Why did 13th century Europeans think that major sixths were dissonant?

In Adam Neely’s new video, he responds to a question about how “the major sixth was illegal in the Renaissance.” This isn’t quite true, they liked major sixths fine in the Renaissance, but it is true that medieval theorists considered them to be dissonant.

Adam quotes an anonymous medieval music theorist who called the sixth a “vile and loathsome discord.” Another 13th century theorist, Johannes de Garlandia, had a more nuanced take; he defined the major sixth as an “imperfect dissonance”, explaining that a dissonance is imperfect “when two voices are joined so that by audition although they can to some extent match, nevertheless they do not concord.” This is weird! If you play C and the A above it on a piano or guitar, they will sound perfectly fine together, so what the heck are these medieval people talking about?

Adam attributes the idea that the sixth is dissonant to the arbitrary and ever-changing nature of musical aesthetic conventions. He also mentions changes in tuning systems, but brushes quickly past that as an explanation. I disagree about that; while cultural conventions are the major factor, I also think we shouldn’t discount tuning as a basis for those conventions. As 12tone likes to say: Fight me, Adam Neely! (No, don’t fight me, I like Adam, I was in one of his videos, he wrote the foreword to our book, he is good people.) 

First, let’s think about how we would ideally want major sixths to sound. Picture a guitar with a string tuned to C, and another string tuned to F a fifth plus an octave lower. The third harmonic of the F string will produce the same pitch as the C string’s fundamental. The fifth harmonic of the F string will produce an A with a frequency that is 5/3 times the C string’s fundamental frequency. It sounds nice! If you add a string to the guitar tuned to the 5/3 A, its third, sixth and twelfth harmonics will produce the same pitches as the C string’s fifth, tenth and fifteenth harmonics. Your ear likes all that harmonic alignment. 

Back in the 13th century, Europeans were using Pythagorean tuning, based on a spiral of pure fifths. They would have tuned their A like so:

• The third harmonic of our imaginary C string produces the pitch G at three times the fundamental frequency of the C string.
• If we add a new string tuned to this G, its third harmonic produces a D at nine times the fundamental frequency of the C string.
• If we add a new string tuned to this D, its third harmonic produces an A at 27 times the fundamental frequency of the C string.

If you bring this Pythagorean A into the same octave as C, then its frequency is 27/16 times the C string’s fundamental. The Pythagorean A’s frequency is 81/80 times sharper than our optimal A, a difference of about 21.5 cents. That is very noticeably sharp, even for an untrained listener. Pythagorean produces pretty grim-sounding thirds, too. You can experience them for yourself in this version I made of the C Major Prelude from Bach’s Well-Tempered Clavier in Pythagorean tuning, through the magic of MTS-ESP.

The first prominent A’s come at 0:16 – they are the highest notes in the arpeggiated pattern. There are also some prominent low A’s at the bottom of the pattern at 0:31.

I know it’s reductive to attribute changing definitions and consonance just to tuning systems, but they play a role. Part of the reason that medieval people liked parallel fifths is that those sound awesome in Pythagorean. And part of the reason that Europeans started not liking them anymore is because they sound awful in later tuning systems like meantone. In 12-TET we get better fifths in exchange for worse thirds, so maybe that’s part of the reason why power chords made such a comeback?

This is an under-taught aspect of music history, because retuning a piano takes two hours. However, retuning a software synth takes two seconds, so now we can make it part of the core curriculum. We should, because the history of music theory would make a lot more sense.