Composing in Bohlen Pierce and Carlos Alpha scales for solo clarinet

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1 Composing in Bohlen Pierce and Carlos Alpha scales for solo clarinet Todd Harrop Hochschule für Musik und Theater Hamburg Nora-Louise Müller Hochschule für Musik und Theater Hamburg ABSTRACT In 2012 we collaborated on a solo work for Bohlen Pierce (BP) clarinet in both the BP scale and Carlos alpha scale. Neither has a 1200 cent octave, however they share an interval of 1170 cents which we attempted to use as a substitute for motivic transposition. Some computer code assisted us during the creation period in managing up to five staves for one line of music: sounding pitch, MIDI keyboard notation for the composer in both BP and alpha, and a clarinet fingering notation for the performer in both BP and alpha. Although there are programs today that can interactively handle microtonal notation, e.g., MaxScore and the Bach library for Max/MSP, we show how a computer can assist composers in navigating poly-microtonal scales or, for advanced composer-theorists, to interpret equaltempered scales as just intonation frequency ratios situated in a harmonic lattice. This project was unorthodox for the following reasons: playing two microtonal scales on one clarinet, appropriating a quasi-octave as interval of equivalency, and composing with non-octave scales. 1. INTRODUCTION When we noticed that two microtonal, non-octave scales shared the same interval of 1170 cents, about a 1/6th-tone shy of an octave, we decided to collaborate on a musical work for an acoustic instrument able to play both scales. Bird of Janus, for solo Bohlen Pierce soprano clarinet, was composed in 2012 during a residency at the Banff Centre for the Arts, Canada. Through the use of alternate fingerings a convincing Carlos alpha scale was playable on this same clarinet. In order to address melodic, harmonic and notational challenges various simple utilities were coded in Max/MSP and Matlab to assist in the pre-compositional work. We were already experienced with BP tuning and repertoire, especially after participating in the first Bohlen Pierce symposium (Boston 2010) where twenty lectures and forty compositions were presented. For this collaboration we posed a few new questions and attempted to answer or at least address them through artistic research, i.e. by the creation and explanation of an original composition. We wanted to test (1) if an interval short of an octave by about a 1/6th-tone could be a substitute, (2) if the Carlos alpha scale could act as a kind of 1/4-tone scale to the BP scale, (3) if alpha could be performed on a BP clarinet, and (4) Copyright: c 2016 Todd Harrop et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License 3.0 Unported, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Figure 1. Bohlen Pierce soprano clarinet by Stephen Fox (Toronto, 2011). Owned by Nora-Louise Müller, Germany. Photo by Detlev Müller, Detail of custom keywork. how would the composer and performer handle its notation? Some of these problems were tackled by computerassisted composition. The final piece grew out of a sketch which was initially composed algorithmically, described in section 3.2. Our composition was premiered in Toronto and performed in Montreal, Hamburg, Berlin and recently at the funeral of Heinz Bohlen, one of the BP scale s progenitors; hence we believe the music is artistically successful considering its unusual demands. The quasi-octave is noticeably short but with careful handling it can be convincing in melodic contexts. Since the scales have radically different just intonation interpretations they do not comfortably coalesce in a harmonic framework. This is an area worth further investigation. Our paper concludes with a short list of other poly-tonal compositions albeit in BP and conventional tunings. 2. INSTRUMENT AND SCALES This project would not have been possible with either a B clarinet or a quarter-tone clarinet since the two scales described in 2.2 have few notes in common with 12 or 24 divisions of the octave. Additionally, our BP clarinet was customized and is able to play pitches which other BP clarinets cannot. 597

