Overtone Series Research Articles

Generating Sonic Phantoms with Quadratic Difference Tone Spectrum Synthesis

Abstract Quadratic difference tones belong to a family of perceptual phenomena that arise from the neuromechanics of the auditory system in response to particular physical properties of sound. Long deployed as “ghost” or “phantom” tones by sound artists, improvisers, and computer musicians, in this article we address an entirely new topic: How to create a quadratic difference tone spectrum (QDTS) in which a target fundamental and harmonic overtone series are specified and in which the complex tone necessary to evoke it is synthesized. We propose a numerical algorithm that solves the problem of how to synthesize a QDTS for a target distribution of amplitudes. The algorithm aims to find a solution that matches the desired spectrum as closely as possible for an arbitrary number of target harmonics. Results from experiments using different parameter settings and target distributions show that the algorithm is effective in the majority of cases, with at least 99% of the cases being solvable in real time. An external object for the visual programming language Max is described. We discuss musical and perceptual considerations for using the external, and we describe a range of audio examples that demonstrate the synthesis of QDTSs across different cases. As we show, the method makes possible the matching of QDTSs to particular instrumental timbres with surprising efficiency. Also included is a discussion of a musical work by composer Marcin Pietruszewski that makes use of QDTS synthesis.

Study on the source of overtone series in a harmonica note

The harmonica reed is a cantilever beam and therefore does not produce integer harmonics. High-speed camera video of the reed and vibrometer measurements both demonstrate this expected behavior at the reed tip when being played. However, the acoustical recording shows an integer overtone series with large amplitudes (<10 dB from the fundamental) all the way past 22 kHz. To understand this phenomenon, we eliminated resonant cavities and other possible acoustic feedback systems by plucking the reed with only the reed plate attached. The acoustical and vibrometer data showed the same discrepancy in this case, but only when the reed enters the reed slot in the reed plate. The overtones don’t show up when the reed is vibrating completely away from the reed slot. In this paper, we will present these data and show how this phenomenon could be a result of vorticity produced by the sharp edge of the plate and the small air layer gap between plate and reed.

Modeling coupled oscillators: Applications for musical sequencing and synthesis

There are many naturally occurring systems of coupled oscillators, e.g., the synchronization of firefly flashes, frog calls, and electrical impulses in the brain. The behavior of a large system of coupled oscillators can be approximated using the Kuramoto-Daido model. The model shows that phase calculations for each oscillator can be simplified by comparing each oscillator’s phase to the average phase of the collection, eliminating the need to sum over the whole group. In this session, we will explore this algorithm in Max/MSP. We will look at simple systems with a small number of low frequency oscillators, both free-running and rhythmically quantized/Euclidean, as a means of generating rhythmic patterns and sequences. Then we will look at larger collections of oscillators within the audible frequency range and explore how the algorithm can be used to modulate audio signals for timbral effects. This example will delve into how the coupling algorithm affects oscillators that have frequency relationships drawing from the overtone series. We will also see how delaying the averaged signal simulates a proximity effect that more closely resembles the localized synchronization that occurs in natural systems.

Singing Planets Don't Sing; They Speak

Ancient Greek philosophers conceived a theory called Music of the Spheres. This ancient theory progressed for almost one thousand years before finally proving itself untrustworthy. However, this examination uncovers an overlooked fact: the large amount of natural order in sound and music existing before the creation of man. Scripture reveals that God is a God of order, and an extensive amount of natural order is found in the universe. Evidence points to God being the creator of the universe. Specific examples of such evidence are the inherent order of sound laid out in pitches, interval ratios, the overtone series, the circle of fifths, and the effect of consonance and dissonance. The structure of music from the ground up reveals a pre-existing model that man developed and turned into music. Man harnessed natural components of sound to create music full of emotion and reaction. Though scientifically inaccurate, Music of the Spheres offers a pathway to discover that music, as a concept and system, exhibits God as the intelligent designer.

