Music critics have compared Bach's music to the precision of mathematics. What "mathematics" and what "precision" are the questions for a curious scientist. The purpose of this short note is to suggest that the mathematics is, at least in part, Mandelbrot's fractal geometry and the precision is the deviation from a log-og linear plot.Music until the 17th century was one ofthe four mathematical disciplines of the quadrivium beside arithmetic, geometry, and astronomy. The cause of consonance, in terms of Aristotelian analysis, was stated to be numerous sonorus, or harmonic number. That the ratio 2:1 produces the octave, and 3:2 produces the fifth, was known since the time of Pythagoras. Numerologists of the Middle Ages speculated on the mythical significance of numbers in music. Vincenzo Galilei, father of Galileo, was the first to make an attempt to demythify the numerology of music (1). He pointed out that the octave can be obtained through different ratios of 2":1. It is 2:1 in terms of string length, 4:1 in terms of weights attached to the strings, which is inversely related to the cross-section of the string, and 8:1 in terms of volume of sound-producing bodies, such as organ pipes.Scientific experiments have revealed the relation between note-interval and vibrational frequency produced by an instrument. We obtain an octave-higher note by doubling the sound frequency, which can be achieved by halving the length of a string. There are 12 notes in an octave in our diatonic music; i.e., the frequency difference is divided by 12 equal intervals (i) so that f /f = (2.0)1/12= 1.05946 = (15.9/15). This relation is well known among musicians, that the ratio of acoustic frequencies between successive notes, f ' and ff, is approximately 16/15. The ratio of acoustic frequencies I between any two successive music notes of an interval i is Ii= 2'/12 = (15 9/15)i, [1] where i in an integer,-ranging from 1 to 12, in the diatonic music. A semitone is represented by i = 1, a tone by i = 2, a small third by i = 3, etc. The numerical value of Ii is approximately a ratio of integers. Some notes have a ratio of small integers. A fourth (i = 5), for example, has an I5 value of 1.3382, or a ratio of about 4/3; a fifth (i = 7) has a value of 1.5036, or a ratio of about 3/2. Those used to be considered consonant tones (1). Others are represented by a ratio of larger integers. A diminished fifth (i = 6), for example, has a ratio of 1.4185 (= 10/7.05). This is not a ratio of small integers; it is not even an accurate approximation of 10/7, and this note has been traditionally considered dissonant (1).Music can be defined as an ordered arrangement of single sounds of different frequency in succession (melody), of sounds in combination (harmony), and of sounds spaced in a temporal succession (rhythm). Melody is supposedly "a series of single notes deliberately arranged in a pattern and chosen from a preexisting series that has been handed down by tradition or is accepted as a convention." Theory of harmony has taught us that the ...

Suggestions have been made that computer musicians should attempt to compose fractal music, and questions have been raised whether there is such a thing as fractal music. Voss and Clark observed that music is scaling, or 1/f noise, as analyzed on the basis ofthe amplitude (loudness) ofthe audio signals; they failed to find a fractal distribution of acoustic frequencies (music notes) in music. Analyzing Bach's and Mozart's compositions, we have shown that the incidence of the frequency intervals, or of the changes of acoustic frequency, has a fractal geometry. Fractal phenomena are characterized by scale-independency. The purpose of this investigation is to demonstrate the self-similarity of music and to explore its implications.ua0e

Maintaining healthy, productive ecosystems in the face of pervasive and accelerating human impacts including climate change requires globally coordinated and sustained observations of marine biodiversity. Global coordination is predicated on an understanding of the scope and capacity of existing monitoring programs, and the extent to which they use standardized, interoperable practices for data management. Global coordination also requires identification of gaps in spatial and ecosystem coverage, and how these gaps correspond to management priorities and information needs. We undertook such an assessment by conducting an audit and gap analysis from global databases and structured surveys of experts. Of 371 survey respondents, 203 active, long-term (>5 years) observing programs systematically sampled marine life. These programs spanned about 7% of the ocean surface area, mostly concentrated in coastal regions of the United States, Canada, Europe, and Australia. Seagrasses, mangroves, hard corals, and macroalgae were sampled in 6% of the entire global coastal zone. Two-thirds of all observing programs offered accessible data, but methods and conditions for access were highly variable. Our assessment indicates that the global observing system is largely uncoordinated which results in a failure to deliver critical information required for informed decision-making such as, status and trends, for the conservation and sustainability of marine ecosystems and provision of ecosystem services. Based on our study, we suggest four key steps that can increase the sustainability, connectivity and spatial coverage of biological Essential Ocean Variables in the global ocean: (1) sustaining existing observing programs and encouraging coordination among these; (2) continuing to strive for data strategies that follow FAIR principles (findable, accessible, interoperable, and reusable); (3) utilizing existing ocean observing platforms and enhancing support to expand observing along coasts of developing countries, in deep ocean basins, and near the poles; and (4) targeting capacity building efforts. Following these suggestions could help create a coordinated marine biodiversity observing system enabling ecological forecasting and better planning for a sustainable use of ocean resources.