by Jonathan A. Kemp RESEARCH Department of Music, University of St Andrews The Great Highland Bagpipe scale and sensory dissonance HIS article aims to illustrate some depends on the frequency) as illustrated in minima in sensory dissonance being introduced of the features of the tuning of the Figure 3.8 of Sethares’ Tuning, Timbre, Spec- with each additional harmonic. THighland bagpipe scale in terms trum, Scale [1]. The shape of the graph for a real bagpipe of the consonance and dissonance of the would depend on the relative amplitude of all chanter against the drones. It presents a Graphs of sensory dissonance the harmonics, which would in turn depend on theoretical plot which estimates how the IN 1995, MacKenzie [2] demonstrated the the instrument, the reeds, the player (includ- sensory dissonance varies for every possible range of tone colours produced by drones and ing moisture build-up effects and fingerings) tuning of the chanter. This is achieved by chanters and considered the implications for and the acoustic of the performance space and charting a steady glissando from low G to intonation. He also reproduced a graph by listening position. While these factors will af- high A, using different colours for the differ- Kameoka and Kuriyagawa [3] covering the fect the relative depth of the minima of sensory ent numbers of harmonics present in chanter, dissonance for harmonic tones in a range of dissonance, they will not alter their frequency. tenor drone and bass drone spectra. Such a less than an octave. As MacKenzie states, such This is determined by the simple ratios for visual representation may be of practical as- a plot represents the dissonance of a chanter coincidence of harmonics of drones against sistance to players, reed makers and bagpipe glissando against another chanter sounding the chanter. makers who are seeking to achieve a better low A, rather than against drones sounding in Since each octave corresponds to a frequency sound. It also offers a new understanding lower octaves. The graph in this article is more ratio of 2, the lowest or fundamental compo- and clearer explanation of why the bagpipe relevant to Highland pipers because it plots sen- nent of the bass drone is a factor of 4 below scale is non-standardised. sory dissonance of the chanter against pitches that of the chanter low A. Labelling the funda- corresponding to the tenor and bass drones. mental frequency of low A as f, the harmonics Harmonics on the bagpipe Using MATLAB programs customised of the bass drone (when tuned accurately) will and dissonance from those of Sethares [1], the proximity of all thus sound at integer multiples of f/4, therefore THE bagpipe produces sound through the air the harmonics of the drones were checked in at 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75 and 2.0 in the bag acting as a high-pressure reservoir, relation to all the harmonics of the (variable) and so on relative to low A. with the reeds partially opening and closing to chanter pitch. The total amount of sensory The horizontal axis on the figure is spaced allow periodic puffs of air into the drones and dissonance was then tallied up and the results logarithmically so that equal distance is given chanter. Each pipe produces a pitched musical are graphed in Figure 1. It should be noted that to scale steps rather than to harmonics. All five sound consisting of a harmonic series (pure no sound synthesis or listening tests on actual plots show minima indicating clear acoustical tone components within the sound that blend bagpipe tones were performed, but rather a reasons for tuning the notes low A, C©, E and together, producing a unique tone colour). The standard method was applied to predict relative high A to achieve coinciding harmonics be- lowest component in a harmonic series is not dissonance levels. tween chanter and drones if it is desired that always the loudest, but is labelled the funda- Different coloured lines on the figure show dissonance should be minimised. Dissonance is mental frequency and this depends principally how the sensory dissonance depends on the not something that should be minimised at all on the length of the air column enclosed. When number of harmonics (or tone colour) of each costs — historically musical taste has dictated multiple frequencies are sounding simultane- sound source: bass drone, tenor drones and preferences for different tunings. A particularly ously (as in chanter and drones), the degree chanter. The horizontal axis shows chanter striking example is that the high A has been of pleasantness or consonance in the sound fundamental frequencies, as measured relative deliberately tuned flat of the