Missing Fundamental Research Articles

The missing fundamentals of harmonic theory: Chord roots and their ambiguity in arrangements of jazz standards

Since Lippius and Rameau, chords have roots that are often voiced in the bass, doubled, and used as labels. Psychological experiments and analyses of databases of Western classical music have not produced clear evidence for the psychological reality of chord roots. We analyzed a symbolic database of 100 arrangements of jazz standards (musical instrument digital interface [MIDI] files from midkar.com and thejazzpage.de ). Selection criteria were representativeness and quality.The original songs had been composed in the 1930s and 1950s, and each file had a beat track. Files were converted to chord progressions by identifying tone onsets near beat locations (±10% of beat duration). Chords were classified as triads (major, minor, diminished, suspended) or seventh chords (major–minor, minor, major, half-diminished, diminished, and suspended) plus extra tones. Roots that were theoretically less ambiguous were more often in the bass or (to a lesser extent) doubled. The root of the minor triad was ambiguous, as predicted (conventional root or third). Of the sevenths, the major–minor had the clearest root. The diminished triad was often part of a major–minor seventh chord; the half-diminished seventh, of a dominant ninth. Added notes (“tensions”) tended to minimize dissonance (roughness or inharmonicity). In arrangements of songs from the 1950s, diminished triads and sevenths were less common, and suspended triads more common, relative to the 1930s. Results confirm the psychological reality of chord roots and their specific ambiguities. Results are consistent with Terhardt’s virtual pitch theory and the idea that musical chords emerge gradually from cultural and historic processes. The approach can enrich music theory (including pitch-class set analysis) and jazz pedagogy.

Innovative methodologies in renewable energy: A review

This paper is concerned with innovative approaches to renewable energy sources computation methodologies, which provide more refined results than the classical alternatives. Such refinements provide additional improvements especially for replacement of fossil energy usages that emit greenhouse gas (GHG) into the atmosphere leading to climate change impact. Current knowledge gap among each renewable energy source calculation is rather missing fundamentals of plausible, rational, and logical explanations for the interpretation of results. In the literature, there are rather complicated and mechanically applicable methodologies, which require input and output measurement data match with missing physical explanations. The view taken in this review paper is to concentrate on quite plausible, logical, rational, and effectively applicable innovative energy calculation methodologies with simplistic fundamentals. For this purpose, a set of renewable energy methodological approaches is revisited with their innovative structures concerning solar, wind, hydro, current, and geothermal energy resources. With the increase in the renewable energy utilizations to combat the undesirable impacts of global warming and climate change, there is a need for better models that will include physical environmental conditions and data properties in the probabilistic, statistical, stochastic, logical, and rational senses leading to refined and more reliable estimations with application examples in the text. Finally, new research directions are also recommended for more refined innovative energy system calculations.

Tone Profiles of Isolated Musical Chords: Psychoacoustic Versus Cognitive Models

We investigated perception of virtual pitches at missing fundamentals (MFs) in musical chords of three chromas (simultaneous trichords). Tone profiles for major, minor, diminished, augmented, suspended, and four other trichords of octave-complex tones were determined. In Experiment 1, 40 musicians rated how well a tone went with a preceding chord; in Experiment 2, whether the tone was in the chord. Mean ratings for nine non-chord tones were compared with predictions of four models: MFs, diatonicity, 5th-interval relations, and tones that complete familiar tetrachords (e.g., 7th chords). Profiles were accounted for by all four models in Experiment 1, and two (MFs, 5th relations) in Experiment 2. Overall, effect size was largest for MFs. In Experiment 3, listeners heard a chord and chose a matching tone from 12 possibilities. Profile peaks were predicted by pitch models (usually, the lower tone of a perfect 5th). Participants who more likely attended to MFs in isolated harmonic complex tones (fundamental listeners) were not more sensitive to MFs in chords, suggesting their responses instead depended on statistical properties of familiar music. We propose a speculative, psychohistoric explanation: MFs influenced the historical development of musical structure, which in turn influenced the perception of enculturated modern listeners.

Infant pitch perception: Missing fundamental melody discrimination.

