Temperate Intervals Research Articles

Purity Ratings of Tempered Fifths and Major Thirds

In this study, the relationship between the degree of tempering of musical intervals and the subjective purity of these intervals was investigated. Subjective purity was determined for fifths and major thirds. The intervals were presented in isolation, that is, they were not given in a musical context. For two simultaneous complex tones the relationship between subjective purity and tempering could be described by exponential functions. These functions were obtained both for ratings on a 10-point equal-interval scale and for subjective distances derived from preference data collected by means of the method of paired comparisons. To verify to what extent subjective purity had been determined by interference of nearby harmonics, the spectral content of the tones was varied. For both the fifths and the major thirds, interference of the various pairs of nearly coinciding harmonics was canceled by deletion of the even harmonics of the higher tone. This deletion resulted in higher purity ratings, the effect being most prominent for the major third. A further reduction of the potential interference between harmonics was still more effective: for simultaneous sinusoidal tones, subjective differences between pure and tempered intervals were much smaller than for complex tones. Purity ratings for simultaneous sinusoids presented at a low sound level were about equal to the ratings for successive tones. The purity ratings were compared with dissonance patterns predicted by models for tonal consonance/dissonance. Only in a few conditions do the patterns predicted by the model of Plomp and Levelt resemble the rating patterns obtained, and the dissonance patterns predicted by the model of Kameoka and Kuriyagawa are at variance with the purity ratings in all conditions. Suggestions for revision of the models are given.

Thresholds for discrimination between pure and tempered intervals: the relevance of nearly coinciding harmonics.

Thresholds for discrimination between pure and tempered musical intervals consisting of simultaneous complex tones (fundamental frequencies f1 and f2) were investigated. For these tones the main clue for the discrimination of pure intervals (f1:f2 = p:q; p and q small integers) from moderately tempered intervals (f1:f2 approximately p:q) is absence versus presence of beats. The strength of the beats (level difference between envelope maximum and minimum or level-variation depth D) was manipulated by introduction of differences in level (delta L) between the two tones. In each of three experiments the discrimination thresholds (DTs) were determined for 13 intervals with different values for p and/or q. Experiment 1 showed that there is a simple relation between frequency-ratio complexity and discriminability: DTs gradually increased (smaller values of delta L) with increasing p + q. Experiment 2, in which tones with harmonics of equal amplitude were used, indicated that level of the interfering harmonics was not responsible for the relation between DT and p + q. Yet, Experiment 3, in which the spectral content of the tones was varied, clearly showed that for all intervals DT had been determined by the interference between nearly coinciding harmonics. Detailed analysis of the results revealed that the relation between DT and ratio complexity might have been the result of masking.

Spectral effects in the perception of pure and tempered intervals: Discrimination and beats

Thresholds for discrimination between pure and tempered musical intervals consisting of simultaneous complex tones have been investigated previously (Vos. 1982). The aim of the present research was to answer three questions: (1) To what extent are these thresholds determined by the interference of just-noncoinciding harmonics? (2) Is the beat frequency of the first pair of these adjacent harmonics equal to the perceived beat frequency of the tempered intervals? (3) Are differences in discriminability at threshold representative of differences between the perceived strength of beats in supraliminal conditions? In the first three experiments, the answer to the first question was sought by investigating the effect of spectral content of the tones on the (Uscrimination thresholds (DTs) for the fifth (Experiment 1) and the major third (Experiments 2 and 3). The results showed that, for moderately tempered fifths, DTs are determined mainly by the interference of the first pair of adjacent harmonics. For major thirds, however, the data suggest that the interference of other harmonics plays a role as well. In Experiment 4, attempts were made to answer the second and third questions: In supraliminal conditions, both the dominantly perceived beat frequency and the perceived beat strength were determined for various spectral conditions and degrees of tempering. For both the fifth and the major third and for all spectral conditions, the dominantly perceived beat frequency was in most cases equal to the frequency difference of the first pair of adjacent harmonics. Comparison of the DTs and the perceived strength of the beats for corresponding spectral conditions revealed that, especially for the fifth, perceived strength of beats and sensitivity to moderately tempered intervals are highly correlated.

Abrupt end of the last interglacial s.s. in north-east France

Close study of past interglacials might indicate how and when the present interglacial will end and whether the limit is heading towards a warming or a cooling. No certain prediction has been possible because of man's interference with the environment. But it is reported that when exploring the records of past temperate intervals frequent signs of abrupt changes of local environment were observed. In particular, abrupt shifts in forest composition end each Pleistocene interglacial whose record was studied in detail. In Grand Pile (north-east France), the Eemian (s.s.) (oxygen isotope substage 5e) temperate forest was replaced by a pine-spruce-birch taiga within ~150+/-75 yr. These results are based on rich pollen content of continuously deposited laminated gyttja and on the assumption of a constant sedimentation rate during the last 11000-yr long interglacial.

