What Pleases Our Ears
What number is halfway between 1 and 9?
You are very likely to give the answer of "five". But if you ever come across a child or a farmer, that has never dealt with math, giving the answer of "three" for the same question; you would better not underestimate his/her mathematical capabilities. It might be because their sense of the nature depends upon a better intuitional basis than yours.
The logic in seeking the answer to that question, if you have given the answer of "five", was probably something like "Five is 4 more than one, and nine is 4 more than five. Hence five must be the halfway between one and nine.".
It is no mistake to expect people's understanding of numbers to be linear, for they are taught of numbers as evenly spaced points on a scale, from the early years of education.
But since it is not easy to manipulate our senses regarding frequencies of sounds, using visual diagrams as we do for numbers; our perception of frequencies of sounds is logarithmic, rather than linear, just like how an uneducated one thinks of numbers. Therefore the relation between, say, 100 hz and 200 hz is the same as that between 200 hz and 400 hz, for what matters to our ears is not the additive difference, but the ratio.
Our aesthetic perception of sounds and the art of arranging sounds in such a way that they satisfy this perception, that is music I've just defined in a fancy way, are widely influenced by our perception of frequencies.
Most of those so-called rules taught in musical composition courses are nothing but statistical data regarding remarkable works of great composers. Yet there are a small but significant number of rules that do not vary too much from person to person or culture to culture. These, let us call, universal rules, are formed around our objective perceptions regarding sound. The simple fact that people agree on the role of frequency ratios over the relation between tones is what enables musicians to define universals of music theory.
The relation between two tones is called an interval. Since some combinations of tones sound sort of pleasant, whereas some do not; we define a hierarchy among intervals. This hierarchy of intervals can be imagined as a scale whose one end is consonance, and the other end is dissonance. The more pleasant it sounds, the more consonant the interval is. The more unpleasant it sounds, the more dissonant it is.
Since Pythagoras, there has been attempts to define a certain hierarchy among the intervals regarding frequencies of tones of which the interval is composed. Pythagoras' conjecture was that the simpler the ratio of frequencies of the tones is, the more consonant the interval would be. If one played two tones with frequencies 200 hz and 300hz, which have 2/3 ratio, that would sound much more satisfactory and stable than the two tones with frequencies 315 hz and 300 hz which have 15/16 ratio.
To Pythagoras, the most consonant interval is the one with the frequency ratio of 1, which is called unison. It is followed by octave with 1/2, perfect fifth with 2/3, perfect fourth with 4/3, major sixth with 5/3, major third with 5/4 and so on. (there is no consensus on the order of the rest of the list.)
Although Pythagoras' theory was simple enough for us to acquire an intuitional insight on intervals and to build intervals and musical scales upon, it was not sufficient enough to explain the whole mystery. How are we supposed to order intervals when the order of simplicity is ambiguous? What about the ratios that can be reduced to a simpler one? What does it have to do with the concept of "ratio" after all, since, say, 1/2 and 2/1 correspond to the same interval?
It was Leonhard Euler (1707-1783) who was bothered by the same sort of problems and tried to figure out the underlying mathematical basis behind consonance and frequencies, with his well known mathematical intellect.
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| Leonhard Euler |
Euler got started by observing the order of intervals in terms of consonance in the common practice and tried to detect a pattern. First, he ranked some of simple intervals in terms of consonance by his ear so that unison with ratio of 1 has the highest rank, followed by the interval with 1/2 ratio, and 1/3, and so on. If that pattern was to be followed, any such interval that has the frequency ratio 1/n would have an order of n.
But according to this pattern 1/5 would be more agreeable than 1/6, which is in contradiction with the practice. Hence he decided that intervals with 1/p ratio, where p is a prime number, should have an order of p.
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*number "1" had been treated as a prime number untill 1935. |
According to him, only prime numbers mattered. Thus, the consonance of the ratios of 1/(p1*p2) type would be in positive correlation with sum of those primes. It also did not matter whether p's are in the denominator or in the nominator, thus p1/p2, p2/p1, (p1*p2), 1/(p1*p2) would all have the same degree. He simply treated the frequencies of the tones as pairs, unlike the Pythagorean ratio approach.
At this stage it is still contradictory with our 1/5 and 1/6 instance since they have the same rank. What is differing for 1/6 is that it has two prime factors, namely, 2 and 3, on the denominator whereas 1/5 has just one. Thus he concluded that the consonance would also be in negative correlation with the number of prime factors.
What remained was to formulise those basic assumptions in such a way that D, the degree of agreeableness (the lower the more consonant), would be dependent upon S, sum of prime factors, positively, and upon N, number of the primes, negatively.
The simplest equation for this relation that might happen to come to anyone's mind is D = S - N and Euler actually went for it, except he added a "+1" to the left hand side so that it meets his first assumptions.
This is Euler's Gradus Function and it also enables us to determine the degree of consonance of chords, too, for it deals with pair of numbers rather than ratios strictly. It was more or less in alignment with practice for simple-ratio intervals but for chords, which unlike intervals are formed of more than two tones, some of its predictions were inconsistent with what musicians felt to be right. The most remarkable instance of that inconsistency was that; to the Gradus Function, the major triad had the same degree of consonance with that of major seventh chord.
At least an equation was formed, and it predicted order consonance of the most-frequently-used intervals, if not all.
The pursuit of an equation in the expectation of explaining the entire phenomena of consonance has never stopped. There has been further trials like Plompt - Levelt's 1965 paper constructing a relation between consonance and beating effect, or Vassilakis's 2004 equation focusing on roughness of pairs of sounds; all of which based upon overtones of sounds. But considering 12-tone-equal-temperament and pure sine waves, the mystery does not seem to be given a thorough solution.
Mathematics is the language in which God has written the universe, quoth Galileo. Sometimes, the way in which math manifests itself is so convincing that what Galileo said seems to be undeniable, as for laws of Physics, Finance, and Statistics. Sometimes, it seems to have little impact on what is going on, as for Course of History, Psychology, and Literature.
But we seldom are at the very centre of a scale whose one end is finding Galileo rather reasonable and the other end is not agreeing on what he said at all, as in, music.
For a mathematician, it is improbable for music not to be affected by the so-called language of the realm to which everything including music belongs. Yet, even the most brilliant minds that might well be attributed as greatest mathematicians are not capable of coming up with an explanation, to the most fundamental building blocks of music, with which everyone can agree.
It is tiresome but gripping at the same time; how math shows itself in a promising way as if it is going to help solve a mystery but runs away once you get closer to it leaving a greater mystery behind.