Xenness
Xenness is a subjective measure of how xenharmonic an interval or chord is. There are many ways to quantify it, but an understanding can be helpful when making xenharmonic music.
Distance from 12edo
| Range ¢ | Closer to |
|---|---|
| -25 ~ 25 | Unison* |
| 25 ~ 75 | Quartertone |
| 75 ~ 125 | Semitone |
| 125 ~ 175 | Neutral second |
| 175 ~ 225 | Whole Tone |
| 225 ~ 275 | Semifourth |
| 275 ~ 325 | Minor Third |
| 325 ~ 375 | Neutral third |
| 375 ~ 425 | Major third |
| 425 ~ 475 | Semisixth |
| 475 ~ 525 | Perfect fourth |
| 525 ~ 575 | Superfourth |
| 575 ~ 625 | Tritone |
| 625 ~ 675 | Subfifth |
| 675 ~ 725 | Perfect fifth |
| 725 ~ 775 | Semitenth |
| 775 ~ 825 | Minor sixth |
| 825 ~ 875 | Neutral sixth |
| 875 ~ 925 | Major sixth |
| 925 ~ 975 | Semitwelfth |
| 975 ~ 1025 | Minor seventh |
| 1025 ~ 1075 | Neutral seventh |
| 1075 ~ 1125 | Major seventh |
| 1125 ~ 1175 | Semifourteenth |
| 1175 ~ 1225 | Octave |
Being the dominant tuning system worldwide, 12edo is the standard against which xenharmonic music is judged. By this metric, all intervals 50¢ from the nearest 12edo step (alternating steps of 24edo) are considered the most xenharmonic.
Intervals can be categorized by whether they are closer to a 12edo interval or an alternating 24edo interval. For example, 75¢ to 125¢ is closer to a minor second and 125¢ to 175¢ is closer to a neutral second. This category could be called "middle second" to avoid ambiguity.
Generally, the most obviously xenharmonic intervals are far from 12edo and the size of typical melodic steps. This includes intervals closer to a quartertone, Neutral second, and semifourth.
* Note that 25¢ is Ground's recommendation for the lower bound of an Aberrisma.
Distance from LCJI
A related metric is to calculate the distance from nearby low-complexity just intonation ratios, such as 4/3 or 9/7. This is not true xenness as there are many LCJI intervals that are simple but very far from 12edo, such as 11/8. This method is also more mathematically complicated, as it may require an infinite sum of distances from various intervals divided by some complexity measure, such as N*D.
Xenness of Tuning Systems
Tuning systems as a whole can also be evaluated for xenness. This could either be based on a lack of "unxen" intervals, the presence of "xen" intervals, or certain structural properties like the stacking of approximations of 3/2.
