A Quantitative Study of Relationships Between Interval-width Preference and Temperament Conditions in Classically-trained Musicians Byron Pillow

Transcription

1 A Quantitative Study of Relationships Between Interval-width Preference and Temperament Conditions in Classically-trained Musicians Byron Pillow Spring 2016 University of South Dakota National Music Museum Rodger Kelly

2 Abstract Ten classically-trained music students were evaluated to explore correlations between extramusical interval width preference, theoretical knowledge of temperament practices, and interval width conditions present in prominent temperament families of Western music. Participants took part in listening and vocalization tests, and completed an exit survey, to generate a quantitative representation of their individual interval width preferences for the primary open intervals: the perfect fifth, perfect fourth, and major third. These individual preferences were then aggregated into two larger groups vocalists, and instrumentalists. The results were analyzed for trends and correlations to various types of temperaments common in the history of Western music. After evaluating the results of the aggregate groups it was determined that the vocalists identified best with the conditions of the family of good temperaments, with no specific temperament aligning completely with their quantitative preferences. The instrumentalist s preferences aligned closely with the conditions of the meantone temperaments, particularly 1/4 syntonic comma meantone. The strongest correlation to interval-width preference was found to be the participant s theoretical knowledge of temperament practices, regardless of their aggregate group. Introduction Since the 17th century, the temperaments used in Western music have been shifting steadily away from the use of mathematically pure interval qualities, in favor of equal interval qualities that allow for greater consistency between different keys. This trend has culminated in the wide-spread implementation of equal temperament, which is used in the composition and performance of most modern Western music. Equal temperament, however, does not affect all musicians equally. Vocalists are entirely free in their range of intonation, and are strictly limited by temperaments only if they perform with certain instruments. Instrumentalists, on the other hand, are generally more limited in their range of intonation. Other than the unfretted string instruments, the slide trombone, and a few other oddities, the intonational range of musical instruments is restricted to the temperament dictated by the morphology of the instrument and its subsequent available notes. Many instruments, such as keyboards, fretted strings, and mallet percussion, have no flexibility whatsoever in regards to intonation and temperament. Since very few instruments are constructed in this day with anything other than equal-temperament in mind, 1

3 instrumentalists exist more so in an equal-tempered world. Regardless of whether it is an unavoidable condition of modern times, deviating from pure intervals has measurable harmonic consequences. Mathematically pure intervals are, mathematically speaking, more consonant than those used in most temperaments. Any deviation within an interval from the pure mathematical ratio results in an audible beating, or wavering quality, the rate of which corresponds precisely to the degree by which the interval is mistuned from pure. This consonance (or lack thereof) is most apparent in the primary open intervals of Western tonal music the pure perfect fifth, perfect fourth, and major third which are the first harmonic intervals encountered in the natural harmonic series. The first interval of the harmonic series is the octave, with a ratio of 2:1. A pure perfect fifth has a frequency ratio of 3:2, and corresponds to the interval between the first and second harmonic. A pure perfect fourth has a ratio of 4:3, and follows the perfect fifth as the third interval of the harmonic series. A pure third has a ratio of 5:4, and follows the fourth as the fourth interval of the harmonic series. Combined, these first four intervals create the consonant major triad, which is the central unit of tonal harmony. Temperaments are designed by adjusting the ratios of these primary open intervals in different relative amounts to achieve differing degrees of purity. In Western music, there are four primary temperament types. The simplest temperament type and also the oldest is just intonation, in which all intervals preserve their pure ratios no matter what their relative starting or ending point is. From there, temperaments split into two distinct lines: temperaments based on preserving purity of major thirds, and temperaments based on preserving purity of perfect fifths. The only prominent temperament family on the major-third side of the tree are the meantone temperaments. In meantone temperaments, thirds are pure (or very nearly pure) in a majority of 2

