Rudolf Rasch

The Perception of Musical Tones

Publisher Summary This chapter provides a research in music perception, which explains how people respond subjectively to musical sound signals. In this respect, music perception is a part of psychophysics, the general denomination for scientific fields concerned with the relationship between the objective, physical properties of sensory stimuli in the environment and the sensations evoked. The chapter explains the phenomena that occur when several tones are presented simultaneously, which is what usually happens in music. This chapter is devoted to psychoacoustics. Psychoacoustics is an empirical or, experimental science. Observations from daily life and informal tryouts are starting points for psychoacoustical knowledge, but the core of the scientific content is the result of laboratory investigations. In this respect, psychoacoustics is an interdisciplinary field of research. Contributions are made both by experimental psychologists and by physicists and acousticians. A psychoacoustical experiment is described most simply in a stimulus response scheme. The procedures used most often in psychoacoustical experiments are choice methods and adjustment methods.

Description of regular twelve‐tone musical tunings

A framework is presented for the systematic description of musical tunings, based on the music-theoretical and on the frequency-ratio concept of musical intervals. All intervals are described as sums of octaves and fifths. Since the octave is always pure (2:1) knowledge of the sizes of the fifths is sufficient to determine the sizes of other intervals. In regular 12-tone tunings, 11 out of the 12 fifths are equal in size, either pure or tempered. Rules are given by which the size of the consonant intervals, and other intervals, can be derived from a single variable, the main fifth. Tunings are evaluated with the help of mean temperings of the consonant intervals in the major triads, mean temperings of all the triads in a tuning, and mean temperings of triads weighted with respect to a certain key. This is a theoretical study, which should be supplemented by observations from practice and perceptual experiments.

Perception of melodic and harmonic intonation of two-part musical fragments.

Short musical fragments consisting of a melody part and a synchronous bass part were mistuned in various ways and in various degrees. Mistuning was applied to the harmonic intervals between simultaneous tones in melody and bass (harmonic mistuning), which caused at the same time a mistuning of the melodic intervals between successive tones in the melody part (melodic mistuning of melody) and/or the bass part (melodic mistuning of bass part). The fragments were presented to musically trained subjects for judgments of the perceived quality of intonation. Results showed that the melodic mistuning of the melody parts had the largest disturbing effects on the perceived quality of intonation, followed closely by the harmonic mistuning. Melodic mistuning of the bass was less influential. It could be reasoned that the deviating interval size was probably of more importance in the perception of harmonic mistuning than the presence of beats.

Theory of Helmholtz-Beat Frequencies

In musical tuning, deviations from the simple frequency ratios of pure consonant intervals are often necessary. These deviations are called temperings. They result in beats in the sounding interval. Rules are developed according to which the beat frequencies can be determined, both exactly and by way of easy integer approximations. Beat frequencies of consonant intervals are most easily expressed as relative beat frequencies, the quotient of the beat frequency and the lower fundamental frequency of the interval. The relative beat frequency is a constant for a certain interval in a certain tuning, whereas the absolute beat frequencies vary with fundamental frequencies. Also described are the relationships between the beat frequencies of the three intervals that make up a consonant triad. Numerical data are given for five model tunings: Pythagorean, equal, Silbermann, meantone, and Salinas.

String Inharmonicity and Piano Tuning

Inharmonicity is a well-known property of stiff strings such as those used in the modern piano. Effects on piano tuning (e.g., the stretched octave) have been suggested but have never been fully investigated. We have measured the inharmonicities of the strings of a medium-sized grand piano. The measured inharmonicities are in excellent correspondence with the predictions by formula from the physical properties of the strings. Strings with higher frequencies usually have higher inharmonicity than strings with lower frequencies; this cancels out part of the effect of inharmonicity on beat frequency.

Farey systems of musical intonation

The distinction between consonant and dissonant intervals, represented in their just forms by frequency ratios expressible in low and high integer numbers respectively, is of prime importance in Western music. It is easy to tell just consonances from tempered ones on the basis of beat perception. Research by Vos & van Vianen (1985) with low‐number frequency‐ratio intervals has shown that beat detection can be predicted by the sum of the ratio numbers in the perfect (just) interval. We investigated the discrimination of perfect and imperfect (tempered) intervals with ratio numbers up to and including 16, thus mainly so‐called dissonances. Beat detection and discrimination are still possible, but are probably determined by other factors that simply the ratio‐numbers magnitude, such as the pattern of interferences in the sound stimulus. We investigated the practical possibilities of the high‐number ratio intervals by describing them as part of a Farey series of musical intervals. These investigations show th...

