In music, pitch space is limited by how we produce sound, such as by voice (a few octaves only) or by musical instruments (pianos have only 88 notes), and by our ability to write the music on sheet music. Although violins, etc., can produce an infinite number of notes (frequencies), violin music is written mostly using the finite number of notes of the chromatic (piano) scale. Here, we explain why we are confined to
the chromatic scale and why, out of the infinity of notes that the human ear can hear and that the violin
can produce, we throw away 99.999999 . . . . % - an infinity - of available pitch space.
Although many musical instruments can produce an infinity of notes, there is no way to notate this infinity so that someone else can reproduce your composition.There are a few exceptions such as the violin glissando (slide), vibrato, etc. Even for these "exceptions", there is no way to notate exactly how to execute them. But, aren't we severely limited by using such a small number of notes and throwing away an infinity?
Animals do not need to notate their songs or produce their music using keyboards. Therefore, they sing a completely different type of music. Having an infinity of notes has its advantages; this may be how a baby penguin can distinguish its parents from thousands of other penguins nearby, by their voices. Thus the piano, with its comparatively minuscule number of notes, has an inordinately huge influence on human music. Does this loss of an infinity of notes restrict us musically?
Mathematicians are well aware of, and have solutions to, this problem of finite bandwidth; it is called
"completeness", which measures the accuracy with which any notation approximates the real thing.
Completeness in music asks the question, "given a specific composition, how well does this music notated using this scale, approximate the composition?" The answer is that it is sufficiently complete in a large number of cases; that is, the chromatic scale can approximate any music fairly accurately. No better system has yet been found; this is analogous to digital photography where you do not need an infinite number of pixels to take a picture, although any real object has an effectively infinite (very large) number of pixels (photons hitting the camera).
But the main reason why we are confined to the chromatic scale is harmony, not completeness. A scale must also contain all the major intervals so that a maximum number of notes will then harmonize with each other, making it possible for the brain to keep track of tonics and chord progressions [see (68) Theory, Solfege for a theoretical explanation]. Unlike vision, the frequency of sound is not calibrated on an absolute scale in the brain. Absolute pitch (perfect pitch) is a memory; not everybody has it and it can change with time. The use of intervals is the only way in which the brain can compare frequencies and the chromatic scale contains all the intervals that the brain needs, as we now show.
The brain's requirement of harmony leads to the piano octave, that must include as many intervals as possible. We need the third, fourth, fifth, sixth, and octave. Starting from C4, we have now placed E4, F4, G4, A4, and C5, a total of 6 notes into the C major scale (all white keys). In order to allow transpositions, two more full tones (white keys) and the black keys need to be added so that the chromatic octave consists of 12 equal semitones. This is made possible by a fortuitous mathematical coincidence that when the octave is divided into twelve semitones, it also contains all the harmonizing intervals to a very good approximation (but not exact, see below).
This requirement to include all the necessary intervals explains why the tonic (C in this case), is the most important note in a scale: it is involved with every interval, and tells us how the brain finds the tonic – by harmonizing every note to it. Thus, after you play a few notes of a scale, the brain figures out the tonic because it is the only note related to all the others by harmony. The prominence of the tonic also explains how the brain keeps track of chord progressions by referencing each new tonic to the tonic of the first scale
used, and why the music must return to that starting scale at the end of the composition; otherwise the brain is left "hanging", having to remember one or more tonics instead of none.
Harmony allows more than one note to be played simultaneously without creating dissonances. The sounds are so scrambled in dissonances that the brain can not figure out what it is which explains why the brain prefers harmonies; however, they don't just sound good — they provide the only mechanism by which the brain can keep track of the frequencies of sound. That is why music with harmony is easier to listen to than music without, and why harmony is the basis of most music, even when only one note is played at a time.
Therefore, harmony, completeness, and practicality are three main reasons for the existence of the chromatic scale. The properties that all intervals are ratios and that the frequency doubles with each ascending octave are properties of a mathematical function called the logarithm. That is, the chromatic scale is a
logarithmic scale which the brain can use for detecting and processing frequencies. The ear evolved a logarithmic detection mechanism (Psychoacoustics,) so as to hear a large frequency range. This logarithmic nature makes it easy to construct a musical instrument that accommodates the entire frequency range of the human ear. If the chromatic scale were linear (not logarithmic), we would need a piano keyboard over half a mile wide!
There is another useful property of a logarithmic scale: scales can be transposed. Starting with any note on the chromatic scale, you can construct another scale with exactly the same interval ratios as C major, without having to add or subtract notes. Although we take transpositions for granted, it is possible only because of the logarithmic nature of the chromatic scale.
The chromatic scale is arbitrarily pegged at A = 440 Hz so that every musician can harmonize with each other. This means that no one is born with absolute pitch; it must be learned. Unlike the auditory system, the visual system is calibrated on an absolute scale based on quantum mechanical chemistry, so that every normal person is born with absolute vision — red is red to everybody, and it never changes with age.
