Dave Benson > Quotes

“A tempered scale is a scale in which adjustments are made to the Pythagorean or just scale in order to spread around the problem caused by wishing to regard two notes differing by various commas as the same note, as in the example of Section 5.8, Exercise 2, and the discussion in Section 5.11.”
― Dave Benson, Music: A Mathematical Offering

“To get good rational approximations, we stop just before a large value of an. So for example, stopping just before the 15, we obtain the well-known approximation π ≈ 22/7. Stopping just before the 292 gives us the extremely good approximation which was known to the Chinese mathematician Chao Jung-Tze (or Tsu Ch’ung-Chi, depending on how you transliterate the name) in 500 AD. The rational approximations obtained by truncating the continued fraction expansion of a number are called the convergents. So the convergents for π are”
― Dave Benson, Music: A Mathematical Offering

“provided that the original analogue signal has no frequency components at half the sample rate or above (this is achieved with a low pass filter), it may be reconstructed exactly from this sampled signal. This rather extraordinary statement is called the sampling theorem,”
― Dave Benson, Music: A Mathematical Offering

“The continued fraction expansion for the base of natural logarithms follows an easily described pattern, as was discovered by Leonhard Euler. The continued fraction expansion of the golden ratio is even easier to describe:”
― Dave Benson, Music: A Mathematical Offering

“the use of the first five harmonics as the starting point for the development of scales.”
― Dave Benson, Music: A Mathematical Offering

“Sophie Germain’s paper, ‘Recherches sur la théorie des surfaces élastiques’, written in 1815 and published in 1821, won her a prize of a kilogram of gold from the French Academy of Sciences in 1816. The paper contained some significant errors, but became the basis for work on the subject by Lagrange, Poisson, Kirchoff, Navier and others. Sophie Germain is probably better known for having made one of the first significant breakthroughs in the study of Fermat’s last theorem. She proved that if x, y and z are integers satisfying x5 + y5 = z5, then at least one of x, y and z has to be divisible by 5. More generally, she showed that the same was true when 5 is replaced by any prime p such that 2p + 1 is also a prime.”
― Dave Benson, Music: A Mathematical Offering

“Modern physics does not lead to any understanding of the design procedures that were used to produce this set of bells.”
― Dave Benson, Music: A Mathematical Offering

“Equal temperament is an essential ingredient in twentieth century twelve tone music, where combinatorics and chromaticism seem to supersede harmony.”
― Dave Benson, Music: A Mathematical Offering

“However, much of Beethoven’s piano music is best played with an irregular temperament (see Section 5.13), and Chopin was reluctant to compose in certain keys (notably D minor) because their characteristics did not suit him.”
― Dave Benson, Music: A Mathematical Offering

“notes are played with a frequency ratio of 3:2, then the third partial of the lower note will coincide with the second partial of the upper note, and the notes will have a number of higher partials in common. If, on the other hand, the ratio is slightly different from 3:2, then there will be a sensation of roughness between the third partial of the lower note and the second partial of the upper note, and the notes will sound dissonant.”
― Dave Benson, Music: A Mathematical Offering

“The binary numbers in a WAV file are always little endian, which means that the least significant byte comes first,”
― Dave Benson, Music: A Mathematical Offering

“strings’. These say that the frequency of a stretched string is inversely proportional to its length, directly proportional to the square root of its tension, and inversely proportional to the square root of the linear density.”
― Dave Benson, Music: A Mathematical Offering

“The particular choice of function is somewhat arbitrary, because of a lack of precision in the data as well as in the subjective definition of dissonance. The main point is to mimic the visible features of the graph.”
― Dave Benson, Music: A Mathematical Offering

“There is one warning that must be stressed at this stage. Namely, it does not make sense to multiply distributions. So, for example, the square of the delta function does not make sense as a distribution. After all, what would be? It would have to be δ(0)g(0), which isn’t a number!”
― Dave Benson, Music: A Mathematical Offering