4 keys while all fifths are noticeably tempered narrow. This has a couple of important side effects: the thirds that are not pure are rendered as wolves, as is one final fifth used to close the circle. These wolves make meantone only usable in certain keys. The perfect fifth side of the tree has more options: the good temperaments and equal temperament. Good temperaments vary, but their underlying similarity is the inclusion of numerous pure perfect fifths, resulting in no pure major thirds, large variance in major third width, and circularity (being usable in all keys, there are no wolves). The thirds in the good temperaments are generally less consonant than those of the meantone temperaments, but not as bad as those in equal temperament. In equal temperament, each of the twelve fifths is tempered by 1/12 th of the cumulative difference between a chain of 12 pure fifths and a chain of 7 pure octaves. This equal distribution of impurity means every interval type is equally deviant from pure in any key. Fifths however, are far less deviant than thirds. Equal-tempered fifths deviate from purity by cents (a cent is a measure equal to 1/100 of an equal-tempered semitone), while major thirds deviate from purity by cents. The apparent incongruity between A) the mathematically superior consonance of pure intervals, and B) the lack of pure intervals (especially thirds) in equal temperament, raises some interesting questions. Even though pure intervals have measurably greater consonance, do musicians trained in the modern environment perceive equal-tempered intervals as preferable? Does the consonance of pure intervals override familiarity and become preferable regardless of the current equal-tempered climate? If participants show a complete preference for neither pure intervals nor equal temperament, is there another temperament that closely mirrors the implications of their preferences? Connecting back to the original matter of different types of 3

5 musicians being influenced by the tempering capabilities of their instrument type, is there any notable difference in tempering preference between vocalists and instrumental musicians? The vocalists, with their flexible intonation, should be more inclined to the pure intervals of justintonation than any other system of temperament. The instrumentalists, operating more frequently in a theoretically tempered system, should show a greater disposition towards tempered intervals than pure ones. Research has previously been conducted in the area of interval-width preference, but all pre-existing studies looked only at harmonic intervals within a musical context. 1 This context makes extracting comparisons to temperaments unnecessarily difficult, as the performers interval-width perception is confounded by factors such as chord progressions, melodic direction, musical emotion, orchestration, within-section-tuning, and others. These studies might be more useful for examining results of ensemble temperament after the fact, but do little to address any underlying preference for interval quality. Underlying preference can be better derived through extra-musical examination, that is, preferences unaffected by musical context. In order to ground comparisons between instrument types, tempering preference, and the types of intervals extant in various temperaments, an experiment was constructed to test participants for their subjective evaluations of consonance in the primary open intervals the major third, perfect fourth, and perfect fifth in an extra-musical setting. These preferences were then evaluated against each other in a number of ways to determine whether or not there are any 1 Brant Karrick, An Examination of the Intonation Tendencies of Wind Instrumentalists Based on Their Performance of Selected Harmonic Musical Intervals, Journal of Research in Music Education, Vol. 46, No. 1 (Spring, 1998), pp ; John Geringer, Tuning Preferences in Recorded Orchestral Music, Journal of Research in Music Education, Vol. 24, No. 4 (Winter, 1976), pp ; Robert Duke, Wind Instrumentalists' Intonational Performance of Selected Musical Intervals, Source: Journal of Research in Music Education, Vol. 33, No. 2 (Summer, 1985), pp

6 correlations between interval-width preference, musical background, and the various temperaments available in Western music. Method Participants The experiment was performed in the library of the National Music Museum (Vermillion, SD), between March 15 th and April 25 th, Participants were ten classically-trained graduate and undergraduate music students from the University of South Dakota. All participants had at least two semesters of aural training or equivalent musical experience. Four of the participants were vocalists, three were wind instrumentalists, two were keyboardists, and one was a fretted string instrumentalist. The procedure consisted of three segments: a vocalization segment, a listening segment, and an exit survey. All three segments were completed by the participant in one session. Materials and Procedures Vocalization segment: The following materials were used during the vocalization task: (1) Røde NT5 small-diaphragm condenser microphone (1) Male-to-female XLR cable (1) Focusrite Scarlett 2i2 USB audio interface (1) Macbook Pro Retina 15 Adobe Audition CC 2016 NCH ToneGenerator v3.22 Sony closed-back headphones For the vocalization segment, each participant was given fifteen computer-generated reference tones through a pair of closed-back headphones. Two separate sets of reference tones were 5

7 available; one for male participants and one for female participants. The reference tones were arranged such that the upper tone of the requested interval would fall within a comfortable range for the typical vocal ranges of their appropriate gender; E3-A3 for males, G4-C5 for females. Before each reference tone was played, a specific interval was announced by a recorded voice. Each participant was instructed to sing the upper tone of the specified interval on the syllable la while the reference tone was playing through their headphones. The specific intervals requested were five perfect fifths, five perfect fourths, and five major thirds, in an alternating order. The audio signal containing the vocalized tones was recorded on its own track as a continuous file for each participant. The segment of the audio file containing the vocalized signal for each requested interval was subsequently analyzed with a 65,536 segment Blackman-Harris Full Fourier Transformation (FFT) to render the average fundamental frequency of the vocalized tone. Listening segment: The following materials were used during the listening task: (1) MacBook Pro Retina 15 (1) Sony closed-back headphones Adobe Audition CC 2016 NCH ToneGenerator v3.22 Listening response answer sheet The listening component involved one task with each participant listening to thirty pre-recorded pairs of harmonic intervals and indicating on the answer sheet which of each pair, if any, they found more consonant. Each pair presented two differently tempered widths of the same interval type, played directly after one another with a four second pause between pairs. There were 6 6

