Equal or non-equal temperament in a capella SATB singing

Equal or non-equal temperament in a capella SATB singing David M Howard Head of the Audio Laboratory, Intelligent Systems Research Group Department of Electronics, University of York, Heslington, York, YO10 5DD, UK Abstract Musical instruments whose pitch control is not fixed to specific notes, such as non-fretted stringed instruments and the human voice, have the freedom to vary the intonation of each note individually. A previous experiment to test whether an SATB a capella quartet made use of an equal or a non-equal tempered tuning system, indicated that the latter was the preference, and that singers tended to a form of just intonation for their tuning. That experiment made use of specially written SATB exercises consisting of a number of modulations between the starting and finishing chords, which themselves were either identical or exactly an octave apart. The intonation difference between the chords in the exercises was measured relative to just intonation and equal tempered tuning. Fundamental frequency data was gathered from each member of the quartet by means of four electrolaryngographs. This paper reports on the results of a second experiment with a different quartet which aims to establish the extent to which the effect might be replicated. The implications for conductors of a capella choirs are discussed. Keywords singing, SATB quartet, tuning, intonation, equal temperament, just intonation, in-tune singing, Introduction The ability to control pitch accurately is a key skill required of any choral singer who wishes to be involved in the highest level of musical singing in an a capella vocal group. Computers can be used to encourage improvement in pitching skills [1-4] and for the measurement of singing pitching accuracy [5]. Apart from the need to develop excellent breath and voice source control to enable fundamental frequency (f0) to be varied with a fine degree of accuracy, excellent listening skills are needed to complete the feedback loop when fine-tuning with the rest of the group. There is more to this process than might be obvious on initial consideration. Singing the note written on the score at the pitch it would be when played on a piano may well be inappropriate. Tuning the notes of a chord for a pleasing or musically consonant result demands that the individual notes are tuned for minimum beats between them, and this was the basis of many keyboard tuning systems before the advent of today s equal tempered tuning [6,7]. Equal Logopedics phoniatrics vocology - page 1 of 13 - printed: 1/9/2016

tempered tuning is used for pianos, pipe organs (although some are tuned in mean tone [8,9]), and electronic keyboards (although some have different tuning options available). The result is that singers (and players of some wind instruments and non-fretted stringed instruments) often produce notes that are not in tune with the piano, although these differences, which are vital for high-level music making, are often rather small [10-16]. A further issue can arise which is often not understood by choral conductors. These differences in tuning of individual notes in comparison with equal tempered tuning can result in an overall pitch shift in the music when it has visited a number of keys during the course of the piece, such that the initial key chord and the final key chord are not in tune with each other. Many conductors do recognise when this happens, but some erroneously attribute it to an inability on the part of the singers to stay in-tune. In fact, the singers might well be singing with excellent tuning as they keep the harmony maximally consonant, thereby incurring the inevitable overall pitch shift. This paper reports on an experiment which follows up a previous one [17] to investigate whether or not the pitch drift previously observed when singing music that visits different key chords takes place with a different a capella vocal quartet. It also provides the basis for consideration of similarities and differences between the strategies adopted by two vocal quartets when singing such music. Method A series of vocal quartet exercises was written for the first experiment, and it turned out that some were unsuitable for this work due to the nature of the musical intervals that were available for making an intonation shift prediction [17]. As a consequence, the present experiment made use of the exercise that was best suited to making a prediction of intonation shift, and it is shown in the upper part of figure 1. The exercise is designed such that the first and last chords consist of the same notes in each part, here spaced an octave apart, so that any overall drift can be readily explored. All the chords in the exercise are linked in terms of their reference for tuning by one of the notes being tied to a note in the following chord. The singers were asked to perform the exercise as illustrated in the lower part of figure 1, with each non-tied minim becoming a crotchet (quarter note) and crotchet rest, such that the tied note could be clearly heard between successive chords, thereby providing an on-going tuning reference. Logopedics phoniatrics vocology - page 2 of 13 - printed: 1/9/2016