2 2.1 BP Clarinet Our instrument is a unique version of a rare contemporary instrument, a Bohlen Pierce soprano clarinet (see fig. 1) with two bespoke keys requested by the owner after a discussion with a colleague in Montreal. These are not specific to producing the Carlos alpha scale, but rather complement the existing BP scale notes with microtonal inflections. The BP clarinets are produced by Stephen Fox following a suggestion made in 2003 by Georg Hajdu [1]. At least ten regular BP soprano clarinets are owned by performers in Canada, the U.S.A. and Germany and we believe our model is unique among them due to its custom keywork. 2.2 Scales Following are brief descriptions of the equal-tempered (e.t.) and just intonation (j.i.) varieties of the BP and alpha scales. 1 We shall express intervals as fractions, e.g., a 5/3 major sixth, chords as a set of frequency ratios, e.g., a 4:5:6 major triad, scales in a shorthand such as 12ed2 for 12 equal divisions of the 2/1 octave, and use c for Ellis s cents value. 2 Furthermore we will look at chromatic-like sets of each scale rather than diatonic-like subsets Bohlen Pierce By way of combination tones, continued fractions and intonational experiments the BP scale was independently discovered between the early 1970s and early 1980s by Heinz Bohlen [2], Kees van Prooijen [3, pp ] and John Pierce et al. [4]. Rather than approximate the 4:5:6 major triad with the 0th, 4th and 7th steps from 12 equal divisions of the 2/1 octave, the e.t. BP scale approximates a 3:5:7 wide triad with the 0th, 6th and 10th steps from 13 divisions of a 3/1 tritave. In other words the core BP triad corresponds with the 3rd, 5th and 7th partials of a harmonic series and sounds like a stable if not consonant combination of pure major sixth and flat minor tenth above a root. Convention dictates that inversions occur at the twelfth rather than the octave, e.g., the first inversion of 3:5:7 is 5:7:9. The e.t. BP step size (we shall avoid the term semitone ) is almost a 3/4-tone (see eq. 1), and since it is more than 100c the BP scale may thus be called a macro-tonal scale. 1200c log 2 log (3 1 ) c (1) Since all of its j.i. intervals are expressed as ratios using combinations of the primes 3, 5 and 7 but not 2, BP is neither a proper 7-limit scale nor is it able to express the familiar intervals of the 3/2 fifth, 4/3 fourth, 5/4 and 6/5 thirds, 8/5 minor sixth, 9/8 and 16/15 seconds, and 15/8 major seventh. On the other hand it expresses astonishingly well other intervals whose simple frequency ratios are composed of odd integers only, e.g., the 9/5 minor seventh and various septimal intervals such as the 9/7 major third, 7/3 minor tenth, 7/5 natural tritone and 15/7 minor 1 Although we will not discuss the tempering out of commas we will use the term e.t. as equal-division of a reference interval, or as equallysized step size. 2 As a frequency ratio, 1c = giving 1200c per octave and 100c per semitone. ninth. Coïncidentally two of BP s j.i. intervals sound like our customary e.t. minor third and major tenth: the 25/21 BP second 301.8c, and 63/25 BP eleventh c. BP s corresponding e.t. intervals, however, are slightly further away at 292.6c and c, respectively Carlos Alpha In contrast to the BP scale the alpha scale was designed to play the conventional 4:5:6 major triad better than in 12ed2 but at the sacrifice of the octave. It was discovered between the late 1970s and mid-1980s, also independently, by Prooijen [3, p. 51] and Wendy Carlos [5]. Equation 2 shows the common, simplest definition of alpha as 9 equal divisions of a perfect fifth though Carlos initially looked at dividing a minor third in half and then in quarters. Both authors give a rounded figure of 78.0c as step size and Benson suggests bringing the thirds more in tune by slightly tempering the fifth and using a step size of c [6, p. 222]. 1200c log 2 log (3 2 ) c (2) There is no standard j.i. version of the scale beyond Carlos s set of target intervals: the 5/4 and 6/5 major and minor thirds, 3/2 perfect fifth, and 11/8 natural eleventh. Her target interval of a 7/4 natural seventh is not achievable in alpha, however an octave-inverted 8/7 septimal major second can be found instead. This accounts for five out of nine intervals within a perfect fifth. For our research we used computation to multi-dimensionally search for candidate j.i. intervals to fill in the harmonic space of the alpha scale and hopefully form bridges between alpha and BP. 3. COMPOSITION 3.1 Scalar Representation Staff The BP scale is always given as thirteen notes spanning a tritave, however, figure 2 shows only the first nine notes of the BP scale up to our experimental interval of equivalency of 1170c. Underneath is an alpha scale beginning on the same tonic of F4 and also ending at 1170c on its sixteenth note. It would appear that the first and last notes are a major seventh apart but this is a shortcoming of our conventional notation system when working with microtonality, especially with non-octave scales. Interpreting the cents deviations above the pitches informs us that F4 is about a 1/5th- or 1/6th-tone