Equal Temperament Research Articles

Dynamic optimal tuning of electronic keyboards as they are being played

The 12-tone equally tempered musical scale can produce noticeably imperfect matching of acoustic harmonics when some combinations of tones are played together. The lower partials (harmonics) which should match, do not. For a 12-tone octave, if a small number of tones in the octave are to be sounded simultaneously, a tuning can be found which produces harmonic errors that are much less than that of equal temperament. Historically, these tunings have been discovered, explored, and abandoned in favor of equal temperament because they generally have much worse harmonic error for tone combinations other than those for which the tuning was optimized. Here, a mathematical description of the total harmonic error of a musical instrument is presented along with a mathematical method which solves for a tuning (tone frequencies) that minimize this total error. The optimal tuning calculations have been implemented in ‘‘C’’ code on a 166-MHZ pentium desktop personal computer equipped with a Creative Labs Sound Blaster AWE 32 soundcard, with a MIDI keyboard attached. The ‘‘C’’ code reads the keys (tones) being played on the keyboard, calculates the optimal tuning, and sounds the tones on the PC soundcard at the optimally tuned fundamental frequencies.

An Examination of the Intonation Tendencies of Wind Instrumentalists Based on Their Performance of Selected Harmonic Musical Intervals

This study is an examination of intonation trends of experienced wind instrumentalists with regard to harmonic intervals. Factors of interest were tuning system, location, interval type, direction of deviation from equal temperament, and group. Subjects (N = 16) were experienced wind instrumentalists (8 professionals, 8 advanced students). Subjects recorded a duet, first playing the melody with a synthesized harmony line, and then vice versa. Target intervals were analyzed, converted to cent distance, and compared. Results indicated that deviation was greatest when compared to just tuning and least when compared to equal tempered tuning. For cent deviation from equal temperament, thirds and sixths were performed slightly less in-tune than fourths, fifths, unisons, and octaves. Location affected direction of deviation as subjects played sharp and less in tune when performing below the stimulus. The student group performed less sharp than the professional group when performing below the stimulus and less in tune when performing above.

‘"Authenticity” and the foreign audience for traditional music in Southwest China

Foreign backpackers have since 1988 flocked to Lijiang, southwest China, to hear amateur urban musicians of the Naxi minority play traditional Dongjing music. Often dismayed by professional troupes’ “adulteration” of Chinese music with Western‐style functional harmony, equal temperament and glitzy staging, such tourists delight in the authenticity of the Naxi musicians’ conservative heterophonic style, local tuning, sober gowns, and dignified demeanor. Yet Lijiang's deliberately traditionalist tourist concerts constitute a new forum for Dongjing music, selectively borrowing from disparate historical contexts to create a staged timelessness. Ironically, truly “authentic” traditional musical gatherings flourish in nearby Naxi villages seldom visited by tourists.

Absolute pitch and sex effect event-related potential activity for a melodic interval discrimination task.

Absolute pitch is a special ability which allows for special perceptual/cognitive strategies. Studies have shown differences in event-related scalp potentials between absolute-pitch (AP) and relative-pitch (RP) subjects of equal musical training. In this study, highly trained musicians (15 females/15 males) performed a melodic interval discrimination task, using intervals on-pitch in equal-tempered tuning (A4 = 440 Hz) and tuned a half-semitone sharp. Subjects identified target intervals (probability 0.2) in a series of 400 randomly transposed intervals. AP subjects were expected to perform differently across intonation conditions, whereas RP subjects were not. Event-related potentials (ERPs) were recorded from three midline sites and two lateral sites. ERPs were analyzed by principal component analysis of variance. Sex was also considered as an independent subject variable. Performance was not significantly different either by absolute pitch or sex. Reaction times did not reveal any significant interactions involving AP or sex, but showed a significant effect by response type (target/nontarget). Strong P3 activity appeared to the target melodic intervals regardless of subject group or intonation. PCA factors with maxima at 352, 511, and 709 ms were sensitive to task relevance. Males showed greater positivity than females along the midline. A significant intonation by response type by sex interaction indicated a greater spread of values for females than males, and greater similarity in response by sex for the sharp than the on-pitch intervals. AP subjects showed reduced P3 activity along the midline, but increased over lateral sites. In a difficult musical task, the ERPs were sensitive to the sex of the listeners, as well as to whether they had absolute pitch.

