sign in sign up
<<
Home , A12 scale, Alpha scale, Chromatic scale, Harmonic Scale

Composing in Bohlen–Pierce and Carlos Alpha scales for solo

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 solo work for Bohlen–Pierce (BP) clarinet in both the BP scale and Carlos . Neither has a 1200 cent , 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 : sounding pitch, MIDI key- board notation for the 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, .., MaxScore and the Bach library for Max/MSP, we show how a computer can assist in navigating poly-microtonal scales or, for advanced composer-theorists, to interpret equal- tempered scales as frequency ratios situ- ated in a 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 clarinet or a quarter-tone clarinet since the two scales de- Copyright: 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. 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 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 [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. 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’ 3. COMPOSITION with the 0th, 6th and 10th steps from 13 divisions of a 3/1 tritave. In other words the core BP triad corresponds 3.1 Scalar Representation with the 3rd, 5th and 7th partials of a harmonic series and sounds like a stable if not consonant combination of pure 3.1.1 Staff major sixth and flat minor tenth above a root. Convention The BP scale is always given as thirteen notes spanning a dictates that inversions occur at the twelfth rather than the tritave, however, figure 2 shows only the first nine notes Figure 3. Score, mm. 20–21. Three representations of same music: octave, e.g., the first inversion of 3:5:7 is 5:7:9. The e.t. of the BP scale up to our experimental interval of equiva- sounding (top), MIDI (middle), and fingering (bottom); in Bohlen–Pierce BP step size (we shall avoid the term ‘’) is almost lency of 1170c. Underneath is an alpha scale beginning (m. 20) and Carlos alpha (m. 21) scales. a 3/4-tone (see eq. 1), and since it is more than 100c the on the same ‘tonic’ of F4 and also ending at 1170c on BP scale may thus be called a macro-tonal scale. its sixteenth note. It would appear that the first and last Figure 3 shows the first modulation in the score from BP notes are a major seventh apart but this is a shortcoming of to alpha using pivot tones. The top staff shows sounding 1200c 3 1 log ( ) 13 146.3c (1) our conventional notation system when working with mi- pitches and has two rows of arabic numerals for scale de- log 2 · 1 ≈ crotonality, especially with non-octave scales. Interpreting grees. Both rows are the same notes but since this passage Since all of its j.i. intervals are expressed as ratios us- the cents’ deviations above the pitches informs us that F4 happens to hover around the ends of the scales, the tonic is ing combinations of the primes 3, 5 and 7 but not 2, BP is is about a 1/5th- or 1/6th-tone sharp and E5 is a 1/6th- or shown both as 0’ or 8 for BP and 0’ or 15 for alpha. BP’s neither a proper 7-limit scale nor is it able to express the 1/5th-tone flat, spanning a range much closer to an octave 7, 9 and 8 correspond with alpha’s 13, 17 and 15. Slight familiar intervals of the 3/2 fifth, 4/3 fourth, 5/4 and 6/5 than a major seventh. discrepancies can be seen between the E5’s and 5’s due thirds, 8/5 , 9/8 and 16/15 seconds, and 15/8 Although the figure shows specific sounding pitches this to the differing reference tones on which the BP and alpha major seventh. On the other hand it expresses astonish- notation was impractical for either composer or performer scales were based, as mentioned earlier. ingly well other intervals whose simple frequency ratios for these reasons: (1) intervals, let alone music, could not are composed of odd integers only, e.g., the 9/5 minor sev- be easily read or written, (2) there appeared to be little con- 3.2 Integer Representation enth and various septimal intervals such as the 9/7 major sistency between pitches, and (3) the performer preferred As seen at the top of figure 3 simple integers proved prac- third, 7/3 minor tenth, 7/5 natural and 15/7 minor another notation based on natural and alternative fingering. tical for composing motifs and chords on paper—without We also discovered that our BP and alpha tonics differed Figure 4. Various BP calculators to convert between sounding, MIDI and 1 Although we will not discuss the tempering out of commas we will by 7 or 6 cents. This was due to the