Why Fifty-Three? |
dividing the pitch space
Here we are concerned with the tonal hierarchy, namely, the stability and embodiment of “the hierarchical relations that accrue to an entire tonal system beyond its instantiation in a particular piece. Such a hierarchy is atemporal in that it represents more or less permanent knowledge about the system rather than a response to a specific sequence of events” (Lerdahl, 2001: p.41) within the pitch space. In this case the pitch space is divided into fifty-three unequal divisions within the octave, calculated by applying a harmonic just intonation (JI) tuning system.
The 53-JI hyperchromatic scale I utilize is based in five-limit tuning. It is the scale Ben Johnston used to demonstrate scalar order where “an interlocking system of triads is set up, such that every tone participates in four different capabilities (as root, as fifth, as major third, and as minor third), and if this derivation is stopped at points where the twelve chromatic regions of the octave would overlap, a fifty-three-tone scale results” (Johnston, 2006: p.24). There are three types of adjacent intervals occurring in this scale, these are 81/80 (21.506 cents), 2048/2025 (19.553 cents), and 3125/3072 (29.614 cents).
The hyperchromatic scale can be subdivided into fifty-three fully functioning 17-tone scales. These scales offer all the recognisable intervals occurring in 12-tone equal temperament (TET): the Root, minor 3rd; Major 3rd; Perfect 4th; Perfect 5th; minor 6th; and Major 6th. Alongside these are the minor 2nd, Major 2nd, the tritone (b5th), minor 7th, and Major 7th. This second set of intervals in the 17-tone scales offer choices. There are two options for each intervallic degree, both a small and large minor/Major 2nd/7th, as well as an augmented 4th and diminished 5th in place of the tritone.
JI intervals are calculated from a fundamental, the 1/1 ratio, and to this I assigned the standard 12-TET note ‘G’ (0 cents, at 48.999Hz, 97.999Hz, 196Hz, 392Hz, etc.). Once the fundamental was assigned it was possible to start creating the other pitch classes in the scale. If we take the common seven note names and assign them to their just intervals, we create the following scale (with ratio and cents values):
G (1/1 - 0)
A (9/8 - 203.9)
B (5/4 - 386.3)
C (4/3 - 498)
D (3/2 - 702)
E (5/3 - 884.4)
F (16/9 - 996.1)
G (2/1 – 1200)
The five accidentals that complete a 12-TET scale commencing from G would fall on 100, 300, 600, 800 and 1100 cents. These are used to demarcate the boundaries between letter names. For example, the two pitches either side of 100 cents fall on 92.2 cents and 111.7 cents, so 92.2 will belong to the G category and 111.7 the A category. So, above the G there are four additional pitches before we reach 100 cents. These will be allocated as sharps. There are four pitches between 100 cents and A at 203.9 cents, these will be allocated as flats. Between the notes B and C, as well as, E and F, there are no accidentals in 12-TET, so the boundaries fall halfway. For example, the pitch class boundary between B and C would fall on 450 cents. There are two additional notes above the B (386.3) before reaching 450 cents and an additional two between 450 cents and C (498). Accidental note naming and related symbols in the 53-JI scale are as follows:
(•) sharp
(••) double sharp
(•••) triple sharp
(x) quadruple sharp
(b) flat
(bb) double flat
(db) triple flat
(d) quadruple flat
If we return to the example between G and A with the 100 cents boundary, there are four notes either side implying the chromatic scale would read:
G / G• / G•• / G••• / Gx (100 cents boundary) Ad / Adb / Abb / Ab / A
The seven alphabetical groupings of pitch are each assigned a colour. These are as follows:
A = blue / B = yellow / C = red / D = green / E = orange / F = pink / G = purple
This system has two major functions; to assist the performers in navigating their way around the instruments and aid the rapid comprehension of scores.
There are four deviation measurement areas encountered throughout the complete cyclic system. The first is 0.1 cents due to the rounding to one decimal place, which is negligible as the human ear is incapable of recognising a difference of under a one cent discrepancy and is marked accordingly as a ‘0’ deviation. The second is approximately 2 cents, perhaps slightly noticeable when using precise electronically generated tones, but again negligible when put into the context of the rich sonic properties created by the acoustical instruments deployed for this work. The third and fourth deviations of approximately 8 and 10 cents are noticeable and can be utilised to create additional variations of psychoacoustic beating patterns when combining adjacent pitch pairings.