Although recent results show that 3-month-olds can discriminate complex tones by their missing fundamental, it is arguable whether they are discriminating on the basis of a perceived pitch. A defining characteristic of pitch is that it carries melodic information. This study investigated whether 3-month-olds, 7-month-olds, and adults can detect a change in a melody composed of missing fundamental complexes. Participants heard a seven-note melody and learned to respond to a change that violated the melodic contour. To ensure that participants were responding on the basis of pitch, the notes in the melody had missing fundamentals and varied in spectral content on each presentation. In experiment I, all melodies had the same absolute pitch, while in experiment II, the melodies were randomly transposed into one of three different keys on each presentation. Almost all participants learned to ignore the spectral changes and respond to the changed note of the melody in both experiments, strengthening the argument that complex tones elicit a sense of musical pitch in infants. These results provide evidence that complex pitch perception is functional by 3 months of age.

Extended Vocal Repertoire inHypsiboas punctatus(Anura: Hylidae)

Male Hypsiboas punctatus emit seven different types of calls: three consist of repeated trains of notes, and four are single-note calls with frequency modulation. Behavioral observations allowed assigning the calls to specific functions: (1) and (2) advertisement calls (short and long duration), (3) territorial signaling, (4) courtship interactions, (5)–(7) aggressive interactions (aggressive, fighting, and release calls). All calls consist of a single note and possess harmonic structure. The harmonic structure from the advertisement call suggests a missing fundamental frequency. Advertisement calls exhibit inter- and intraindividual variation in the dominant frequency. Our results indicate that previous comparisons among vocalizations of H. punctatus were based on different types of calls.

Vibrational and Ghost-Vibrational Resonances in a Modified Chua's Circuit Model Equation

The role of the number of breakpoints N in the sawtooth form of the characteristic function in the modified Chua's circuit model equation on vibrational and ghost-vibrational resonances is investigated in this paper. To observe vibrational resonance, the system should be driven by two periodic forces of frequencies ω and Ω, with Ω ≫ ω. Resonance occurs at the frequency ω when the amplitude of the high-frequency force is varied. When the system is subjected to an input signal containing multifrequencies which are of higher-order than a certain (missing) fundamental frequency, then a resonance at the missing fundamental frequency is induced by the high-frequency input signal and is called ghost-vibrational resonance. In both types of resonances, the number of resonances is N and hysteresis occurs in each resonance region. There are some similarities and differences in these two resonance phenomena. We report in detail the influence of the role of number of breakpoints N on the features of vibrational and ghost-vibrational resonances.

Evaluation of a posteriori Wiener filtering applied to frequency-following response extraction in the auditory brainstem

The main goal of the present study was to determine the best preprocessing method for extracting the frequency-following response (FFR) in the auditory brainstem. The a posteriori Wiener filtering (APWF) method was first applied in FFR preprocessing and then compared with the standard method of conventional averaging with artifact rejection (MeanAR). Two other methods, sub-band optimal weighted averaging (SubBand) and median averaging (Median), were also investigated. FFRs were recorded from 10 normal-hearing subjects. A harmonic complex tone with a missing fundamental frequency was used as the sound stimulus. Comprehensive and quantitative indices were constructed to evaluate the quality of FFRs processed by the four methods. The indices in the time domain included the root mean square (RMS) of the residual background noise, RMS of the FFR, and autocorrelation function, and the indices in the frequency domain included the signal-to-noise ratios (SNRs) of the harmonics. The results revealed that the APWF method achieved the best performance in FFR extraction. Additionally, the effect of sweep number on FFR quality was studied. Paired t-tests indicated that APWF required far fewer sweeps compared with other methods in obtaining equivalent high-quality FFRs. In conclusion, APWF is a more suitable method for FFR preprocessing than the existing methods because of its advantages in improving SNR and experiment efficiency.

Perception of the pitch of unresolved harmonics by 3- and 7-month-old human infants.

Three-month-olds discriminate resolved harmonic complexes on the basis of missing fundamental (MF) pitch. In view of reported difficulty in discriminating unresolved complexes at 7 months and striking changes in the organization of the auditory system during early infancy, infants' ability to discriminate unresolved complexes is of some interest. This study investigated the ability of 3-month-olds, 7-month-olds, and adults to discriminate the pitch of unresolved harmonic complexes using an observer-based method. Stimuli were MF complexes bandpass filtered with a -12 dB/octave slope, combined in random phase, presented at 70 dB sound pressure level (SPL) for 650 ms with a 50 ms rise/fall with a pink noise at 65 dB SPL. The conditions were (1) "LOW" unresolved harmonics (2500-4500 Hz) based on MFs of 160 and 200 Hz and (2) "HIGH" unresolved harmonics (4000-6000 Hz) based on MFs of 190 and 200 Hz. To demonstrate MF discrimination, participants had to ignore spectral changes in complexes with the same fundamental and respond only when the fundamental changed. Nearly all infants tested categorized complexes by MF pitch suggesting discrimination of pitch extracted from unresolved harmonics by 3 months. Adults also categorized the complexes by MF pitch, although musically trained adults were more successful than musically untrained adults.

