625.78125 707.81250 758.20312 925.78125 976.17188 1129.68750 2/1 This 12-note system, while not, of course, a Moment of Symmetry or MOS because of the various sizes of adjacent steps, nevertheless has a pleasing symmetry, with a diverse complement of septimal and other neomedieval interval sizes (e.g. thirds at 422 and 287 cents). It provides a basis for the next step of expansion -- Goya-17. ---------------------------------------------- 2. Goya the Friendly Giant Rodent: 12 + 5 = 17 ---------------------------------------------- While the 12-note set provides a complement of major and minor septimal approximations above our 1/1 at Db, and also the augmented fourth Db-G, we might wish to add a set of neutral intervals, thus arriving at a 17-note system. In this system, Goya-17, we add in addition to a neutral second, third, sixth, and sixth also a diminished fifth or superfourth at Db-Gb*, a bit larger than 11:8 at its size of around 554 cents. N2 N6 N3 N7 dim5 min2 min6 min3 min7 171 867 363 1059 554 50 758 267 976 Ebb/D Bbb/A Fb/E Cb/B* Gb* Db* Ab* Eb* Bb* | | | | | | | | | | Gb Db Ab Eb Bb F C G 504 0 708 217 925 434 1130 626 4th 1/1 5th maj2 maj6 maj3 maj7 aug4 In some ways, Goya-17 much resembles an unequal 17-tone circle. Thus above the 1/1, we have a sequence of minor, neutral, and major intervals, often with sizes comparable to those found within such a circle (for example, George Secor's 17-WT). This familiar pattern is one of the appeals of the system. ! zest24-Goya17_Db.scl ! Near-septimal major/minor intervals (12) plus 171 363 554 867 1059 cents 17 ! 50.39062 171.09375 216.79688 267.18750 363.28125 433.59375 503.90625 554.29688 625.78125 707.81250 758.20312 867.18750 925.78125 976.17188 1059.37500 1129.68750 2/1 At the same time, in other aspects Goya is radically different from a 17-tone circle. While the latter has some kind of small semitone, or "thirdtone," for each adjacent step, often with sizes in the range of about 55-80 cents (64-78 cents in Secor's tuning, here from Db to Eb, for example, we have a sequence of steps (Db-Db*-D*-Eb) at 50-121-46 cents. Also, the name Goya might especially fit the "giant" neutral second step of Db-D* at 171 cents, rather larger than 11:10 (about 165 cents) or the 22-EDO chromatic semitone at around 164 cents. Like the latter, this interval can represent the septimal ratio of 54:49 (168 cents), the difference, for example, between a septimal minor third at 7:6 and a major third at 9:7. Indeed, we can cite an example where the 171-cent "Goya" step nicely plays this role, marking the difference between a virtually just 14:9 minor sixth at 766 cents and a somewhat wide 12:7 major sixth at 938 cents (just sizes 765 and 933 cents). 766 937 Db -- D* F F The dramatic effect of this giant neutral second can be felt in this realization of a two-voice progression from Marchettus of Padua (1318), where this famous medieval theorist recommends that the motion of the upper voice divide the whole tone (here Db-Eb) into a "chroma" equal to "four parts" plus a cadential "diesis" equal to "one part." while these "five parts" need not be taken as equal, the kind of radical contrast between the sizes of the chroma and diesis which this description seems to imply is realized yet more dramatically in Goya-17 than in a 17-tone circle: Db -- +171 -- D* -- +46 -- Eb 696 938 1200 Gb -- -70 -- F -- -217 -- Eb In this neomedieval context, typical of Goya-17, the motion of the lowest voice may be regarded as routinely diatonic: a small semitone at 70 cents followed by a whole tone at 217 cents. The upper voice moves by the Goya-size chroma of 171 cents, followed by a small semitone or diesis of 46 cents, resolving the 938-cent major sixth to a 1200-cent octave. Here the 171-46 cent division of the tone in the upper voice happens to be not too far from a literal ratio of 4:1. This progression also suggests how our perception of the category of an interval may depend on context. In many settings, a 46-cent step would be musically as well as theoretically quite distinct in effect from a "semitone." Here, however, it plays much the same role as a more usual 28:27 step in septimal JI, facilitating the expansion from a 12:7 major sixth to an octave while the lower voice descends a tone. Similarly, while in a wide range of contexts we could consider 27:25 at 133 cents as a small neutral second, in a 5-limit JI system it can play the role of a large semitone or "limma" (to borrow a Pythagorean term for a diatonic semitone at 256:243 or 90 cents, about 43 cents smaller!). The Goya-17 set is a fascinating one, but can use a bit of help from a few more steps. Let us see musically why Goya-17 invites expansion to the very happy system of Supergoya-17plus3. ------------------------------------------------- 3. Freedom of Mobility and Supergoya: 17 + 3 = 20 ------------------------------------------------- Like a 17-tone circle, Goya-17 provides over our 1/1 at Db a full complement of major, minor, and neutral intervals. However, from a viewpoint of dynamic polyphony, more notes are needed in order