clarinet using two ref- 3 We use the term ‘quarter-tone–like’ to imply a scale whose notes fit fingering notations of a given pitch. use the term e.t. as equal-division of a reference interval, or as equally- erence tones on which to build each scale: the BP scale ex- exactly in between another scale’s, like 24ed2 to 12ed2. If BP is 13ed3 sized step size. then 26ed3 would be the more accurate choice. Incidentally there may be 2 As a frequency ratio, 1c = 1200√2 giving 1200c per octave and 100c tended from 442 or 443 Hz (A4), whereas the alpha scale more prospect in 39ed3 [7, “Erlich’s Triple BP Scale”] or 65ed3 [8], i.e. Eventually music had to be written so a series of calcu- per semitone. extended from 261.63 Hz (C4). Although 5 differs by splitting BP into third- or fifth-tones. lators in Max/MSP assisted the composer in transcribing

599 Proceedings of the International Computer Music Conference 2016 Proceedings of the International Computer Music Conference 2016 600 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. 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’ 3. COMPOSITION with the 0th, 6th and 10th steps from 13 divisions of a 3/1 tritave. In other words the core BP triad corresponds 3.1 Scalar Representation with the 3rd, 5th and 7th partials of a harmonic series and sounds like a stable if not consonant combination of pure 3.1.1 Staff major sixth and flat minor tenth above a root. Convention The BP scale is always given as thirteen notes spanning a dictates that inversions occur at the twelfth rather than the tritave, however, figure 2 shows only the first nine notes Figure 3. Score, mm. 20–21. Three representations of same music: octave, e.g., the first inversion of 3:5:7 is 5:7:9. The e.t. of the BP scale up to our experimental interval of equiva- sounding (top), MIDI (middle), and fingering (bottom); in Bohlen–Pierce BP step size (we shall avoid the term ‘semitone’) is almost lency of 1170c. Underneath is an alpha scale beginning (m. 20) and Carlos alpha (m. 21) scales. a 3/4-tone (see eq. 1), and since it is more than 100c the on the same ‘tonic’ of F4 and also ending at 1170c on BP scale may thus be called a macro-tonal scale. its sixteenth note. It would appear that the first and last Figure 3 shows the first modulation in the score from BP notes are a major seventh apart but this is a shortcoming of to alpha using pivot tones. The top staff shows sounding 1200c 3 1 log ( ) 13 146.3c (1) our conventional notation system when working with mi- pitches and has two rows of arabic numerals for scale de- log 2 · 1 ≈ crotonality, especially with non-octave scales. Interpreting grees. Both rows are the same notes but since this passage Since all of its j.i. intervals are expressed as ratios us- the cents’ deviations above the pitches informs us that F4 happens to hover around the ends of the scales, the tonic is ing combinations of the primes 3, 5 and 7 but not 2, BP is is about a 1/5th- or 1/6th-tone sharp and E5 is a 1/6th- or shown both as 0’ or 8 for BP and 0’ or 15 for alpha. BP’s neither a proper 7-limit scale nor is it able to express the 1/5th-tone flat, spanning a range much closer to an octave 7, 9 and 8 correspond with alpha’s 13, 17 and 15. Slight familiar intervals of the 3/2 fifth, 4/3 fourth, 5/4 and 6/5 than a major seventh. discrepancies can be seen between the E5’s and F 5’s due thirds, 8/5 minor sixth, 9/8 and 16/15 seconds, and 15/8 Although the figure shows specific sounding pitches this to the differing reference tones on which the BP and alpha major seventh. On the other hand it expresses astonish- notation was impractical for either composer or performer scales were based, as mentioned earlier. ingly well other intervals whose simple frequency ratios for these reasons: (1) intervals, let alone music, could not are composed of odd integers only, e.g., the 9/5 minor sev- be easily read or written, (2) there appeared to be little con- 3.2 Integer Representation enth and various septimal intervals such as the 9/7 major sistency between pitches, and (3) the performer preferred As seen at the top of figure 3 simple integers proved prac- third, 7/3 minor tenth, 7/5 natural tritone and 15/7 minor another notation based on natural and alternative fingering. tical for composing motifs and chords on paper—without We also discovered that our BP and alpha tonics differed Figure 4. Various BP calculators to convert between sounding, MIDI and 1 Although we will not discuss the tempering out of commas we will by 7 