Furious Feedback and the Revolutionary Ode to Noise

Furious Feedback and the Revolutionary Ode to Noise

Inferring pitch from coarse spectral features

Fundamental frequency (F0) has long been treated as the physical definition of “pitch” in phonetic analysis. But there have been many demonstrations that F0 is at best an approximation to pitch, both in production and in perception: pitch is not F0, and F0 is not pitch. Changes in the pitch involve many articulatory and acoustic covariates; pitch perception often deviates from what F0 analysis predicts; and in fact, quasi-periodic signals from a single voice source are often incompletely characterized by an attempt to define a single time-varying F0. In this paper, we find strong support for the existence of covariates for pitch in aspects of relatively coarse spectra, in which an overtone series is not available. Thus linear regression can predict the pitch of simple vocalizations, produced by an articulatory synthesizer or by human, from single frames of such coarse spectra. Across speakers, and in more complex vocalizations, our experiments indicate that the covariates are not quite so simple, though apparently still available for more sophisticated modeling. On this basis, we propose that the field needs a better way of thinking about speech pitch, just as celestial mechanics requires us to go beyond Newton's point mass approximations to heavenly bodies.

A Microtonal Analysis of Igor Stravinsky’s Concept of Pitch and Its Resulting Scale

Igor Stravinsky towers as the most prominent composer of the 20th century. When asked on a television broadcast what the Maestro believed to be the next direction in music, Stravinsky claimed that it would be microtonality. The author of the article, who is a prominent specialist in microtonality, proposes a microtonal interpretation in performances of Stravinsky’s musical compositions. He substantiates the validity of such an approach from Stravinsky’s particular spelling of certain chromatic intervals in his musical works, which clearly infer certain intervals from the overtone series, as well as the composer’s words pronounced in his written texts. What Stravinsky had in common with other musical geniuses of the 20th century and earlier times was the inner hierarchical ordering of a constellation of tonal relationships. However, each of these composers had a completely different model of idealized intonation constituted in their respective minds. The author’s additional experience with microtonality was his thorough study of historical European and non-European alternate tunings and temperaments, as the result of which he has been able to determine which of them are appropriate for being employing in performing music by composers from past epochs, such as J.S. Bach, Mozart and Beethoven. This provides additional substantiation for microtonal interpretations of performances of Stravinsky’s music. Citing the northern Lithuanian folk music tradition as expressed in the Sutartines, as being close in its essence to the microtonal interpretation of the great composer’s oeuvres, the author presents the 128-note-per-octave scale created by him for use in his own musical compositions, based on the extension of the overtone scale to the eighth octave as a viable tuning to apply to Stravinsky’s music.

Musikalische Selbstauslegung: eine sichere Quelle historischer Musiktheorie?. Überlegungen zu Skrjabin und Schönberg

To conceptualize music in terms of the historical context of its creation seems to offer a highly authentic theoretical perspective towards understanding musical works. This essay aims to show that, in contrast, composers’ self-interpretations must be considered with enhanced scepticism, even where they seem to provide substantial insights into the relation between compositional technique and theoretical conceptualization. By discussing two prominent examples from early twentieth-century music, the text points at the necessity of a critical assessment of such sources. While Alexander Scriabin supported Leonid Sabanejew’s deduction of his “Prometheus-chord” from the overtone series, a continuous evolution from Romantic harmonic tonality (particularly Chopin) to the complexity of the “Prometheus-chord” can be traced in his stylistic developement. Beside these historical facts, the significance of Sabanejew’s deduction is debatable, for nearly every chord of sufficient complexity can be traced back to the overtone series. Furthermore, Scriabin’s harmony cannot be reduced to one governing principle. In his late works, an interaction of different scales or modes, including whole tone and half tone-whole tone-scales, can be observed. Briefly summarized, Scriabin’s own explanation of the “Prometheus-style” is misleading and reflects the high social status associated with “scientific” justifications in the arts at the beginning of the 20th century, rather than a historically adequate interpretation. A similar scepticism is advisable for a reading of the self-interpretations included in Arnold Schönberg’s Harmonielehre. In one of its final chapters, Schönberg discusses the whole tone scale as a common stylistic element in the music of his time. Although as a composer he used this scale occasionally and its contribution to the destabilization of tonal harmony in his works is obvious, in the Harmonielehre he deduces the scale from an altered dominant chord. This marginalisation of the whole tone scale as an independent structural entity without a clear tonal centre is governed by Schönberg’s intention to delineate his own stylistic development in the years 1907–1911 from the music of Claude Debussy, Alexander Scriabin and other contemporaries and to stress the uniqueness of his abandonment of tonality.

