The Derivation of Beatless M3 and P5

The beatless tuning of M3 and P5 can be derived from the beatless tuning of a major triad, which in turn can be derived from the harmonic structure of tones. For this derivation, we introduce the notion of a schematic spectrogram, a series of lines that indicate the frequency components of a sound without reference to their amplitude or absolute location on a frequency scale. A tone with harmonics 1 through 20 is illustrated in Figure 3.5.

  

Figure 3.5: A schematic spectrogram of a tone

If we listened to harmonics 1 through 5 of this sound melodically, it would sound like a rolled major chord with the root appearing at two octave transpositions. If the root were C4, it would sound like a rolled chord of C4-C5-G5-C6-E6. Why stop at harmonic 5? Well, harmonic 6 is just an octave transposition of harmonic 3, and then harmonic 7 sounds like an m7 above harmonic 4 (albeit a very flat m7 to ears accustomed to 12TET). In terms of pitch equivalents, harmonic 6 would be a G6 and harmonic 7 would be a funny-sounding B6. Since our goal is to derive the major triad from the harmonic structure of tones, we should stop at harmonic 5. (Nonetheless, it is worth mentioning that many theorists have investigated the possibility of using harmonic 7 as a basis for tuning the 7th of dominant seventh chords.)

Discarding the octave transpositions of the root, if we actually form a chord from tones based on harmonics 1, 3, and 5, it has the schematic spectrogram that appears in Figure 3.6. (In this and subsequent figures, example pitches are given for the fundamentals of the tones illustrated.) Since all the harmonics of the constituent tones are aligned, this chord is beatless and extremely consonant. In fact, it could be considered a spectrally shaped version of the root, not a chord of distinct tones.

  

Figure 3.6: A chord from harmonics 1, 3, and 5 of a tone

Transposing the fundamentals of the tones of this chord down to be within an octave of each other, we get a major triad. Harmonic 5 is transposed down two octaves to form an M3, which therefore has ratio 5/4. Harmonic 3 is transposed down one octave to form a P5, which therefore has ratio 3/2.

This major triad is illustrated in Figure 3.7, and, to be able to see the alignment of harmonics more clearly, it appears in overlapped form in Figure 3.8.

  

Figure 3.7: A beatless major triad

  

Figure 3.8: A beatless major triad, overlapped

Although this chord looks like a mess compared to the untransposed version, it does not sound like a mess. In fact it is still considered beatless. There are a few reasons for this.

The harmonics that do not align would cause beating only if the root were fairly low. Let a be the frequency of the root, which is also the distance between harmonics of the root. If the root were middle C, $a \approx 265$ Hz. In this situation, two harmonics would have to be within about a/20 of each other in order to beat. But no such close spacing exists in the major triad. The closest any two harmonics come to each other is a/4. The observation that a major triad is not so consonant at low frequencies is in accord with conventional musical wisdom, which would prohibit any close voicing of a chord at low frequencies.

Another reason that a major chord doesn't sound as messy as it looks is that the combined waveform is periodic with a period two octaves below the root. In other words, if the root is C3, the combined waveform is periodic with the same period as C1. In fact, due to the perceptual phenomenon of difference tones, this C1 may actually be heard along with such a chord.

We have derived the ratios for a beatless M3 and P5 and can now proceed to study diatonic tunings with reference to how they treat these intervals.