An Experimental Study of Internal Interval Standards in Javanese and Western Musicians

Six Javanese and six Western musicians performed a magnitude-estimation task using 36 melodic intervals ranging from 60 to 760 cents at 20-cent increments. Several musicians displayed well-defined regions of confusion in which a range of intervals was assigned approximately equal magnitude estimates. The results suggest that these listeners assimilate the intervals to a set of internal interval standards. No evidence for assimilation was found for other musicians in both groups, some of whom made highly accurate estimates. For the Javanese musicians who showed assimilation to internal interval standards, the regions corresponded to the two Javanese tuning systems, slendro and pelog. For the Western musicians, the regions corresponded to the equal-tempered scale. The relatively wider regions of confusion for the Javanese musicians may reflect the greater variability of intonation in Java. In addition, the Javanese musicians seemed able to choose between internal interval standards based on the two tuning systems.

Perceptual integrality of major chord components.

In the present study, an accuracy, rather than a reaction time, version of the Garner paradigm was used to evaluate the integrality or separability of major chord components. Tuned (prototype, or P) and mistuned (nonprototype, or NP) sets of root position C-major triads were constructed by holding the C constant in all stimuli and varying the E and G frequencies in 2- and 4-Hz steps. The P stimuli represent small systematic mistunings in the E and G notes relative to an equal-tempered C-major chord. The NP stimuli represent an equivalent range of frequency variation, but relative to a significantly out-of-tune C-major triad. In different experimental sessions, a same-different (AX) task was used to separately evaluate discrimination performance for the E and G frequencies as a function of whether the nontarget frequency (G or E) was fixed or varied in either a correlated or an orthogonal fashion (with the C frequency always held constant). Compared with a fixed baseline condition where only the target frequency changed, both chord components exhibited a significant redundancy gain in the correlated conditions and, to varying degrees, significant interference effects in the orthogonal condition, indicating that the chord components function largely in an integral fashion. Relative to the discrimination of G, discrimination of the E frequency was less influenced by variation in the nontarget (G) frequency, showing that attention, to some degree, could be selectively allocated to the E chord component. In addition, the results were consistent with previous findings that the functional prototype for the major chord category seems to act as a perceptual anchor, rather than as a magnet, and appears to be located in the physiologically defined area of just temperament, as opposed to the more experientially defined area of equal temperament.

Some Thoughts Concerning the Effects of Tuning on Selected Musical Works (From Landini to Bach)

RILM abstract: "The type of tuning system in use during any particular period in history sometimes had a profound influence on the musical works produced. The impure intervals of the Pythagorean, quasi-Pythagorean, and later meantone temperaments might well have been used and heard as dissonances in certain contexts. Composers such as Louis Couperin, Francois and Bach exploited the distinctive tonal colors of different keys arising from the French and German irregular temperaments. However, systematically varied intervallic nuances for different keys were not a characteristic feature of the music for Baroque fretted instruments; for these works, equal temperament was the norm."

Early Flemish harpsichord string scalings and their acoustical implications

Measurements of string lengths and diameters in historical harpsichords such as those made by the Ruckers family, active in Antwerp from 1579 to the 1680s, are commonly used to calculate: (1) the pitch at which they were intended to be tuned, based on the assumption that strings were stressed to the practical limits of their tensile strength; and (2) the resulting tensions. Consideration of early music theory and workshop design practices have led to the proposal of alternative assumptions that some pre-Ruckers’ makers employed shorter string lengths. With other variables remaining constant, higher calculated coefficients of inharmonicity result. The strings, however, were probably thinner, therefore resulting in inharmonicities similar to those of Ruckers’ instruments. Further, coupling with sympathetically vibrating generally undamped strings of the earliest harpsichords might have reduced the theoretically calculated inharmonicities of the upper partials of bass and tenor strings. Other aspects to consider in light of early instrument makers’ use of traditional geometrical methods of design include: (1) the Ruckers’ slight lengthening of virginal (rectangular harpsichord) E, F-sharp, and G-sharp strings intended to be tuned in meantone temperament, in which these pitches are flatter than in equal temperament; and (2) the development of reasonable plucking points.