Grande Pile peat bog: A continuous pollen record for the last 140,000 years

In the Southern Vosges Mountains, Northeastern France, the Grande Pile peat bog (47° 44′ N, 6°30′14″ E, 330-m altitude, about 20 m deep) gives a continuous pollen sequence for the last 140,000 years, contrary to others in Northwestern and Central Europe which are all truncated. For the first time, in a region close to the type locatity for the Eem deposits and close to the Würm and Riss stratotypes, palynology demonstrates a complete “glacial-interglacial cycle” offering the possibility of studying the rapid degradation of vegetation at the end of the Last Interglacial, perhaps in sufficient detail to be useful soon in long-term global climate forecasting. The Grande Pile pollen sequence shows, between the classical Eemian Interglacial and the Last (Würm) Glaciation, two temperate intervals interpreted as interglacials (palynological definition): St. Germain I and St. Germain II. These are separated by two very cold phases, probably glacial: Melisey I and Melisey II. This sequence, not easily correlated with the classical European chronology of Woldstedt, agrees well with Frenzel's chronology and, therefore, makes the synchrony of the Alpine glaciations with those of Northern Germany questionable. An attempt is made to correlate the Grande Pile pollen sequence with other chronologies (e.g., deep sea curves based on foraminiferal fauna, oxygen isotopes, and carbonate content, Barbados sea levels, Rocky Mountains sequence) that span the period between 140,000 and 70,000 years BP.

A New Integral Ratio Chromatic Scale

The IRC scale presented herein is neither a Pythagorean scale, nor a just scale, nor an equally tempered scale, as those terms are defined in Acoustical Terminology, but it is shown to possess the major advantages of all three. In a D-D octave, all of the notes are interconnected by a perfectly symmetrical arrangement of just fourths, and the note frequencies of the octave containing C4=256 cps are 144, 153, 162, 171, 180, 192, 204, 216, 228, 240, 256, 272, 288. A distinguishing characteristic of the IRC scale is that four ascending octaves commencing with a C embrace exactly four successive just fifths (C-G-D-A-E) immediately followed by four successive just fourths (E-A-D-G-C). The IRC scale, like the conventional just scale, is based on alternately conjunct and disjunct just tetrachords, but the IRC tetrachord intervals are the smoothly decreasing ones of 9:8, 10:9, 16:15, instead of the just scale irregularly arranged intervals of 16:15, 9:8, 10:9. The major and minor tones of the IRC scale are divided by the known superparticular proportion, giving, in addition to the just semitone 16:15, semitone ratios of 17:16, 18:17, 19:18, and 20:19. These five semitones are located in smoothly progressing order, forming three perfect harmonic series embracing a just major third, fourth, and minor third. The IRC scale is shown to be superior to the just and Pythagorean scales and to excel the equally tempered scale in modulation. If, as the author believes, Helmholtz was correct in his thesis that just intervals are musically better than equal tempered intervals, then the IRC scale is also superior to the equally tempered scale, possessing, among many other advantages, greater ease of tuning pianos and organs, and the ability to tune pianos, violins, and cellos exactly alike and play them in the same temperament.

MINIMIZING DISCREPANCIES OF INTONATION IN VALVE INSTRUMENTS

If the crooks for the first, second, and third valves of a brass instrument are made of the proper lengths to produce respectively a tempered whole tone, a tempered half tone, and a tempered tone and a half, then the notes produced by the simultaneous use of any two or more valves are sharper than the corresponding tempered intervals. The problem of tempering the scale of a valve instrument thus becomes a matter of making each valve crook too long for tempered intervals when the valves are used singly, so that the sharpness of notes produced by the combined use of valves shall be reduced as much as possible. The precise lengths of the several valve crooks for minimizing discrepancies of intonation in a valve instrument appear to be unknown both to writers on acoustics and to makers of valve instruments. The paper presents the method of minimizing such discrepancies.

Minimizing Discrepancies of Intonation in Valve Instruments

If the crooks for the first, second, and third valves of a brass instrument are made of the proper lengths to produce respectively a tempered whole tone, a tempered half tone, and a tempered tone and a half, then the notes produced by the simultaneous use of any two or more valves are sharper than the corresponding tempered intervals. The problem of tempering the scale of a valve instrument thus becomes a matter of making each valve crook too long for tempered intervals when the valves are used singly, so that the sharpness of notes produced by the combined use of valves shall be reduced as much as possible. The precise lengths of the several valve crooks for minimizing discrepancies of intonation in a valve instrument appear to be unknown both to writers on acoustics and to makers of valve instruments. The paper presents the method of minimizing such discrepancies.