Equal temperament

An equal temperament is a musical temperament or tuning system that approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, which gives an equal perceived step size, as pitch is perceived roughly as the logarithm of frequency. In classical music and Western music in general, the most common tuning system since the 18th century has been 12 equal temperament (also known as 12-tone equal temperament, 12-TET or 12-ET, informally abbreviated as 12 equal), which divides the octave into 12 parts, all of which are equal on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That resulting smallest interval, the width of an octave, is called a semitone or half step. In Western countries the term equal temperament, without qualification, generally means 12-TET. In modern times, 12-TET is usually tuned relative to a standard pitch of 440 Hz, called A440,

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On the Richter-Thomassen Conjecture about Pairwise Intersecting Closed Curves

János Pach, Gábor Tardos

A long-standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any pair of them intersect and no three curves pass through the same point, is at least (1 - o(1)) n(2). We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class touches every curve from the second class. (Two closed or open curves are said to be touching if they have precisely one point in common and at this point the two curves do not properly cross.) An important ingredient of our proofs is the following statement. Let S be a family of n open curves in R-2, so that each curve is the graph of a continuous real function defined on R, and no three of them pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is Omega(nt root logt/log log t).

Cambridge Univ Press2016

Ab initio calculations of mode selective tunneling dynamics in (CH3OH)-C-12 and (CH3OH)-C-13

Thomas Rizzo

A simplified formulation of the harmonic reaction path Hamiltonian (RPH) approach is used to calculate mode specific tunneling splittings and stereomutation times in (CH3OH)-C-12 and (CH3OH)-C-13. The experimental torsional spectrum is very well reproduced, as well as the few known isotope shifts. The mode specific changes in tunneling splitting are investigated for the excitation of fundamentals and OH stretching overtones. Good agreement between experiment and the RPH model is obtained, except for excitations of modes which are perturbed by anharmonic resonances. The inverted tunneling splittings (E level below A) experimentally observed for the fundamental transitions of the CH-stretching modes nu(2) and nu(9) and of the CH-rocking mode nu(11) are shown to result from a pure symmetry effect and not from a breakdown of vibrational adiabaticity. Introducing a proper geometrical phase factor but retaining the adiabatic separation of the torsional dynamics yields calculated values of Delta(ν) over tilde (2)=-3.6 cm(-1), Delta(ν) over tilde (9)=-3.2 cm(-1), and Delta(ν) over tilde (11)=-8.2 cm(-1) that are in satisfactory agreement with experimental data. Negative tunneling splittings are also predicted for the asymmetric CH-bending modes nu(4) and nu(10) and the CH3-rocking mode nu(7). A smooth decrease of the tunneling splitting is calculated for increasing OH stretching excitation [Delta(ν) over tilde(nu(1))=6.2 cm(-1),...,Delta(ν) over tilde (6nu(1))=1.5 cm(-1)] in quantitative agreement with experiment [Delta(ν) over tilde(nu(1))=6.3 cm(-1),...,Delta(ν) over tilde (6nu(1))=1.6 cm(-1)]. The effect is shown to result in about equal parts from the increase of the effective torsional barrier and the effective lengthening of the OH bond. (C) 2003 American Institute of Physics.

2003

Data Compression via Logic Synthesis

Luca Gaetano Amarù, Andreas Peter Burg, Giovanni De Micheli, Pierre-Emmanuel Julien Marc Gaillardon

Nowadays, most software and hardware applications are committed to reduce the footprint and resource usage of data. In this general context, lossless data compression is a beneficial technique that encodes information using fewer (or at most equal number of) bits as compared to the original representation. A traditional compression flow consists of two phases: data decorrelation and entropy encoding. Data decorrelation, also called entropy reduction, aims at reducing the autocorrelation of the input data stream to be compressed in order to enhance the efficiency of entropy encoding. Entropy encoding reduces the size of the previously decorrelated data by using techniques such as Huffman coding, arithmetic coding, and others. When the data decorrelation is optimal, entropy encoding produces the strongest lossless compression possible. While efficient solutions for entropy encoding exist, data decorrelation is still a challenging problem limiting ultimate lossless compression opportunities. In this paper, we use logic synthesis to remove redundancy in binary data aiming to unlock the full potential of lossless compression. Embedded in a complete lossless compression flow, our logic synthesis based methodology is capable to identify the underlying function correlating a data set. Experimental results on data sets deriving from different causal processes show that the proposed approach achieves the highest compression ratio compared to state-of-art compression tools such as ZIP, bzip2 and 7zip.

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In music theory, an interval is a difference in pitch between two sounds. An interval may be described as horizontal, linear, or melodic if it refers to successively sounding tones, such as two adj

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In music, a note is the representation of a musical sound. Notes can represent the pitch and duration of a sound in musical notation. A note can also represent a pitch class. Notes are the building

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Discrete mathematics is a discipline with applications to almost all areas of study. It provides a set of indispensable tools to computer science in particular. This course reviews (familiar) topics as diverse as mathematical reasoning, combinatorics, discrete structures & algorithmic thinking.

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