“In other words, as long as θ(x) never gets large, the motion of the string is essentially determined by the wave equation (3.2.2) where D’Alembert2 discovered a strikingly simple method for finding the general solution to equation (3.2.2). Roughly speaking, his idea is to factorize the differential”
― Dave Benson, Music: A Mathematical Offering

“So what kind of a function is δ(t)? The answer is that it really isn’t a function at all, it’s a distribution, sometimes also called a generalized function. A distribution is only defined in terms of what happens when we multiply by a function and integrate. Whenever a delta function appears, there is an implicit integration lurking in the background.”
― Dave Benson, Music: A Mathematical Offering

“The earliest known advocates of the 5:4 ratio as a consonant interval are the Englishmen Theinred of Dover (twelfth century) and Walter Odington (fl. 1298–1316),26 in the context of early English polyphonic music. One of the earliest recorded uses of the major third in harmony is the four part vocal canon sumer is icumen in, of English origin, dating from around 1250. But for keyboard music, the question of tuning delayed its acceptance.”
― Dave Benson, Music: A Mathematical Offering

“Adding musical intervals corresponds to multiplying frequency ratios. So, for example, if an interval of an octave corresponds to a ratio of 2:1 then an interval of two octaves corresponds to a ratio of 4:1, three octaves to 8:1, and so on. In other words, our perception of musical distance between two notes is logarithmic in frequency, as logarithms turn products into sums.”
― Dave Benson, Music: A Mathematical Offering

“The theory of Fourier series shows that such a sound can be decomposed as a sum of sine waves with various phases, at integer multiples of the frequency ν, as in Bernoulli’s solution (3.2.7) to the wave equation. The component of the sound with frequency ν is called the fundamental. The component with frequency mν is called the mth harmonic, or the (m – 1)st overtone.”
― Dave Benson, Music: A Mathematical Offering

“For example, when tones of 400 Hz and 800 Hz are presented to the two ears with opposite phase, about 99% of subjects experience the lower tone in one ear and the higher tone in the other ear. When the headphones are reversed, the lower tone stays in the same ear as before. See her 1974 article in Nature for further details.”
― Dave Benson, Music: A Mathematical Offering

“The dominant Western tuning system – equal temperament – is merely a 200 year old compromise that made it easier to build mechanical keyboards.”
― Dave Benson, Music: A Mathematical Offering

“Keyboard instrument tuners tended to colour the temperament, so that different keys had slightly different sounds to them.”
― Dave Benson, Music: A Mathematical Offering

“Every real number has a unique continued fraction expansion, and it stops precisely when the number is rational.”
― Dave Benson, Music: A Mathematical Offering

“This allows us, for example, to make sense of the plucked string, where the initial displacement is continuous, but not even once differentiable.”
― Dave Benson, Music: A Mathematical Offering

“For irrational numbers the continued fraction expansion is unique.”
― Dave Benson, Music: A Mathematical Offering

“So the Pythagorean system does not so much have a circle of fifths, more a sort of spiral of fifths as in Figure 5.4.”
― Dave Benson, Music: A Mathematical Offering

“By the mid- to late fifteenth century, especially in Italy, many aspects of the arts were reaching a new level of technical and mathematical precision. Leonardo da Vinci was integrating the visual arts with the sciences in revolutionary ways.”
― Dave Benson, Music: A Mathematical Offering

“In this and other irregular temperaments, different key signatures have different characteristic sounds, with some keys sounding direct and others more remote. This may account for the modern myth that the same holds in equal temperament.17”
― Dave Benson, Music: A Mathematical Offering

“It is probably the high common harmonic which causes us to associate minor chords with sadness.”
― Dave Benson, Music: A Mathematical Offering

“We saw in the last chapter that for notes played on conventional instruments, where partials occur at integer multiples of the fundamental frequency, intervals corresponding to frequency ratios expressable as a ratio of small integers are favoured as consonant.”
― Dave Benson, Music: A Mathematical Offering