The scope of this paper is to introduce and present two recent pieces I made for solo keyboard, Contracting triads in temperaments from 12 - 24 (2011) and Similarly spaced triads in temperaments from 12 - 24 (2011), and discuss the manner in which I used various equal-temperaments within a simple iterative framework to explore transformation of a chord pattern.
As a means of providing context for this discussion, it seems useful to present my current compositional concerns:
gradual pitch transformation, occurring through stepwise iterations and the audibility of such a process;
using combinations of tones, rather than what might be termed ‘a single line’, as the material of such a process;
the sonic and other results of such a process;
building in some sort of audible closure, most often through pitch movement reaching a clearly defined endpoint;
having human interpreters realise my music;
the sonic and other results of this realisation;
the occasional use of employing pitch transformations within various tuning systems.
Prompted by a commission from the Amsterdam-based Trio Scordatura, who specialise in performing music in different tuning systems, I developed the idea of feeding pitch material through a series of related tunings which would make audible the gradually pitch transformation as the material ‘passed through’ one system to the next. I was interested in seeing how the methodical nature of the process would explore intervallic space, and whether certain tunings would yield unique sonorities.
I had made other recent pieces which consisted of the circle of fifths (the harmonic construct which neatly returns to its starting point when applied to twelve-tone equal temperament (12-TET)) being fed into a process constructed around the nature of the progression, including Logical Harmonies 1 & 2(2011), Organ Harmonies (2012) and Ascending scale with piano accompaniment and Descending scale with piano accompaniment (2013); it seemed like appropriate material to employ again, this time to be fed through varying tuning environments. Moving through the circle of fifths anti-clockwise in twelve-tone temperament, this construct allows each of the 12 pitches to be heard in all three positions of a triadic chord, thus providing a good demonstration of the temperament’s capabilities. There are other patterns which do this just as well, but there seemed a certain appropriateness about employing such an harmonic construct designed entirely around the 12 pitches of the commonly-used equal temperament, in systems consisting of more than 12 pitches. This choice of harmonic material did not follow the concept of the processual tuning systems per say, but rather was a part of it. To be clear, my predilection for the circle of fifths at the time is undoubtedly a result of my own cultural conditioning in this respect.
In Contracting triads in temperaments from 12 - 24 and Similarly spaced triads in temperaments from 12 - 24, I decided that the circle of fifths would be heard initially in 12-TET, then in 13-TET, then 14-TET all the way through to 24-TET (since we start in 12-TET, another 12 seemed appropriate) thus yielding thirteen different temperaments in total. It seemed to me that there were two options to explore how the chords would be worked out when switching from one temperament to the next. As as result I made the two keyboard pieces with each exploring either the contraction of - or the similar spacing of -triadic materials.
Contracting triads in temperaments from 12 to 24
In this first piece, in each iteration, the chord shapes retain the same proportions (that is to say the same intervals in between the three notes of each chord) as in 12-TET, so that as the divisions of the octave become smaller and greater in number, the intervals between the pitches of the chords become smaller and smaller. One of the key principles of the circle of fifths – that the top note of each root triad is the root of the preceding triad – is then continued within each tuning system, and this results in a version of the circle of fifths heard in all thirteen of the different temperaments.
Figure 1 presents the first three chords from the circle of fifths in 12-TET (the chords alter by inversions, as I wanted to keep the chords within the same octave so comparison could be more readily made between temperaments). The figures below show all twelve notes in the octave (in the two pieces presented here, number 1 is always middle C, but can theoretically begin on any pitch) with highlighted numbers indicated those included in each chord.
Chord 1: 1 2 3 4 5 6 7 8 9 10 11 12 (notes C, E and G)
Chord 2: 1 2 3 4 5 6 7 8 9 10 11 12 (notes C, F and A)
Chord 3: 1 2 3 4 5 6 7 8 9 10 11 12 (notes D, G and B)
Figure 1: First three chords from the circle of fifths in 12-TET.
