Mathematics of the Chromatic Scale and Chords
Three octaves of the chromatic scale are shown in Table 2.2a using the A, B, C, . . . notation. Black keys on the piano are shown as sharps, e.g. the # on the right of C represents C#, etc., and are shown only for the highest octave. Each successive frequency change in the chromatic scale is called a semitone and an octave has 12 semitones. The major chords and the integers representing the frequency ratios for those chords are shown above and below the chromatic scale, respectively. The word chord is used here to mean two notes whose frequency ratio is a small integer. Except for multiples of these basic chords, integers larger than about 10 produce chords not readily recognizable to the ear. In reference to Table 2.2a, the most fundamental chord is the octave, in which the frequency of the higher note is twice that of the lower one. The interval between C and G is called a 5th, and the frequencies of C and G are in the ratio of 2 to 3. The major third has four semitones and the minor third has three. The number associated with each chord, 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. Note that the word "scale" in "chromatic scale", "C-major scale", and "logarithmic or frequency scale" (see below) has different meanings; the second is a subset of the first.
TABLE 2.2a: Frequency Ratios of Chords in the Chromatic Scale
|--Octave-- |
|--5th-- |
|--4th-- |
|-Maj.3rd- |
|-Min.3rd- |
| |
| |
We can see from the above that a 4th and a 5th "add up" to an octave and a major 3rd and a minor 3rd "add up" to a 5th. Note that this is an addition in logarithmic space, as explained below. The missing integer 7 is also explained below.
The "equal tempered" (ET) chromatic scale consists of "equal" half-tone or semitone rises for each successive note. They are equal in the sense that the ratio of the frequencies of any two adjacent notes is always the same. This property ensures that each note is the same as any other note (except for pitch). This uniformity of the notes allows the composer or performer to use any key without hitting bad dissonances, as further explained below. There are 12 equal semitones in an octave of an ET scale and each octave is an exact factor of two in frequency. Therefore, the frequency change for each semitone is given by
semitone^{12} = 2 or
semitone = 2^{1/12} = 1.05946. . . . . . . . . . . . . . . . . . Eq. (2.1)
Eq. (2.1) defines the ET chromatic scale and allows the calculation of the frequency ratios of "chords" in this scale. How do the "chords" in ET compare with the frequency ratios of the ideal chords? The comparisons are shown in Table 2.2b and demonstrate that the chords from the ET scale are extremely close to the ideal chords.
TABLE 2.2b: Ideal Chords versus the Equal Tempered Scale
Chord |
Freq. Ratio |
Eq. Tempered Scale |
Difference |
Min.3rd: |
6/5 = 1.2000 |
semitone^{3} = 1.1892 |
+0.0108 |
Maj.3rd: |
5/4 = 1.2500 |
semitone^{4} = 1.2599 |
-0.0099 |
Fourth: |
4/3 = 1.3333 |
semitone^{5} = 1.3348 |
-0.0015 |
Fifth: |
3/2 = 1.5000 |
semitone^{7} = 1.4983 |
+0.0017 |
Octave: |
2/1 = 2.0000 |
semitone^{12} = 2.0000 |
0.0000 |
The errors for the 3rds are the worst, over five times the errors in the other chords, but are still only about 1%. Nonetheless, these errors are readily 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 learn to live with, if we are to adopt the ET scale. 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.
The integer 7, if it were included in Table 2.2a, would have represented a chord with the ratio 7/6 and would correspond to a semitone squared. The error between 7/6 and a semitone squared is over 4% and is too large to make a musically acceptable chord and was therefore excluded from Table 2.2a. It is just a mathematical accident that the 12-note chromatic scale produces so many ratios close to the ideal chords. Only the number 7, out of the smallest 8 integers, results in a totally unacceptable chord. The chromatic scale is based on a lucky mathematical accident in nature! It is constructed by using the smallest number of notes that gives the maximum number of chords. 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 chords until the numbers become quite large, making that approach impractical for most musical instruments.