sharp and E5 is a 1/6th- or 1/5th-tone flat, spanning a range much closer to an octave than a major seventh. Although the figure shows specific sounding pitches this notation was impractical for either composer or performer for these reasons: (1) intervals, let alone music, could not be easily read or written, (2) there appeared to be little consistency between pitches, and (3) the performer preferred another notation based on natural and alternative fingering. We also discovered that our BP and alpha tonics differed by 7 or 6 cents. This was due to the clarinet using two reference tones on which to build each scale: the BP scale extended from 442 or 443 Hz (A4), whereas the alpha scale extended from Hz (C4). Although D 5 differs by 599

3 Figure 2. Bohlen Pierce (top) and Carlos alpha (bottom) scales encompassing nearly one octave. Numerals indicate deviation from standard pitch (top) and interval from tonic (bottom), in cents. only 3c between scales, E5 was chosen as tonic (or F4 by quasi-octave transposition) as this allowed for neighbouring notes to be better matched and function as pivot tones between the two scales Quarter-tone like Pivots Despite the discrepancy in the tonics they and their neighbours were used as melodic pivot tones for modulating between scales. The first BP notes above and below the tonic were close enough to the second alpha notes above and below, thereby giving three common pitches at each tonic. More would have been welcome but beyond these the scales diverged and re-converged at the next tonic 1170c away. Therefore the usefulness of alpha as a quarter-tone like scale to BP was limited. 3 musical staves by assigning a number to each scale note. This was also useful in an early sketch which was algorithmically composed. The two sets of scale notes were interwoven in a 120 note super-scale (of 9.75 cent steps) with two modes: every 15th step made a BP scale, and every 8th step made an alpha scale. Harmonic movement was achieved by cross-fading the probabilities of chord notes being played rather than their volumes or dynamics. This often resulted, however, in some mingling of BP and alpha whenever a cross-fade was not instantaneous. Therefore the method was abandoned in order to keep the two tunings separate for the benefit of the listeners and performer. Figure 3. Score, mm Three representations of same music: sounding (top), MIDI (middle), and fingering (bottom); in Bohlen Pierce (m. 20) and Carlos alpha (m. 21) scales. Figure 3 shows the first modulation in the score from BP to alpha using pivot tones. The top staff shows sounding pitches and has two rows of arabic numerals for scale degrees. Both rows are the same notes but since this passage happens to hover around the ends of the scales, the tonic is shown both as 0 or 8 for BP and 0 or 15 for alpha. BP s 7, 9 and 8 correspond with alpha s 13, 17 and 15. Slight discrepancies can be seen between the E5 s and F 5 s due to the differing reference tones on which the BP and alpha scales were based, as mentioned earlier. 3.2 Integer Representation As seen at the top of figure 3 simple integers proved practical for composing motifs and chords on paper without 3 We use the term quarter-tone like to imply a scale whose notes fit exactly in between another scale s, like 24ed2 to 12ed2. If BP is 13ed3 then 26ed3 would be the more accurate choice. Incidentally there may be more prospect in 39ed3 [7, Erlich s Triple BP Scale ] or 65ed3 [8], i.e. splitting BP into third- or fifth-tones. Figure 4. Various BP calculators to convert between sounding, MIDI and fingering notations of a given pitch. Eventually music had to be written so a series of calculators in Max/MSP assisted the composer in transcribing 599

4 from one notation style to another (see fig. 4). Sometimes up to five staves were necessary, each with a different notation style. Grouped into four types they are: (4a) desired pitch, (4b) sounding pitch notated in 1/8th-tones for convenience, (4c) MIDI keyboard pitches for compositional work, and (4d) so-called fingering pitches easiest for a clarinet player to read and execute (e.g., see fig. 3, bottom staff). Only one staff sounded as it looked and the inclusion of four others, each with its own notation style, made the compositional process laborious. 