or 6 cents. This was due to the clarinet using two ref- 3 We use the term ‘quarter-tone–like’ to imply a scale whose notes fit fingering notations of a given pitch. use the term e.t. as equal-division of a reference interval, or as equally- erence tones on which to build each scale: the BP scale ex- exactly in between another scale’s, like 24ed2 to 12ed2. If BP is 13ed3 sized step size. then 26ed3 would be the more accurate choice. Incidentally there may be 2 As a frequency ratio, 1c = 1200√2 giving 1200c per octave and 100c tended from 442 or 443 Hz (A4), whereas the alpha scale more prospect in 39ed3 [7, “Erlich’s Triple BP Scale”] or 65ed3 [8], i.e. Eventually music had to be written so a series of calcu- per semitone. extended from 261.63 Hz (C4). Although D5 differs by splitting BP into third- or fifth-tones. lators in Max/MSP assisted the composer in transcribing

598 Proceedings of the International Computer Music Conference 2016 Proceedings of the International Computer Music Conference 2016 599 from one notation style to another (see fig. 4). Sometimes in the end, separate settings were used for BP and alpha B2 25/21 up to five staves were necessary, each with a different no- otherwise far too many candidate intervals would be found B5 A2 tation style. Grouped into four types they are: (4a) desired to depict in the lattice (see fig. 5). In other words there can 75/49 35/32 pitch, (4b) sounding pitch notated in 1/8th-tones for con- be more than one ratio appropriate for most steps of alpha. A15 55/28 venience, (4c) MIDI keyboard pitches for compositional And although j.i. BP has a reference set of standard ratios B6 A5 A14 5/3 5/4 15/8 work, and (4d) so-called ‘fingering pitches’ easiest for a some of its lesser-known are actually closer clarinet player to read and execute (e.g., see fig. 3, bottom to corresponding e.t. scale steps. E.g., the standard 27/25 B7 49/27 A8 10/7 staff). Only one staff sounded as it looked and the inclu- is 13c away from the e.t. BP first, but the 49/45 alternative A7 11/8 sion of four others, each with its own notation style, made is only 1c away [10, p. 190]. 6 The same holds true for A6 21/16 the compositional process laborious. their inversions (at the tritave of course). A1 A10 11/7 22/21 1/1 A9 We wanted harmonic compactness, i.e. not having ratios AB 0 3/2 3.3 Harmonic Representation floating untethered in space despite their sounding closer to an e.t. interval. Often these ratios were quite complex A12 B3 After the composition was completed we were still curious A3 9/7 therefore there was a trade-off between simplicity and ac- B1 49/45 8/7 12/7 A2 A11 as to how to represent the union of two microtonal scales in 12/11 18/11 curacy. Tenney’s harmonic distance function (see eq. 3) 7/5 A1 harmonic space, beyond identifying affinities between su- 7 was one of the criteria for filtering candidate ratios. B4 21/20 perficial melodic pivot points, in case of future work with For the Bohlen–Pierce set of ratios a tolerance of 9.35 A5 44/35 9/5 these or other combinations of scales. A4 cents was required for the BP second (25/21) and a har- 6/5 A13, B7 3.3.1 BP Lattice monic distance of 11.85 was needed to catch the complex BP fifth (75/49). This yielded ten pitches including the A15, B8 49/25 The j.i. BP scale is customarily shown on a square lattice complex 49/27 as an alternative to 9/5. with its 13 chromatic pitches delineated within a symmetri- For the alpha set the allowable prime numbers expanded Figure 5. Lattice showing 11-limit ratios, attempting to combine alpha (black ‘A’s) with BP (red ‘B’s). cal diamond. This arrangement can be tiled to show groups from [3,5,7] to [2,3,5,7,11]. The tolerance was set much of ‘extended’ BP pitches or enharmonics which are higher lower, to 6.6 cents, and the harmonic distance was set to or lower than the reference pitches by various multiples of either 10.59 to catch an accurate 55/28 quasi-octave or sim- An advantage to visualizing ratios on a lattice is to see ge- be best since the denominator is a power of two, suggesting 7 cents. This lattice is a 3-d space projected onto a 2-d ometric patterns which correspond with intervals or entire a fundamental nearly six below. 4 ply 10.26 to co¨ıncide with BP’s standard 49/25. Other en- plane with axes representing primes 5 and 7, and the axis harmonic pairs were the first and second intervals of alpha chords in j.i. When scale steps are uneven it is not obvious Furthermore since BP theory states that the 3/1 tritave re- tritave, for prime 3 flattened because the interval 3/1, the as can be found in the chart. which sets of intervals will sound the same. In e.t. scales of places the 2/1 octave as interval of equivalency, but our is the interval of equivalency in BP theory. 