Figure 2 presents the first three chords of the progression in 13-TET, with each new chord taking the root of the previous chord as its highest note (in root position), and the two subsequent notes being three and seven notes lower respectively, following the shaping of the triads in 12-TET:
Chord 1: 1 2 3 4 5 6 7 8 9 10 11 12 13
Chord 2: 1 2 3 4 5 6 7 8 9 10 11 12 13
Chord 3: 1 2 3 4 5 6 7 8 9 10 11 12 13
Figure 2: First three chords from the circle of fifths in 13-TET.
In Figure 3 the initial starting chords in the four initial tunings are presented for comparison, in order to clarify the contracting nature of the chords as each tuning system includes an extra division of the octave:
In 12-TET: 1 2 3 4 5 6 7 8 9 10 11 12
In 13-TET: 1 2 3 4 5 6 7 8 9 10 11 12 13
In 14-TET: 1 2 3 4 5 6 7 8 9 10 11 12 13 14
In 15-TET: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Figure 3: First chord outlined in the four initial tunings to demonstrate intervallic contraction.
Whereas the 12-TET circle of fifths comprises 12 chords, each new additional tuning version has one extra chord to accommodate the additional tone added in the octave, such that, 13-TET has 13 chords and 22-TET has 22 chords in each cycle. There are two deviations from this norm resulting from the seven-note intervallic gap between the lowest and highest note of the root position major chords in the harmonic construct. The first is 14-TET, in which there are only two chords in the progression (see Figure 4):
Chord 1: 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Chord 2: 1 2 3 4 5 6 7 8 9 10 11 12 13 14
Figure 4: The two chords in the 14-TET progression.
The following chord in this progression would theoretically be Chord 1 again, so the cycle loops back on itself after only two chords; this is simply because the seven-note gap between the lowest and highest notes divides the scale into two, and so lands back on note 1 again after only two chords. Similarly, 21-TET has three chords in the progression, as 21 tones divided by intervals of a seventh results in only three chords (see Figure 5):
Chord 1: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Chord 2: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Chord 3: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Figure 5: The three chords in the 21-TET progression.
Had the source major chord included only a six-note gap rather than seven, then 12-TET would result in a two chord progression, 18-TET in three chords and 24-TET in four chords; as it is, the seven-note interval from a major chord in 12-TET results from specific acoustic-historical reasons.
Similarly spaced triads in equal temperaments from 12 to 24
In this second piece, the notes of the first chord in each tuning retain as similar a pitch-distance as possible to those in 12-TET, with each proceeding chord in that tuning following the intervallic shape set by its first chord. Thus, whilst all chords do not share the exact same intervals between their three notes as with Contracting triads in temperaments from 12 to 24, there is a greater similarity in how the chords cover the octave register and use pitch space. In Similarly spaced triads in equal temperaments from 12 to 24, the first chord in 17-TET has a much more similar pitch-spacing to the first chord in 12-TET than in Contracting triads in temperaments from 12 to 24. Figure 6 presents the first three chords in 12-TET and 17-TET:
Figure 6: Comparison of the first three chords of Similarly spaced triads in 17-TET with 12-TET triads.
At play in these pieces is the demonstration, and exploration, of the nature of dividing the octave into equal steps – or at least a window into that, from 12 divisions up to 24 inclusive. To that end, as much as I may conceive of the circle of fifths fulfilling the role of material, in a manner it is simply the carrier of the experience, of a journey through these temperaments. I hear a residual effect from the perceptual draw of the circle of fifths which to some degree spills over into the succeeding systems, but that may simply be my own hearing doing its best to, on occasion, customise the auditory material into a form of twelve-tone temperament.
As various chords are compared to earlier chords in 12-TET, some seem closer than others, as we would expect. This act of comparing then prompts me to interrogate my own pitch perception abilities; how much can I distinguish between one chord and another? How clearly can I recognise that we are back to the first chord in the progression? Can I find similarities between some chords in a tuning system? How well can I familiarise my listening?