Three octaves of the chromatic scale are shown in Table 3.1. Each successive frequency change in the chromatic scale is called a semitone and an octave has 12 semitones. Black keys on the piano are shown as sharps, e.g. the # on the right of C represents C#; all the semitones are shown only for the highest octave. The major intervals and the integers representing the frequency ratios for those intervals are shown above and below the chromatic scale, respectively. The number associated with each interval, e.g. four in the 4th, is the number of white keys, inclusive of the two end keys, for the C-major scale and has no further
mathematical significance.
TABLE 3.1 Frequency Ratios of Intervals in the Chromatic Scale Octave 5th 4th Maj.3rd Min.3rd
CDEFGAB CDEF GAB C # D # E F # G#A#B C
1 2 3 4 5 6 8
The frequency ratios (bottom row) are the desired ratios for perfect harmony, not the exact numbers from the chromatic scale, as explained below. The missing integer 7 is also explained below.
The ratio of the frequencies of any two adjacent notes of the chromatic scale, called a semitone, is always the same. There are 12 semitones in an octave and each octave is a factor of two in frequency. Therefore, the frequency change for each semitone is given by
Semitone12 = 2, or
Semitone = 21/12 = 1.05946... Eq. (3.1)
Eq. (3.1) defines the chromatic scale and allows the calculation of the frequency ratios of intervals in this scale. How do these intervals compare with the frequency ratios of the ideal intervals (bottom row) of Table 3.1? The comparisons are shown in Table 3.2 and demonstrate that the intervals from the chromatic scale are extremely close to the ideal intervals.
TABLE 3.2: Ideal versus Equal Tempered Intervals
Interval Freq. Ratio Eq. Temp. Scale Error
Minor 3rd 6/5 = 1.2000 Semitone3 = 1.1892 -0.0108
Major 3rd 5/4 = 1.2500 Semitone4 = 1.2599 +0.0099
Fourth: 4/3 = 1.3333 Semitone5 = 1.3348 +0.0015
Fifth: 3/2 = 1.5000 Semitone7 = 1.4983 -0.0017
Octave: 2/1 = 2.0000 Semitone12 = 2.0000 +0.0000
The errors for the 3rds are the worst, over five times the errors in the other intervals, but are still only about 1%. Nonetheless, these errors are audible, and some piano aficionados have generously dubbed them "the rolling thirds" while in reality, they are unacceptable dissonances. It is a defect that we must live with, if we are to adopt this scale (there is no better choice). The errors in the 4ths and 5ths produce beats of about 1 Hz near middle C, which is barely audible in most pieces of music; however, this beat frequency doubles for every higher octave.
It is a mathematical accident that the 12-note chromatic scale produces so many interval ratios close to the ideal ratios. Only the number 7, out of the smallest 8 integers (Table 3.1), results in a totally
unacceptable dissonance. The chromatic scale is based on a lucky mathematical accident in nature! No wonder early civilizations believed that there was something mystical about this scale. Increasing the number of keys in an octave does not result in much improvement of the intervals until the numbers become quite large, making that approach impractical.
Note that the frequency ratios of the 4th and 5th do not add up to that of the octave (1.5000 + 1.3333 = 2.8333 vs. 2.0000). Instead, they add up in logarithmic space because (3/2)x(4/3) = 2. In
logarithmic space, multiplication becomes addition; that is why when you add a fourth to a fifth on the piano, you get an octave. Why might this be significant? The geometry of the cochlea has a logarithmic component. Analyzing ratios of frequencies therefore becomes simple because instead of multiplying or dividing two frequencies, you only need to add or subtract their logarithms. For example, if C3 is detected by the cochlea at one position and C4 at another position 2mm away, then C5 will be detected at a distance of 4 mm, exactly as in the slide rule calculator. Therefore, intervals are simple to analyze in a logarithmically structured cochlea.
Although we are not born with absolute pitch, we are born to recognize harmonies because of the logarithmic human hearing; another consequence is that the ear hears a large difference in pitch between 40 and 42.4 Hz (a semitone or 100 cents), but hears almost no difference between 2000Hz and 2002.4 Hz (about 2 cents), for the same difference of 2.4 Hz. Because the chromatic scale is logarithmic, and the brain is equipped to compute in logarithms, everyone can recognize relative pitch, unlike absolute pitch, for which the brain has no absolute calibration. The only way to acquire absolute pitch is to remember it in memory.
Eq. 3.1 is not the way in which the chromatic scale was historically developed. Musicians first started with intervals and tried to find a music scale with the minimum number of notes that would produce those intervals. The requirement of a small number of notes is obvious since it determines the number of keys, strings, holes, etc. needed to construct a musical instrument. This minimum number turned out to be 12 notes per octave.
When we play intervals, we are performing mathematical computations in logarithmic space on a mechanical computer called the piano, as was done in the 1950's using the slide rule. Thus the logarithmic nature of the chromatic scale has more consequences than just providing a wide frequency range. It is also related to how the brain identifies, processes and interprets music [(68) Theory, Solfege]. When you play a piano, a similar mathematical process takes place in the brain!