Figure 1. Score (upper) for the exercise sung and indication of how it was performed (lower) showing where singers were asked to include rests during the performances. The lines sung by the soprano (S), alto (A), tenor (T) and bass (B) are indicated for convenience. The quartet that took part in this experiment was a group of undergraduate music students from the University of York who practise and perform together as a quartet on a regular basis. Prior to the recording, no indication was given to the singers as to the purpose of the experiment. At the start, the tonic of the first chord (C) was provided as a tuning reference from a Korg electronic tuner. An electrolaryngograph [18] was connected to each singer, and appropriate electrode placement was ensured by checking the amplitude of the output waveform (Lx) on an oscilloscope. The four electrolaryngographs were obtained from Laryngograph Ltd (www.laryngograph,com) and specially adjusted so that they would operate without interfering with each other in terms of any beating between their RF carriers. Four microphones were used (two cardioid Rode NT5 and two AKG 414 set to cardioid response), each of which was placed off-axis out of the air stream approximately 30cm from the lips of a singer. The eight outputs (four microphones and four electrolaryngographs) were connected to a Korg D-1600 eight-channel digital hard disk recorder. Phantom power and microphone amplifiers are available on four of the Korg s input channels via which the microphones were connected. Results The score was analysed in terms of the musical intervals between the notes within each chord with respect to equal tempered tuning and just intonation in order to derive a predicted frequency ratio for each note of the exercise. A spreadsheet program was used for this analysis. These were normalised to the tonic of the first chord, in this case the first note of the alto part (middle C) for calculation convenience. The values used to calculate the frequency ratios in just intonation and equal temperament for the intervals in the chords in this exercise are shown in table 1; a detailed derivation of these values can be found in [6]. The intervals for just intonation are integer ratios because they are taken from the natural harmonic series, and those for equal temperament are based on each semitone having an identical frequency ratio; the twelfth root of 2. Logopedics phoniatrics vocology - page 3 of 13 - printed: 1/9/2016

Interval Just intonation Equal temperament Octave 2/1 12 ( 2) (12 semitones) Fifth 3/2 12 ( 2) (12 semitones) Fourth 4/3 12 ( 2) (12 semitones) Major third 5/4 12 ( 2) (12 semitones) Minor third 6/5 12 ( 2) (12 semitones) Minor sixth 8/5 12 2 ( ) 8 (12 semitones) Table I. Frequency ratios used in the prediction calculations for the intervals in the chords of Exercise 3 (see Figure 1) in just temperament and equal temperament (for their derivation, see (6)). The values shown are for rising intervals; take the reciprocal for falling intervals. In order to make the analysis musically appropriate and relevant to how pitch is perceived, the predicted frequency ratios for each note of each chord were converted into cents (one cent = one hundredth of a semitone) using equation 1 (a derivation can be found in appendix 2 of [6]). c 3986.3137 log10 [ ] = (1) where f ratio c = cents f ratio = frequency ratio Based on the hypothesis that singers singing a capella tend towards just intonation, the difference in cents between the calculated just intonation and equal tempered cent values gives the predicted pitch drift. Figure 2 shows the result of this prediction, and it can be seen that over the 13 chords in exercise 3, the predicted pitch drift is flat by a little over one semitone; a somewhat surprising result. Logopedics phoniatrics vocology - page 4 of 13 - printed: 1/9/2016

Figure 2. Expected deviation from equal temperament for each of the four parts (SATB) in Exercise 3 (see Figure 1) assuming that the singers tend towards just intonation. (Notes: The vertical lines equate to the bar-lines in Exercise 3 (see Figure 1). Maximum and minimum values on the y-axis have been fixed to enable direct comparison between all the figures provided.) The Lx waveforms for each singer were analysed using the SPEAD software from Laryngograph Ltd. to find the mean f0 value for each note of each chord in the exercise. The measurement of fundamental period (Tx) measurement, from which f0 is found as (1/Tx), is made to 1µs accuracy within the laryngograph hardware interface. The f0 contour for each part was examined separately, and the SPEAD software allows the mouse to be used to select an individual note from which the mean f0 can be recorded. The f0 mean values for each note of each chord were entered onto the spreadsheet, and the difference in cents between each measured value and the associated predicted equal tempered value was found. Figure 3 shows these results for the two performances of exercise 3 by the a capella quartet, and these can be compared directly with the predicted variation plotted in figure 2. Logopedics phoniatrics vocology - page 5 of 13 - printed: 1/9/2016

