Music Quotes

“The entropy of a signal is defined to be the logarithm to base two of the number of different possibilities for the signal. The less random the signal, the fewer possibilities are allowed for the data in the signal, and hence the smaller the entropy. The entropy measures the smallest number of binary bits the signal could be compressed into.”
― 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

“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

“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

“For irrational numbers the continued fraction expansion is unique.”
― 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

“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

“In the twentieth century, the dominance of chromaticism and the advent of twelve tone music have pretty much forced the abandonment of unequal temperaments, and piano tuning practice has reflected this.”
― 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

“Keyboard instrument tuners tended to colour the temperament, so that different keys had slightly different sounds to them.”
― 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

“But when the French imitated the sound of the parallel chords, they use the top line rather than the middle line as the melody, giving what is referred to as Faux Bourdon.”
― 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

“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

“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

“We begin at the end. Most music in the western world imparts a sense of finality through the sequence V–I, or variations of it (V7–I, vii0–I).13 It is not fully understood why V–I imparts such a feeling of finality, but it cannot be denied that it does.”
― Dave Benson, Music: A Mathematical Offering

“Euler’s Monochord (Leonhard Euler, Tentamen novæ theoriæ musicæ, St. Petersburg, 1739)”
― Dave Benson, Music: A Mathematical Offering

“the Pythagorean comma is defined to be the difference between twelve perfect fifths and seven octaves,”
― 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

“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

“To convert from a frequency ratio of r:1 to cents, the value in cents is To convert an interval of n cents to a frequency ratio, the formula is”
― 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

“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

“the use of the first five harmonics as the starting point for the development of scales.”
― 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

“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

“as light sources usually don’t contain harmonics.”
― 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

“So if some part of the auditory system is behaving in a nonlinear fashion, a quadratic nonlinearity would correspond to the perception of doubles of the incoming frequencies, which are probably not noticed because they look like overtones, as well as sum and difference tones corresponding to the terms cos(m + n)t and cos(m – n)t.”
― Dave Benson, Music: A Mathematical Offering