This spectrogram shows the harmonic distortion of TIA210 amplifier at 1.6 watts, and was made by Matt Kamna with a Tektronix SG505 signal generator and a Hewlett-Packard 8535A spectrum analyzer. If the amplifier were distortionless, all you would see would be a single 1kHz tone extending to the top of the display, and the "grass" or noise floor at the bottom of the display. The vertical axis is amplitude (0 to -100dB), and the horizontal axis is frequency (0 to 10kHz).
The display is calibrated with horizontal lines that show 10dB intervals, and vertical lines that show 1kHz intervals. The 2nd harmonic is one interval to the right of the big signal, being at 2kHz. The smaller it is, the less the distortion. Since the top of the little bump is 7 intervals below the top of the display, that means it is 70dB below the 1kHz fundamental. -70dB corresponds to 0.03%, so the 2nd harmonic is around 0.03%.
The next little bump to the right of that is about 6.5 intervals down from the top of the display, which is 65dB below the fundamental, or about 0.05%. That's the 3rd harmonic. A little bit of 5th is peeking out from the noise, and it's around 8.2 intervals down from the top, which is -82dB down, or about 0.008%. The rest appears to be noise at the -92 to -95dB level ... this is the limit of the measuring system, not necessarily amplifier noise.
The next set of spectrograms were measured by Harry F. Olson (no relation as far as I know) and show the harmonic distortion spectra of a single-ended 6F6 pentode and a single-ended 2A3 triode. (Vertical intervals at 20 dB instead of 10 dB)
Note that H.F. Olson's measurements are for single-ended circuits, not push-pull. If he had measured a perfectly balanced PP circuit, the even harmonics (2nd, 4th, 6th, etc.) would be completely cancelled, leaving no change in the magnitudes and proportions of the odd harmonics (3rd, 5th, 7th, etc.). Since perfectly balanced PP circuits don't exist in the real world, there is always some residue of even harmonic, which is in direct proportion to the degree of imbalance. A circuit in 10% imbalance (quite large) would have a 20dB cancellation of even harmonics, 5% imbalance (more typical) would have 26dB of even-harmonic cancellation, and 1% imbalance would have 40dB of even-harmonic cancellation.
In practice, 1% gain-matching is not going to realized under dynamic conditions. Moreover, what good is accomplished by almost entirely removing even harmonics while leaving odd harmonics alone? With well-behaved triodes, the typical 5% imbalance results in even and odd harmonics conveniently having the same levels. With nonlinear triodes, though, the proportion of odd harmonics is higher, so they will dominate in a PP circuit, resulting in a more "transistor-like" sound. This is a subtle reason to seek out triodes with low proportions of 3rd-harmonic distortion when building a PP circuit (no 6DJ8's, in other words). The same qualities that make a triode desirable in a SE circuit (moderate 2nd harmonic, very low 3rd) make it even more desirable in a PP circuit.
Despite the passage of a half-century and totally different instrumentation, H.F. Olson's 1 watt SE 2A3 spectrograms are remarkably similar to the Amity PP amplifier at 1.6 watts; the 3rd harmonic is almost exactly at the same level below the fundamental, or -60dB down. Note the PP circuit gives a 26dB reduction of the 2nd harmonic compared to single-ended; this is almost exactly what would be predicted from a PP circuit with a residual 5% imbalance. With zero feedback (local or global), a Class A PP direct-heated triode amplifier has the same (or less) distortion as a Williamson-style pentode amplifier with 20dB of feedback, which is a comment on the impressive linearity of direct-heated triodes. Not only that, the distortion spectrum is much cleaner, with almost no high-order harmonic components. This is an important reason that triodes have that hard-to-describe "direct" and "fresh" sound, while other devices sound more "canned" and "electronic" in character. The ear is not fooled by feedback; what we hear are the actual characteristics of the amplifying devices themselves.
This is why we feel that linearity right at the device level is the most important quality of an amplifier; this preference comes from a background in speaker design, where driver build quality sets the upper limit on the sonic potential of the entire system. In a similar way, amplifying elements themselves set the upper limit on the Sonics of the system.