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Figure 1. Spectrogram 0. One can see that evolving musical practice as well as a different evaluation of actual sensation and perception of music were factors that, by about —, began to challenge traditional views. These fields were part also of music theory, and connected in various ways to musical practice. For example, Zarlino, who still preferred numbers and proportions as principles for rational deduction of musical scales and intervals, offers a discourse on how notes either high or low in pitch apparently relate to the speed of vibration of a string as well as to its diameter and the tension with which it is stressed.
Beeckman, Descartes, Galilei and others noted that this was actually not the case. The correct solution, which needs a theory of isochronous periods of vibration, was found only in the course of the 17 th and 18 th centuries, respectively. Hence, the ear is viewed as a factor relevant to finding practical solutions that, however, may not be acceptable to the musical mind seeking rational analysis.
To Kepler, the ear alone cannot provide the degree of exactitude required for confirming models of musical scales and intervals, which thus is left to mathematical justification. Music theory, in this respect, was part of fundamental research Grundlagenforschung , and also intrinsically tied to philosophy. Though one may point to various if not many writings treating aspects of musical syntax, musical rhetoric, etc. In the following sections, I will further explore relations between theoretical reasoning, speculation touching on both heuristics and imagination , and empiricism in regard to music theory.
Accepting the limitations of an article, a selection of case studies will be presented, each of which seems suited to discuss issues that have been of relevance to music theory. The first case study considers relations between harmonically differentiated compositional structure as found in chromatic and enharmonic settings from the 16 th century onwards , tunings, and intonation.
In the third of the case studies, some steps in the development of harmonic dualism are retraced in brief, in order to illustrate certain problems. One can study the relation of theory to practice, for example, in the area of chromatic and enharmonic tone systems, scales, and intonation. Meanwhile, the whole issue of chromaticism and ultra-chromaticism as well as enharmonic experimentation has been subjected to scrutiny including reconstruction of instruments, performance practice, and empirical investigation.
Sweelinck and M. In this respect, the correct tuning certainly is an intrinsic part of the work in question, and not just an attribute from the exterior. Still far in the 19 th century, Anton Bruckner, famous as a composer, recognized that compositional structures such as modulations between distant keys should be executed in a musically differentiated way namely, in just intonation . Microtonal systems, as they emerged in the course of the 16 th century, became executable both on a variety of instruments as well as for vocalists.
Music written by Nicola Vicentino that comprises enharmonic tone steps in melody and some uncommon simultaneous intervals can be alternatively performed in the diatonic, in the chromatic, or in the enharmonic genus. One can check the quality of intonation either by ear or by empirical measurement based on digital signal processing. The overall pitch pattern realized by the five voices can be seen when tracking their fundamental frequencies, which was done by a LPC algorithm. Figure 2. According to the sound analysis, the soprano, the tenor and also the bass are exceptionally stable in their intonation once they have matched their voices to the fundamental frequencies required for the G-major chord.
Looking into details of pitch adjustments one finds, for example, the bass succeeds in producing a stable fundamental frequency of the note G for ca. Bass, tenor and soprano in this chord realize fundamental frequencies representing just intonation pitches in a perfect manner the other two voices being a few cents sharp. Figure 3. Tracking of the bass voice in the final chord of O dolorosa gioia.
Recordings by singers and ensembles trained to perform in highly chromatic or even enharmonic genres  demonstrate that intonation according to an elaborate scale and chord structure is possible. However, the singer must know which pitch representing a certain note she or he is expected to realize is the correct one in a given structure of intervals, chords and sonorities.
Depending on a weighting of the two factors counterpoint and harmony , different solutions in regard to intonation for a certain piece or of some part of a piece are feasible. Moreover, music theory, beginning perhaps in the Renaissance era, often included a developmental momentum in that theorists designed systemic features and techniques they thought could or should be relevant for contemporary as well as for future composers when creating music. Parallels between the harmonic ratios of tones making up a major chord, and a series of harmonic partials as found in a single vibrating string are all too obvious.
Both will give rise to a sensation of consonance though the quality may vary somehow. A single thin brass string of a harpsichord when plucked contains many harmonics. The spectrum Figure 4 taken from the sound of the lowest string of a French harpsichord  offers no less than 19 strong harmonics in the frequency range from 50 Hz to 1 kHz. A trained musician like Rameau, when analyzing the sound radiated from such an instrument, may well detect a number of the low harmonics that can be resolved by the ear:.
The fact that a series of harmonics as observed in a vibrating string results in a sound which listeners experience as consonant, follows from the spectral harmonicity, which in turn implies strict periodicity of the sound signal in the time domain. This in turn results in a periodic neural excitation pattern, which, finally, results in a sensation of consonance. The problem Rameau and legions of later music theorists were faced with is the minor tonality, for which no easy acoustical explanation is at hand.