3.3 Harmonic Representation After the composition was completed we were still curious as to how to represent the union of two microtonal scales in harmonic space, beyond identifying affinities between superficial melodic pivot points, in case of future work with these or other combinations of scales BP Lattice The j.i. BP scale is customarily shown on a square lattice with its 13 chromatic pitches delineated within a symmetrical diamond. This arrangement can be tiled to show groups of extended BP pitches or enharmonics which are higher or lower than the reference pitches by various multiples of 7 cents. This lattice is a 3-d space projected onto a 2-d plane with axes representing primes 5 and 7, 4 and the axis for prime 3 flattened because the interval 3/1, the tritave, is the interval of equivalency in BP theory. 5 It would be possible then to show the extended-bp pitch closest to any given alpha pitch, however, we found that this representation did not treat each scale fairly, not when most alpha pitches were plotted far from the tonic origin without contiguous intervallic connections to the BP core; nor was this model appropriate for our artificial scale since the standard BP set includes five more ratios, between the octave and tritave, which we had rejected when we limited our experiment to the interval of 1170 cents. Therefore the extended BP lattice was not used limit Lattice In her alpha, beta and gamma scales Carlos sought to express ratios involving primes 2 through 11 not just 3, 5 and 7 therefore a 4-dimensional lattice with prime 2 flattened seemed more appropriate to us in depicting alpha and BP ratios together within a single model. A Matlab program was made to calculate the thousands of ratios located as points in a hypercube and reject those that did not meet the following criteria: the ratio needs to be (1) within one quasi-octave of about 1170c, (2) within a tolerance range of error from the nearest e.t. scale step, and (3) factorially simple according to Tenney s harmonic distance metric [9] given in equation 3. HD(f a,f b ) log(a)+log(b) =log(ab) (3) Setting the range was simple, allowing for a little stretch of a few cents tolerance at either end. Setting the tolerance and harmonic distance involved more tweaking and, 4 The axes are actually 7/5 and 5/3 for the diamond configuration.[7] 5 For comparison the regular chromatic scale can also be depicted on a lattice with axes for primes 3 and 5, and the axis for prime 2 flattened because 2/1 is the octave and taken for granted as interval of equivalency. in the end, separate settings were used for BP and alpha otherwise far too many candidate intervals would be found to depict in the lattice (see fig. 5). In other words there can be more than one ratio appropriate for most steps of alpha. And although j.i. BP has a reference set of standard ratios some of its lesser-known enharmonics are actually closer to corresponding e.t. scale steps. E.g., the standard 27/25 is 13c away from the e.t. BP first, but the 49/45 alternative is only 1c away [10, p. 190]. 6 The same holds true for their inversions (at the tritave of course). We wanted harmonic compactness, i.e. not having ratios floating untethered in space despite their sounding closer to an e.t. interval. Often these ratios were quite complex therefore there was a trade-off between simplicity and accuracy. Tenney s harmonic distance function (see eq. 3) was one of the criteria for filtering candidate ratios. 7 For the Bohlen Pierce set of ratios a tolerance of 9.35 cents was required for the BP second (25/21) and a harmonic distance of was needed to catch the complex BP fifth (75/49). This yielded ten pitches including the complex 49/27 as an enharmonic alternative to 9/5. For the alpha set the allowable prime numbers expanded from [3,5,7] to [2,3,5,7,11]. The tolerance was set much lower, to 6.6 cents, and the harmonic distance was set to either to catch an accurate 55/28 quasi-octave or simply to coïncide with BP s standard 49/25. Other enharmonic pairs were the first and second intervals of alpha as can be found in the chart. Compared to typical lattices this structure in black, which represents our particular interpretation of Carlos alpha in j.i., appeared somewhat sparse and pokey. Unfortunately the BP scale, whose ratios are depicted in red, did not satisfyingly coalesce in this model. A more sophisticated presentation could probably be put together using our quasi-octaves as unison vectors after gleaning articles by Fokker and Erlich. The interested reader is directed to Erlich s Partch s 43-tone scale as a periodicity block, 1999, Onelist Tuning Digest ; and to study Joe Monzo s Lattice Diagram of 11-Limit Tonality Diamond, 1998; both at We attempted to show in figure 5 lines in red corresponding to BP s 5/3 and 7/5 axes in contrast to alpha s axes in black, which are 3/2, 5/4, 7/4 and 11/8. The problem was that in our model the vectors representing 5/3 and 7/5 were not in a ninety degree angle to each other, or any angle acute enough