to enjoy a very attractive feature of such circles: freedom of mobility in making directed cadences from a given unstable sonority to alternative stable goals for the resolution. This imperative of mobility gives us a prime motivation for expanding the system, and in the process adding extra features and options beyond those present in a circulating 17-note tuning. The issue of mobility hit me squarely when I was considering resolutions for this fine sixth sonority Db-F-Bb, which at 0-434-926 cents is not far from a just 7:9:12 (0-435-933 cents), the accuracy being close to that of 22-EDO. In a 17-tone circle, such a sonority should have three available standard resolutions. The first choice is an _intensive_ resolution where the lower voice descends by a whole tone while the upper voices ascend by semitones. The second choice is a _remissive_ resolution where the lowest voice descends by a semitone and the upper voices ascend by whole tones. The third alternative is an _equable_ resolution, as George Secor has well named it borrowing from the language of Harry Partch and ultimately Ptolemy, where each voice moves by some kind of neutral second. In Goya-17, the remissive and equable resolutions are easy to find: remissive equable Bb -- +204 -- C Bb -- +134 Cb*/B* F -- +192 -- G F -- +121 Gb*/F#* Db -- -70 -- C Db -- -141 Cb*/B* However, to find an intensive resolution of Db-F-Bb, we need more notes than are at hand in this 17-note subset. In a full Zest-24, the obvious solution would be present within the lower keyboard: Bb -- +83 -- Cb/B F -- +70 -- Gb/F# Db -- -191 -- Cb/B An alternative solution in keeping with the septimal theme of the Goya family of tunings within Zest-24 is to add the note F* on the upper keyboard, thus facilitating an intensive resolution with a characteristically "Goyan" color: Bb -- +50 -- Bb* F -- +50 -- F* Db -- -224 -- Bb* Here both the large whole-tone step in the lower voice, approaching 8:7 in size, and the very compact 50-cent semitone steps in the upper voices give the progression a distinct flavor. In a mode like medieval or neomedieval Lydian on Bb*, these small steps, rather as in 22-EDO, might take on the quality of "diatonic quartertones": Bb* Db Eb F F* Ab Bb Bb* 0 224 441 657 708 932 1150 1200 224 217 217 50 224 217 50 At the same time as we have provided an intensive resolution for the routine neomedieval sixth sonority Db-F-Bb, we have also added a new septimal approximation with respect to Db as the 1/1: F* at 484 cents, or somewhat wide of 21:16 (471 cents), but with a similar "narrow fourth" color. This new interval can be used very effectively, for example, in a mathematically rough but musically telling approximation of the beautiful 16:21:24:28 sonority (0-471-704-969 cents) at Db-F*-Ab-Bb* (0-484-708-976 cents). If we continue this expansion process by adding not only F*, but C* and G*, then we have included all 12 notes of the upper keyboard, each contributing some septimal approximation within about 14 cents of just with respect to the 1/1 at Db -- if we take D* to represent 567/512, as shown in this diagram of the chains of fifths, or 54/49 (168 cents), which may best fit both its size and its typical role: |---------------------------------------------------------------------------------| 7168/ 3584/ 6561 2187 896/729 448/243 112/81 28/27 14/9 7/6 7/4 21/16 63/32 189/128 567/512 Ebb* Bbb* Fb* Cb* Gb* Db* Ab* Eb* Bb* F* C* G* D* 171 867 363 1059 554 50 758 267 976 484 1180 676 171 +18 +12 +6 0.3 -7 -13 -7 +0.3 +7 +13 +7 +1 -6 | | 4/3 1/1 3/2 8/7 12/7 9/7 27/14 81/56 Gb Db Ab Eb Bb F C G 504 0 708 217 926 434 1130 625 +6 Just +6 -14 -7 -2 -7 -12 In addition to the notes of Goya-17, we thus have the three "comma keys" F*, C*, and G*, representing 21/16, 63/32, and 189/128 -- respectively a septimal comma (64:63, 27.26 cents) smaller than 4:3, 2:1, and 3:2. The narrow fourth is very attractive in a range of septimal applications, and also serves as an essential element in the tempered 6-7-9 duodene with respect to Db. In gamelan styles, both this narrow fourth and the narrow fifth around 676 cents are most congenial sizes, while the narrow or wide octaves introduced into the system by these three "comma keys" can, in appropriate timbres, capture something of the shimmering sonority of Balinese or Javanese ensembles. As shown in the diagram, Ebb* or D* on the upper keyboard might be interpreted in linear chain-of-fifths terms as either a quite inaccurate 7168/6561 septimal neutral third at 153.19 cents, wider than 12/11 by 19712:19683 or about 2.55 cents; or as a less inaccurate 567/512 at 176.65 cents, which like the actual 171.09-cent interval can nicely serve as an "equable heptatonic" step near 1/7-octave. This septimal ratio of 567:512 less 1029:1024 (about 8.43 cents) yields 54:49, to which our 171-cent Goya/Supergoya step is closest. Here, as the diagram shows, Db-G* is very close to a just 189:128 (674.69 cents); given that G*-D* is about 7 cents narrow of 3:2, an amount tempering close to the ratio of 1029:1024, Db-D* is thus a good approximation of 