Figure 3. Measured deviation from equal temperament for each of the four parts (SATB) in Exercise 3 (see Figure 1) for performances 1 and 2 of the exercise by quartet 2. (Notes: The vertical lines equate to the bar-lines in Exercise 3 (see Figure 1). Maximum and minimum values on the y-axis have been fixed to enable direct comparison between all the figures provided.) The first point to note is the value for the last note of the soprano part in the first performance, which is clearly out compared with the rest of the group and the general trend of the data. Listening to the performance reveals that rather than rising by another semitone from Eb to E natural, she sang an Eb again by mistake making the last chord C minor rather than C major. The semitone error against the predicted values can be seen in the positions of the points for the last two soprano notes in the first performance, which is close to a semitone. The actual values for these points are -80.1 cents for the penultimate value and -182.6 cents for the final value (i.e. the last note was exactly a semitone lower than the previous note when compared with the predicted value). Logopedics phoniatrics vocology - page 6 of 13 - printed: 1/9/2016

Overall, the drift follows the pattern in all four parts for the predicted just intonation. The repetition indicates that this effect is both consistent and repeatable. Following the first performance, the quartet were aware that they had drifted in pitch (by comparing the last chord with the starting note which was available for starting the second performance). They expressed surprise at the drift, and on being asked what their intention was with respect to overall pitching, they stated that they are very aware when singing of staying in tune. They were asked in the second performance to pay close attention to the relative tuning of successive chords by careful listening to the tied notes (see figure 1) and to forget about trying to maintain the overall pitch by allowing it to drift. In their first performance, all parts drifted flat by approximately one semitone (-100 cents) overall, which is close to the just intonation prediction shown in figure 2. There are differences between the parts themselves and in terms of the relative tuning of each chord which is generally rather wider that the prediction which shows the parts to be within 20 cents of each other in each chord. The prediction is of a steady flattening of the pitch during the exercise with each bar exhibiting the same pattern; the chord structure in the exercise itself forms a harmonic sequence. The pitch drift is not uniform throughout the first performance. Rather, the quartet drift together in a manner that is closest to the prediction during the last six chords, where they uniformly flatten more rapidly than predicted. In the second performance, the pitch drift was uniform throughout the exercise and the relative tuning between the parts in each chord was more consistent within each chord relative to the prediction. The flattening trend was more uniform throughout the exercise, and again they were closer to the prediction. However, the overall pitch change is larger than that predicted, indicating that they were able to let the pitch drift further once they decided (a) to listen carefully to the relative tuning of each chord via the tied notes, and (b) to allow the overall pitch to drift. Discussion It is clear from these results that singers will modify their tuning in relation to each other, and that this modification appears to be towards just intonation as suggested elsewhere in the literature as discussed in the introduction. A number of points worthy of note arise from this experiment. Firstly, the extent of the tuning variation that can be associated with a relatively small number of chords (here the prediction was for a semitone variation over 13 chords) is somewhat surprising; indicating something of the potential impact of this tuning issue in a capella choral performance. Secondly, here is a quartet who have a tuning strategy to remain in pitch; something that was not evident in their first performance, where their pitch did shift in a way that was fairly consistent with the prediction of flattening by one semitone. This suggests that their ability to sing as an ensemble is to some degree overriding their desire to remain in pitch. Herein lies the dichotomy for a capella choral singers and a capella choir conductors: to stay in tune throughout or to keep the harmony consonant. Another way of looking at this is to suggest that an a capella group has to stray in pitch to stay in tune. Logopedics phoniatrics vocology - page 7 of 13 - printed: 1/9/2016