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. Why might this be significant? The answer is because the geometry of the cochlea of the ear seems to have a logarithmic component. Detecting acoustic frequencies on a logarithmic scale accomplishes two things: you can hear a wider frequency range for a given size of cochlea, and analyzing ratios of frequencies becomes simple because instead of dividing or multiplying two frequencies, you only need to subtract or add 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. To show you how useful this is, given F5, the brain knows that F4 will be found 2mm back! Therefore, chords (remember, chords are frequency divisions) are particularly simple to analyze in a logarithmically constructed cochlea. When we play chords, we are performing mathematical operations in logarithmic space on a mechanical computer called the piano, as was done in the 1950's with the slide rule. Thus the logarithmic nature of the chromatic scale has many more consequences than just providing a wider frequency range. The logarithmic scale assures that the two notes of every chord are separated by the same distance no matter where you are on the piano. By adopting a logarithmic chromatic scale, the piano keyboard is mathematically matched to the human ear in a mechanical way! This is probably one reason for why harmonies are pleasant to the ear - harmonies are most easily deciphered and remembered by the human hearing mechanism.
Suppose that we did not know Eq. 2.1; can we generate the ET chromatic scale from the chord relationships? If the answer is yes, a piano tuner can tune a piano without having to make any calculations. These chord relationships, it turns out, completely determine the frequencies of all the notes of the 12 note chromatic scale. A temperament is a set of chord relationships that provides this determination. From a musical point of view, there is no single "chromatic scale" that is best above all else although ET has the unique property that it allows free transpositions. Needless to say, ET is not the only musically useful temperament, and we will discuss other temperaments below. Temperament is not an option but a necessity; we must choose a temperament in order to accommodate these mathematical difficulties. No musical instrument based on the chromatic scale is completely free of temperament. For example, the holes in wind instruments and the frets of the guitar must be spaced for a specific tempered scale. The violin is a devilishly clever instrument because it avoids all temperament problems by spacing the open strings in fifths. If you tune the A(440) string correctly and tune all the others in 5ths, these others will be close, but not tempered. You can still avoid temperament problems by fingering all notes except one (usually A-440). In addition, the vibrato is larger than the temperament corrections, making temperament differences inaudible.
The requirement of tempering arises because a chromatic scale tuned to one scale (e.g., C-major with perfect chords) does not produce acceptable chords in other scales. If you wrote a composition in C-major having many perfect chords and then transposed it, terrible dissonances can result. There is an even more fundamental problem. Perfect chords in one scale also produce dissonances in other scales needed in the same piece of music. Tempering schemes were therefore devised to minimize these dissonances by minimizing the de-tuning from perfect chords in the most important chords and shifting most of the dissonances into the less used chords. The dissonance associated with the worst chord came to be known as “the wolf”.
The main problem is, of course, chord purity; the above discussion makes it clear that no matter what you do, there is going to be a dissonance somewhere. It might come as a shock to some that the piano is a fundamentally imperfect instrument! We are left to deal forever with some compromised chords in almost every scale.
The name "chromatic scale" generally applies to any 12-note scale with any temperament. Naturally, the chromatic scale of the piano does not allow the use of frequencies between the notes (as you can with the violin), so that there is an infinite number of missing notes. In this sense, the chromatic scale is incomplete. Nonetheless, the 12-note scale is sufficiently complete for the majority of musical applications. The situation is analogous to digital photography. When the resolution is sufficient, you cannot see the difference between a digital photo and an analog one with much higher information density. Similarly, the 12-note scale apparently has sufficient pitch resolution for a sufficiently large number of musical applications. This 12-note scale is a good compromise between having more notes per octave for greater completeness and having enough frequency range to span the range of the human ear, for a given instrument or musical notation system with a limited number of notes.
There is healthy debate about which temperament is best musically. ET was known from the earliest history of tuning. There are definite advantages to standardizing to one temperament, but that is probably not possible or even desirable in view of the diversity of opinions on music and the fact that much music now exist, that were written with particular temperaments in mind. Therefore we shall now explore the various temperaments.