Знали ли древние греки натуральный обертоновый ряд?

Music theory, which developed in the course of a single musical-mathematical-astronomical synthesis, does not give a direct answer to the question posed – "did the ancient Greeks know the natural overtone series" as a reasonable system. Nevertheless, the musical theory in the preserved sources-original works, their fragments, compilations – came quite close to understanding of the existening of the phenomenon much later called "natural overtone series". The article attempts to find a correspondence between individual mathematically calculated intervals, the main system-forming generic tetrachordal structures (diaton, chroma, enharmony) and segments of the natural overtone series as evidence of the unity of the scientific and empirical (auditory) approach to achieving the best sound of Music, universal Harmony. The article is a revised and expanded version of the Lecture for students and teachers of conservatories "Musical acoustics in the works of ancient scientists" (Novosibirsk, 2009).

Meditations for organ: parallels to musical creativeness

The purpose of the article is to determine genre parameters of meditation for the organ on the example of the pieces of composers of the early 21st century. It is proposed to concretize the prerequisites of meditation for organ as an independent genre and to compare the strategies of reading the genre by representatives of different national schools of composition (Italy, Ukraine, Brazil). Among the research methods there were used: historiographic (restoring the historical retrospective of genre formation), the method of genre analysis (confirming the genre status of meditation), the method of style determination in combination with the comparative method (comparing the strategies of reading the genre by composers of different nationalities). The novelty of the proposed topic lies in the identification of the genre status of meditation for the organ and the introduction into the musicological discourse of works that have not previously come to the attention of Ukrainian researchers. Conclusions. Meditations “Shiva” by K. Ferrari, “And there was night, and there was morning, and there were quiet heavenly flutes...” by M. Shukh and Prelude-Meditation by F. Costa make it possible to define meditation as an independent genre of organ art with a constant set of stable indicators. Meditation for the organ is a concert piece that belongs to the genres of cult-ritual music and is characterized by an introverted structure of the communicative act. All works with a similar genre name are united by a single semantic field of religious contemplation. The compositional foundations of meditation as a genre consist in the multiple repetition of structures (of different scales) with a clear “looseness” of the form, which guarantees the tightness of the same sound environment and the monochromatic text. Stylistic characteristics became variable components of the meditation: the meditative profile of “Shiva” by C. Ferrari is emphasized by the using of techniques of minimalism, in the piece by M. Shukh the emphasis is transferred to the timbre of the organ with appeal to the intonation of oriental music, and in the Prelude-meditation by F. Costa attempts to build a new sound universe, as extended scale of the overtone series.

Is consonance attractive to budgerigars? No evidence from a place preference study

Consonant tone combinations occur naturally in the overtone series of harmonic sounds. These include sounds that many non-human animals produce to communicate. As such, non-human animals may be attracted to consonant intervals, interpreting them, e.g., as a feature of important social stimuli. There is preliminary evidence of attraction to consonance in various bird species in the wild, but few experimental studies with birds. We tested budgerigars (Melopsittacus undulatus) for attraction to consonant over dissonant intervals in two experiments. In Experiment 1, we tested humans and budgerigars using a place preference paradigm in which individuals could explore an environment with multiple sound sources. Both species were tested with consonant and dissonant versions of a previously studied piano melody, and we recorded time spent with each stimulus as a measure of attraction. Human females spent more time with consonant than dissonant stimuli in this experiment, but human males spent equal time with both consonant and dissonant stimuli. Neither male nor female budgerigars spent more time with either stimulus type. In Experiment 2, we tested budgerigars with more ecologically relevant stimuli comprised of sampled budgerigar vocalizations arranged into consonant or dissonant chords. These stimuli, however, also failed to produce any evidence of preference in budgerigar responses. We discuss these results in the context of ongoing research on the study of consonance as a potential general feature of auditory perception in animals with harmonic vocalizations, with respect to similarities and differences between human and budgerigar vocal behaviour, and future methodological directions.