8 different paired combinations used throughout the process: a pure (P) interval 2 against an equaltempered (ET) interval 3 ; an ET interval against an inversely equal-tempered (IET) interval 4 ; an IET interval against a P interval; an ET interval against a Pythagorean (PY) interval 5 ; and a PY interval against a P interval. One pairing of a P interval against a wolf (W) interval was included for each type of interval as a control. There were ten pairs of perfect fifths, ten pairs of perfect fourths, and ten pairs of major thirds. Table 1. Listening segment interval pair conditions Perfect Fifths Perfect Fourths Perfect Thirds Interval A Interval B Interval A Interval B Interval A Interval B Pair 1 P ET P ET P ET Pair 2 ET IET IET P ET PY Pair 3 IET P ET IET PY P Pair 4 P ET P ET P ET Pair 5 ET IET IET P ET PY Pair 6 IET P ET IET PY P Pair 7 P ET P ET P ET Pair 8 ET IET IET P ET PY Pair 9 IET P ET IET PY P Pair 10 P W P W P W 2 Pure intervals are constructed of simple-number ratios: 5/4 for major thirds, 4/3 for perfect fourths, and 3/2 for perfect fifths. This equates to widths, in cents, of , , and respectively. 3 In equal-temperament, major thirds measure 400 cents wide, perfect fourths 500 cents wide, and perfect fifths 700 cents wide. 4 An IET interval is an interval which is tempered by the same amount as an ET interval, but in the inverse direction. For example, an equal-tempered fifth is 1.96 cents narrower than pure. Subsequently, an IET fifth is tempered 1.96 cents wider than pure. 5 A Pythagorean major third is the size of major third that results when a fixed-pitch instrument is tuned by a chain of consecutive pure fifths and fourths. It deviates further from pure than equal-tempered major thirds, measuring cents. 7

9 The tones used for the harmonic intervals were generated using NCH ToneGenerator v3.22, with every fifth tone checked for accuracy using the Blackman-Harris FFT. The tones were comprised of sawtooth waveforms with a low-pass filter applied at 10,000hz to minimize any distortion or digital artifacts. 6 Exit survey: For the final segment, each participant was given a response form with 24 questions. They responded to the questions on a scale ranging from strongly disagree to strongly agree. The questions addressed the participant s musical background, extent of musical education, and their general awareness and understanding of the concept of temperament as it applies in the musical world. The participant s coded answers to the survey were used to compute an arbitrary Temperament Awareness Index (TAI) score. TAI scores are calculated on a scale of 0-100%, with 50% equating to roughly average knowledge of temperaments. 0% would equate to no knowledge of temperaments, and 100% to excellent knowledge of temperaments. Results (For detailed figures of individual segment results, refer to Appendix I. For graphs of cross-segment comparisons, refer to Appendix II.) Temperament Awareness Index (TAI) scores Population TAI scores had a mean of 65.4%, with a range of 29% and a standard deviation of 8%. Vocal TAI scores had a mean of 61.5%, while instrumental TAI scores had a mean of 68.1%. Vocalization Segment 6 Sawtooth waveforms were used because they contain all even and odd-number harmonics above a given fundamental frequency. It is the interference of coincidental harmonics between two notes that causes the beating effect audible when intervals deviate from pure. The complete harmonic inclusion of the sawtooth waveform allows for any coincidental harmonics, and therefore audible mistunings, to be readily apparent to the participants. 8