Discrimination of musical chord components

The current study uses an accuracy version of the Garner paradigm to evaluate the nature of the roles played by the E and G frequencies in the perception of root position C major triads. Triads differed in the E and/or G frequencies, relative to an equal-tempered triad (prototype, or P set) or an out-of-tune triad (nonprototype, or NP set). Thus, the P stimuli were more representative of major chords than the NP stimuli. Significant redundancy gains and interference effects were found for both frequencies in both contexts, thus demonstrating the components to be integral, as might be expected for tuning based upon intervals. The amount of redundancy gain found was dependent on the type of correlation, with the negatively correlated stimuli experiencing a greater redundancy gain than the positively correlated stimuli. Finally, better overall discrimination was found in the P context, thus replicating previous work [Acker, Pastore, and Hall, Percept. & Psychophys. (in press)] showing that a prototype functions as a perceptual anchor. In addition, major chord prototypes appear to be located in the physiologically motivated area of just temperament, as opposed to the more experientially defined area of equal temperament. [Work supported by AFOSR.]

The Effect of Musical Experience on the Conception of Accurate Tuning

The present study investigates the relationship between musical experience and subjects' conception of accurate tuning. In a paired comparisons experiment, 7 violinists, 7 pianists, and 10 nonmusicians evaluated the tuning of computer-generated, ascending and descending eight-tone diatonic scales of C major. Subjects were required to indicate which member of the pair was "most accurately tuned." The subjects were unaware that all scales were perfectly tuned in the Pythagorean, just, or equal-tempered intonation, respectively. Results showed that (1) violinists, as a group, preferred Pythagorean to equal-tempered scales more frequently than vice versa (p< .01), (2) pianists preferred equaltempered to Pythagorean scales more frequently than vice versa (p < .01), (3) violinists and pianists judged just intoned scales to be less accurately tuned than either Pythagorean or equal-tempered scales (p < .01), and (4) nonmusicians did not show any preference for any of the three intonation models. These findings confirm the claim that subjects' conception of accurate tuning is determined by musical experience rather than by characteristics of the auditory system. Relevance of the results to assessment of tonal perception is discussed.

Religious music: Its time and reality

I) What is Religious music? Sacred music. Purpose and use make music religious. It does not exist in historical time but is always actually present. Imitation of older styles is of no use for religious music. A historical style, modernized or not, is not at all religious. II) Comparison with the Christian mosaics which show the invisible. What is real is not attached to form and characteristics of style, does not recall its origin, does not perish. The spiritual, complete value is lost in the world. III) Music in itself is not religion. In the world it becomes secular. An artificial world is created by way of background, to be able to give music a habitation. Music as a bridge between the humane and the divine, is a means for reconciliation. Man and all that he creates belongs to the created world. Music testifies to the real proportions. IV) The natural order, not to be neglected, is the basis of human creation; no power would emanate from building on unreal ground. Equal temperament is an artificial system. A living rhythm is not built on an artificial foundation. It finds its origin and essence in poetry and dance.

Within-category discrimination of musical chords: perceptual magnet or anchor?

Recent speech research has begun to evaluate the internal structure of categories. In one such study, Kuhl (1991) found that discrimination was poorer for vowel stimuli that were more representative of the category (prototype, or P, set) than it was for less representative stimuli (nonprototype, or NP, set). This finding was interpreted as indicating that a category prototype may function as a "perceptual magnet", effectively decreasing perceptual distance, and thus discriminability, between stimuli. The present study examines the function of prototypes in a musical category--another natural, but nonspeech category. Paralleling the Kuhl study, representative (P) and less representative (NP) sets of major triad stimuli were constructed, based on equal temperament. Musically experienced subjects rated the stimuli in each set for a goodness as a major triad, with the highest rated stimulus serving as a prototype standard for a subsequent discrimination task. Results from the discrimination task demonstrated better performance in the P context than in the NP context. The current nonspeech results indicate that a prototype functions as an anchor rather than a magnet. In addition to providing a natural, nonspeech standard for comparison with speech findings, the results provide some important insights into the nature of musical categories.