Overtone series in variable density strings

The vibrational mode frequencies of a taut string depend on the form of its mass distribution. To lowest order, deviations away from a uniform mass density that are even about the center of the string cause shifts in the overtones away from harmonic, while deviations that are odd do not. The problem of determining the overtone series for a given mass distribution is formulated and a full solution is derived for variations in the inverse mass density that are expressed as a Fourier cosine series. Variations arising from a single Fourier component predominantly shift the relative frequency of a single mode, with the 2nth Fourier component shifting the frequency of the nth mode. For a physically reasonable range of mass density variations, the frequency of a mode can typically be shifted by two semitones. Wrapped metal strings with variations in inverse mass density with different Fourier components have been fabricated, characterized, and tested. Light plucking produces overtones in agreement with the model; ordinary plucking produces incommensurate tones at sum and difference frequencies. Overtone series with several different major and minor tones are possible for variations involving more than one Fourier component.

From Aural Teaching to Musical Literacy in the Elementary Horn Class

Name: Klaske de Haan Main Subject: Master Music Education According to the Kodály concept. Research Supervisors: Suzanne Konings and Herman Jeurissen Title of research: From Aural Teaching to Musical Literacy in the Elementary Horn Class First steps to a Kodály inspired horn method. Research Question: How to develop a methodology for the beginning horn player, from an aural approach to musical literacy? Summary of Results: The Netherlands and Germany are almost the only countries where children start playing the B flat horn instead of the F horn because of the rich culture of wind bans. There are almost no horn methods based on the harmonics of the B flat horn. The traditional instrumental music lesson is mainly based on reproducing: a certain image on paper matches a fitting fingering thus producing the matching sound. When the sound is not matching the image than the teacher corrects this sound. I noticed that the children could play well when playing by themselves, but when they played together with other instruments they could easily play the whole phrase on the wrong pitch (wrong harmonics) but with the right fingerings. The horn is an instrument based on harmonic overtone series. Finding the logical steps on the valves of the horn is very difficult. Woodwind players have a logical system on the instrument playing diatonic series like do re mi. For the horn, with 3 valves, there is not a logical sequence playing diatonic series. This made me realize that some changes in the methodology are necessary for the horn to make an aural approach more logical for the beginning horn player. It also did me realize how extremely important it is for the young horn player to develop the inner hearing very well and make the logical steps from sound to symbol. My research paper investigates the possibilities to develop musical literacy in the horn lesson but also in the instrumental group lesson with transposing instruments. My main conclusion is that it is very important to make singing and moving a part of the horn lesson in order to develop inner hearing in relation to the horn. In that way the music will be understand and musical literacy can be developed. Biography: Klaske de Haan studied horn at The Royal Conservatoire in The Hague and at the Fontys Conservatoire in Tilburg with Herman Jeurissen. She graduated in 2000 for her diploma Uitvoerend Musicus. As a free lance horn player Klaske worked with several professional orchestras in the Netherlands for example The Residentie Orchestra. Klaske is a horn teacher at the Music school, Scholen in de Kunst, in Amersfoort and she works at the Royal Conservatoire at The Hague with BASIS. A new music program for young wind players. From 2014 Klaske specializes in Music education and pedagogy by attending several course such as “muziek als vak”.

„nicht tonal und nicht atonal“ Zur Bedeutung der Quinten in Ligetis Etüde Nr. 8 „Fém“

Ligeti’s claim that his etudes are neither tonal nor atonal can be demonstrated with the functions of the fifths in the etude No. 8 “Fém.” The fifths are integrated into the musical relationships partly in a traditional and partly in a modern manner. Customarily, the interval of a fifth is added to single tones in order to establish tonal spaces and implying fundamentals. A peculiarly modern aspect is the use of fifths as parts of the overtone series.