10 A total of 150 intervals were recorded for the vocalization segment. Of those 150, 136 fell within the acceptable error range of ±50 cents from a target interval. This included 61 perfect fifths, 32 perfect fourths, and 34 major thirds. The remaining 14 intervals were excluded from the sample-set, since any mistuning of over 50 cents (half a semitone) would make an interval aurally identifiable as a different interval entirely, instead of as a mistuned target interval. Mean interval widths of the entire sample-set for thirds, fourths, and fifths measured at (R=31.87, σ =11.91), (R=35.83, σ =13.69), and cents (R=49.34, σ = 16.73) respectively. For vocalists, the mean widths were (R=32.17, σ = 11.89), (R=34.20, σ = 14.11), and (R=53.44, σ = 17.79) cents. For instrumentalists, the mean widths were (R=31.97, σ = 11.93), (R=37.76, σ = 13.16), and cents. This equates to total aggregate deviations from pure of -2.99, -7.02, and 3.06 cents; vocalists aggregate deviations of 8.67, -5.20, and cents; and instrumentalist aggregate deviations of 0.05, -6.38, and cents. Participants inadvertently vocalized 22 perfect fifths, 3 perfect fourths, and 0 major thirds. Listening segment A total of 300 interval-width preferences were compiled in the listening segment. Within that sample-set, participants chose pure intervals 57.62% of the time (50% for vocalists and 62.7% for instrumentalists). 0% of vocalists indicated pure intervals were preferable 100% of the time, compared to 33.3% of instrumentalists. No participants indicated that any of the non-pure tempering conditions were preferable 100% of the time. Participants also showed no distinct preference when two non-pure intervals were presented in a pair, being unsure of which tempered option they preferred 70% of the time, whether vocal or instrumental. No participants indicated that they preferred a wolf interval when it was available. The preference for pure 9

11 thirds, fourths, and fifths over any other available degree of tempering was: 39.4%, 53.6%, and 57.1% for vocalists, and 66.7%, 59.5%, and 61.2% for instrumentalists. Participants were unsure of their preference for 1.1% of P-E pairings (0% vocal, 1.85% instrumental), 73.3% of ET-IET pairings (87.5% vocal, 63.89% instrumental), 0% of IET-P pairings, 10% of P-PY pairings (0% vocal, 16.7% instrumental), 66.7% of ET-PY pairings, and 3.3% of P-W pairings (0% vocal, 5.6% instrumental). Cross-segment comparisons Total aggregate TAI scores and percent pure listening preference (all interval types) showed a moderate positive correlation (higher TAI score = higher pure listening preference), r 2 = See Graph 2. Total aggregate TAI scores and absolute mean deviation (from pure) of vocalized intervals (all interval types) showed no measurable correlation, r 2 = See Graph 3. Total aggregate pure listening preference and absolute mean deviation (from pure) of vocalized intervals (all interval types) showed a very weak positive correlation (higher pure listening preference = vocalization mean further from pure), r 2 = See Graph 4. Total aggregate pure listening preference and absolute mean deviation (from pure) of vocalized intervals (major thirds) showed a very weak positive correlation (higher pure listening preference = vocalization mean further from pure), r 2 = See Graph 5. Total aggregate pure listening preference and absolute mean deviation (from pure) of vocalized intervals (perfect fifths) showed a very weak positive correlation (higher pure listening preference = vocalization mean further from pure), r 2 = See Graph 6. 10

12 Vocalist aggregate pure listening preference and absolute mean deviation (from pure) of vocalized intervals (major thirds) showed a very weak positive correlation (higher pure listening preference = vocalization mean further from pure), r 2 = See Graph 7. Instrumentalist aggregate pure listening preference and absolute mean deviation (from pure) of vocalized intervals (major thirds) showed a very weak negative correlation (higher pure listening preference = vocalization mean closer to pure), r 2 = See Graph 8. Vocalist aggregate pure listening preference and absolute mean deviation (from pure) of vocalized intervals (perfect fifths) showed a weak positive correlation (higher pure listening preference = vocalization mean further from pure), r 2 = See Graph 9. Instrumentalist aggregate pure listening preference and absolute mean deviation (from pure) of vocalized intervals (perfect fifths) showed a strong negative correlation (higher pure listening preference = vocalization mean closer to pure), r 2 = See Graph 10. Discussion Although there were varying results among individual participants, the total sample-set indicated that there was no strict preference for mathematically pure or equal-tempered intervals, in either the listening or vocalization segments, among either group. In the listening segment, only 20% of participants selected a pure interval as preferable 100% of the time it was available, with 0% of participants doing the same for equal-tempered intervals. In the vocalization segment, no participants consistently vocalized pure or equaltempered intervals. Immediately two of the original questions raised in this study can be answered. Did the participants as a whole show a distinct preference for equal-tempered intervals? No. Concurrently, they also showed no distinct preference for pure intervals. The remaining questions is there a temperament that aligns closely to the tested preferences of the 11

