Normally, treaties contain express provision on duration and termination. These can take a variety of forms, including indefinite duration with a right to terminate, or a conditional right to terminate. Various types of clauses are examined, together with the situation where a treaty contains no provision for termination or withdrawal. A treaty may also be terminated by consent or by conclusion of a later treaty. The chapter examines the relationship between treaty provisions and countermeasures, and analyses the right of one or more parties to terminate or suspend a treaty for material breach. It also examines other grounds for termination, including supervening impossibility of performance and fundamental change of circumstances (rebus sic stantibus), the procedure for termination, and special circumstances such as the severance of diplomatic relations or outbreak of hostilities.

Frequency, as events per second measured in hertz (Hz), is introduced as one type of rate that is important for music. Tones associated with music are typically found to correspond to a few hundred events per second. Other rates are considered as examples. Two types of rates are distinguished: additive and multiplicative. For an additive rate, a quantity is added for each interval. A multiplicative rate involves a multiplicative change, such as a percentage change, for each interval. Multiplicative rates lead to exponentially increasing and decreasing behavior. Exponential behavior is often described using logarithms. It is found that the frequency of tones is an additive rate, but the change in the frequency going up and down the keyboard is multiplicative. In particular, octaves are a factor of 2 in frequency.

The speed of sound in gases is derived in a manner similar to what was done for the speed of waves on a string. The mass of the gas molecules provides the inertial contribution. The gas laws, along with approximation techniques, are used to estimate the return force. The appropriate process is adiabatic, and so the speed of sound depends on the adiabatic constant. The adiabatic constant, in turn, depends largely on the shape of the molecules. Excellent agreement is found between the rather simple theory and measured results. The speed of sound in a gas is found to depend on the (average) mass of the molecules, temperature, and the adiabatic constant (the shape of the molecules); however, there is very little pressure dependence when the pressure is near 1 atmosphere.

Newton’s laws and consideration of units are used to present and discuss the mass on a spring as an example of a harmonic oscillator, a mechanical oscillator with a sinusoidal time dependence. Both the transient solution, where the oscillator is started away from equilibrium and left on its own, and the driven solutions, where a sinusoidal driving force is applied, are presented. The quality factor, Q, is introduced, which characterizes the relative amount of damping forces present, such as those of friction or air drag. The quality factor is related to the number of oscillations that are made when left on its own and to the excitation bandwidth, the range of frequencies over which resonance is observed. Musically relevant examples include the ocarina, tuning fork, some speaker enclosures, and the phenomenon of sympathetic resonance.

How electronics can be used to alter and/or create musical signals is considered. Emphasis is on those techniques where physics is evident, recognizing that much is now done using programming languages. Basic effects done using simple physics include distortion, reverberation, and tremolo. Various signal-processing and playback techniques are also used, along with psychoacoustics, to provide surround sound and other effects. Electronic music can be created using the Hammond organ, a theremin, and electronic synthesizers. A basic synthesizer will combine a carrier tone and a time-dependent amplitude for that tone in order to create different sounds. Although synthesizers are often controlled by keyboards, other user interfaces have also been developed that, for example, resemble a drum kit or a woodwind instrument.

Non-legally binding instruments do not constitute treaties but they are widely used by states and international organisations as a form of political commitment. They have various titles, the most common of which is memorandum of understanding, although some states use this title also for treaties with a certain subject matter. In the multilateral context, non-binding instruments are often used for plans of action, guidelines and other forms of standard setting. To ensure that an instrument is non-legally binding, care must be taken in the drafting to evidence that intention, including use of the right terminology. An examination of state practice reveals broadly similar principles. Why do states and international organisations use non-legally binding instruments in preference to treaties? Reasons include confidentiality, lack of formality, and ease of amendment and termination. But there are risks, including possible lack of respect for commitments and lack of care in drafting. Non-legally binding instruments, both bilateral and multilateral instruments, may in some contexts be regarded as a form of soft law. They may also exceptionally give rise to estoppel. Non-legally binding instruments may also be evidence of a state’s decision or policy in the context of judicial review by domestic courts.

This chapter explains the role played by reservations, the application of the rules of the Vienna Convention on the Law of Treaties, and practice in dealing with reservations. It defines reservations and distinguishes them from interpretative declarations and political declarations. It examines whether reservations are permissible or impermissible, including the test whether a reservation is incompatible with the object and purpose of the treaty. It also discusses the rules governing acceptance and objection, the legal effect of reservations, the question whether a prohibited reservation can be severed, and special cases such as human rights treaties, reservations relating to internal law, vague or general reservations, reference to Islamic law, late reservations, and the role of treaty monitoring bodies. It analyses the procedure for making or objecting to reservations or withdrawing them, and the functions of the depositary in relation to reservations.

The harmonic vibrational frequencies of an ideal string under tension, fixed at the ends, are developed, starting with the results from the harmonic oscillator. The corresponding resonances result in sinusoidal standing waves. The end conditions restrict the possible solutions to be those with an integer number of half wavelengths between the ends of the string. A general vibration of the string, such as from a pluck, can be described as the sum of sinusoidal solutions. More general end conditions include a free end, in which case there is an odd number of quarter wavelengths between the ends.

When a treaty prescribes how disputes can be settled, it usually provides for a preliminary phase involving direct negotiations or consultations. In the case of bilateral agreements, more binding mechanisms tend to be found in agreements with more specific obligations, or those directly affecting individual interests, such as trade. Otherwise, bilateral agreements will often specify that disputes are to be resolved by the diplomatic channel or other avenue of voluntary settlement. Both bilateral and multilateral treaties may provide for conciliation or mediation. The parties may, however, wish to resort to compulsory binding settlement. The two principal features are a prior agreement to submit disputes to a third party, and a provision that the decision of the third party is legally binding on the parties in dispute. Procedures of arbitration and judicial settlement, including the International Court of Justice, are analysed. In respect of remedies, the chapter examines countermeasures and the principles underlying them, sacrosanct obligations and necessity.