54:49. Note that the sizes of 7168:6561 and 567:512 (153 and 177 cents) differ by a Pythagorean comma at 531441:524288, or 23.46 cents -- an interval tempered out in the 12-note circle on the upper keyboard whose circumnavigation brings us from Ebb* to the equivalent D*. One feature revealed by this diagram is the "ebb and flow" of narrow and wide septimal approximations as we encounter regions on either keyboard with narrow or wide fifths. There are a few notable points with near-just approximations: 1/1 (of course) and 9/7 on the lower keyboard; and on the upper, 7/6, 448/243, and 189/128. Interestingly, the near-just 9:7 is shared in common with 22-EDO; and the near-just 448:243 neutral seventh at around 1059 cents with 17-EDO. Further, in Supergoya as in 17-EDO, the difference of 141 cents or so between this interval and the octave is divided into two equal semitone steps of around 70 or 71 cents (70.588 cents in 17-EDO; 70.312 cents here), thus Cb*/B*-C-Db. An advantage of Supergoya is that 19 of the 20 steps have perfect fifths reasonably close to 3:2 -- all except G on the lower keyboard. Likewise, Gb on the lower keyboard has no close approximation of 4:3, although it has a near-just 21:16 narrow fourth (472.27 cents, or not quite 1.5 cents wide). The other 18 steps have reasonable approximations of both 4:3 and 3:2. ! zest24-supergoya17plus3_Db.scl ! Goya-17 plus 484, 676, and 1180 cents 20 ! 50.39062 171.09375 216.79688 267.18750 363.28125 433.59375 483.98437 503.90625 554.29688 625.78125 676.17187 707.81250 758.20312 867.18750 925.78125 976.17188 1059.37500 1129.68750 1180.07812 2/1 ----------------------------- 4. Two Supergoya applications ----------------------------- The Supergoya system of "17plus3" has at least two sides. We can focus on types of intervals also typically approximated by a 17-tone circle like Secor's 17-WT; or on varieties of septimal intervals (e.g. 21:16) not found in such a circle, but invaluable for "quasi-JI" styles or gamelan music, for example. The following version of a medieval Near Eastern tuning known as Buzurg illustrates the first approach. The Buzurg pentachord, as described by the Persian theorists Safi-ad-Din al-Urmavi and Qutb-al-Din al-Shirazi in the era of around 1250-1300, has a scheme of ratios such as 1/1-14/13-16/13-4/3-56/39-3/2 (0-128-359-498-626-702 cents). Here a lower Buzurg pentachord is complemented by an upper tetrachord designed to maximize "consonance" -- that is, the number of intervals available near 3:2 or 4:3. Cb* Db Eb Fb* F* Gb* Ab Bb Cb* 0 141 357 504 625 695 848 1066 1200 141 217 146 121 70 154 218 134 While the relative symmetry between lower and upper tetrachords (steps of 141-217-146 and 154-218-134 cents) produces a kind of mode lending itself to many polyphonic cadences featuring a resolution to a stable fifth on various degrees, I should emphasize that more asymmetrical tetrachords are very typical of both the Arabic/Turkish/Kurdish maqam systems and the dastgah system of Persian music. Thus a Buzurg pentachord might serve as a basis for various modes. A 10-note subset within the subset of Supergoya exemplifies a "quasi-JI" approach to septimal sonority: a "trihexany" or "quadriquad" combining three tempered variations on the 1-3-7-9 hexany (1/1-9/8-7/6-21/16-3/2-7/4). We find different flavors of this hexany, an example of the Combination Product Set (CPS) approach explored by Erv Wilson and others, on the steps of Gb, Ab, and Db. 254 963 472 1180 676 Ab* ----- Eb* ----- Bb* ----- F* ------ C* | 963 | 976 | 976 | 963 | | 254 267 | 267 268 | 268 267 | 267 254 | | 442 | 441 | 441 | 442 | Gb ------ Db ------ Ab ------ Eb ------ Bb 0 696 204 913 422 This trihexany system may also be called a "quadriquad" because it includes four septimal "quads" or four-voice sonorities approximating more or less closely 12:14:18:21 (~0-267-702-969 cents), a favorite neomedieval sonority characteristically inviting intensive, remissive, or equable resolutions. Here the chains of fifths are shown horizontally, and approximations of the 7:6 minor third vertically. Here close septimal approximations are mixed with other types of neomedieval intervals: for example, the 254-cent minor thirds at Gb-Ab* and Bb-C* which might be termed "interseptimal," since they occupy the territory between two septimal ratios, here 8:7 and 7:6; and also major and minor thirds at 422 and 287 cents. For more on "interseptimal" and other regions of the interval spectrum, see . The diagram above shows the four quads, with the numbers within the square representing a given quad showing the sizes of the outer minor seventh, two minor thirds, and major third between the middle pair of voices forming it. As in a septimal JI system, we have intervals at (or here very close to) such ratios as 21:16 and 189:128; and as in Supergoya, or more generally Zest-24, we have the characteristic variations of unabashed unequal temperament. Possibly Supergoya is especially intriguing in the way it brings together these traits. Most appreciatively, Margo Schulter mschulter@calweb.com 16 December 2007