13 participants, and is there a notable difference between the vocalists and instrumentalists are open for further exploration. Since both pure and equal-tempered intervals have been dismissed as overriding preferences, the temperaments that consist of only those types of intervals just-intonation and equal temperament can be eliminated as universally preferable. There was also no major inclination towards pure or equal-tempered intervals in either the vocalist or instrumentalist groups. This leaves two possible options for common temperament families that could align with the total sample-set preference; meantone temperaments and good temperaments, which both use different combinations of pure and tempered major thirds and perfect fifths. Evaluating the experimental results to determine which of these two temperament families further corresponds to the preference of the participants requires first examining the listening and vocalization results separately. In the vocalization results alone, the instrumentalists preference clearly aligned with a known temperament while the vocalists preference was less conclusive. In both vocalists and instrumentalists, the perfect fourths were out of line with the results of both the major thirds and perfect fifths. The unexpected aggregate deviation of all recorded fourths of cents does not stimulate any clear explanation; fourths that deviate that far from pure are unknown in any prominent temperaments. 7 However, the results of the perfect fourths and perfect fifths, since those intervals complementally must total a pure octave, can be used to create a combined fifth deviation. By averaging the absolute value of the fourth/fifth deviations, and applying that average to each individual interval in the appropriate direction (narrow for fifths, wide for fourths), the resulting figure represents the best approximation of preference for each interval 7 Perfect fourths were also one of the most frequently missed intervals in the vocalization segment, implying that singing a perfect fourth is in some way more challenging than a major third or perfect fifth. 12

14 type. In both groups, major thirds had a smaller range of widths than perfect fifths (32.17 cents to cents for vocalists, to cents for instrumentalists) as well as a smaller standard deviation (11.89 cents to cents for vocalists, cents to cents for instrumentalists). This shows that regardless of the relative distance from purity, in both groups, major thirds were vocalized with more consistency than perfect fifths. This consistency suggests that overall, participants are less flexible in their preferences with major thirds than perfect fifths. Within the instrumentalist group, the aggregate deviation was 0.05 cents for major thirds and the combined fifth deviation was cents. These figures align nearly exactly with the conditions of the most common meantone temperament, 1/4 syntonic comma (s.c.) meantone, with an error of 0.05 cents for major thirds, perfect fourths, and perfect fifths. Fifths narrow by the same amount exist in a number of good temperaments, but pure major thirds are strictly limited to 1/4 s.c. meantone and just intonation. Since just intonation was already eliminated by lack of universal pure-interval preference, 1/4 s.c. meantone is the only viable option to fit the vocalized widths of the instrumentalists Open Interval Widths - Meantone Temperaments and Aggregate Instrumentalist Vocalization Inst Avg 3 1/4c 3 1/5c 3 1/6c 3 Inst Avg 5 1/4c 5 1/5c 5 1/6c 5 Figure 1. Comparative deviations from pure (in cents) of thirds and fifths in the most common meantone temperaments, along with the average deviation of major thirds and the combined fifth average of instrumentalist participant vocalization. 13

15 The vocalists showed an aggregate deviation of 8.67 cents wide for major thirds, and a combined fifth deviation of cents. These results alone are not enough to definitely fit either the meantone or good temperament families. Major thirds over 8 cents wide are unknown in any of the meantone temperaments, but only by an error of roughly 1 cent, which is less than ½ the difference in major third deviation between any two meantone variants. Perfect fifths with a deviation near cents are found in 1/6 s.c. meantone (-3.60 cents). With the vocalist s mean deviation falling within ±1.5 cents of 1/6 s.c. meantone, this is a close possible fit. However, these same figures are close to the deviations of fifths and thirds in a number of keys in Vallotti/Young s good temperament, with roughly the same margin of error as that of 1/6 s.c. meantone. Establishing which temperament more closely fits the preferences of the vocalists requires analysis of the listening segment along with the vocalization segment. Open Interval Widths - Meantone Temperaments and Vocalist Vocalizations Vocal Avg 3 1/4c 3 1/5c 3 1/6c 3 Vocal Avg 5 1/4c 5 1/5c 5 1/6c 5 Figure 2. Comparative deviations from pure (in cents) of thirds and fifths in the most common meantone temperaments, along with the total average deviation of recorded participant vocalization. 14

