I've sung most of my life, yet only recently have I started to ask the question "How do unaccompanied vocalists stay in tune?" The process sounds simple enough at first: vocalists stay in tune because they have good relative or absolute pitch memory -- they remember what the 5th of the scale sounds like, and can sing it whenever they like. This explanation works at some level, but a few nagging questions remain.
Music of the western hemisphere is based on harmonic consonance and dissonance: that two tones sounded together will seem relatively static (consonant) or dynamic (dissonant). This consonance or dissonance is based on the ratio of the frequency (vibrations per second) of one tone to the other.
The ancient Greeks came up with a musical system of tones based on simple ratios. For example, two tones making an Octave have the ratio of 2 to 1, and two tones making a Fifth have the ratio of 3 to 2. The modern western system of tuning has its roots in this ancient Greek tuning system.
In the late 1800's the physicist Helmholz showed that any periodic sound (like that produced by a musical instrument) can be described as a series of Harmonics (or "Overtones"). The first Harmonic is at the lowest, or Fundamental frequency of the sound; the second harmonic is at 2 times the fundamental's frequency; the third harmonic is at 3 times the fundamental frequency; and so on.
Helmholz provided a physical explanation for the tuning system of the ancient Greeks. Figure 1 shows the relationship of the harmonics of a note to the notes of the scale. It also shows several intervals and their relationship to the harmonic series.
Figure 1: We define our intervals in terms of the Harmonic Series
The Octave is the most consonant interval (outside the Unison). It has a ratio of 2 to 1. The Perfect Fifth is next, with a ratio of 3 to 2. Next comes the Perfect Fourth, at the ratio of 4 to 3. Next comes the Major Third, at the ratio of 5 to 4. Finally, the Minor Third has a ratio of 6 to 5.
Going back to the harmonic series of figure 1, we can now show how a pair of vocalists can make very precise adjustments to the pitches they sing. When the pair sings an octave, the second harmonic of one singer is at the same frequency as the fundamental of the other singer. If the two singers are even slightly out of tune, they will hear beats. (See Figure 2)
Figure 2: Tuning the octave using beats
Similarly, when two singers sing an interval of a fifth, the 3rd harmonic of the lower note aligns with the 2nd harmonic of the higher note.
Figure 3: Tuning the fifth
In the interval of the fourth, the 4th harmonic of the lower note aligns with the 3rd harmonic of the higher note.
Figure 4: Tuning the fourth
In a major third, the 5th harmonic of the lower note aligns with the 4th harmonic of the higher note.
Figure 5: Tuning the major third
Finally, in a minor third, the 6th harmonic of the lower note aligns with the 5th harmonic of the higher note. The 6th harmonic is clearly audible, it being only two octaves and a fifth above the lower note being sung.
Figure 6: Tuning the minor third
The modern piano is tuned to a system called "Equal Temperament" or "Equal-tempered tuning". The big advantage of equal-tempered tuning is that you don't have to retune your piano to play in a different key: the intervals sound equally out of tune in any key. To put a more positive spin on it, the music sounds equally pleasant in every key.
In Equal Tempered tuning, the Octave is divided into 12 equal semitones. Mathematically, this means that each semitone's frequency is the 12th root of 2 (or about 1.059463094) higher than the semitone before it. Each semitone is further divided into 100 equal parts, called "Cents", allowing tuners to talk about very precise differences in pitch.
Table 1: Equal tempered tuning
|Tone||Ratio to 'C'|
You can see how far out of tune equal temperament is by comparing it to the pure tunings described above. The octaves, fifths, and fourths are in pretty good shape, but the major thirds and minor thirds are barely tolerable.
Table 2: Errors in equal temperament
|Interval||Pure Ratio||Equal Tempered Ratio||Difference (in cents)|
Fortunately, vocalists don't have to change their tuning much or very often. Although there is no solution for an 8-tone scale, there are a number of "nearly-correct" solutions: scales in which only one interval is badly out of tune.
One such scale is Just Intonation (or "Ptolemaic Temperament"), shown in Table 3.
Table 3: Just Intonation (for a C-Major scale)
|Tone||Ratio to 'C'|
|D||9 / 8|
|E||5 / 4|
|F||4 / 3|
|G||3 / 2|
|A||5 / 3|
|B||15 / 8|
Because the intervals of the Just-tempered scale are not equal, checking the tuning of that scale requires a little more effort than in the equal-temperament case. Here, we need to look at the tuning of each of the chords based on a step of the scale, as shown in Table 4.
Table 4: Errors in Just intonation
|Triad||Notes||M3 Ratio||m3 Ratio||M3 Error||m3 Error|
|I||C E G||1.25||1.2||0||0|
|IV||F A C||1.25||1.2||0||0|
|V||G B D||1.25||1.2||0||0|
|Triad||Notes||m3 Ratio||M3 Ratio||m3 Error||M3 Error|
|ii||D F A||1.18518||1.25||-21||0|
|iii||E G B||1.2||1.25||0||0|
|vi||A C E||1.2||1.25||0||0|
As you can see from table 4, the only bad chord is the ii chord. Traditionally, such a badly out of tune chord is called a "wolf", because it "howls". This chord can be repaired by re-tuning the D of the scale from a ratio of 9 / 8 to one of 10 / 9. Of course this change will throw off a lot of the other chords, so the D must be tuned to a ratio of 10 / 9 for the ii chord and 9 / 8 for all other chords.
Alternatively, instead of changing the D of the ii chord, we could change the F and A of that chord.
Since changing the tuning of a sustained note is very noticeable (and difficult to do precisely), I suspect that vocalists tend to follow just one rule: If two chords in a progression share a tone, keep the tuning of that shared tone constant -- that is, tune the other notes in the second chord to the tone that is in common with the first.
Now all I have to do is examine some traditional vocal music and see what happens to the tuning during the piece if I follow the rule above.
Wish me luck!