Historical temperaments analysis system

This paper describes in a succinct form an historical temperaments analysis system designed for musicians with advanced knowledge of intervalic theory. By introducing the concept of positivetrajectoryreduction in the fifth circle diagram of the temperament to be analyzed, it is possible to do a very fast calculation of the size of any interval inside that temperament, within an error of ±1 cent. The corresponding exact frequency ratio becomes obvious in many cases and it is easy to summarize all the intervalic information in one analysis sheet. In addition to the fifth circle diagram, this sheet includes frequency ratios for the chromatic, diatonic, and hexatonic scales (especially their size in cents), intervals with respect to the tonic, and the deviation in cents of each note from equal temperament. With this system, piano tuning students can make comparisons of different temperaments, classify them, and understand their practicability for different types of keyboard music. Besides, the practical tuning instructions for each temperament can be comprehended much better. Also, with this analysis system, composers and keyboardists are able to program any temperament in an electronic synthesizer with user-tuning option, and the system is even helpful for creating new temperaments.

The Social Construction of the Senario and the Septimal Heresy: Response to Duncan

Berger and Luckmann's (1966) penetrating grasp of the social construction of reality has, ironically, rarely been employed to address constructions in disciplines such as music or physics, where the boundaries are maintained by means of mastering an arcane body of problems, solutions, and applied knowledge. Likewise, neither Gushee's (1973) analysis of social genre by the observed function of historical musical treatises, nor Power's (1981) deployment of emic and etic methods to reconstruct theoretically conceived versus empirically identified tonal and modal types in music, have entered sociological discourse. These systematic omissions and the academic custom of measuring the significance of a paper by its length, perhaps explain how the social theorist might have overlooked Dudley Duncan's achievement in dismantling Weber's Musiksoziologie in an earlier issue of Sociological Theory. In three pages Duncan not only undermines the Weberian edifice, but also frames a challenge to Western equal temperament manifest in the iiberteilige Bruche (superparticular ratios) which Weber postulated as the earliest steps to the rationality of Occidental, artistic, tonal music. One is left to wonder not how and why a mere set of notes by the great scholar came to be published-since the manuscript might well have been rejected if penned by someone of lesser repute-but whether the research question Weber posed has been so badly misunderstood that it can no longer be raised without the taint of Weber's misconstruction.

Local consonance and adaptive tunings: Sound demonstrations

A relationship between tuning (or scale) and timbre (or spectrum) based on the perceptual results of Plomp and Levelt [J. Acoust. Soc. Am. 38, 548–560 (1965)] was presented by Sethares [J. Acoust. Soc. Am. 94, 1218–1228 (1993)]. This relationship can be used to determine an appropriate scale in which a given timbre sounds most consonant, or to design timbres for play in specified scales. This presentation gives sound demonstrations using stretched and compressed timbres, scales related to FM (and other) timbres, and timbres designed specifically for play in 10-tone equal temperament. The notion of an adaptivetuning is an extension in which the tuning of notes are allowed to vary in real time so as to iteratively maximize a measure of consonance. Sound examples that demonstrate the evolution of notes in an adaptive tuning are presented. Examples include complex tonal clusters that slowly evolve towards consonance, adaptation of nonharmonic notes to related scales, and automatic ‘‘correction’’ of out-of-tune harmonic notes. The sound demonstrations provide concrete auditory evidence of the validity of the theoretical results.

Low Energy and Equal Spacing; The Multifactorial Evolution of Tuning Systems

Abstract Studies of tuning systems established in different historical and ethnic cultures suggest that their evolution is mainly dependent on two different, mathematically incompatible, principles: equidistance, based on a mostly logarithmic perception of the audible frequency range, and harmonic consonance generated by simple integer pitch ratios (SIPR). Since for tuning systems the number of relevant SIPRs was overestimated in older theories (often mixing a scientific approach with irrational numerology), a new perceptual model based on both, harmonic consonance and pitch strength, the latter is closely related to timbre, is presented here. A method for numeric evaluation of the harmonic consonance of any interval in pitch space allows a mathematical and graphical representation of harmonic energy. The quality and stability of tuning systems thus can be calculated by combining the energy values for each element of their interval set. The combinatorial energy for tempered systems is plotted in order to determine the system with the lowest value, yielding different systems for different pitch strengths. The high pitch‐strength plots are dominated by the most common system, 12‐tone equal temperament, whereas the low pitch‐strength plots give overwhelming priority to the equal tempered pentatonic system, a finding which suggests a strong relationship between the historical evolution of tuning systems and a culturally determined focus on different musical parameter like melody or rhythm.