Figure 4. Measured deviation from equal temperament for each of the four parts (SATB) in Exercise 3 (see Figure 1) for performances 1 and 2 of the exercise by quartet 1 (reported in Howard (17)). (Notes: The vertical lines equate to the bar-lines in Exercise 3 (see Figure 1). Maximum and minimum values on the y-axis have been fixed to enable direct comparison between all the figures provided.) This experiment was conducted as a follow-up to an earlier experiment [17] with another vocal quartet, which established the usefulness of exercise 3 for testing the hypothesis that An SATB a capella vocal quartet will drift in pitch with modulation if it tends to non-equal temperament, in order to ascertain whether or not another quartet would exhibit a similar trend. The results for the quartet in that earlier experiment (who will be termed quartet 1 for the purposes of this discussion) are shown in figure 4 in the same format as those for the quartet in this experiment (who will be termed quartet 2 for the purposes of this discussion) shown in figure 3 to allow direct comparison. It can be seen in figure 4 that the pitch variation for quartet 1 is consistent over two performances, but that in neither do the singers drift a full semitone flat as suggested by the prediction. In their second performance, the tuning between the four parts within each chord is somewhat closer in the second performance (as it was in this experiment see figure 3), Logopedics phoniatrics vocology - page 8 of 13 - printed: 1/9/2016

indicating increased familiarity with the music and/or greater attention being paid to the relative tuning of notes in each chord between the parts. Both quartets show a tendency to allow the tuning to start to drift earlier in the exercise in the second performance as compared to the first. In the second performance by quartet 2, after they have accepted that their pitch can drift, the pitch drift is uniform on a chord-by-chord basis throughout the exercise as suggested by the prediction and due to the sequence nature of the chord structure, but their gradient is too steep. They end up approximately 1.5 semitones flat against the prediction of 1 semitone flat. Figure 5. Overall absolute pitching accuracy in cents with respect to equal temperament and just intonation for each member (SATB) of each quartet (Q1: the quartet reported in Howard (17); Q2: the quartet reported in this experiment) individually, averaged for all four members of each quartet (Q1-All, Q2-All) and averaged for all singers (ALL). In order to make a quantitative comparison between the data for each quartet, absolute average pitch variations against equal temperament and just intonation were calculated. Absolute values were used because the pitch drift was not uniformly flat in every case; some were sharp. Not to take absolute values would mean that a sharpening on one note would cancel out the equivalent flattening on another note. The results are shown in figure 5 for each member (SATB) of each quartet (Q1, Q2) averaged across the two performances against equal temperament and just intonation (left and right bars in each case respectively), for each quartet as a whole (Q1-All, Q2- All) and for all eight singers over both performances (ALL). The overall average absolute values for each quartet (Q1-All, Q2-All), for all eight singers (ALL) are given along with the predicted values in table 2 relative to equal temperament and just intonation for convenience. Absolute average pitching (cents) relative to: Equal temperament Just intonation Q1-All 35.1 22.7 Q2-All 62.4 25.5 ALL 48.7 24.1 Logopedics phoniatrics vocology - page 9 of 13 - printed: 1/9/2016