5 Compared to typical lattices this structure in black, which course all steps and combinations of steps are even. Our j.i. experiment stops short of the octave, it is difficult to con- It would be possible then to show the extended-BP pitch represents our particular interpretation of Carlos alpha in alpha scale had a septimal second of 8/7 between A0–3, 1– solidate both scales into a cohesive harmonic lattice. Were closest to any given alpha pitch, however, we found that j.i., appeared somewhat sparse and pokey. Unfortunately 4, 2–5, not 3–6 nor 4–7, but 5–8, 6–9, 7–10 etc. It was also we to experiment further in combining BP with another this representation did not treat each scale fairly, not when the BP scale, whose ratios are depicted in red, did not sat- easily apparent that there were two identical tetrads when scale we might consider cheating with any interval that was most alpha pitches were plotted far from the tonic origin isfyingly coalesce in this model. recognized as parallelograms: (0358) and (1469), whose perceptually close enough to a BP interval, i.e. including without contiguous intervallic connections to the BP core; A more sophisticated presentation could probably be put ratio sequence is 28:32:35:40, the obvious subset of which BP ratio doppelgangers¨ from an 11-limit j.i. system. Us- nor was this model appropriate for our artificial scale since together using our quasi-octaves as unison vectors after is a 7:8:10 ‘wide second–diminished fifth’ triad. ing our limits described earlier this would yield alterna- the standard BP set includes five more ratios, between the gleaning articles by Fokker and Erlich. The interested reader tives listed in table 1. These are less than 5 cents out from octave and tritave, which we had rejected when we limited is directed to Erlich’s “Partch’s 43-tone scale as a period- equal-tempered BP, would probably sound the same to any our experiment to the interval of 1170 cents. Therefore the 3.4 Rhythmic Tiling Canon icity block”, 1999, Onelist Tuning Digest 463–65; and to listener, and be better connected in our 4-dimensional har- extended BP lattice was not used. study Joe Monzo’s “Lattice Diagram of 11-Limit A chance encounter with the mathematician Herbert Spohn monic 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 3.3.2 11-limit Lattice Diamond”, 1998; both at http://tonalsoft.com. during our residency encouraged us to include a rhythmic We attempted to show in figure 5 lines in red correspond- tiling canon which occurs near the end of the composition. second. In her alpha, beta and gamma scales Carlos sought to ex- ing to BP’s 5/3 and 7/5 axes in contrast to alpha’s axes It is a three-voice passage for one acoustic clarinet built press ratios involving primes 2 through 11—not just 3, 5 on a three-note motif, and each voice is related by aug- Interval BP Alternate in black, which are 3/2, 5/4, 7/4 and 11/8. The problem 1 49/45 12/11 and 7—therefore a 4-dimensional lattice with prime 2 flat- was that in our model the vectors representing 5/3 and 7/5 mentation corresponding to the rhythm of the motif such tened seemed more appropriate to us in depicting alpha and that no simultaneous onsets of pitches would occur nor 2 25/21 32/27 were not in a ninety degree angle to each other, or any 5 75/49 32/21 BP ratios together within a single model. angle acute enough to easily distinguish them. We could any overlap of notes. Thereby the clarinettist can perform A Matlab program was made to calculate the thousands also imagine lines connecting BP ratios B5–6–7, 0–1, 3–4 the passage without multiphonics, singing, delay effects of ratios located as points in a hypercube and reject those and 7–8 but these are related by the ratio 49/45, not one of or pre-recorded parts. Although this paper is focused on Table 1. Sample of alternate ratios within 5c of e.t. BP steps. that did not meet the following criteria: the ratio needs to our primary ratios described above therefore no lines were poly-microtonal aspects we mention this canon as a rel- be (1) within one quasi-octave of about 1170c, (2) within evant adjunct within the framework of computer-assisted drawn in the figure. Both BP and alpha are modern scales with atypical chal- a tolerance range