Infants' ability to perceive the pitch of unresolved harmonics

An important phenomenon for models of pitch perception is that adult listeners’ can extract pitch from complexes containing only unresolved harmonics. Although 3-month-olds discriminate resolved harmonics on the basis of missing fundamental (MF) pitch, their ability to discriminate unresolved harmonics is unknown. This study investigated the ability of adults, 7- and 3-month-olds, to perceive the pitch of unresolved harmonics using an observer-based method. Stimuli were MF complexes that were bandpass filtered with a -12 dB/octave slope, combined in random phase, and presented at 70 dB SPL for 650 ms with a 50 ms rise/fall and with a pink noise to mask distortion products. The experiment consisted of two conditions: (1) “low” unresolved harmonics between 2500 and 4500 Hz based on MFs of 160 Hz (H17–H26) and 200 Hz (H13–H22) and (2) “high” unresolved harmonics between 4000 and 6000 Hz based on MFs of 190 Hz (H22–H31) and 200 Hz (H20–H29). To demonstrate MF pitch discrimination, participants were required to ignore spectral changes in complexes with the same fundamental and to respond only when the fundamental changed. Interestingly, variable performance in the “high” condition was observed with adult participants. However, nearly all infants tested categorized complexes by MF pitch in both conditions, suggesting discrimination of unresolved harmonics at 3 months.

Infants' ability to perceive the pitch of unresolved harmonics.

An important phenomenon for models of pitch perception is that adult listeners can extract pitch from complexes containing only unresolved harmonics (UH). Although 3-month-olds discriminate resolved harmonics on the basis of missing fundamental (MF) pitch, their ability to discriminate UH is unknown. This study investigated the ability of adults, 7- and 3-month-olds to perceive the pitch of UH using an observer-based method. Stimuli were MF complexes that were bandpass filtered with a -12 dB/octave slope, combined in random phase, and presented at 70 dB SPL for 650 ms with a 50 ms rise/fall and a pink noise to mask distortion products. The experiment had two conditions: 1) "low" UH between 2500-4500 Hz based on MFs of 160 Hz (H17-H26) and 200 Hz (H13-H22) and 2) "high" UH between 4000-6000 Hz based on MFs of 190 Hz (H22-H31) and 200 Hz (H20-H29). To demonstrate MF pitch discrimination, participants were required to ignore spectral changes in complexes with the same fundamental and respond only when the fundamental changed. Interestingly, variable performance in the "high" condition was observed with adults. However, nearly all infants tested categorized complexes by MF pitch in both conditions, suggesting discrimination of unresolved harmonics at 3 months.

Patterns of individual differences in the perception of missing-fundamental tones.

Recent experimental findings suggest stable individual differences in the perception of auditory stimuli lacking energy at the fundamental frequency (F0), here called missing fundamental (MF) tones. Specifically, some individuals readily identify the pitch of such tones with the missing F0 ("F0 listeners"), and some base their judgment on the frequency of the partials that make up the tones ("spectral listeners"). However, the diversity of goals and methods in recent research makes it difficult to draw clear conclusions about individual differences. The first purpose of this article is to discuss the influence of methodological choices on listeners' responses. The second goal is to report findings on individual differences in our own studies of the MF phenomenon. In several experiments, participants judged the direction of pitch change in stimuli composed of two MF tones, constructed so as to reveal whether the pitch percept was based on the MF or the partials. The reported difference between F0 listeners and spectral listeners was replicated, but other stable patterns of responses were also observed. Test-retest reliability is high. We conclude that there are genuine, stable individual differences underlying the diverse findings, but also that there are more than two general types of listeners, and that stimulus variables strongly affect some listeners' responses. This suggests that it is generally misleading to classify individuals as "F0 listeners" or "spectral listeners." It may be more accurate to speak of two modes of perception ("F0 listening" and "spectral listening"), both of which are available to many listeners. The individual differences lie in what conditions the choice between the two modes.

Perception of missing fundamental pitch by 3- and 4-month-old human infants.