to easily distinguish them. We could also imagine lines connecting BP ratios B5 6 7, 0 1, 3 4 and 7 8 but these are related by the ratio 49/45, not one of our primary ratios described above therefore no lines were drawn in the figure. Instead we show dashed lines which connected the alpha and BP lattices and were interesting as potential harmonic bridges between them. Black lines link BP and alpha, red lines link BP and alpha or BP. 6 Krantz and Douthett use Bohlen s hekts instead of cents, where 1 hekt equals 1/100th of a BP step. 7 For comparison with the conventional 12-note chromatic scale we would set a Tenney harmonic distance of 10.5 in order to catch a tritone (45/32) and a rather high tolerance of 15.7 cents to capture the traditional thirds and sixths. These limits would also catch two enharmonics for the minor third and major sixth: 32/27 as alternative for 6/5, and 27/16 for 5/3. Although these ratios are more complex they are also ten cents closer to e.t. scale steps of 300 and 900 cents. 601

5 B2 25/21 A15 55/28 B6 5/3 B5 75/49 A5 5/4 A2 35/32 A14 15/8 B7 49/27 A1 22/21 A8 10/7 A7 11/8 A10 11/7 AB 0 1/1 A9 3/2 A6 21/16 B1 49/45 A5 44/35 A3 8/7 A12 12/7 7/5 B4 B3 9/7 A2 12/11 A1 21/20 9/5 A4 6/5 A13, B7 A11 18/11 A15, B8 49/25 Figure 5. Lattice showing 11-limit ratios, attempting to combine alpha (black A s) with BP (red B s). An advantage to visualizing ratios on a lattice is to see geometric patterns which correspond with intervals or entire chords in j.i. When scale steps are uneven it is not obvious which sets of intervals will sound the same. In e.t. scales of course all steps and combinations of steps are even. Our j.i. alpha scale had a septimal second of 8/7 between A0 3, 1 4, 2 5, not 3 6 nor 4 7, but 5 8, 6 9, 7 10 etc. It was also easily apparent that there were two identical tetrads when recognized as parallelograms: (0358) and (1469), whose ratio sequence is 28:32:35:40, the obvious subset of which is a 7:8:10 wide second diminished fifth triad. 3.4 Rhythmic Tiling Canon A chance encounter with the mathematician Herbert Spohn during our residency encouraged us to include a rhythmic tiling canon which occurs near the end of the composition. It is a three-voice passage for one acoustic clarinet built on a three-note motif, and each voice is related by augmentation corresponding to the rhythm of the motif such that no simultaneous onsets of pitches would occur nor any overlap of notes. Thereby the clarinettist can perform the passage without multiphonics, singing, delay effects or pre-recorded parts. Although this paper is focused on poly-microtonal aspects we mention this canon as a relevant adjunct within the framework of computer-assisted composition. 4. DISCUSSION We are satisfied with Bird of Janus as a musical composition however it can be seen that a harmonic representation of non-octave scales, which might potentially show patterns not otherwise apparent, is not easily achieved. Neither the Bohlen Pierce nor the Carlos alpha scale has a 2/1 octave and the closest interval of 1170 cents can be interpreted as 49/25 as it is in BP or as 63/32 or even 55/28 as we considered when graphing alpha. Perhaps 63/32 would be best since the denominator is a power of two, suggesting a fundamental nearly six octaves below. Furthermore since BP theory states that the 3/1 tritave replaces the 2/1 octave as interval of equivalency, but our experiment stops short of the octave, it is difficult to consolidate both scales into a cohesive harmonic lattice. Were we to experiment further in combining BP with another scale we might consider cheating with any interval that was perceptually close enough to a BP interval, i.e. including BP ratio doppelgängers from an 11-limit j.i. system. Using our limits described earlier this would yield alternatives listed in table 1. These are less than 5 cents out from equal-tempered BP, would probably sound the same to any listener, and be better connected in our 4-dimensional harmonic lattice. In fact the 11-limit alternative for a 49/45 BP first can already be found in figure 5 as a 12/11 alpha second. Interval BP Alternate 1 49/45 12/ /21 32/ /49 32/21 Table 1. Sample of alternate ratios within 5c of e.t. BP steps. Both BP and alpha are modern scales with atypical challenges for composers and performers. There are several works for alpha using synthesizer, notably by Carlo Serafini, but we are not aware of any for acoustic instruments. There is a growing body of acoustic concert music in BP especially from Germany and North America, for clarinets and various other acoustic instruments but none combining BP with alpha. Instead there is a handful of works that combine BP with the standard scale, i.e. 13ed3 with 12ed2: Night Hawks by Fredrik Schwenk, Pas de deux by Sasha Lino Lemke, and Re: Stinky Tofu by Roger Feria. An appropriate lattice could be designed to accentuate harmonic structures bridging these two scales as a tool, e.g., 601