The Song of the Sirens

In Homer’s account of the adventurous journey of Odysseus, the song of the sirens was so appealing and tempting that it lured sailors to their deaths. Warned by the goddess Kirke, Odysseus overcame the trap by plugging his crew’s ears with wax. An archaeo-acoustical research expedition undertaken by members of Humboldt University Berlin made sound propagation experiments at the supposedly historical scene at the Galli Islands where it’s said that the sirens originally sung. At the site we broadcasted both synthetic signals and natural voices via loudspeakers in the direction Odysseus most probably should have approached the Siren’s island. Subjective listening as well as objective acoustic analysis of the recorded signals revealed evidence for a combination of site-specific acoustic effects, which may explain the nature and origin of the song of the sirens in Homer. The local arrangement of the three islands deforms the acoustic signals by amplification and by changes in timbre. Two female singers from the Berlin State Opera were asked to sing differently pitched musical intervals to be tested in the Li Galli environment. The experiment evinced that the first overtones (octave, fifth, and fourths) would be merged by the echo of the rocks; yet when singing pure thirds and less consonant intervals, which yield higher orders in the overtone series, the voices appear recognisable as being two. As a result, and particularly because Homer stresses the number of exactly two sirens several times, the evidence of our research supports the musicological theory for a rather early existence of enharmonic tunings and most prominently a two-part polyphonic singing of Greek songs. Given that the rocky formation of the Galli Islands most likely didn’t change during the geological tick of just 2,700 years, we conclude that there has been a real acoustic basis for the myth reported by Homer and that a “song of the Sirens”, most probably based on natural voices, was transformed by the particular acoustic conditions of the landscape in such a way that signals were amplified and sent out in one concrete direction. Based on these results, we continue to discuss further leading acoustic theories that offer new insights into the mythology and which were essential to motivate our expedition in the first place. After all, the question remains open what kind of beings the first emitters of the song might have been.

Anharmonic Vibrational Treatment Exclusively in Curvilinear Valence Coordinates: The Case of Formamide.

A highly correlated approach using curvilinear valence coordinates is applied to calculate the vibrational fundamentals and some combination modes of the formamide molecule with high accuracy. A series of potential energy surfaces (PESs) has been generated by AGAPES, a program for adaptive generation of adiabatic PESs, at various electronic structure qualities until excellent nonaccidental agreement with the experimentally assigned fundamental transitions was reached at the CCSDT(T)-F12a/aug-cc-pVTZ level of theory using the improved relaxation method of the Heidelberg multiconfiguration time-dependent Hartree (MCTDH) package in connection with an exact expression for the kinetic energy in valence coordinates generated by the TANA program. By comparison of the overtone series ν1-3ν1 to experiment, we demonstrate that the known problems concerning the floppy ν1 wagging motion are solved within this approach. The potential energy coupling as well as the vibrational coupling in curvilinear coordinates is discussed together with the efficiency of this approach.

MICROTONAL INTONATION FROM COMPOSER’S, PERFORMER’S AND CONCERT-AUDIENCE’S PERSPECTIVE: THE CASE OF GYÖRGY LIGETI’S Hora lungă FOR VIOLA SOLO