of error from the nearest e.t. scale step, Instead we show dashed lines which connected the alpha composition. and (3) factorially simple according to Tenney’s harmonic lenges for composers and performers. There are several and BP lattices and were interesting as potential harmonic works for alpha using synthesizer, notably by Carlo Ser- distance metric [9] given in equation 3. bridges between them. Black lines link BP and alpha, red 4. DISCUSSION afini, but we are not aware of any for acoustic instruments. lines link BP and alpha or BP. There is a growing body of acoustic concert music in BP HD(fa,fb) log(a)+log(b)=log(ab) (3) ∝ 6 We are satisfied with Bird of Janus as a musical composi- especially from Germany and North America, for clarinets Setting the range was simple, allowing for a little stretch Krantz and Douthett use Bohlen’s hekts instead of cents, where 1 hekt equals 1/100th of a BP step. tion however it can be seen that a harmonic representation and various other acoustic instruments but none combin- of a few cents’ tolerance at either end. Setting the toler- 7 For comparison with the conventional 12-note we of non-octave scales, which might potentially show pat- ing BP with alpha. Instead there is a handful of works ance and harmonic distance involved more tweaking and, would set a Tenney harmonic distance of 10.5 in order to catch a tritone terns not otherwise apparent, is not easily achieved. Nei- that combine BP with the standard scale, i.e. 13ed3 with (45/32) and a rather high tolerance of 15.7 cents to capture the traditional 4 ther the Bohlen–Pierce nor the Carlos alpha scale has a 2/1 12ed2: Night Hawks by Fredrik Schwenk, Pas de deux by The axes are actually 7/5 and 5/3 for the diamond configuration.[7] thirds and sixths. These limits would also catch two enharmonics for the octave and the closest interval of 1170 cents can be inter- Sasha Lino Lemke, and Re: Stinky Tofu by Roger Feria. 5 For comparison the regular chromatic scale can also be depicted on minor third and major sixth: 32/27 as alternative for 6/5, and 27/16 for a lattice with axes for primes 3 and 5, and the axis for prime 2 flattened 5/3. Although these ratios are more complex they are also ten cents closer preted as 49/25 as it is in BP or as 63/32 or even 55/28 as An appropriate lattice could be designed to accentuate har- because 2/1 is the octave and taken for granted as interval of equivalency. to e.t. scale steps of 300 and 900 cents. we considered when graphing alpha. Perhaps 63/32 would monic structures bridging these two scales as a tool, e.g.,

601 Proceedings of the International Computer Music Conference 2016 Proceedings of the International Computer Music Conference 2016 602 from one notation style to another (see fig. 4). Sometimes in the end, separate settings were used for BP and alpha B2 25/21 up to five staves were necessary, each with a different no- otherwise far too many candidate intervals would be found B5 A2 tation style. Grouped into four types they are: (4a) desired to depict in the lattice (see fig. 5). In other words there can 75/49 35/32 pitch, (4b) sounding pitch notated in 1/8th-tones for con- be more than one ratio appropriate for most steps of alpha. A15 55/28 venience, (4c) MIDI keyboard pitches for compositional And although j.i. BP has a reference set of standard ratios B6 A5 A14 5/3 5/4 15/8 work, and (4d) so-called ‘fingering pitches’ easiest for a some of its lesser-known enharmonics are actually closer clarinet player to read and execute (e.g., see fig. 3, bottom to corresponding e.t. scale steps. E.g., the standard 27/25 B7 49/27 A8 10/7 staff). Only one staff sounded as it looked and the inclu- is 13c away from the e.t. BP first, but the 49/45 alternative A7 11/8 sion of four others, each with its own notation style, made is only 1c away [10, p. 190]. 