6 for microtonal music theory analysis. The lattice has been enormously helpful to artists such as Ben Johnston, Jim Tenney and Erv Wilson to name but a few. Given that microtonal notation of higher-limit harmony and subsets thereof is daunting for many composers we believe that a harmonic lattice and notation software could be of interest to those willing to give it a try. For scale analysis there already exist superb programs such as Tonescape by Joe Monzo, L il Miss Scale Oven by X.J. Scott and Scala by Manuel Op de Coul. 5. CONCLUSIONS Musical staff notation remains a delicate issue for composers and performers however there is at least one promising solution for the Max/MSP environment: MaxScore by Georg Hajdu and Nick Didkovsky. It features the ability to change the notation style for each staff with a few clicks [11]. E.g., one line of music can be shown in 1/4- to 1/12th-tone resolution, as nearest j.i. ratios, in Extended Helmholtz Ellis or Sagittal notation, in BP clarinet fingering notation, or even in a 6-line staff notation specifically designed for BP music [12]. The user may also customize his or her own notation style particular to a project. This may be the killer app for allowing composers and performers each to read in their preferred notation style. These tools will certainly be welcome when we create another work in mixed tunings, with or without octaves. Although the interval of 1170c is noticeably short of an octave we believe it is possible to stand in as interval of equivalency especially in a busy, melodic context. If two pitches this far apart are sounding simultaneously then the result will be quite dissonant however, as was done in Bird of Janus, motifs that repeat a quasi-octave away do not dwell on the false note but instead continue to move through the melody and dispel any discomfort the listener might have on those moments. Of more practical concern was making sure that the alpha notes, played on an instrument not designed to play alpha, were consistent and stable enough in tone quality, and this often lifted our focus away from too much mathematical, intonational concern. Finally, although Carlos alpha does not function as a 1/4- tone like scale to BP the investigation was most welcome. Obviously 26ed3 would double BP s 13ed3 scale. Nevertheless we believe poly-microtonality to be a fertile area for new music creation, whether with one or both of the scales presented here or with others. [2] H. Bohlen, 13 Tonstufen in der Duodezime, Acustica, vol. 39, no. 2, pp , [3] K. v. Prooijen, A Theory of Equal-Tempered Scales, Interface, vol. 7, pp , [4] M. V. Mathews, L. A. Roberts, and J. R. Pierce, Four new scales based on nonsuccessive-integer-ratio chords, Journal of the Acoustical Society of America, vol. 75, p. S10, [5] W. Carlos, Tuning: At the Crossroads, Computer Music Journal, vol. 11, no. 1, pp , [6] D. Benson, Music: A Mathematical Offering. Cambridge University Press, [7] H. Bohlen, The Bohlen Pierce Site: Web place of an alternative harmonic scale. [Online]. Available: [8] B. McLaren, The Uses and Characteristics of Nonoctave Scales, Xenharmonikôn: An Informal Journal of Experimental Music, vol. 14, pp , [9] J. Tenney, Soundings 13: The Music of James Tenney. Frog Peak, 1984, ch. John Cage and the Theory of Harmony. [10] R. Krantz and J. Douthett, Algorithmic and computational approaches to pure-tone approximations of equal-tempered musical scales, Journal of Mathematics and Music: Mathematical and Computational Approaches to Music Theory, Analysis, Composition and Performance, vol. 5, no. 3, pp , Dec [11] G. Hajdu, Dynamic notation - a solution to the conundrum of non-standard music practice, in TENOR 2015: International Conference on Technologies for Music Notation & Representation, Université Paris- Sorbonne and IRCAM. Paris: Institut de Recherche en Musicologie, IReMus, 2015, pp [12] N.-L. Müller, K. Orlandatou, and G. Hajdu, 1001 Mikrotöne 1001 Microtones. Bockel, 2015, ch. Starting Over Chances Afforded by a New Scale, pp Acknowledgments The authors acknowledge the support of the Conseil des arts et des lettres du Québec, the Canada Council for the Arts and the Goethe Institut during the collaboration period, and of the Claussen-Simon-Stiftung during the preparation of this paper. 6. REFERENCES [1] S. Fox, The Bohlen Pierce clarinet project. [Online]. Available: 603