Predicted 52.6 0.0 Table II. Absolute average pitching deviations in cents relative to equal temperament and just intonation for the prediction and for each quartet over both performances (Q1-All, Q2- All) and for all eight singers (ALL). These data are plotted in Figure 5. It can be seen that all individual singers except the soprano in quartet 1showed on average a tendency to prefer just intonation as opposed to equal temperament, and that the preference was more marked for quartet 2 than for quartet 1. Bearing in mind that the overall deviation from equal temperament during the exercise is essentially linear over 100 cents (see figure 2), the expected overall absolute average predicted deviation from equal temperament should be close to 50 cents (and that from just intonation should be 0 cents). The calculated absolute average predicted deviation from equal temperament is 52.6 cents. Neither quartet exhibited this value in practice in any performance, but quartet 2 were closer than quartet 1 in terms of their deviation from equal temperament. However, quartet 1 exhibited a slightly smaller deviation from just intonation which indicates that on average, they were closer to just intonation. The overall average absolute deviation from equal temperament (48.7 cents) does suggest an appropriate variation based on the prediction of 52.6 cents. The average overall variation from just intonation (24.1 cents) is considerably different from the predicted value of 0 cents, but this might be explained by the ability of singers to pitch notes in tune, which was found for a group of ten trained adult singers to be close to 20 cents [19]. Other factors affect accurate pitching including the fact that our perception of pitch varies not only as a function of changing f0, but also with timbre and loudness [6]. For example, the effect of timbre variation on pitch is familiar to many singers when a vowel is made brighter (more like [i:]) or darker (more like [u:]), and a professional singer will vary their f0 to keep the vowel in-tune. Whilst these effects are small relative to the pitch changes that are associated with f0 variation, they become potentially significant when considering intonation accuracy for in-tune singing in a capella choirs. Conclusions Singers in a capella choirs appear to prefer to sing in just intonation which is based on the use of integer ratios to derive the frequency ratios between the main musical intervals that make up a scale - in just intonation these are the octave (2:1; the perfect fifth (3:2) and the major third (5:4). It is not possible to tune a keyboard in just intonation throughout unless the consonance of the octave is sacrificed; something that is not acceptable to Western ears. Today s keyboards are tuned in equal temperament where each semitone is equal to the twelfth root of 2, with the result that no interval is consonant except the octave. The premise underlying this work is that for singers to remain in just intonation, the overall pitch must shift if the music changes key which presents a dichotomy to singers in terms of whether to sing in-tune or whether to allow the overall pitch to drift. The results from an experiment in which the f0 values for every note of every chord in a specially written four part exercise were measured by means of an electrolaryngograph demonstrate that singers tend to sing in just intonation and that the pitch does drift. Indeed, the quartet in this experiment claimed that they do endeavour not to allow the pitch to drift when they sing, but the results suggest otherwise. When given permission to let the pitch drift, they tended to over-compensate in terms of being close to just intonation. Logopedics phoniatrics vocology - page 10 of 13 - printed: 1/9/2016

Comparison with another quartet who took part in an earlier experiment shows that this effect is repeatable but that each group responds somewhat differently, although it should be noted that further replications of the experiment with the same quartet would be appropriate in the future to confirm the repeatability of the effect. It is also worth mentioning that the tenor line is quite high which might in itself be a reason for him to drift flat; however, the tenor in this quartet was highly trained and experienced, and he exhibited no signs of vocal strain on his upper notes during the experiment. There may be other effects causing tuning variation, such as any tendency the singers might have to tune specific intervals flat or sharp, and future experiments will investigate this. Producing an exercise without such an extreme would be an advantage, but producing a sequential musical exercise without movement towards some extreme range in at least one part is not a trivial activity. It is also worth noting that some allowance is taken of these effects in KTH's automatic music rule generation system [20]. Achieving tuning in just intonation when singing a capella invokes a number of facets, some of which tend to work towards success, including:! singers do seem able to change their tuning, even sub-consciously! singers are not completely locked to equal temperament! some consistency in approach has been demonstrated! natural shift is towards just intonation! presenting one note to start a piece; not an equal tempered chord! good listening skills and some of which tend to work against success, including: " choir practice with a piano or other keyboard tuned in equal temperament " listening to recorded and live music which is in equal temperament " working with choral conductors who are unaware of the implications " presenting an equal tempered chord to start a piece rather than one note " being trained to remain in-tune " poor listening skills. The main issue is the ubiquity of equal temperament for those brought up within the Western musical tradition. It is interesting to speculate on the skills that are required to tune intervals in equal temperament, since there is no readily available physical guide to be derived such as the absence of beating. This suggests that tuning notes in equal temperament must probably rely on memory fed by the ubiquity of equal temperament in the exposure of Western ears to music from an early age. However, given the generally poor ability of singers to pitch intervals accurately after hearing a reference when singing alone (excluding those with any form of so-called perfect pitch ), such a memory is probably not available. Given, on the other hand, the ability of singers to pitch intervals accurately when singing a capella in a group, it is likely that they are using some psychoacoustic basis for this such as the degree of beating or musical consonance. To sum up then, it would then appear that an a capella group does indeed have to stray in pitch in order to stay in tune. Choral conductors need to understand this and make appropriate pitching strategy decisions. Logopedics phoniatrics vocology - page 11 of 13 - printed: 1/9/2016