It was during the 18 th century that string vibration was investigated from both a mathematical and experimental perspective; the results demonstrate that a string set to longitudinal and transversal vibrations produces a series of harmonics above a fundamental. Harmonic minor is found in so many works of the 17 th and early 18 th centuries, respectively, that theorists had to deal with minor chords and the minor tonality in a reasonable way.
In general, theorists  elaborated on musically usable consonant and dissonant intervals before discussing chords and chord progressions. Tartini , one of the pioneers in research on combination tones, saw that the minor tonality was cumbersome to derive from the series of harmonic partials, and also less coherent than major when viewed from the combination tone perspective. His solution, based on elaborate geometric operations, indeed leads to a minor chord, in which the tones C-E b -G-c stand in the relation Zarlino had discussed arithmetic and harmonic division of intervals, which can be ordered in reciprocal series.
Riemann, translating and interpreting Parte II, c. Figure 5. The virtual pitches found for the major and the two minor chords might be of different salience. Figure 6. Virtual pitches found by autocorrelation algorithm for the C-major, the a-minor and the c-minor chord 61, 20, 25 Hz. With respect to another relevant perceptual feature, spectral brightness, major and minor chord based on identical fundamental frequency differ but little.
For the C-major and the c-minor chord used as examples, the average spectral centroid is ca. Figure 7. Oettingen took great pains to reinstall music theory on a basis provided by musical acoustics and psychoacoustics.
When Riemann, in , began to publish on music theory, he could draw from Hauptmann , Oettingen , and Helmholtz , as he in fact did in various contexts approving as well as contradicting points of view issued by his masters. Apparently, things are not as simple as to dismiss the case altogether. The spectrum obviously contains a considerable energy below f 1 ; among the many components is one about one octave below f 1. Also, there are b many components between f 1 and f 2 that are certainly not partials of the note c, which has been played, but partials of strings whose fundamental frequency is below that of the note c.
This point of view is maintained in the Ideen zu einer Lehre von den Tonvorstellungen — Another simplification Riemann undertook concerned the two-dimensional Tonnetz.
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Riemann had adopted this scheme while others had simplified it early on to six tone relations per centre tone. From a phenomenological point of view, the position of the ideal listener in a two-dimensional tonal space has been identified with the base note of the major and minor triad. The argument is that this position is implied by the structural composition of triads thirds piled up on a base note acting as an anchor as well as by our listening experience.
According to a prediction model, perceptual roots of root-position diatonic triads in a minor key lie in the bass. The cadences were taken from Riemann and other textbooks on harmony, and recorded with two synthesizers Yamaha TX 81 Z in just intonation tuning. Each cadence was presented A in original form, B with one voice added in the bass, and C one voice added on top of the soprano. To avoid marked shifts in spectral brightness, these additional voices were recorded -8dB below the level of the main 4-part texture.
The resulting sound pattern for cadence no. The actual ratings, however, did not exhibit significant differences for the three versions: . Table 3. The only correlation that is not significant is r 26 for musical conclusiveness between the original version A and version C. Hence, the three versions seem to appear quite similar to listeners. Interestingly, version B , which gained a little in harmonic coherence by adding a voice to the bottom, loses as much against the other two versions in conclusiveness.
Taking the data from five treatments, average ratings in general increased from the original A to versions B, C on both variables when either an additional voice below the bass or above the soprano were offered. Also, ratings increased when in one complete I-IV-I-V-I cadence a supporting voice was added, in the same sound stimulus, below the bass and another one above the soprano voice.
In certain works, or parts thereof, listeners would be forced, in a strict dualistic approach, to shift between anchor points identified with the base note for major chords , and the fifth for minor chords. The degree of differentiation and exactitude musically trained subjects may achieve in regard to apprehending and conceptualizing tonal relations, of course varies according to certain objective and subjective factors.
Apparently, advanced conceptualizations of intricate chord and key relationships as are found in many works of the 19 th and early 20 th century, can be developed by intensive training including listening, performance, and analysis. His own work, impressive as it is, would certainly have benefited from empirical investigations relevant to music perception and cognition. Riemann was lacking support from psychologists and acousticians, and he may have been too idiosyncratic to cooperate in a team should such an opportunity have existed.
Oettingen, by contrast, notwithstanding his strong theoretical claims, was more of an empiricist,  and carried out experiments on perception of major and minor chords including aspects of tuning and intonation . Systematic research in areas of perception and cognition relevant to music theory remained scarce for decades to come. This line of tradition can be traced well into the 20 th century  up to the present. This field has been combined with music analysis in order to foster understanding of techniques of composition that were of importance in the past.