Tuning, Containing the Perfection of Eighteenth-Century Temperament, the Lost Art of Nineteenth-Century Temperament, and the Science of Equal Temperament, Complete with Instructions for Aural and Electronic Tuning

Tuning, Containing the Perfection of Eighteenth-Century Temperament, the Lost Art of Nineteenth-Century Temperament, and the Science of Equal Temperament, Complete with Instructions for Aural and Electronic Tuning

The Structural Relationships of Fifths and Thirds in Equal Temperaments

Despite the long history of equal temperaments (henceforth ET's), few theorists have examined their properties systematically. Some recent studies have focused on several individual ET's or on ET's in general (e.g., Mandelbaum 1961, Regener 1973, Blackwood 1985, Rasch 1985). But despite some in-depth treatments (e.g., Blackwood 1985, Rasch 1985),' we rarely find ways of thoroughly interrelating a large number of ET's through their basic generating intervals. This article explores ET's in just this way. Although it discusses only ET's of the octave and implications based on harmonics no higher than 5, the principles presented here can be expanded to accommodate higher harmonics. The article is in three sections which explore, in turn, the perfect fifth, the major third, and relationships between the two intervals. My aim is to describe and explain a multitude of interrelated properties among a number of ET's. Exploring these properties appears timely, as the extensive use of digital music synthesizers has enabled many composers to discover equal temperaments. Admittedly, in some cases this discovery was occasioned by the capacity of even some earlier analog synthesizers to

Nonlinear vibrations in the air column of a clarinet artificially blown

A clarinet with all toneholes closed has been blown artificially and the respective variations of mouthpiece sound pressure, reed opening, and volume flow from the artificial mouth into the instrument have been measured. For a fixed lip adjustment (lip–reed contact position and height of the reed opening under lip pressure), several different vibratory states were found over the complete range of the blowing pressure. The states were stable within their respective regions, but sudden transitions took place from one state to the other when the blowing pressure was varied beyond the stable region. Multiple vibratory states were excitable in some ranges of the blowing pressure, the state excited being dependent upon the manner in which the blowing pressure was varied. The experiment was repeated for different lip adjustments. For the examples discussed, the closed tonehole fundamental of this instrument (D3 on the equally tempered scale) was inevitably produced, but other stable vibratory states were also excited. The vibratory states found for the lip adjustments presented were both periodic states emitting tones of pitch D3 or of much higher pitches, and aperiodic states most of which had an attractor of torus T2. At least one state of torus T3 was identified. Excitation regions of these vibratory states are described.

Intonation of solo violin performance with reference to equally tempered, Pythagorean, and just intonations

The purpose of this study was to determine which musical scale best models the solo performances of violinists when they play diatonic scales of C major very slowly and without frequency vibrato as accurately as possible. Eight professional violinists played without stopping three adjacent scales in ascending order (from C4 to C7), followed immediately by an analogous return to the initial note C4 in descending order. Results show that when violin performances are analyzed taking into account the context of the scale in which they were played, Pythagorean and equally tempered scales are equally good models for the description of performances. Just intonation fits the data significantly less well. When the data analysis is limited to the intervals between separate pairs of notes, not taking into consideration the context of the scale in which they were played, observed interval sizes are almost identical to the arithmetic means of the corresponding interval sizes as defined in Pythagorean and equally tempered intonations. Octave intervals calculated by summing the sizes of performed adjacent major and minor seconds within octaves, show an average stretching of 3.9 cents. However, the arithmetic mean of computed interval sizes between two C’s, deviates only 0.3 cents from the theoretical tonal extent of 1200 cents. Results suggest that when scales of C major are played, the tonic C is adopted as an absolute cognitive reference point.