Relations between multiple divisions of the octave and the traditional tonal system

Abstract Multiple divisions of the octave are described as extensions of the traditional tonal system, by building them from chromatic and diatonic semitones with certain numbers of steps assigned to them. Except for some multiple division systems with relatively few tones per octave, this procedure always works. The number of circles of fifths in the system is equal to the largest common divisor of the numbers of steps contained in the two types of semitones. The description provides a consistent notation and a method for evaluating the qualities of the fifths and the major thirds in the systems. A list of systems with 5 up to 129 tones per octave is given, with summary references to instruments, compositions, and literature.

Eine neue Quelle zu Johann Jacob Frobergers Claviersuiten

Es geschieht nicht oft, dass nach mehr als einem Jahrhundert intensiver musikwissenschaftlichen Forschungen uber einen bedeutenden Komponisten noch eine wichtige, bisher unbekannte Quelle seiner Werke ans Licht kommt. Eines dieser seltenen Ereignisse ist die mit Strassburg, den 15. Marz 1675 datierte Handschrift, die vierzehn Claviersuiten von Johann Jacob Froberger (1616–1667) enthalt sowie sieben weitere Claviersuiten verschiedener Komponisten, welche zumeist Beziehungen mit Strassburg aufweisen. Die Sachsische Landesbibliothek — Staats- und Universitatsbibliothek, Dresden, hat die Neuentdeckung vor kurzem aus Privatbesitz erworben und bewahrt sie jetzt unter der Signatur 1-T-595 auf.

The Unification of Tonal Systems, or About the Circle of Fifths

During the past centuries a great many ways have been proposed to divide the octave into smaller units. The twelve semitones have been by far the most often proposed and used, but many systems with sometimes much more steps in the octave have been developed as well. There are basically two ways to construct these systems with multiple steps: either subdividing the octave into an arbitrary number of equal steps (the ensuing systems are often called multiple divisions), or building intervals by addition from basic intervals. The basic intervals then follow the order of prime numbers in their frequency ratios: 2:3 (perfect fifth), 4:5 (just major third), 4:7 (‘harmonic seventh’), etc. This method lies behind the just intonations. In this paper it will be shown that in both cases the various systems can be most efficiently described in terms of the circle of fifths. Methods are given to derive from the basic premises of the systems one or more circles of fifths, which encompass all the notes of the systems and relate all the notes to one another. Of course, the main property of a circle of fifths is the number of fifths comprised in it. In both classes there are certain systems which seem to be preferred both by theorists and by composers because in them the consonant intervals (perfect fifth, major third,...) deviate least from their just form (2:3,4:5,...). It is shown that these preferred systems have certain numbers of tones in their circles of fifths, such as 12, 19, 24, 31, 34, 41, 46, and 48, and that these preferred numbers of tones in the circles of fifth occur both regarding the multiple divisions and regarding the just intonations. It appears as well that the number of circles of fifth in a system with a given number of notes is usually the same, regardless whether the system is a multiple division or a just intonation. In this way the two classes, which seem quite apart from the point of view of their construction, are tied together by one single concept: the circle of fifths.

Perception of melodic and harmonic intonation of two‐part musical fragments

Short musical fragments consisting of a melody part and a bass part were mistuned in various ways and to various degrees. They were presented to a group of subjects for a judgment of the quality of intonation and for an identification of the mistuned part. Mistuning was applied to the melodic frequency intervals of the melody part and/or the bass part (melodic mistuning) as well as to the frequency intervals of simultaneous tones in melody and bass (harmonic mistuning). Results indicate that the melodic mistuning of the melody part has the largest influence on the perception of the intonation, followed by the harmonic mistuning, while the melodic mistuning of the bass is less important. When mistuning is present in both melody and bass part, mostly the melody is considered to be the mistuned part. When the sound‐pressure levels of the two parts are unequal, there is a tendency to consider the louder part as the mistuned part. In the perception of harmonic mistuning, the deviating interval size is probably...