6 The same holds true for A6 21/16 the compositional process laborious. their inversions (at the tritave of course). A1 A10 11/7 22/21 1/1 A9 We wanted harmonic compactness, i.e. not having ratios AB 0 3/2 3.3 Harmonic Representation floating untethered in space despite their sounding closer to an e.t. interval. Often these ratios were quite complex A12 B3 After the composition was completed we were still curious A3 9/7 therefore there was a trade-off between simplicity and ac- B1 49/45 8/7 12/7 A2 A11 as to how to represent the union of two microtonal scales in 12/11 18/11 curacy. Tenney’s harmonic distance function (see eq. 3) 7/5 A1 harmonic space, beyond identifying affinities between su- 7 was one of the criteria for filtering candidate ratios. B4 21/20 perficial melodic pivot points, in case of future work with For the Bohlen–Pierce set of ratios a tolerance of 9.35 A5 44/35 9/5 these or other combinations of scales. A4 cents was required for the BP second (25/21) and a har- 6/5 A13, B7 3.3.1 BP Lattice monic distance of 11.85 was needed to catch the complex BP fifth (75/49). This yielded ten pitches including the A15, B8 49/25 The j.i. BP scale is customarily shown on a square lattice complex 49/27 as an enharmonic alternative to 9/5. with its 13 chromatic pitches delineated within a symmetri- For the alpha set the allowable prime numbers expanded Figure 5. Lattice showing 11-limit ratios, attempting to combine alpha (black ‘A’s) with BP (red ‘B’s). cal diamond. This arrangement can be tiled to show groups from [3,5,7] to [2,3,5,7,11]. The tolerance was set much of ‘extended’ BP pitches or enharmonics which are higher lower, to 6.6 cents, and the harmonic distance was set to or lower than the reference pitches by various multiples of either 10.59 to catch an accurate 55/28 quasi-octave or sim- An advantage to visualizing ratios on a lattice is to see ge- be best since the denominator is a power of two, suggesting 7 cents. This lattice is a 3-d space projected onto a 2-d ometric patterns which correspond with intervals or entire a fundamental nearly six octaves below. 4 ply 10.26 to co¨ıncide with BP’s standard 49/25. Other en- plane with axes representing primes 5 and 7, and the axis harmonic pairs were the first and second intervals of alpha chords in j.i. When scale steps are uneven it is not obvious Furthermore since BP theory states that the 3/1 tritave re- tritave, for prime 3 flattened because the interval 3/1, the as can be found in the chart. which sets of intervals will sound the same. In e.t. scales of places the 2/1 octave as interval of equivalency, but our is the interval of equivalency in BP theory. 5 Compared to typical lattices this structure in black, which course all steps and combinations of steps are even. Our j.i. experiment stops short of the octave, it is difficult to con- It would be possible then to show the extended-BP pitch represents our particular interpretation of Carlos alpha in alpha scale had a septimal second of 8/7 between A0–3, 1– solidate both scales into a cohesive harmonic lattice. Were closest to any given alpha pitch, however, we found that j.i., appeared somewhat sparse and pokey. Unfortunately 4, 2–5, not 3–6 nor 4–7, but 5–8, 6–9, 7–10 etc. It was also we to experiment further in combining BP with another this representation did not treat each scale fairly, not when the BP scale, whose ratios are depicted in red, did not sat- easily apparent that there were two identical tetrads when scale we might consider cheating with any interval that was most alpha pitches were plotted far from the tonic origin isfyingly coalesce in this model. recognized as parallelograms: (0358) and (1469), whose perceptually close enough to a BP interval, i.e. including without contiguous intervallic connections to the BP core; A more sophisticated presentation could probably be put ratio sequence is 28:32:35:40, the obvious subset of which BP ratio doppelgangers¨ from an 11-limit j.i. system. Us- nor was this model appropriate for our artificial scale since together using our quasi-octaves as unison vectors after is a 7:8:10 ‘wide second–diminished fifth’ triad. ing our limits described earlier this would yield alterna- the standard BP set includes five more ratios, between the gleaning articles by Fokker and Erlich. The interested reader tives listed in table 1. These are less than 5 cents out from octave and tritave, which we had rejected when we limited is directed to Erlich’s “Partch’s 43-tone scale as a period- equal-tempered BP, would probably sound the same to any our experiment to the interval of 1170 cents. Therefore the 3.4 Rhythmic Tiling Canon icity block”, 1999, Onelist Tuning Digest 463–65; and to listener, and be better connected in our 4-dimensional har- extended BP lattice was not used. study Joe Monzo’s “Lattice Diagram of 11-Limit Tonality A chance encounter with the mathematician Herbert Spohn monic 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 3.3.2 11-limit Lattice Diamond”, 1998; both at http://tonalsoft.com. during our residency encouraged us to include a rhythmic We attempted to show in figure 5 lines in red correspond- tiling canon which