Then there are music psychologists with an interest in music theory as well as theorists who contribute to the psychology of music,  not to forget investigations the neurosciences have done in music perception and cognition that also have implications for music theory.
Hindemith As Handschin , f. See the volume edited by Christensen The German project Geschichte der Musiktheorie Zaminer ff. Much of the relevant research has been summarized in The Garland Encyclopedia of World Music 10 volumes. New York: Garland Publishing — See Barker , Not to complicate matters, I leave out the issue of origins parts of Greek music theory may have had in the Orient for example Babylon and in Egypt.
Kittler , chapter 2 provides insights in the cultural context of Greek music-mathematics as developed, in the main, in the cities of Magna Graecia from Pythagoras to Archytas. It is of no particular interest here whether or not Euclid is the author of the work in question. See Becker , 6 and the review by Kurt von Fritz in Gnomon 33, , 7. One field where Pythagorean proportions including 3 : 4 : 5 certainly played a role was architecture.
For recent findings and empirical data, see Birnbaum Aristotle met. See Schneider , ff. The view Zarlino , 78 outlines is that strings for high notes undergo fast movements, and strings for low notes slow movements equal plucking force assumed. Zarlino also correctly notes that the sound of strings for high notes lasts much shorter than the sound of strings tuned to low notes. He infers from this that the speed of vibration of the string, and the interaction between string and surrounding air, determines the duration of the sound emitted from the respective string this aspect relates, implicitly, to radiation and consumption of energy as well as to viscous damping.
Intervals and cents are also given as cumulated. There are very small errors in the cent values resulting from interpolation that, in sum, amount to 0. For a comprehensive survey, see Barbieri and also the Proceedings of the colloquium Chromatic und enharmonic Music and musical instruments in the 16 th and 17 th centuries.
Bern: P. Lang As to details of signal processing applied to musical sound, see Schneider a, , Besides vocal polyphony of the 16 th and 17 th centuries, respectively, one has to take into account also vocal music of the 19 th and 20 th century including microtonal compositions. See sound examples provided by Sethares , Cordes , Barbieri Also material available on the Internet, e.
This consideration is still found in the 20 th century, in works on microtonal systems.
Sauveur — preserved as a manuscript that seems to contain the material Sauveur had presented as lectures. See the Introduction of Rasch in Sauveur II, Dallas , p. XV , Pischner and Christensen The spectrum was taken from the C string tuned to See Handschin , ff. Handschin, one of the most thoughtful and erudite musicologists ever, was familiar with publications on perception and music psychology of his time.
Rameau , Riemann of course translated only those portions he believed to contain the most relevant information with respect to harmony. Fourier synthesis was done with Mathematica 7. They are identical in level, which is An interesting difference to be reported elsewhere becomes obvious in a formant analysis of the two sounds. Helmholtz , See Rehding , chapter 1. It is in fact possible to produce subharmonics in strings, by parametric excitation.
See Meyer and Guicking , See Schneider a, I will discuss this concept and its Kantian, Herbartian, etc.
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The data, which are between ordinal and interval scale in quality, are processed as interval scaled. A more detailed discussion of empirical data must be given elsewhere. He wanted to demonstrate, by means of different intonation patterns, several alternative solutions to harmonic problems posed by complex chord progressions see Bister For example, Fay , ; Lerdahl and Jackendorff See also Krumhansl , ; Seifert Dogmatic in a philosophical or other theoretical context means that a body of theory is regarded as authoritative and valid.
See Rexroth Melody beyond notes: a study of melody cognition. Albersheim, Gerhard. Zur Musikpsychologie. Wilhelmshaven: Heinrichshofen. Translated by Friedrich Wilhelm Marpurg. Leipzig: Breitkopf. Translated by Hermann Bonitz ed. Edited by H. Carvallo and E. Reinbek: Rowohlt Edited by N.
Tsouyopoulos and E. Bader, Rolf.
Book Music And Schema Theory Cognitive Foundations Of Systematic Musicology
Edited by Manfred Stahnke. Hamburg: von Bockel: — Bader, Rolf ed. Barbieri, Patrizio. Enharmonic Instruments and music Latina: Levante. Barker, Andrew. Greek Musical Writings. Volume 2: Harmonic and acoustic theory. The Science of Harmonics in classical Greece. Becker, Oskar. Zwei Untersuchungen zur antiken Logik. Wiesbaden: Harrassowitz. Benson, David J. Music: a mathematical offering. Beurmann, Andreas, and Albrecht Schneider. Edited by Constantin Floros et. Laaber: Laaber: — Birnbaum, Jens.