A hallmark of complex pitch perception is that the pitch of a harmonic complex is the same whether or not the fundamental frequency is present. By 7 months, infants appear to discriminate on the basis of the pitch of the missing fundamental (MF). Although electrophysiological cortical responses to MF pitch changes have been recorded in infants younger than 7 months, no psychophysical studies have been published. This study investigated the ability of 3- and 4-month-olds to perceive the pitch of MF harmonic complexes based on fundamentals of 160 Hz and 200 Hz using an observer-based method. In experiment I, to demonstrate MF pitch discrimination, 3- and 4-month-olds were required to ignore spectral changes in complexes with the same fundamental and to respond only when the fundamental changed. In experiment II, a 60-260 Hz noise was presented with complexes to mask combination tones at the fundamental frequency. In experiment III, complexes were bandpass filtered with a -12 dB/octave slope to limit use of spectral edge cues and presented with a pink noise to mask all distortion products. Nearly all infants tested categorized complexes by MF pitch in these experiments, suggesting perception of the missing fundamental at 3 months.

Influence of image‐quality degradation on accommodation mechanism in human vision: Conditions necessary for objective image‐quality evaluation metrics

AbstractThis paper clarifies the relationship between image quality and accommodation in human vision through two types of experiments. One examines how image‐quality degradation influences the accommodation mechanism, and the other examines which type of information of quality‐degraded images activates the accommodation mechanism. Actually, accommodative responses are measured using an infrared optometer while subjects are subjectively evaluating sharpness, noise, and pseudo‐contours and while they are observing sine, square, and missing fundamental (MF) square waves. The following results were derived: (1) the accommodation lag increases as the degree of sharpness is degraded regardless of the tone‐reproducing methods; (2) the accommodation lag decreases considerably in the existence of noise or pseudo‐contours, whereas it increases for uniform or gently curved planes; (3) the spatial features of presented images activate the accommodation mechanism. These results suggest that accommodative responses influence human subjective judgments as well as being a human factor related closely to image quality and that the spatial features of quality‐degraded images underlie human subjective judgments. In other words, they imply that objective image‐quality evaluation metrics should satisfy the following two conditions: the incorporation of the accommodation characteristics into such metrics and the formulation of such metrics in the spatial region.

Theoretical study of the molecular force field and vibration analysis of the hydrogen cyanide addition compound with boron tribromide

An extensive HF, MP2, B3LYP and CCSD study of the molecular structure, force field and normal vibrations has been carried out for the hydrogen cyanide compound with boron tribromide. Most of the calculations agree that the HCN-BBr(3) molecule belongs to C(3v) point group and has a N-B length of 1.55-1.70 A and a N-B-Br angle of 103-105 degrees. These calculations also have estimated the missing low-wavenumber fundamentals in the 190 (nu(5)), 150 (nu(9)) and 105 cm(-1) (nu(10)) regions, instead of the supposed fundamentals at 207, 188 and 150 cm(-1), respectively, based on the combination bands. The quantum chemical force constants, by the B3LYP/6-31G and CCSD/3-21G calculations, have been adjusted by the scaling factors to reproduce the fundamentals in the literature [3] to include the bands below 200 cm(-1) proposed in this study. Normal coordinate analysis using the scaled force constants has been performed to interpret the molecular vibrations of four isotopic molecules, HCN-(10)BBr(3), HCN-(11)BBr(3), DCN-(10)BBr(3) and DCN-(11)BBr(3).

Pitch Detection/Tracking Strategy for Musical Recordings of Solo Bowed-String and Wind Instruments

A pitch detection/tracking strategy for solo bowed-string and wind musical instrumental recordings is presented. To avoid the missing fundamental problem, we adopted the greatest common divisor method and modified it with a weighted-and-voting technique that can reveal more information of strong partials in the target signal. Moreover, a frame-based correction method with consideration of the performing aspects of the instruments is also proposed to emendate possible misjudgments in the transition from one note to the next note. Experimental results showed that the proposed strategy is superior to three popular methods for a pitch extraction/tracking task. The proposed method was also tested when the sound source is reverberant and the results were compared with other methods, too.

A new strategy on pitch estimation in polyphonic musical audio

In this paper, a new perceptually motivated approach for pitch estimation in polyphonic musical audio is presented. The proposed algorithm is based on the pair-wise analysis of spectral peaks. Starting from a spectral representation of the audio signal, audible spectral peaks are selected with the help of a simplified psychoacoustic model. The idea of the technique lies in the identification of partials with a successive harmonic number (1 and 2, 2 and 3, etc.). Since two successive harmonics of a harmonic sound have a well-defined frequency ratio, a possible fundamental can be derived from the instantaneous frequencies of the two peaks. Furthermore, under certain conditions the harmonic integer ratios 5/3, 7/5, and 9/7 are also considered. Consecutively, the identified harmonic pairs are evaluated according to a perceptually motivated rating scheme, and the resulting pitch strengths are added to a pitch spectrogram. In addition to the efficient estimation of the fundamental frequency, the presented approach results in a quantitative prediction of the perceived pitch strength, allowing the identification of the predominant voices (e.g., melody and bass line) in music recordings. The method is robust in regard to the missing fundamental frequency phenomenon and works well with quasi-harmonic sounds.