16 For both vocalists and instrumentalists, the results of the listening segment reveal an obscure picture that cannot be simplified to a single average deviation figure. The slight total preference of pure intervals in the total listening sample-set indicates that although pure thirds or fifths might be a consonant bonus in an ideal temperament, they are not mandatory. There was, however, a notable difference in preference for pure intervals between the vocalists and instrumentalists. As seen in the results, the instrumentalists indicated a pure interval as preferable 12.7% more readily than the vocalists. The difference between the groups was most apparent in regards to the major thirds, where instrumentalists indicated a pure interval as preferable more than 1.5 times as often (66.7% to 39.4%). Vocalists also indicated a Pythagorean major third as preferable almost 4 times as often as instrumentalists did (50% to 13.9%), choosing Pythagorean over pure major thirds 83.3% of the time. This suggests that while purity is slightly more preferable to all participants throughout the total set of intervals, pure major thirds are measurably more preferable to instrumentalists than they are to vocalists. The inverse of course is also implied, such that perfect fifths and fourths are more important to vocalists in determining consonance than major thirds. This seems a bit backwards, given that the flexible intonation of vocalists permits them to perform even in just intonation, with pure thirds and fifths, if so desired. 8 The instrumentalists attraction to pure major thirds with a higher degree of frequency than found in the fourths and fifths aligns with the principles of the meantone family of temperaments, where the quality of fifths is sacrificed in order to preserve pure or nearly-pure major thirds. Their weaker preference for pure over tempered fifths indicates that although pure 8 It is explained, perhaps, by the higher general TAI scores of the instrumental group, supporting the notion that although vocalists have greater flexibility than most instrumentalists, they are less likely to be aware of, or sensitive to, the qualities that differentiate pure from nearly-pure intervals. 15

17 fifths are perceived as being more consonant, their consonance is less crucial than that of the major thirds. In accordance with the expected conditions of a meantone temperament, instrumentalists chose tempered fourths as preferable more frequently than pure fourths. However, if their listening results were to directly indicate the meantone temperaments, the instrumentalists would have had to show a preference for tempered over pure fourths and fifths, which was not the case. The vocalists showed the exact opposite the purity of fourths and fifths was more often recognized and perceived as consonant than in major thirds. This suggests that a temperament built using pure fifths would better suit the tastes of the vocalists that one using pure thirds, which points to the family of good temperaments. Their likelihood of indicating equal tempered or Pythagorean thirds as preferably consonant further indicates the good temperaments, where thirds of these qualities are often utilized. The failure of all participants to reliably distinguish between the ET and IET fifths and fourths suggests that the direction of tempering applied to an interval is not significant; the only discernible, acknowledged effect is the beats generated by the degree of mistuning, regardless of direction of change. The only absolute finding of the listening segment confirmed that wolf intervals of any type are unacceptable when consonance is the presiding measure of approval. Interpreting the results of the listening segment alongside the vocalization segment reinforces some of the independently reached conclusions for each segment alone but also further obscures others. In both the listening and vocalization segment, the instrumentalist group showed a clear preference for pure major thirds with less concern for the quality of the fifths and fourths. The results of the two segments reinforce the previously-reached conclusions that a meantone temperament whether it is one with pure thirds or slightly compromised thirds in 16

18 favor of less egregious fifths aligns closely with the consonance preferences of the instrumentalists. The very weak correlation across the segments showed that to a small degree, instrumentalists who showed a greater preference for pure major thirds in the listening portion also had lower deviations from pure for their vocalized major thirds. A much stronger correlation of the same type arose for the perfect fifths, indicating that the listening and vocalized preferences of the instrumentalists were in line across the different interval types. For the vocalists, the results of the listening segment indicate that although the vocalization segment yielded a result roughly equivalent to either a good temperament or 1/6 s.c. meantone, the vocalists in general preferred pure fifths more often than pure thirds. This shifts their ideal temperament away from the meantone family and distinctly into the family of good temperaments. It is difficult to say which temperament then best fits the vocalists preferences, as all the good temperaments have varying widths of thirds and fifths, unlike the meantone temperaments which have only two sizes available (good and wolf). Oddly, the vocalists results generated correlations opposite the expected for both major thirds and perfect fifths, with a higher pure listening preference corresponding to a greater vocalized deviation from purity. The correlations were, however, weak at best, placing this odd outcome within the threshold of error and less telling than those found in the instrumentalist group. Numerous sources of potential error became apparent through the experimental process. First and foremost, all of the reference tones used in both segments were computer generated. This yielded tones with absolute precision in pitch, but it also presented sounds to the participants that were unlike anything that would be heard in classical music. It is a known practice to intentionally detune tones on synthesizers to improve the perceived tone of the 17