Acknowledgements The author would like to thank the eight singers who took part in the experiments (Quartet 1: Hilary Layton soprano, Anna Burr Alto, Andrew Bunney Tenor, Charles Allenby Bass; quartet 2: Catharine Jackson Soprano, Isabel Alto, Chris O Gorman Tenor, Matthew Jelf Bass), and Laryngograph Ltd for supplying and setting up the four field electrolaryngographs such that they could function in close proximity to each other. References [1] Howard, D.M., Welch, G.F., Brereton, J., Himonides, E., DeCosta, M., Williams, J. and Howard, A.W. WinSingad: A real-time display for the singing studio, Logopedics Phoniatrics Vocology, 2004; 29; 3; 135-144. [2] Garner, P.E., and Howard, D.M. Real-time display of voice source characteristics, Logopedics Phoniatrics Vocology, 1999; 24; 19-25. [3] Thorpe, C.W., Callaghan, J., and van Doorn, J. Visual feedback of acoustic voice features for the teaching of singing, Australian Voice, 1999; 5; 32-39. [4] Nair G: Voice - Tradition and technology, San Diego: Singular Publishing Company, 1999. [5] Howard, D.M. Measuring the tuning accuracy of thousands singing in unison: An English Premier Football League table of fans singing tunefulness, Logopedics Phoniatrics Vocology, 2004; 29; 2; 77-83. [6] Howard, D.M. and Angus, J.A.S. Acoustics and psychoacoustics, 3 rd Ed., Oxford: Focal Press, 2006. [7] Meffen, J. A guide to tuning musical instruments, Newton Abbot, Devon: David and Charles, 1982. [8] Carlsson, A., Davidsson, H., Ruiter-Feenstra, P., Dunthorne, S, and Speerstra, J. Tracing the organ s masters secret, Göteborg University: GOart publications No 2, 2000. [9] Padgham, C.A. The well tempered organ, Oxford: Positif Press, 1986. [10] Nordmark, J., and Ternström, S. Intonation preferences for major thirds with nonbeating ensemble sounds, TMH-QPSR, KTH, 1996; 1; 57-62. [11] Hagerman, B., and Sundberg, J. Fundamental frequency adjustment in barbershop singing, Journal of Research in Singing, 1980; 4; 1; 3-17. [12] Ternström, S. Choir acoustics: An overview of scientific research published to date, International Journal of Research in Choral Singing, 2003; 1; 1; 3-12. [13] Howard, D.M. A capella SATB quartet in-tune singing: evidence of intonation shift, Proceedings of the Stockholm Music Acoustics Conference, SMAC-03, August 6-9, Stockholm, 2003; 2; 462-466. [14] Bohrer, J,C.S. Intonational Strategies in Ensemble Singing, University of London: PhD Thesis, 2002. [15] Teresawa, H. Pitch drift in choral music, Final Paper for Music 221A, available at: http://www-ccrma.stanford.edu/~hiroko/pitchdrift/paper221a.pdf, 2004. Logopedics phoniatrics vocology - page 12 of 13 - printed: 1/9/2016

[16] Vurma, A., and Ross, J. Intonation accuracy when singing in ensemble, Proceedings of the Conference on Interdisciplinary Musicology, CIM04, Graz, 15-18 April, 2004; 160-161. [17] Howard, D.M. Intonation drift in a capella SATB quartet singing with key modulation, Journal of Voice, accepted and in press, 2006. [18] Abberton, E.R.M., Howard, D.M., and Fourcin, A.J. Laryngographic assessment of normal voice: A tutorial, Clinical Linguistics and Phonetics, 1989; 3; 3; 281-296. [19] Howard, D.M., and Angus, J.A.S. A comparison between singing pitching strategies of 8 to 11 year olds and trained adult singers, Logopedics Phoniatrics Vocology, 1998; 22; 4; 169-176. [20] http://www.speech.kth.se/music/publications/thesisaf/sammfa2nd.htm#mixint Logopedics phoniatrics vocology - page 13 of 13 - printed: 1/9/2016