However, the process for me here affords a continuous examination of the tuning environments, or rather what the constructions of the piece suggests are single, distinct environments. It was important not to build in any kind of audible signal to raise awareness of a change to a new temperament (the performer plays consistently across each change of tuning, without making any distinction), so that each listener deals with the transforming temperaments in their own way. For this reason, as with other uses of systematic process in which I am interested, the listener does not notice the gears changing between different iterations, but rather perceives ever-changing colour, or of transformation of new sonic relationships made by their auditory systems, which is simply searching for whatever patterns it can find. When I listen to these pieces, it is not important to me whatsoever to appreciate whether the chords currently heard are in the 16- or 17-tone temperament; rather it is the continual transformation of the sonorities of the chords, afforded by these successively changing tuning environments, to which I attend.
Despite the prevalence of the twelve-tone temperament here, as mentioned before, I am also intrigued by the potential of having a version in, say, thirteen-tone equal temperament, with enough repetition at the beginning and a pitch or chordal pattern designed specifically within that temperament to take advantage of its own pitch relationships.
As a means of contextualising the approach to multiple tuning systems in these pieces, we can turn to the notion of ‘Polymicrotonality’ which has been propagated by composer and bassoonist Johnny Reinhard since the 1990s. The term describes his approach of utilising various tunings systems within a piece, either simultaneously or in some form of succession, so as to exploit certain intervallic relationships and create specific sonic environments for the purposes of exploring the “emotional plane”  in music. Whilst I am also combining various tuning systems, I feel that I am approaching things in a somewhat different way. I am less interested in selecting a combination of tuning systems based upon the specific colouration they produced in reference to emotional, expressive concerns; rather, for me it is the audibility of the process that makes this fascinating - the sonorities of the changing tunings speak for themselves, and I as the composer do not feel I need to exploit specific intervals within them for expressive gain. I believe what expression there is lies within the process itself. I am strongly drawn towards these logical, stepwise processes, which do not divert but play themselves out and result in a certain parity in construction regarding the nature of the different systems. Violinist Mira Benjamin describes similar process-based approaches as “implied microtonality” , in which the beginning and ending pitches are fixed, and microtonality necessarily emerges from the transition between the two; whilst the two keyboard pieces here have each of the discrete steps in the process notated down (i.e. each of the separate tunings), they were both conceived as traversing the tuning plane between 12-TET and 24-TET, and so in this way they are fulfilling the notion of implied microtonality.
I feel I have no particular allegiance to any specific tuning system, except to point out a cultural conditioning towards twelve-tone equal temperament. I have made pieces using Just Intonation, numerous divisions of the octave, open microtonal colourings and plenty of pieces in which I do not consider tunings per say, but I have no intention of advocating one system over another. My outlook for the type of processes I am advocating here is that the choice of tuning system should be entirely open. Equal divisions of the octave lend themselves well to a systematic process, but as long as the process itself is based around the mechanics of the tuning system, the piece will then explore the nature of the system itself. To clarify, here are some further examples:
the organ in the Phipps Hall performance space at the University of Huddersfield is in Valotti tuning, an 18th century system based upon pure fifths between F through Ab to B in the circle of fifths, and fifths which are a Pythagorean comma narrower from F through D to B. Thus, a process which plays lines or chords through different Valotti tunings, where each iteration is out by for example a Pythagorean comma, would lead to results centred around the nature of the Valotti tuning itself;
a just intonation environment, in which material based upon the harmonic series is fed through a system of JI tunings whose fundamentals derive from subsequent pitches played in the initial material;
a 29-note equal tempered chromatic scale, which is then played alongside a 30-note equal-tempered scale chromatic scale beginning on the same pitch over the same duration, and then a 31-note equal tempered chromatic scale, and on.