19 instrument, a practice which is entirely ineffective in acoustic instruments. 9 Several participants commented to the proctor that during the listening portion, it was difficult to decide between which interval they found more consonant, and which they found more timbrally pleasing, due to the effect of detuning the sawtooth waveforms of the synthesized tone. The study also only examined preference in isolated intervals, not interval placement within the context of a triad. It is possible that the participants would have indicated different preferences if they had been asked to choose between two triads in which one note was altered slightly in its tuning. The sample size was small, limiting the usefulness of cross-segment comparisons, as well as comparisons of other factors such as education level, specific instrument families, and ensemble participation, which could have some effect on perceived ideal interval width. If this experiment was to be repeated, the following alterations would likely improve the resultant data and create a more accurate picture of what temperament, if any, follows the natural inclinations of consonance among modern, classically-trained musicians. A much larger samplewould likely increase the accuracy of results. Repeating the process using tones that either emulate an acoustic instrument, or were actually generated by an acoustic instrument such as a harpsichord or piano, could present more cohesive results with stronger differentiation of preference between levels of tempering, eliminating the obscuring effect of the synthetic detuning phenomenon. Instead of looking directly at extra-musical intervals, extra-musical triads would be more indicative of correlations to temperament types. Increasing the scope of the listening portion to include justly-intoned triads, equal tempered triads, meantone triads, and other variations would further reduce preference to a single prominent temperament. 9 Jerry Kovarsky, Synth Oscillator Tuning Tricks, Keyboard Mag (April 2013), accessed April 28, 2016, 18

20 Concluding Remarks All things considered, this study generated an interesting look at temperament preference among a set of classically trained musicians. While the method employed and the subsequent sources of error were not sufficient to pinpoint any particular temperament preference with finality, some interesting results still arose. There is not an absolute subjective superiority of pure intervals even though they are mathematically less discordant and equal temperament, for all its pervasive implementation, has not completely overridden the preferable consonance of mathematically pure intervals. There is, however, a distinct dichotomy in preference between the vocalists and the instrumentalists, with instrumentalists gravitating towards the meantone temperaments with more consonant thirds, and the vocalists gravitating towards the good temperaments with more consonant fifths. This is the opposite of the expected deviation, as pure thirds are more readily attainable by vocalists than most instrumentalists, suggesting that it is not the ease of interval-width production that determines preference. Instead, it seems more correlated to the individual s apparent knowledge of the theory and practice of temperament. Since temperament is a concept derived from the limitations of musical instruments, instrumentalists are already pre-disposed to have to address the issues that temperaments present while performing. In this regard, they are more likely to hold a heightened theoretical understanding of temperament and tempered intervals. This understanding requisites greater exposure and experience with differences between pure and tempered intervals, giving those with the heightened understanding an increased chance to prefer consonant purity. It seems, for this group of participants at least, that the more one knows about temperaments, the more they understand the purity of different open intervals, and the more their preferences align to any 19

21 given temperament in particular regardless of what limitations their instruments do or do not impose. 20

22 Appendix I Experimental Segment Results Listening Segment: Table 2. Listening segment responses for all subjects. Answer coding: 0=P, 1=ET, 2=IET, 3=PY, 4=W, A=Not sure/p-et, B=Not sure/et-iet, C=Not sure/iet-p, D=Not sure/p-py, E=Not sure/py-e, F=Not sure/p-w. A B C D E F G H I J INST INST INST VOC INST INST INST VOC VOC VOC B 1 B B 1 B B B B B B 1 B B 1 1 B 2 2 B B 1 B B 1 B B B B B B 1 B B 2 B B B B B B 1 B B 2 B B B B B B 1 B B 2 B B 1 B B A E E E E E E E D D E 1 E E 1 E 1 3 E E D E 1 E E E E 1 1 E E F

23 Table 3. Percent indicated preference, by subject, for each interval condition when available. Total of thirds, fourths, and fifths. % PURE % ET % IET % PY % W A B C D E F G H I J VOCAL INST TOTAL Table 4. Percent indicated preference, by subject, for each interval condition when available. Perfect fifths only. % PURE % ET % IET % W A B C D E F G H I J VOCAL INST TOTAL II

24 Table 5. Percent indicated preference, by subject, for each interval condition when available. Perfect fourths only. %PURE %EQUAL %IET %W A B C D E F G H I J VOCAL INST TOTAL Percent indicated preference, by subject, for each interval condition when available. Major thirds only. %PURE %EQUAL %PY %W A B C D E F G H I J VOCAL INST TOTAL III