Virtual pitch in a computational physiological model

A computational model of nervous activity in the auditory nerve, cochlear nucleus, and inferior colliculus is presented and evaluated in terms of its ability to simulate psychophysically-measured pitch perception. The model has a similar architecture to previous autocorrelation models except that the mathematical operations of autocorrelation are replaced by the combined action of thousands of physiologically plausible neuronal components. The evaluation employs pitch stimuli including complex tones with a missing fundamental frequency, tones with alternating phase, inharmonic tones with equally spaced frequencies and iterated rippled noise. Particular attention is paid to differences in response to resolved and unresolved component harmonics. The results indicate that the model is able to simulate qualitatively the related pitch-perceptions. This physiological model is similar in many respects to autocorrelation models of pitch and the success of the evaluations suggests that autocorrelation models may, after all, be physiologically plausible.

Reversed Café Wall illusion with missing fundamental gratings

The direction of Café Wall illusion was measured for ordinary Café Wall figures comprised of blocks with a square wave profile and for those with a missing fundamental (MF) profile. For the MF version, it was found that the illusion direction alternates according to the patterns’ main component frequency when the shift between adjacent rows was systematically varied. The direction of illusion for the MF version was opposite to that for the original version when the phase shifts of adjacent rows were between 60 and 120 degrees of the original grating. The results indicate that the illusion depends on the physical component rather than the appearance of the pattern. The possibility of the Café Wall illusion being mediated by low-level luminance mechanisms each tuned to a different spatial frequency is discussed based on the results.

Dominance of missing fundamental versus spectrally cued pitch: individual differences for complex tones with unresolved harmonics.

In a two-alternative, forced-choice experiment, subjects had to compare the pitches of two sounds, A and B. Each sound was composed of four successive harmonics of a fundamental frequency between 100 to 250 Hz, added in cosine or Schröder phase. The harmonic frequencies of A were lower than those of B; the missing fundamental frequency of A was higher than that of B. The dominance of the missing fundamental versus the spectrally cued pitch--a pitch percept corresponding to spectral components--was measured as a function of nA, the lowest harmonic in A. The pitch percept is dominated by the missing fundamental if the harmonics are resolved (nA<7). If the harmonics become unresolved and are added in Schröder phase, the dominance shifts to a spectrally cued pitch (7<nA<13; 75% of the subjects). In the cosine phase condition, many subjects could detect the fundamental pitch well into the unresolved harmonic range (nA>20). For others, the transition was in the realm of partly resolved harmonics. This shows that the temporal envelope modulation of stimuli with only four unresolved harmonics can give a relatively clear fundamental pitch percept. However, this percept varies considerably among subjects.

Perception of harmonics in the combination long call of cottontop tamarins, Saguinus oedipus

A number of nonhuman primates produce vocalizations with time-varying harmonic structure. Relatively little is known about whether such spectral information plays a role in call type classification. We address this problem by utilizing acoustic analyses and playback experiments on cottontop tamarins‘ combi nation long call, a species-typical vocalization with a characteristic harmonic structure. Specifically, we used habituation–discrimination experiments to test whether particular frequency components, as well as the relationship between components, have an effect on the perception and classification of long calls. In Condition 1, we show that tamarins classify natural and synthetic exemplars of the long call as perceptually similar, thereby allowing us to use synthetics to manipulate components of this signal precisely. In subsequent conditions, we tested the perceptual salience and discriminability of long calls in which we deleted (1) the second harmonic, (2) the fundamental frequency, or (3) all frequencies above the fundamental; we also examined the effects of frequency mistuning by shifting the second harmonic by 1000Hz. Following habituation to unmanipulated long calls, tamarins did not respond (transferred habituation) to long calls with either a missing fundamental frequency or the second harmonic, but responded (discriminated) to long calls with the upper harmonics eliminated or with the second harmonic mistuned. These studies reveal the importance of harmonic structure in tamarin perception, and highlight the advantages of using synthetic signals for understanding how particular acoustic features drive perceptual classification in nonhuman primates. Copyright 2002 The Association for the Study of Animal Behaviour. Published by Elsevier Science Ltd. All rights reserved.