"A perfect fourth or fifth tends to be beatless, so you have a little area in which to play around. The same with octaves. I’ve narrowed the fourths and fifths slightly in order to improve the thirds and sixths dramatically.”
What does he mean by this? A perfect fourth or fifth is beatless? Fourths and fifths have a beat rate that is in the audible range; in the case of the fifth it’s one-third of the frequency of the higher note and half the frequency of the lower note; the fourth has a beat rate a quarter of the higher note and a third the frequency of the lower note. In both cases they manifest as difference tones (link will open in new window) and it’s the difference tone’s modulation of the summed output of the two waveforms that give the interval its character. Only true unisons and octaves have a zero beat rate.
The claim to improve the thirds by narrowing the fourths and fifths is particularly bizarre because what he is suggesting is not only retrograde, but actually impossible.
The guitar is tuned in four steps of a fourth and one of a third but if we follow the whole number ratios which our ears tell us is “correct” we arrive at a top E which is noticeably flat relative to the low E. This is because the interval ratios of 4:3 for the fourths and 5:4 for the third, when multiplied together in five steps are too “short” to resolve into the two octave interval of 4:1 (the interval created,in fact, has a ratio of 320:81).
The guitarist often attempts to resolve the problem by retuning the top E only to find that the interval between the B and E has opened up and is now dissonant. Attempting to close the gap opens the major third between the B and G strings (this is why people often refer to difficulties in tuning these two strings). Tuning the guitar with natural harmonic fourths between the E,A,D and G, and the B and top E strings results in a stretched third between the G and B, which always sounds out of tune. Equal temperament stretches the fourths by a tiny amount reducing the stretching of the major third needed to close the gap and keep the integrity of the octave intervals. It’s fairly obvious from this that shortening the fourths would only result in the third being stretched more, especially since Feiten also offsets the the bass strings slightly, opening out the octave!