The aim of this study is to explore the microtonal intonation, i.e., pitch lowered by a quarter-tone, in György Ligeti’s (1923-2006) „Hora lungă” for viola solo. It is examined from three points of view: composer’s, performer’s, and concert-audience’s. The artistic reasons for the use of microtonality in this piece were explored in Ligeti’s own and other written sources. The difficulties performer encounters hitting the microtonal pitch on viola (corresponding to the 11th harmonic in the overtone series) were investigated practicing the piece by the author of this paper herself. Two methods of practicing microtonal intonation were tested. Firstly, the out-of-tune (“false”) sounding pitch was searched by ear. Secondly, the validity of the second method – listening to harmonics of open C string was proofed, measuring pitch frequencies of open string harmonics in 16 recordings, including the author’s. Finally the perception of microtonal intonation by 26 subjects in live concert setting was tested. Pitch frequencies of performed excerpts were also analysed. Additionally, 17 subjects were interviewed about the perception of microtonal intonation. It was concluded that the appropriate way to practice microtonal intonation is to search the out-of-tune sounding pitch by ear, while other method – listening to open string harmonics, could be questionable as frequency measurements of harmonics showed that 11th harmonic deviated largely from the target. Perception test results have shown that both musicians and non-musicians were able to hear by a quartertone lowered pitch in the author’s performance. Frequency analysis has shown that microtonal intonation in the author’s performance was reliable. From answers in questionnaires and additional interviews it could be concluded that the composer’s intention to create the impression of a „strange, magical, nostalgic world” through the use of microtonal intonation worked, although it remains unclear, if the result would be similar without preparatory explanations before the interviews.

A combined DFT/HREELS study of the vibrational modes of terphenylthiol SAMs

Self-assembled monolayers of p-terphenylthiol (TPT, HS-(C6H4)2-C6H5) deposited onto gold can serve as model systems for aromatic lithography resists. Such thin molecular films are suitably probed using high resolution electron energy loss spectroscopy, due to its high surface sensitivity. Extended energy loss spectra were measured at different probing energies. The TPT monolayer overlapping ν(CH) stretching modes could be modelled by a single effective anharmonic oscillator sustained by a Morse potential energy curve, thanks to the resonant excitation of the associated overtone series at 6 eV. A remarkably good agreement was obtained between the TPT monolayer energy loss spectrum and the computer-simulated infrared vibrational spectrum of the isolated TPT molecule. Density Functional Theory calculations for TPT, fully deuterated TPT and benzenethiol isolated molecules were performed with the exchange correlation functional B3LYP and a dispersion correction, using a triple ζ+ polarisation basis set. By comparing the vibrational patterns obtained for these parent systems, (re-)assignments of all the features observed in the TPT self-assembled monolayer energy loss spectrum are discussed. The obtained vibrational assignments can be confidently transposed to other related systems, such as benzenethiol and biphenyl self-assembled monolayers.

The harmonic vistaphone

Percussion instruments offer a wide range of frequencies from the lowest gongs to the highest bells. The natural modes of vibrations of all percussion instruments produce overtones, which are inharmonic. Builders of keyboard percussion instruments must carve bars in specific ways to include a few harmonic overtones of their choosing. Garry Kvistad, founder/owner of Woodstock Percussion, Inc., has built a set of tubes/rods, which he calls the Vistaphone, carefully tuned to the first 32 partials of the harmonic overtone series. In this presentation, Mr. Kvistad will demonstrate how polyrhythmic patterns played slowly can be sped up by means of a proprietary software program to yield musical intervals that correspond to the ratios of just intonation. After this introduction illustrating the relationship of pulse and pitch, the Vistaphone will be played to reinforce the phenomena of pure harmonic relationships. The listener will be able to experience the pure harmonic overtone series in just intonation which yields a rich bouquet of sustained sound with a multitude of harmonically related difference/summation tones. The only way to obtain this effect acoustically is to build a unique instrument of this type.

Models and findings of synchronization in musical acoustics and music psychology

Synchronization is a crucial mechanism in music tone production and perception. With wind instruments, the overtone series of notes synchronize to nearly perfect harmonic relations due to nonlinear effects and turbulence at the driving mechanism although the overblown pitches of flutes or horns may differ considerably from such a simple harmonic relation. Organ pipes close to each other synchronize in pitch by interaction of the sound pressures. With violins, the sawtooth motion appears because of a synchronization of the stick/slip interaction with the string length. All these models are complex systems also showing bifurcations in terms of multiphonics, biphonation or subharmonics. On the subjects perception and music production side models of synchronization, like the free-energy principle modeling perception by minimizing surprise and adaptation to physical parameters of sound production, neural nets of timbre, tone, or rhythm perception or synergetic models of rhythm production are generally suited much better to model music perception than simplified linear models.