Naughty and nice harmonics

A common assumption made by just intonation composers and theorists is that intervals that occur lower on the harmonic series are always more consonant than those involving higher harmonics.

However, in practice, this may be an oversimplification. It might be better to view the decrease in consonance as a general trend, but not a constant one. In particular, some higher harmonics (particularly the 19th) seem to provide more consonant intervals than their position in the harmonic series might suggest, while lower harmonics (like the 11th) aren't so good. An analogy could be made with atomic nuclei; nuclei heavier than iron get less stable with increasing atomic number, but certain numbers of protons and neutrons are "magic" and yield more stable nuclei than the overall trend would suggest, while other numbers (such as 43 protons, corresponding with technetium) are unstable.

With the harmonic series, 19 could be considered a "magic" or "nice" harmonic because its intervals sound good relative to its position in the series, whereas 11 is "naughty" because its intervals sound bad relative to its position in the series.

The concept of naughtiness and niceness is rather subjective, and some listeners and composers might disagree with my (Mason Green's) analysis and actually enjoy the sound of undecimal intervals. However, as a general rule, I would argue that

• Harmonics that can form strong triads or tetrads are "nice". The 16:19:24 triad sounds very good (it is close to the 12edo minor triad), 17 and 13 can also be used to build triads which sound reasonably good (14:17:21 and 10:13:15, respectively) but 11 cannot. Triads such as 8:11:12 have a lot of "tension" due to the small size of the 11:12.
• Harmonics whose intervals are extremely close to one another are "naughty", since this results in higher harmonic entropy. 17 fails this test because 12:17 and 17:24 are extremely close to one another. 11 also runs into problems; 9:11 and 22:27 are very close to one another, as are 8:11 and 11:15. Meanwhile, 13 and 19 perform better in this regard; their intervals are spaced apart more evenly.

Overall, while individual opinions may vary, it might be best to consider the 11th harmonic unusually "naughty" or discordant relative to its position, while the 19 is unusually "nice", and the 13 and 17 are somewhere in between.

Applications

It is possible to derive a generalized measure of harmonic entropy by weighting certain harmonics more than others. For instance, if we have designated the 11th harmonic as "naughty" (and there are good reasons to do so as given above) we can recalculate the harmonic entropy by simply ignoring all possible matches that contain a factor of 11. This causes 12edo to appear even stronger as a tuning system, since it avoids the 11th harmonic. It could be that over time, listeners have gradually been doing this mentally; since Western music does not use the 11th harmonic, our brains can automatically rule it out, thereby improving the ease with which other intervals are identified. Conversely, this also means that when someone used to 12edo actually does hear an undecimal interval, they will perceive it as something very unsettling or discordant.

Because the 11 is naughty, damping or omitting the 11th harmonic partial from the sound of synthesized tones or physical instruments may result in a more pleasant timbre. Similarly, the 19th harmonic could be amplified louder than it would otherwise be.

It might be a good idea to use a modified Z function when analyzing higher edos, removing the component corresponding to the 11th harmonic (thus making it a no-elevens Z function) while increasing the weighting of the component corresponding to 19. Using such a function even more firmly establishes 12edo's position as supreme among small edos, since 12edo matches the 19th harmonic very closely while avoiding the 11th.

Among higher edos, the one that has the most to gain is 53edo, which closely matches the 13 and 19, while matching the 17 less well and completely avoiding the 11. Even using the unmodified Z function, 53edo is already a zeta peak, integral, and gap edo (and it shares that rare distinction with 12edo). Using the modified Z-function improves its standing even further.

Of the other two "triple zeta edos" in this range (19edo and 31edo), 19edo fares fairly well, too; while it does not match the "nice" 19th or the 17th, it does match the 13th and avoids the "naughty" 11th. 31edo, on the other hand, does not look nearly as good when using the modified Z-function, since it closely matches the 11th "naughty" harmonic while not matching the 13th, 17th, or "nice" 19th nearly as well.

Thus, if "naughtiness and niceness" of higher harmonics is taken into account, 53edo looks like one of the best candidates for extending 12edo. It sounds similar to 12edo in many respects (both tunings closely match the perfect fifth, and while 53edo is not a meantone tuning, it does closely approximate Pythagorean tuning, as does 12edo). 53edo does not contain 12edo as a subset, but 60edo (which is consistent as a no-elevens, no-seventeens 27 limit system) does, making that tuning a good option as well.