true octave ratio depends on the extent to which harmonics to a low A that is perfectly in tune with the bass of 2 by some players in order “not to lose the coincide. drone two octaves below and tenor drones one note in the drones”[2]. When pure tone harmonics from two dif- octave below. Dips on each line show frequen- The line including six harmonics in the ferent sounds are very close to each other but cies that my theoretical modelling predicts analysis also has broad, shallow dissonance not matching then beating may be perceived in would give a less dissonant (and therefore more minima close to the low G and B pitches, and the sound (as observed during tuning). If the consonant) sound against the drones. a single broad minima covering the region in- difference is slightly too large for beats to be The blue line assumes that each sound source cluding F© and high G. Interestingly, the pitch observed then a very rough, dissonant sound contributes six equal amplitude harmonics. A D is shown to be relatively dissonant due to is perceived. Listening tests have concluded good reason for only including six harmonics the fifth harmonic of the bass drone (sounding that the sensory dissonance has a maximum is that the higher harmonics are closer together around C©) lying close enough (around 40 Hz which typically occurs when the difference in in pitch and thus are not so well resolved by away for typical playing frequencies) to clash frequency between two pure tone components the ear [4]. Lines for including up to ten equal with the fundamental frequency of the chanter. in the sound is around a semitone (but this amplitude harmonics in the analysis show extra A sharper pitch for D would reduce the extent PAGE 18 PIPING TODAY ISSUE 75 • 2015 RESEARCH The Great Highland Bagpipe scale and sensory dissonance Nearest equal temperament values low G low A B C# D E F# G A 10 No. of harmonics considered 10 9 9 8 7 8 6 7 6 5 4 3 Sensory dissonance between chanter and drones 2 1 1 1.25 1.5 1.75 2 2.25 Frequency of chanter relative to low A Figure 1. Theoretical plots of sensory dissonance across the range of the chanter, taking into consideration equal amplitude harmonics for bass drone, tenor drone and chanter spectra. of this clash. Analysis by Brown of the late 17th in the pipe scale are labelled low G, low A, B, the equally tempered value. This is relatively century chanter of Iain Dall MacKay shows C, D, E, F, high G and high A as is standard in consonant because the second harmonic of the that the D was “colourful”, or sharp of the material written by and for pipers, but it should chanter matches the seventh harmonic of the interval that would be expected from modern be noted that those labelled C and F are tuned bass drone. When nine harmonics are included, instruments by approximately 20 cents[5]. closer to the equally tempered pitches C© and F© the ninth harmonic of the tenor drones matches Interestingly this tuning was common until respectively (allowing the A Mixolydian mode the fifth harmonic of the chanter to give a small around the 1950s[6]. to sound rather than the A Aeolian mode). dip in dissonance at a value 18 cents sharp of an Notes of the Pipe Scale In addition to this, the pitch standard for the equally tempered low G. These two options for IT is helpful to set out the implications of definition of A has changed over the years to be a more consonant low G correspond to the fre- sensory dissonance for each note in the bagpipe increasingly sharp of concert pitch. quency ratios 7/8 and 9/10 (i.e. the frequency of scale in turn, from lowest pitch to highest. In Low G Note low G over the frequency of low A). It is possible doing so, I will refer to the cent scale in which WHEN seven harmonics are included in the that the preference for one or other tuning may an equally tempered semitone is divided into analysis, a dissonance minimum appears cor- depend on the relative amplitude of the seventh 100 cents. The headings for the different notes responding to a low G tuned 31 cents flat of and ninth harmonics in the drone spectra. PIPING TODAY ISSUE 75 • 2015 PAGE 19 RESEARCH Low A Note responds to a pure major 6th, 16 cents flat of and low G. The harmonic interval for B requires THE first harmonic of the chanter (its funda- an equally tempered F©. This minimum is weak the ninth harmonic to be perceived. mental frequency) matches the fourth harmonic compared to others because the chanter funda- The pitch F© has a contribution to dis- of the bass drone and the second harmonic of mental clashes with the seventh harmonic of the sonance associated with its proximity to the the tenor drones at the position marked 1.0 bass drone.