The system creates a multitude of 17-tone scales with slight to distinct variations of intervallic pitch variations. There are scales with no deviations such as the scales with roots of Ab - 10/9 and Bdb - 6/5; to scales that contain multiple deviations such as C•• - 512/375 and Ebb - 625/384. The modulation of key in this system is comparable to music composed in ‘well temperament’, where all scales “are usable, but they do not all sound the same” (Gann, 2019: p.90). Like the composers who used this tuning system I also undertake explorations of the different qualities each key centre can offer.
As Harry Partch poetically describes when comparing the artist’s palette with the composer’s scale, “the composer yearns for the streaking shades of sunset. He gets red. He longs for geranium, and gets red. He dreams of tomato, but he gets red. He doesn’t want red at all, but he gets red, and is presumed to like it” (Partch, 2000: p.159-160). Partch satisfied his yearning for tonal colourings, shadings, and contrasts with the construction of his 43-JI scale; a hyperchromatic scale based in eleven-limit tuning. The 53-JI scale gives me access to these important features but also something extra that Partch’s scale and many other just intonation scales would not, processual pitch procedures.
Using Partch’s scale to demonstrate a simple example of the differences in pitch functionality to that of the 53-JI scale we need look no further than the Root to PER5 interval which has a cents measurement of 702. Both 43 and 53-JI scales contain some deviations from the systematic just intervals when modulating key, but there is an insurmountable predicament when concerned with forming processual pitch procedures.
The 12/11 Root has its PER5 interval fulfilled by 18/11 with 0 cents deviation. The following chromatic pairing of fifths features 11/10 as the Root and 5/3 as PER5, with a cents deviation of +17.4. If we pair 11/10 with 18/11 the deviation is less at -14.4, but that ratio is already occupying a PER5 function with the 12/11 pairing. The following chromatic pairing of fifths features 10/9 as the Root and 27/16 as PER5, with a large deviation of +21.5 cents (81/80, a syntonic comma). If we pair 5/3 with 10/9, we get 0 cents deviation, then pair 27/16 with the Root 9/8 we also get 0 cents deviation. A problem arises in that there is now no ratio available to function as fifth for the 11/10 as both possibilities are occupied in other couplings.
When working with processual pitch procedures using just intonation, this ‘pitch void’ would produce an immediate halt to any trajectory the composer chooses to navigate through the hyperchromatic scale. 53-JI is absent of any ‘cul-de-sacs’ within the intervallic relations and is one reason why I choose to use this scalic structure. I could have chosen to use an equal division of the octave (EDO) system, but this would have limited the intervallic pitch relations into a singular standardized format with no variation when modulating key centres. If, for example, I utilized 53-EDO all adjacent pitches would measure 22.641 cents, and all beating pattern rates would be standardized into respective interval distances. I find it preferable to embrace the variations of intervallic combinations that arise through just intonation.
53-JI is integral to my compositional methods and there is an intrinsic relationship between the tuning system and how it allows me to create processual pitch procedures. Having an enlarged pitch gamut allows me to inhabit extended temporal durations that feature a regularity of harmonic movement and variation that would not be possible otherwise.
A further reason I choose to use 53-JI relates to the concept of pitch space itself and how I regard it in some sense as a materiality, or a smooth space that has a surface on which one makes striations. Campbell characterizes this sense of pitch space as “interior spatiality” (Campbell, 2013: p.68) where, when referring to Boulezian concepts, he surmises that “striated space is marked by a standard, regular measure, which creates clear perceptual landmarks for the ear to orient itself, whereas smooth space is free, irregular and dispenses with all points of reference” (Ibid. p.72). The works presented here undoubtably inhabit striated space as the fifty-three striations are divided, discontinuous, and thoroughly perceptible to the ear.
Having the pitch space divided into fifty-three striations is already many divisions, far more than is commonly encountered in western music making, and the “smaller the partitions or the micro-intervals within a striated space, the closer it will be to being conceived of as an unbroken smooth continuum” (Ibid. p.73). To attain an approximation of smooth space the composer Wyschnegradsky “considers the interval of 1/12 of a tone as the limit to ultrachromatic possibility, almost imperceptible to the human ear” (Ibid. p.96). This would create a hyperchromatic scale in 144-EDO. Fifty-three striations are an acceptable compromise as practicality comes into play here. Using fixed-pitch acoustic instruments, I am limited in the possibilities of exterior spatiality in which to store them and rehearse an ensemble, but through compositional means the works aim to extend a sense of a smooth space that is undivided and continuous.
​