occurs near the end of the composition. second. In her alpha, beta and gamma scales Carlos sought to ex- ing to BP’s 5/3 and 7/5 axes in contrast to alpha’s axes It is a three-voice passage for one acoustic clarinet built press ratios involving primes 2 through 11—not just 3, 5 on a three-note motif, and each voice is related by aug- Interval BP Alternate in black, which are 3/2, 5/4, 7/4 and 11/8. The problem 1 49/45 12/11 and 7—therefore a 4-dimensional lattice with prime 2 flat- was that in our model the vectors representing 5/3 and 7/5 mentation corresponding to the rhythm of the motif such tened seemed more appropriate to us in depicting alpha and that no simultaneous onsets of pitches would occur nor 2 25/21 32/27 were not in a ninety degree angle to each other, or any 5 75/49 32/21 BP ratios together within a single model. angle acute enough to easily distinguish them. We could any overlap of notes. Thereby the clarinettist can perform A Matlab program was made to calculate the thousands also imagine lines connecting BP ratios B5–6–7, 0–1, 3–4 the passage without multiphonics, singing, delay effects of ratios located as points in a hypercube and reject those and 7–8 but these are related by the ratio 49/45, not one of or pre-recorded parts. Although this paper is focused on Table 1. Sample of alternate ratios within 5c of e.t. BP steps. that did not meet the following criteria: the ratio needs to our primary ratios described above therefore no lines were poly-microtonal aspects we mention this canon as a rel- be (1) within one quasi-octave of about 1170c, (2) within evant adjunct within the framework of computer-assisted drawn in the figure. Both BP and alpha are modern scales with atypical chal- a tolerance range of error from the nearest e.t. scale step, Instead we show dashed lines which connected the alpha composition. and (3) factorially simple according to Tenney’s harmonic lenges for composers and performers. There are several and BP lattices and were interesting as potential harmonic works for alpha using synthesizer, notably by Carlo Ser- distance metric [9] given in equation 3. bridges between them. Black lines link BP and alpha, red 4. DISCUSSION afini, but we are not aware of any for acoustic instruments. lines link BP and alpha or BP. There is a growing body of acoustic concert music in BP HD(fa,fb) log(a)+log(b)=log(ab) (3) ∝ 6 We are satisfied with Bird of Janus as a musical composi- especially from Germany and North America, for clarinets Setting the range was simple, allowing for a little stretch Krantz and Douthett use Bohlen’s hekts instead of cents, where 1 hekt equals 1/100th of a BP step. tion however it can be seen that a harmonic representation and various other acoustic instruments but none combin- of a few cents’ tolerance at either end. Setting the toler- 7 For comparison with the conventional 12-note chromatic scale we of non-octave scales, which might potentially show pat- ing BP with alpha. Instead there is a handful of works ance and harmonic distance involved more tweaking and, would set a Tenney harmonic distance of 10.5 in order to catch a tritone terns not otherwise apparent, is not easily achieved. Nei- that combine BP with the standard scale, i.e. 13ed3 with (45/32) and a rather high tolerance of 15.7 cents to capture the traditional 4 ther the Bohlen–Pierce nor the Carlos alpha scale has a 2/1 12ed2: Night Hawks by Fredrik Schwenk, Pas de deux by The axes are actually 7/5 and 5/3 for the diamond configuration.[7] thirds and sixths. These limits would also catch two enharmonics for the octave and the closest interval of 1170 cents can be inter- Sasha Lino Lemke, and Re: Stinky Tofu by Roger Feria. 5 For comparison the regular chromatic scale can also be depicted on minor third and major sixth: 32/27 as alternative for 6/5, and 27/16 for a lattice with axes for primes 3 and 5, and the axis for prime 2 flattened 5/3. Although these ratios are more complex they are also ten cents closer preted as 49/25 as it is in BP or as 63/32 or even 55/28 as An appropriate lattice could be designed to accentuate har- because 2/1 is the octave and taken for granted as interval of equivalency. to e.t. scale steps of 300 and 900 cents. we considered when graphing alpha. Perhaps 63/32 would monic structures bridging these two scales as a tool, e.g.,

