On the Mathematics of Music: From Chords to Fourier Analysis

Mathematics is a far reaching discipline and its tools appear in many applications. In this paper we discuss its role in music and signal processing by revisiting the use of mathematics in algorithms that can extract chord information from recorded music. We begin with a light introduction to the theory of music and motivate the use of Fourier analysis in audio processing. We introduce the discrete and continuous Fourier transforms and investigate their use in extracting important information from audio data

Applications of Fourier Analysis to Audio Signal Processing: An Investigation of Chord Detection Algorithms

The discrete Fourier transform has become an essential tool in the analysis of digital signals. Applications have become widespread since the discovery of the Fast Fourier Transform and the rise of personal computers. The field of digital signal processing is an exciting intersection of mathematics, statistics, and electrical engineering. In this study we aim to gain understanding of the mathematics behind algorithms that can extract chord information from recorded music. We investigate basic music theory, introduce and derive the discrete Fourier transform, and apply Fourier analysis to audio files to extract spectral data

Fundamental Limits of Detection in the Near and Mid Infrared

The construction of the James Webb Space Telescope has brought attention to infrared astronomy and cosmology. The potential information about our universe to be gained by this mission and future infrared telescopes is staggering, but infrared observation faces many obstacles. These telescopes face large amounts of noise by many phenomena, from emission off of the mirrors to the cosmic infrared background. Infrared telescopes need to be designed in such a way that noise is minimized to achieve sufficient signal to noise ratio on high redshift objects. We will investigate current and planned space and ground based telescopes, model the noise they encounter, and discover their limitations. The ultimate goal of our investigation is to compare the sensitivity of these missions in the near and mid IR and to propose new missions. Our investigation is broken down into four major sections: current missions, noise, signal, and proposed missions. In the proposed missions section we investigate historical and current infrared telescopes with attention given to their location and properties. The noise section discusses the noise that an infrared telescope will encounter and set the background limit. The signal section will look at the spectral energy distributions (SED) of a few significant objects in our universe. We will calculate the intensity of the objects at various points on Earth and in orbit. In the final section we use our findings in the signal and noise sections to model integration times (observation time) for a variety of missions to achieve a given signal to noise ratio (SNR)

An Introduction to Fourier Analysis with Applications to Music

In our modern world, we are often faced with problems in which a traditionally analog signal is discretized to enable computer analysis. A fundamental tool used by mathematicians, engineers, and scientists in this context is the discrete Fourier transform (DFT), which allows us to analyze individual frequency components of digital signals. In this paper we develop the discrete Fourier transform from basic calculus, providing the reader with the setup to understand how the DFT can be used to analyze a musical signal for chord structure. By investigating the DFT alongside an application in music processing, we gain an appreciation for the mathematics utilized in digital signal processing

Recommended temperature metrics for carbon budget estimates, model evaluation and climate policy

Recent estimates of the amount of carbon dioxide that can still be emitted while achieving the Paris Agreement temperature goals are larger than previously thought. One potential reason for these larger estimates may be the different temperature metrics used to estimate the observed global mean warming for the historical period, as they affect the size of the remaining carbon budget. Here we explain the reasons behind these remaining carbon budget increases, and discuss how methodological choices of the global mean temperature metric and the reference period influence estimates of the remaining carbon budget. We argue that the choice of the temperature metric should depend on the domain of application. For scientific estimates of total or remaining carbon budgets, globally averaged surface air temperature estimates should be used consistently for the past and the future. However, when used to inform the achievement of the Paris Agreement goal, a temperature metric consistent with the science that was underlying and directly informed the Paris Agreement should be applied. The resulting remaining carbon budgets should be calculated using the appropriate metric or adjusted to reflect these differences among temperature metrics. Transparency and understanding of the implications of such choices are crucial to providing useful information that can bridge the science–policy gap

Uncertainty and Predictability of Seasonal-to-Centennial Climate Variability

The work presented in this dissertation is driven by three fundamental questions in climate science: (1) What is the natural variability of our climate system? (2) What components of this variability are predictable? (3) How does climate change affect variability and predictability? Determining the variability and predictability of the chaotic and nonlinear climate system is an inherently challenging problem. Climate scientists face the additional complications from limited and error-filled observational data of the true climate system and imperfect dynamical climate models used to simulate the climate system. This dissertation contains four chapters, each of which explores at least one of the three fundamental questions by providing novel approaches to address the complications. Chapter 1 examines the uncertainty in the observational record. As surface temperature data is among the highest quality historical records of the Earth’s climate, it is a critical source of information about the natural variability and forced response of the climate system. However, there is still uncertainty in global and regional mean temperature series due to limited and inaccurate measurements. This chapter provides an assessment of the global and regional uncertainty in temperature from 1880-present in the NASA Goddard Institute for Space Studies (GISS) Surface Temperature Analysis (GISTEMP). Chapter 2 extends the work of Chapter 1 to the regional spatial scale and monthly time scale. An observational uncertainty ensemble of historical global surface temperature is provided for easy use in future studies. Two applications of this uncertainty ensemble are discussed. First, an analysis of recent global and Arctic warming shows that the Arctic is warming four times faster than the rest of the global, updating the oft-provided statistic that Arctic warming is double that of the global rate. Second, the regional uncertainty product is used to provide uncertainty on country-level temperature change estimates from 1950-present. Chapter 3 investigates the impacts of the El Niño-Southern Oscillation (ENSO) on seasonal precipitation globally. In this study, novel methodology is developed to detect ENSO-precipitation teleconnections while accounting for missing data in the CRU TS historical precipitation dataset. In addition, the predictability of seasonal precipitation is assessed through simple empirical forecasts derived from the historical impacts. These simple forecasts provide significant skill over climatological forecasts for much of the globe, suggesting accurate predictions of ENSO immediately provide skillful forecasts of precipitation for many regions. Chapter 4 explores the role of initialization shock in long-lead ENSO forecasts. Initialized predictions from the CMIP6 decadal prediction project and uninitialized predictions using an analogue prediction method are compared to assess the role of model biases in climatology and variability on long-lead ENSO predictability. Comparable probabilistic skill is found in the first year between the model-analogs and the initialized dynamical forecasts, but the initialized dynamical forecasts generally show higher skill. The presence of skill in the initialized dynamical forecasts in spite of large initialization shocks suggest that initialization of the subsurface ocean may be a key component of multi-year ENSO skill. Chapter 5 brings together ideas from the previous chapters through an attribution of historical temperature variability to various anthropogenic and natural sources of variability. The radiative forcing due to greenhouse gas emissions is necessary to explain the observed variability in temperature nearly everywhere on the land surface. Regional fingerprints of anthropogenic aerosols are detected as well as the impact of major sources of natural variability such as ENSO and Atlantic Multidecadal Variability (AMV)

Scaling up phenotypic plasticity with hierarchical population models

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