The possibilities for processes derived from the machinery of a tuning system are many, and present an opportunity to treat tunings not as a means to exploit certain intervals so as to enhance expressive, dramatic features within the music (and likewise, combining two or more tunings to create greater gestural-dramatic impact), but as a means of gradual transformation, of methodically varying appropriate material resulting in a continually changing pitch. The high predictability of these pieces in construction leads to a surplus of nuanced change in the auditory outcome. Each tuning systems reveals aspects about itself, its own pitch relationships, and relationships between one iteration and the next; iterative processes reveal these colours and their combinations in what I perceive as a satisfyingly continuous form.
Currently, I feel it is still important for my own music to be performed by human interpreters, as I enjoy the sense of ownership with which they are able to assume over the nature of the piece. Whilst I very much enjoy the music of a number of artists’ work without that performance step in the process, it is something which I continue to align with as a content creator in my own work. For the two present pieces, however, this causes issues in translating the concept onto instruments to be performed live, in which performers need to navigate easily through these various tuning environments. Thus, the approach to composition for this new piece was prompted particularly by the opportunity offered by the Trio Scordatura commission: Bob Gilmore plays keyboards which have the ability to switch easily between different tuning systems. This is a paradigm away from custom-made instruments in specific tunings, for instance the 19-division trumpet, the orchestral xylophone and the quartet-tone bass flute; these instruments are designed to reveal certain aspects within one single tuning system, and shifting between others, or combining two or more, is made very difficult.
A comparative instrumental approach is provided by guitars with moveable frets for each string, which allow for tuning system changes but there is less flexibility to switch quickly between tunings. The magnetic replaceable fretboards by guitarist John Schneider called ‘Switchboards’ (see Figure 7) provide a wide capability of tunings, but less flexibility mid-performance; pieces/sections can only explore one tuning, before having to pause whilst attaching a new fretboard.
Figure 7: ‘Switchboard’ Guitar by John Schneider, (reproduced from BillaIves.com at which there is more information on the history of flexibility in tuning for guitars.)
In contrast, I purposefully required the most minimal change affordable by the performer so that there is no, or little, demarcation of the move into a new tuning environment; a keyboard, tuning software such as Scala, and a foot switch solve the problem, by granting the performer an easy flexibility to switch between the various tunings. The two pieces were designed entirely around performance at the keyboard, and as the interpreter, Gilmore had free reign over which samples he used to perform. The harp sounds he chose were informed by a desire for perceptual clarity, so as to bring the various pitch relationships to the fore.
I believe we can conceive of tuning systems, or orderings of discernible frequencies, as compositional objects. Relationships between specific frequencies in individual systems become the grain of these systems, and the relationships between systems become a structural tool. For a music-maker interested in simple gradual pitch processes, the elegance of aligning tuning systems together in this way provides me with in an extraordinary exploration of pitch space.
There are plenty more ways in which process can act as a structuring tool for exploring successive tuning systems, and the choice of pitch material leaves options for directing the listener towards audible, predictable effects of the tunings, or into entirely unpredictable, varied sonic environments, or somewhere in the vast region between.
Alves, B. (n.d.). The Tuning of Lou Harrison - Por Gitaro: Suites for Tuned Guitars, performed by John Schneider. [Online]. [Accessed 28/11/2014]. Available from:http://www.billalves.com/porgitaro/porgitarotuning.html
 Reinhard, J. (1997). Composing Polymicrotonally. [Online]. [Accessed 28/11/2014]. Available from: http://www.stereosociety.com/jrpolymi.shtml
 In conversation. My pieces Imperfect Harmony for double bass and Chords and Transformations for violin are examples of such an approach.
About the Author:
ichard Glover is a composer and writer whose work focuses on the use of sustained tones and processual music structures. As a writer he has published articles on minimalist music, the work of Phill Niblock as well as co-authoring the book ‘Temporality and experimental music’ with Bryn Harrison (2013). Glover’s paper reveals the specific processes that underpin two of his recent keyboard pieces for Bob Gilmore, commissioned by Trio Scordatura.