600 Proceedings of the International Computer Music Conference 2016 Proceedings of the International Computer Music Conference 2016 601 for microtonal analysis. [2] H. Bohlen, “13 Tonstufen in der Duodezime,” Acus- The lattice has been enormously helpful to artists such tica, vol. 39, no. 2, pp. 76–86, 1978. as , Jim Tenney and Erv Wilson to name but a few. Given that microtonal notation of higher-limit har- [3] K. v. Prooijen, “A Theory of Equal-Tempered Scales,” mony and subsets thereof is daunting for many composers Interface, vol. 7, pp. 45–56, 1978. we believe that a harmonic lattice and notation software [4] M. V. Mathews, L. A. Roberts, and J. R. Pierce, could be of interest to those willing to give it a try. For “Four new scales based on nonsuccessive-integer-ratio scale analysis there already exist superb programs such chords,” Journal of the Acoustical Society of America, as Tonescape by Joe Monzo, L’il Miss’ Scale vol. 75, p. S10, 1984. Oven by X.J. Scott and Scala by Manuel Op de Coul. [5] W. Carlos, “Tuning: At the Crossroads,” Computer Music Journal, vol. 11, no. 1, pp. 29–43, 1987. 5. CONCLUSIONS [6] D. Benson, Music: A Mathematical Offering. Cam- Musical staff notation remains a delicate issue for com- bridge University Press, 2008. posers and performers however there is at least one promis- ing solution for the Max/MSP environment: MaxScore [7] H. Bohlen, “The Bohlen–Pierce Site: Web place of by Georg Hajdu and Nick Didkovsky. It features the abil- an alternative .” [Online]. Available: ity to change the notation style for each staff with a few http://www.huygens-fokker.org/bpsite/ clicks [11]. E.g., one line of music can be shown in 1/4- [8] B. McLaren, “The Uses and Characteristics of Non- to 1/12th-tone resolution, as nearest j.i. ratios, in Extended octave Scales,” Xenharmonikon:ˆ An Informal Journal Helmholtz–Ellis or Sagittal notation, in BP clarinet finger- of Experimental Music, vol. 14, pp. 12–22, 1993. ing notation, or even in a 6-line staff notation specifically designed for BP music [12]. The user may also customize [9] J. Tenney, Soundings 13: The Music of James Tenney. his or her own notation style particular to a project. This Frog Peak, 1984, ch. “John Cage and the Theory of may be the ‘killer app’ for allowing composers and per- ”. formers each to read in their preferred notation style. These tools will certainly be welcome when we create [10] R. Krantz and J. Douthett, “Algorithmic and com- another work in mixed tunings, with or without octaves. putational approaches to pure-tone approximations of Although the interval of 1170c is noticeably short of an oc- equal-tempered musical scales,” Journal of Mathemat- tave we believe it is possible to stand in as interval of equiv- ics and Music: Mathematical and Computational Ap- alency especially in a busy, melodic context. If two pitches proaches to Music Theory, Analysis, Composition and this far apart are sounding simultaneously then the result Performance, vol. 5, no. 3, pp. 171–194, Dec 2011. will be quite dissonant however, as was done in Bird of [11] G. Hajdu, “Dynamic notation - a solution to the co- Janus, motifs that repeat a quasi-octave away do not dwell nundrum of non-standard music practice,” in TENOR on the false note but instead continue to move through the 2015: International Conference on Technologies for and dispel any discomfort the listener might have Music Notation & Representation, Universite´ Paris- on those moments. Sorbonne and IRCAM. Paris: Institut de Recherche Of more practical concern was making sure that the alpha en Musicologie, IReMus, 2015, pp. 241–248. notes, played on an instrument not designed to play alpha, were consistent and stable enough in tone quality, and this [12] N.-L. Muller,¨ K. Orlandatou, and G. Hajdu, 1001 often lifted our focus away from too much mathematical, Mikrotone¨ 1001 Microtones. Bockel, 2015, ch. “Start- intonational concern. ing Over – Chances Afforded by a New Scale”, pp. Finally, although Carlos alpha does not function as a 1/4- 127–173. tone–like scale to BP the investigation was most welcome. Obviously 26ed3 would double BP’s 13ed3 scale. Never- theless 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.

Acknowledgments The authors acknowledge the support of the Conseil des arts et des lettres du Quebec,´ the Canada Council for the Arts and the Goethe Institut during the collaboration pe- riod, and of the Claussen-Simon-Stiftung during the prepa- ration of this paper.

6. REFERENCES [1] S. Fox, “The Bohlen–Pierce clarinet project.” [Online]. Available: http://www.sfoxclarinets.com/bpclar.html

603 Proceedings of the International Computer Music Conference 2016 604