Der Apollontempel von Didyma. Analyse einer pythagoreisch-platonischen Entwurfskonstruktion. Bister, Heribert. Scala enigmatica armonizzata a 4 voci miste. Festschrift Martin Vogel zum Bad Honnef: G. Blackwood, Easley. The Structure of recognizable diatonic tunings. Princeton: Princeton University Press. Braun, Werner. Deutsche Musiktheorie des Edited by Frieder Zaminer. Darmstadt: Wissenschaftliche Buchgesellschaft. Bruhn, Herbert.
Harmonielehre als Grammatik der Musik. Propositionale Schemata in Musik und Sprache. Busch, Oliver. Logos Syntheseos. Cannon, John, and Sigalia Dostrovsky. The Evolution of dynamics: vibration theory from to New York: Springer. Capellen, Georg. Leipzig: Kahnt. Christensen, Thomas. Rameau and musical thought in the Enlightenment. Christensen, Thomas ed. Cambridge: Cambridge University Press. Clarke, Eric. Ways of listening. An ecological approach to the perception of musical meaning. Oxford: Oxford University Press. Cohen, H. Quantifying music. The Science of music at the first stage of the scientific revolution Cordes, Manfred.
Nicola Vicentinos Enharmonik. Graz: Akademische Druck- und Verlagsanstalt. Crombie, Alistair. Dahlhaus, Carl. Darmstadt: Wissenschaftliche Buchgesellschaft: 8— Dostrovsky, Sigalia, and John Cannon. Darmstadt: Wissenschaftliche Buchgesellschaft: 7— Die Harmonielehre des Klaudios Ptolemaios. Darstellung und Interpretation seines Denkens. Heidelberg: Winter.
Eberlein, Roland, and Jobst Fricke. Regionale maqam-Traditionen in Geschichte und Gegenwart 2 volumes. Berlin: no publisher. Erpf, Hermann. Form und Struktur in der Musik. Mainz: Schott. Falcon, Andrea. Aristotle and the science of nature. Unity without uniformity. Music and Mathematics. From Pythagoras to fractals. Fay, Thomas. Perceived hierarchic structure in language and music. Journal of Music theory — Federhofer, Hellmut.
Fend, Michael. Zarlino: Theorie des Tonsysems. Das erste und zweite Buch der Istitutione harmoniche. Fokker, Adriaan D. Rekenkundige Bespiegeling der Muziek. Gorinchem: Noorduijn. English Translation: New Music with 31 Notes. Frieler, Klaus. Mathematik und kognitive Melodieforschung. Hamburg: Dr. Galilei, Vicenzo. Dialogo della musica antica et della moderna.
Florence: G. Goldbach, Karl Traugott Mainz: Schott: — Handschin, Jacques. Der Toncharakter. Harburger, Walter. Die musikalische Logik: Geometrie der Empfindungen. Harrison, Daniel. Harmonic function in chromatic music: a renewned dualist theory and an account of its predecents. Chicago: University of Chicago Press. Hauptmann, Moritz. Die Natur der Harmonik und Metrik 2 nd edition Heinichen, Johann. Der Generalbass in der Komposition. Reprint Hildesheim: Olms Helmholtz, Hermann von. Lecture, Bonn Braunschweig: Vieweg. Hindemith, Paul. Unterweisung im Tonsatz I: Theoretischer Teil.
Neue, erweiterte Ausgabe. Huovinen, Erkki. Return to Book Page. This book presents a coherent state-of-the-art survey on the area of systematic and cognitive musicology which has enjoyed dynamic growth now for many years. It is devoted to exploring the relationships between acoustics, human information processing, and culture as well as to methodological issues raised by the widespread use of computers as a powerful tool for theory con This book presents a coherent state-of-the-art survey on the area of systematic and cognitive musicology which has enjoyed dynamic growth now for many years.
It is devoted to exploring the relationships between acoustics, human information processing, and culture as well as to methodological issues raised by the widespread use of computers as a powerful tool for theory construction, theory testing, and the manipulation of musical information or any kind of data manipulation related to music. Get A Copy. Paperback , pages. More Details Original Title. Other Editions 1. Friend Reviews. To see what your friends thought of this book, please sign up.
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Showing Rating details. More filters. Sort order. Jun 13, Jon rated it really liked it. The book contains 35 chapters on music psychology and computational analysis of the perception of music, concerning, e. This book is a pioneering work on the field of music information retrieval, with a much de The book contains 35 chapters on music psychology and computational analysis of the perception of music, concerning, e.
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