25 Table 7. Percent preference for each specific interval condition pairing. Inverse conditions have inverse preference rates. Total of thirds, fourths, and fifths. %ET > IET %ET > P %IET > P %PY > ET %PY > P A B C D E F G H I J VOCAL INST TOTAL Table 8. Percent preference for each specific interval condition pairing. Inverse conditions have inverse preference rates. Perfect fifths only. %P > E %E > IET %IET > P A B C D E F G H I J VOCAL INST TOTAL IV

26 Table 9. Percent preference for each specific interval condition pairing. Inverse conditions have inverse preference rates. Perfect fourths only. %P>E %E>IET %IET>P A B C D E F G H I J VOCAL INST TOTAL Table 10. Percent preference for each specific interval condition pairing. Inverse conditions have inverse preference rates. Major thirds only. %P>E %E>PY %PY>P A B C D E F G H I J VOCAL INST TOTAL V

27 Table 11. Percent unsure responses in preference of possible interval condition pairs. Total of thirds, fourths, and fifths. %N/P-E %N/E-I %N/I-P %N/P-PY %N/PY-E %N/P-W A B C D E F G H I J VOCAL INST TOTAL Table 12. Percent unsure responses in preference of possible interval condition pairs. Perfect fifths only. %N/ P-E %N/ E-IET %N / IET-P %N / W A B C D E F G H I J VOCAL INST TOTAL VI

28 Table 13. Percent unsure responses in preference of possible interval condition pairs. Perfect fourths only. %N/P-E %N/E-IET %N/IET-P &N/W A B C D E F G H I J VOCAL INST TOTAL Table 14. Percent unsure responses in preference of possible interval condition pairs. Major thirds only. %N/P-E %N/E-PY %N/PY-P &N/W A B C D E F G H I J VOCAL INST TOTAL VII

29 Vocalization Segment: Table 15. Mean deviation of vocalized intervals from pure for individual subjects, and aggregate groups. R = range, σ = standard deviation. All values presented in cents. FIFTHS FOURTHS THIRDS Mean R σ Mean R σ Mean R σ Deviation Deviation Deviation A B n/a n/a C n/a n/a n/a n/a n/a D E F G H I J VOCAL INST TOTAL Graph 1. Aggregate vocalized deviations from pure (in cents) for all groups, with equal temperament shown for reference. X=0 represents no deviation from a pure interval. Aggregate Interval-Witdh Deviations Major Thirds Perfect Fourths Perfect Fifths Deviation from Purity (cents) Inst Subject Average Vocal Subject Average Equal Temperament Total Subject Average VIII

30 Appendix II Cross-Segment Analysis Graph 2. Relationship of Temperament Awareness Index to Percent Pure Interval Listening Preference for all participants 120 Percent Pure Preference Temperament Awareness Index R² = Graph 3. Relationship of Temperament Awareness Index to the absolute mean deviation of vocalization from pure for all participants. Absolute Mean Deviation (cents) Temperament Awareness Index R² = 1.6E-05 IX

31 Graph 4. Relationship of percent pure interval listening preference and absolute mean deviation values for all participants Absolute Mean Deviation (cents) Percent Pure Preference R² = Graph 5. Relationship of percent pure major third preference to absolute mean deviation of major thirds for all participants. Absolute Mean Deviation (cents) Percent Pure Major Third Preference R² = Graph 6. Relationship of pure perfect fifth preference to absolute mean deviation of perfect fifths for all participants. Absolute Mean Deviation (cents) Percent Pure Perfect Fifth Preference R² = X

32 Graph 7. Relationship of pure major third preference to absolute mean deviation of major thirds for instrumentalist participants Abstolute Mean Deviation (cents) Percent Pure Major Third Preference R² = Graph 8. Relationship of percent pure major third preference to absolute mean deviation of major thirds for vocalist participants. Absolute Mean Deviation (cents) Percent Pure Major Third Preference R² = Graph 9. Relationship of percent pure perfect fifth preference to absolute mean deviation of perfect fifths for instrumentalist participants. Absolute Mean Deviation (cents) Percent Pure Perfect Fifth Preference R² = XI

33 Graph 10. Relationship of percent pure perfect fifth preference to absolute mean deviation of perfect fifths for vocalist participants. Absolute Mean Deviation (cents) Percent Pure Perfect Fifth Preference R² = XII