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Composing in Bohlen–Pierce and Carlos Alpha Scales for Solo Clarinet

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Composing in Bohlen–Pierce and Carlos Alpha scales for solo clarinet Todd Harrop Nora-Louise Muller¨ Hochschule fur¨ Musik und Theater Hamburg Hochschule fur¨ Musik und Theater Hamburg [email protected] [email protected] 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 in- terval of 1170 cents which we attempted to use as a sub- stitute for motivic transposition. Some computer code as- sisted us during the creation period in managing up to five staves for one line of music: sounding pitch, MIDI key- board 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 in- teractively 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 equal- tempered scales as just intonation frequency ratios situ- ated 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. Figure 1. Bohlen–Pierce soprano clarinet by Stephen Fox (Toronto, 2011). Owned by Nora-Louise Muller,¨ Germany. Photo by Detlev 1. INTRODUCTION Muller,¨ 2016. Detail of custom keywork. 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 how would the composer and performer handle its nota- work for an acoustic instrument able to play both scales. tion? Some of these problems were tackled by computer- Bird of Janus, for solo Bohlen–Pierce soprano clarinet, was assisted composition. The final piece grew out of a sketch composed in 2012 during a residency at the Banff Centre which was initially composed algorithmically, described in for the Arts, Canada. Through the use of alternate finger- section 3.2. ings a convincing Carlos alpha scale was playable on this Our composition was premiered in Toronto and performed same clarinet. In order to address melodic, harmonic and in Montreal, Hamburg, Berlin and recently at the funeral notational challenges various simple utilities were coded in of Heinz Bohlen, one of the BP scale’s progenitors; hence Max/MSP and Matlab to assist in the pre-compositional we believe the music is artistically successful considering work. its unusual demands. The quasi-octave is noticeably short We were already experienced with BP tuning and reper- but with careful handling it can be convincing in melodic toire, especially after participating in the first Bohlen–Pierce contexts. Since the scales have radically different just in- symposium (Boston 2010) where twenty lectures and forty tonation interpretations they do not comfortably coalesce compositions were presented. For this collaboration we in a harmonic framework. This is an area worth further in- posed a few new questions and attempted to answer or at vestigation. Our paper concludes with a short list of other least address them through artistic research, i.e. by the poly-tonal compositions albeit in BP and conventional tun- creation and explanation of an original composition. We ings. 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, 2. INSTRUMENT AND SCALES (3) if alpha could be performed on a BP clarinet, and (4) This project would not have been possible with either a B clarinet or a quarter-tone clarinet since the two scales de- Copyright: c 2016 Todd Harrop et al. This is an open-access article scribed in 2.2 have few notes in common with 12 or 24 Figure 3. A screenshot of the main patch, showing the first measures of the meta-score modeling Vortex Temporum. distributed under the terms of the Creative Commons Attribution License divisions of the octave. Additionally, our BP clarinet was 3.0 Unported, which permits unrestricted use, distribution, and reproduc- customized and is able to play pitches which other BP clar- tion in any medium, provided the original author and source are credited. inets cannot. 596 Proceedings of the International Computer Music Conference 2016 Proceedings of the International Computer Music Conference 2016 597 2.1 BP Clarinet ninth. Co¨ıncidentally two of BP’s j.i. intervals sound like our customary e.t. minor third and major tenth: the 25/21 Our instrument is a unique version of a rare contemporary BP second 301.8c, and 63/25 BP eleventh 1600.1c. instrument, a Bohlen–Pierce soprano clarinet (see fig. 1) BP’s corresponding≈ e.t. intervals, however, are slightly≈ fur- with two bespoke keys requested by the owner after a dis- ther away at 292.6c and 1609.3c, respectively. cussion with a colleague in Montreal. These are not spe- cific to producing the Carlos alpha scale, but rather com- 2.2.2 Carlos Alpha plement the existing BP scale notes with microtonal in- flections. The BP clarinets are produced by Stephen Fox In contrast to the BP scale the alpha scale was designed following a suggestion made in 2003 by Georg Hajdu [1]. to play the conventional 4:5:6 major triad better than in At least ten regular BP soprano clarinets are owned by per- 12ed2—but at the sacrifice of the octave. It was discovered 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. formers in Canada, the U.S.A. and Germany and we be- between the late 1970s and mid-1980s, also independently, lieve our model is unique among them due to its custom by Prooijen [3, p. 51] and Wendy Carlos [5]. Equation 2 keywork. shows the common, simplest definition of alpha as 9 equal divisions of a perfect fifth though Carlos initially looked only 3c between scales, E5 was chosen as tonic (or F4 by musical staves—by assigning a number to each scale note. at dividing a minor third in half and then in quarters. Both quasi-octave transposition) as this allowed for neighbour- This was also useful in an early sketch which was algorith- 2.2 Scales authors give a rounded figure of 78.0c as step size and Ben- ing notes to be better matched and function as pivot tones mically composed. The two sets of scale notes were inter- Following are brief descriptions of the equal-tempered (e.t.) son suggests bringing the thirds more in tune by slightly between the two scales. woven in a 120 note super-scale (of 9.75 cent steps) with 1 two modes: every 15th step made a BP scale, and every and just intonation (j.i.) varieties of the BP and alpha scales. tempering the fifth and using a step size of 77.965c [6, p. 3.1.2 Quarter-tone–like Pivots We shall express intervals as fractions, e.g., a 5/3 major 222]. 8th step made an alpha scale. Harmonic movement was sixth, chords as a set of frequency ratios, e.g., a 4:5:6 ma- Despite the discrepancy in the tonics they and their neigh- achieved by cross-fading the probabilities of chord notes 1200c 3 1 bours were used as melodic pivot tones for modulating being played rather than their volumes or dynamics. This jor triad, scales in a shorthand such as 12ed2 for 12 equal log ( ) 9 77.995c (2) divisions of the 2/1 octave, and use ‘c’ for Ellis’s cents log 2 · 2 ≈ between scales. The first BP notes above and below the often resulted, however, in some mingling of BP and alpha 2 tonic were close enough to the second alpha notes above whenever a cross-fade was not instantaneous. Therefore value. Furthermore we will look at chromatic-like sets There is no standard j.i. version of the scale beyond Car- of each scale rather than diatonic-like subsets. and below, thereby giving three common pitches at each the method was abandoned in order to keep the two tun- los’s set of target intervals: the 5/4 and 6/5 major and mi- tonic. More would have been welcome but beyond these ings separate for the benefit of the listeners and performer. 2.2.1 Bohlen–Pierce nor thirds, 3/2 perfect fifth, and 11/8 natural eleventh. Her the scales diverged and re-converged at the next tonic 1170c target interval of a 7/4 natural seventh is not achievable in away. Therefore the usefulness of alpha as a quarter-tone– By way of combination tones, continued fractions and into- alpha, however an octave-inverted 8/7 septimal major sec- like scale to BP was limited. 3 national experiments the BP scale was independently dis- ond can be found instead. This accounts for five out of covered between the early 1970s and early 1980s by Heinz nine intervals within a perfect fifth. For our research we Bohlen [2], Kees van Prooijen [3, pp. 50–51] and John used computation to multi-dimensionally search for candi- Pierce et al. [4]. date j.i. intervals to fill in the harmonic space of the alpha Rather than approximate the 4:5:6 major triad with the scale and hopefully form bridges between alpha and BP.
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